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Analysis and control of nonlinear multiple input systemswith coupled dynamics by the method of normal forms

ndashApplication to nonlinear modal interactions andtransient stability assessment of interconnected power

systemsTian Tian

To cite this versionTian Tian Analysis and control of nonlinear multiple input systems with coupled dynamics bythe method of normal forms ndashApplication to nonlinear modal interactions and transient stabilityassessment of interconnected power systems Electric power Eacutecole Nationale Supeacuterieure drsquoArts etMeacutetiers 2017 English tel-01616932

Ndeg 2009 ENAM XXXX

Arts et Meacutetiers ParisTech - Campus de Lille L2EP LSIS

2016-ENAM-00XX

Eacutecole doctorale ndeg 432 Sciences des Meacutetiers de lrsquoingeacutenieur

preacutesenteacutee et soutenue publiquement par

Tian TIAN

le 15 septembre 2017

Analysis and control of nonlinear multiple input sy stems with coupled dynamics by the method of normal forms

Doctorat ParisTech

T H Egrave S E pour obtenir le grade de docteur deacutelivreacute par

lrsquoEacutecole Nationale Supeacuterieure dArts et Meacutetiers

Speacutecialiteacute ldquo geacutenie eacutelectrique rdquo

Directeur de thegravese Xavier KESTELYN Co-encadrement de la thegravese Olivier THOMAS

T

H

Egrave

S

E

Jury Mme Manuela SECHILARIU Professeur AVENUES EA 7284 UTC Compiegravegne Preacutesidente M Claude LAMARQUE Professeur LTDS UMR 5513 ENTPE Vaux-en-Velin Rapporteur M Bogdan MARINESCU Professeur LS2N UMR 6004 Eacutecole Centrale de Nantes Rapporteur M Olivier THOMAS Professeur LSIS UMR 7296 Arts et Meacutetiers ParisTech Examinateur M Xavier KESTELYN Professeur L2EP EA 2697 Arts et Meacutetiers ParisTech Examinateur

2

Eacutecole Doctorale n432 Sciences des Meacutetiers de lingeacutenieurs

Doctorat ParisTech

T H Egrave S Epour obtenir le grade de docteur deacutelivreacute par

Arts et Meacutetiers ParisTech

Speacutecialiteacute Geacutenie Eacutelectrique


TianTIAN


Analysis and control of nonlinear

multiple input systems with coupled

dynamics by the method of normal forms

Application to nonlinear modal interactionsand transient stability assessment of

interconnected power systems

Jury

Preacutesidente Olivier THOMAS Professeur des Universiteacutes Arts et Meacutetiers ParisTechRapporteur Bogdan MARINESCU Professeur des Universiteacutes Eacutecole Centrale de NantesRapporteur Claude-Henri LAMARQUE Professeur ENTPE Eacutecole Nationale des Travaux Publics de lEacutetat Manuela SECHILARIU Professeur des Universiteacutes Universiteacute de Technologie de CompiegravegneExaminateur Xavier KESTELYN Professeur des Universiteacutes Arts et Meacutetiers ParisTech

2

Analyse et controcircle de systegravemes non lineacuteaires agrave entreacutees multiples etagrave dynamiques coupleacutees par la meacutethode des formes normales

Application aux interactions modales non lineacuteaires et eacutevaluation dela stabiliteacute des reacuteseaux eacutelectriques

RESUME Les systegravemes composeacutes dune somme de sous-systegravemes interconnecteacutesorent les avantages majeurs de exibiliteacute dorganisation et de redondance syn-onyme de abiliteacute accrue Une des plus belles reacutealisations baseacutee sur ce conceptreacuteside dans les reacuteseaux eacutelectriques qui sont reconnus agrave ce jour comme la plus grandeet la plus complexe des structures existantes jamais deacuteveloppeacutees par lhommeLes pheacutenomegravenes de plus en plus non lineacuteaires rencontreacutes dans leacutetude des nou-veaux reacuteseaux eacutelectriques amegravenent au deacuteveloppement de nouveaux outils permettantleacutetude des interactions entre les dieacuterents eacuteleacutements qui les composent Parmi lesoutils danalyse existants ce meacutemoire preacutesente le deacuteveloppement et lapplication dela theacuteorie des Formes Normales agrave leacutetude des interactions preacutesentes dans un reacuteseaueacutelectrique Les objectifs speacuteciques de cette thegravese concernent le deacuteveloppement dela meacutethode des Formes Normales jusquagrave lordre 3 lapplication de cette meacutethodeagrave leacutetude des oscillations preacutesentes dans des reacuteseaux tests et lapport de la meacutethodedeacuteveloppeacutee dans leacutetude de la stabiliteacute des reacuteseaux

Mots cleacutes Systegravemes interconnecteacutes oscillations non-lineacuteaires theacuteorie des formesnormales

Analysis and control of nonlinear multiple input systems with cou-pled dynamics by the method of normal forms

Application to nonlinear modal interactions and transient stabilityassessment of interconnected power systems

ABSTRACT Systems composed with a sum of interconnected sub-systems oerthe advantages of a better exibility and redundancy for an increased reliability Oneof the largest and biggest system based on this concept ever devised by man is theinterconnected power system Phenomena encountered in the newest interconnectedpower systems are more and more nonlinear and the development of new tools fortheir study is od major concern Among the existing tools this PhD work presentsthe development and the application of the Normal Form theory to the study ofthe interactions existing on an interconnected power system The specic objectivesof this PhD work are the development of the Normal Form theory up to the thirdorder the application of this method to study power system interarea oscillationsand the gain of the developed method for the study of stability of power systemsKeywords interconnected systems nonlinear oscillations normal form theory

ii

iii

Acknowledgements

Thanks for the initiator Prof Xavier KESTELYN who introduced me to the sci-entic research He is a parent-like team-mate And he gives me great condencegreat patience and a lot of freedom in the scientic exploration We started fromvery few resources containing only two reference papers but we has obtained a lotof achievements a transaction already published an another transaction in redac-tion 3 IEEE international conference papers and a PhD dissertation that lays thefoundation for several future PhD dissertations Without the encouragements andthe scientic vision of Xavier we will just end by some fruitless explorations asthe pity that we have seen from some previous researches Thanks for Prof OlivierTHOMAS for all the mathematical formulations and theoretical foundations Sincethis PhD topic is multi-disciplinary which mix the electrical engineering mechan-ical engineering and automatic engineering it is the mathematics that link all thedomains

Thanks to the committee members Mrs Manuela SECHILARIU kindly acceptedto be the president of the committee and designed very interesting questions MrClaude-Henri LAMARQUE and Mr Bogdan MARINESCU evaluated carefullymy PhD dissertation and gave deeper insights than I had expected Thanks toMr Lamarque who gives insightful remarks to complete the theoretical work Forme it is a great pleasure to see the person that has achieved a milestone work 26years ago and with two important papers cited in this PhD dissertation Thanks toBogdan for his insightful comments and inspirations that I have never read hearedor thought before I hope that we can collaborate to make some good discoveriesThanks to Mrs Manuala SECHILARIU for your encouragements and especiallyfor your care and suggestions for my future career During my PhD thesis defenseI was too glad and too excited that I forgot to take a photo with the committeemembers It is a great pity I hope that we can get together again someday

I would also like to thank an unknown reviewer of PES GM16 who helps us to nda pile of references papers Before that (a period of 16 months) we found very fewreference papers we were alone and we work in blind

I would also like to thank our international collaborators Mr Hiroyuki AMANOfrom Tokyo University for your inspiration in the nonlinear modal interaction andmeticulous scrutiny Thanks for Mr Arturo Romaacuten MESSINA you held the torch inthe darkness and lighted the way Im looking forward to our further collaborations

ENSAM is a wonderful place to work Many thanks go to all my colleagues andfriends in labo-L2EP and Labo-LSIS Ngac Ky Nguyen Adel OLABI JulietteMORIN Hussein ZAHR Tiago MORAES Martin LEGRYJeacuterocircme BUIRE GabrielMARGALIDA Oleg Gomozov Thank you all for making my PhD journey memo-rable and enjoyable In addition I would like to thank the support from our ocethe colleagues from LSIS Emmanuel COTTANCEAU David BUSSON Pierre BES-

iv

SET Miguel SIMAO Alexandre RENAULT Michel AULELEY Arthur GIVOISMartin LEGRY Joris GUEacuteRIN I am grateful for your help and support both formy research and my daily life A special thank goes to David who is always keento help and oered tremendous help for me during my defence preparation daysThank you all for all your support over the years You are truly wonderful

Thanks to the school Not only for the support in scientic work by the Scolariteacuteoce but also in the private life Mr Pierre GUIOL Mrs Amandine GRIMON-PREZ and Mr Jean-Michel GARCIA Just one month before my landlady doesntrespect the contract and makes the living unbearable It is the school that providesme one-month apartment and the association CNL(Mrs Perrine Boulanger) thatensures me some peace

Thanks to my parents and grand-parents for your continuous support in spirit andmaterial Thanks for your ever-lasting love I hope that I dont let you down

To the end I would like to thankWeiyi DENG my sincere sister-like friend Withoutyou I were not to be what I am now I may do other jobs but I may never meetanother person like you

Simplicity is the ultimate sophistication

Leonardo da Vinci

vi

List of Abbreviations

bull Dynamical System a system in which a function describes the time depen-dence of a point in a geometrical space

bull Normal Forms a simpler form of the original system where all the systemproperty is kept

bull Method of Normal Forms (MNF) the method that study the dynamicalsystem based on its normal forms

bull Normal Form Transformation the transformation of coordinates of state-variables to obtain the normal forms

bull Normal Form Coordinates the coordinates where the dynamics of thestate-variables are described by the normal forms

bull Normal Dynamics the dynamics of the normal forms

bull DAEs dierential algebraic equations

bull Invariant if motion (or oscillation) is initiated on a particular mode theywill remain on this mode without transfer to others

bull Modal Suppersition

bull Decoupled the dynamics of one state-variable is independent of that ofothers

bull Amplitude-dependent Frequency Shift it describes a phenomenon thatin system exhibiting nonlinear oscillations the free-oscilation frequency willvary with respect to the amplitudes

bull Normal Mode a basic particular solution of systems in free-oscillationsof motion that move independently to others the most general motion of asystem is a combination of its normal modes

bull LNM Linear Normal Mode

bull NNM Nonlinear Normal Mode

bull MS method the Modal Series method

bull SP method the Shaw-Pierre method

vii

viii

Contents

List of Figures xix

List of Tables xxi

Introductions geacuteneacuterales 1

1 Introductions 1911 General Context 20

111 Multiple Input System with Coupled Dynamics 20112 Background and Collaborations of this PhD Research Work 21113 What has been achieved in this PhD research 24

12 Introduction to the Analysis and Control of Dynamics of Intercon-nected Power Systems 25121 Why interconnecting power systems 25122 Why do we study the interconnected power system 26123 The Questions to be Solved 26124 The Principle of Electricity Generation 27125 The Focus on the Rotor Angle Dynamics 28126 Assumptions to Study the Rotor Angle Dynamics in this PhD

dissertation 2913 A Brief Review of the Conventional Small-singal (Linear Modal)

Analysis and its Limitations 29131 Small-Signal Stability 32132 Participation Factor and Linear Modal Interaction 32133 Linear analysis fails to study the system dynamics when it

exhibits Nonlinear Dynamical Behavior 34134 How many nonlinear terms should be considered 36135 Linear Analysis Fails to Predict the Nonlinear Modal Interaction 36

14 Reiteration of the Problems 39141 Need for Nonlinear Analysis Tools 39142 Perspective of Nonlinear Analysis 39143 The Diculties in Developing a Nonlinear Analysis Tool 41

15 A Solution Methods of Normal Forms 41151 Several Remarkable Works on Methods of Normal Forms 42152 Why choosing normal form methods 43

16 A Brief Introduction to Methods of Normal Forms 46161 Resonant Terms 46162 How can the methods of normal forms answer the questions 47163 Classication of the Methods of Normal Forms 47

17 Current Developments by Methods of Normal Forms Small-SignalAnalysis taking into account nonlinearities 47

ix

x Contents

171 Inclusion of 2nd Order Terms and Application in NonlinearModal Interaction 48

172 Inclusion of 3rd Order Terms 49173 What is needed 50

18 Objective and Fullment of this PhD Dissertation 50181 Objective and the Task 50

19 A Brief Introduction to the Achievements of PhDWork with A SimpleExample 51191 An example of Two Machines Connected to the Innite Bus 51192 The proposed method 3-3-3 51193 Theoretical Improvements of Method 3-3-3 Compared to ex-

isting methods 52194 Performance of Method 3-3-3 in Approximating the System

Dynamics Compared to the Existing Methods 52195 The Proposed Method NNM 53196 Applications of the Proposed Methods 54

110 Scientic Contribution of this PhD Research 551101 Contribution to Small Signal Analysis 551102 Contribution to Transient Stability Assessment 561103 Contribution to Nonlinear Control 561104 Contribution to the Nonlinear Dynamical Systems 561105 A Summary of the Scientic Contributions 56

111 The Format of the Dissertation 57

2 The Methods of Normal Forms and the Proposed Methodologies 5921 General Introduction to NF Methods 60

211 Normal Form Theory and Basic Denitions 60212 Damping and Applicability of Normal Form Theory 62213 Illustrative Examples 62

22 General Derivations of Normal Forms for System Containing Polyno-mial Nonlinearities 64221 Introduction 64222 Linear Transformation 64223 General Normal Forms for Equations Containing Nonlineari-

ties in Multiple Degrees 65224 Summary analysis of the dierent methods reviewed and pro-

posed in the work 66225 General Procedures of Applying the Method of Normal Forms

in Power System 6723 Derivations of the Normal Forms of Vector Fields for System of First

Order Equations 68231 Class of First-Order Equation that can be studied by Normal

Form methods 68232 Step1Simplifying the linear terms 69

Contents xi

233 Step 2 Simplifying the nonlinear terms 69234 Analysis of the Normal Dynamics using z variables 74235 Other Works Considering Third Order Terms Summary of

all the Normal Form Methods 74236 Interpretation the Normal Form Methods by Reconstruction

of Normal Dynamics into System Dynamics 75237 The signicance to do 3-3-3 77238 Another Way to Derive the Normal Forms 77239 Comparison to the Modal Series Methods 77

24 Nonlinear Indexes Based on the 3rd order Normal Forms of VectorFields 80241 3rd Order Modal Interaction Index 81242 Stability Index 81243 Nonlinear Modal Persistence Index 82244 Stability Assessment 82

25 Derivations of Normal Forms of Nonlinear Normal Modes for Systemof Second-Order Dierential Equations 83251 Class of Second-Order Equations that can be studied by Meth-

ods of Normal Forms 83252 Linear Transformation 84253 Normal Modes and Nonlinear Normal Modes 84254 Second-Order Normal Form Transformation 85255 Third-order Normal Form Transformation 88256 Eliminating the Maximum Nonlinear Terms 89257 Invariance Property 91258 One-Step Transformation of Coordinates from XY to RS 92259 Comparison with Shaw-Pierre Method where Nonlinear Nor-

mal Modes are Formed 932510 Possible Extension of the Normal Mode Approach 942511 A Summary of the Existing and Proposed MNFs Based on

Normal Mode Approaches 9426 Nonlinear Indexes Based on Nonlinear Normal Modes 94

261 Amplitude Dependent Frequency-Shift of the Oscillatory Modes 95262 Stability Bound 95263 Analysis of Nonlinear Interaction Based on Closed-form Results 96

27 The Performance of Method NNM Tested by Case Study 97271 Interconnected VSCs 97272 Analysis based on EMT Simulations 100273 Coupling Eects and Nonlinearity 100274 Mode Decomposition and Reconstruction 101275 Amplitude-dependent Frequency-shift 101276 Analytical Investigation on the Parameters 102277 Closed Remarks 106

28 Issues Concerned with the Methods of Normal Forms in Practice 106

xii Contents

281 Searching for Initial Condition in the Normal Form Coordinates106282 The Scalability Problem in Applying Methods of Normal

Forms on Very Large-Scale System 109283 The Dierential Algebraic Equation Formalism 109

29 Comparisons on the two Approaches 110291 Similarity and Dierence in the Formalism 110292 Applicability Range and Model Dependence 111293 Computational Burden 111294 Possible Extension of the Methods 112

210 Conclusions and Originality 112

3 Application of Method 3-3-3 Nonlinear Modal Interaction andStability Analysis of Interconnected Power Systems 11531 Introduction 116

311 Need for inclusion of higher-order terms 11632 A Brief Review of the existing Normal Forms of Vector Fields and

the Method 3-3-3 118321 Modeling of the System Dynamics 119322 Method 2-2-1 120323 Method 3-2-3S 120324 Method 3-3-1 121325 Method 3-3-3 121

33 Chosen Test Benches and Power System Modeling 12234 Assessing the Proposed Methodology on IEEE Standard Test system

1 Kundurs 4 machine 2 area System 123341 Modelling of Mutli-Machine Multi-Area Interconnected Power

System 124342 Network Connectivity and Nonlinear Matrix 127343 Programming Procedures for Normal Form Methods 133344 Cases Selected and Small Signal Analysis 134345 Case 1-2 Benets of using 3rd order approximation for modal

interactions and stability analysis 136346 Nonlinear Analysis Based on Normal Forms 138

35 Applicability to Larger Networks with More Complex Power SystemModels 140351 Modelling of the System of Equations 142352 Case-study and Results 145

36 Factors Inuencing Normal Form Analysis 147361 Computational Burden 147362 Strong Resonance and Model Dependence 149

37 Conclusion of the Reviewed and Proposed Methods 149371 Signicance of the Proposed Research 149372 Future Works and Possible Applications of Method 3-3-3 149

Contents xiii

4 Application of Method NNM Transient Stability Assessment 15141 Introduction 15242 Study of Transient Stability Assessment and Transfer Limit 152

421 Existing Practical Tools 156422 Advantages of the Presented Methods 159423 Challenges for the Presented Methods 159424 A Brief Review of the Proposed Tool Method NNM 161425 Technical Contribution of the Method NNM 163426 Structure of this Chapter 163

43 Network Reduction Modelling of the Power System 163431 Modelling of SMIB System 163432 Modeling of Multi-Machine System 164433 Taylor Series Around the Equilibrium Point 164434 Reducing the State-Variables to Avoid the Zero Eigenvalue 165

44 Case-studies 166441 SMIB Test System 166442 Multi-Machine Multi-Area Case IEEE 4 Machine Test System 168

45 Conclusion and Future Work 170451 Signicance of this Research 170452 On-going Work 170453 Originality and Perspective 173

5 From Analysis to Control and Explorations on Forced Oscillations17551 From Nonlinear Analysis to Nonlinear Control 17652 Two Approaches to Propose the Control Strategies 17753 Approach I Analysis Based Control Strategy 17754 Approach II the Nonlinear Modal Control 181

541 State-of-Art The Applied Nonlinear Control 181542 The Limitations 182543 Decoupling Basis Control 182544 A Linear Example of Independent Modal Space Control Strategy184545 Nonlinear Modal Control 185546 How to do the nonlinear modal control 185

55 Step I Analytical Investigation of Nonlinear Power System Dynamicsunder Excitation 185551 Limitations and Challenges to Study the System Dynamics

under Excitation 186552 Originality 186553 Theoretical Formulation 187554 Linear Transformation Modal Expansion 187555 Normal Form Transformation in Modal Coordinates 187556 CaseStudy interconnected VSCs under Excitation 188557 Signicance of the Proposed Method 192

56 Step II Implementing Nonlinear Modal Control by EMR+IBC 193

xiv Contents

561 The Control Schema Implemented by Inversion-Based ControlStrategy 193

562 Primitive Results 19457 Exploration in the Control Form of Normal Forms 196

571 A Literature Review 196572 A Nouvel Proposed Methodology to Deduce the Normal Form

of Control System 19758 Conclusion 197

6 Conclusions and Perspectives 19961 A Summary of the Achievements 20062 Conclusions of Analysis of Power System Dynamics 200

621 Analysis of Nonlinear Interarea Oscillation 201622 Transient Stability Assessment 201623 A Summary of the Procedures 202624 Issues Concerned with the Methods of Normal Forms in Practice202

63 Conclusions of the Control Proposals 20264 Conclusions of the Existing and Proposed Methodologies 204

641 Validity region 204642 Resonance 204643 Possibility to do the Physical Experiments 204

65 Multiple Input Systems with Coupled Dynamics 205651 Classes of Multiple-input Systems with Coupled Dynamics

Studied by the Proposed Methodologies 205652 System with Coupled Dynamics in the First-Order Dierential

of State-Variables of System 205653 System with Coupled Dynamics in the Second-Order Dier-

ential of State-Variables of System 20666 The End or a new Start 206

List of Publications 207

A Interconnected VSCs 209A1 Reduced Model of the Interconnected VSCs 209

A11 A Renewable Transmission System 209A12 Methodology 210A13 Modeling the System 211A14 The Disturbance Model 214

A2 Parameters and Programs 216A21 Parameters of Interconnected VSCs 216A22 Initial File for Interconnected VSCs 216A23 M-le to Calculate the Coef 220

A3 Simulation Files 230

Contents xv

B Inverse Nonlinear Transformation on Real-time 231B1 Theoretical Formulation 231B2 Linearization of the Nonlinear Transformation 232

C Kundurs 4 Machine 2 Area System 235C1 Parameters of Kundurs Two Area Four Machine System 235C2 Programs and Simulation Files 236

C21 SEP Initialization 236C3 Calculation of Normal Form Coecients 244

C31 Calculating the NF Coecients h2jkl of Method 2-2-1 244C32 Calculation the NF Coecients of Method 3-3-3 245C33 Calculation of Nonlinear Indexes MI3 247C34 Normal Dynamics 249C35 Search for Initial Conditions 250

C4 Simulation les 257

D New England New York 16 Machine 5 Area System 259D1 Programs 259D2 Simulation Files 290

Bibliography 291

xvi Contents

List of Figures

1 Examples of Multiple-input with Coupled Dynamics 12 Supergrid Transfering the Renewable Energy from Remote Sites to

the Cities 23 Kundurs 2 Area 4 Machine System 64 Equal Area Criteria (E-A) 10

11 Examples of Multiple-input with Coupled Dynamics 2112 EMR elements (a) source (b) accumulation (c) conversion and (d)

coupling 2313 Illustration of the Inversion-Based Control Strategy 2414 Super-grid Transferring the Renewable Energy from Remote Sites

to the Cities 2515 The magnetic eld of rotor and stator 2716 A Disturbance Occurs On the Transmission Line 2817 The Trajectory of Dynamics 2818 Kundurs 2 Area 4 Machine System 3019 Example Case 1 Linear Small Signal Analysis Under Small Disturbance 31110 Linear Oscillations after Small Disturbance 35111 Linear analysis fails to approximate the nonlinear dynamics 35112 Linear analysis fails to predict the stability 36113 Dierent Order Analysis and the Operational Region of the Power

System 37114 Example Case 2 Linear Small Signal Analysis of Stressed Power

System Under Large Disturbance 38115 Example Case 2 Linear Small Signal Analysis of Stressed Power

System Under Large Disturbance 38116 An idea using nonlinear change of variables to simplify the nonlin-

earities 42117 The People Working on Methods of Normal Forms 43118 Procedures to Obtain the Normal Forms 46119 A system of Two Machines that are Interconnected and Connected

to the Innite Bus 51120 Meaning of the name of method 3-3-3 52121 Method 3-3-3 approximates best the system dynamics 53122 N-dimensional System modeled Second-order Dierential Equations 53123 Linear system dynamics can be decoupled by linear transformation 54124 Nonlinear system dynamics can be approximately decoupled by non-

linear transformation 54125 System dynamics are approximately decoupled into two modes 55

21 The Interconnected VSCs 98

xvii

xviii List of Figures

22 Comparison of LNM and NNM models with the exact system (EX)Full dynamics (left) and Detail (right) 100

23 The Eect of the Cubic and Coupling Terms in the Nonlinear ModalModel 101

24 Mode Decomposition An Approximate Decoupling of Nonlinear Sys-tem 102

25 Amplitude-dependent Frequencies of the Nonlinear Modes 10326 X1 X2 vary from 01 to 11 while X12 keeps 001 10327 X1 X2 keep 07 while X12 varies from 0001 to 01 10328 Dynamics Veried by EMT when X12 keeps 001 10429 Dynamics Veried by EMT when X1 X2 keeps 07 105

31 The Need of Inclusion of Higher-Order Terms 11832 The Relationships Between the Generator Exciter and PSS 12233 IEEE 4 machine test system Kundurs 2-Area 4-Machine System 12334 The Rotating Frame and the Common Reference Frame of 4 Machine

System 12435 Control Diagram of Exciter with AVR 12636 Comparison of dierent NF approximations Case 1 13637 FFT Analysis of dierent NF approximations Case 1 13738 Comparison of dierent NF approximations Case 2 13739 Dynamics of Dominant State Variables Associated to Positive Eigen-

value Pairs Case 2 138310 New England New York 16-Machine 5-Area Test System [75] 142311 The Rotating Frame and the Common Reference Frame of 16 Machine

System 143312 Exciter of Type DC4B [78] 144313 The Electromechanical Modes of the 16 Machine System When No

PSS is Equipped [75] 145314 The Electromechanical Modes of the 16 Machine System When all 12

PSSs are Equipped [75] 145315 FFT Analysis of Generator Rotor Angles Dynamics when G1 to G12

are equipped with PSSs 147316 FFT Analysis of Generator Rotor Angles Dynamics when G1 to G12

are equipped with PSSs 148

41 The Spring Mass System for the Illustration of the Transient ProcessWhen the System Experiences Large Disturbances 153

42 Typical Swing Curves [79] 15443 New England and New York 16-Machine 5-Area Test System [75] 15544 Example Case 4 Simulation of Rotor Angles Dynamics of the New

England New York 16-Machine 5-Area System 15545 Representative Example of SMIB System 156

List of Figures xix

46 Example Case 3 The Unstressed SMIB System under Large Distur-bance 156

47 Representative Example of SMIB System 16448 A Case Where the System Works to its limit δSEP = π

24 16749 A Case Where the System Works to its limit δSEP = π

12 167410 When the oscillation amplitude is beyond the bound δmax = 29798

for δep = π12 169

411 When the oscillation amplitude is beyond the bound δmax = 29798

for δep = π12 169

412 IEEE 4 machine test system 170413 Dierent Models to Approximate the Reference Model 171414 At the Stability Bound 171415 Exceeding 2 Bound 172416 Exceeding 5 Bound 172

51 Control and Analysis 17652 The Controller Composed of Cascaded Loops 17853 The Sensitivity-Analysis Based Algorithm of Tuning the Controllers 17954 The Proposed Nonlinear Sensitivity-Analysis Based Algorithm for

Tuning the Controllers 18055 The Topology of the Linear Modal Control 18456 The Topology of the Nonlinear Modal Control 18557 Test System Composed of interconnected VSCs 18958 System Dynamics under Step Excitation 19059 System Dynamics Decomposed by NNM 191510 Decomposition the Modes under Excitation R1R2 191511 The Control Schema of the Nonlinear Modal Control Implemented

by Inversion-Based Control Strategy 193512 An Example of Gantry System 194513 The Simplied System of the Gantry System 195514 The Topology of the Nonlinear Modal Control 195

61 The Flowchart of Applying Methods of Normal Forms in Analysis ofSystem Dynamics 203

A1 Renewable Distributed Energy Located in Remote Areas [99] 210A2 The Basic Unit to Study the Inter-oscillations in Weak Grid 210A3 Grid-side-VSC Conguration [29] 211A4 EMR and Inversion Based Control Strategy 213A5 The cascaded control loops [29] 213A6 How the three-loops system can be reduced to one-loop 215

xx List of Figures

List of Tables

11 Oscillatory Modes Example Case 1 3312 Theoretical Improvements of Method 3-3-3 Compared to existing

methods 52

21 List of Existing and Proposed MNFs with Vector Fields Approach 6722 List of Existing and Proposed MNFs with Normal Modes Approach 6723 Performance Evaluation of the Reviewed and Proposed MNFs 7724 A Summary of the Presented Normal-Mode-Based MNFs 9525 The Nonlinear Properties Quantied by Method 3-3-3 and Method

NNM 113

31 Oscillatory Modes Case 1 13532 Oscillatory Modes Case 2 13533 Initial Conditions 13934 3rd Order Coecients for Mode (17 18) 13935 Resonant Terms associated with Mode (17 18) 14036 Stability Indexes of Mode (5 6)(7 8) 14137 Inter-area Modes of the 16 Machine 5 Area System with PSSs Present

on G6 and G9 14738 Search for Initial Conditions in the NF Coordinates 14839 Performance evaluation of the studied NF approximations 150

41 Comparisons of the Presented Methods 16042 The Proposed Stability Bound E-A Bound and Bound without Con-

sidering the Nonlinear Scaling Factor 168

51 Comparison of Existing and Proposed Normal Form Approximations 186

A1 The parameters of the Interconnected VSCs 216

C1 Generator Data in PU on Machine Base 235C2 Power Flow Data for Case 1 and Case 2 in Steady State after the

fault is cleared Load 235C3 Power Flow Data for Case 1 and Case 2 in Steady State after the

fault is clearedGenerator 236C4 Mechanical Damping and Exciter Gain of Generators for case 1 case

2 and case 3 236

xxi

xxii List of Tables

Introductions geacuteneacuterales

Cadre geacuteneacuteral

Les systegravemes eacutelectromeacutecaniques industriels agrave haute performance sont souvent com-poseacutes dune collection de sous-systegravemes interconnecteacutes travaillant en collaborationdont la stabiliteacute dynamique est tregraves preacuteoccupante dans lopeacuteration Plusieurs exem-ples sont preacutesenteacutes dans la gure 1

(a) un parc eacuteolien (b) un avion plus eacutelectrique

(c) un sou-marin (d) une veacutehicule eacutelectrique

Figure 1 Examples of Multiple-input with Coupled Dynamics

Du point de vue des systegravemes dynamiques ces systegravemes eacutelectromeacutecaniques sontdans le cadre systegraveme agrave entreacutees multiples avec dynamique coupleacutee qui sont om-nipreacutesents dans tous les aspects de notre vie quotidienne Leur stabiliteacutes meacuteritentdecirctre eacutetudieacutees de maniegravere intelligente et meacuteticuleuse pour exploiter leur capaciteacuteavec moins de travail humain Cela donne naissance agrave lavenir des technologies avecdes bienfaits tels que lauto-conduite (mieux controcircleacute) respect de lenvironnement(pour reacuteduire la consommation deacutenergie) et etc

Ce travail de thegravese sinscrit dans le cadre geacuteneacuteral de lanalyse et la commande dessystegravemes non-lineacuteaires agrave entreacutees multiples ayant dynamiques coupleacutees qui donneun aperccedilu sur les comportements dynamiques non-lineacuteaires dune grande classe dessystegravemes agrave entreacutees multiples comme illustreacute sur la gure 1 Ce travail de thegraveseest multidisciplinaire et la meacutethodologie systeacutematique proposeacutee analyse et controcircle

1

2 INTRODUCTIONS GENERALES

les systegravemes non-lineacuteaires en simpliant les non-lineacuteariteacutes par la theacuteorie des formesnormales Des applications sont dabord reacutealiseacutees sur lanalyse de la stabiliteacute et lacommande des grands reacuteseau eacutelectriques

Analyse de la dynamique des grands reacuteseaux eacutelectriques

Un grand reacuteseau eacutelectrique (appeleacute aussi systegraveme de puissance interconnecteacutes) secompose deacuteleacutements (geacuteneacuterateurs transformateurs lignes ) plus ou moins nom-breux selon la taille du reacuteseau interconnecteacutes formant un systegraveme complexe capablede geacuteneacuterer de transmettre et de distribuer leacutenergie eacutelectrique agrave travers de vasteseacutetendues geacuteographiques

Lanalyse de la dynamique des reacuteseau eacutelectriques daujourdhui est un problegravemedicile et un problegraveme agrave forte intensiteacute de calcul Dans la dynamique de nombreuxpheacutenomegravenes sont inteacuteressants et restent encore agrave relever

Cela sexplique en partie par le fait que chacun services deacutelectriciteacute nest pas des ilesde geacuteneacuteration et de controcircle indeacutependants Les interconnexions des grandes reacutegionset son utilisation intensive de ces interconnexions faire les ingeacutenieurs de systegravemede puissance daronter un systegraveme large pousseacute agrave la limite et non-lineacuteaire pouranalyser et faire fonctionner

Par exemple de plus en plus deacutenergie renouvelable a eacuteteacute incluse dans les grandsreacuteseaux eacutelectriques agrave ce jour Leacutenergie est geacuteneacutereacutee dans des sites distants puis trans-feacutereacutee aux zones urbaines avec une longue ligne de transport deacutelectriciteacute Ses cocircteacutesde leacutenergie sont interconnecteacutes pour assurer une reacutepartition optimale de leacutenergieUn exemple du systegraveme de puissance interconnecteacute agrave grande eacutechelle sappelle supergrid qui attire de plus en plus dattention nationale et globale comme illustreacute surla gure 2

Figure 2 Supergrid Transfering the Renewable Energy from Remote Sites to theCities

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Dautre part selon lexigence de transfeacuterer de leacutenergie intelligemment il sagitdexploiter la capaciteacute de transfert en reacuteponse agrave la crise de leacutenergie et deacuteconomiserle travail humain ce qui conduit agrave un systegraveme de puissance stresseacute dont la dy-namique devrait ecirctre examineacutee an dassurer une opeacuteration stable Alors leacutetude dela dynamique du systegraveme de puissance est neacutecessaire surtout la stabiliteacute des grandsreacuteseaux eacutelectriques

La dynamique du systegraveme est souvent caracteacuteriseacutee par des oscillations eacutelectrique-meacutecaniques Analyser la dynamique du systegraveme est de reacutepondre agrave deux questions

bull Comment les composants du systegraveme interagissent lun sur lautre lesinteractions modales

bull Est-ce que le systegraveme arrive nalement un eacutetat de stabiliteacute eacutevaluation destabiliteacute

Les deux questions sont reacutepondu par leacutetude de la stabiliteacute du systegraveme de puissance

Leacutetude de la stabiliteacute des grands reacuteseaux eacutelectriques

La stabiliteacute du reacuteseau eacutelectrique peut se deacutenir de maniegravere geacuteneacuterale commela proprieacuteteacute dun systegraveme qui lui permet de rester en eacutetat deacutequilibre opeacuterationneldans des conditions de fonctionnement normales et de retrouver un eacutetat acceptabledeacutequilibre apregraves avoir subi une perturbation (Eacutetape post-deacutefaut) La stabiliteacute dusystegraveme deacutepend de la condition de fonctionnement initiale ainsi que de la nature dela perturbation Le systegraveme est initialement supposeacute ecirctre agrave une condition deacutetat depreacute-deacutefaut reacutegie par les eacutequations

Cette thegravese se concentre sur la dynamique des angles du rotor quand il y a des per-turbations sur la ligne de transport deacutelectriciteacute puisque les oscillations eacutelectriques-meacutecaniques sont souvent contasteacutees dans la dynamique des angles rotoriques Et ladynamique des angle rotoriques caracteacuterise la capaciteacute du systegraveme de transfeacuterer deleacutelectriciteacute

Pour eacutetudier la dynamique des angles rotoriques on suppose que la stabiliteacute detension et la stabiliteacute de freacutequencie sont deacutejagrave assureacutees

bull Stabiliteacute de tension

On peut deacutenir la stabiliteacute de tension comme la capaciteacute dun systegravemedeacutenergie eacutelectrique agrave maintenir des tensions stables agrave tous ses ndivideuds apregravesavoir eacuteteacute soumis agrave une perturbation agrave partir dune condition initiale de fonc-tionnement de ce systegraveme Dans un certain nombre de reacuteseaux linstabiliteacutede tension est consideacutereacutee comme une importante contrainte dexploitation

bull Stabiliteacute de freacutequence


Dans un grand systegraveme interconnecteacute la freacutequence subit des variations rel-ativement faibles mecircme lors dincidents seacutevegraveres Linstabiliteacute de freacutequenceconcerne essentiellement les situations ougrave la perte de plusieurs lignes de trans-port conduit agrave un morceacutelement du systegraveme Si un bloc se deacutetache du reste dusystegraveme il eacutevolue vers une freacutequence propre et le controcircle de celle-ci peut ecirctredicile en cas de deacuteseacutequilibre important entre production et consommation ausein de ce bloc En cas de deacutecit de production la chute de la freacutequence peutecirctre arrecircteacutee par un deacutelestage de charge (en sous-freacutequence) Par contre encas de surplus de production la hausse de la freacutequence du systegraveme est arrecircteacuteepar une deacuteconnexion rapide de certaines uniteacutes de productions de sorte queleacutequilibre production consommation soit reacutetablie

En eet selon la nature de la perturbation provoquant linstabiliteacute on distinguedeux types de stabiliteacute des angles rotoriques que lon explique ci-apregraves

bull Stabiliteacute angulaire aux petites perturbations Dans les reacuteseauxmodernes linstabiliteacute angulaire aux petites perturbations prend la formedoscillations rotoriques faiblement amorties voire instables En eet ces os-cillations du rotor qui sajoutent au mouvement uniforme correspondant agrave unfonctionnement normal ont des freacutequences qui se situent entre 01 et 2Hz

suivant le mode doscillation

Lextension dune interconnexion et lincorporation agrave celle-ci de systegravemesmoins robustes peut faire apparaicirctre de telles oscillations Les modesdoscillation les plus diciles agrave amortir sont les modes interreacutegionaux danslesquels les machines dune reacutegion oscillent en opposition de phase avec cellesdune autre reacutegion

bull Stabiliteacute angulaire transitoire Linstabiliteacute angulaire aux grandes per-turbations concerne la perte de synchronisme des geacuteneacuterateurs sous leet duncourt-circuit eacutelimineacute trop tardivement (rateacute de protection) ou de la perte deplusieurs eacutequipements de transport La perte de synchronisme se solde par ledeacuteclenchement des uniteacutes concerneacutees

Traditionnellement les eacutetudes de la dynamique angulaire aux petites perturba-tions utilisent le modegravele lineacuteaire et les eacutetudes de la dynamique angulaire transitoireutilisent le modegravele non-lineacuteaire par contre Cependant il est dicile de isoler cesdeux parties Par exemple les oscillations interreacutegionaux qui est un sujet importantde la dynamique angulaire aux petites perturbations sont observeacutees dans lanalysetransitoire (apregraves avoir un deacutefaut de court-circuit triphaseacute dans la ligne de transportdeacutelectriciteacute) mais sa proprieacuteteacute est eacutetudieacutee par lanalyse de la stabiliteacute aux petitesperturbations

En outre lorsque le systegraveme eacuteprouve de grandes perturbations non seulement leslimites de transfert sont concerneacutees mais aussi les sources et les freacutequences desoscillations devraient ecirctre eacutetudieacutees Pour ce problegraveme lanalyse de la stabiliteacute auxpetites perturbations est plus approprieacutee

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Par conseacutequent dans ce travail de thegravese leacutetude de la dynamique angulaire auxpetites perturbations est actuellement eacutetendu au domaine non lineacuteaire pour eacutetudierdes oscillations non-lineacuteaires dans des reacuteseaux eacutelectriques lorsque le systegraveme subiune grande perturbation Et lanalyse de la stabiliteacute transitoire se concentre sur leslimites de transfert ougrave une stabiliteacute analytique approximative est proposeacutee

Une analyse bibliographique des recherches de la dynamique angulaire sont preacutesenteacuteedans les parties suivantes

Analyse de la stabiliteacute angulaire aux petites perturbations baseacutee surla lineacutearisation du modegravele

La quasi-totaliteacute des systegravemes dynamiques reacuteels possegravede des caracteacuteristiques non-lineacuteaires Le comportement dynamique dun systegraveme de puissance peut ecirctre deacutecritpar un ensemble deacutequations dieacuterentielles et algeacutebriques (EDA) (1)

x = f(xu) (1)

g(xu) = 0

Compte tenu que le systegraveme de puissance eacutevolue geacuteneacuteralement autour dun pointde fonctionnement donneacute lors des petites perturbations il est possible de lineacuteariserses eacutequations EDA autour de ce point

Le point de fonctionnement normal du systegraveme se deacutenit comme un point deacutequilibreou une condition initiale Les deacuteriveacutees des variables deacutetat en ce point sont donceacutegales agrave zeacutero

x0 = f(x0u0) = 0 (2)

Ougrave x0 est le vecteur des variables deacutetat correspondantes aux points deacutequilibre

Si une petite perturbation se superpose aux valeurs deacutequilibre la seacuterie de taylor de(1) en point x0 est la seacuterie de fonctions suivante

4x = A4x+O(2) (3)

avec matrice deacutetat A (ntimes n) dont Akl = partfkpartxl

et u = u0 O(2) ramasse les termesdordre supeacuterieur

Neacutegliger tout les termes dordre supeacuterieur et diagonaliser A par des transformationsde similariteacute unitaires x = Uy Nous obtenons

4y = Λ4y (4)

Ougrave λi est une iegraveme valeur propre Ui est le iegraveme valeur propre agrave droite associeacute agrave λiVi est le iegraveme valeur propre agrave droite associeacute agrave λi


Figure 3 Kundurs 2 Area 4 Machine System

La solution de (4) est une seacuterie de e(λi)t i = 1 2 middot middot middot N Cest-agrave- dire que ladynamique du systegraveme est la supposition des modes indeacutependants se caracteacuterise parλi = σi + jωi La stabiliteacute de mode i se caracteacuterise par σi Si σi lt 0 le systegraveme eststable en reacutegime dynamique Et moins σ est plus stable cest

Analyse des interactions modales baseacutee sur la lineacutearisation du mod-egravele

Vecteur propre agrave droite U mesure linuence relative de chaque variable deacutetat xkdans un iegraveme mode et quun vecteur propre agrave gauche Vi indique la contribution delactiviteacute de xk dans le iegraveme mode Par conseacutequent une quantiteacute caracteacuteristiquedun mode donneacute peut ecirctre obtenue par produit eacuteleacutement par eacuteleacutement dun vecteurpropre agrave droite et dun vecteur propre agrave gauche correspondant Cette quantiteacuteappeleacutee le facteur de participation est calculeacutee par la relation suivante

pki = UkiVik (5)

Ainsi le facteur de participation peut fournir des informations nes sur le problegravemeil repreacutesente une mesure relative de la participation de la kegraveme variable deacutetat dansle iegraveme mode et vice versa [1]

Par le facteur de participation nous constatons que il y a quatre type doscillationdans des grands reacuteseaux eacutelectriques

bull Les oscillations des modes locaux Les modes locaux sont les modes les plusrencontreacutes dans les systegravemes de puissance Ils sont associeacutes aux oscillationsentre un geacuteneacuterateur (ou un groupe des geacuteneacuterateurs) dune centrale eacutelectriqueet le reste du systegraveme Le terme local est utiliseacute car les oscillations sontlocaliseacutees dans une seule centrale ou une petite partie du systegraveme (par exemple les geacuteneacuterateurs G1 par rapport au geacuteneacuterateur G2 trouveacute dans la mecircme

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reacutegion (Area 1) gure 3) La gamme de freacutequence de ces oscillations estgeacuteneacuteralement de 1 agrave 2 Hz Lexpeacuterience montre que ces oscillations tendent agrave seproduire lors de lutilisation des reacutegulateurs de tension posseacutedant une reacuteponserapide et quand le lien de transmission entre une centrale et ses charges esttregraves faible Pour assurer un bon amortissement de ces modes des sourcesdamortissement tel le stabilisateur de puissance peuvent ecirctre ajouteacutees auxgeacuteneacuterateurs agrave lorigine de ces modes

bull Les oscillations des modes globaux Les oscillations des modes globaux ouoscillations interreacutegionales sont associeacutees agrave loscillation entre certains geacuteneacutera-teurs dune partie du systegraveme et certains geacuteneacuterateurs dune autre partie dusystegraveme (par exemple les geacuteneacuterateurs G1 et G2 de la reacutegion (area 1) oscillentensemble par rapport au geacuteneacuterateur G3 et G4 de la reacutegion (area 2) gure 3)

Les modes associeacutes agrave ces oscillations preacutesentent geacuteneacuteralement des amortisse-ments tregraves faibles et si ces derniers sont neacutegatifs de petites perturbationspeuvent exciter des oscillations divergentes

Les freacutequences de ces oscillations se trouvent geacuteneacuteralement dans la gammede 01 agrave 1 Hz Cette gamme est infeacuterieure agrave celle des modes locaux car lesreacuteactances des liens entre les systegravemes de puissance sont eacuteleveacutees Geacuteneacuterale-ment la freacutequence naturelle et le facteur damortissement dun mode interreacute-gional deacutecroissent lorsque limpeacutedance dune ligne dinterconnexion ou la puis-sance transmise augmente Le systegraveme dexcitation et les caracteacuteristiques descharges aectent eacutegalement les oscillations des modes interreacutegionaux Ainsices modes preacutesentent des caracteacuteristiques plus complexes que ceux des modeslocaux Eacutetant donneacute que les modes interreacutegionaux impliquent plusieurs geacuteneacutera-teurs un bon amortissement de tels modes peut exiger lutilisation de stabil-isateurs de puissance pour un grand nombre des geacuteneacuterateurs

bull Les oscillations des modes de controcircle Les oscillations associeacutees auxmodes de controcircle sont dues

soit aux controcircleurs des geacuteneacuterateurs (mauvais reacuteglage des controcircleurs dessystegravemes

soit aux autres dispositifs controcircleurs (convertisseurs HVDC SVC)

La freacutequence de ces oscillations est supeacuterieure agrave 4Hz

bull Les oscillations des modes de torsion Ces oscillations sont essentiellementrelieacutees aux eacuteleacutements en rotation entre les geacuteneacuterateurs et leurs turbines Ellespeuvent aussi ecirctre produites par linteraction des eacuteleacutements de rotation avec lecontrocircle dexcitation le controcircle de gouverneur les lignes eacutequipeacutees avec descompensateurs de condensateurs en seacuterie La freacutequence de ces oscillationsest aussi supeacuterieure agrave 4Hz

Un systegraveme ayant plusieurs geacuteneacuterateurs interconnecteacutes via un reacuteseau de transport se


comporte comme un ensemble de masses interconnecteacutees via un reacuteseau de ressortset preacutesente des modes doscillation multiples Parmi tout les types de oscillationsles oscillations des modes globaux nous inteacuteressent le plus Dans les oscillationsde modes interreacutegionaux plusieurs geacuteneacuterateurs de dieacuterentes reacutegions sont impliqueacuteset ils peuvent provoquer des deacutefauts en cascade Eacutetudier les interactions entre lescomposantes sont tregraves important pour comprendre les comportements dynamiquedu systegraveme et puis ameacuteliorer sa performance dynamique par exemple lamortissentdes oscillations de modes interreacutegionaux par lemplacement optimal de PSS (powersystem stabilizer)

Eacutevaluation de la stabiliteacute transitoire

Le processus transitoire peut durer longtemps et ce que nous nous inteacuteressonscest le processus de post-perturbation cest-agrave-dire si le systegraveme va revenir agrave leacutetatdeacutequilibre apregraves que la perturbation est eaceacutee Il sagit de leacutevaluation de lastabiliteacute transitoire (TSA)

Une varieacuteteacute dapproches permettant leacutevaluation de la stabiliteacute des reacuteseaux deacutenergieeacutelectrique a eacuteteacute proposeacutee dans la litteacuterature Les meacutethode les plus reacutepandue peuventecirctre classeacutees en deux cateacutegories distinctes

bull Meacutethodes indirectes dinteacutegration numeacuterique

bull Meacutethodes directes eacutenergeacutetiques

Il y a des meacutethodes pour TSA ougrave une solution analytique nest pas fournie Voyezchapitre 1 pour un examen de la litteacuterature plus deacutetailleacute des recherches sur ce sujet

Meacutethodes indirectes dinteacutegration numeacuterique

Cest lapproche la plus largement appliqueacutee dans le marcheacute est linteacutegrationnumeacuterique eacutetape pas-agrave-pas mais geacuteneacuteralement neacutecessite beaucoup de calcul pourlapplication en ligne Les algorithmes dinteacutegration numeacuterique sont utiliseacutes pourreacutesoudre lensemble des eacutequations dieacuterentielles de premier ordre qui deacutecrivent ladynamique dun modegravele de systegraveme Linteacutegration numeacuterique fournit des solutionsrelatives agrave la stabiliteacute du systegraveme en fonction des deacutetails des modegraveles utiliseacutes

Dans de nombreux cas cela sut pour sassurer que le systegraveme restera synchroniseacutepour tous les temps apregraves le retour

Cependant dans dautres cas la dynamique du systegraveme peut ecirctre telle que la pertede synchronisme ne se produit pas jusquagrave ce que les geacuteneacuterateurs aient connu demultiples oscillations Alors cest dicile de deacutecider quand sarrecircter une simulationet cest impossible dobtenir la marge de stabiliteacute

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Meacutethodes directes eacutenergeacutetiques

Meacutethodes graphiques (Critegravere deacutegaliteacute des aires) Consideacuterons un deacutefaut telun deacutefaut sur la ligne de transmission appliqueacute au systegraveme preacuteceacutedent disparaissantapregraves quelques peacuteriodes du systegraveme Ceci va modier leacutecoulement de puissance etpar conseacutequent langle de rotor δ Retraccedilons la courbe (P minus δ) en tenant comptede ce deacutefaut gure 4 En dessous de cette courbe nous pouvons consideacuterer deuxzones [1]

1 La premiegravere zone (zone A1 zone dacceacuteleacuteration) se situe au-dessous de la droitehorizontale correspondante au point de fonctionnement initial (la droite decharge) Elle est limiteacutee par les deux angles de rotor (δ0 et δ1) correspondantsagrave lapparition et agrave la disparition de deacutefaut Cette zone est caracteacuteriseacutee parleacutenergie cineacutetique stockeacutee par le rotor du fait de son acceacuteleacuteration Pm gt Pe

2 La deuxiegraveme zone (zone A2 zone de deacuteceacuteleacuteration) qui commence apregravesleacutelimination du deacutefaut se situe en dessus de la droite de charge elle estcaracteacuteriseacutee par la deacuteceacuteleacuteration du rotor Pm lt Pe

Si le rotor peut rendre dans la zone A2 toute leacutenergie cineacutetique acquise durant lapremiegravere phase le geacuteneacuterateur va retrouver sa stabiliteacute Mais si la zone A2 ne per-met pas de restituer toute leacutenergie cineacutetique la deacuteceacuteleacuteration du rotor va continuerjusquagrave la perte de synchronisme (Voyez [1 2] pour une deacutemonstration matheacutema-tique)

Durant les trois derniegraveres deacutecennies les meacutethodes eacutenergeacutetiques directes ont sus-citeacute linteacuterecirct de plusieurs chercheurs [3] AM Lyapunov a deacuteveloppeacute une struc-ture geacuteneacuterale pour leacutevaluation de la stabiliteacute dun systegraveme reacutegi par un ensem-ble deacutequations dieacuterentielles an dobtenir une eacutevaluation plus rapide Lideacutee debase des nouvelles meacutethodes deacuteveloppeacutees est de pouvoir conclure sur la stabiliteacute oulinstabiliteacute du reacuteseau deacutenergie sans reacutesoudre le systegraveme deacutequations dieacuterentiellesreacutegissant le systegraveme apregraves leacutelimination du deacutefaut Elles utilisent un raisonnementphysique simple baseacute sur leacutevaluation des eacutenergies cineacutetique et potentiel du systegraveme

Contrairement agrave lapproche temporelle les meacutethodes directes cherchent agrave deacuteterminerdirectement la stabiliteacute du reacuteseau agrave partir des fonctions deacutenergie Ces meacutethodesdeacuteterminent en principe si oui ou non le systegraveme restera stable une fois le deacutefauteacutelimineacute en comparant leacutenergie du systegraveme (lorsque le deacutefaut est eacutelimineacute) agrave unevaleur critique deacutenergie preacutedeacutetermineacutee

Les meacutethodes directes eacutenergeacutetiques non seulement permettent de gagner un tempsrequis au calcul pas agrave pas que neacutecessite lanalyse temporelle mais donnent eacutegale-ment une mesure quantitative du degreacute de stabiliteacute du systegraveme Cette informationadditionnelle rend les meacutethodes directes tregraves inteacuteressantes surtout lorsque la stabil-iteacute relative de dieacuterentes installations doit ecirctre compareacutee ou lorsque les limites destabiliteacute doivent ecirctre eacutevalueacutees rapidement Un avantage cleacute de ces meacutethodes est leurhabiliteacute dans leacutevaluation du degreacute de stabiliteacute (ou dinstabiliteacute) Le second avantage


Figure 4 Equal Area Criteria (E-A)

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est leur capaciteacute agrave calculer la sensibiliteacute de la marge de stabiliteacute agrave divers paramegravetresdu reacuteseau permettant ainsi un calcul ecient des limites dexploitation

Dans la pratique il existe encore des problegravemes non reacutesolus et des inconveacutenientsde cette approche Lecaciteacute de cette meacutethode deacutepend des simplications choisiesdes variables du systegraveme Linteacutegration de la faute sur les eacutequations du systegraveme estneacutecessaire pour obtenir la valeur critique pour eacutevaluer la stabiliteacute Il est dicilede construire la fonction approprieacutee de Lyapunov pour reeacuteter les caracteacuteristiquesinternes du systegraveme La meacutethode nest rigoureuse que lorsque le point de fonction-nement est dans la reacutegion de stabiliteacute estimeacutee

Problegraveme

Depuis une trentaine danneacutees les grands reacuteseaux eacutelectriques se trouvent obligeacutesde fonctionner agrave pleine puissance et souvent aux limites de la stabiliteacute Faire lalineacutearisation dun systegraveme deacutequations entraicircnerait beaucoup derreurs de modeacutelisa-tion et eacutechouerait pour preacutedire des comportements dynamiques tell que oscillationsnon-lineacuteaires agrave freacutequence variable [4] et la stabiliteacute [5] Il a entraicircneacute la neacutecessiteacutedune meilleure compreacutehension des non-lineacuteariteacutes dans le comportement dynamiquedu reacuteseaux eacutelectrique

Quelques caracteacuteristiques propres aux grands reacuteseaux eacutelectriques qui contribuentaux comportements non lineacuteaire en reacutegime dynamique ou transitoire peuvent ecirctreciteacutees

bull la relation puissance-angle qui est une relation non-lineacuteaire en sinus

bull eacutetendue geacuteographique des sites jusquagrave plusieurs dizaines dhectares composeacuteedune collection de geacuteneacuterateurs

bull longueur des connexions lignes et cacircbles jusquagrave centaines de kilomegravetres pourles dieacuterents niveaux de tension

bull une plus forte demande en eacutelectriciteacute tel que le systegraveme est deacutejagrave pousseacute agrave lalimite

Alors il faut trouver une meacutethodologie systeacutematique pour eacutetudier les comportementsnon-lineacuteaire en utilisant les modeleacutes non-lineacuteaire

Il aussi neacutecessite de trouver une meacutethode an

bull didentier les interactions modales nonlineacuteaires entre les composants du sys-tegraveme et les modes doscillations non-lineacutear

bull dobtenir une eacutevaluation de la stabiliteacute plus rapide et plus facile en tenantcompte les non-lineacuteariteacutes

Cependant garder tous les termes non-lineacuteaires est laborieux et nest pas neacutecessaireDans mes travaux il sagit de trouver une meacutethode ougrave les non-lineacuteariteacutes principales


sont gardeacutees an de conserver lessentiel du comportement non-lineacuteaire alors quedautres sont ignoreacutees pour simplier les calculs

Nous conservons notamment les termes qui contribuent agrave la preacutecision la stabiliteacute etla freacutequence des oscillations Cette meacutethode est baseacutee sur la theacuteorie des formes nor-males de Poincareacute qui eacutelimine des termes non-lineacuteaire par le biais de transformationsnon-lineacuteaires

Meacutethode des formes normales

Theacuteorie des formes normales

Theacuteoregraveme de Poincareacute-Dulac Soit un systegraveme dynamique deacutependant de X X =

AX +N(X) A est lineacuteaire N(X) est nonlineacuteaire Le changement de variable X =

Y + h(Y ) permet deacuteliminer tous les monocircmes non-reacutesonnants Y = AY + An(Y )qui est sous sa forme la plus simple

Cette meacutethode est inteacuteressante pour des systegravemes susamment dieacuterentiables Enprenant successivement les changements de variables X = Y + h2(Y ) puis Y =

Z+h3(Z) etc on stabilise bien un terme suppleacutementaire de degreacute xeacute agrave chaque eacutetapeet obtient les formes normales avec une certaine preacutecision On a deacutejagrave obtenu desreacuteussites sur lanalyse dun systegravemes en meacutecanique des structures Nous consideacuteronslutiliser sur systegravemes eacutelectromeacutecaniques agrave dynamiques coupleacutees non-lineacuteaires

Analyse de la dynamique des grands reacuteseaux eacutelectriques par latheacuteorie des formes normales

Des formes normales agrave lordre 2 sont proposeacutee et bien appliqueacutees dans lanalyse desdynamiques de grands reacuteseaux eacutelectriques [615] Cependent il y a des pheacutenomegravenesnon-lineacuteaire restent agrave relever par des formes normales agrave lorder 3 Mais des formesnormales agrave lorder 3 nest pas compleacuteteacutee et nest pas susamment preacutecise [5 16]

Analyse en meacutecanique des structures baseacutee sur la theacuteorie des formesnormales

Notion de mode normal

Pour un systegraveme oscillatoire agrave plusieurs degreacutes de liberteacute un mode normal ou modepropre doscillation est une forme spatiale selon laquelle un systegraveme excitable (microou macroscopique) peut osciller apregraves avoir eacuteteacute perturbeacute au voisinage de son eacutetatdeacutequilibre stable une freacutequence naturelle de vibration est alors associeacutee agrave cetteforme Tout objet physique comme une corde vibrante un pont un bacirctiment ou

13

encore une moleacutecule possegravede un certain nombre parfois inni de modes normaux devibration qui deacutependent de sa structure de ses constituants ainsi que des conditionsaux limites qui lui sont imposeacutees Le nombre de modes normaux est eacutegal agrave celui desdegreacutes de liberteacute du systegraveme

Le mouvement le plus geacuteneacuteral dun systegraveme est une superposition de modes nor-maux Le terme normal indique que chacun de ces modes peut vibrer indeacutepen-damment des autres cest-agrave-dire que lexcitation du systegraveme dans un mode donneacutene provoquera pas lexcitation des autres modes En dautres termes la deacutecompo-sition en modes normaux de vibration permet de consideacuterer le systegraveme comme unensemble doscillateurs harmoniques indeacutependants dans leacutetude de son mouvementau voisinage de sa position deacutequilibre stable

Mode non-lineacuteaire baseacutee sur des formes normales

Le comportement des systegravemes oscillatoires agrave dynamiques coupleacutees non-lineacuteaires nepeut pas ecirctre traiteacutes par des modes normaux lineacuteaires car des proprieacuteteacutes non-lineacuteairesseraient perdues On propose la notion de Mode non-lineacuteaire qui conserve varieacuteteacuteinvariante de lespace des phases [1719] Tous les modes non-lineacuteaires sont lesformes normales dun systegraveme oscillatoire calculeacutes en une seule opeacuteration baseacuteesur la theacuteorie des formes normales [1719] Valideacute par lexpeacuterimentation les modesnon-lineacuteaires preacutedisent plus exactement le comportement dynamique non lineacuteairedun systegraveme meacutecanique

Ce que nous proposons dans cette thegravese

Dans cette thegravese des meacutethodes des formes normales sont propoeacutees par deux ap-proches

Dans une premiegravere approche nous avons employeacute la meacutethode des champs vectorielsLeacutequation dun systegraveme est sous la forme comme suit

x = f(x) (6)

Lexpeacuterimentation est reacutealiseacutee dans leacutetude de cas IEEE 2-area 4-machine systegraveme(g 3) Le travail suscite un inteacuterecirct et une collaboration au niveau internationalHIROYUKI Amano (Japonais) et MESSINA Arturo Roman (Mexicain) Une pub-lication est accepteacutee par une revue internationale IEEE Transactions on Power

Systems

Dans une deuxiegraveme approche nous avons modieacute les modes non-lineacuteaire normauxpour eacutetudier le dynamique dun systegraveme eacutelectromeacutecanique tell que convertisseurs


interconnecteacutes Leacutequation dun systegraveme est sous la forme de celle dun oscilla-teur comme (7) an dobtenir une meilleur compreacutehension physique par rapportagrave la meacutethode des champs vectorielles Une publication dans revue est en cours dereacutedaction

M q +Dq +Kq + fnl(q) = 0 (7)

En reacutealisant ce travail nous consideacuterons la geacuteneacuteralisation de leacutetude du comporte-ment dynamique non-lineacuteaire du systegraveme eacutelectromeacutecanique comme suit

1 de lanalyse agrave la commande et agrave la conception A partir des reacutesultats fournispar les analyses nonlineacuteaire il est possible de ddeacutevelopper ensuite des meacutethodespour ameacuteliorer la performance dynamique du systeacuteme de puissance par exem-ple lemplacement optimal de PSS (power system stabilizer) selon les reacutesultatsde lanalyse des formes normales [20] eacutetudier linuences de paramegravetres prin-cipaux pour le reacuteglage de controcircleurs [16] et la conception des grands reacuteseauxeacutelectriques [21]

Ailleurs des explorations tregraves originales dans la commande des entraicircnementseacutelectriques utilisant des formes normales ont eacuteteacute eectueacutees Ces approchesconsistent agrave deacutecoupler le systeacuteme non-lineacuteaire agrave sous-systegravemes qui peut ecirctrecontrocircleacute de faccedilon indeacutependante

2 de la theacuteorie agrave la pratique comment faire lanalyse et la commande du sys-tegraveme industriel Raner la meacutethodologie prenant en compte des apportsindustriels par exemple appliquer les meacutethodologies sur le systegraveme agrave grandeeacutechelle

Objectif organisation et position de cette recherche de

doctorat

Objectif et la tacircche

Lobjectif de cette thegravese est de

bull eacutelaborer une meacutethodologie systeacutematique pour analyser et controcircler les sys-tegravemes agrave entreacutees multiples non lineacuteaires avec une dynamique coupleacutee baseacutee surdes meacutethodes de formes normales agrave lordre 3

bull appliquer la meacutethodologie systeacutematique pour eacutetudier le comportement dy-namique non lineacuteaire des grands reacuteseaux eacutelectriques

bull analyser les facteurs que inuencent la performance de la meacutethodologie pro-poseacutee et comprendre ses limites et la faccedilon de lameacuteliorer

15

bull tirer de conclusions geacuteneacuterales pour lanalyse de systegravemes dynamiques non-lineacuteaires et deacutevelopper la strateacutegie de commande pour ameacuteliorer leur perfor-mances dynamiques

La reacutealisation des objectifs se compose de

bull le deacuteveloppement des meacutethodes des formes normales (MNF) jusquagrave lordre 3

bull la mise en divideuvre de programmes geacuteneacuteriques de MNF applicables aux systegravemesdynamiques avec des dimensions arbitraires

bull la modeacutelisation matheacutematique des grands reacuteseaux eacutelectriques sous forme deseacuterie Taylor jusquagrave lordre 3

bull lapplication des meacutethode proposeacutee pour eacutetudier les oscillations des modesinterreacutegionaux et la stabiliteacute des grands reacuteseaux eacutelectriques

L2EP et LSIS

Cette thegravese est issu dune collaboration entre deux laboratoires le L2EP (Labora-toire dElectrotechnique et dElectronique de Puissance de Lille) et le LSIS-INSM(Laboratoire des Sciences de lInformation et des Systegravemes-Ingeacutenierie Numeacuteriquedes Systegravemes Meacutecaniques) de lENSAM sur le campus agrave Lille Le directeur de thegraveseest Xavier KESTELYN (L2EP) et le co-directeur est Olivier THOMAS (LSIS) donttous les deux sont professeurs agrave lENSAM-Lille Il sagit eacutegalement dune collabo-ration entre les domaines eacutelectriques et les domains meacutecaniques ainsi quune collab-oration entre le commande et lanalyse et tous sappartiennent agrave matheacutematiquesappliqueacutes

Cette thegravese est baseacutee sur certaines compeacutetences et expertises deacutejagrave accumuleacutees agravelinteacuterieur du laboratoire L2EP et LSIS L2EP a eectueacute beaucoup de pratiquesdu controcircle non-lineacuteaire dans le domaine eacutelectrique LSIS-INSM a deacutejagrave accumuleacutebeaucoup dexpertise dans lanalyse non-lineacuteaire des structures exibles et de lavalidation expeacuterimentale

Une application de la theacuteorie des formes normales dans lanalyse des structures dansle domaine meacutecanique consiste agrave tirer de les modes normals non-lineacuteaire (nonlinearnormal mode NNM) pour le cas speacutecique ougrave des systegravemes vibratoires preacutesentantdes non-lineacuteariteacutes de type polynomial NNM est un outil theacuteorique utiliseacute pourreacuteduire le systegraveme de degreacutes nis en un seul degreacute tout en conservant la proprieacuteteacuteessentielle du systegraveme et preacutedire son comportement sous excitation

Dans lanalyse de la dynamique du systegraveme les principaux avantages bien connuset largement utiliseacutes des modes normaux sont 1) Les modes et les freacutequencesnaturelles reacutesultent de la reacutesolution (theacuteorique ou numeacuterique) dun problegraveme desvaleurs propres (freacutequences) et des vecteurs propres (modes) qui peuvent simultaneacute-ment indiquer les oscillations modales ainsi que la stabiliteacute ou linstabiliteacute asympto-


tique 2) Le modegravele peut ecirctre reacuteduit pour ne consideacuterer que les modes dominantsce qui peut largement simplier lanalyse et le calcul

LSIS Analyse en meacutecanique des structures baseacutee sur la theacuteorie desformes normales


Pour un systegraveme oscillatoire agrave plusieurs degreacutes de liberteacute un mode normal ou modepropre doscillation est une forme spatiale selon laquelle un systegraveme excitable (microou macroscopique) peut osciller apregraves avoir eacuteteacute perturbeacute au voisinage de son eacutetatdeacutequilibre stable une freacutequence naturelle de vibration est alors associeacutee agrave cetteforme Tout objet physique comme une corde vibrante un pont un bacirctiment ouencore une moleacutecule possegravede un certain nombre parfois inni de modes normaux devibration qui deacutependent de sa structure de ses constituants ainsi que des conditionsaux limites qui lui sont imposeacutees Le nombre de modes normaux est eacutegal agrave celui desdegreacutes de liberteacute du systegraveme




L2EP laboratoire se concentre sur le systegraveme eacutelectrique agrave multiple en-treacutees ayant dynamiques coupleacutees

L2EP a accumuleacute beaucoup dexpertise dans la modeacutelisation lanalyse des systegravemesde puissance (mais principalement en fonction de lanalyse lineacuteaire des petits sig-naux) et du controcircle baseacute sur le modegravele Cela contribue agrave la modeacutelisation et aux

17

eacuteleacutements de controcircle dans le deacuteveloppement dune meacutethodologie pour analyser etcontrocircler le systegraveme non lineacuteaire

Contributions scientiques

Des contributions scientiques de cette thegravese sont reacutesumeacutes comme suit

bull Du point de vue des domaines disciplinaires cette thegravese est multidisciplinaireassocieacutee aux matheacutematiques appliqueacutees agrave la dynamique non-lineacuteaire au geniemeacutecanique et au genie meacutecanique eacutelectrique Et il seorce de tirer des pro-prieacuteteacutes non-lineacuteaires essentielles qui sont communes aux systegravemes dynamiquesnon-lineacuteaires ayant le type particulier de dynamique coupleacutee quel que soit lesystegraveme meacutecanique ou le systegraveme eacutelectrique Par exemple le pheacutenomegravene ougravela freacutequence varie avec lamplitude des oscillations libres est courant pour unestructure meacutecanique [17] et des grands reacuteseaux eacutelectriques [22] Les reacutesultatspreacutesenteacutes dans cette theegravese peuvent ecirctre utiliseacutes pour interpreacuteter la dynamiquenon-lineacuteaire des autres systegravemes

bull Du point de vue des recherches sur le systegraveme dynamique

1 il a eectueacute lanalyse non lineacuteaire sur loscillation libre ainsi que surloscillation forceacutee

2 il seorce de prendre lanalyse ainsi que le controcircle du systegraveme dy-namique non-lineacuteaire qui relie leacutecart entre lanalyse non-lineacuteaire et lecontrocircle non-lineacuteaire Il propose le controcircle non lineacuteaire baseacute sur lanalyseet le controcircle de la base de deacutecouplage

bull Du point de vue de lapplication

1 cest la premiegravere fois que toutes les meacutethodes existantes des formulairesnormaux jusquau lordre 3 ont eacuteteacute reacutesumeacutees et que leurs performancesont eacuteteacute eacutevalueacutees

2 cest la premiegravere fois quune meacutethode des formes normales en forme demodes normaux est appliqueacutee pour eacutetudier la stabiliteacute transitoire du sys-tegraveme

3 cest la premiegravere fois que la modeacutelisation matheacutematiques du systegraveme depuissance dans la seacuterie Taylor est fournie

4 cest la premiegravere fois que les programmes geacuteneacuteriques des meacutethodes desformes normales preacutesenteacutees dans cette thegravese sont partageacutes en ligne pourfaciliter les futurs chercheurs Ces programmes sont applicables agrave dessystegravemes agrave dimension arbitraire


Corps de la thegravese

Cette thegravese est composeacutee de 6 chapitres

bull Chapitre 1 Dans ce chapitre le sujet de cette dissertation est introduit avecdes exemples concrets sans utilisant de mots diciles ou complexes pour fa-ciliter la compreacutehension Il commence agrave partir des concepts et de la theacuteorieclassiques et bien connus Ensuite il utilise des matheacutematiques simples etdes exemples tregraves concrets et reacutealistes pour montrer les limites des meacuteth-odes conventionnelles et introduit graduellement la probleacutematique Les ap-proches dans la litteacuterature sont examineacutee est pour expliquer la neacutecessiteacute deproposer de nouvelles meacutethodologies et ce quon peut attendre de cette nou-velle meacutethodologie Il explique eacutegalement pourquoi il sagit de la meacutethodedes formes normales qui devrait ecirctre utiliseacutee plutocirct que dautres meacutethodesanalytiques telles que les meacutethodes de la seacuterie-modale et les meacutethodes Shaw-Pierre Lobjectif lorganisation et le positionnement du travail de doctoratsont eacutegalement indiqueacutes

bull Chapitre 2 Dans ce chapitre deux meacutethode sont proposeacutees avec des pro-grammes geacuteneacuteriques qui sont applicables aux systegravemes dynamiques avec desdimensions arbitraires

bull Chapitre 3-4 Ce chapitre preacutesente lapplication des meacutethode proposeacutee (meacuteth-ode 3-3-3 et meacutethode NNM) pour eacutetudier les oscillations des modes interreacute-gionaux et eacutevaluer la stabiliteacute transitoire grands reacuteseaux eacutelectriques avec desmodeacutelisation matheacutematiques sous forme de seacuterie Taylor jusquagrave lordre 3 Lesmeacutethodes sont valideacutee sur le reacuteseau test eacutetudieacute (reacuteseau multimachines inter-connecteacute)

bull Chapitre 5 Ce chapitre tire des conclusions geacuteneacuterales sur lanalyse de ladynamique non-lineacuteaire du systeacuteme et se propose dameacuteliorer la performancedynamique par deux approches

1 le reacuteglage des paramegravetres du controcircleur ou lemplacement de PSS baseacuteesur lanalyse le modegravele non-lineacuteaire du systegraveme

2 la commande des entraicircnements eacutelectriques utilisant des formes normales

Les recherches sont encore loin decirctre naliseacutees dans 3 trois sans mais ilsidentient de problegravemes inteacuteressants qui peut devenir les sujets dautres dis-sertations

bull

bull Chapitre 6 Les conclusions et perspectives sont preacutesenteacutees dans ce chapitre

Chapter 1

Introductions

In this chapter the topic of this dissertation is introduced with basic denition and

daily examples without using dicult or complex words to facilitate comprehension

It introduces the problems and the conventional linear methods Then it uses simple

mathematics and very concrete and realistic examples from the experimental tests

to show the limitations of the conventional methods and the call for nonlinear tools

A state-of-art of nonlinear tools then explain the need to propose new methodologies

and what should be expected from this new methodology It also explains why it is the

method of Normal Forms that should be used rather than other analytical nonlinear

methods such as the Modal Series Methods and Shaw-Pierre Methods The objective

organization and positioning of the PhD work is also stated

Keywords Power System Stability Rotor Angle Stability Small-signalStability Transient Stability Methods of Normal Forms Modal SeriesMethods

19

20 Chapter 1 Introductions

11 General Context

111 Multiple Input System with Coupled Dynamics

High-performance industrial electromechanical systems are often composed of a col-lection of interconnected subsystems working in collaboration to increase the exi-bility and reliability Several daily examples are shown in Fig 11

Fig 11 (a) shows an interconnected power system composed of numerous wind-turbine generators to capture the maximum wind energy and the generators areelectrically coupled Since the wind energy captured by each generator varies drasti-cally in a day using a collection of generators can smooth the output power Fig 11(b) shows a popular European projectmore electric aircrafts that involves moreand more electric motors in the aircrafts which are either electrically coupled or me-chanically coupled Those motors work in collaboration to increase the autonomyof the aircrafts and the comfort of the ight Fig 11 (c) shows a submarine that isdriven by multi-phase electrical drives where the phases are coupled magneticallyWhen there is fault in one phase the electrical drive can still work Fig 11 (d)shows a electric vehicle where the drives are coupled energetically to give higherexibility and reliability in driving

bull Interconnected renewable energy generators Fig1 (a)

bull More electric aircrafts Fig1 (b)

bull Multi-phase electrical drives Fig1 (c)

bull Electric vehicles Fig1 (d)

(a) Wind Farm (b) More Electric Aircrafts

11 General Context 21

(c)Submarine (d) Electric vehicles


From the perspective of dynamical systems those electromechanical systems canbe generalized as multiple input system with coupled systems dynamics whichis omnipresent in all aspects of our daily life and the analysis and control of itsdynamic performance are of great concern

112 Background and Collaborations of this PhD Research Work

This PhD research work based on two national laboratories L2EP (LaboratoiredElectrotechnique et dElectronique de Puissance de Lille) and LSIS-INSM (Lab-oratoire des Sciences de lInformation et des Systegravemes - Ingeacutenierie Numeacuterique desSystegravemes Meacutecaniques) in the campus of ENSAM in Lille The director of this PhDthesis is Xavier KESTELYN (L2EP) and the co-director is Olivier THOMAS(LSIS)both of whom are professors at ENSAM-Lille This PhD topic is multidisciplinaryIt is also a collaboration of people working in electrical and mechanical domainsand a collaboration between control and analysis The common foundation is theapplied mathematics and the essential physical properties of system dynamics

This PhD topic Analysis and Control of Nonlinear Multiple Input Systems withCoupled dynamics is proposed to solve the common problems encountered by L2EPand LSIS L2EP has performed a lot of nonlinear control practices in electricaldomain LSIS-INSM has already accumulated a lot of expertise in nonlinear analysisof exible structures and experimental validation

1121 LSIS Expertise in Normal Form Analysis of Mechanical Struc-tures

An application of Normal Form Theory in structural analysis in mechanical domainis to derive Nonlinear Normal Mode (NNM) for the specic case of vibratory systemsdisplaying a polynomial type of nonlinearities NNM is a theoretical tool used toreduce the nite-degree system into single-degree while keeping the essential physicalproperty of the system and predict its behaviour under excitation


Nonlinear normal mode is to extend the notion of linear normal mode In the linearanalysis the system dynamics is a composition of independent modes which arenamed as normal modes And the NNM proposed by Touzeacute are invariant modes Itis invariant because when the motion is only initialized in one mode j other modeswill keep zero and the system dynamics are only contained in the mode j

In the mechanical domain it is used to reduce the model and the system dynam-ical behaviour can be characterized by a single NNM Experiments show that asingle NNM based on NF predicts the correct type of nonlinearity (hardeningsoft-ening behaviour) [17] whereas single linear mode truncation may give erroneousresult Another application is focused on how all the linear modal damping termsare gathered together in order to dene a precise decay of energy onto the invariantmanifolds also dened as nonlinear normal modes (NNMs) [18]

1122 L2EP and EMR

L2EP is a Laboratory working on multiple input electrical system with coupleddynamics It has accumulated a lot of expertise in power system modelling andanalysis (though mainly based on the linear small signal analysis) [Team Reacuteseau]and control of multiple input electrical system with coupled dynamics such as theelectric vehicles multiphase electrical machines [Team Commande] and etc

Although those researches proposed various methodologies to solve the practicalproblems most of their researches are based on the Energetic Macroscopic Repre-sentation (EMR) and Inversion-Based Control (IBC) Strategy

Proposed by Prof Alain BOUSCAYROL the leader of the control team of labL2EP EMR contributes to modelling and control of multi-inputs system with highlycoupled dynamics

EMR is a multi-physical graphical description based on the interaction principle(systemic) and the causality principle (energy) by Prof Alain BOUSCAYROL Ithighlights the relations between the subsystems as actions and reactions decidedby the inputs and outputs owing between each other It classies subsystemsinto sources accumulationconversion and distribution elements by their energeticfunctions It uses blocks in dierent colour and shapes to represent subsystems withdierent functions correspondingly

A real system having multiple inputs and strong energetic couplings can be mod-elled by a set of decoupled ctitious single-input systems The control structurewhich is deduced from the model facilitates the control of stored energies helps torun the system in fault mode and makes it possible to use simple and easy-to-tunecontrollers since each of them is assigned to a unique objective

The EMR classies the subsystems according to their energetic function in the sys-tem dynamics into 4 categories source accumulation conversion and coupling


Source elements are terminal elements which deliver or receive energy they are de-picted by green oval pictographs as in Fig 12 (a) Accumulation elements storeenergy eg a capacitor a ywheel an inductance etc they are depicted by or-ange rectangular pictogram with an oblique bar inside Fig 12 (b) Conversionelements transform energy without accumulating it eg an inverter a lever a gear-box etc they are depicted by an orange square for a mono-physical conversion egelectrical-electrical or by an orange circle if the conversion is multi-physical egelectrical-mechanical as shown in Fig 12 (c) Finally coupling elements allow dis-tributing energy among subsystems they are depicted by interleaved orange squaresfor a mono-physical coupling and by interleaved orange circles for a multi-physicalcoupling see Fig 12 (d)

Figure 12 EMR elements (a) source (b) accumulation (c) conversion and (d)coupling

The objective of a control structure is to dene an adapted input so to produce theexpected output see Fig 13 In fact the control has to express the tuning inpututun(t) as a function of the output set-point yref (t) In consequence the controlcan be dened as the inverse of the approximate behavior model describing therelationships between the inputs and outputs

The eectiveness of EMR and IBC has been proved through many practices of thecontrol team members

Numerous applications of the proposed methodology have been experimented onprototype test benches which are equipped with electromechanical systems havingmultiple inputs and having strong magnetic couplings (eg multiphase synchronousmachines) electric couplings (eg multi-leg voltage source inverters) and mechanicalcouplings (eg multi-actuated positioning gantry systems)

Previously EMR and IBC are used for linear system in this PhD research work ithas been extended to the modelling and the control parts in developing methodologyto analyze and control the nonlinear system


Figure 13 Illustration of the Inversion-Based Control Strategy

1123 International Collaboration from Japan and Mexico

This PhD research work has also gained international collaboration from Japan andMexico

bull Hiroyuki Amano Japanese Tokyo University Tokyo who rstly proposedMethod 3-2-3S

bull Arturo Roman Messina Mexican Centre for Research and Advanced Studiesof the National Polytechnic Institute Mexico City Mexico who has achieve-ments in the application of Method 2-2-1 Method 3-3-1

113 What has been achieved in this PhD research

In this PhD thesis a systematic methodology for the general topic Analysis andControl of Nonlinear Multiple-Input System with Coupled Dynamics has been pro-posed which casts insights on nonlinear dynamical behaviours of a large class of themultiple-input systems as shown in Fig 11 This PhD research is multidisciplinaryand the proposed systematic methodology can analyze and control the nonlinear sys-tems by simplifying the nonlinearities based on the normal form theory In this PhDdissertation applications of the proposed methodology are presented on the nonlin-ear modal interaction and stability assessment of interconnected power system

12 Introduction to the Analysis and Control of Dynamics ofInterconnected Power Systems 25

12 Introduction to the Analysis and Control of Dynam-

ics of Interconnected Power Systems

121 Why interconnecting power systems

The analysis of the dynamics of todays electric power systems is a challenging andcomputationally intensive problem that exhibits many interesting and yet to beexplained phenomena

This is due in part to the fact that individual electric utilities are no longer islandsof independent generation and control The interconnections of large regions andthe heavy use of these interconnections present to the power system engineer with alarge stressed nonlinear system to analyse and operate The physical indicators ofstress include the heavy loading of transmission lines generators working to real orreactive power limits and the separation of generation and loads by long distances

Interconnecting the power systems can optimize the exploit of energetic sources andcan provide us the electricity with

bull Better generation

bull Better transmission

bull Better distribution

For example nowadays more and more renewable energy has been included in thepower grids The energy is generated in remote sites and then transferred to theurban areas through long transmission lines Those energy sites are interconnectedwith each others as the super grid which receives more and more attention in theworld as shown in Fig 14 Imagine that we can use the electricity generated in the

Figure 14 Super-grid Transferring the Renewable Energy from Remote Sites tothe Cities

remote areas by the renewable energies in the North Europe while we are in our


comfortable home in France Whats more as shown in Fig 14 an area benetsfrom the electricity transferred from several electrical sites therefore to have abetter distribution and more secured electricity supply

122 Why do we study the interconnected power system

bull Its own value studying the system dynamics of interconnected power systemcan serve

bull Its value as a reference to other systems

123 The Questions to be Solved

To analyze and control the dynamic performance it is to answer two questions

bull Q1 In the microscopic view How system components interact

with each other Modal Interaction

bull Q2 In the macroscopic view Will the system come to the

steady state Stability Assessment

Q1 focuses on the details of the system dynamics such as the source of oscillations(oscillation is a motion that occurs often in system dynamics and normally harmfulfor system operation) The answer to Q1 can aid us to damp the oscillations (oreven to use the oscillations in some cases)

Q2 focuses on the global tendency of the dynamical behaviour of the system Theanswer to Q2 can aid us to enhance the stability and to exploit the working capacityof the system

Normally to study the system dynamics there are both the numerical or experimen-tal approach and the analytical approach By the step-by-step numerical simulationor the experimental measurements we can observe the system dynamics such asthe waveforms of the oscillations but we cannot identify the sources of the oscilla-tions and predict the stability And thereby we cannot develop the control strategyto enhance the stability or damp the oscillations Therefore analytical analysistools based on the mathematical model are needed to give physical insights of theoscillations

If the system is linear or can be linearized those two questions have been alreadysolved by small-signal (linear modal) analysis

However the nonlinearities and couplings in power system make the small-signal(linear modal) analysis fails to study the system dynamics when the system workin a bigger range Where do the nonlinearities and couplings come from To


understand the nonlinearities and couplings we should rst understand the principleof electricity generation

124 The Principle of Electricity Generation

The nonlinearities and couplings originate from the electricity generation The elec-tricity is generated due to the interaction between the stator magnetic eld and therotor magnetic eld A generator is composed of stator and rotor The stator iscomposed of windings and the rotor is equivalent to a magnet The currents inducemagnetic elds and the alternating currents in the stator windings induce a rotatingstator magnetic eld If we put the rotor (which is equivalent to a magnet) in thestator magnetic eld The interaction between the stator magnetic eld and rotormagnetic eld will generate electro-magnetic torque The rotor is driven by theelectro-magnetic torque and the mechanical torque therefore the mechanical en-ergy is converted into electricity (generator) or inversely (motor) Fig 15 indicatesthe interaction between the stator and rotor magnetic elds

Figure 15 The magnetic eld of rotor and stator

where ~BS is the vector representing stator magnetic eld and ~BR is the vectorreprenting rotor magnetic eld Therefore the electrical power is a vector crossproduct of the stator and rotor magnetic eld as Eq (11) If the angular displace-


ment between the stator magnetic eld and rotor magnetic eld is dened as RotorAngle with the symbol δ Then the electric power is a nonlinear function of δ asindicated by Eq (11)

Pe = k ~BS times ~BR = kBSBRsin δ

δ = ωsω

ω = (Pm minus Pe minusDω(ω minus 1))(2H)

(11)

To study the power dynamics we focus on the dynamics of rotor angles Next weillustrate the nonlinearities and couplings in the dynamics To start we introduceour focus on the rotor angle dynamics and basic assumptions

125 The Focus on the Rotor Angle Dynamics

If we connect the stator to the great grid (which is often modelled as the innite bus)the electricity then can be transferred to the users The power can be transferredsmoothly when the rotor angle is constant However when a disturbance occurs onthe transmission line the rotor angle will have oscillations And its dynamics areshown in Fig 17

Eprime 6 δVinfin = 16 0

Innite Bus

Great Grid

G1

Pe

jXe

minusrarr

E

Figure 16 A Disturbance Occurs On the Transmission Line

t(s)0 5 10 15 20 25

δ(degree)

30

40

50

60Predisturbance

PostdisturbanceDisturbance

13 A Brief Review of the Conventional Small-singal (Linear Modal)Analysis and its Limitations 29

Figure 17 The Trajectory of Dynamics

As shown in Fig 17 the disturbance will make the rotor leave the steady-state andcause oscillations It will then reach to a steady state or become unstable What westudy rst is the system dynamics after the fault is cleared ie the postdisturbancestage From the point view of study of dynamical systems what we focus rst isthe free-oscillation In the Chapter 5 we talk about the system dynamics of forcedoscillations

126 Assumptions to Study the Rotor Angle Dynamics in thisPhD dissertation

Several assumptions and simplications are made in modelling the system dynamicsto study the rotor angle dynamics with acceptable complexity and accuracy

1 Steady-State Operating Condition in the steady state operating condi-tion we assume that all the state-variables keep constant or change periodi-cally Though power systems are in fact continually experiencing uctuationsof small magnitudes for assessing stability when subjected to a specied dis-turbance it is usually valid to assume that the system is initially in a truesteady-state operating condition

2 Disturbance the disturbance occurs on the transmission lines in the formof three-phase short circuit fault The fault is supposed to be cleared with orwithout the line tripped o

3 Voltage Stability large volume capacitor is installed to avoid the voltagecollapse

4 Frequency Stability the generation and load are balanced to avoid thefrequency instability the load is assumed as constant impedance during thefault

In the next two subsections we will rst introduce the linear analysis and to showthat linear analysis fails to study the system dynamics when there exists nonlin-earities and couplings in power system dynamics Based on which we illustratethe necessities to develop nonlinear analysis tools and why we choose methods ofNormal Forms

13 A Brief Review of the Conventional Small-singal

(Linear Modal) Analysis and its Limitations

The system dynamics of rotor angle in the interconnected power system when thereis a disturbance in the transmission line are usually caracterized by oscillations


The numerical simulation tools such as EMTP Maltab or the monitoring measure-ments have been widely used to obtain a picture of the system dynamics especiallythe oscillations in the interconnected system However those tools fail to predictthe system dynamical behaviour and identify the sources of oscillations

Small signal analysis is always an important tool in power system research tostudy the modal interaction and to assess the stability

To give the reader an idea of the small-signal analysis Kundurs 4 machine 2 areasystem is taken as an example of interconnected power systems with a tie line powerow from Area 1 to Area 2 as shown in Fig 18 All the generators are equippedwith exciters (Automatic Voltage Regulators VARs) and large capacitance banksto avoid voltage collapose

Example Case 1


When there is a disturbance in Bus 7 all the generators as well as their excitersare involved For example if there is a three-phase fault at bus 7 and cleared after015s oscillations can occur in the rotors of generators shown by the black curvein Fig 19 coming from the time-domain step-by-step numerical simulation of thevalidated demo pss_Power (no PSS is equipped) Data for the system and theselected case are provided in the Appendix section


0 2 4 6 8 10 12

δ1(degree)

40

60

80

100

120

140

t(s)0 2 4 6 8 10 12

δ2(degree)

0

20

40

60

80

100

Exact

Linear

Figure 19 Example Case 1 Linear Small Signal Analysis Under Small Disturbance

Since the corresponding mathematical model is a set of high-dimensional nonlineardierential equations whose analytical solution is not available in most cases acommon practice is to linearize this system around the equilibrium point and to usethe linearized model to approximate the system dynamics This is the principle ofthe conventional small-signal analysis or linear small-signal analysis With the lin-earized system the eigenvalue analysis can be computed Each eigenvalue indicatesa mode of oscillation that the system can exhibit in the dynamics and the systemdynamics is a superposition of those modes local stability properties are given bythe eigenvalues of the linearized system at each equilibrium point

A brief review is given here starting by the dynamical system modeled as

x = f(x) (12)

Linearizing (12) it leads4x = A4x (13)

with A collects all the coecients of rst-order terms in Taylor series Aij = partfixj

andapply the eigen analysis which is started by the similarity transformation x = Uy


it leads toyj = λjyj (14)

λj = σj + jω is the jth eigenvalue of matrix A U and V are the matrices of theright and left eigenvectors of matrix A respectively

131 Small-Signal Stability

In the sense of small-signal stability the stability of mode j can be predicted by σjfor j = 1 2 middot middot middot n if the sign of σj is negative it is stable if possitive it is unstableAnd the stability margin is decided by the eigenvalue closet to the imaginary axiswhich is normally named as the critical eigenvalue One method to improve thesystem dynamic performance is to tune the controller parameter to obtain the bestcritical values

132 Participation Factor and Linear Modal Interaction

The eigenvectors transform the original system into jordan form while the linearpart is decoupled into independent modes characterized as eigenvalues Thereforethe eigenvectors contain the information that can be used to quantify the contribu-tion of state-values to the modes The contribution of k-th state variable to the i-thmode can be quantied as participation factor pki which is dened as [1]

pki = ukivik (15)

The matrix P with pki in the k-th row and i-th column is the participation matrixNormally the column vectors of pki are scaled to have the largest element equals to1 or are normalized to have a sum of 1 to provide more intuitive physical meanings

The small-signal stability and the linear modal interaction are the fundamentals ofconventional small-signal analysis by which the frequency of the oscillation and theparticipation of state-variables keep to the modes can be identied Based on theparticipation of state-variables keep the oscillations fall into four broad categorieswhich will help us to understand the complex dynamics in power system oscillations

1 Local mode oscillations these oscillations generally involve nearby powerplants in which coherent groups of machines within an area swing againsteach other The frequency of oscillations are in the range of 1 to 2 Hz Thedominant state-variables keep are the rotor angle and rotor angular speeds ofnearby power plants

2 Inter-area mode oscillations these oscillations usually involve combina-tions of many synchronous machines on one part of a power system swingingagainst machines on another part of the system Inter-area oscillations arenormally of a much lower frequency than local machine system oscillations in


the range of 01 to 1 Hz These modes normally have wide spread eects andare dicult to control

3 Control modes oscillations these oscillations are associated with generat-ing units and other controls Poorly tuned exciters speed governors HVDCconverters and static var compensators are the usual causes of instability ofthese modes the frequency is normally higher than 4 Hz

4 Torsional modes oscillations these oscillations are associated with theturbine-generator shaft system rotational components Instability of torsionalmodes may be caused by interaction with excitation controls speed governorsHVDC controls and series-capacitor-compensated lines the frequency is alsohigher than 4 Hz

Those modes of oscillations can be again illustrated by the example of Kundurs 2area 4 machine system with the dynamics as shown in Fig 19 The generators aremodelled using a two-axis fourth-order model and a thyristor exciter with a TransientGain Reduction Loads L1 and L2 are modeled as constant impedances and no PowerSystem Stabilizer is used The eigenvalues and eigenvectors are calculated by theopen source Matlab toolbox PSAT [23] v1210 and all the oscillatory modes arelisted in Tab 11

Table 11 Oscillatory Modes Example Case 1

Mode Eigenvalue Pseudo-Freq Damping Time Constant Dominant (Hz) Ratio () t = 1

ζωnStates

56 minus404plusmn j9209 147 439 0247 Eprimeq1 E

primeq2 E

primeq3 E

primeq4

78 minus653plusmn j87132 139 75 0153 Control unit G2910 minus1859plusmn j61621 102 289 0054 Control unit (G3G4)1112 minus1764plusmn j6367 105 267 0057 Control unit G1(δ1 δ2 ω1 ω2)1314 minus208plusmn j674 112 295 0481 Local Area 2(δ3 δ3 ω4 ω4)1516 minus239plusmn j653 111 344 0418 Local Area 1(δ1 δ2 ω1 ω2)1718 minus054plusmn j283 045 187 185 Interarea (δ1 δ2 δ3 δ4)

As observed from Tab 11 the system dynamics may contain the local mode oscil-lation interarea mode oscillation and control mode oscillation By the small signalanalysis it can be identied that the system dynamics in Fig 19 exhibit interareaoscillations at the moment

By small-signal analysis the system dynamics can be predicted and quantied with-out performing the step-by-step time-domain numerical simulation For example itcan be deduced from the 2nd row in the Tab 11 that when there is disturbance inthe controller such as the parameter variation it will cause a control mode oscil-lation at the frequency around 139Hz By small-signal analysis it can also foundout that the electromechanical oscillations (whose modes are dominated by the rotorangle and rotor angular speed) are either local mode oscillations or interarea oscil-lations Among them the interarea oscillation is a unique phenomenon presentedin interconnected power systems leading to global problems and having widespread


eects Their characteristics are very complex and signicantly dier from those oflocal plant mode oscillations

The eigenvalues and eigenvectors obtained by the linear small signal analysis canprovide quantitative results and have several applications as follows

bull Describing the systems response The eigenvalues of the system indicatethe decaying speed and frequencies that may be observed in the oscillations ofsystem variables For a given initial condition the linearized systems responsecan be expressed in a closed form using the eigenvalues and eigenvectors of thesystem The eigenvalues are also a tool to analyse the stability of linearizedsystems Even the system dynamics can be very complex when several modesare excited under disturbance it can be approximated by the superposition ofthe dynamics of several modes Thus linear analysis completely describes thelinearized systems response with reduced complexity

bull Aiding in the location and design of controls When it comes to inter-connected power systems one would like to gure out the interactions betweenthe system components which are often referred to as modal interactions insome literatures Moreover although the number of inertial modes is equalto the number of system states (2[n minus l]) typically only a few modes domi-nate the system response Under stressed conditions modes related to groupsof machines from dierent regions begin to dominate The participation anddominance of some modes will inuence the performance of controls the infor-mation being contained in the eigenvectors The eigenvectors themselves havebeen used to aid in location of controls [24] and various forms of participationfactors [2527] (which are derived using the eigenvectors) have been used todetermine the eectiveness of controls on the system modes This is referredas modal analysis [1 2427] and control techniques such as observability andcontrollability are well developed Since 1980s linear modal analysis has beenextensively applied to the power system participation factors and measures ofmodal dominances and are used extensively to characterize linearized powersystem behaviour

133 Linear analysis fails to study the system dynamics when itexhibits Nonlinear Dynamical Behavior

Although the linear analysis can provide answers to modal interaction and stabilityassessment and therefore we can improve the system dynamic performance How-ever it fails to study the system dynamics when it exhibits nonlinear dynamicalbehavior We can illustrate this using a SMIB system that has presented in theprevious sections

When the disturbance is small the oscillations are almost linear as shown in Fig 110and linear analysis can well approximate the system dynamics


Figure 110 Linear Oscillations after Small Disturbance

What happens if the disturbance is bigger

When the disturbance is bigger as shown in Fig 111 the oscillations are predom-inantly nonlinear which is unsymmetrical and the frequency doesnt coincide withthe imaginary part of the eigenvalue

t(s)0 05 1 15 2 25

δ(rad)

-1

0

1

2

Exact Linear Equilibrium Point

Figure 111 Linear analysis fails to approximate the nonlinear dynamics

When the disturbance is even bigger as shown in Fig 112 the system dynamicsindicated by the black curve is unstable while the linear analysis still predicts asmall-signal stability

This is because that the electric power Pe is a nonlinear function of δ as Eq 16

Pe =EprimeV0X

sin δ (16)


Figure 112 Linear analysis fails to predict the stability

When the disturbance is small sin δ asymp δ and 4x = A4x The system dynamicscan be studied by linear analysis When the disturbance is bigger the higher-orderterms in Taylor series should be taken into account

134 How many nonlinear terms should be considered

If nonlinear terms should be kept then how many

Since Taylors expansion of sin δ around the stable equilibrium point (SEP)δsep is

sin δ = sin δsep + cos δsep(δ minus δsep) (17)

minus sin δsep2

(δ minus δsep)2 minuscos δsep

6(δ minus δsep)3 + middot middot middot

If we plot the power VS rotor angle in Fig 113 we can observe

bull To exploit the power transfer system has to work in the nonlinear region

bull System dynamics such as the stability should be accurately studied by con-sidering Taylor series terms up to order 3

135 Linear Analysis Fails to Predict the Nonlinear Modal Inter-action

Linear small signal analysis is accurate in the neighborhood of the equilibrium butthe size of this neighbourhood is not well dened There is also evidence that the


Power Angle δ(degrees)10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

Pow

erf(δ)

02

05

08

1

Linear Analysis

2nd Order Analysis

3rd Order Analysis

Regionmaybeunstable

Figure 113 Dierent Order Analysis and the Operational Region of the PowerSystem

nonlinear interactions of the inertial modes increase under stressed conditions [28]With the emergence of the interarea mode and other problems not explained bylinear analysis there is a need to extend the analysis to include at least some ofthe eects of the nonlinearities For example when for a poorly damped case of theKundurs 2 area 4 machine system the linear small-signal analysis fails to approxi-mate the interarea oscillations as shown in Fig 114 and frequency of oscillationsdo not corroborate with the results obtained by the linear small-signal analysis asshown in Fig 115

Example Case 2

The interarea oscillations occur as two groups of generators in dierent regions os-cillate with respect to each other Under a large disturbance (nonlinearities cannotbe ignored) interarea-mode response can be described as follows initially followingthe large disturbance generators closed to the disturbance are accelerated as the


0 5 10 15 20 25 30

δ1(degree)

40

70

100

130140

t(s)0 5 10 15 20 25 30

δ2(degree)

0

30

60

90100

ExactLinear

Figure 114 Example Case 2 Linear Small Signal Analysis of Stressed Power SystemUnder Large Disturbance

0 036 072 108 144 180 212 252Hz

101

102

Mag

(

of F

unda

men

tal)

FFT Analysis of the Power Angle ResponseExactLinear

Fundamental Frequency

2nd OrderFrequency

3rd OrderFrequency


14 Reiteration of the Problems 39

transient continues other generators which may be far from the disturbance alsobecome adversely aected This system wide response involving a large number ofgenerators is in contrast with the more local response involving only a few gener-ators and typical of unstressed power systems In addition these interarea-modeoscillations may cause cascaded instabilities that occur at some time after the initialswing caused by the fault

In a word the interrearea oscillations are due to the interactions between the systemcomponents Understanding the interarea oscillations is to understand the interac-tions between the system components which is of great signicance in optimizing theoperation of the whole system As shown in Fig 114 when the system is stressedthe linear analysis fails to approximates the system dynamics As observed fromFig 115 there are components at higher-order frequencies And those higher-orderfrequency components correspond to nonlinear modal interactions that fails to becaptured by linear small-signal analysis

14 Reiteration of the Problems

141 Need for Nonlinear Analysis Tools

Therefore considering the nonlinearities and couplings and in the power systemthe questions of Modal Interaction and Stability Assessment are essentially

bull Nonlinear Modal Interaction how the system components interact with eachother nonlinearly

bull Nonlinear Stability Assessment how the nonlinearities inuence the stabilityof the system dynamics

Since the linear analysis fails to do

142 Perspective of Nonlinear Analysis

The answers needed to improve the system dynamic performance ie to develop themetho

bull Nonlinear Modal Interaction =rArr Damp the oscillations

bull Nonlinear Stability Assessment =rArr Enhance the system stability or increasethe power transfer limit

To give the reader a clear idea of how the control strategy can be implemented sev-eral examples in the linear control are introduced and this PhD work will point thathow those applications can also be used to improve the system dynamic performancewhen it works in the nonlinear region


1 Aiding in the location and design of controls

2 Tuning Controller Parameters

3 Optimizing the Power Transfer

1421 Aid in the Location and Design of Controls

When it comes to interconnected power systems one would like to gure out theinteractions between the system components which is often referred as modal in-teractions in some literatures Moreover although the number of inertial modes isequal to the number of system states (2[N minus l]) typically only a few modes dom-inate the system response Under stressed conditions modes related to groups ofmachines from dierent regions begin to dominate The participation and domi-nance will inuence the performance of controls The information is contained inthe eigenvectors The eigenvectors themselves have been used to aid in location ofcontrols [24] and various forms of participation factors [2527] (which are derivedusing the eigenvectors) have been used to determine the eectiveness of controlson the system modes This is referred as modal analysis [1 2527] and controltechniques such as observability and controllability are well developed Since 1980slinear modal analysis has been extensively applied to power system participationfactors and measures of modal dominances are used extensively to characterizelinearised power system behaviour

1422 Automatic Tuning Controller Parameters

Sensibility method based on the small-signal analysis make possible to automati-cally tuning the controller parameters The objective is to enhance the small-signalstability Numerous examples can be found A recent example can be found inthe voltage source converters controlled by strategy of virtual synchronous machine(VSM) where three controller loops are employed and the frequency domain design(appropriate for a maximum number of 2 loops) [29]


Once the power transfer limit of each generator can be gured out a global op-timization of power transfer in the power grid is possible and we can obtain themaximum economic benets from the whole power grid

The applications are not conned to the above examples

15 A Solution Methods of Normal Forms 41

143 The Diculties in Developing a Nonlinear Analysis Tool

As pointed out in Section 134 we should consider nonlinear terms in Taylor seriesup to order 3 to be accurate enough Therefore for a N -dimensional system thenonlinear terms should be in the scale of N4 How to deal with such an enormousquantity of nonlinear terms

15 A Solution Methods of Normal Forms

Extending the small-signal analysis to the nonlinear domain by including the higherorder polynomial terms in Taylor series may lead to cumbersome computationalburden since analytical results for nonlinear dierential equations are not ensuredAnd the strong couplings in the nonlinear terms make it dicult to extract usefulinformations about how the components interact with each other

One solution is to simplify the nonlinear terms to obtain a simplest form of thesystem dynamics where system property is maintained with reduced complexityThis simplest form is called as the normal forms as rstly proposed by Poincareacutein 1899 To solve the problem the rst step is to have the mathematical formulationof the system dynamics as Fig 18

x = f(xu) (18)

Doing the Taylors expansion around the Stable Equilibrium Point (SEP)

4x = A4x+ f2(4x2) + f3(4x3) (19)

If linear analysis uses a linear change of state variables (linear transformation) tosimplify the linear terms

4x = A4x 4x=Uy=====rArr yj = λjyj (110)

As indicated by Fig 116 if the system dynamics are nonlinear after decouplingthe linear part the system of equation is as Eq(a) with decoupled linear part andcoupled nonlinear part What if we add a nonlinear change of variables(nonlineartransformation) as Eq(b) Will we obtain an equation with simplied nonlinearitiesas Eq(c)


yj = λjyj + F2j(y2) + F3j(y3) + middot middot middot (a)

zj = λjzj + simpler nonlinearities+ middot middot middot (c)

y = z + h2(z2) + h3(z3) + middot middot middot (b)

Figure 116 An idea using nonlinear change of variables to simplify the nonlinear-ities

The answer is YES And it is answered by H Poincareacute in 1899 in his book NewMethods of Celestial Mechanics And the equation of simplied nonlinearities iscalled as normal form(des formes normales)

Poincareacute introduced a mathematical technique for studying systems of nonlinear dif-ferential equations using their normal forms [30] The method provides the means bywhich a dierential equation may be transformed into a simpler form (higher-orderterms are eliminated) It also provides the conditions under which the transforma-tion is possible Although the normal form of a vector eld or dierential equationmay be expressed using a number of forms the polynomial form is selected becauseof its natural relationship to the Taylor series The polynomial normal form of asystem of equations contains a limited number of nonlinear terms which are presentdue to resonances of system eigenvalues This normal form is obtained via a non-linear change of variable (nonlinear transformation) performed on Taylor series ofthe power systems nonlinear dierential equations The nonlinear transformationis derived by cancelling a maximum of nonlinearities in the normal forms

Once the normal form is obtained the quantitative measures can be used to answerthe modal interaction and assess the stability

151 Several Remarkable Works on Methods of Normal Forms

During the last decades there are several remarkable works on methods of normalforms in both electrical engineering and mechanical engineering Those methodswill be reviewed in this PhD dissertation

Electrical Engineering

Method 2-2-1 [Vittal et all 1991] [28] Method 3-3-1 [Martinez 2004 [31]Huang2009 [32]] Method 3-2-3S [Amano 2006] [5]


Figure 117 The People Working on Methods of Normal Forms

Mechanical Engineering Method NNM [Jezequel et Lamarque1991 [33] Touzeacute etThomas 2004 [17]]

152 Why choosing normal form methods

The existing analytical and statistical tools for nonlinear system dynamics are re-viewed in this Section to show the previlege of methods of normal forms

1521 Modal Series Methods for Modal Interaction

Other methods such as the Modal Series (MS) [3436] and the Perturbation Tech-nique (PT) [37] have been used to study the power system dynamic performanceThese papers have concluded that in the stressed power systems the possibility ofnonlinear interactions is increased and thereby the performance of power systemcontrollers will be deteriorated As concluded in [37] the PT and MS method areidentical And as concluded in [35 38] the MS method exhibits some advantagesover the normal form method Its validity region is independent of the modes res-onance does not require nonlinear transformation and it can be easily applied onlarge power systems leading the MS method once more popular in recent years

The principle of the MS method is to approximate the system dynamics by a non-linear combination of linear modes 1) all the higher-order terms are kept 2) thestability is predicted by the linear modes

Therefore the MS method 1) is not appropriate for nonlinear stability analysis 2)the complexity increases largely as the order of the nonlinear terms increase


1522 Shaw-Pierre Method

Another method Shaw-Pierre method which originated from the center-manifoldtheory is popular in studying the nonlinear dynamics of mechanical systems [39]In [40] nonlinear normal mode is formulated by SP method to study the nonlineardynamics of interconnected power system However the nonlinearities studied isonly up to 2nd order and this approach itself has some drawbacks making it onlyapplicable in very few cases

Detailed discussion on MS method and on SP method with mathematical formula-tion can be found in Chapter 2 (Section 239)

1523 Methods for Transient Stability Assessment for Power Systems

Among all the methods the direct methods arouse the biggest interest since it pro-vides the analytical results which quanties the stability margin Direct MethodsThese approaches are also referred to as the transient energy function (TEF) meth-ods The idea is to replace the numerical integration by stability criteria The valueof a suitably designed Lyapunov function V is calculated at the instant of the lastswitching in the system and compared to a previously determined critical value VcrIf V is smaller than Vcr the postfault transient process is stable (Ribbens-Pavellaand Evans 1985 [41]) In practice there are still some unresolved problems anddrawbacks of this approach The eciency of this method depends on the chosensimplications of the system variables keep The integration of the fault on systemequations is needed to obtain the critical value for assessing stability It is dicult toconstruct the appropriate Lyapunov function to reect the internal characteristicsof the system The method is rigorous only when the operating point is within theestimated stability region

Probabilistic Method (Anderson and Bose 1983 [42]) With these methodsstability analysis is viewed as a probabilistic rather than a deterministic problembecause the disturbance factors (type and location of the fault) and the condition ofthe system (loading and conguration) are probabilistic in nature Therefore thismethod attempts to determine the probability distributions for power system sta-bility It assesses the probability that the system remains stable should the specieddisturbance occur A large number of faults are considered at dierent locations andwith dierent clearing schemes In order to have statistically meaningful results alarge amount of computation time is required (Patton 1974 [43]) Therefore thismethod is more appropriate for planning Combined with pattern recognition tech-niques it may be of value for on-line application

Expert System Methods (Akimoto 1989 [44]) In this approach the expertknowledge is encoded in a rule-based program An expert system is composed oftwo parts a knowledge base and a set of inference rules Typically the expertisefor the knowledge base is derived from operators with extensive experience on a


particular system Still information obtained o-line from stability analyses couldbe used to supplement this knowledge

The primary advantage of this approach is that it reects the actual operation ofpower systems which is largely heuristic based on experience The obvious drawbackis that it has become increasingly dicult to understand the limits of systems undertodays market conditions characterized by historically high numbers of transactions

Database or Pattern Recognition Methods The goal of these methods is toestablish a functional relationship between the selected features and the location ofsystem state relative to the boundary of the region of stability (Patton 1974 [43]Hakim 1992 [45] Wehenkel 1998 [46]) This method uses two stages to classifythe system security (a) feature extraction and (b) classication The rst stageincludes o-line generation of a training set of stable and unstable operation statesand a space transformation process that reduces the high dimensionality of the initialsystem description The second stage is the determination of the classier function(decision rule) using a training set of labeled patterns This function is used toclassify the actual operating state for a given contingency Typically the classierpart of this approach is implemented using articial neural networks (ANNs)

1524 Limitations of the Above Methods

Although there are other methods that can provide analytical solution they havesome limitations

bull Modal Series nonlinearities are not simplied

bull Shaw-Pierre Method only for system with non-positive eigenvalues

bull Direct Methods based on Lyapunov theory and other existing methods forTSA only for stability assessment not for nonlinear modal interaction

1525 Advantages of Methods of Normal Forms

bull It keeps the concept of modes for a better understanding of nonlinear modalinteraction

bull It is for both stable and unstable system for not only the nonlinear modalinteraction but also stability assessment

bull The nonlinearities are simplied in the mathematical model which may con-tribute to the control


16 A Brief Introduction to Methods of Normal Forms

The principle of methods of normal forms is to obtain the normal forms Theprocedures are shown in Fig 118

x = f(xu)Original Systemx is a vector ofstate variables

4x = A4x+ f2(4x2) + f3(4x3) + middot middot middot

Expansion by Taylor series

Perturbationmodel

yj = λjyj + F2j(y2) + F3j(y3) + middot middot middot

Linear change of state-variables4x = Uy

Linear decouplingj = 1 2 middot middot middot NLinear Modeswith all Nonlin-earities

zj = λjzj + gj2(z2) + gj3(z

3)

Nonlinear change of state-variablesyj = zj +hj2(zkzl) +hj3(zpzqzr) + middot middot middot

Nonlinear Modeswith simpliednonlinearities

Figure 118 Procedures to Obtain the Normal Forms

The goal of methods of normal forms is to carefully choose the coecients of thenonlinear transformation h2h3middot middot middot to simplify the nonlinearities in the normal formas much as possible The most ideal condition is that all the nonlinearities arecancelled However nonlinear terms that satisfy the resonance must be kept Theresonance occurs if the nonlinear term has a frequency equivalent to the frequencyof the dominant modes

161 Resonant Terms

For the weakly damped modes λj = σj + jωj σj asymp 0 the resonance occurs whenthe terms have the same frequency of the dominant mode j

sumNk=1 ωk = ωj For

mode j

zkzl ωj = ωk + ωl ωj = 2ωk (2nd Order Resonance)

zpzqzr ωj = ωp + ωq + ωr ωj = 3ωk (3nd Order Resonance)

terms zkzl and zpzqzr are resonant termsIn this PhD dissertation the focus is not on resonant terms and resonances areavoided when selecting the case studies However there is a type of resonance thatcannot be avoidedSpecial Case trivially resonant termsIf there are conjugate pairs λ2kminus1 = λlowast2kω2kminus1+ω2k = 0 then ω2k+ω2kminus1+ωj equiv ωj


For mode j terms zjz2kz2kminus1 are trivially resonant terms which intrinsic and inde-pendent of the oscillatory frequenciesTrivially resonant terms must be kept

162 How can the methods of normal forms answer the questions

Normal forms = Linear mode + resonant terms

zj = λjzj +Nsum

ωkωl

NsumisinR2

gj2klzkzl +Nsum

ωpωq ωr

NsumisinR3

gj3pqrzpzqzr (111)

bull Stability Assessment of mode j can be indicated by λj gj2 g

j3

The system dynamics is to add a nonlinear combination of modes to the linearmodes

yj = zj + hj2(zkzl) + hj3(zpzqzr) (112)

bull Nonlinear Modal Interaction indicated by h2 and h3Normal form is a simplest form with the most information

163 Classication of the Methods of Normal Forms

Dierent methods of rnormal forms are proposed in the last decades to cater fordierent requirements To make their dierences obvious we can classify and namethem by the order of Taylor series order of nonlinear transformation order ofnonlinear terms in normal forms whether nonlinear terms in the normal forms aresimplied (S) or not

3rd order of Taylor series keeping all terms in normal forms

Fn hn gn S or

order of Nonlinear transformation order of normal forms

17 Current Developments by Methods of Normal

Forms Small-Signal Analysis taking into account

nonlinearities

Conventionally the study of small-signal stability employs small-signal analysiswhere the power system is not stressed and the disturbance is small And assess-ment of transient employs transient analysis where the power system experiences


large disturbance However small-signal analysis are not isolated from the transientanalysis For example the interarea oscillations which is a big concern in the smallsignal analysis is rstly observed in the transient analysis (when there is a threephase short circuit fault in the transmission line of the interconnected power sys-tem and cleared without tripping o the line) but its property is studied by thesmall-signal analysis Also as nowadays the power system is a more stressed sys-tem exhibiting predominantly nonlinear dynamics compared to before When thesystem experiences large disturbances not only the transfer limits are concernedbut also the sources and frequencies of oscillations should be studied For this issuethe small-signal analysis is more appropriateTherefore in this PhD work the small-signal stability is extended to the nonlin-ear domain to study the nonlinear oscillations in the interconnected power systemwhen the system experiences large disturbance (which is conventionally studied bytransient analysis)Therefore it essentially leads tobull Small-signal analysis taking into account nonlinearities

171 Inclusion of 2nd Order Terms and Application in NonlinearModal Interaction

bull Taylor Series yj = λjyj + F2j(y2)

bull Nonlinear Transformation y = z + h2(z2)

bull Normal Form zj = λjzj

2nd order of Taylor series keeping all rst-order terms in normal forms

2 2 1

2rd order Nonlinear transformation 1st order normal forms

Gradually established and advocated by investigators from Iowa State University inthe period 1996 to 2001 [615] it opens the era to apply 2nd Order Normal Formsanalysis in studying nonlinear dynamics in power system Its eectiveness has beenshown in many examples [691214152047] to approximate the stability boundary[9] to investigate the strength of the interaction between oscillation modes [61047]to dealt with a control design [1112] to analyze a vulnerable region over parameterspace and resonance conditions [14 15] and to optimally place the controllers [20]This method is well summarized by [48] which demonstrated the importance ofnonlinear modal interactions in the dynamic response of a power system and theability of inclusion of 2nd order termsIn all the referred researches the general power-system natural response (linear ornonlinear) is often considered to be a combination of the many natural modes of


oscillation present in the system The linear modes (eigenvalues) represent thesebasic frequencies that appear in the motions of the systems machines Even forlarge disturbances when nonlinearities are signicant the linear modes play an im-portant role in determining the dynamic response of the machine variables Thenonlinear eects should be considered as additions to the linear-modal picture notas replacements for it [10]In another word those referred researches mainly employ the information of 2ndorder modal interactions to better describe or improve the system dynamic per-formance This is the main advancement of method 2-2-1 compared to the linearanalysisFor the stability it predicts a small-signal stability as the linear analysis

172 Inclusion of 3rd Order Terms

Later the researchers found that inclusion of nonlinear terms only up to order 2fails to explain some phenomenon Therefore explorations are done in inclusion of3rd order terms in the small-signal analysis

1721 Method 3-3-1 to study the 3rd order modal interaction

bull Taylor Series yj = λjyj + F2j(y2) + F3j(y3)

bull Nonlinear Transformation y = z + h2(z2) + h3(z3)

bull Normal Form zj = λjzj + simpler nonlinearities3rd order of Taylor series keeping all rst-order terms in normal forms

3 3 1


The mathematical formulation of method 3-3-1 is rstly proposed by Martiacutenez in2004 and then by Huang in 2009 [32] the nonlinear indexes are proposed to quantifythe 3rd order modal interaction

1722 Method 3-2-3S to study the nonlinear stability



bull Normal Form zj = λjzj + trivially resonant terms for oscillatory modes


3rd order of Taylor series 3rd order terms are simplied in normal forms

3 2 3 S

2nd order Nonlinear transformation 3rd order normal forms

This method is proposed by HAmano in 2007 [16] It keeps the trivially resonantterms in the normal forms to study the contribution of the nonlinearities to thesystem stability A nonlinear index called stability bound is proposed based onthe normal forms which serves as a criteria to tune the controller parameters toimprove the system dynamic performance

173 What is needed

We need a method to include nonlinear terms up to order 3 to study both thenonlinear modal interaction and nonlinear stability

18 Objective and Fullment of this PhD Dissertation

181 Objective and the Task

The objective of this PhD research work can be sprecied asbull Develop a systematic methodology to analyze and control nonlinear multiple-input systems with coupled dynamics based on methods of normal forms upto 3rd orderbull Apply the systematic methodology to study the nonlinear dynamical be-haviour of interconnected power systemsbull Analyze the factors inuencing the performance of the proposed methodologyand gure out its limitations and the way to improve it laying the foundationfor future researches

The fullment of the objectives is composed ofbull Derivation of methods of normal forms (MNFs) up to order 3bull Implementation of generic programs of MNFs applicable for dynamical systemswith arbitrary dimensions wherebull Mathematical modelling of interconnected power system in the form of Taylorseries up to 3rd orderbull Application of MNFs to study the inter-area oscillations and assess the tran-sient stability of interconnected power system

19 A Brief Introduction to the Achievements of PhD Work with ASimple Example 51

19 A Brief Introduction to the Achievements of PhD

Work with A Simple Example

To give the reader a clear idea of the methods of normal forms and the proposedmethod 3-3-3 and method NNM we use a concrete example to illustrate those meth-ods

191 An example of Two Machines Connected to the Innite Bus

G1 δ1

G2 δ2

jX12jXe

ZZPe

Innite Bus

E

Figure 119 A system of Two Machines that are Interconnected and Connected tothe Innite Bus

Pe1 = V1V2X12

sin(δ1 minus δ2) + V1VinfinXe

Xe sin δ1ω1 = (Pm1 minus Pe1 minusD(ω1 minus 1))(2H)

δ1 = ωs(ω1 minus 1)

Pe2 = V1V2X12




(113)

4 state-variables x = [δ1 δ2 ω1 ω2]

Taylor series around equilibrium point

4x = A4x+ a2x2 + a3x

3 (114)

Decoupling the linear part it leads j = 1 2 3 4

yj = λjyj +4sum

k=1

4suml=1

ykyl +4sump=1

4sumq=1

4sumr=1

ypyqyr (115)

It has at least 28 nonlinear terms how to solve it

192 The proposed method 3-3-3

Adding a nonlinear transformation yj = zj + hj2(z2) + hj3(z

3)

Since the maximum number of conjugate pairs are two λ1 λ2 and λ3 λ4


A total maximum of 8 nonlinear terms in the normal forms

zj = λjzj + g3jj12zjz1z2 + g3jj34zjz3z4

Trivially resonant terms

(116)


3 2 3 S


Figure 120 Meaning of the name of method 3-3-3

193 Theoretical Improvements of Method 3-3-3 Compared to ex-isting methods

Tab 12 lists the nonlinear transformations and the normal forms of each method

Table 12 Theoretical Improvements of Method 3-3-3 Compared to existing methods

j=12 yj = zj + hj2(z2) yj = zj + hj2(z

2) + hj3(z3)

zj = λjzj method 2-2-1 method 3-3-1zj = λjzj + gj3zjz1z2 method 3-2-3S method 3-3-3

+gj3zjz3z4

As observed from Tab 12 only method 3-3-3 has both two characteristics as belowbull Method 3-3-3 is up to 3rd orderbull Trivially resonant terms are kept in method 3-3-3

Having both the two characteristics render method 3-3-3 a better performance instudy of system dynamics compared to the existing methods

194 Performance of Method 3-3-3 in Approximating the SystemDynamics Compared to the Existing Methods

As shown in Fig 121 method 3-3-3 approximates best the system dynamics com-pared to the existing methodsBy method 3-3-3 Nonlinear terms are reduced from 28 to 8 while keeping all non-linear propertiesTherefore we can sayMethod 3-3-3 accurately and largely simplies the system


Figure 121 Method 3-3-3 approximates best the system dynamics

195 The Proposed Method NNM

Other than method 3-3-3 another method is proposed which deals with directlyof system modelled by second-order equations It decouple a N -dimensional systemcomposed of coupled second-order oscillators into a series of second-order oscillatorsIts principle can be illustrated by EMR as followingA N -dimensional system with coupled dynamics can be modelled by second-orderdierential equations and represented as Fig 122

Figure 122 N-dimensional System modeled Second-order Dierential Equations

If system dynamics are linear it can be decoupled by a linear transformation asshown in Fig 123If system dynamics are nonlinear it can be approximately decoupled by a nonlineartransformation as shown in Fig 124By Method NNM the nonlinear system dynamics can be decomposed into twononlinear modes as shown in Fig 125


Figure 123 Linear system dynamics can be decoupled by linear transformation

Figure 124 Nonlinear system dynamics can be approximately decoupled by non-linear transformation

196 Applications of the Proposed Methods

In this PhD work it has shown how method 3-3-3 and method NNM can be usedin nonlinear modal interaction and stability assessment The method 3-3-3 canquantify the 3rd order nonlinear modal interaction And by taking into account thetrivially resonant terms it provides a more accurate stability assessment by whichthe transfer limit can be further exploited The method NNM (a simplied NNM)has shown its ability to extract the nonlinear properties and predict the stabilitylimit

110 Scientic Contribution of this PhD Research 55

0 05 1 15 2minus05

0

05

1

15

t(simulation time)

δ1(rad)

Mode Decomposition

EXMode 1Mode 2

0 05 1 15 2minus05

0

05

1

15

t(simulation time)

δ2(rad)

Figure 125 System dynamics are approximately decoupled into two modes

110 Scientic Contribution of this PhD Research

In this PhD work inclusion of 3rd order terms in small-signal analysis is madepossible by the proposed method of normal forms The quantication of 3rd ordermodal interactions and nonlinear stability assessment is guaranteed by the proposednonlinear indexes based on normal forms These points are detailed in Chapter 2and Chapter 3

1101 Contribution to Small Signal Analysis

The increased stress in power system has brought about the need for a better un-derstanding of nonlinearities in power-system dynamic behaviour The higher-orderapproximation of the system of equations contain signicantly more informationthan the traditionally-used linear approximationHowever taking into higher-order terms is extremely complex due to couplings be-tween the terms It therein desires a methodology to simplify the analysis whileproviding the most accurate result leading to

1 Analytical solution is needed to explain the qualitative properties2 Quantitative measures of interactions and nonlinear eects (eg stability

boundary amplitude-dependent frequency-shift) within the system compo-nents are needed

3 Stability analysis


1102 Contribution to Transient Stability Assessment

For transient stability analysis this PhD work studies on the power swing andproposes a stability criterion which is based on the

1103 Contribution to Nonlinear Control

As pointed out in the previous sections the anaylsis of system dynamics can help usnd out the essentials of system physical properties Once a clear picture of systemdynamics is obtained control strategies can be proposed to improve the

1104 Contribution to the Nonlinear Dynamical Systems

Power system dynamics is similar to dynamics of other interconnected system andhas fundamental mathematical underpinningsDetailed mathematical models have been derived and concrete applications aredemonstrated to show the physical insights rendered by the proposed methodol-ogy Therefore the signicance of this PhD dissertation is not conned in powersystem stability analysis and control but can also be used for multiple-input systemwith dierentiable nonlinearities in the mathematical model

1105 A Summary of the Scientic Contributions

To the authors knowledge its scientic contributions can be summarized asbull From the aspect of academic disciplines this PhD research is multi-discplinary associated to applied mathematics nonlinear dynamics mechan-ical engineering and electrical enigneering And it endeavours to extract theessential nonlinear properties which are common for nonlinear dynamical sys-tems having the particular type of coupled dynamics whatever the mechan-ical system or electrical system For example the phenomenon where thefrequency varies with the amplitude in free-oscillations is common for a me-chanical structure [17] and interconnected power systems [22] The results pre-sented in this PhD research can be used to interpret the nonlinear dynamicsof other dynamical systems inside or out of the eld of electrical engineeringbull From the aspect of researches on dynamical system

1 It conducts the nonlinear analysis on both the free-oscillation and theforced-oscillation

2 It endeavours to take the analysis and control of nonlinear dynamicalsystem as a whole which bridges the gap between nonlinear analysis andnonlinear control It proposes the analysis-based nonlinear control andthe decoupling basis control

bull From the aspect of application and implementation1 It is for the rst time that all the existing methods of normal forms up to

3rd order in power system have been summarized and their performanceshave been evaluated


2 It is for the rst time that a real-form normal form method is applied tostudy the transient stability of electric power system

3 It is for the rst time that the mathematical modelling of interconnectedPower System in Taylor series is provided from which the origins of thenonlinearities of the system are detected

4 It is for the rst time that the generic programs of the presented methodsof normal forms are shared on-line to facilitate the future researchersThose programs are applicable for systems with arbitrary dimensions

bull From the theoretical partbull It summarizes the existing MNFs and propose methods of both vector eldsand nonlinear normal modesbull It deduces the normal forms under forced-oscillation and proposes the directionto do the normal forms of the control system (see Chapter 5)

111 The Format of the Dissertation

This PhD dissertation is composed of 6 Chapters In Chapter 2 a general introduc-tion to Normal Form theory is made and two methods are proposed a novel method3-3-3 and method NNM In Chapter 3 method 3-3-3 has been applied to study thenonlinear modal interaction and to predict the nonlinear stability of the intercon-nected power system In Chapter 4 method NNM which is Nonlinear Normal-Modebased approach is applied to study the transient stability analysisChapter 6 concludes on the nonlinear analysis and points out two approaches todo the nonlinear control one is the nonlinear analysis based control the other oneis the control on nonlinear model decoupling basis nonlinear modal control TheChapter 6 contributes mainly in the study of dynamical system and Normal FormtheoryTo make this PhD dissertation as pedagogical as possible each chapter has anabstract in its front page


Chapter 2

The Methods of Normal Formsand the Proposed Methodologies

This chapter is composed of 4 parts

Part I the Normal Form theory with its basic denitions and general procedures to

derive the normal forms are introduced in Section 21 amp 22

Part II 3rd order normal forms of vector elds (method 3-3-3) in Section 23 and

its nonlinear indexes in Section 24

Part III 3rd order normal forms of nonlinear normal modes (method NNM) in

Section 25 and its nonlinear indexes in Section 26

Part IV Practical issues of implementing the methods of normal forms (MNF) are

dealt with in Section 28 The comparisons on vector elds approach and normal

modes approach are conducted in Section 29

Based on the Normal Forms and Normal Form transformation coecients non-

linear indexes are proposed in Section 24 and Section 26 which can be used to

study the nonlinear modal interaction and to assess the nonlinear stability as well

as predict and quantify the parameters inuences on the amplitude-dependent fre-

quency shift and the predominance of the nonlinearity in the system dynamics The

programming procedures and computational techniques are also documented in Ap-

pendix To explain why MNFs are chosen to develop the tools to analyse and control

the dynamical systems the competitors Modal Series (MS) method and Shaw-Pierre

(SP) method are reviewed with mathematical formulation in Sections 239 259

and their limitations are pointed out

This chapter lays the theoretical foundation of the applications presented in Chap-

ter 3 and Chapter 4 The proposed methodologies can also be generally applied to

multiple-input systems with nonlinear coupled dynamics Unlike the previous liter-

atures focused on the bifurcation analysis this chapter focuses on the analysis of

nonlinear modal interaction and nonlinear stability All the nonlinear indexes are

at the rst time proposed by this PhD work

Keywords Methods of Normal Forms System of First-order EquationsSystem of Second-order Equations Nonlinear Indexes Nonlinear ModalInteraction Nonlinear Stability Assessment

59

60Chapter 2 The Methods of Normal Forms and the Proposed

Methodologies

21 General Introduction to NF Methods

The methods of normal forms can be applied to nd out the simplest form of anonlinear system making it possible to predict accurately the dynamic behaviourwhere a complete linearization will lose the nonlinear properties

211 Normal Form Theory and Basic Denitions

Normal form theory is a classical tool in the analysis of dynamical systems andgeneral introductions can be found in many textbooks see eg [30] mainly in themechanical domain It is mainly applied in bifurcation analysis [49] to analyzethe qualitative inuence of parameters on system properties (Bifurcation analysiscomes from the phenomenon that the system properties may be totally dierentwhen there is a only small change of parameters at the point of bifurcation)It is based on Poincareacute-Dulac theorem which provides a way of transformation ofa dynamical system into the simplest possible form which is called Normal Form(formes normales in French [50]) A normal form of a mathematical object broadlyspeaking is a simplied form of the object obtained by applying a transformation(often a change of coordinates) that is considered to preserve the essential featuresof the object For instance a matrix can be brought into Jordan normal form byapplying a similarity transformation This PhD research focuses on Normal Formfor non-linear autonomous systems of dierential equations (rst-order equation onvector elds and second-order equations in form of modes) near an equilibrium pointThe starting point is a smooth system of dierential equations with an equilibrium(rest point) at the origin expanded as a power series

x = Ax+ a2(x) + a3(x) + middot middot middot (21)

where x isin Rn or Cn A is an ntimesn real or complex matrix and aj(x) is a homogeneouspolynomial of degree j (for instance a2(x) is quadratic) The expansion is taken tosome nite order k and truncated there or else is taken to innity but is treatedformally (the convergence or divergence of the series is ignored) The purpose isto obtain an approximation to the (unknown) solution of the original system thatwill be valid over an extended range in time The linear term Ax is assumed to bealready in the desired normal form usually the Jordan or a real canonical form

2111 Normal Form Transformation

A transformation to new variables y is applied which is referred to as NormalForm Transformation and has the form

x = y + h2(y) + h3(y) + middot middot middot (22)

where hj is homogeneous of degree j This transformation can also be done suc-cessively such x = y + h2(y) y = z + h3(z) z = w + h4(w) middot middot middot to render the

21 General Introduction to NF Methods 61

same normal forms Up to order 3 the coecients of one-step transformation andsucessive transformation will be the same a short proof is given as follows

x = y + h2(y)

= z + h3(z) + h2(z + h3(z)) (23)

= [z + h2(z)] + h3(z) +Dh2(z)h3(z)

= z + h2(z) + h3(z) + (O)(4)

2112 Normal Form

This results in a new system as which is referred to asNormal Form in this thesis

y = λy + g2(y) + g3(y) + middot middot middot (24)

having the same general form as the original system The goal is to make a carefulchoice of the hj so that the gj are simpler in some sense than the aj Simplermay mean only that some terms have been eliminated but in the best cases onehopes to achieve a system that has additional symmetries that were not present inthe original system (If the normal form possesses a symmetry to all orders thenthe original system had a hidden approximate symmetry with transcendentally smallerror)

2113 Resonant Terms

Resonant terms cannot be cancelled in Normal FormIf (24) has N state-variables and λj = σj + jωj for the resonance occurs whensumN

k=1 ωk = ωj For example for the 2nd order and 3rd order resonance

ωk + ωl minus ωj = 0 (2nd Order Resonance)

ωp + ωq + ωr minus ωj = 0 (3nd Order Resonance)

terms ykyl and ypyqyr are resonant terms

2114 Normal Dynamics

The solution to the Normal Form (24) is normal dynamics By (22) we canreconstruct the solution of x ie the original system dynamics

2115 Reconstruction of the System Dynamics from the Normal Dy-namics

The system dynamics can be reconstructed from the normal dynamics by x =

h2(y) + h3(y) + middot middot middot This is a method to verify if the normal dynamics containthe invariance propertyHowever this is not an ecient manner to use the Normal Forms In the engineeringpractice indexes can be proposed based on the Normal Forms and Normal Form


Methodologies

transformation to predict interpret and quantify the system dynamics

212 Damping and Applicability of Normal Form Theory

The normal forms are initially proposed for undamped system ie σj = 0 And theresonant terms is dened if only σi = 0 [30 50] the no- damped case However wecan still extend it to weakly-damped case as

λj = minusξjωj plusmn jωjradic

1minus ξ2jasymp plusmnjωj minus ξjωj +O(ξ2j ) (25)

(25) shows that the inuence of damping on the oscillatory frequency is at leasta second-order eect For computing the normal form the general formalism canbe adopted excepting that now the eigenvalues are complex number with real andimaginary parts Physical experiments in [19] proved that approximation of (25)when ξ lt 04 In fact the oscillations of interest in electric power system is ξ le 01therefore Normal Form theory can be extended to study the dynamics of electricpower system

213 Illustrative Examples

2131 Linear System

In fact the eigenvalue matrix is the simplest Normal Form For a linear system asbefore

x = Ax (26)

We can use x = Uy to simplify the equation into

y = Λy (27)

where Λ is a diagonal matrix and the jth equation reads

yj = λjyj (28)

This is the simplest form and the Normal Dynamic of (28) y(t) = y0j eλjt There-

fore transformation x = Uy renders the system in the simplest form a linearNormal Form

2132 Nonlinear System

Let us consider a dynamical system [19]

X = sX + a2X2 + a3X

3 + middot middot middot = sX +sumpgt1

apXp (29)


where X isin R (phase space of dimension n = 1) X = 0 is a hyperbolic point aslong as s 6= 0 otherwise a marginal case is at hand (bifurcation point)Let us introduce a Normal Form transformation

X = Y + α2Y2 (210)

where α2 is introduced in order to cancel the quadratic monom of the original systemie a2X

2 in (29) Here Y is the new variable and the goal of the transformationis to obtain a dynamical system for the new unknown Y that is simpler than theoriginal one Dierentiating (210) with respect to time and substituting in (29)gives

(1 + 2α2Y )Y = sY + (sα2 + a2)Y2 +O(Y 3) (211)

All the calculation are realized in the vicinity of the xed point One can multiplyboth sides by

(1 + 2α2Y )minus1 =+infinsump=0

(minus2)pαp2Yp (212)

Rearranging the terms by increasing orders nally leads to as

Y = sY + (a2 minus sα2)Y2 +O(Y 3) (213)

From that equation it appears clearly that the quadratic terms can be cancelled byselecting

α2 =a2s

(214)

which is possible as long as s 6= 0It leads to the Normal Form

Y = sY +O(Y 3) (215)

Compared to (29) (215) is simpler in the sense that the nonlinearity has beenrepelled to order three And the terms O(Y 3) arise from the cancellation of 2ndorder terms and the 3rd order terms of the original system According to the specicrequirements all the terms or some terms of O(Y 3) will be neglected And it rendersa simpler form for which the analytical solution is available or numerical simulationhas less computational burdenContinuing this process it is possible to propel the nonlinearity to the fourth orderThe reader who is curious can be addressed to the reference [51] where NormalForm of a dynamical system is deduced up to order 11However the computational burden increases extraordinarily Therefore the Nor-mal Form are always kept up to order 2 or order 3 in practice And only 2nd Orderand 3rd Order Normal Form Transformation will be introduced and deduced in thisthesis for for higher order Normal Form interested readers may refer to somereferences as [171851] or deduce it by oneself similar to order 2 or order 3


Methodologies

22 General Derivations of Normal Forms for System

Containing Polynomial Nonlinearities

Although the methods of normal forms can be applied to system with dierenttypes of continuous and dierentiable nonlinearities its applications are primarily insystems containing polynomial nonlinearities or nonlinearities can be approximatedby polynomials after Taylor series expansion In this PhD work we focus on systemwith polynomials up to order 3 and the proposed methodology can be extended tosystem with higher-order polynomialsIn this section the general derivations of normal forms up to order 3 will be deducedwhich lays the basis for the next two sections where normal forms are derived fortwo particular types of systems system of rst order dierential equations on vectorelds and system of second-order equations on modal space

221 Introduction

If the system dynamics can be formalized as

u = Au+ εF2(u) + ε2F3(u) + (216)

where u and Fm are column vectors of length n A is an n times n constant matrixand ε is a small nondimensional parameter which is just a bookkeeping deviceand set equal to unity in the nal result In this PhD dissertation it functions asbookkeeping device to separate terms into dierent degrees In the literature thenormalization is usually carried out in terms of the degree of the polynomials in thenonlinear terms [313248] However some concepts are confused and befuddled intheir works [3132] By introducing the bookkeeping device ε their mistakes can bedeviled

222 Linear Transformation

A linear transformation u = Px is rst introduced where P is a non-singular orinvertible matrix in (216) and it obtains

P x = APx+ εF2(Px) + ε2F3(Px) + middot middot middot (217)

Multiplying (217) by Pminus1 the inverse of P it reads

x = Λx+ εf2(x) + ε2f3(x) + middot middot middot (218)

whereΛ = Pminus1AP and fm(x) = Pminus1Fm(Px)

To assure the accuracy of following transformation P should be chosen so that Λ

has a simple real form [30] or pure imaginary form [33] If A is in complex-valuedform extra procedures should be applied to separate the real parts and imaginary

22 General Derivations of Normal Forms for System ContainingPolynomial Nonlinearities 65

parts [33] However to make it as simple as possible in the previous researches inpower system nonlinear analysis A is always treated as pure imaginary even if it iscomplex-valued and the researches focus on the oscillatory characteristics broughtby the complex conjugate pairs However as separating real parts and imaginaryparts is technically dicult at present the previous researches deduced the formulain the assumption that A is in pure imaginary form This will denitely lead tounexpected errors when try to balance the right side and left side when deducingthe normal form transformation coecients This issue associated to A has beendiscussed deeply by the literature in mechanic domain and will not be addressedin this dissertation (Dumortier1977 [52] Guckenheimer and Holemes 1983 [53])The practice of this PhD work is to assume that A is pure imaginary (as assumedin [313248]) and pointed out the limitations

223 General Normal Forms for Equations Containing Nonlinear-ities in Multiple Degrees

In our analysis both quadratic and cubic nonlinearities are concerned It is provedthat a single transformation and successive transformations can produce the sameresults both theoretically [30] and experimentally on power system [32] up to order3Thus instead of using a sequence of transformations a single normal form transfor-mation

x = y + εh2(y) + ε2h3(y) + middot middot middot+ εkminus1hk(y) + middot middot middot (219)

is introduced into (218) and it leads to the so-called normal form (220)

y = Λy + εg2(y) + ε2g3(y) + middot middot middot+ εkminus1gk(y) + middot middot middot (220)

hm is chosen so that gm takes the simplest possible form The minimum terms keptin gm are the resonant terms In the best case (no resonance occurs) gn is null and(220) turns into y = Λy whose solution is ready at hand Otherwise gn is referredto as resonance and near-resonance termsSubstituting (219) into (218) yields

[I+D(εh2(y)) +D(ε2h3(y)) + middot middot middot ]y (221)

=Λy + εΛh2(y) + ε2Λh3(y) + middot middot middot+ εf2[y + εh2(y) + ε2h3(y) + middot middot middot ]+ ε2f3[y + εh2(y) + ε2h3(y) + middot middot middot ] + middot middot middot

Where Taylor series can be applied

εkminus1fk[y + εh2(y) + ε2h3(y) + middot middot middot ] (222)

= εkminus1fk(y) + εkDfk(y)h2(y) + εk+1Dfk(y)h3(y) + middot middot middot


Methodologies

and

[I+D(εh2(y)) +D(ε2h3(y)) + middot middot middot ]minus1 (223)

= Iminus[D(εh2(y)) +D(ε2h3(y)) + middot middot middot

]+[D(εh2(y)) +D(ε2h3(y)) + middot middot middot

]2+ middot middot middot

Substituting (222) on the right side of (221) multiplying (223) on both twosides of (221) and equating coecients of like powers of ε with the aid of (222)it satises

ε0 y = Λy (224)

ε1 g2(y) +Dh2(y)Λy minus Λh2(y) = f2(y) (225)


+Df2(y)h2(y)minusDh2(y)g2(y)

εkminus1 gk(y) +Dhk(y)Λy minus Λhk(y) = fkminus1(y) (227)

+

ksuml=2

[Dfl(y)hk+1minusl(y)minusDhl(y)gk+1minusl(y)]

where termssumk

l=2[Dfl(y)hk+1minusl(y)minusDhl(y)gk+1minusl(y)] arise from the cancella-tion of terms lower than order k hm is chosen so that normal dynamics of y will bein the simplest form Normally hm is chosen so that gm only contain the resonantterms which cannot be cancelled by Normal Form transformationsIt is seen from (225) that the transformation at degree ε will add terms to degree ε2Df2(y)h2(y) Dh2(y)g2(y) which are important terms obviated in the previousresearches in deducing 3rd order normal forms [3254]We would like to remind the reader that it is the degree of nonlinearity rather thanthe order of nonlinear terms that matters in the normal form transformation1

224 Summary analysis of the dierent methods reviewed and pro-posed in the work

Based on Normal Form theory dierent Normal Form methods are developed de-pending on hn the order of Normal Form Transformation fn the order of termsin Taylor series and gnthe order of the Normal Form As some or all terms of n-thorder will be neglected in gn it can be specied as Full (F) meaning that all theterms are kept and Simplied(S) meaning that some terms are neglectedThe methods proposed in this work can be compared on four features Each method

1Note we would like to notify the dierence between the order of terms and the degrees ofnonlinearities which is not always the same For example if a system of equations only containcubic nonlinear terms and no quadratic nonlinear terms its degree of nonlinearity is ε whenequating the cubic terms it is (225) instead of (226) should be employed


is then labeled using three digits and one optional letter where1 The rst digit gives the order of the Taylor expansion of the systems dynamic

ie the order of fn2 The second digit gives the order of the Normal Form transformation used ie

the order of hn3 The third digit gives the order of the Normal Forms ie the order of g(n)4 The optional letter indicates the fact that some terms have not been taken

into account in g(n) (S for Simplied)In this chapter Normal Forms are deduced by two approaches the vector eld ap-proach for system of rst-order equations and the normal mode approach for systemof second-order equations All the methods are listed in Tab 21 and Tab 21In this PhD dissertation the vector eld methods are labelled as 2-2-1 3-2-3S andetc To avoid the ambiguity the normal mode methods are labelled as LNM NM2FNNM NNM instead of 1-1-1 2-2-1 3-3-3 and 3-3-3STab 21 lists of the four vector eld Normal Forms methods proposed in this work

Table 21 List of Existing and Proposed MNFs with Vector Fields ApproachMethod Taylor NF Normal nth order Related Cited

series Trans Form NF Terms Equation Reference2-2-1 2nd order order 2 order 1 (241) [48]3-2-3S 3rd order order 2 order 3 S (256) [5 16]3-3-1 3rd order order 3 order 1 (258) [3132]3-3-3 3rd order order 3 order 3 (251)(252)

A summary of the existing and proposed normal modes approaches can be foundin Tab 22 Since all those methods are the rst time to be used in interconnectedpower system therefore all those methods are assessed in the Section 27 on inter-connected VSCs to to give the reader a concrete example

225 General Procedures of Applying the Method of NormalForms in Power System

As illustrated in the previous sections the Normal Form theory is dened for equa-tion x = f(x) however in the engineering the x will be constrained by algebraicequations and normally the dierential algebraic equations (DAEs) are formed By

Table 22 List of Existing and Proposed MNFs with Normal Modes ApproachMethod Taylor NF Normal nth order Related Cited

series Trans Form NF Terms Equation ReferenceLNM (1-1-1) 1st order order 1 order 1 (292) [55]NM2 (2-1-1) 2nd order order 2 order 1 (2105) [56]FNNM (3-3-3) 3rd order order 3 order 3 (2123) [1718]NNM (3-3-3S) 3rd order order 3 order 3 S (2126) [2122]


Methodologies

expanding the equation around the stable equilibrium point (SEP) into Taylor se-ries and substituting by xminus xSEP by 4x a disturbance model can be formed as4x = A4x + Higher-order Terms The DAE formalism will be discussed furtherin Section 283The general procedures is well documented in [30 51] and can be adapted to thepower system analysis consisting of eight major steps

1 Building the dierential algebraic equations (DAEs) of the power system thedierential equations and the power ow constraints

2 Solving the power ow to obtain the stable equilibrium point (SEP) for thepost-fault system ie the operating point

3 Expanding the system of equations around the SEP into Taylor series up tothird-order

4 Simplifying the linear part of the system by the use of a linear transformation5 Simplifying the non-resonant terms of higher-order terms by successive Normal

Form (NF) transformations6 Simplifying the Normal Forms by neglecting (if possible) some resonant terms

that can not be annihilated by NF transformations7 Reconstructing the original systems dynamic from the Normal Forms dynam-

ics in order to determine the order of the Taylor series and the NF transfor-mations to be selected according to the expected accuracy

8 Using the chosen Normal Forms for dynamic and stability analysisFollowing the procedures above the Normal Form formula are deduced in the nextsections Those sections can be roughly divided into two parts The rst part isdedicated to system represented by a set of rst order equations the second part isdedicated to system represented by a set of second-order equations the oscillators

23 Derivations of the Normal Forms of Vector Fields for

System of First Order Equations

The vector elds is a mapping where the dynamics of the system are characterizedby a set of rst order dynamical equations

231 Class of First-Order Equation that can be studied by NormalForm methods

The class of systems that can be studied by MNF are usually modeled using Dier-ential Algebraic Equations (DAEs) [1] By subsituting the algebraic equations intothe dierential ones one transforms those DAEs in a dynamical system which canbe written

x = f(xu) (228)

where x is the state-variables vector u is the systems inputs vector and f isa nonlinear vector eld Expanding this system in Taylor series around a stable

23 Derivations of the Normal Forms of Vector Fields for System ofFirst Order Equations 69

equilibrium point (SEP) u = uSEP x = xSEP one obtains

4x = H1(4x) +1

2H2(4x) +

1

3H3(4x) +O(4) (229)

where Hq gathers the qth-order partial derivatives of f ie for j = 1 2 middot middot middot nH1jk = partfjpartxk H2jkl =

[part2fjpartxkpartxl

] H3jklm =

[part3fjpartxkpartxlpartxm

]and O(4) are

terms of order 4 and higher

232 Step1Simplifying the linear terms

The linear part of (229) is simplied using its Jordan form

y = Λy + F2(y) + F3(y) + middot middot middot (230)

supposed here to be diagonal where the jth equation of (232) is

yj = λjyj +Nsumk=1

Nsuml=1

F2jklykyl +Nsump=1

Nsumq=1

Nsumr=1

F3jpqrypyqyr + middot middot middot (231)

λj is the jth eigenvalue of matrixH1 and j = 1 2 middot middot middot n U and V are the matri-ces of the right and left eigenvectors of matrix H1 respectively UV = I F2j =12

sumNi=1 vji

[UThj3U

] F3jpqr = 1

6

sumNi=1 vji

sumNk=1

sumNk=1

sumNm=1H3ikLNMu

lpumq u

Nr vji

is the element at j-th row and i-th colomn of Matrix V ulp is the element at thep-th row and l-th column of matrix U The fundamentals of linear small-signal analysis uses the sign of the real parts σjof the eigenvalues λj = σj + jωj to estimate the system stability U is used toindicate how each mode yj contribute to the state-variable x and V indicates howthe state-variables of x are associated to each mode yj

233 Step 2 Simplifying the nonlinear terms

After the linear transformation to decouple the linear part and obtain the EigenMatrix the system equation reads

y = Λy + F (y) (232)

Λ is the Eigen Matrix and F (y) gathers the higher-order terms of Taylor seriesafter the linear transformation

2331 Method 2-2-1 Cancelling all the 2nd Order Terms

When only 2nd order terms are kept in the Taylor series of the systems dynamics(232) becomes

y = Λy + F2(y) (233)


Methodologies

where the jth equation of (233) writes

yj = λjyj +Nsumk=1

Nsuml=1

F2jklykyl (234)

To eliminate the second-order terms in (233) the following second-order transfor-mation is applied

y = z + h2(z) (235)

The jth equation of (235) being

yj = zj +Nsumk=1

Nsuml=1

h2jklzkzl (236)

Applying (235) to (233) it leads to

z + ˙h2(z) = Λz + Λh2(z) + F2(z + h2(z)) (237)

z(I +Dh2(z)) = Λz + Λh2(z) + F2(z) +DF2(z)h2(z) +HF2(z)(h2(z))2

z = (I +Dh2(z))minus1(Λz + Λh2(z) + F2(z) +DF2(z)h2(z) +HF2(z)(h2(z))2)

z = (I minusDh2(z) + (Dh2(z))2 +O(3))(Λz + Λh2(z) + F2(z) +DF2(z)h2(z) +O(4))

Expanding the equation it obtains

z = Λz (order 1)

minus (Dh2(z)Λz minus Λh2(z)minus F2(z)) (order 2)

+Dh2(z)(Dh2(z)Λz minus Λh2(z)minus F2(z)) +DF2(z)h2(z) (order 3)

+ (Dh2(z))2(Λh2(z) + F2(z))minusDh2(z)DF2(z)h2(z) +O(4) (order 4)

+ (Dh2(z))2DF2(z)h2(z) (order 5)

and keeping terms up to the second order leads to

z = Λz minusDh2(z)Λz + Λh2(z) + F2(z) (238)

where Dh2 is the Jacobin matrix of h2 In order to eliminate the second-order termsfrom (238) transformation (235) must satisfy the following equation

Dh2(z)Λz minus Λh2(z) = F2(z) (239)

If no internal resonance occurs h2jkl is given by [48]

h2jkl =F2jkl

λk + λl minus λj(240)

Neglecting all terms with order superior to 2 the transformed equation is a set of


decoupled rst-order linear dierential equations

z = Λz +DF2(z)h2(z) +O(4) (241)

= Λz +O(3)

= Λz

2332 Method 3-2-3S Keeping some third order terms

Although the second-order method gives a more accurate picture than the linearsmall-signal analysis keeping some third-order terms in the Normal Dynamics afterthe 2nd-order Normal Form transformation can improve the stability analysis [5]and serve as a new criteria to design the system controllers in order to improve thetransfer limit [16]Keeping 3rd order terms in the Taylor series the j-th state equation of (232)becomes

yj = λjyj +

Nsumk=1

Nsuml=1

F2jklykyl +

Nsump=1

Nsumq=1

Nsumr=1

F3jpqrypyqyr (242)

Applying the 2nd Order NF transformation y = z + h2(z) to (242)it leads to

z + ˙h2(z) = Λz + Λh2(z) + F2(z + h2(z) (243)


z = (I +Dh2(z))minus1(Λz + Λh2(z) + F2(z) +DF2(z)h2(z)

+HF2(z)(h2(z))2 + F3(z) +DF3zh2(z))

z = (I minusDh2(z) + (Dh2(z))2 +O(3))(Λz + Λh2(z) + F2(z)

+DF2(z)h2(z) + F3(z) +O(4))

Expanding (243) it leads

z = Λz (order 1)


+Dh2(z)(Dh2(z)Λz minus Λh2(z)minus F2(z)) +DF2(z)h2(z) + F3(z) (order 3)

+ (Dh2(z))2(Λh2(z) + F2(z))minusDh2(z)F3(z)minusDh2(z)DF2(z)h2(z) +O(4)

(order 4)

+ (Dh2(z))2(DF2(z)h2(z) + F3(z)) (order 5)

As order 2 has been deleted by the 2nd NF transformation andDh2(z)ΛzminusΛh2(z)minusF2(z) = 0 keeping terms to order 3 (243) reduces to

z = Λz +DF2(z)h2(z) + F3(z) +O(4) (244)


Methodologies

wherebull DF2(z)h2(z) are third order terms coming from the 2nd Order NF transfor-mation used to cancel the 2nd order termsbull F3(z) are the original 3rd order terms from the system (242)

For the jth variable it reads

zj =λjzj (245)

+

Nsump=1

Nsumq=1

Nsumr=1

(Nsuml=1

(F2jpl + F2jlp)h2pqr + F3jpqr

)zpzqzr

=λjzj +Nsump=1

Nsumq=1

Nsumr=1

Cjpqrzpzqzr

where Cjpqr =sumN

l=1(F2jpl + F2jlp)h2pqr + F3jpqr

2333 Method 3-3-3 Elimination of a maximum of third-order terms

To simplify as much as possible the third order terms without neglecting importantterms as [16] did the following third-order NF transformation can be applied to(244) [31]

z = w + h3(w) (246)

where h3 is a polynomial of w containing only 3rd-order terms Applying transfor-mation (246) to (244) it leads to

w + ˙h3(w) = Λw + Λh3(w) +DF2(w + h3(w))h2(w + h3(w)) + F3(w + h3(w))

(247)

w(I +Dh3(w)) = Λw + Λh3(w) + (DF2(w)h2(w) +O(5)) + (F3(w) +O(5))

w =(I +Dh3(w))minus1(Λz + Λh3(w) +DF2(w)h2w + F3(w) +O(5))

w =(I minusDh3(w) + (Dh3(w))2

+O(5))(Λw + Λh3(w) +DF2(w)h2(w) + F3(w) +O(5))


w = Λw (order 1)

+DF2(w)h2(w) + Λh3(w)minusDh3(w)Λw + F3(w) (order 3)

minusDh3(w)(DF2(w)h2(w) + Λh3(w)minusDh3(w)Λw + F3(w)) +O(5)

(order 5)

and keeping only terms up to order 3 it leads to the Normal Dynamics

w =Λw +DF2(w)h2(w) (248)

+ Λh3(w)minusDh3(w)Λw + F3(w) +O(5)


It should be emphasized that there are neither terms of order 2 nor terms of order4 in (248) and that the terms of order 3 arebull DF2(w)h2(w) that comes from the 2nd-order NF transformation used tocancel the 2nd-order termsbull Λh3(w) minusDh3(w)Λw that comes from the use of transformation (246) inorder to cancel the 3rd-order termsbull F3(w) the original 3rd order terms of system (242)

For the j-th variable (248) can be written as

wj =λjwj

+

Nsump=1

Nsumq=1

Nsumr=1

[Cjpqr minus (λp + λq + λr minus λj)h3jpqr]wpwqwr

+O(5) (249)

The next step is dedicated to the elimination of a maximum number of 3rd-orderterms The elimination of third-order terms deserves careful attention because res-onant terms are systematically present when the system exhibits undamped (orweakly damped) oscillating modes (pairs of complex eigenvalues close the the realaxis) (249) shows that third order terms can not be eliminated if the conditionλp + λq + λr minus λj asymp 0 holds If we consider a weak-damped oscillating mode com-posed of two conjugated poles λ2l and λ2lminus1 the necessary condition for eliminatingthe associated third-order term w2kw2lw2lminus1 will not be met for j = 2k Indeed inthat case λj = λ2k and the condition ω2l + ω2lminus1 asymp 0 holds leading to the impos-sibility of computing the coecient h32k2k2l2lminus1As a consequence term w2kw2lw2lminus1cannot then be eliminated and must be kept in the normal dynamics [5157]If we consider that the system possesses M weakly-damped oscillatorymodes M third-order terms can thus not be eliminated from the NormalDynamics of the jth variable (j isin M) Apart from internal resonances due tocommensurability relationships between eigenvalues as for example when ω1 = 3ω2ω1 = 2ω2 + ω3 or ω1 = ω2 + ω3 + ω4 all the other third-order terms are eliminatedby third-order transformation (246) where coecients h3jpqr are computed by

h3jpqr =F3jpqr +

sumNl=1(F2jpl + F2jlp)h2pqr

λp + λq + λr minus λj(250)

The normal dynamics of a system composed ofM weakly-damped oscillatory modesare then

wj = λjwj +

Msuml=1

cj2lwjw2lw2lminus1 +O(5) 2k isinM (251)

wj = λjwj +O(5) j isinM (252)


Methodologies

where w2lminus1 is the complex conjugate of w2l and coecients cj2l are dened as fork 6= l

cj=2k2l =C2k

(2k)(2l)(2lminus1) + C2k(2k)(2lminus1)(2l) + C2k

(2l)(2k)(2lminus1)

+ C2k(2l)(2lminus1)(2k) + C2k

(2lminus1)(2l)(2k) + C2k(2lminus1)(2k)(2l)

cj=2k2k =C2k

(2k)(2k)(2kminus1) + C2j(2k)(2kminus1)(2k) + C2k

(2kminus1)(2k)(2k) (253)

cj=2kminus12l =C2kminus1

(2kminus1)(2l)(2lminus1) + C2k(2kminus1)(2lminus1)(2l) + C2kminus1

(2l)(2kminus1)(2lminus1)

+ C2kminus1(2l)(2lminus1)(2kminus1) + C2k

(2lminus1)(2l)(2kminus1) + C2kminus1(2lminus1)(2kminus1)(2l)

cj=2kminus12k =C2kminus1

(2kminus1)(2k)(2kminus1) + C2j(2k)(2kminus1)(2kminus1) + C2k

(2kminus1)(2kminus1)(2k) (254)

(251) shows that considering third-order terms in the Normal Dynamic changesthe way of making the stability analysis As shown in [5 16] the third order termscan have a stabilizing or a destabilizing eect and the inspection of the sign of theeigenvalue real parts is not sucient to predict the stability of the system

234 Analysis of the Normal Dynamics using z variables

To evaluate the gain aorded by the 3rd order method over the 2nd order methodthe Normal Dynamics (251) and (252) have to be reconstructed with the z coordi-nates The change of variables gives

zj =wj +

Nsump=1

Nsumq=1

Nsumr=1

h3jpqrwpwqwr j isinM (255)

It can be seen from this reconstruction that 3rd order terms are separated into twosets one set is wj whose stability is inuenced be coecients c2k2l in 251 that canbe used for stability analysis [516] and the other set with coecients h32kpqr that canbe used to quantify the 3rd order modal interaction [32]The most important issue of Normal Form methods is how they contribute to theapproximation of system dynamics which will be compared with a brief introductionof other works in the next Section 235

235 Other Works Considering Third Order Terms Summary ofall the Normal Form Methods

In [5] and [16] third-order terms are present in the Taylor series but only thesecond-order transformation is used Some 3rd order terms are kept on the Normal


Dynamics and leads for the oscillatory modes to

z(j=2k) = λ2kz2k +Msuml=1

c2k2l z2kz2lz2lminus1 +O(3) 2k isinM (256)

where z2lminus1 is the complex conjugate of z2lFor non-oscillatory modes the Normal Dynamics keeps the same as expressed by(241) Compared to the proposed Normal Dynamics using the third order transfor-mation only terms with frequency ωm+n are involved in the zj=2k expression and3rd-order modal interaction can not be studiedIn [31] and [32] a 3rd order NF transformation is proposed where the termsDF2(w)h2(w) of (248) are not taken into account leading to coecients givenby

h3jpqr =F3jpqr


Since some 3rd order terms are omitted the accuracy of the proposed normal dynam-ics is worse than the one proposed by the dynamic deduced with using coecientsgiven by (250)Moreover the considered Normal Dynamics are

wj = λjwj +O(3) (258)

and since all resonant terms in (258) are neglected it fails to give more informationabout the system stability than the linearized small-signal stability analysis

236 Interpretation the Normal Form Methods by Reconstructionof Normal Dynamics into System Dynamics

2361 2-2-1 and 3-2-3S

The reconstruction for 2-2-1 and 3-2-3S is the same as only the 2nd NF transfor-mation is done

y = z + h2(z) (259)

As indicated in [48] the initial condition of z and coecients h2jkl can be usedto develop nonlinear indexes to quantify the second-order modal interaction Theapplication can be found in the placement of stabilizers [20]For 2-2-1

yj = zj︸︷︷︸ωj

+

Nsumk=1

Nsuml=1

h2iklzkzl︸︷︷︸ωk + ωl

+O(3) (260)

For 3-2-3S


+

Nsumk=1

Nsuml=1

h2jklzkzl︸︷︷︸ωk + ωl

+O(3) (261)


Methodologies

2362 3-3-3 and 3-3-1

In fact successive transformation is equivalent to one-step transformation to thesame order Transformation y = z + h2(z) z = w + h3(w) is equivalent toy = w + h2(w) + h3(w) As

y =Λw + h2(z) (262)

Λw + h3(w) + h2(w + h3(w))

=Λw + h3(w) + h2(w)

+Dh2(w)h3(w) +Hh2(w)(h3(w))2 + middot middot middot=Λw + h2(w) + h3(w) +O(4)

Therefore the reconstruction can be simplied as

y = w + h2(w) + h3(w) (263)

For 3-3-1

yj = wj︸︷︷︸ωj

+Nsumk=1

Nsuml=1

h2jklwkwl︸︷︷︸ωk + ωl

+Nsump=1

Nsumq=1

Nsumr=1

h3jpqrwpwqwr︸︷︷︸ωp + ωq + ωr

(264)

For 3-3-3


+

Nsumk=1

Nsuml=1


+

Nsump=1

Nsumq=1

Nsumr=1


(265)

(265) indicates that for the oscillatory modes the assumed linear modes will bechanged by cj2l This violates the assumption in 2-2-1 and 3-3-1 that taking intononlinearities by NF will just add nonlinear modes on the linear modes obtained bythe small signal analysisStability and Higher-order InteractionMethod 3-2-3S can be used to suggest more information of system stability both λjand cj2l will contribute to the stability of oscillatory mode j isinM and therefore thestability of the whole system [516]Method 3-3-1 can be used to quantify the 3rd order interaction (if the coecienth3M has enough resolution) [32]Method 3-3-3 can do both jobs the stability analysis by c2k2l and the 3rd order modalinteraction by h3jpqr The insight of physical properties of the system dynamics canbe exploited more deeply And in both aspects it has a resolution up to 5 whilebothAccuracyIndicated by Eqs (261) (264) the accuracy of both method 3-2-3s and 3-3-1 isup to order 3 since some 3rd order terms are neglected in the Normal DynamicsAnd the method 3-3-3 is up to order 5


237 The signicance to do 3-3-3

The discussed methods for systems of rst equations in this dissertation are Linear2-2-1 3-2-3S 3-3-1 3-3-3Their performances can be evaluated from three theoretical aspects

1 Describing the power system dynamics with less computational burden com-pared to the full numerical simulation

2 Modal interaction linear modal interaction and nonlinear modal interactionto see how the system components interact with each other important forplacement of stabilizers

3 Transient Stability analysis to predict the stability of the power system ex-ploit the power transfer capacity and maximize the economical benets

Table 23 Performance Evaluation of the Reviewed and Proposed MNFsMethod Accuracy Order of Modal Transient Applicability to

Interaction Stability 3rd Order ModesLinear O(2) 1st Linear None2-2-1 O(3) 2nd Linear None3-2-3S O(3) 2nd Non-linear Oscillatory3-3-1 O(3) 3rd Linear Non-oscillatory3-3-3 O(5) 3rd Non-linear All

238 Another Way to Derive the Normal Forms

In some literature [3058]another way to deduce the NF transformation is presentedthough less rigorous in mathematicsInstead of inverting (I +Dh2(z)) by (I +Dh2(z))minus1 = I minusDh2(z) + (Dh2(z))2 ittakes

(z + h2(z)) = z +Dh2(z)z = z +Dh2(z)Λz (266)

Therefore there is no need to inverse the nonlinear transformation However (266)is wrong because it will cause error in the deducing Normal Forms higher than 2ndorder

239 Comparison to the Modal Series Methods

Despite the Methods of Normal Forms (MNF) several other methods are used by theresearchers to study the nonlinear dynamics of interconnected power system suchas Perturbation Techniques (PT) and Modal Series (MS) Methods (As concludedin [37] the PT techniques and MS methods are identical) Having proposed byShanechi in 2003 [35] the 2nd MS method has declared a better method thanMNF [3537385960] since it can suggest the closed form solution when resonanceoccurs The reader may have a questionWhy MNF rather than MS is used to study the 3rd order nonlin-earities


Methodologies

As it is dicult to nd a satisfying answer in the literatures the author will answerthis question by deducing the 3rd order MS methods

2391 A Brief Review of the MS Methods

Lets start by a brief review of the MS methods The princple of MS method is toapproximate the solution of the nonlinear dierential equation by power seriesLet the solution of (242) for initial condition y0 be yj(y0 t) Let ψ denote theconvergence domain of Maclaurin expansion of yj(y0 t) in term of y0 for all t isin[0 T ] sub R micro isin Cn is the convergence domain of y when expanding the Taylorseries If θ = microcapψ is nonempty then the solution of (242) for y0 isin θ and for someinterval t isin [0 T ] sub R is

yj(t) = f1j (t) + f2j (t) + f3j (t) + middot middot middot (267)

where the right hand side of (267) can be found by solving the following series ofdierential equations with initial conditions f1(0) = [f11 (0) f12 (0) middot middot middot f1N (0)]T =

y0 and fkj (0) = 0 for each j isin 1 2 middot middot middot N

f1j = λjf1j (268)

f2j = λjf2j +

Nsumk=1

Nsuml=1

Cjklf1kf

1l (269)

f3j = λjf3j +

Nsumk=1

Nsuml=1

Cjkl(f1kf

2l + f2kf

1l ) +

Nsumk=1

Nsuml=1

Nsumk=1

Djpqrf

1p f

1q f

1r (270)

The solution of (268) can be easily found as

f1j (t) = y0j eλjt (271)

Substituting (271) into (269) reads

f2j = λjf2j +

Nsumk=1

Nsuml=1

Cjkly0ky

0l e

(k+l)t (272)

By Multi-dimensional Laplace Transform

f2j (s1 s2) =

Nsumk=1

Nsuml=1

Cjkl1

s1 + s2 minus λjf1k (s1)f

1l (s2) (273)


the solution of f2j can be found as

f2j =Nsum

k=1

Nsuml=1

Cjklf1k (0)f1l (0)Sjkl(t) (274)

=Nsum

k=1

Nsuml=1

Cjkly0ky

0l S

jkl(t)

where

Sjkl(t) =1

λk + λl minus λj(e(λk+λl)t minus eλjt)

Sjkl(t) = teλjt for (k l j) isin R2

and the set R2 contains all the 2nd order resonancesAs observed from (274) the solution of f2j is a polynomial series of e(λj)t e(λk+λl)tSimilarly substituting (272) into (270) using three-dimensional Laplace transform

f3j (s1 s2 s3) =Nsum

k=1

Nsuml=1

Cjkl1

s1 + s2 + s3 minus λj(f2k (s1 + s2)f

1l (s3) + f1k (s1)f

2l (s2 + s3))+

(275)Nsum

k=1

Nsuml=1

Nsumr=1

Djpqr

1

s1 + s2 + s3 minus λjf1p (s1)f

1q (s2)f

1r (s3)

Substituting (273) into (275) it leads

f3j (s1 s2 s3) =

Nsumk=1

Nsuml=1

Cjkl1

s1 + s2 + s3 minus λj

[Nsum

k=1

Nsuml=1

Cjkl1


1l (s2)

]f1l (s3)

(276)

+ f1k (s1)

[Nsum

k=1

Nsuml=1

Cjkl1


1l (s3)

]

+Nsum

k=1

Nsuml=1

Nsumr=1

Djpqr

1


1q (s2)f

1r (s3)

As observed from the (272) and (275) the solution of f3j will be a polynomial seriesof eλjt e(λk+λl)t and e(λp+λq+λr)tContinuing the above procedures to order n the solution of yj(t) is a polynomialseries of eλjt e(λk+λl)t and eλp+λq+λr e(λ1+λ2+middotmiddotmiddot+λn)tAs observed from the procedures above it can be concluded that

1 The solution is a linear combination of eλjt e(λk+λl)t e(λp+λq+λr)t or equiv-alently speaking it is a nonlinear combination of linear modes since the basiccomponent is e(λi)t with constant eigenvalue Therefore the nonlinearities will


Methodologies

not change the decaying speed and oscillatory frequency of the fundamentalmodes

2 For each mode j all the nonlinear terms are taken into account when solvingthe equations in the scale of (Cj)2

3 Closed form results are always available

2392 Advantages and Disadvantages

Therefore compared to MS method the MNF is preferred since1 the MNF can take the minimum nonlinear terms while the MS method has

to keep all the nonlinear terms since for N -dimensional system the scalabil-ity for 2nd order nonlinear terms is N3 and for 3rd order modal series N5

(N times dim(Cj)2) this makes the MS method nonapplicable for large-scale sys-tem while MNF is more capable for study of large-scale problem and can beextended to take into account higher-order nonlinear terms

2 the MNF can predict the systems behaviour without oering closed-form re-sults therefore oering the closed form results when resonance occurs is nota privilege of MS method

3 the Normal Forms makes possible to assess the nonlinear stability while theMS methods only work on the convergence domain

Though nonlinear indexes can be proposed without solving the modal series basedon Cjkl and Dj

kl and etc they can only quantify the nonlinear modal interactionand cannot perform the nonlinear stability assessment

24 Nonlinear Indexes Based on the 3rd order Normal

Forms of Vector Fields

The objective of NF methods is to quantify modal interaction and to predict thesystem stability To evaluate the gain obtained by the 3rd order method over the2nd order method normal dynamics (251) and (252) have to be reconstructed withthe z coordinates as

zj = wj +Nsump=1

Nsumq=1

Nsumr=1

h3jpqrwpwqwr (277)

with |λj minus λp minus λq minus λr| 0It can be seen from the reconstruction that zj is separated into two sets a rst setwith coecients h3jpqr that can be used to quantify the 3rd order modal interactions[32] a second set is wj which may be inuenced by cj2l that can be used for stabilityanalysis [5 16]Based on the Normal Forms and h3jpqr c

j2l nonlinear indexes are proposed to quan-

tify the 3rd order modal interaction and nonlinear stability margin

24 Nonlinear Indexes Based on the 3rd order Normal Forms of VectorFields 81

241 3rd Order Modal Interaction Index

As observed from (255) the 3rd order oscillation is caused by h3jpqrwpwqwr Thenthe third order Modal Interaction index MI3jpqr can be dened as

MI3jpqr =|h3jpqrw0

pw0qw

0r |

|w0j |

(278)

which indicates the participation of 3rd order modal interaction with the frequencyωp+ωq+ωr in the mode j IntroducingMI3jpqr leads to a more clear picture of howthe system components interact with each other and a more precise identication ofthe source of the oscillations Those additional informations are crucial especiallywhen some fundamental modal interaction 2nd order modal interactions and 3rdorder modal interactions may exhibit the same oscillatory frequency (as indicated inthe Chapter 3 the 16 machine 5 area test system) and it is dicult to identify theorder of modal interaction from the FFT analysis frequency spectrum let alone toidentify the source of oscillation NF analysis plays a role more than that of time-domain simulation plus signal processing in the sense that FFT can only identifythe oscillatory frequency of the frequency spectrum while NF analysis can alsoidentify the source of oscillatory frequency

242 Stability Index

Since the stability of oscillatory modes in (229) is consistent with (251) (252) thestability can be assessed without performing the time-domain simulation assessmentFor the non-oscillatory modes wj = λjwj the stability is determined by λj Andfor the oscillatory modes j isinM

wj = λjwj +

Msuml=1

cj2lwjw2lw2lminus1 (279)

= (λj +Msuml=1

cj2l|w2l|2)wj

(280)

where |w2l| is the magnitude of w2lA stability interaction index can be dened as

SIIj2l = cj2l|w02l|2 j isinM (281)

Since

λj +Msuml=1

cj2l|w2l|2 = σj + real(Msuml=1

SIIj2l) + j(ωj + imag(Msuml=1

SIIj2l)) (282)


Methodologies

the real part of SII will contribute to the stability of the mode and the imaginarypart of SII will contribute to the fundamental frequency

243 Nonlinear Modal Persistence Index

The nonlinear indexes proposed in the previous sections just indicate the nonlinearinteraction at the point when the disturbance is cleared To quantify the 3rd ordermodal interaction in the overall dynamics a persistence index is dened to indicatehow long a nonlinear interaction will inuence the dominant modes Similar to (23)(24) in [48] the Tr3 is the ratio of the time constants to the combination modesand the dominant mode

Tr3 =Time constant for combination mode(λp + λq + λr)

Time constant for dominant mode(λj)(283)

A small Tr3 indicates a signicant presence of the combination mode For exampleif Tr3 = 1 it means that the nonlinear interactions decay at the same speed as thedominant mode if Tr3 is very large it means that the inuence of combinationmode (p q r) decays too quickly compared to the dominant mode A relatively highvalue of the product SIItimesTr3MI3timesTr3 tends to reveal a condition of persistentmodal interaction

244 Stability Assessment

As indicated from (279) it is both the eigenvalues λj and the stability indexesSIIj2l contribute to the system stability However λj will keep constant duringthe system dynamics while SIIj2l will decay as time goes by with time constant ofTr3jj(2lminus1)(2l) If the SII

j2l is large but SII

j2l times Tr

jj(2l)(2lminus1) is small the 3rd-order

terms may stabilize the system at the beginning but leaves a long-term instabilityThis will lead to a wrong stability assessment Therefore the time constant ofSIIj2l must be taken into account when assessing the overall stability and an indexof stability assessment can be dened as

SIj = σj +Msuml=1

Re(SIIj2l)times Trjj(2lminus1)(2l) j isinM (284)

with σ coming from λj = σj + jωj It immediately follows from this denition that

SIj gt 0 unstableSIj lt 0 stable

(285)

when SIj = 0 the system may stay in limit cycles or switch between the stable andthe unstable phase Dierent cases should be discussed and further investigationsshould be madeInversely letting SIj ge 0 forallj isinM a set of equation will be formed with the stabilitybound |wj |forallj isinM as the variables solving this set of equation the stability bound

25 Derivations of Normal Forms of Nonlinear Normal Modes forSystem of Second-Order Dierential Equations 83

can be obtainedCompared to [5] this proposed method is more accurate as 1) it takes into accountthe time pertinence 2) it is appropriate to case where σj = 0

25 Derivations of Normal Forms of Nonlinear Nor-

mal Modes for System of Second-Order Dierential

Equations

In the previous section derivation of normal forms for rst-order equations hasbeen performed A drawback of this method is that the state variables may havecomplex form values in the normal form coordinates and lose the physical meaningsFor example the displacement x is naturally a real form number however by thenormal form transformations it may be in complex form fails to direct correspondto physical phenomaIn this section derivation of normal forms for second-order equations will be per-formed The normal forms for second-order equations have been found in nonlinearstructure analysis [1719] and will be applied to transient analysis of power systemin Chapter 4This approach is mainly used to deduce the nonlinear normal modes in the mechan-ical engineering and it can be referred to as Normal Mode approach Or it canalso be referred to as normal mode approach since in the normal transformationsthe state-variables always have real form values

251 Class of Second-Order Equations that can be studied byMethods of Normal Forms

The nonlinear electromechanical oscillations in interconnected power system can bemodeled as second-order coupled oscillators If the variables of the power systemaround a given equilibrium point are gathers in a N dimensional vector q the basicmodel under study writes

Mq +Dq +Kq + fnl(q) = 0 (286)

In the above equation M and D are constant symmetric inertia and dampingmatrix whose values depend on the physical parameters of the power system andcontroller parameters K and fnl indicate the coupling between the variables whereK is a constant matrix including the linear terms and fnl gathers the nonlineartermsIf fnl is developed in a Taylor series up to the third order it comes

fpnl(q) =Nsum

i=1jgeif2pijqiqj +

Nsumi=1jgeikgej

f3pijkqiqjqk (287)


Methodologies

where fpij and fpijk (i j k p = 1 N) are coecients of the quadratic and cubicterms Components of K and fnl depend on the chosen equilibrium point and thesystem structure for example the network connectivity of the power gridsIt is seen that (286) is a N -dimensional nonlinear dynamical problem


Viewed from (286) the dynamics physical property (oscillatory frequency damp-ing friction etc) is obscured by the mathematical modelIn order to decouple the equation into second-order oscillators a linear transforma-tion 4q(t) = Φx(t) can be used To obtain Φ is to solve the following eigen-valueproblem (

KS minus Ω2M)Φ = 0 (288)

where Ω2 is a diagonal matrix collecting the natural frequency of the oscillatorsAfter decoupling the linear terms the system dynamics is therefore characterizedby a group of oscillators with coupled nonlinearities

Xp+2ξpωpXp + ω2pXp (289)

+Nsumi=1

Nsumjgei

gpijXiXj +Nsumi=1

Nsumjgei

Nsumkgej

hpijkXiXjXk = 0

where ξp is the p-th modal damping ratio And gpij and hpijk are quadratic and cubic

nonlinearities coming from decoupling of KS which are given as

[gp] =

Nsumi=1

Φminus1pi (ΦTKp2SΦ)

Φminus1i MiΦ[Φminus1MΦ]pp (290)

hpijk =

Nsumi=1

Φminus1pi (

NsumP=1

NsumQ=1

NsumR=1

Kp3SijkΦiPΦjQΦkR)[Φminus1MΦ]pp (291)

Φpi is the element of p-th row and i-th column of matrix Φ

253 Normal Modes and Nonlinear Normal Modes

Neglecting all the nonlinearities the (289) reads a Linear Normal Mode (LNM)

Xp + 2ξpωpXp + ω2pXp = 0 (292)

The physical meaning and physical properties of normal mode bull The basic particular solution of the motion of oscillations of system and thesystem dynamics is a combination of normal modesbull The free motion described by the linear normal modes takes place at the xedfrequencies These xed frequencies of the normal modes of a system are


known as its natural frequenciesbull All the normal modes are orthogonal to each other ie when the motion isinitialized at one normal mode its motion will always contains in that mode

The most general motion of a linear system is a superposition of its normal modesThe modes are normal in the sense that they can move independently that is tosay that an excitation of one mode will never cause motion of a dierent modeIn mathematical terms normal modes are orthogonal to each other By normalmodes the complex system dynamics are decomposed into normal modes with xedfrequencyHowever in oscillations with large amplitudes the system dynamics are predomi-nantly nonlinear where the theorem of supposition of normal modes fails and thenonlinear property of system dynamics cannot be extracted from the normal modesIn order to extend the physical insights of normal modes to nonlinear domain theconcept of nonlinear normal modes are proposed [61 62] Rosenberg [61] rstlydene nonlinear normal modes as specic periodic solutions in which the modalcoordinates exhibit particular features Using center manifold theory Shaw andPierre [62] dene a NNM as an invariant manifold in phase space tangent at theorigin to their linear counterpart This denition is used in this PhD thesis Thederivation of nonlinear normal modes by normal form theory are based the works[1719] which have been technically veried within the mechanical context for anassembly of nonlinear oscillators thanks to a real formulation of the normal form

254 Second-Order Normal Form Transformation

Taking into account of the quadratic nonlinearities of (289) it reads

Xp + 2ξpωpXp + ω2pXp +

Nsumi=1

Nsumjgei

gpijXiXj = 0 (293)

which can be rewritten as

Xp = Yp (294)

Yp = minus2ξpωXp minus ω2Xp minusNsumi=1

Nsumjgei

gpijXiXj (295)

(the mention forallp = 1 middot middot middotN will be omitted when not confusing)The rst step consists of dening a second-order polynom with 2N variables (Npairs displacement-velocity) It is chosen tangent to the identity and is written

X = U + ε(a(U2) + b(V 2) + c(UV ))

Y = V + ε(α(U2) + β(V 2) + γ(UV ))

(296)


Methodologies

where ε is a small nondimensional parameter which is just a bookkeeping device toequal the terms in the same degree in the normal form transformation For examplepolynomial terms with ε are in the same polynomial degree and polynomial termswith ε2 indicate a higher polynomial degreeforallp = 1 N Yp = Xpthen

Xp = Up +Nsumi=1

Nsumjgei

(apijUiUj + bpijViVj) +Nsumi=1

Nsumj=1

cijUiVj

Yp = Vp +Nsumi=1

Nsumjgei

(αpijUiUj + βpijViVj) +Nsumi=1

Nsumj=1

γijUiVj (297)

(Up Vp) are the new variables The polynoms (297) are written in this form to takecommuting and non-commuting terms into accountThe unknown of the problem are now the 2N2(2N + 1) coecientsapij b

pij c

pij α

pij β

pij γ

pij

They are determined by introducing (297) to (293) This

generates terms of the form UiUj ViVj UiVj which can be remedied by observingthat at lower order

Uj = Vj +O(U2i V

2i ) (298)

Vj = minusω2jUj +O(U2

i V2i ) (299)

It therefore renders

D(apijUiUj) = apijUiUj + apijUiUj = (apij + apji)UiVj +O(U3i V

3i ) (2100)

D(bpijViVj) = bpijViVj + bpij ViVj = minusω2j (b

pij + bpji)UiVj

D(cpijUiVj) = cpijUiVj + cpijUiVj = minusω2j cpijUiUj + cpijViVj

D(αpijUiUj) = αpijUiUj + αpijUiUj = (αpij + αpji)UiVj + ViUj +O(U3i V

3i )

D(βpijViVj) = βpijViVj + βpij ViVj = minusω2j (β

pij + βpji)UiVj

D(γpijUiVj) = γpijUiVj + γpijUiVj = γpijUi(minusω2jUj) + γpijViVj

They are cumbersome for the identication of the dierent monomials becauseinvolving the derivative of a variable against time This is remedied by observingthat at lower orderLetting Vp = UpEquating the 2nd order terms there is

D(a+ b+ c) = α+ β + γ (2101)

D(α+ β + γ) + 2ξωD(a+ b+ c) (2102)

= minusω2(a+ b+ c)minus g(U2)


To degree ε1 Da(U2)minus ω2Db(V 2) = γ (UV )

Dc(UV )V = α (U2)

Dc(UV )U = β (V 2)

(2103)

Dα(U2)minus ω2Dβ(V 2) = minus2ξωDa(U2) + 2ξω3Db(V 2)minus ω2c(UV ) (UV )

minusω2Dγ(UV )V = 2ξω3Dc(UV )V minus ω2a(U2)minus g(U2) (U2)

Dγ(UV )U = minusω2b(V 2)minus 2ξωc(UV )U (V 2)

(2104)where Dc(UV )V indicates partc(UV )

partV There are six equations with six variables solv-

ing (2103)(2104)apij b

pij c

pij α

pij β

pij γ

pij

can be solved The coecients are listed

in the Appendix for case where no resonance occurs When there occurs internalresonance occurs ie ωk + ωl 6= ωj the corresponding second-order terms will bekept in the normal forms and corresponding coecients will be zeroIf all the 2nd-order terms can be cancelled then the Normal Forms is

Up + 2ξpωpUp + ω2pUp +O(3) = 0 (2105)

with

O =Nsumi=1

Nsumjge1

Nsumkgej

(Hpijk +Apijk)UiUjUk (2106)

+Nsumi=1

Nsumjgei

Nsumkgej

BpijkUiUjUk +

Nsumi=1

Nsumjgei

Nsumkgej

CpijkUiUjUk

Apijk Bpijk Cpijk are coecients of terms DG(U)a(U) DG(U)b(U V ) and

DG(U)c(V ) arising from the cancellation of 2nd order terms

Apijk =

Nsumlgei

gpilaljk +

sumllei

gplialjk (2107)

Bpijk =

Nsumlgei

gpilbljk +

sumllei

gplibljk (2108)

Cpijk =

Nsumlgei

gpilcljk +

sumllei

gplicljk (2109)


Methodologies

Therefore by cancelling the quadratic nonlinearities and neglecting all the cubicnonlinearities the normal dynamics is obtained in the 2nd order normal mode(2105) is labelled as NM2

255 Third-order Normal Form Transformation

However when the disturbance is predominantly nonlinear a second approximationis not sucient O(3) can contain terms that contribute to the nonlinear stabilityof the system dynamics In Refs [17 18 21 22] a 3rd order NF transformationis proposed by which the state-variables are transformed into Nonlinear NormalModes represented as RS

U = R+ ε2(r(R3) + s(S3) + t(R2 S) + u(RS2))

V = S + ε2(λ(R3) + micro(S3) + ν(R2 S) + ζ(RS2))

(2110)

Equating the terms at dierent degrees by assuming R = S S = minus2ξωRminus ω2R+

O(2) the expressions for the coecients can be obtainedTo cancel the 3rd order terms there is

D(r + s+ t+ u) = λ+ micro+ ν + ζ (2111)

D(λ+ micro+ ν + ζ) + 2ξωD(r + s+ t+ u) (2112)

= minusω2(r + s+ t+ u) +Dg(R2)(a+ b+ c)minus h(R3)

To degree ε2

DrR minus ω2Dus = ν (R2 S)

minusω2DsS + 2DtR = ζ (RS2)

minusω2DtS = λ (R3)

DuR = micro (S3)

(2113)

DλR minus ω2DζS = minus2ξω(DrR minus ω2DuS minus ω2t+Dg(R2)c (R2 S)

minusω2DmicroS +DvR = minus2ξω(minusω2DsS +DtR)minus ω2u+Dg(R2)b (RS2)

minusω2DνS = minus2ξω(minusω2DtS)minus ω2r +Dg(R2)aminus h(R)3 (R3)

DζR = minus2ξω(minusω2DuR)minus ω2s (S3)

(2114)


There are 8 coecients described by 8 dierent equations and the coecients canbe solved by symbolic programming with the coecients of trivially resonant termsset as zero

forallp = 1 middot middot middotN

rpppp = spppp = tpppp = upppp = 0

λpppp = micropppp = νpppp = ζpppp = 0

forallj ge p middot middot middotN

rppjj = sppjj = tpjpj = tppjj = uppjj = upjpj = 0

λppjj = microppjj = vppjj = vpjpj = ζppjj = ζpjpj = 0

foralli le p

rpiip = spiip = tpiip = tppii = upiip = uppii = 0

λpiip = micropiip = vpiip = vppii = ζpiip = ζppii = 0

(2115)

The detailed calculation of the coecients and their values can be found in [18 19]and are listed in Appendix

256 Eliminating the Maximum Nonlinear Terms

To eliminate the maximum nonlinear terms the issue is to identify the resonantterms

2561 Internal Resonance

Like the approach for systems of rst order equations all the terms can be eliminatedwhen there is no internal resonances between the eigenfrequencies of the systemFor example order-two internal resonances reads for arbitrary(p i j)

ωp = ωi + ωj ωp = 2ωi (2116)

while third-order (linked to cubic nonlinear coupling terms) writes (p i j k)

ωp = ωi + ωj plusmn ωk ωp = 2ωi plusmn ωj ωp = 3ωi (2117)

There are also two characteristics of internal resonant terms 1)dependent of theoscillatory frequency2)can be brought by either the quadratic terms or cubic termsor both


Methodologies

2562 Trivial Resonance Introduced by Conjugate Pairs

Indeed if only one oscillator equation is considered the one can always fulll thefollowing relationship between the two complex conjugate eigenvalues+iωminusiω

+iω = +iω + iω minus iω (2118)

ie a relationship of the form (289) with order p = 3 Coming back to the oscillatorequation this means that the Dung equation

X + ω2X + αX3 = 0 (2119)

is under its normal form the cubic terms (monom associated with the resonancerelation (2118)) cannot be canceled through a nonlinear transform On a physicalviewpoint this stands as a good news Indeed one of the most important observedfeature in nonlinear oscillations is the frequency dependence upon oscillation am-plitude If the system could be linearized this would mean that the underlyingdynamics is linear hence the frequency should not change with the amplitude Andthis will violate the well-established common-sense in mechanical dynamics that thesolution of (2119) is X(t) = a cos(ωNLt+ φ) +O(a2)

Neglecting the trivially resonant terms the proposed Normal Transform renders theequations too simple to contain the basic characteristics of nonlinear dynamics itis uselessIn fact the normal dynamics contain the trivially resonant terms which exist what-ever the values of the frequencies of the studied structures areIn the general case whereN oscillator-equations are considered numerous resonancerelationships of the form

+iωp = +iωp + iωk minus iωk (2120)

are possible for arbitrary p k isin [1 N ]2 This means that the original system can besimplied but numerous terms will remain at the end of the process in the normalform following Poincaregrave-Dulac theorem However as it will be shown next thiseort is worthy as numerous important terms will be canceled and also because theremaining terms can be easily interpretedThe resonance relationships put forward through the (2118) (2119) (2120) aredenoted as trivial and there are two characteristics 1) independent of the values ofthe frequencies of the studied oscillations 2) only for the cubic terms

2563 Damping and Resonance

The internal resonance is rstly dened for conservative oscillation ie forallp =

1 2 middot middot middotN ξp = 0 However they can be extended to case with viscous dampingas the oscillatory frequency writes [1819]

λplusmnp = minusξpωp plusmn iωpradic

1minus ξ2p (2121)


Besides the real part of (2121) which controls the decay rate of energy along thepth linear eigenspace the imaginary part shows that the damping also have an eecton the oscillation frequency However in the oscillatory case we study the dampingratio ξp is always very small compared to ωp so that the assumption of a lightlydamped system could be considered In that case a rst-order development of(2121) shows that

λplusmnp = plusmniωp minus ξpωp +O(ξ2p) (2122)

Therefore the denition of internal resonance and trivial resonance relationship canbe extended to damped oscillators

257 Invariance Property

Like the case of system of rst-order equations all the higher-order terms are ne-glected in the previous researches [30] when no internal resonance occurs And thenormal dynamics are depicted by linear dierential equations However the non-linear terms contain qualitative eect of the original nonlinear dynamics such asthe amplitude-dependent frequency shift which cannot be represented by the linearequations In fact there exists resonant terms that cannot be neglected in normalnormals and an complete linearization is impossibleIn fact the normal dynamics contain the trivially resonant terms which exist what-ever the values of the frequencies of the studied structures are By keeping thetrivially resonant terms the invariance property is conservedIf no internal resonances are present in the eigen spectrum this nonlinear transfor-mation weakens the couplings by shifting all the nonlinearities to the cubic termsThe new dynamical system writes for all p = 1 N

Rp+2ξpωpRp + ω2pRp (2123)

+ (hpppp +Apppp)R3p +Bp

pppRpR2p + CppppR

2pRp

+RpP(2)(Ri Rj) + RpQ(2)(Ri Rj) = 0

where P(2) and Q(2) are second order polynomials in Rj and Rj j 6= p j 6= p andwrite

P(2) =

Nsumjgep

[(hppjj +Appjj +Apjpj)R

2j +Bp

pjjR2j + (Cppjj + Cpjpj)RjRj

](2124)

+sumilep

[(hpiip +Apiip +Appii)R

2i +Bp

piiR2i + (Cppii + Cpipi)RiRi

]

Q(2) =

Nsumjgep

(BpjpjRjRj + CpjjpR

2j ) +

Nsumilep

(BpiipRiRi + CpiipR

2i ) (2125)

and the coecients (Apijk Bpijk C

pijk) arise from the cancellation of the quadratic


Methodologies

terms [17 18] and are dened in (2109) This model can be called as full NNM(FNNM)Compared to (289) all the oscillators are invariants in the new dynamical FNNMsystem (2123)Observed from (2123) all the nonlinear terms can be classied into two types cross-coupling nonlinearities (terms with P(2) and Q(2)) and self-coupling nonlinearities(the other nonlinear terms with coecients hpppp +Apppp B

pppp C

pppp)

If the dynamics is initiated on a given mode (say the p-th Xj = xj = 0 forallj 6= p att = 0) the system dynamics will only be inuenced by this given mode the systemoscillates as an invariant second-order oscillator Equivalently speaking in thiscase P(2) and Q(2) are null leading to the property of invarianceFor the cases where several modes have non zero initial conditions cross-couplingnonlinearities are not zero However their eect on the dynamics can be weak com-pared to the any self-coupling nonlinearities Thus a series of decoupled subsystemis obtained which can be generally expressed as



pppRpR2p + CppppR

2pRp = 0

Model ((2126)) is labelled as NNM and can be viewed as a type of nonlinear normalmode as all the modes are decoupled and nonlinear The solution of each modecan be approximated by an analytical perturbation method

258 One-Step Transformation of Coordinates from XY to RS

As indicated in Section 22 sequential transformations render the same results asone-step transformation therefore once the coecients are calculated the normalforms in RS coordinates can be obtained by

Xp = Rp +

Nsumi=1

Nsumjgei

(apijRiRj + bpijRiRj) +

Nsumi=1

Nsumj=1

cijRiSj

+Nsumi=1

Nsumjgei

Nsumkgej

(rpijkRiRjRk + sNijkSiSjSk)

+

Nsumi=1

Nsumjgei

Nsumkgej

(tpijkSiRjRk + upijkSiSjRk)


Yp = Sp +Nsumi=1

Nsumjgei

(αpijRiRj + βpijSiSj) +2sumi=1

2sumj=1

γijRiSj

+

Nsumi=1

Nsumjgei

Nsumkgej

(λpijkRiRjRk + micropijkSiSjSk)

+Nsumi=1

Nsumjgei

Nsumkgej

(νpijkSiRjRk + ζpijkSiSjRk) (2127)

259 Comparison with Shaw-Pierre Method where Nonlinear Nor-mal Modes are Formed

Another method to form nonlinear normal mode is the Shaw-Pierre method rstlyproposed by Shaw amp Pierre [39] and applied in both structural analysis in mechanicalengineering [39] and power system analysis in electrical engineering [40]This method originates from the master-slave coordinates transformation of the Cen-ter Manifold method However it only works for slave-coordinates where Re(λs) lt 0

ie the linearized system must be stableIts principle is to reduce the N -dimensional nonlinear system with N variables xinto Nc dimensional systems with Nc variables xc And the other Ns variables xs

expressed as static functions of xs = f(xc) Ns + Nc = N With the SP methodsome dynamics of xs are lostTo illustrate such a point a simple example of a 2-dimensional nonlinear system ispresented [63] x

y

=

micro 0

0 minus1

x

y

+

xy + cx3

bx2

(2128)

where the micro is a parameter dependent of the point at which the system dynamicswould be studied the critical point For example in power system dynamics microdepends on operating pointIf the critical point critical state variable and stable coordinates are as

microc = 0 xc = x xs = y (2129)

The equation of center manifold is y = h(x micro) substituting it into y = minusy + bx2

and equating like power terms it leads to

y = bx2 + (|x micro|3) (2130)


Methodologies

and (2128) can be reduced into the bifurcation equation

x = microx+ (b+ c)x3 (2131)

As observed from (2131) the system dynamic is dominated by state-variable xand the dynamics of y is under the slave of x Therefore the coordinates of x isalso referred to as master coordinates and y is also referred to as slave coordinatesAs observed in (2131) the dynamics of y totally depend on the x y is the passivestate-variables and x the active state-variables (2131) fails to study the case when

y is excited or matrix

micro 0

0 minus1

has positive eigenvalues

Center manifold theorem has been successfully applied to bifurcation analysis [49]where only a few of critical state-variables are concerned in the mechanical systems[39] However in power system dynamic analysis it is dicult to decide which one isthe critical state-variable as all state-variables can be excited simultaneously in somescenario and positive eigenvalue may exist even when the system is stable [5 16]Therefore the SP method which originates from center-manifold method is notapplicable for a lot of cases In [40] nonlinear normal modes are formed to studythe inter-area oscillations in power system when only one inter-area mode is excitedhowever in interconnected power systems several inter-area modes can be excitedsimultaneously even when only one system component is disturbed [64] In thosecases the center-manifold method is not that ecient

2510 Possible Extension of the Normal Mode Approach

Compared to (229) from a mathematical view some generalities are lost in (286)for example nonlinear couplings such as YiXj and YiYj are not considered Howeverthe similar normal form transformations can be applied to deduce the normal forms

2511 A Summary of the Existing and Proposed MNFs Based onNormal Mode Approaches

The reviewed and proposed normal-mode-based MNFs are summarized in Tab 24which are assessed on an example of interconnected VSCs in Section 27

26 Nonlinear Indexes Based on Nonlinear Normal

Modes

From the normal forms several nonlinear properties can be extracted such as theamplitude-dependent frequency-shift [2122] and the stability assessment

26 Nonlinear Indexes Based on Nonlinear Normal Modes 95

Table 24 A Summary of the Presented Normal-Mode-Based MNFs

Method Transformation DecoupledInvariant Normal Dynamics Related Equation

LNM [55] linear decoupled linear (292)

NM2 [48] order 2 decoupled linear (2105)

FNNM [18] order 3 invariant order 3 (2123)

NNM ( [2122] order 3 decoupled order 3 (2126)

261 Amplitude Dependent Frequency-Shift of the OscillatoryModes

Rp(t) = Cpeminusσpt cos(ωnlp t+ ϕ) = a cos(ωnlp t+ ϕ) (2132)

Where the parameter Cp represents the amplitudes of the nonlinear oscillationsrespectively which is a function of the initial conditions R0

p S0p [18] And R0

p S0p

satises X0p = R0

p +R(3)(R0i S

0i ) Y 0

p = S0p + S(3)(R0

i S0i )

ωnlp is the damped natural frequency of the oscillation Compared to the previousresearch [40] it is not a constant value but amplitude-dependent which reads

ωnlp = ωp

(1 +

3(Apppp + hpppp) + ω2pB

pppp

8ω2p

a2

)(2133)

The frequency-shift is an important property of the nonlinear dynamical system(2132) (2133) indicate how the nonlinear interactions between the state variablesbrought by nonlinear couplings inuence both the amplitude and frequency of theRp Unlike the previous researches where all the coupling terms are canceled afternormal from transformation the NNM approach concentrates nonlinearities intothe self-coupling terms while weakens the cross-coupling terms This simplies thesystem while reserving the nonlinear essence to the maximum

262 Stability Bound

On the stability bound the amplitude of Rp reaches the maximum and Rp = 0therefore (2126) reads

Rp = minusω2pRp + (hpppp +Apppp)R

3p

= minus[ω2p + (hpppp +Apppp)R

2p

]Rp (2134)

Therefore for the p-th mode to ensure the stability the Rp should meet

ω2p + (hpppp +Apppp)R

2p ge 0 (2135)


Methodologies

with

Rp le

radicminus

ω2p

hpppp +Apppp(2136)

As Rp is the nonlinear projection of Xp in the Normal-Mode coordinate we shouldcalculate the bound in the physical coordinate taking into account of the scaling ofthe coordinatesTo predict the stability bound in the physical coordinate the scaling problem in-troduced by the nonlinear transformation should be consideredStrictly speaking the stability bound in XY coordinates is a nonlinear mapping ofthat in RS coordinatesSince there is no an inverse nonlinear transformation a nonlinear factor is intro-duced to facilitate the calculation which is as followsSince when only p-th mode is excited Ri 6= 0 Rj 6= 0 when i 6= 0 j 6= 0

Xp = Rp +R3p + apppR

2p + rppppR

3p (2137)

The nonlinear can be calculated as XpRp when assuming Rp = 1 which reads

Knl =RpXp

=1

1 + apppp + rpppp(2138)

Therefore the proposed stability bound (PSB) can be dened as

PSB = RpKnlΦ + qSEP (2139)

263 Analysis of Nonlinear Interaction Based on Closed-form Re-sults

The dierence between the nonlinear approach and linear approach are demon-strated in two aspectsbull Vibration amplitudesbull Amplitude-dependent frequency-shift

Thus when analyzing the nonlinear interaction in the system nonlinear index shouldbe dened both for vibration amplitude and vibration frequency Compared to theprevious research which assumes that there is no frequency-shift the proposedmethod is more accurate in predicting nonlinear dynamicsBased on the work of [59 65] the nonlinear index for vibration amplitude can bedened as (2140)

NIp =

radicsumNk=1

sumNj=1

∣∣∣Rp0 + apijRi0Rj0 minusXp0

∣∣∣2|Xp0|

(2140)

Where Rp0 Xp0 is the initial value of LNM and NNM correspondingly which canbe calculated algebraically

27 The Performance of Method NNM Tested by Case Study 97

NIp can measure the biggest error caused by the linear approach to study thenonlinear dynamics with respect to the nonlinear vibration amplitudes In fact thedenition of NIp is based on the assumption that the vibration frequency capturedby Xp and Rp is the same Thus when there is a big frequency-shift in the nonlineardynamics NIp may lead to wrong predictionsAs the nonlinear frequency of each mode is indicated in (2133) [18] the frequency-shift index of each mode p is therefore dened which writes

FSp =3(Apppp + hpppp) + ω2

pBpppp

16πωp(2141)

If the cubic terms in (2126) are neglected it reads a linear equation whichfails to capture the amplitude-dependent frequency-shift in nonlinear dynamics Inthe previous researches the nonlinearities are truncated in the equations after thenormal form transformation [5965] which is correct only when FSp is smallThe impacts of parameters on the nonlinear interaction can be studied analyticallyin avoidance of time-consuming and high-cost numerical simulations When thenonlinear indices are small it means that linear analysis is not that far from beingaccurate whose technique is more mature and simple

27 The Performance of Method NNM Tested by Case

Study

A small example of the interconnected VSCs is used to show the physical insightsbrought by the NNM and the proposed nonlinear indexes

271 Interconnected VSCs

The interconnection of power systems and injection of wind energy into the trans-mission grid are realized by large capacity Voltage Source Converters (VSCs) Theproposed case study is composed of two VSCs V SC1 and V SC2 which are inter-connected by a short connection line with the reactance X12 and are both connectedto the transmission grid by long transmission lines with reactance X1 X2 as shownin Fig 21To study the nonlinear interactions of the power angle δ1 δ2 under disturbances thedetailed mathematical model is rst made which consists of the physical structureas well as the current voltage and power loops To design the power loop sincethe conventional PLL controller can cause synchronization problems when facingconnections using long transmission lines [66] the Virtual Synchronous Machine(VSM) control strategy is adopted [29]Linear stability analysis shows that this system works well under dierent operatingpoints However when there is a large disturbance the linear analysis tools fail tooer satisfying resultsThe overall control is composed of three cascaded loops where only the power loop


Methodologies

Figure 21 The Interconnected VSCs

is considered since voltage and current loops dynamics decays to steady state rapidlycompared to the power loops After reducing the modelthe system can be expressedas (2142) The procedures to reduce the model and all the parameters can be foundin Appendix

J1d2δ1dt2

+D1(δ1 minus ωg) +V1VgL1

sin δ1 + V1V2L12

sin(δ1 minus δ2) = P lowast1

J2d2δ1dt2


sin δ2 + V1V2L12


(2142)

Where J1 D1 J2 D2 are the controller parameters of the power loop δ1 δ2 arethe power angles V1 V2 are the voltages P lowast1 P

lowast2 are the references values of the

power of the two VSCs ωg is the angular frequency of the grid and L1 L2 is theimpedance of the cable connecting VSC to the grid and L12 is the impedance ofthe transmission line connecting the two VSCs to each other L1 L2 is very high(07pu) leading a very big power angle of VSC to transfer the electricity L12 isvery low due to high voltage of the transmission system leading to strong couplingbetween the two VSCs When there is a large disturbance on one VSC there willbe drastic nonlinear oscillations between the two VSCsPerturbing (2142) around the stable equilibrium point (SEP) and applying thetransformations of coordinates it reads in RS coordinates


R1 + 2ξ1ω1R1 + ω21R1 + (h1111 +A1

111)R31 +B1

111R1R12

+ C1111R

21R1

+R1((A1212 +A1

122 + h1122)R22 +B1

122R22

+ (C1122 + C1

212)R2R2) + R1(B1212R2R2 + C1

221R22) = 0

R2 + 2ξ2ω2R2 + ω22R2 + (h2222 +A2

222)R32 +B2

222R2R22

+ C2222R

22R2

+R2((A2112 +A2

211 + h2112)R22 +B2

211R12

+ (C2211 + C2

121)R1R1)) + R2(B2112R1R2 + C2

112R22) = 0

(2143)

(2143) appears longer on the page than (289) but it is much simpler in essenceTo analyze the relation between the two modes we can rewrite the equation intothe form as (2144) where Q2

p P2p represent quadratic polynomials

R1 + 2ξ1ω1R1 + ω21R1 + (h1

111 +A1111)R

31 +B1

111R1R12+ C1

111R21R1 +R2P

(2)1 (Ri Ri) + R2Q

(2)1 (Ri Ri) = 0

R2 + 2ξ2ω2R2 + ω22R2 + (h2

222 +A2222)R

32 +B2

222R2R22+ C2

222R22R2 +R1P

(2)2 (Ri Ri) + R1Q

(2)2 (Ri Ri) = 0

(2144)

(2144) highlights the couplings between R1 and R2 Although this equation stillhas nonlinear coupling it puts all the nonlinearities to cubic terms and the strengthof coupling is largely reduced What is more when only one mode is excited forexample R1 6= 0 but R2 = 0 R2 = 0 there will be no couplingsThis is shown in (2145)

R1 + 2ξ1ω1R1 + ω21R1 + (h1111 +A1

111)R31 +B1

111R1R12

+ C1111R

21R1 = 0

(2145)

Now the dynamic of the system will just exhibit vibration under one single modeThe complex dynamics are decomposed This is the invariance In the (289) evenwhen X2 = 0 X2 at t = 0 the motion of X1 will denitely inuence the value ofX2 X2making it impossible to analyze and control X1 X2 independentlyAlso if we make some approximations when R1 6= 0 R2 6= 0Equation ((2143)) willbe

R1 + 2ξ1ω1R1 + ω21R1 + (h1111 +A1

111)R31 +B1

111R1R12

+ C1111R

21R1 = 0

R2 + 2ξ2ω2R2 + ω22R2 + (h2222 +A2

222)R32 +B2

222R2R22

+ C2222R

22R2 = 0

(2146)

Now we can see that the system is decoupled Although it has some quantitativeerror compared to the original equation it represents the nonlinear behavior of thesystem with great accuracy and can be used to decompose the complex dynamicsfor analysis and decouple the system for controlThe 2-dimension second-order problem is reduced to 1-dimension second-order prob-lem The complexity of the problem is largely reduced We can solve these two inde-pendent nonlinear equations by numerical integration with much lower complexityor perturbation methods to obtain analytical results of R1 R2 And then by (2127)


Methodologies

the system dynamics can be obtainedThen in our case the dynamics of R1 R2can be studied independently and analyti-cal results can be obtained by perturbation method Thus the nonlinear behavior canbe predicted by the analytical results (discussed in Section 275) and the dynamicsare decomposed (Section 274) and it is possible to analyze the nonlinear inuencesof parameters to suggest the analysis based control (discussed in Section 276 andChapter 6)

272 Analysis based on EMT Simulations

The proposed system is simulated using a EMT (Electromagnetic transient) soft-ware The test consists in experiencing at t = 0 a large disturbance in the powerangle of VSC2 Before the disturbance the two VSCs are working under dierentoperating points

0 02 04 06 08 10

05

1

15

δ1(rad)

EXNNMLNM

0 02 04 06 08 1minus05

0

05

1

15

t(s)

δ2(rad)

EXNNMLNM

0 005 01 015 020

05

1

15

δ1(rad)

EXNNMLNM

0 005 01 015 02minus05

0

05

1

15

t(s)

δ2(rad)

EXNNMLNM

Figure 22 Comparison of LNM and NNM models with the exact system (EX) Fulldynamics (left) and Detail (right)

The nonlinear dynamics of the system are shown in Fig 22 where the Exact Model(named EX and based on (286)) is compared to the LNM and NMM models Itshows that the large disturbance imposed on only one VSC causes drastic power an-gle oscillations in both VSCs This shows the signicance of studying the nonlinearoscillations caused by the interconnection of VSCs in weak grid conditions More-over the NNM model match the exact model much more better than the LNMnot only on the vibration amplitudes but also on the frequency shift due to thenonlinearities

273 Coupling Eects and Nonlinearity

The accuracies of Model FNNM (2123) NNM (2126) and NM2 (2105) to the exactmodel are compared in Fig 23 to study the coupling eect and nonlinear essence ofthe studied system It seen that NNM is almost as good as FNNM convincing that


0 02 04 06 08 1minus02

minus01

0

01

02

Errorin

δ1(rad)

Error

FNNMNNMNM2

0 02 04 06 08 1minus02

minus01

0

01

02

Errorin

δ2(rad)

Figure 23 The Eect of the Cubic and Coupling Terms in the Nonlinear ModalModel

the coupling can be neglected The LNM cannot track the nonlinear dynamicsbecause the cubic terms are neglected which are responsible of the amplitude-dependent frequency-shifts [17] The present NNM method appears very accurateand keep the advantage of linear normal modes as it approximately decomposes thedynamics into independent (nonlinear) oscillators

274 Mode Decomposition and Reconstruction

As NNM method makes possible to model the nonlinear dynamics by decomposingthe model into two independent nonlinear modes R1 R2 whose nonlinear dynamicsare described by (2126) the complex dynamics can be viewed as two independentmodes Fig 24 shows those two modes projected on the generalized coordinatesδ1 and δ2 As with linear modes that decompose a N -dimensional linear systeminto a linear sum of 1-dimensional linear systems nonlinear modes make possibleto decompose a N -dimensional nonlinear system into a nonlinear sum of nonlinear1-dimensional nonlinear system

275 Amplitude-dependent Frequency-shift

Another important characteristic in nonlinear dynamics is the amplitude-dependentfrequency-shift shown in Fig 22 not captured by Model LNM and NM2 nor evenby FFT analysis The natural frequencies of the two linear modes (X1 X2) areextracted as ω1 = 114rads ω2 = 1374rads at the equilibrium point There isslight frequency-shift in the mode frequency according to the vibration amplitudes


Methodologies

0 05 1 15 2minus05

0

05

1

15

t(simulation time)

δ1(rad)

Mode Decomposition

EXMode 1Mode 2

0 05 1 15 2minus05

0

05

1

15

t(simulation time)

δ2(rad)

Figure 24 Mode Decomposition An Approximate Decoupling of Nonlinear System

Based on (2133) it can be predicted that if nonlinear modal frequency ω2 is almostconstant frequency ω1 experiences variations around 1 rads as shown by Fig 25

276 Analytical Investigation on the Parameters

In our case the nonlinear interaction is mainly determined by the reactances oftransmission and connection lines It is seen from Fig 26 that the reactancesof transmission lines X1 X2 mainly inuence the nonlinear vibration amplitudesWhile the reactance of connection line is more responsible for the frequency-shift ofthe modes shown in Fig 27Furthermore it is also seen from Fig 26 that Mode minus 1 is more sensitive to thereactance of the transmission lines than Mode minus 2 This corresponds to the factshown in Fig 24 that the vibration amplitude is decided by Modeminus 1It is seen from Fig 27 thatNIp keeps aLNMost constant while Modeminus2 shows greatvariance in the frequency-shift indicated by FS2 When only NIp is considered thischaracteristic cannot be captured and this will lead to great error

2761 Verication by EMT

The predictions of nonlinear dynamics by nonlinear indices are veried by the EMTsimulation Both the dynamics and errors are plotted out shown in Fig 28 andFig 29 To make the plots more clear and with the limitation of space the dynamicsare just shown for an interval 02s while the errors are plotted out in a longerinterval 1s to give more information in comparing the performance of nonlinearanalysis and its linear counterpartPlots in Fig 28 and Fig 29 verify the predictions made by the nonlinear indexesAs NIp indicates the performance of linear analysis in cases of X12 = 0001 and


0 02 04 06 08 1

005

1105

11

115

Disturbance in δ1

Mode Frequency

Disturbance in δ2

ω1(rad)

0 02 04 06 08 1

005

1137

1375

138

Disturbance in δ1

Disturbance in δ2

ω2(rad)

Figure 25 Amplitude-dependent Frequencies of the Nonlinear Modes

01 03 07 09 110

005

01

015

02

025

03

035

04

045

05

Reactance of transmission linesX1 X2

NI

Mode1Mode2

01 03 07 09 110

05

1

15

2

25

3

35


FS

Mode1Mode2

Figure 26 X1 X2 vary from 01 to 11 while X12 keeps 001

0001 0005 001 005 010

005

01

015

02

025

03

035

Reactance of Connection LineX12

NI

Mode1Mode2

0001 0005 001 005 010

1

2

3

4

5

6

7

8

9

10


FS

Mode1Mode2

Figure 27 X1 X2 keep 07 while X12 varies from 0001 to 01


Methodologies

0 005 01 015 02minus01

0

01

02

δ1(rad)

Dynamics

0 005 01 015 02minus02

minus01

0

01

02

t(s)

δ2(rad)

EXNNMLNM

(a) X1 = X2 = 01

0 005 01 015 0205

1

15

2

δ1(rad)

Dynamics

EXNNMLNM

0 005 01 015 020

05

1

15

t(s)

δ2(rad)

(b) X1 = X2 = 11

0 02 04 06 08 1minus5

0

5x 10

minus4

Errorin

δ1(rad)

Error

0 02 04 06 08 1minus5

0

5x 10

minus4

Errorin

δ2(rad)

NNMLNM

0 02 04 06 08 1

minus05

0

05

Errorin

δ1(rad)

Error

0 02 04 06 08 1

minus05

0

05

Errorin

δ2(rad)

NNMLNM

Figure 28 Dynamics Veried by EMT when X12 keeps 001


0 005 01 015 020

05

1

15

δ1(rad)

Dynamics

EXNNMLNM

0 005 01 015 02minus1

0

1

2

t(s)

δ2(rad)

(a) X12 = 0001

0 005 01 015 020

05

1

15

δ1(rad)

Dynamics

EXNNMLNM

0 005 01 015 02minus1

0

1

2

t(s)

δ2(rad)

(b) X12 = 01

0 02 04 06 08 1minus05

0

05

Errorin

δ1(rad)

Error

NNMLNM

0 02 04 06 08 1minus05

0

05

Errorin

δ2(rad)

0 02 04 06 08 1minus01

0

01

Errorin

δ1(rad)

Error

NNMLNM

0 02 04 06 08 1minus01

0

01

Errorin

δ2(rad)

Figure 29 Dynamics Veried by EMT when X1 X2 keeps 07


Methodologies

X12 = 01 should be same However it is seen from Fig 29 the error of linearanalysis in case of X12 = 0001 can be as big as 05 while in case of X12 = 01the error is less than 01 Compared to the methodology proposed in the previousresearch it is convinced that our method is more accurate by considering bothvibration amplitudes and frequency-shift

277 Closed Remarks

Veried by EMT simulation this example shows that the NNM has a good perfor-mance in approximating the nonlinear dynamics of the system And it weakens thenonlinear couplings by normal form transformations and approximately decouplesthe nonlinear system reducing the complexity of the systemThird-order based analytical investigation on the inuence of the parameters onthe nonlinear dynamics taking into accounts of both vibration amplitudes andfrequency-shift in avoidance of numerical simulations In recent works [59 65]nonlinear index is dened as the criteria to quantify the inuence of controllerparameters on nonlinear interaction However failing to take into account thefrequency-shift their prediction can be wrong in some cases Thus in this PhDwork nonlinear indices are for the rst time dened both for the vibration ampli-tude and the frequency-shift rendering the nonlinear predictions more precise

28 Issues Concerned with the Methods of Normal

Forms in Practice

The methods of normal forms are rstly proposed by Poincareacute in 1899 its applicationin analysis of system dynamics Though the methods of normal forms theoreticallyrender a simplest form of nonlinear dynamical systems several issues are concernedin the practice

281 Searching for Initial Condition in the Normal Form Coordi-nates

The analytical expression of normal form transformation normal forms and nonlin-ear indexes are derived in the previous sections However the analytical expressionsof the initial condition in the normal form coordinates are not availableSince there is no analytical inverse of the normal form transformation the numericalsolution is obtained by formulating a nonlinear system of equations of the form

f(z) = z minus y0 + h(z) = 0 (2147)

The solution z (which is w for method 3-3-1 and method 3-3-3 respectively) to theabove equation (2147) provides the initial condition z0 This system of equationsis directly related to the nature of the second-order nonlinear terms andor third-order nonlinear terms And it describes how the z-variable dier from y0 as a result

28 Issues Concerned with the Methods of Normal Forms in Practice107

of the nature of the higher-order terms The numerical solution of these equationsis complicated It requires a robust algorithm and is sensitive to the choice of theinitial conditions for the solutionThe most popular algorithm is the Newton-Raphson method which is adopted inthis PhD work A more robust algorithm is proposed in [47] to circumvent somedisadvantages of NR method which can be adopted also for the 3rd order NFmethodsIt is recommended that y0 be selected as the choice of the initial condition based onthe nature of the system of (2147) where the variables z dier from y0 according tothe higher-order nonlinear terms ie z0 = y0 where there is no higher-order termFrom the analysis on a variety of test systems the choice of z0 = y0 provides themost robust results The pseudo-code of searching for the initial condition of 3rdorder normal form methods is listed as follows which is an extension of the searchalgorithm for method 2-2-1 [48]

2811 The Search Algorithm of Initial Condition for method 3-3-3

The algorithm begins1 x0 x0 is the initial condition of the system dynamics at the end of the distur-bance eg when the fault is cleared in the power system this value can be obtainedfrom measurement or time-domain simulation

2 y0 y0 = Uminus1x0 the initial condition in the Jordan coordinate

3 z0 the system of (2147) is solved for z0 using the Newton-raphson method asfollows

a formulate a nonlinear solution problem of the form j = 1 2 middot middot middot N

fj(w) = wj +Nsumk=1

Nsuml=1

h2jklh2jklwkwl +Nsump=1

Nsumq=1

Nsumr

h3jpqrh3jpqrwpwqwr = 0

b choose y0 as the initial estimate for w0 Initialize the iteration counters = 0

c compute the mismatch function for iteration s j = 1 2 middot middot middot N

fj(w(s)) = w

(s)j +

Nsumk=1

Nsuml=1

h2jklh2jklw(s)k w

(s)l +

Nsump=1

Nsumq=1

Nsumr

h3jpqrh3jpqrw(s)p w(s)

q w(s)r = 0


Methodologies

dcompute the Jacobian of f(z) at w(s)

[A(ws)] =

[partf

partw

]w=w(s)

ecompute the Jacobian of f(z) at w(s)

4w(s) = minus [A(ws)]minus1 f(w(s))

f determine the optimal step length ρ using cubic interpolation or any other ap-propriate procedure and compute

w(s+1) = w(s) + ρ4w(s)

g continue the iterative process until a specied tolerance is met The value of ws

when the tolerance is met provides the solution w0

The algorithm ends

2812 Search for Initial Condition of the Normal Mode Approach

When it comes to the normal mode approach the search for initial condition isthe same except the nonlinear solution problem should be formulated as forallp =

1 2 middot middot middot N

fp = Rp +Nsumi=1

Nsumjgei

apijRiRj +Nsumi=1

Nsumjgei

Nsumkgej

rpijkRiRjRk minusXp (2148)

with Y 0 = 0 S0 = 0In fact there are well validated Newton-Raphson software package on line [67] whichcan determine the optimal step length to avoid the non-convergence and what isneeded in practice is just formulate the nonlinear solution problem in the form of(2147)The codes for all the methods presented in this PhD thesis can be found inthe Appendix


282 The Scalability Problem in Applying Methods of NormalForms on Very Large-Scale System

One concern of the methods of normal forms is the scalability problem when isapplied on the large-scale system or huge dimensional system One solution is toreduce the model by focusing on only the critical modes which will be illustratedin details in Chapter 3

283 The Dierential Algebraic Equation Formalism

In practice the dynamic power ow models are often formulated as DAE withnonlinear algebraic part (for instance due to constant power loads with small timeconstants) And generalizations of normal form approach that would be applicableto these systems are availableIn fact the dierential equations x = f(x) of system dynamics essentially comesfrom a set of DAE equation composed of the dierential equations representing themachine dynamics and the algebraic equations representing the network connectivityand power ow constraints Using the power ow analysis the stable equilibriumpoint (SEP) is obtained and the dierential equation is the DAE perturbed aroundthe SEPThere is an alternate formalism in [54] which transforms DAEs into an explicit setof dierential equations by adding an appropriate singularly perturbed dynamicsIt points out the interests to take DAE into normal form methods as

1 Sparsity can be taken into account2 DAE models are compatible with current small signal stability commercial

software ie PSSE DSAT tolos etc


Methodologies

29 Comparisons on the two Approaches

291 Similarity and Dierence in the Formalism

For methods of normal mode approach (289) can be rewritten into an equivalentrst-order form with state-variables [Xp Yp]

X1

Y1

Xp

Yp

XN

˙YN

=

0 1

minusω21 minus2ξ1ω1

0 0 middot middot middot 0

0

0 1

minusω2p minus2ξpωp

0

0

0 0 middot middot middot

0 1

minusω2N minus2ξNωN

X1

Y1

Xp

Yp

XN

YN

+

g(X1) + h(X1)

0

g(Xp) + h(Xp)

0

g(XN) + h(XN)

0

(2149)


As observed from (2149) the linear matrix is composed with N [2times2] blocks at theprinciple diagonal In such a way Xp are independent of Xj(j 6= p) but dependentof YpDiagonalizing the linear matrix the diagonal element may have complex conjugatevalues and Xp will be independent of Yj which is exactly the vector eld approachAs observed from (2149) and (230) the dierence between the two approacheslays in the linear transformation ie the way to decouple the linear termsIn the vector eld approach the basic unit is dierential equations of each state-variable yp yp is linearly uncoupled with yj j 6= p in the normal mode approachthe basic unit is dierential equations of the state-variable pair [Up Vp] [Up Vp] islinearly uncoupled with [Up Vp] but Up are coupled with VpAs indicated in Section 2911 since the simplied linear matrix should have asimple real form [30] or pure imaginary form [33] to ensure the accuracy the vectoreld approach which involves the complex form numbers is less accurate than thenormal mode approach


The essentials of the normal form transformation of those two approaches are equat-ing the like power systems as generalized in Eqs (219) to (227)The dierence in the normal form transformation is that in the vector eld ap-proach yj is canceled by the replacement yj = λjyj +

sumNk=1

sumNl=1 F2jklykyl +sumN

p=1

sumNq=1

sumNr=1 F3jpqrypyqyr + middot middot middot the equating of like power terms involves only

29 Comparisons on the two Approaches 111

the state-variable ypin the normal mode approach Up is canceled by letting Up = Vpand Vp is canceled by letting Vp = minusω2

pUpminus2ξpωpVp the equating of like power termsinvolves the state-variable pair [Up Vp]

292 Applicability Range and Model Dependence

Second-order (or order higher than 2nd order) equations denitely have equivalentrst-order equations but the equivalent second-order (or order higher than 2ndorder) equations of rst-order equations are not guaranteed From the mathematicalview the applicability range of the normal mode approach is a subset of that of thevector eld approachHowever from the engineering view the normal mode approach may do a better jobsince it represents better the physical properties of inter-oscillations and highlightsthe relations between the state variables Also the complex-form will cause someerror in the computation compared to the normal mode approach which will beillustrated by case studies in Chapter 4The vector eld approach is the most widely used approach in analysis and control ofnonlinear dynamical systems and aLNMost applicable for all types of power systemmodels with dierentiable nonlinearities First applied and advocated by investiga-tors from Iowa State University in the period 1996 to 2001 [615] it opens the erato apply Normal Forms analysis in studying nonlinear dynamics in power systemIts eectiveness has been shown in many examples [6 912141520314754]The normal mode approach proposed in this PhD work is found mostly in me-chanical engineering [171956] and can be adopted to study the electromechanicaloscillations in interconnected power system which will be illustrated in Chapter 4

293 Computational Burden

When comes to the issue of computational burden the normal mode approach isadvantaged over the vector eld approach since calculating the complex-form valuesis much heavier than that of the real-form values when

1 Formulating the nonlinear matrices calculating F2 F3 may be quite laboriouscompared to calculation of H2 H3 by both two possible approachesbull approach 1 rstly calculating the matrix H2 H3 (H2 H3 are cal-culated by doing the Hessian matrix and 3rd order dierentiation of

f(x)) secondly calculating F2 F3 by F2j = 12

sumNi=1 vji

[UThj3U

]

F3jpqr = 16

sumNi=1 vji

sumNk=1

sumNk=1

sumNm=1H3ikLNMu

lpumq u

Nr

bull approach 2 rstly calculating the nonlinear equation F (y) on the vectorelds by F (y) = Uminus1f(Uy) secondly doing the Hessian matrix and3rd order dierentiation of F (y) to obtain F2(y) F3(y)

Both those two approaches involve complex-form matrix multiplication lead-ing to heavy computational burden

2 Searching for the initial conditions The state-variables x0 are real-form valuesand the state-variables y0 contain complex-form values as far as there exists


Methodologies

oscillations in the system dynamics and the terms h2 h3 are in complex-formtherefore the mismatch function contain complex-values this will add to thecomputational burden in searching for the initial conditions

294 Possible Extension of the Methods

As indicated in Section 22 both method 3-3-3 and method NNM can be extendedto include terms higher than 3rd orderThe method NNMmay be also extended to system with state-variable pair consistingofM state-variables [q1 q2 middot middot middot qM ]M gt 2 ie system of higher-order equations Insuch a case when represented by rst-order equations the linear matrix is composedof [M timesM ] blocks at the principle diagonal

210 Conclusions and Originality

In this chapter two approaches are proposed to deduce the normal forms of systemof nonlinear dierential equations One is the most classical vector-eld approachmethod 3-3-3 working for system of rst-order equations The other approachis the normal mode approach method NNM working for system of second-orderequationsMethod 3-3-3 makes possible the quantication of 3rd order modal interaction andthe assessment of nonlinear stability and will be applied to the interconnected powersystem in Chapter 3By method NNM nonlinear interaction can be described by a N -dimensional cou-pled second-order problem which involves highly computational numerical simu-lation and analytical results are rare In this chapter a methodology is proposedto solve this problem by decoupling it into a series of 1-dimensional independentsecond-order problems whose analytical solutions are available and by which com-plex dynamics are decomposed into several simpler ones Moreover nonlinear be-havior can be accurately predicted by the dened nonlinear indexes in avoidance ofnumerical simulation The application of this method may be

1 Providing information in designing system parameters by quantifying theirinuence on the nonlinearities of the system with the nonlinear indexes asshown in Section 27

2 Transient stability analysis of interconnected power systems which will beillustrated in Chapter 4

The similarities and dierences between the vector-eld approach and normal modeapproach have been made to go further in illustrating the principle of normal formtheory And possible extensions of method 3-3-3 and method NNM have beenpointed out to cater for further demands in future researchesAll the nonlinear properties can be predicted by one approach can be revealed bythe other which are summarized in Tab 25 for mode jWhats more the competitors of methods of normal forms (MNF) has been reviewedsuch as the MS method in the vector eld approach and Shaw-Pierre (SP) method in


Table 25 The Nonlinear Properties Quantied by Method 3-3-3 and Method NNM

Properties Nonlinear Modal Amplitude-dependent Nonlinear

Interaction Frequency-shift Stability

3-3-3 h2jkl h3jpqr Im(SIIj) SIj

NNM aj bj cj middot middot middot FS PSB

forming the NNM As it is dicult to nd literature comparing MNF to MS methodand SP method we hope that the comparisons made in this PhD thesis bridge a bitthe gaps in the literatureConcerning the originality it is for the rst time that method 3-3-3 is proposed tostudy the system dynamics And for method NNM though its FNNM has beenproposed in mechanical domain it is mainly for bifurcation analysis The nonlinearproperties extracted from the method NNM such as the amplitude-frequency shiftand nonlinear mode decomposition are rstly found in power system Whats moreall the nonlinear indexes are at the rst time proposed by this PhD workIn the previous researches the methods of normal forms are conventionally used forbifurcation analysis [49] And their derivations of normal forms focus more on thevariation of parameters ie the parameters can be viewed as state-variables butin dierent-scales with the physical state-variables of the system In this chapterwe focus on the inuence of physical state-variables on the system dynamics


Methodologies

Chapter 3

Application of Method 3-3-3Nonlinear Modal Interaction and

Stability Analysis ofInterconnected Power Systems

The inclusion of higher-order terms in Small-Signal (Modal) analysis has been an

intensive research topic in nonlinear power system analysis Inclusion of 2nd or-

der terms with the Method of Normal Forms (MNF) has been well developed and

investigated overcoming the linear conventional small-signal methods used in the

power system stability analysis and control However application of the MNF has

not yet been extended to include 3rd order terms in a mathematically accurate form

to account for nonlinear dynamic stability and dynamic modal interactions Due to

the emergence of larger networks and long transmission line with high impedance

modern grids exhibit predominant nonlinear oscillations and existing tools have to

be upgraded to cope with this new situation

In this chapter the proposed method 3-3-3 is applied to a standard test system the

IEEE 2-area 4-generator system and results given by the conventional linear small-

signal analysis and existing MNFs are compared to the proposed approach The

applicability of the proposed MNF to larger networks with more complex models has

been evaluated on the New England New York 16 machine 5 area system

Keywords Interconnected Power System Power System Dynamic Non-linear Modal interaction Stability Methods of Normal Forms

115

116Chapter 3 Application of Method 3-3-3 Nonlinear Modal Interaction

and Stability Analysis of Interconnected Power Systems

31 Introduction

Todays standard electrical grids are composed of several generators working inparallel to supply a common load An important problem associated with intercon-nected power systems is the presence of oscillations that could have dangerous eectson the system The multiplication of distributed generation units usually composedof renewable-energy-based-generators and the increase of energy exchanges throughlong distance lead to highly stressed power systems Due to the large amount ofpower owing through the lines the low-frequency oscillations called in classicalpower system studies electromechanical oscillations exhibit predominant nonlin-ear behaviors Since these oscillations are essentially caused by modal interactionsbetween the system components after small or large disturbances they are callednonlinear modal oscillations higher order modes or higher order modal interactionsinaccurately modelled by the linear analysis based on a linearized modelAlthough intensive research has been conducted on the analytical analysis of non-linear modal oscillations based on the Normal Form Theory with inclusion of 2ndorder terms in the systems dynamics this chapter proposes to show that in certainstressed conditions as modern grids experience more and more inclusion of 3rdorder terms oer some indubitable advantages over existing methodsThe Method of Normal Forms (MNF) being based on successive transformations ofincreasing orders the proposed 3rd order-based method inherits the benets of thelinear and the 2nd order-based methods ie

1 Analytical expressions of decoupled (or invariant) normal dynamics2 Physical insights keeping the use of modes to study the contribution of system

components to inter-area oscillations3 Stability analysis based on the evaluation of the system parameters

311 Need for inclusion of higher-order terms

Small-signal analysis is the conventional analysis tool for studying electromechani-cal oscillations that appear in interconnected power systems It linearizes the powersystems equations around an operating point by including only the rst-order termsof the Taylor series of the systems dynamics The eigenanalysis is made to obtainanalytical expressions of the systems dynamic performances and the stability anal-ysis is realized on the basis of the rst Lyapunov Method (Analysis of the real partsof the poles) Besides modal analysis uses the eigenvectors to give an insight ofthe modal structure of a power system showing how the components of the powersystem interact Thanks to modal analysis power system stabilizers can be placedat the optimal location in order to stabilize the whole system ensuring then asmall-signal stability [1]Later researchers suggested that in certain cases such as when the system is severelystressed linear analysis techniques might not provide an accurate picture of thepower system modal characteristics From 1996 to 2005 numerous papers [615]have been published proving that higher order modal interactions must be studied

31 Introduction 117

in case of certain stressed conditions MNF with the inclusion of 2nd order termsshows its great potential in power system stability analysis and control design Thoseachievements are well summarized in the Task-force committee report [48] whoseformalism is referred to as the method 2-2-1 in Chapter 2The existing 2nd-order-based method gives a better picture of the dynamic perfor-mance and the mode interactions than the classical linear modal analysis Howeverit fails to take benet of the systems nonlinearities for studying the stability wherethe conventional small-signal stability analysis fails Based on this [5] proposed tokeep a second order transformation but with including some of the third-order termsin order to improve the system stability analysisThe emergence of renewable energy (unbalanced energy generation and weak grid)and fast control devices [59] increases the nonlinearity and inclusion of 3rd orderterms is demandedFinally some 3rd-order-based MNFs have been proposed in [3132] but have not beenfully developed yet not leading to a more useful tool than the ones using linear-basedand second-order-based methods For nonlinear mechanical systems that ofteninclude lightly damped oscillatory modes and possible internal resonances NormalForms up to third order are widely used either to classify the generic families ofbifurcations in dynamical systems [5368] or to dene Nonlinear Modes of vibrationand to build reduced-order models [171833]Taking into account the 3rd order nonlinearities will increase the expected modelaccuracy in most cases while keeping the overall complexity at a relatively lowlevel This can be illustrated by a simple example Primitive as the example is itvisualizes the need of inclusion of higher-order terms and the increased stress dueto the increased power transferredAs an illustration of the gains aorded by inclusion of higher-order-terms thetransferred active power through a power grid as a function of the power angleδ P = V 2

X sin δ is analyzed in steady state Considering the voltage V at the ter-minals and the reactance X of the line both constant the transmitted power canbe simplied as P = k sin δ k = constant The exact system (Exact) and its Taylorseries up to the rst order (Linear) up to the second order (Tayl(2)) and up tothe third order (Tayl(3)) are plotted in Fig 31 Depending on the power anglevalues inclusion of higher-order terms by nonlinear analysis especially up to 3rdorder terms is necessary to maximize the transfer capacity of the modern powersystem with ever-increasing stress




Powerf(δ)

02

04

06

08

1

12

14

16Different Order Analysis and the Operational Region of the Power System

Exact

Linear

Tayl(2)

Tayl(3)

Regionmaybeunstable

3rd Order Analysis

2nd Order Analysis

Linear Analysis

Figure 31 The Need of Inclusion of Higher-Order TermsThe Chapter 3 is organized as follows A brief review of the existing normal formson vector eld and proposed method 3-3-3 is conducted in Section 32 The testbenches are presented in Section 33 with mathematical modelling The applicationsof method 3-3-3 will be conducted on interconnected power systems in Section 34and Section 35 where the performance of reviewed and proposed MNFs on vectorelds will be assessed And the benets promised in Chapter 2 will be demonstratedSection 36 discusses the factors inuencing the Normal Form analysis Section 37proposes some conclusions and possible applications of the proposed method aresuggested

32 A Brief Review of the existing Normal Forms of Vec-

tor Fields and the Method 3-3-3

The Methods of Normal Forms (MNF) was initially developed by Poincareacute [50] tosimplify the system dynamics of nonlinear systems by successive use of near-identitychanges of coordinates The transformations are chosen in such a way as to eliminatethe nonresonant terms of a corresponding order The principle is to transform thesystem dynamics into invariant normal dynamics The word invariant means thatqualied property is the same in the system dynamics and normal dynamics It was

32 A Brief Review of the existing Normal Forms of Vector Fields andthe Method 3-3-3 119

proved by Poincareacute that linear invariants are eigenvalues and nonlinear invariantsare the homogeneous resonant terms [69]In Chapter 2 the existing methods linear small-signal analysis method 2-2-1method 3-2-3S method 3-3-1 and the proposed method 3-3-3 have already illus-trated with detailed derivations In this section for the compactness a simplereview is made and the assessment of those methods is performed in the followingsections

321 Modeling of the System Dynamics

The class of systems that can be studied by MNF are usually modeled by DierentialAlgebraic Equations (DAEs) [1] By subsituting the algebraic equations into thedierential ones one transforms those DAEs in a dynamical system which can bewritten

x = f(xu) (31)

where x is the state-variables vector u is the systems inputs vector and f isa nonlinear vector eld Expanding this system in Taylor series around a stableequilibrium point (SEP) u = uSEP x = xSEP one obtains

4x = H1(4x) +1

2H2(4x) +

1

3H3(4x) +O(4) (32)



] H3jklm =


]and O(4) are


3211 The Linear Small Signal Analysis

The linear part of (32) is simplied into its Jordan form



yj = λjyj +

Nsumk=1

Nsuml=1

F2jklykyl +

Nsump=1

Nsumq=1

Nsumr=1



sumNi=1 vji

[UThj3U

] F3jpqr = 1

6

sumNi=1 vji

sumNk=1

sumNk=1

sumNm=1H3iklmu

lpumq u

nr vji is

the element at j-th row and i-th colomn of Matrix V ulp is the element at the p-throw and l-th column of matrix U The fundamentals of linear small-signal analysis uses the sign of the real parts σjof the eigenvalues λj = σj + jωj to estimate the system stability U is used toindicate how each mode yj contribute to the state-variable x and V indicates how



the state-variables of x are associated to each mode yj

x = Uy rArr yj = λjyy +O(2) (35)

322 Method 2-2-1

Gradually established and advocated by investigators from Iowa State Universitythe period 1996 to 2001 [615] opens the era to apply Normal Forms analysis instudying nonlinear dynamics in power system This method is labelled 2-2-1 ac-cording to our classicationIts eectiveness has been shown in many examples [6 912 14 15 20 47] to ap-proximate the stability boundary [9] to investigate the strength of the interactionbetween oscillation modes [61047] to dealt with a control design [1112] to analyzea vulnerable region over parameter space and resonance conditions [14 15] and tooptimally place controllers [20]This method is well summarized in [48] which demonstrated the importance ofnonlinear modal interactions in the dynamic response of a power system and theutility of including 2nd order terms Reference [70] propose an extension of themethod to deal with resonant cases It has become a mature computational tooland other applications have been developed [3654] or are still emerging [71]Method 2-2-1 can be summarized as

yj = zj + h2jklzkzl rArr zj = λjzj +O(3) (36)

with h2jkl =F2jkl

λk+λlminusλj where O(3) = (DF2)h2 + F3It is very important to note that the normal dynamics given by (36) uses the linearmodes of the linearized system The nonlinearities are then taken into accountonly through the nonlinear 2nd-order transform y = z + h2(z) and results in aquadratic combination of linear modes leading to 2nd-order modal interactionsThen the stability analyses conducted using this linear normal dynamics give thesame conclusion than the analysis that can be conducted using the linear dynamics

323 Method 3-2-3S

Although the second-order-based method gives a more accurate picture of the sys-tems dynamics than the linear small-signal analysis keeping some of the 3rd orderterms in (DF2)h2 + F3 contributes some important features to the method Asexamples improvements in the stability analysis have been proved in [5] and newcriteria to design the system controllers in order to improve the transfer capacity oftransmissions lines have been established in [16] where some 3rd order terms arekept in (37)The normal forms of a system composed ofM weakly-damped oscillatory modes are


then

y = z + h2(z)rArr

zj = λjwj +

sumMl=1 c

j2lwjw2lw2lminus1 +O(5) j isinM

wj = λjwj +O(5) j isinM

(37)

where cj2l is the sum of coecients associated with term wjw2lw2lminus1 of DF2h2+F3more details are found in Chapter 2It should be noted that the normal dynamics given by (37) is nonlinear contributedby the third-order terms cj2lwjw2lw2lminus1 depending on oscillatory amplitude |w2l|Therefore 3rd order terms contain the nonlinear property such as the stabilitybound [5 5] and amplitude-frequency dependent [21 22 55] However it is not thesimplest form up to order 3

324 Method 3-3-1

Later to study the 3rd order modal interaction method 3-3-1 has been proposed[3132] and 3rd order indexes are proposed [32]

y = w + h2(w) + h3(w)rArr wj = λjwj (38)

where h3jpqr =F3jpqr

λp+λq+λrminusλjHowever it is not an accurate 3rd order normal form since 1) it neglects the termDF2h2 and may lead to a big error in quantifying the 3rd order modal interaction2) it fails to study the oscillatory interaction between oscillatory modes and maylead to strong resonance

325 Method 3-3-3

To overcome those disadvantages the proposed method 3-3-3 takes into accountboth the M oscillatory modes and N minusM non-oscillatory modes

y = w + h2(w) + h3(w)rArr

wj = λjwj +

sumMl=1 c


wj = λjwj +O(5) j isinM(39)

As observed from (39) method 3-3-3 approximates the system dynamics by anonlinear combination of nonlinear modes Based on the normal forms nonlinearindexes MI3 and SI are proposed to quantify the 3rd order modal interaction andassess the stability of the system dynamicsIn this chapter all the presented methods have been assessed on the IEEE standardtest systems



33 Chosen Test Benches and Power System Modeling

In this PhD work two IEEE standard test systems are chosen the rst is theKundurs 4-machine 2-area system whose mechanism is easy to understand andwhich serves as a classic example for small-signal analysis and modal analysis Itserves as a didactic example to show how the proposed method 3-3-3 does a betterjob in quantifying the nonlinear modal interaction and assessing the stability ofsystem dynamics than the existing methods The second test bench is the NewEngland New York 16 machine 5 area system used to assess the applicability of theproposed method 3-3-3 to large scale power systemFor both two test benches the chosen mathematical model is built with reasonablesimplication and is well validated Power systems are nonlinear dynamic systemswhose behaviors are usually modelled by dierential-algebraic equations(DAEs)

Figure 32 The Relationships Between the Generator Exciter and PSS

The study of a general power system model might be dicult and unnecessary sincethe time constants of dierent dynamic elements are quite dierent ranging from10minus3s eg the switching time constant of power electronic devices like FACTSvoltage source converters (VSCs) to 10s eg the governor system for controllingthe active power input to generators Thus for a specic study purpose a com-mon handling is to consider the fast and slow dynamics separately which meansonly part of the dynamic equations corresponding to the elements with interest-ing behaviors will be remained In particular for studying the electromechanicaloscillation and angle stability of a power system which decides the overall powertransfer limit equations associated with the generator exciter and power systemstabilizer models are usually remained while the amplitude of terminal voltages andthe network parameters are assumed as constant The generator which is the sourceof the electro-mechanical oscillations the exciter which largely contributes to thetransient and sub-transient process in the system dynamics and the power systemstabilizer (PSS) which is used to stabilize the electromechanical oscillations Theirrelationships are shown in Fig 32For any stable interconnected power system the rotor angles of all generators are

34 Assessing the Proposed Methodology on IEEE Standard Testsystem 1 Kundurs 4 machine 2 area System 123

kept still to each other but constantly changing at a certain synchronized speedfre-quency which is slightly oating around 60Hz50Hz Since the electromechanicaloscillation will last for tens of seconds which is caused by the relative motions amongdierent power angles of concern a rotating coordinate system with d and q axesat the synchronized speedfrequency is commonly used to represent the generatorequations where the rotor is placed at the d axisFor power ow analysis the model used in this work is a phasor model wherethe voltages and currents are placed in the synchronization frame with x axes andy axes This simplies the power ow analysis and facilitates the analysis of thesystem dynamicsFor the 4 machine system the transient generator model is adopted and for the16 machine system the sub-transient generator model is adopted The transientgenerator model and sub-transient generator model are among the most widelyused generator models with a relatively high accuracy and low complexity

34 Assessing the Proposed Methodology on IEEE Stan-

dard Test system 1 Kundurs 4 machine 2 area Sys-

tem

The chosen test system to assess the proposed method is a well known IEEE standardsystem the Kundurs 2area 4machine system shown in Fig 33 It is a classicalsystem suitable for the analysis of modal oscillations for the validation of small-signalanalysis [1] and for the validation of 2nd order Normal Form analysis [48]

Figure 33 IEEE 4 machine test system Kundurs 2-Area 4-Machine System

The generators are modelled using a two-axis fourth-order model and a thyristorexciter with a Transient Gain Reduction Loads L1 and L2 are modeled as constantimpedances and no Power System Stabilizer (PSS) are used To make the systemrobust each area is equipped with large capacitor banks to avoid a voltage collapseThe data for the system and the selected case are provided in the Appendix sectionThe full numerical time domain simulation is used to assess the performance of theMNFs in approximating the nonlinear system dynamics based on the well validated



demo power_PSS in Matlab 2015aThe test case is modeled as a 27th order system with the rotor angle of G4 takenas the reference

341 Modelling of Mutli-Machine Multi-Area InterconnectedPower System

The diculty in modelling of Multi-Machine Multi-Area interconnected power sys-tem lays in the need of unifying the angle reference frameFor each generator the rotating frame is dierent therefore a common referenceframe with xy orthogonal axes is demanded to do the power ow analysis Thediculty lays in the transformation between the common reference frame with xyorthogonal axes and the rotating frame shown as Fig 34 which consists of the i-thdq rotating frame for generator i the dq rotating frame for generator k and thecommon reference frame xy All those three frames rotate at the synchronizationspeed ωsThere are dierent dq rotating frames and depending on which the expression oftransient and sub-transient models are established In this chapter two most widelyused common reference frames are introduced One is presented with the 4 machinecase and other other is presented with the 16 machine case

di

qi

x

y

dk

qk

δiδk

ωs

Figure 34 The Rotating Frame and the Common Reference Frame of 4 MachineSystem


3411 Transient Generator Model

Exciters (automatic voltage regulators ) are considered for the IEEE 4 machinesystem to represent the industrial situation to the maximum Therefore the classicalgenerator model is not appropriate and the 4th order generator model is introducedalong with the 3th order exciter model containing the voltage regulatorThe 4th order generator model in i-th dq rotating frame reads

δi = ωsωk

ωi = 12Hi

(Pmi minus Pei minusDiωi)

˙eprimeqi = 1

Tprimed0i

(Efqi minus (Xdi minusXprimedi)idi minus e

primeqi)

˙eprimedi = 1

Tprimeq0i

((Xqi minusXprimeqi)iqi minus e

primedi)

(310)

where δiωieprimeqi and e

primedi are the rotor angle rotor speed transient voltages along q

and d axes respectively of generator i They are state-variables for each generatorwhich are crucial in the transient analysis and rotor angle stability Pmi is themechanical power of generator and ωs is the synchronized frequency of the systemThen the transformation matrix 4i

dqminusxy from i-th dq rotating frame to the commonreference frame and its inverse can be dened as

4idqminusxy =

sin δi cos δi

minus cos δi sin δi

4ixyminusdq =

sin δi minus cos δi

cos δi sin δi

(311)

with 4idqminusxy = (4i

xyminusdq)T The generator parameters consist of Hi Di T

primed0T

primeq0

Hi and Di are inertia and damping coecient of generator Tprimeq0 and T

primed0 are the

open-circuit time constants Xd and Xq are the synchronous reactance Xprimed and X

primeq

are the transient reactance respectively for d and q axes of generator iThe power ows are indicated by Pei idi and iqi which are the electric power statorcurrents of d and q axes respectively of generator i which depend on both the statevariables of i-th generator as well as the rotor angle of other generators and thenetwork connectivity

3412 Modelling of Exciter (Automatic Voltage Regulators)

To enhance the stability of power system exciter which serves as Automatic VoltageRegulator (AVR) is added as shown as Fig 35 Efqi becomes a state variabledescribed by the dierential equation associated with the exciter And the voltage



11+sTR

times 1+sTC1+sTB

KA1+sTA

VT

+

Vref

minusXE1 XE2 Efd

Figure 35 Control Diagram of Exciter with AVR

regulators will add up to the exciter model which reads

˙Efdi = KAiXE2iTAi

minus 1TAi

Efdi

˙XE1i = minus 1TRi

XE1i + 1TRi

VTi

˙XE2i = 1TBi

(TCiTRiminus 1)XE1i minus 1

TBiXE2i minus TCi

TBiTRiVT i + 1

TBiVrefi

(312)

whereVT generator terminal voltage VT is given by

VT i =radic

(eprimeqi +X primediidi)2 + (eprimedi minusX

primeqiiqi)

2 i = 1 2 middot middot middot m (313)

Vref exciter reference voltage

3413 Modelling of the Whole System

As indicated in Chapter 1 the power angle stability of the interconnected powersystem which is composed of generators equivalently means the synchronism of therotor angles of the generators And lacking of innite bus in the multi-machine sys-tem leads to zero eigenvalue In the conventional small signal analysis the solutionis to neglect the zero eigenvalues or taking the rotor angle of one generator as theHowever when taking into account the nonlinearities the zero eigenvalues can not beneglected since the zero eigenvalues may have non-zero nonlinearities and becomesaddle pointsTherefore we choose the new state-variables δiN = δiminus δN with the N -th generatoras reference instead of δiAnd the equation of the rotor angle is

δiN = ωs(ωi minus ωN ) (314)

For a detailed model considering the transient voltages and inuence ofexciter and voltage regulators the i-th generator has 7 state-variables[δiN ωi e

primed e

primeq Efdi XE1i XE2i

]


3414 Possibility to Add Other Components

As the time constant of turbo governor is much larger than the electro-mechanicaltime constants the model of turbo governor is not included However it can beadded by taken into account the dierential equations related to Tm

342 Network Connectivity and Nonlinear Matrix

Although the dynamics are described by (310) (312) it is still far from forming amulti-machine dynamic system for interconnected power systemThe question ishow the interconnections inuence the dynamicsThe answer lays in the network connectivity and the reference frame of the genera-tors The biggest challenge in dealing with the interconnected power system is thenetwork connectivityTo calculate Pei a common reference frame (with synchronous speed) is used wherevoltages and currents are placed in the xminus y frameThe expressions of Pei idi and iqi are a system having n generators are [72]

eprimexieprimeyi

= 4idqminusxy

eprimedieprimeqi

It = Y

[eprimex1 e

primey1 middot middot middot e

primexn e

primeyn

]T(denote)It = [ix1 iy1 middot middot middot ixn ixn]

Pei = eprimexiixi + e

primeyiiyi

(315)

where eprimeqi e

primeqi are the transient internal bus voltages of generator i e

primexi e

primeyi are

their projections in the common reference frame Y is the admittance matrix of thereduced network to the internal bus of each generator i including source impedancesof generators and constant-impedance loads It is the terminal current of generatorsViewed from all the equations above assume that all the parameters are constantthe nonlinearities of the power system comes from sin δcos δ(315) indicates the network constraints of the 4th order generator model for whichthe key point is to derive the admittance matrix Y The calculation of Y is in next Section with programming procedures Even with allthe equations above the modelling for the power system is not ready for computer-based analysis since the matrix Y still needs a lot of work and the terms Taylorpolinomial terms H1(x) H2(x) H3(x) in (31) are not availableSince it is dicult to nd a reference to present the details in modelling the inter-connected power system with nonlinearities by a systematic procedures (althoughinsightful mathematical modellings are presented in Refs [1 73] some technique



details are not suciently illustrated ) this job is done in this PhD book

3421 Calculation of Y Matrix

Calculation of Y matrix is composed of two stepsbull Step1 Calculate Yt from Ybus reducing the bus admittance matrix to theterminals of the generatorsbull Step2 Calculate Y from Yt reducing the terminal admittance to the internalbus of generators

Step1 from Ybus to YtIf a system has m + n nodes among which n nodes are terminals of generatorsYbus is a [(m+ n)times (m+ n)] matrix describing the relation between the currentsand voltages at each node and Yt is a [ntimes n] matrix describing the Current-Voltagerelation at each terminalYbus is dened by node voltage equation using the ground potential as referencegives

Ibus = YbusVbus (316)

I1

I2

Im

Im+1

Im+n

=

Y11 middot middot middot Y1m Y1(m+1) middot middot middot Y1(n+m)


Ym1 middot middot middot Y2m Ym(m+1) middot middot middot Y2(n+m)

Y(m+1)1 middot middot middot Y(m+1)m Y(m+1)(m+1) middot middot middot Y(m+1)(m+n)

Y(m+n)1 middot middot middot Y(m+n)m Y(m+n)(m+1) middot middot middot Y(m+n)(m+n)

V1

V2

Vm

Vm+1

Vm+n

(317)

where Ibus is the vector of the injected bus currents Vbus is the vector of bus voltagesmeasured from the reference node Ybus is the bus admittance matrix Yii (diagonalelement) is the sum of admittances connected to bus i Yij (o-diagonal element)equals the negative of the admittance between bus i and jThe admittance matrix depicts the relation between the injected currents and thenode voltage The injected bus currents may come from the generators loadsbatteries etc Taking the load at node k as constant impedance it adds the shuntimpedance to the element Ykk


Since only the generators inject currents into the grid rewriting (316) into (318)then all nodes other than the generator terminal nodes are eliminated as follows 0

In

=

Ymm Ymn

Ynm Ynn

Vm

Vtn

(318)

Therefore from (318) it obtains

In =[Ynn minus YnmY minus1mmYmn

]Vtn (319)

Then the admittance matrix of the reduced network to the terminal bus of generatoris dened as

Yt =[Ynn minus YnmY minus1mmYmn

](320)

However Yt is not appropriate for computer-aided calculation as Yt is a complexmatrix and Vt is also a complex matrix

As Iti = ixi + jiyi =Nsumj=1

Ytijej =Nsumj=1

(Gtij + jBtij)(exi + jeyi) =

Nsumj=1

[(Gtijexi minusBt

ijeyi) + j(Btijexi +Gtijeyi)

]we can therefore dene a [(2N)times (2N)] matrix Yr as

Yr =

Gt11 minusBt11

Bt11 Gt11

middot middot middot

Gt1n minusBt1n

Bt1n Gt1n

Gtn1 minusBt

n1

Btn1 Gtn1


Gtnn minusBtnn

Btnn Gtnn

(321)

By Yt we calculate It once the Vtn = [e1 e2 middot middot middot en] is known

However as in the dynamics the state variables are[eprime1 e

prime2 middot middot middot e

primen

] an admittance

matrix Y is needed which denes as

It = Yr [exi eyi middot middot middot exn eyn] = Y[eprimexi e

primeyi middot middot middot e

primexn e

primeyn

](322)

Step2 Obtaining Y from YtTherefore to obtain Y the transformation between [exi eyi middot middot middot exn eyn][eprimexi e


primexn e

primeyn

]should be found at rst



Since edieqi

=

eprimedieprimeqi

minusRai minusX primeqiXprimedi Rai

idiiqi

=

eprimedieprimeqi

minus Ziidiiqi

(323)

multiplying both sides by the matrix 4idqminusxy

4idqminusxy

edieqi

= 4idqminusxy

eprimedieprimeqi

minus4idqminusxyZk

idiiqi

(324)

it therefore leads exieyi

=

eprimexieprimeyi

minus4idqminusxyZk4i

xyminusdq

ixiiyi

(325)

Dening T1 = diag[41dqminusxy42

dqminusxy middot middot middot 4idqminusxy middot middot middot 4n

dqminusxy

] T2 =

diag [Z1 Z2 middot middot middot Zi middot middot middot Zn] and using (322) it leads to

ex1

ey1

ex2

ey2

exn

eyn

=

eprimex1

eprimey1

eprimex2

eprimey2

eprimexn

eprimeyn

minus T1T2Tminus11

ix1

iy1

ix2

iy2

ixn

iyn

=rArr Y minus1r

ix1

iy1

ix2

iy2

ixn

iyn

= Y minus1

ix1

iy1

ix2

iy2

ixn

iyn

minus T1T2Tminus11

ix1

iy1

ix2

iy2

ixn

iyn

(326)

=rArr Y minus1r + T1T2Tminus11 = Y minus1

Therefore it obtainsY = (Y minus1r + T1T2T

minus11 )minus1 (327)

If Xprimeq = X

primed then 4i

dqminusxyZi = Zi4idqminusxy then T1T2 = T2T1 so

Y = (Y minus1r + T2)minus1 (328)


In our test systems Xprimedi = X

primeqi for simplicity and the calculation of matrix Y is as

(328)

3422 Calculating the Nonlinear Matrix

The next step is to calculate the nonlinear matrix H1 H2 H3 in (32)Since Y is obtained the function of Idq = [id1 iq1 middot middot middot idn iqn] on e

prime=[

eprimed1 e

primeq1 middot middot middot e

primedn e

primeqn

]can be obtained as

[id1 iq1 middot middot middot idi iqi middot middot middot idn iqn

]T= Tminus11 Y T1

[eprimed1 e


primedi e

qi middot middot middot eprimedn eprimeqn

]T(329)

Therefore Yδ is dened asYδ = Tminus11 Y T1 (330)

where Yδ is a nonlinear matrix and is time-variant in the system dynamics assin δi cos δi varies as the rotor oscillates In (329) only state-variables will exist in(310) (312) And the system can be formed as

δi = ωs(ωk minus 1)

ωi = 12Hi


eprimeqi = 1

Tprimed0i


primeqi)

eprimedi = 1

Tprimeq0i


primedi)

˙Efdi = KAiXE2iTAi

minus 1TAi

Efdi

˙XE1i = minus 1TRi

XE1i + 1TRi

VT i

˙XE2i = 1TBi


TBiXE2i minus TCi

TBiTRiVT i + 1

TBiVrefi

(331)



which can be rewritten as (331)

δi

ωi

eprimeqi

eprimedi

˙Efdi

˙XE1i

˙XE2i

=

0 ωs 0 0 0 0 0

0 minus Di2Hi

0 0 0 0 0

0 0 minus 1

Tprimed0i

0 1

Tprimed0i

0 0

0 0 0 minus 1

Tprimeq0i

0 0 0

0 0 0 0 minus 1TAi

0 KAiTAi

0 0 0 0 0 minus 1TRi

0

0 0 0 0 0 1TBi

(TCiTRiminus 1) minus 1

TBi

δi

ωi

eprimeqi

eprimedi

Efdi

XE1i

XE2i

+

0

minus eprimediidi+eprimeqiiqi

2Hi

minusXdiminusXprimedi

Tprimed0i

idi

XqiminusXprimeqi

Tprimeq0i

iqi

0

VTiTRi

minus TCiTBiTRi

VT i

+

0

Pmi2Hi

0

0

0

0

VrefiTBi

(332)

when there is no disturbance in the voltage reference and power reference

As [Idq] = Yδ

[eprimedq

] are nonlinear functions of δi eprimedi e

primeqi (331) can be rewritten as

xi =

δi

ωi

eprimeqi

eprimedi

Efdi

XE1i

XE2i

Agi =

0 ωs 0 0 0 0 0

0 minus Di2Hi

0 0 0 0 0

0 0 minus 1

Tprimed0i

0 0 0 0

0 0 0 minus 1

Tprimeq0i

0 0 0

0 0 0 0 minus 1TAi

0 KAiTAi


0

0 0 0 0 0 1TBi


TBi

Ni(X) =

0


2Hi


Tprimed0i

idi

XqiminusXprimeqi

Tprimeq0i

iqi

0

VTiTRi

minus TCiTBiTRi

VT i

Ui =

0

Pmi2Hi

0

0

0

0

VrefiTBi

(333)

Therefore the whole system can be written as

x1

x2

xn

=

Ag1 0 middot middot middot 0

0 Ag2 middot middot middot 0

0 0 middot middot middot Agn

x1

x2

xn

+

N1(x)

N2(x)

Nn(x)

+

U1

U2

Un

(334)

x = Agx+N(x) + U (335)

where Ni(x) is the ith row of N(x) which can be calculated by (329)Therefore doing the Taylor series with the stable equilibrium point (SEP) as thepoint of expansion The SEP is calculated by the power ow analysis


Free OscillationIf free oscillation is studied (ie U = 0) (335) becomes

4x = Ag4x+N1(4x) +1

2N2(4x) +

1

3N3(4x) (336)

where N1N2N3 are rst-order second order and third-order derivative of N(x)Then for the denition in (32) the nonlinear coecients can be extracted as

H1 = Ag +N1 H2 = N2 H3 = N3 (337)

As observed from (337) the linearization matrix is Ag+N1 for the multi-machinesystem instead of Ag for the single machine system N1 counts for the inuence ofpower ows between the generators on the eigenvalue analysis N2 N3 account forthe second order nonlinear interactions and third order nonlinear interactionsTherefore all the procedures of Normal From transformations can be followed

343 Programming Procedures for Normal Form Methods

3431 How to Initialize the Normal Form Methods Calculation

1 Forming the system dynamic equation it is a 28th modelbull For each generator station it is 4th for generator 3 for the AVR andThyristor Exciterbull For the network the systems of equations of 4 generator are concatenatedby the Bus voltage and Flow currents in the network

2 Equilibrium Points Doing the Power Flow Analysis for the steady state andobtaining the equilibrium points xeq to calculate all the normal form coe-cients (Software PSAT)

3 Perturbation Condition Recording all the disturbances of the state-variableswhen the generator power angle reaches the maximum xinit

3432 Programming Steps of the Normal Form Methods

The steps can be detailed as1 Forming the system of equations with the rst Taylor polynomial term 2nd

Taylor polynomial term F2jkl and 3rd Taylor polynomial term F3jpqr calculatedwith the equilibrium point

2 Calculate Eigen-matrix Λ3 Normal Form Coecients Calculation h2jkl and h3jpqr4 Using xinit search the initial values for zinitwinit by Newton-Raphson Method5 Calculating the Normal Dynamics by Runge-Kutta Method6 Reconstruction the system using dynamic coecients h2jkl and h3jpqr

3433 The Procedures of Calculating the Matrices

1 Initialize the parameters and input the matrix Ybus



2 Calculate Ybus rarr Yt rarr Y by Eqs (320) (328)3 Calculate Ydelta from (330)4 Calculate N(x) from Ag from (334)5 Calculate N1(x) N2(X) N3(x) from (336)6 Calculate coecients H1H2H3 from (337)

344 Cases Selected and Small Signal Analysis

The oscillatory modes for the selected cases are listed in Tab 31 and 32 with theassociated pseudo frequency time-constant and dominate states It gives a clearpicture of the physical property of the system dynamics by small-signal analysisSome stable real modes are not listed for the sake of compactnessThe cases analyzed in this section have been selected to highlight information pro-vided by the 3rd order Normal Form analysis The selected system operating condi-tions for the study are highly stressed cases where the system is close to the voltagecollapse characterized by a tie line ow of 420MW from Area1 to Area2 To con-sider the emergence of renewable energy based generators some modications aremade compared to the conventional small-signal and 2nd-order-based NF analysisthat have been already conducted on the same test case The powers generated bygenerators G1 and G2 in Area1 are unbalanced to consider the production of energyfrom distant renewable energy based generators in large areasThe results of conventional small-signal analysis are shown in Tab 31 and Tab 32where the damping frequency time-constant and the dominant states of the oscil-latory modes are listedThose results come from two cases that are considered

1 Case 1 This case represents a poorly damped situation where the dampingratio of the inter-area mode is only 59 and the system is at its limit ofstability according to a linear analysis A three phase short-circuited fault isapplied at Bus 7 and after 041s line B and line C are tripped The excitergain Ka is set as 150 for all generators

2 Case 2 This case represents a situation where the damping ratio of two os-cillatory modes is negative (modes (56) and (78) as indicated in Tab 32)The negative damping is introduced by changing the gain of the 4 thyristorexciters by a higher value (Ka = 240) A three phase short-circuited fault isapplied at Bus 7 and after 010s line B and line C are tripped

The transient analysis of the overall system based on the NF analysis is presentedin the next section where G1 and G2 exhibit predominant electromechanical oscil-lations


Table 31 Oscillatory Modes Case 1

Mode Eigenvalue Pseudo-Freq Damping Time Constant Dominant

(Hz) Ratio () t = 1ζωn

States


primeq2 E

primeq3 E

primeq4

78 minus594plusmn j883 140 671 0168 Control unit G2

910 minus181plusmn j628 999 277 0055 Control unit (G3G4)


1314 minus151plusmn j673 107 219 0662 Local Area 2(δ1 δ2 ω1 ω2)


1718 minus0135plusmn j228 036 59 741 Interarea (δ1 δ2 δ3 δ4)




States

56 361plusmn j1079 172 -334 -0277 Eprimeq1 E

primeq2 E

primeq3 E

primeq4

78 173plusmn j1041 166 -166 -579 Control unit G2








345 Case 1-2 Benets of using 3rd order approximation formodal interactions and stability analysis

0 5 10 15 20 25 30

δ1(degree)

40

70

100

130140

t(s)0 5 10 15 20 25 30

δ2(degree)

0

30

60

90100

Exact 3-2-3S 2-2-1 3-3-1 3-3-3 Linear

Figure 36 Comparison of dierent NF approximations Case 1When the system is close to its limits of stability as depicted by Case 1 Fig36 showsthat the linear analysis gives wrong predictions concerning the dynamic behavior ofthe system 2nd-order and 3rd-order Normal Form approximations can both bettermodel the system dynamic response than the Linear MethodThe proposed third-order approximation called 3-3-3 is superior over the othermethods since it makes possible to model the interactions of oscillatory of non-oscillatory modes up to order 3 contributing to a better modeling of the frequencyvariation of the oscillations as recently formulated in [4] [74]Concerning Case 2 the system is such that the linear analysis leads to the compu-tation of two oscillatory modes with a negative damping (See modes (56) and (78)given by Tab 32) According to the conventional small-signal stability analysisthe overall system is then considered as unstable However as shown in Fig 38the systems nonlinearities contribute to the overall stability of the system [5] Thecomparison of the dierent NF approximations shows that only methods keeping3rd-order terms in the normal dynamics are able to predict the stability of thesystem (3-2-3S and proposed 3-3-3 approximations)The inear analysis predicts that Gen 1 loses its synchronization (the power anglereaches 180) at around 19s In the physical meaning it indicates that the synchro-nization torque is weak if only its linear part is considered From the eigineeringview considering that the power system works in the nonlinear domain its working


Hz0 036 072 108 144 180 212 252

Mag

(

of F

unda

men

tal)

101

102

Exact3-2-3S2-2-13-3-13-3-3Linear

Figure 37 FFT Analysis of dierent NF approximations Case 1

0 5 10 15 20 25 30

δ1(degree)

-100

0

100

180

t(s)0 5 10 15 20 25 30

δ2(degree)

-100

-50

0

50

100

150180200

Exact 3-2-3S 3-3-3 2-2-1 3-3-1 Linear

Figure 38 Comparison of dierent NF approximations Case 2



capacity can be exploited to the maximumAlthough 2-2-1 and 3-3-1 have done a better job in considering the higher-ordermodal participation they cannot predict the stability as their Normal Dynamic islinear and the stability only depends on the eigenvaluesBy keeping trivially resonant 3rd order terms in the Normal Dynamics 3-2-3S and3-3-3 can better predict the stability of the system Shown in all the cases method3-3-3 is more accurate and the results are more realistic

t(s)0 10 20 30

(pu)

085

09

095

1

105

11

115Eprime

q1 Eprime

q2 Eprime

q3 Eprime

q4

Eq1

Eq2

Eq3

Eq4

t(s)0 5 10 15 20 25 30

(pu)

-55Vf

min

-4

-3

-2

-1

0

1

2

3

4

Vfmax

55Variables of Control Unit of Gen 2

Vmex2

Vfex2

Figure 39 Dynamics of Dominant State Variables Associated to Positive EigenvaluePairs Case 2

The dynamic responses by numerical simulation of the state-variables associatedwith the modes having a negative damping are shown in Fig 39 (the left forEprimeq1 E

primeq2 E

primeq3 E

primeq4 and the right for the Control Unit of Gen2) Even if some eigen-

values have a negative real part the damping aorded by the nonlinearities have tobe taken into account for a correct stability analysis

346 Nonlinear Analysis Based on Normal Forms

It has been validated by the time-domain simulation in the previous section that thenormal dynamics well approximate the system dynamics In this section the MNFis used to make quantitative analyses of the system dynamics using the nonlinearindexes

3461 Initial Condition and Magnitude at Fundamental Frequency

The initial conditions in the Jordan form (y0) and for methods 2-2-1 (z0221) 3-3-1(z0331) and 3-3-3 (z0333) are listed in Tab 33 It has to be noticed that method3-2-3S has the same initial conditions as method 2-2-1 ie z0323 = z0221 Fromthe initial condition it can be seen that the magnitude of the ratio of y017 z022117z033117 z033317 over z033317 is respectively equals to 124 1078 1077 and100 which approximately matches the ratios at the fundamental frequency of thedierent curves in Fig 37


Table 33 Initial Conditions

j y0 z0221 z0331 z0333

5 514 + j293 389 + j238 388 + j238 03105 + j03263

7 minus636minus j273 minus324minus j095i minus324minus j095 minus036minus j249

9 minus135minus j139 048minus j314 058minus j23 075minus j159

11 minus984minus j425 minus607minus j458 minus627minus j458 minus054minus j384

13 minus025minus j021 minus027minus j008 minus0226minus j0085 minus020minus 022i

15 032 + j010 004minus 00012i 005minus j00012 0072 + j005

17 087 + j252 082 + j216 084 + j210 170 + j131i

Table 34 3rd Order Coecients for Mode (17 18)

p q r (DF2h2)jpqr F3jpqr h3jpqr

(171717) 0046minus j0118 0029 + j0034 0017minus j0019

MI3jpqr h317171717 MI3jpqr Tr3

01220 00068 + j0008 00560 033

3462 Distribution of the Frequency Spectrum

The third order spectrum can be found by MI3jpqr Among all the MI3jpqrMI317(17)(17)(17) and MI318(18)(18)(18) contribute most to the 3rd order spectrum ofthe rotor angle oscillations The 3rd order coecients for mode (17 18) are listed

in Tab 34 with MI3jpqr calculated by h3jpqrComparing results of Tab 34 with data extracted frm Fig 37 it is seen that 1)MI317171717 = 011 = 11 approximately matches the magnitude ratio of the 3rd

order component at approximately 1Hz 2) ignoringDF2h2 h3jpqr is small and leads

to a too modest prediction of the 3rd order interaction MI3jpqr = 0056 = 56A deeper analysis can be made using NF methods compared to FFT analysis Forexample MI317172627 = 045 indicates a strong nonlinear interaction however sinceTr = 67341 times 10minus4 and MI3 times Tr = 303 times 10minus4 such a short duration will bedicult to be captured by FFT analysis

3463 Frequency Shift of the Fundamental Component

It can be also noted that there is a shift in the fundamental frequency As alreadymentioned it is indicated by the imaginary part of index SIIj2l Among all those co-ecients coecients SIIj18 are predominant and Tr3 indicates a long-time inuenceas listed in Tab 35 Although the real part of SIIj18 are too small to contribute



to the stability compared to λj its imaginary part indicates that the there is afrequency shift added to the eigen-frequency

Table 35 Resonant Terms associated with Mode (17 18)

j SIIj18 λj + λ2l + λ2lminus1 Tr3

5 minus130minus j259 minus373 + j932 093

7 048 + j096 minus621 + j883 096

9 00117 + j017 minus1837 + j628 098


13 0074minus j025 minus178 + j673 085

15 089 + j0532 minus196 + j651 086

17 012minus j0678 minus0405 + j228 033

This analysis corroborates with the analysis of the frequency spectrum in Fig 37If there are no resonant terms in the normal forms the fundamental frequency ofthe system oscillatory dynamics will be exactly as ωj


Seen from Tab 36 the nonlinear interactions can enhance (egMI35556) or weaken

the stability (eg MI3j71718) Only taking into account one specic MI3 withoutconsidering its time pertinence will lead to a wrong prediction In this sense thestability assessment proposed by method 3-3-3 is more rigorous and comprehensivethan method 3-2-3 As SI5 0 SI7 0 modes (56) and (78) are essentiallystable as validated by the time domain analysis as shown in Fig 38The proposed nonlinear stability indicators make possible to better use the powertransfer capability of the power grid For example in the studied case the high-gainexciter can improve the system response to a fault which is not shown using theconventional eigen-analysis

35 Applicability to Larger Networks with More Com-

plex Power System Models

For larger networks with more complex models method 3-3-3 is more demandingsince the modeling of modal interactions is more complex such as for the case of theNew England New York 16 machine 5 area system [75] whose typology is shown in

35 Applicability to Larger Networks with More Complex PowerSystem Models 141

Table 36 Stability Indexes of Mode (5 6)(7 8)

2l minus 1 SII52l Tr3 real(SII)times Tr35

5 minus5354minus j10977 033 -176682

9 minus337minus j634 minus018 minus063

13 minus306minus j756 750 minus2295



SI5 = 361minus 176682minus 063 + 2295minus 14625minus 172 = minus1856082






17 029 + j131 11801 034

SI7 = 173minus 2106minus 041minus 399minus 462 + 034 = minus2801

Fig 310 It is composed of ve geographical regions out of which NETS and NYPSare represented by a group of generators whereas the power import from each of thethree other neighboring areas are approximated by equivalent generator models (G14to G16) G13 also represents a small sub-area within NYPS Generators G1 to G8and G10 to G12 have DC excitation systems (DC4B) G9 has a fast static excitation(ST1A) while the rest of the generators (G13 to G16) have manual excitation asthey are area equivalents instead of being physical generators [75] The realisticparameters and well validated Simulink models can be found in [75 76] where thegenerators are modeled with the sub-transient models with four equivalent rotorcoils There are 15 pairs of electromechanical modes among which there are 4 inter-area modes This system is unstable when no PSS or only one PSS is installed [75]



Figure 310 New England New York 16-Machine 5-Area Test System [75]

351 Modelling of the System of Equations

3511 The Sub-transient Generator Model

A frequently used generator model is the sub-transient model with four equivalentrotor coils as per the IEEE convention The slow dyanmics of the governors areignored and the mechanical orques to the generators are taken as constant inputsStandard notations are followed in the following dierential equations which repre-sent the dynamic behavior of the ith generator [77]

dδidt = ωs(ωi minus 1) = ωsSmi

2HiSmidt = (Tmi minus Tei)minusDiSmi

Tprimeq0i

dEprimedidt = minusEprimedi + (Xqi minusX

primeqi)minusIqi +

(XprimeqiminusX

primeprimeqi)

(XprimeqiminusXlsi)2

((Xprimeqi minusXlsi)Iqi minus E

primedi minus ψ2qi)

Tprimed0i

dEprimeqi

dt = Efdi minus Eprimeqi + (Xdi minusX

primedi)Idi +

(XprimediminusX

primeprimedi)

(XprimediminusXlsi)2

(ψ1di + (Xprimedi minusXlsi)Idi minus E

primeqi)

Tprimeprimed0i

dψ1didt = E

primeqi + (X

primedi minusXlsi)Idi minus ψ

prime1di)

Tprimeprimeq0i

dψ2qi

dt = minusEprimedi + (Xprimeqi minusXlsi)Iqi minus ψ

prime2qi)

TcidEprimedcidt = Iqi(X

primeprimedi minusX

primeprimeqi)minus E

primedci

(338)


where Tei = EprimediIdi

(XprimeprimeqiminusXlsi)

(XprimeqiminusXlsi)

+ EprimeqiIqi

(XprimeprimediminusXlsi)

(XprimediminusXlsi)

minus IdiIqi(Xprimeprimedi minus X

primeprimeqi) +

ψ1diIqi(XprimeprimediminusX

primeprimedi)

(XprimediminusXlsi)

minus ψ2qiIdi(XprimeprimeqiminusX

primeprimeqi)

(XprimeqiminusXlsi)

and Iqi+jIdi = 1(Rai+jXprimeprimedi)E

primeqi


(XprimediminusXlsi)

+ψprime1di

(XprimediminusX

primeprimedi)

(XprimediminusXlsi)

minusVqi+j[Eprimedi


(XprimeqiminusXlsi)

minus

ψprime2qi

(XprimeqiminusX

primeprimeqi)

(XprimeqiminusXlsi)

minus Vdi + Eprimedci]

qi

di

x

y

qk

dk

δiδk

ωs


The reference frame transformation from the DQ frame to dq frame is

I = Iq + jId = ejδi(Ix + Iy)Vg = Vx + jVy = ejδi(Vqi + jVdi)

3512 Excitation Systems (AVRs)

Two types of automatic voltage regulators (AVRs) are used for the excitation ofthe generators The rst type is an IEEE standard DC exciter (DC4B) (shown inFig 312) and the second type is the standard static exciter (ST1A)The dierential equations governing the operation of the IEEE-DC4B excitationsystem are given by (339) while for the IEEE-ST1A are given by (340)

TedEfd

dt = Va minus (KeEfd + EfdAexeBexEfd)

TadVadt = KaVPID minus Va

VPID = (Vref + Vss minus Vr minusKf

Tf(Efd minus Vf ))(Kp + Ki

s + sKdTd+1)

TrdVrdt = Vt minus Vr

TfdVfdt = Efd minus Vf

(339)



where Efdmin le Va le EfdmaxEfdminKa le VPID le EfdmaxKa

Figure 312 Exciter of Type DC4B [78]

Efd = KA(Vref + Vss minus Vr)


(340)

where Efdmin le Efd le EfdmaxHere Efd is the eld excitation voltage Ke is the exciter gain Te is the exciter timeconstant Tr is the input lter time constant Vr is the input lter emf Kf is thestabilizer gain Tf is the stabilizer time constant Vf is the stabilizer emf Aex andBex are the saturation constants Ka is the dc regulator gain Ta is the regulatortime constant Va is the regulator emf Kp Ki Kd and Td are the PID-controllerparameters KA is the static regulator gain Vref is the reference voltage and Vss isoutput reference voltage from the PSS

3513 Modeling of Power System Stabilizers (PSSs)

Besides the excitation control of generators we use PSSs as suppelements to dampthe local modes The feedback signal to a PSS may be the rotor speed (or slip)terminal voltage of the generator real power or reactive power generated etc Signalis chosen which has maximum controllability and observability in the local modeIf the rotor slip Sm is used as the feedback signal then the dynamic equation of thePSS is given by (341)

Vss = KpsssTw

1 + sTw

(1 + sT11)

(1 + sT12)

(1 + sT21)

(1 + sT22)

(1 + sT31)

(1 + sT32)Sm (341)


Here Kpss is the PSS gain Tw is the washout time constant Ti1 and Ti2 are the ith

stage lead and lag time constants respectively

352 Case-study and Results

Using (314) to avoid the zero eigenvalues and neglecting the zero eigenvalues causedby the exciter and PSS controller parameters (the time constant of exciter and PSScontroller are much more smaller than that of the electromechanical modes) theeigenvalue analysis shows that this system has 15 pairs of electromechanical modesamong which there are 4 inter-area modes as shown in Tab 313 and Tab 314

Figure 313 The Electromechanical Modes of the 16 Machine System When No PSSis Equipped [75]

No Damping Frequency State Participation State Participation State Participation State Participation

Ratio() (Hz) State Factor Factor State Factor State Factor

1 -0438 0404 δ(13) 1 ω(13) 0741 ω(15) 0556 ω(14) 0524

2 0937 0526 δ(14) 1 ω(16) 0738 ω(14) 05 δ(13) 0114

3 -3855 061 δ(13) 1 ω(13) 083 δ(12) 0137 ω(6) 0136

4 3321 0779 δ(15) 1 ω(15) 0755 ω(14) 0305 δ(14) 0149

5 0256 0998 δ(2) 1 ω(2) 0992 δ(3) 0913 ω(3) 0905

6 3032 1073 δ(12) 1 ω(12) 0985 ω(13) 0193 δ(13) 0179

7 -1803 1093 δ(9) 1 ω(9) 0996 δ(1) 0337 ω(1) 0333

8 3716 1158 δ(5) 1 ω(5) 1 ω(6) 0959 δ(6) 0958

9 3588 1185 ω(2) 1 δ(2) 1 δ(3) 0928 ω(3) 0928

10 0762 1217 δ(10) 1 ω(10) 0991 ω(9) 0426 δ(9) 042

11 1347 126 ω(1) 1 δ(1) 0996 δ(10) 0761 ω(10) 0756

12 6487 1471 δ(8) 1 ω(8) 1 ω(1) 0435 δ(1) 0435

13 7033 1487 δ(4) 1 ω(4) 1 ω(5) 0483 δ(5) 0483

14 6799 1503 δ(7) 1 ω(7) 1 ω(6) 0557 δ(6) 0557

15 3904 1753 δ(11) 1 ω(11) 0993 Ψ1d(11) 0056 ω(10) 0033

Figure 314 The Electromechanical Modes of the 16 Machine System When all 12PSSs are Equipped [75]




Ratio () (Hz) Factor Factor State Factor State Factor

1 33537 0314 ω(15) 1 ω(16) 0982 δ(7) 0884 δ(4) 0857

2 3621 052 δ(14) 1 ω(16) 0645 ω(14) 0521 δ(15) 0149

3 9625 0591 δ(13) 1 ω(13) 083 ω(16) 0111 ω(12) 0096

4 3381 0779 δ(15) 1 ω(15) 0755 ω(14) 0304 δ(14) 0148

5 27136 0972 ω(3) 1 δ(3) 0962 ω(2) 0924 δ(2) 0849

6 18566 108 ω(12) 1 δ(12) 0931 Eprimeq(12) 0335 PSS4(11) 0196

7 23607 0939 δ(9) 1 ω(9) 0684 PSS3(9) 0231 PSS2(9) 0231

8 30111 1078 δ(5) 1 ω(5) 0928 ω(6) 0909 δ(6) 0883

9 28306 1136 ω(2) 1 δ(2) 0961 δ(3) 0798 ω(3) 0777

10 13426 1278 ω(1) 1 δ(1) 0969 Eprimeq(1) 0135 ω(8) 0111

11 1883 1188 δ(10) 1 ω(10) 0994 Eprimeq(10) 0324 PSS3(10) 019

12 32062 1292 δ(8) 1 ω(8) 0935 Eprimeq(8) 061 PSS2(8) 0422

13 39542 1288 δ(4) 1 ω(4) 0888 Eprimeq(4) 0706 PSS4(4) 0537

14 33119 1367 δ(7) 1 ω(7) 0909 δ(6) 0653 ω(6) 0612

15 2387 5075 ω(11) 1 Eprimeq(11) 0916 Ψ1d(11) 0641 PSS4(10) 0464

This system is unstable with positive eigenvalues of electromechanical modes asshown in Tab 313 And the PSS is therefore used to stabilise this system Whenplacing PSSs on G1 to G12 (the maximum number of possible PSSs) all the localmodes are damped and 3 inter-modes are poorly damped as shown in Tab 314No more information is available from the linear analysis to damp the inter-areamodes [75] When a three-phase fault is applied to G13 and cleared in 025s themachines exhibit nonlinear inter-area oscillations and nally damp out to a steady-state equilibrium in 50s as shown by the frequency spectrum in Fig 315 As allthe local-modes are well damped the components at frequency higher than 1Hz areexpected as nonlinear interactionsWhen placing PSSs only on G6 and G9 (the minimum number of PSSs to ensurethe stability of the system) all inter-area modes are poorly damped as listed inTab 37 When there is a three-phase short circuit fault applied near the end ofG13 generators in all the 5 areas exhibit poorly damped electromechanical oscilla-tions and nally damp out to a steady-state equilibrium in a large time Since thelinear participation factor of G13 is [00075 07997 00214 06871] on the inter-area modes 1-4 only Mode 2 and Mode 4 will be eectively excited while Mode 1and Mode 3 are trivially excited However as observed from the frequency spectrumin Fig 316 there are signicant components at 05Hz (G14 the fundamental fre-quency of Mode 3) at 08Hz (G15 the fundamental frequency of Mode 1) around10Hz (G6G9G13G14G15 the frequency is not corroborated with any inter-areamode) Therefore SII and MI3 are expected to give additional information to

36 Factors Inuencing Normal Form Analysis 147

Frequency (Hz)0 02 04 06 08 10 12 14 16 18 2

Mag

( o

f Fun

dam

enta

l)

100

101

102G1G12G13G14G15

Figure 315 FFT Analysis of Generator Rotor Angles Dynamics when G1 to G12are equipped with PSSs

Table 37 Inter-area Modes of the 16 Machine 5 Area System with PSSs Presenton G6 and G9

Mode 1 2 3 4

damping ratio 335 055 158 276

frequency (Hz) 07788 06073 05226 03929

identify the modal interactionsIn this case method 3-3-3 may provide more information to 1) reduce the numberof PSSs 2) damp the inter-area modes The siting of PSSs based on NF analysis isnot the issue to be dealt with in this chapter and it can be found in [20]

36 Factors Inuencing Normal Form Analysis


The essence of NF analysis is to calculate the nonlinear indexes to predict themodal interactions and to give information on the parameters inuencing the systemstability which is composed of two phases of computations

1 the SEP Initialization in order to obtain the eigen matrix λ and matrices F2

and F3 which depends on the post-fault SEP2 the Disturbance Initialization in order to obtain the initial points in the NF

coordinates which depend on the disturbances the system experiencesThe values of nonlinear indexes depend both on the SEP and the disturbances Item2 has been discussed in detail in [47 48] while Item 1 is somewhat neglected in the



Frequency (Hz)0 02 04 06 08 10 12 14 16 18 20

Mag

(

of F

unda

men

tal)

100

101

102G6G9G13G14G15


Table 38 Search for Initial Conditions in the NF Coordinates

Method Case 1 Case 2

Iterations Resolution Iterations Resolutions

3-3-3 12 1294e-12 17 1954e-6

2-2-13-2-3 3 1799e-11 6 1838e-06

3-3-1 7 1187e-09 11 7786e-09

literature

3611 Search for the Initial Conditions

In this chapter the search for the initial conditions is performed using the Newton-Raphson (NR) method [67] the starting search point is y0 and it converges in severaliterations as shown in Tab 38 This is because the normal coecients h2 and h3

are small in size When these coecient are large in size (near strong resonancecase) the search for z0 w0 can be extremely slow and even fails to convergeA more robust algorithm is proposed in [47] to circumvent some disadvantages ofNR method which can be adopted also for the 3rd order NF methods

37 Conclusion of the Reviewed and Proposed Methods 149

3612 SEP Initialization

If the power load changes the SEP initialization must be restarted The calculationof the nonlinear matrices F2 and F3 can be extremely tedious The problem isthat matrices F2 and F3 are composed of complex values Using the same Matlabfunction to calculate matrices A H2 H3Λ F2 and F3 for the IEEE 4 machine caseit leads to dierences from 200s to 2000s Reducing the time needed to multiplycomplex matrix can be a direction to optimize the programOnce the perturbation model around the SEP is established in Jordan form calcu-lation of nonlinear indexes can be computed in a short time

362 Strong Resonance and Model Dependence

Validated using 4th order or higher order generator models the proposed method 3-3-3 also inherits the applicability to systems modeled by 2nd order generator modelsas method 3-3-1 used in [54] and 3rd order generator as method 3-2-3S used in [5]The technique works where method 3-3-1 is not applicable For example using clas-sical models with zero mechanical dampings the eigenvalues will be pure imaginary

ie σj=0 In this case method 3-3-1 fails to be applied since h3jj2lminus12l =infin (Mode(2l-12l) being a conjugate pair that leads to a strong resonance)In addition as the eigenvalues are purely imaginary the stability boundary proposedin [516] (see Eqs(13) and (22)) will be wrongly predicted as the stability boundary

will be calculated as Rj =

radicminus real(λj)

real(cj=2k2k )

= 0

Method 3-3-3 is more complete as it can make possible to predict both the impor-tance of modal interactions and to proposes nonlinear stability indexes for a broaderranges of implemented models

37 Conclusion of the Reviewed and Proposed Methods

371 Signicance of the Proposed Research

With the nonlinear indexes the proposed method 3-3-3 makes possible to quantifythe third-order modal interactions and oers some pertinent information for stabilityanalysis providing a better tool compared to the linear or other existing normalforms methods The indication given by the nonlinear indexes are validated bytime-domain simulation and FFT analysisFactors inuencing NF analysis such arethe computational burden and the model dependency are discussedBesides this chapter gives a good review of existing MNFs along with a performanceevaluation of the dierent NF approximations studied in this work (see Tab 39)

372 Future Works and Possible Applications of Method 3-3-3

Based on the proposed 3rd-order NF approximations potential applications couldbe emerged such as



Table 39 Performance evaluation of the studied NF approximations

Method Resolution Order of Modal Transient Applicability to

Interaction Stability 3rd Order Modes

Linear O(2) 1st Linear None

2-2-1 O(3) 2nd Linear None

3-2-3S O(3) 2nd Non-linear Oscillatory

3-3-1 O(3) 3rd Linear Non-oscillatory

3-3-3 O(5) 3rd Non-linear All

1 Methods to nd the optimal location of Power System Stabilizers2 Sensibility analysis based on the higher-order participation factors to aid in

the design of the power system structure and controllers parameters tuning3 Nonlinear stability analysis to better predict the power transfer limits of the

system Transferring as much as possible power with an existing grid is ofmajor interest

Chapter 4

Application of Method NNMTransient Stability Assessment

Transient stability assessment is one of the most important issue for power sys-

tems composed of generators The existing popular tools are either the time-domain

simulation or direct methods Time-domain simulation stability analysis is time-

consuming and therefore makes the on-line assessment impossible The current direct

methods involving the energy function formulation to calculate the energy threshold

of rotor angle is dicult to implement due to the quadratic nonlinearities In this

chapter analytical stability bound is provided by method NNM for transient stability

assessment Taking into account the 3rd order nonlinearities from the Taylor series

the proposed method NNM cancels the quadratic nonlinearities and conserves only

the self-conserving cubic nonlinearities in the nonlinear normal modes by nonlinear

transformations leading to a conservative prediction of the stability bound

The proposed stability bound has been assessed in SMIB system and Kundurs 2 area

4 machine system Conclusions and Perspectives are given

Keywords Analytical Transient Stability Analysis Stability Bound 3rdNormal Forms Nonlinear Normal Mode

151

152Chapter 4 Application of Method NNM Transient Stability

Assessment

41 Introduction

Nowadays the electricity is transferred and distributed in a more smart way leadingto interconnected power grid which is a complex and large-scale system with richnonlinear dynamics Local instabilities arising in such a power network can triggercascading failures and ultimately result in wide-spread blackoutsThe detection and rejection of such instabilities will be one of the major challengesfaced by the future smart power grid The envisioned future power generation willrely increasingly on renewable energies such as wind and solar power Since theserenewable power sources are highly stochastic there will be an increasing number oftransient disturbances acting on an increasingly complex power grid Thus an im-portant issue of power network stability is the so-called transient stability which isthe ability of a power system to remain in synchronism when subjected to large tran-sient disturbances (event disturbances) such as faults or loss of system componentsor severe uctuations in generation or loadThe power system under such a large disturbance can be considered as going throughchanges in conguration in three stages from prefault to fault-on and then topostfault systems The prefault system is usually in a stable equilibrium point(SEP) The fault occurs (eg a short circuit) and the system is then in the fault-oncondition before it is cleared by the protective system operation Stability analysisis the study of whether the postfault trajectory will converge (tend) to an acceptablesteady-state as time passes [3]Transient stability has become a major operating constraint of the interconnectedpower system The existing practical tools for transient stability analysis are eithertime-domain simulation or direct methods based on energy-function and Lyapounovdirect method For other tools a literature review can be found in Section 152

42 Study of Transient Stability Assessment and Trans-

fer Limit

The transient stability analysis focus more on the electromechanical oscillationscaused by large disturbances The classical models of generators that are used inthat case are

δ = ω (41)

2Hω = Tm minus Te minusD(ω minus ωs)

In alternating current (AC) systems the generators must all operate in synchronismin steady state When a fault occurs on the system the electrical power output ofsome generators Te (usually those near the fault) will tend to decrease Since theturbine power input Tm does not change instantaneously to match this these gen-erators will accelerate above the nominal synchronous speed At the same timethe electrical power output of other generators may increase resulting in decelera-

42 Study of Transient Stability Assessment and Transfer Limit 153

Figure 41 The Spring Mass System for the Illustration of the Transient ProcessWhen the System Experiences Large Disturbances

tion below the nominal synchronous speed As a fundamental property of rotatingequipment the generators must all reverse their trends before the energy imbalancesbecome so large that return to synchronous operation is impossibleTransient stability analysis focuses on this phenomenon which is analogues to theexperience with a spring-mass system as shown in Fig 41When we pull the mass away from its equilibrium point x = 0 the spring will give

a resistance that once we lease it it will be pulled back shown as Fig 41 ccopy the

further it deviates from the equilibrium point the bigger the resistance When it

returns back to the equilibrium the force is zero but the speed x is not zero there-

fore the mass continues to move and the spring gives it a resistance to decrease

x until x = 0 shown as Fig 41b When x = 0 the resistance reaches to the

maximum and it pushes the mass to move from Fig 41 bcopy to Fig 41 ccopy If there

is a damping after several oscillations the mass will stop at the equilibrium after

several oscillations and the spring-mass system comes to steady state If there is

no damping the spring-mass system will continue the periodic oscillation Both the

damped oscillation and the periodic oscillation indicate a transient stability How-

ever when the displacement x is too much big the mass will not go back to the

equilibrium point and the system is not stable anymore In the power system the


Assessment

Figure 42 Typical Swing Curves [79]

resistance is the synchronizing torque and the damping is equivalent to mechanical

damping damping winding etc The objective is to gure out the stability bound of

x which is the transfer limit of the interconnected power system

When generators accelerate or decelerate with respect to each other their speeddeviations (and the corresponding angle deviations) constitute swings If two ormore generators swing apart in speed and then reverse their return to synchronismcould be considered rst-swing stable if the analysis concludes at the point ofreturn The transient stability and the the rst-swing unstable is illustrated inFig 42 (However it happens that the system becomes unstable after the rstswing [3] as shown in Fig 43 and Fig 44 which is dicult to explain This maybe explained by the proposed stability assessment index and time pertinance index)

A time domain simulation is performed on the well validated matlab simulink model[76] to assess the transient stability When there is a three-phase short circuitfault applied on Bus 13 and cleared in 030s generators in all the 5 areas exhibitoscillations with the dynamics of rotor angles shown in Fig 44 The oscillationsrstly damp and nearly converge to the steady state in 32s but then they begin todissipate later For such a case it is dicult to decide when to stop the simulationand analytical tools are neededCompared to the nonlinear analysis conducted the Chapter 3 the biggest dierencebetween the small-signal analysis and the transient analysis is thatThe nonlinearity in the small signal analysis is mainly contributed byxSEP and the nonlinearity in the transient analysis is mainly contributedby 4xExample Case 3Taking the single-machine innite-bus (SMIB) system as an example with classicalmodel as (42)

δ = ωsω (42)



Figure 43 New England and New York 16-Machine 5-Area Test System [75]

t(s)0 5 10 15 20 25 30 35 40 45 50

δ(rad)

-1

-05

0

05

1

15

Gen6Gen9Gen13Gen14Gen15

Figure 44 Example Case 4 Simulation of Rotor Angles Dynamics of the NewEngland New York 16-Machine 5-Area System


Assessment

Figure 45 Representative Example of SMIB System

with

Pmax =EprimeqE0

XPe = Pmax sin δ Pm = Pmax sin δep (43)

where Pm is the mechanical input in the machine Pe is the electrical power suppliedby the machine δep is the equilibrium point Dω is the damping on the rotor speedThe parameters are set as E0 = 1 Eprimeq = 17 X = 1 2H = 3 D = 0When there is a three-phase short circuit fault applied on the transmission line asobserved from Fig 46 the system behavior is predominantly nonlinear that thelinear analysis fails to approximate though this system is unstressed When the

t(s)0 05 1 15 2 25 3 35 4

δ(rad)

-3

-2

-1

0

1

2

3

Exact Linear SEP

Figure 46 Example Case 3 The Unstressed SMIB System under Large Disturbance

disturbance is big enough the system will be unstable The objective of TransientStability Assessment (TSA) is to predict the maximum disturbance that the systemcan experience while maintaining the stability

421 Existing Practical Tools

4211 Time-domain simulation tools

Until recently transient stability analysis has been performed in power companiesmainly by means of numerical integrations to calculate generator behaviours rela-tive to a given disturbance EMTP software such as PSCAD phasor-type softwaresuch as MatlabSymPower have to be used By those time-domain simulation pro-grams a clear picture of the post-fault trajectory can be obtained The time-domain


simulation is advantaged in the sense that1 it is directly applicable to any level of detail of power system models2 all the information of state variables during transient process as well as steady-

state is available3 the results are visual and can be directly interpreted by system operators

However the main disadvantages are that1 it is time-consuming2 it fails to oer analytical results ie it tells whether the system is stable but

fails to tell the degrees of the stability eg the stable margin [3] which is crucialto exploit the power transfer limit and maximize the economic opportunities

Therefore analytical methods are required to do the on-line stability assessment andserve as a preventive tool

4212 Direct Methods(Ribbens-Pavella and Evans 1985 [41] Chiang1995 [3])

An alternative approach is the direct methods employing the energy functions (Lya-pounovs second method for stability) It avoids the step-by-step time-domain sta-bility analysis of the post-fault system and also provides a quantitative measureof the degree of system stability to provide guide in the power system planing andoperatingThere are three main steps needed for direct methods

1 Constructing an energy function for the post-fault system say V (δ ω)

2 Computing the critical energy value Vcr for a given fault-on trajectory (saybased on the controlling uep method)

3 Comparing the energy value of the state when the fault is cleared sayV (δcl ωcl) with the critical energy value Vcr If V (δcl δcl) lt Vcr the postfaulttrajectory will be stable Otherwise it may be unstable

By direct methods conservative transient stability is foundFor the single-machine innite-bus(SMIB) direct methods are easy to be imple-mented one method is named the equal-area (E-A) criterion [1] with good visual-ization and good interpretationHowever for the high-dimension systems (a multi-swing case which is widespreadin the inter-area oscillations in the interconnected large-scale problem) formulatingthe energy function (multi-variable function) is a challenging work demanding veryskilful interpretors and operatorsWhats more the Lyapounov second method for stability analysis requires the equi-librium point hyperbolic (there is no eigenvalue having zero-real part) making it notappropriate for the weakly-damping or zero-damping caseThose drawbacks make the direct methods academically prospective but impracticalfor industry until 1990sAfter 1990s with the development of computing algorithms the applicability rangeof direct methods is enlarged However the computing development benets morethe time-domain simulation tools than the direct methods tools The current market


Assessment

is still a small-signal analysis + time-domain analysisHowever it is also dicult to interpret the results obtained by direct methods asthe physical meanings are obscured by the energy concept The recently proposedtool such as the F-A curve [4] is an eort to make the direct method visual byusing the direct methods to establish the one-to-one mapping between the oscillationamplitude and frequency However it is based on the assumption that the nonlinearbehaviour of the global system is the sum of the individual behaviour of each systemcomponent which is not mathematically proved and far from our experience withthe physical systemsWhen interpreting the physical motion by energy concept the direct methods arenot that direct

4213 Problem Formulating

In essence the problem of transient stability analysis can be translated into thefollowing given a set of non-linear equation with an initial condition determinewhether or not the ensuing trajectories will settle down to a desired steady-stateIn this sense time-domain simulation is too time-consuming in providing unneces-sary details and direct-methods employing the energy functions not to highlight thephysical motion of the power dynamicsLets go to the fundamentals and we will note that the couplings in the nonlinear-ities make the analytical solutions unavailable To settle this problem a methodbased on Normal Form (NF) Theory is proposed to simplify the nonlinearities bythe use of nonlinear transformations The procedure is well documented in [30 51]and consists of 4 major steps

1 Simplifying the linear part of a system by the use of a linear transformation2 Simplifying the nonresonant terms of higher-order terms by successive Normal

Form (NF) transformations3 Simplifying the Normal Dynamics by neglecting (if possible) some resonant

terms that can not be annihilated by NF transformations4 Using the chosen Normal Dynamics for dynamic and stability analysis

4214 Existing Normal Form Methods

In [51648] the formulas of dierent NF methods are derived the normal dynamicsare in the Jordan form where complex numbers are involved For [48] the normaldynamics are linear and the stability analysis is a linear analysis plus higher-ordeructuations In [5 16] some 3rd order terms are kept to render a more accurateprediction of the nonlinear stability based on which the stability bound is proposedHowever as the stability bound is the threshold of the rotor angle which is areal number extra eort is made to transform the complex number to the realnumber leading to additional computational errors Those methods are summarizedas methods on vector elds in Chapter 2Another approach is to keep the dierential equation in the second-order form andto decouple the linear matrix in [2times 2] blocks to avoid the computation of complex


numbers It originates from the Touzeacutes method to formulate the nonlinear normalmode [17] The nonlinear normal mode (NNM) is the basic particular solution ofthe nonlinear dynamics By studying NNM the nonlinear properties of the systemdynamics are revealed and quantiedIn [40] it also formulates the nonlinear normal mode to study the power systemdynamics However it is based on the Shaw-Pierre method and fails to suggestaccurately the nonlinear system stability Discussions on SP method can be foundin Section 259The comparisons between the time-domain simulation direct methods employingenergy functions and the proposed NF method are listed in Tab 41 The advantagesand disadvantages of each method are summarized in the Sections 422 423

422 Advantages of the Presented Methods

4221 Advantages of Time-domain Simulations

1 it is directly applicable to any type of power system models2 all the information of state variables during transient as well as steady-state

is available3 the results are visual and can be directly interpreted by system operators

4222 Advantages of TEF Methods

1 It avoids time-consuming numerical integration of a postfault power system2 It makes quantitative measure of the degree of system stability available

bull It provides information to designing network conguration plans to pushthe power system to its operating limitsbull The derivation of preventive control actions


1 Analytical conservative stability bound is predicted2 It provides clear physical insights notion of modes is kept in the normal dy-

namics which is better to predict the motion compared to the energy equationbased methods

3 It is easy to be implemented with clear physical meanings that anyone whohas knowledge with nonlinear dierential equations can implement it

4 It is applicable for high-dimensional system the complexity does not increasedrastically with the high-dimensional systems

423 Challenges for the Presented Methods

4231 Challenges for Time domain simulations

1 It is dicult to be applied for large-dimensional system2 It is dicult to decide when to stop the simulation


Assessment

Table 41 Comparisons of the Presented Methods

Methods Time-domainStep-by-stepIntegration

Direct Methodsemploying energyfunctions

NF methods em-ploying NormalModes

Principle The post-faulttrajectory ob-tained by step-by-step simulationtells the stability

The potential en-ergy decides thestability

The NormalForms of thesystem (obtainedby nonlineartransformationconserving theinvariance prop-erty) tells thestability

Trajectories anddynamics

Prefault trajec-tory fault-ontrajectory post-fault trajectory

trajectory is un-known

postfault tra-jectory of elec-tromechanicaloscillations

State-variables rotor angle cur-rents voltages

Energy V (δcl) Rotor angle δ

Computation1 Numerical

o-line2 Step-by-

step inte-gration

3 The com-putationaltime istypicallybetween10-30s

4 Qualitativestabilityassessment

1 Mostly-Numerical

2 No integra-tion of post-fault trajec-tory but

3computationaltime mil-liseconds

4 CalculationofV (cr) V (X(t+cl))ifV (X(t+cl)) ltVcrthenx(t) isstableotherwiseunstable

1 Explicitlyanalyticalresults

2 Systematicprocedure

3 stabilitybound canbe system-aticallycalculatedout in mi-croseconds


4232 Challenges for TEF Methods

1 The modelling not every (post-fault) transient stability model admits anenergy function eg the model contains the quadratic nonlinearities

2 The function only applicable to rst swing stability analysis of power systemtransient stability models described by pure dierential equations

3 The reliability the reliability of a computational method in computing thecontrolling UEP for every study contingency

4233 Challenges for Methods of Normal Forms

1 The modelling real-form approach is just for network-reduction modelvector-eld approach can cover network-preserving model [48]

2 The functiononly applicable to rst swing stability analysis3 The reliabilitythe reliability of a computational method in computing the

controlling UEP for every study contingency

424 A Brief Review of the Proposed Tool Method NNM

The nonlinear electromechanical oscillations in interconnected power system can bemodelled as second-order coupled oscillators If the variables of the power systemaround a given equilibrium point are gathers in a N dimensional vector q the basicmodel under study writes

Mq +Dq +Kq + fnl(q) = 0 (44)


fpnl(q) =

Nsumi=1jgei

f2pijqiqj +

Nsumi=1jgeikgej

f3pijkqiqjqk (45)


4241 LNM

As viewed from (286) the dynamics physical property (oscillatory frequency damp-ing friction etc) is obscured by the mathematical model


Assessment

In order to decouple the equation into second-order oscillators a linear transforma-tion 4q(t) = Φx(t) can be usedAfter decoupling the linear terms the system dynamics is therefore characterizedby a group of oscillators with coupled nonlinearities


+

Nsumi=1

Nsumjgei

gpijXiXj +

Nsumi=1

Nsumjgei

Nsumkgej

hpijkXiXjXk = 0


nonlinearities coming from decoupling of KS which are given in Chapter 2Neglecting all the nonlinearities the (46) reads a Linear Normal Mode (LNM)


4242 NM2

Using a second order NF transformation to cancel all the 2nd-order terms theNormal Forms in the new coordinate U minus V is

Up + 2ξpωpUp + ω2pUp +O(3) = 0 (48)

4243 NNM

Using a 3rd order NF transformation to cancel all the 2nd-order terms and thenon-resonant 3rd order terms the Normal Forms in the new coordinate Rminus S is



pppRpR2p + CppppR

2pRp = 0

with the coecients (Apijk Bpijk C


terms [1718] and are dened in (2109)

4244 Analytical Stability Bound for Transient Stability Assessment

Since the bound of Rp indicates the bound the Xp in Section 262 the proposedstability bound (PSB) is dened as


with

Rp le

radicminus

ω2p

hpppp +Apppp(411)

43 Network Reduction Modelling of the Power System 163

and Knl is dened to consider the nonlinear scaling bround by the NF transforma-tion

Knl =RpXp

=1


425 Technical Contribution of the Method NNM

The proposed method NNM uses nonlinear transformation up to 3rd order to cancelall the 2nd order and part of 3rd order nonlinearities in the normal dynamicsIt deals with the second-order equation directly the complex numbers are avoidedand the computational burden is reduced while better physical meanings are pre-served Compared to the reviewed methods it has the advantages as

1 it provides analytical conservative stability bound2 it keeps the notion of modes which is better to predict the motion compared

to the energy equation based methods3 it is easy to be implemented with clear physical meanings anyone who has

knowledge with nonlinear dierential equations can implement it4 its complexity does not increase drastically with the high-dimensional systems

426 Structure of this Chapter

In Section 41 an introduction to the problematic is made with a bibliographicalreview of the existing practical toolsThe mathematical modelling of test systems are established in Section 43In Section 44 it shows how the normal-mode based MNFs can work on the transientstability assessment of interconnected power systemsThe conclusions and perspectives are made in Section 45

43 Network Reduction Modelling of the Power System

431 Modelling of SMIB System

The classic electro-mechanical model of one single machine connected to the gridwith voltage V0 (as shown in Fig 47) with constant amplitude emf Eprimeq and thestate variables are δ and ω which can be descibed by the dierential equations areas follows

δ = ωsω (413)

ω = (Pm minus Pe minusDω(ω minus 1))(2H) (414)

with

Pmax =EprimeqV0Xe

Pe = Pmax sin δ Pm = Pmax sin δep (415)

where Pm is the mechanical input in the machine Pe is the electrical power suppliedby the machine δep is the equilibrium point Dω is the damping coecient on therotor speed


Assessment


432 Modeling of Multi-Machine System

By reducing the network the interconnected power system exhibiting oscillationscan be essentially characterized as a multi-machine model

2Hi

ωbδi = Pmi minus Pei minus

Dωi

ωbδi (416)

where

Pei = E2iGgii +

Nsumk=1k 6=i

EiEk [Ggik cos δik +Bgik sin δik] (417)

Ei is the terminal voltage of the i-th machine and Ggik is the conductance betweenmachine i and k Bgik is the susceptance between machine i and k

433 Taylor Series Around the Equilibrium Point

For a perturbation around the equilibrium point (δ0i ) the state-variable is changedinto 4δ = δ minus δ04ω = ω minus ω0 As in the free-oscillation case ω0 = 1 (416)can therefore be written as (44)

4331 Synchronization Torque for SMIB System

The synchronization torque is given as the rst second and third derivative of Pe

KS = cos δep K2S = minussin δep2

K3S = minuscos δep6

(418)

4332 Synchronization Torque for Multi-Machine System

[KSij ] =partPeipartδj

=

sumN

k=1k 6=i

EiEj[minusGgik sin δ0ik +Bgik cos δ0ik

] j = i

EiEj[Ggik sin δ0ik minusBgik cos δ0ik

] j 6= i

(419)


For i-th machine Ki2Sij ] = 1

2part2Peipartδiδj

reads

Ki2Sii = minus

sumNk=1k 6=i

EiEk[Ggik cos δ0ik +Bgik sin δ0ik

]Ki

2Sij = EiEj

[Ggij cos δ0ij +Bgij sin δ0ij

] j 6= i

Ki2Sjj = minusEiEj


] j 6= i

(420)

For i-th machine Ki3Sijk = 1

6part3Peipartδiδjδk

it reads

Ki3Siii = 1

6

sumNk=1k 6=i

EiEk[Ggik sin δ0ik minusBgik cos δ0ik

]Ki

3Siij = minus12EiEj

[Ggij sin δ0ij minusBgij cos δ0ij

] j 6= i

Ki3Sijj = 1

2EiEj


] j 6= i

Ki3Sjjj = minus1

6EiEj


](421)

434 Reducing the State-Variables to Avoid the Zero Eigenvalue

In the multi-machine system composed of N generators if no innite bus existsthere is zero-eigenvalue in such a case the proposed Stability Bound is not ecientIn another word if all the variables δiδj foralli j isin N are unstable the global systemcan still be stable if δij is stable ie the stability is ensured once the synchronizationis ensuredTo avoid the zero-eigenvalue we can use δin δjn instead of δi δj therefore δij =

δin minus δjnStarting from the dierential equations

2Hi

ωbδi minus

Dwi

ωbδi = Pmi minus Pei (422)

2Hn

ωbδn minus

Dwn

ωbδn = Pmn minus Pen (423)

When Hi = Hj = H Di = Dj = D Eq (422)-Eq (423) leads to

2H

ωb(δi minus δn)minusD(δi minus δn) = Pmi minus Pmn minus (Pei minus Pen) (424)

Calculating Taylor series around the equilibrium point foralli = 1 2 middot middot middot nminus 1

2H

ωb4δN +

D

ωb4 ˙δN +KprimeS4δN +Kprime2S4δ2N +Kprime3S4δ3N = 0 (425)


Assessment

where4δN = 4δiminus4δn and4δ2N contains the quadratic terms such as4δiN4δjNand 4δ3N contains the cubic terms such as 4δiN4δjN4δkNThen the coecients KS K2S K3S are replaced as

K primeSij =part(Pei minus Pen)

partδjN= KSij + EnEi

[Ggni sin δ0ni minusBgni cos δ0ni

](426)

For i-th machine Kprimei2Sij ] = 1

2part2(PeiminusPen)partδiN δjN

reads

Kprimei2Sii = Ki

2Sii + EnEi[Ggni cos δ0ik +Bgik sin δ0ik

]Kprimei2Sij = Ki

2Sijj 6= i

Ki2Sjj = Ki

2Sjj + EnEj

[Ggnj cos δ0nj +Bgnj sin δ0nj

]j 6= i

(427)



it reads

Kprimei3Siii = Ki

3Siii + EnEi[Ggni sin δ0ni minusBgni cos δ0ni

]Kprimei3Siij = Ki

3Siij j 6= i

Kprimei3Sijj = Ki

3Sijj j 6= i

Kprimei3Sjjj = Ki

3Sjjj + 16EnEj

[Ggnj sin δ0nj minusBgnj cos δ0nj

](428)

44 Case-studies

441 SMIB Test System

4411 Nonlinearity and Asymmetry

Fig 48 and Fig 49 show how the dierent methods approximate the rotor an-gle dynamics in the transient process where the NNM captures the two nonlinearproperties of the electromechanical oscillations 1)asymmetry 2)amplitude depen-dent frequency-shift [4 2122]In all the gures the curve LNM is symmetric as it is in the linear caseFor δSEP = π

24 the oscillatory frequency is f = 0836hZ and the nonlinear oscil-latory frequency calculated by (2133) is f = 078hZ is closer to the exact systemdynamicsFor δSEP = π

12 the oscillatory frequency is f = 162hZ and the nonlinear oscilla-tory frequency calculated by (2133) is f = 124hZ is closer to the exact systemdynamics

44 Case-studies 167

t(s)0 05 1 15 2 25 3 35 4

δ(rad)

1

12

14

16

EX LNM NM2 NNM SEP

Figure 48 A Case Where the System Works to its limit δSEP = π24

t(s)0 02 04 06 08 1 12 14 16 18 2

δ(rad)

-3

-2

-1

0

1

2

3

EX LNM NM2 NNM SEP



More results are listed in Tab 42 where the stability bound predicted by themethod NNM is labeled as PSB and the stability bound predicted by the E-Acriteria is labelled as EAB To illustrate the inuence of the nonlinear scalingfactor Knl the stability bound predicted in the R minus S coordinates by Rp is alsolistedAs observed from Tab 42 the PSB is smaller than EAB while R1max can bebigger than EAB Therefore by considering the nonlinear scaling the prediction ofstability bound is conservative and more accurate The stability bound predictedby PSB is veried by time-domain simulation as shown in Fig 410 and Fig 411Fig 410 shows that when the δ goes beyond the boundary NNM can predict theinstability in the system dynamics while LNM and NM2 fails Fig 411 shows thatwhen the δ reaches the stability bound predicted by EAB the rotor angles keeps stilland the frequency of oscillations is zero PSB indicates that the system is unstablewhile LNM and NM2 indicates that the system is still stableTherefore corroborated with the EAB and the time-domain simulations the PSBconducts a conservative transient stability assessment for the SMIB system


Assessment

Table 42 The Proposed Stability Bound E-A Bound and Bound without Consid-ering the Nonlinear Scaling Factor

δ0 R1max PSB EAB Error1 Error2

+δ0 Bound Rbound+δ0minusEABEAB

PSBminusEABEAB

π2 15708 15708 15708 0 0π21 16790 16380 16456 203 -046π22 17766 1 6985 17136 368 -088π23 18643 17526 17757 499 -130π24 19430 18010 18326 602 -173π25 20136 18440 18850 682 -217π26 20769 18823 19333 743 -264π27 21337 19164 19780 787 -312π28 21848 19467 20196 818 -361π29 22307 19763 20583 838 -411π3 22719 19976 20944 848 -462π4 25174 22700 23562 684 -366π5 26100 23960 25133 385 -467π6 26449 24587 26180 103 -608π7 26557 24917 26928 -138 -747π8 26557 25096 27489 -339 -871π9 26508 25192 27925 -507 -979π10 26438 25241 28274 -650 -1073π11 26359 25262 28560 -771 -1155π12 26278 25267 28798 -875 -1226π13 26200 25261 28999 -965 -1289

442 Multi-Machine Multi-Area Case IEEE 4 Machine Test Sys-tem

In the multi-machine case the E-A criterion doesnt work any more However thePSB still worksThe test system selected for this work is a well known IEEE standard systemKundurs 2area 4machine system shown in Fig 412It is a classical system suitable for the analysis of modal oscillations for the validation

44 Case-studies 169

t(s)0 05 1 15 2 25 3

δ(rad)

-4

-2

0

2

4

6

EX LNM NM2 NNM SEP

Figure 410 When the oscillation amplitude is beyond the bound δmax = 29798for δep = π

12

t(s)0 05 1 15 2 25 3

δ(rad)

-4

-2

0

2

4

EX LNM NNM2 NNM3 SEP


12

of small-signal analysis [1] and for the validation of 2nd order Normal Form analysis[48] In the renewable energy generation case the power generated by G1 and G2 isunbalanced due to the unied distribution of wind energy And there is a power owaround 415MW through the tie line between Area 1 and Area 2 The mechanicaldamping is weak to obtain the fast response to capture the maximum wind energyThe generators are modelled using classical model [73] assuming that the excitationvoltage is constant The loads L1 and L2 are modelled as constant impedance andno PSS is equipped The data for the system and selected case are provided inAppendix Open source Matlab toolbox PSAT v1210 is used to do the eigenanalysis and the full numerical time domain simulation based on the well validateddemo d_kundur1As indicated before dynamics of state-variable δi can not reveal the stability prop-erty of the whole system Because there is no innite bus it is the synchronizationdetermines the stability ie the relative motion between δi and δj


Assessment

Figure 412 IEEE 4 machine test system

In this sense rotor angle of the 4th generator is served as the reference And thestability is decided δ14 δ24 δ34 It is a weakly damped and highly stressed large-scaleinterconnected power system It has one inter-area mode and two local modesThe case of concern is interarea oscillations

Case There is a three-phase short-circuit fault at Bus 5 after a certaintime the fault is cleared the rotor angles are measured as δmax In thiscase only the inter-area mode is excited

Fig 413 shows that when the interarea oscillations are under the stability boundamong all the methods NNM approximates the system dynamics (ÈX) to the bestAs observed from Figs 414 415 and 416 the PSB predicts a conservativestability bound

45 Conclusion and Future Work

451 Signicance of this Research

The PSB can suggest a conservative stability bound for both the SMIB system andthe interconnected system ie this proposed method can be used to study not onlythe case where the fault is caused by loss of a machine but also the loss of one area

452 On-going Work

Two things are due to done to complete the testsbull Testing the PSB on larger networks eg the New England New York 16 Ma-chine 5 Area Systembull Comparing the performance of method NNM with the TEF methods

45 Conclusion and Future Work 171

0 5 10 15 20

δ14(rad)

0

05

1

15

2

25

t(s)0 5 10 15 20

δ24(rad)

-05

0

05

1

15

2EX LNM NM2 NNM SEP

Figure 413 Dierent Models to Approximate the Reference Model

0 5 10 15 20

δ14(rad)

0

1

2

3

EX LNM NM2 NNM SEP

t(s)0 5 10 15 20

δ24(rad)

0

1

2

Figure 414 At the Stability Bound


Assessment

0 5 10 15 20

δ14(rad)

-2

0

2

4

EX LNM NM2 NNM SEP

t(s)0 5 10 15 20

δ24(rad)

-2

0

2

4

Figure 415 Exceeding 2 Bound

0 5 10 15 20

δ14(rad)

-2

0

2

4

EX LNM NM2 NNM SEP

t(s)0 5 10 15 20

δ24(rad)

-2

0

2

4



453 Originality and Perspective

It is for the rst time that the transient stability is assessed by taking the nonlin-earities simplied by normal form transformations In the sense of simplest form(the principle of MNFs) it makes possible to calculate a minimum of nonlinearitiesand therefore develop a tool more applicable for larger networks composed withmore generators The current work is to test the applicability of the proposed toolon larger networks and to work in the engineering part The perspective of thisresearch task is to make possibleOn-Line Transient Stability Assessment of Large Interconnected PowerSystems


Assessment

Chapter 5

From Analysis to Control andExplorations on Forced

Oscillations

In previous chapter nonlinear analysis using the proposed methodologies are made

with investigations on the advantages and limitations The objective of nonlinear

analysis is to have a clear picture of the system dynamics

The next step is to improve the system dynamic performances ie to the control

techniques to enhance the system stability Concerning nonlinear control two ap-

proaches are proposed One is the nonlinear analysis based control the other is the

nonlinear model decoupling basis control

The rst approach is to improve the system dynamic response based on the extra

information oered by the nonlinear analysis which is recently popular The prin-

ciple of the second approach is to decouple the nonlinear system and is named as

Nonlinear Modal Control The second approach is for the rst time proposed in

this PhD work and it is very original and innovative

Besides the explorations on the control analytical investigations on the forced os-

cillations are conducted which is the rst step to implement the nonlinear modal

control And the derivation of Normal Forms under forced oscillations illustrates

further the philosophy of normal form theory

For both approaches a literature review of similar works are conducted to show the

necessity and expected scientic contribution of the proposals

As several novel concepts concerning the nonlinear modal control are proposed in this

chapter and in order to facilitate the readers comprehension their linear counterpart

are rstly illustrated and then extended to the nonlinear domain This PhD research

may kick the door for future researches or another couple of PhD dissertations

In the nal section it also provides an innovative proposal in research of normal

form theory the derivation of the control form of normal form

Keywords Conclusion of Nonlinear Analysis On-going Works NonlinearModal Control

175

176Chapter 5 From Analysis to Control and Explorations on Forced

Oscillations

51 From Nonlinear Analysis to Nonlinear Control

This PhD research works on the very fundamental problems concerning nonlin-ear systems having multiple input with coupled dynamics Multiple input systemwith coupled dynamics almost cover all the modern industrial systems since high-performance industrial electromechanical systems are often composed of a collectionof interconnected subsystems working in collaboration These dynamical systemscan be found everywhere and contribute largely to our daily life to the nationaleconomy and global developmentIn the previous chapters of this PhD dissertation methodologies to analyze the dy-namics of some types of multiple input systems with coupled dynamics are proposedand validated with case-studies of interconnected power systemsHowever the more important thing in this work is to point out the way to use theinformation obtained from the analysis to improve the dynamic performance of thesystem such as enhancing the system stability damping the interarea oscillationsand etc This can be partly done by designing the parameters of the system usingthe analysis results as a reference partly done by controlling the existing systemIn Chapter 2 with the case study of interconnected VSCs method NNM has alreadyshown its ability to quantify the inuence of physical parameters the nonlinearityin the system Therefore a rather simple but rude way is to design the physicalparameters such that the system can work in a region where it can be viewed asa linear system Then all the advantages of linear systems would be granted (Ofcourse method 3-3-3 can do a similar job)However it is very dicult or expensive to design parameters in such a way orimpossible especially for the electric power system Therefore the main techniqueto improve the system dynamics is to control itBased on EMR + IBC (which is introduced in Chapter 1) once the systems be-haviours can be quantied by the analytical analysis the control strategies canbe developed to improve the system dynamic performance since the relationshipbetween control and analysis can be described as Fig 51

Figure 51 Control and Analysisbull For the analysis it is to predict the output from the given inputbull For the control it is Analysis + Feedback

52 Two Approaches to Propose the Control Strategies 177

where the output is system dynamic performance such as the motion the oscil-lation and etc and the input can be the design parameter of the physical systemsuch as the parameters of the controller the placement of the stabilizers and theexternal force

52 Two Approaches to Propose the Control Strategies

There are innumerable approaches to develop the control strategies to improve thesystem dynamic performances Based on the accumulated experience in linear con-trol there are two typical approachesbull Analysis based control strategy the principle of this type of control strategyis to obtain the best analysis results Examples can be found in tuning thecontroller parameters The mathematical model of the system including thecontrollers is rstly formed and secondly the indexes quantifying the systemdynamics are calculated thirdly the search algorithm is lanced to search tooptimizes the indexes to ensure a best dynamic performance Examples oflinear control can be found in tuning the controllers and aid in the location ofcontrollersbull Decoupling basis control strategy the principle of this type of control strategyis to decouple the multiple input system with coupled dynamics into inde-pendent subsystems and control strategies are proposed to control those sub-systems independently by imposing the command force The command forceare calculated by inverse the mathematical model of the system dynamics andplus the compensation Examples of linear control strategies belonging to thistype can be found in vector control and modal space control

In this chapter the linear control is extended to the nonlinear domain To illustratethis the concerned linear control strategies are reviewed and then the methods toextend the linear controls to the nonlinear domain are outlined

53 Approach I Analysis Based Control Strategy

5301 Tuning the controllers

a Tuning the controller or stabilizers by linear small signal analysisIn the linear analysis and control analysis-based control strategies are widely usedto tune the controller parameters such as tuning the controller parameters basedon sensitivity-analysis The model is based on x = Ax since the eigenvalues of Adecides the stability of the system and the time needed to reach steady state theobjective is to obtain the wanted eigenvalues This method is especially ecient forautomatic tuning of controller composed of cascaded loops as shown in Fig 52Fig 52 indicates an example coming from the Virtual Synchronous Machine controlof voltage source converter (VSC) [29] The objective is to reduce the time constantof the critical eigenvalues (the eigenvalues with the biggest time constant) thereforethe system will have fastest possible dynamic response It is a heuristic process


Oscillations

to tune the parameters with the highest sensibility factor (ie the parameters thatcontribute the most to the critical eigenvalues) as shown in Fig 53 More detailscan be found in [2980]

Figure 52 The Controller Composed of Cascaded Loops

b Tuning the controllers or stabilizers based on the nonlinear indexes Asindicated in the previous sections the stability and the oscillation of the nonlinearsystems can be quantied by the nonlinear indexes And it is the nonlinear indexessuch as SI and MI3 that predict accurately the system dynamic performance (such as stability and modal oscillations) rather than the eigenvalues Thereforethe principle to tune the controller parameters taking into account the nonlinearproperties is to obtain the desired nonlinear indexes To enhance the stability isto reduce the real part of SI and to damp the oscillation is to reduce MI3 Anexample can be found in [16] where the parameters of PSS are tuned to obtain thebest nonlinear stability performanceThe nonlinear sensitivity-analysis based algorithm for tuning the controllers is shownin Fig 54

5302 Aid in the Location of Stabilizers

The stabilizer can damp the oscillations in the dynamical system such as the PowerSystem Stabilizer (PSS) for power systems However the stabilizers can be expen-sive and dicult to be equipped An important issue in power system is to damp theoscillations to the maximum with a minimum number of PSSs One choice to placethe stabilizer on the symtem component that contributes the most to the oscillationsin the system dynamics And the principle of this type of control strategies is todamp the oscillations by locating the stabilizer on the system component containingthe states that have largest participation factorsa Locating the stabilizer using linear participation factor We take NewEngland New York 16 machine 5 area system as an example which is presented in

53 Approach I Analysis Based Control Strategy 179

Figure 53 The Sensitivity-Analysis Based Algorithm of Tuning the Controllers


Oscillations

Figure 54 The Proposed Nonlinear Sensitivity-Analysis Based Algorithm for Tun-ing the Controllers

54 Approach II the Nonlinear Modal Control 181

Chapter 1 amp Chapter 3 The PSSs are equipped on the generators with the highestparticipation factor in the oscillations As in Chapter 3 by locating the PSSs onthe G6 and G9 (whose electromechanical states contributes much to the oscillationsin the interconnected power system)b Locating the stabilizer based on the nonlinear indexes Since the nonlinearoscillations can be quantied by analytical nonlinear sollutions nonlinear partici-pation factor can be proposed And the nonlinear oscillations can be damped tothe maximum by placing the stabilizers on the system component with the highestnonlinear participation factorSecond participation factors have been proposed by several literatures [32 48 58]And the third participation factors can be similarly proposed For sake of compact-ness the third participation factors are not listed in this PhD dissertation and thereader can easily deduce them under the light of [4858]

5303 Assigning the Power Generation

Indicated by Chapter 3 amp Chapter 4 the power transfer limit can be predictedmore accurately by taking into account of the nonlinearities In the interconnectedpower system there may be numerous generators or electric sources which work incollaboration to deliver the load Knowing the power transfer limit can help us tooptimize the power generation assignments to exploit the systems working capacityto the maximum which is crucial for optimal operation and planning processes ofmodern power systems This problem is more economical than technical

54 Approach II the Nonlinear Modal Control

The rst approach is almost an o-line approach is there an approach to do the

541 State-of-Art The Applied Nonlinear Control

In the milestone book Applied Nonlinear Control whose rst edition is publishedin 1991 by Slotinea professor in MIT the nonlinear control strategies in practiceare reviewed [81] and can be classied into three groups [82]

1 The most common control method is to ignore the nonlinearities and assumetheir eects are negligible relative to the linear approximation and then designa control law based on the linear model Despite numerous other methods de-vised for nding equivalent linear systems the most popular procedure is theTaylor series of the equations of motion around the stable equilibrium pointand abandoning the nonlinear terms It is the most common method sinceit is the simplest method to be implemented and can be applied to systemswith large numbers of degrees of freedom However as Nayfeh and Mook men-tioned a number of detrimental eects can be over looked when nonlinearitiesare ignored [55] von Kaacutermaacuten observed that certain parts of an airplane canbe violently excited by an engine running at an angular speed much larger


Oscillations

than their natural frequencies and Lefschetz described a commercial aircraftin which the propellers induced a subharmonic vibration in the wings whichin turn induced a subharmonic vibration in the rudder The oscillations wereviolent enough to cause tragic consequences

2 The second most common control method is to apply linearizing control tothe system such that the system acts as if it is linear By changing the statevariables andor applying a control which linearizes the system the nonlinearequations of motion are transformed into an equivalent set of linear equationsof motion such as the feedback linearization control The main benets of lin-earizing control are that the full range of powerful linear control methodologiescan be applied However major limitations exist for linearizing control [83]Firstly it cannot be used for all systems Secondly strict controllability andobservability conditions must be met And nally robustness of the controlledsystem is not guaranteed in the presence of parameter uncertainty Any errorin the nonlinear part of the model will prevent the linearizing control fromlinearizing the dynamics of the structure

3 A third method is to apply robust control such as Hinfin and include the nonlin-earities as uncertainties Although this will lead to robust stability it will notprovide robust performance Clearly the best control law should come fromdesign using all of the available model information which is not applicable inindustry practice [82]

542 The Limitations

Although the development of computation technology and microprocessors broughtnumerous new methods to deal with nonlinear control their principles still belongto the above three groupsAs concluded from the their principles these control methods either neglect thenonlinearities or only add some compensations in the amplitudes And they allfail to consider the qualitative aspect of the non-linear behaviour accurately butjust endeavor increase the quantitative resolution Those applied controls werevery practical because at the moment when those nonlinear control strategies wereproposed the analytical analysis results hadnt been availableHowever nowadays by the nonlinear analysis nonlinear properties are exacted suchas the amplitude-dependent frequency-shift (which will inuence the frequency of theoscillations) the higher-order modal interaction and the nonlinear stability (whichwill inunce the decaying of the oscillations) Therefore control techniques maybe proposed to accurately take into account the nonlinearities

543 Decoupling Basis Control

The decoupling basis control methods are very popular since they decompose thecomplex multiple-input system having coupled dynamics into decoupled single inputsingle output (SISO) subsystems One then needs only to control the SISO system


which is a solved problem The techniques are easy to comprehend since it has clearphysical meanings

5431 Vector Control in Electrical Domain

Among the decoupling basis control methods one famous example is the dq decou-pling vector control based on the Park transformation proposed in 1929 by RobertH Park It is rstly applied to analyze the dynamics of electric drives by decompos-ing the physical state variables into dq components where d component contributesto the magnetic ux of the motor the q component contributes the torque By inde-pendently controlling the dq components the dynamic performance of electric driveshave been largely improved Nowadays it has been generally applied in modellinganalyzing and controlling of electrical systems such as electrical drives generatorspower electronics and etc Even in this PhD dissertation the test systems are mod-elled in the dq synchronous frame such as indicated by the equations in Chapter 3And another example can be found in Fig 52 The vector control is almost themost popular classic and well konwn control method in electrical engineeringIn the vector control the N -dimensional dynamical system are decoupled into rst-order 1-dimensional dierential equations and the control strategies are implementedindependently based on each equation

5432 Modal Space Control in Mechanical Engineering

Similar to the analysis part there are decoupling control methods for system mod-elled as second-order dierential equations which are implemented in the modalspaceModal space control is very popular in the mechanical engineering which can beclassied into Independent Modal Space Control and Dependent Modal Space Con-trol(DMSC) The modal approach has a more clear physical meaning as it decouplesthe N dimensional system (the original system) into N 1-dimensional subsystemand then all the subsystems are second-order oscillators The solution in the modalform can indicates the physical motion of the system which is called as normalmode The modes are normal in the sense that they can move independently thatis to say that an excitation of one mode will never cause motion of a dierent modeIn mathematical terms linear normal modes are orthogonal to each other The mostgeneral motion of a linear system is a superposition of its normal modes Thereforethe original N -dimensional problem is reduced to a series of 1-dimensional problemsThe problem complexity is largely reducedInversely speaking when each mode can be independently studied the contributionof each input to the dynamics of this system can be gured out and by imposingdierent inputs desired response of the system can be obtained The precision ofpositioning control is largely increased This is the principle of modal control To theauthors knowledge the linear modal control is rstly realized by Balas in 1978 [84]And the idea of decoupling and the concept independent modal space controlwas rst proposed by [85] After that linear modal control has been widely studied


Oscillations

and applied Normally there are two types of linear modal control IndependentModal Space Control (IMSC) and Dependent Modal Space Control (DMSC) [86]Both of them belong to linear modal controlCompared to vector control the advantage of modal space control is that it can beimplemented with fewer controllers and fewer loops Controlling a system composedof one second-order equation two control loops should be implemented if vectorcontrol is adopted however only one PID controller may be needed if modal spacecontrol is adoptedIn this PhD dissertation some explorations are made to generalize the linear modalcontrol into the nonlinear modal control

544 A Linear Example of Independent Modal Space ControlStrategy

An example can be found in [87] whose principle can be shown in Fig 55

Figure 55 The Topology of the Linear Modal Control

It decouples the system under excitation represented by (51)

M q +Dq +Kq = F (51)

into a set of 1-dimensional second-order equations as (52)

Xj + 2ξjωjXj + ω2jXj =

ΦTj Fj

Mj(52)

Therefore by intelligently choosing the Fj the dynamics of Xj will be controlled asdesired To compensate the time-delay and uncertainties in the system dynamicsthe feedfoward and feedback controllers are implemented to improve the controlperformance [87] Those technical details will not be addressed here and the readercan refer to [87]As shown in Fig 56 the IMSC which is based on the equation of the dynamicalsystem is a linear version of the equation studied in Chapter 2 and Chapter 4

55 Step I Analytical Investigation of Nonlinear Power SystemDynamics under Excitation 185

Therefore it can be generalized to nonlinear domain

545 Nonlinear Modal Control

A schema of the proposed nonlinear modal control can be found in Fig 56 Theprinciple is to decouple the nonlinear system into several nonlinear subsystems andindependently control those subsystems or damp the desired modes

Figure 56 The Topology of the Nonlinear Modal Control

546 How to do the nonlinear modal control

As shown in Fig 56 we control the system dynamics by imposing the nonlinearmodal force To implement the nonlinear modal control there are two questions toanswerbull What is the nonlinear modal force for each decoupled nonlinear mode bull How to impose the force

To answer the rst question we need to investigate the system dynamics under theexcitation This is the rst stepTo answer the second question we use the inversion-based control (IBC) strategyThis is the second step

55 Step I Analytical Investigation of Nonlinear Power

System Dynamics under Excitation

As presented in the Chapter 4 we essentially study the system dynamics modelledby Eq (53)

Mq +Cq +Kq + fnl(q) = 0 (53)


Oscillations

Table 51 Comparison of Existing and Proposed Normal Form Approximations

Method Transformation DecoupledInvariant Normal Dynamics Free Oscillations Excited systems

LNM [55] linear decoupled linear free and forced

NNM [2122] order 3 decoupled order 3 free

FDNF3 order 3 decoupled order 3 free and forced

with M and D are constant symmetric inertia and damping matrix whose valuesdepend on the physical parameters of the power system and controller parametersK and fnl indicate the coupling between the variables where K is a constantmatrix including the linear terms and fnl gathers the nonlinear termsAs observed from (62) and (53) the input is null Therefore the methodologiespresented in Chapters 2 3 4 focus on the case that the system is not excited or theexcitation is constant Or speaking from the perspective of mechanical engineeringwhat we studied before are free-oscillationsIf we want to inuence the systems nonlinear dynamics by imposing the forceanalytical investigation should be conducted on system dynamics under excitation

551 Limitations and Challenges to Study the System Dynamicsunder Excitation

Nonlinear analysis can replace the linear analysis as it does a better job in allthe three issues describing the dynamic response modal interactions and stabilityanalysis However the existing nonlinear analysis tools only deal with the case offree oscillations (eg the post-fault case after the fault is cleared) and not withexcited systems ( ie the oscillations caused by variable power references or variableloads) This largely limits the scope of the NF analysis This is a big disadvantagecompared to the linear analysis which can deal with the case when there are changesin the input (voltage reference power reference load etc) as well as when there areno excitation (free oscillations)The challenge is to take into account the force vector when doing the normal formtransformations A methodology is proposed in Section 55 to overcome such achallenge

552 Originality

In this PhD work a method is introduced to study the power system nonlinear forcedoscillations To the authors knowledge it is for the rst time that an analyticalanalysis tool based on NF theory is proposed to analyze the nonlinear power systemdynamic under excitation This enlarges the scope of NF analysis and lays thefoundation for the lately proposed nonlinear modal control


553 Theoretical Formulation

The oscillatory modes of power system can be modeled by (54) If the variables ofthe system are gathers in a N dimensional vector q the motion equation writes as

Mq +Cq +Kq + fnl(q) = F (54)

where M C are constant diagonal inertia and damping matrices whose valuesdepend on the physical and controller parameters of the system K and fnl indicatethe coupling between the variables whereK is a constant matrix including the linearterms and fnl gathers the nonlinear terms coming from the 2nd and 3rd order termsof the Taylor series F is the excitation vector

554 Linear Transformation Modal Expansion

A modal expansion q(t) = ΦTx(t) can transform (54) into modal coordinates withdecoupled linear part [55]


+

Nsumi=1

Nsumjgei

gpijXiXj +

Nsumi=1

Nsumjgei

Nsumkgej

hpijkXiXjXk = Fp

where ξp is the p-th modal damping ratio and gpij and hpijk are quadratic and cubic

nonlinearities coming from fnl in Taylor series Neglecting the nonlinear terms in(55) a linear model called LNM [55] is obtained and can be used both for analysisand control of multi-input systems

555 Normal Form Transformation in Modal Coordinates

A normal form transformation is proposed in [1719] which reads forallp = 1 N

Xp = Rp +R(2)(Ri Si) +R(3)(Ri Si) Yp = Sp + S(2)(Ri Si) + S(3)(Ri Si)(56)

where Yp = Xp Sp = Rp and R(3)S(3) and R(3)S(3) are second order polynomialsand third order polynomials in Ri ans Si respectively fully dened in [17 19] andare functions of the gpij and h

pijk coecients present in (55)


Up to order 3 and keeping all the terms after the transformation it leads to a setof invariant oscillators (if no internal resonance occurs) dened as FNNM [17 19]FNNM separates the nonlinear terms into cross-coupling terms and self-couplingterms Neglecting the cross-coupling terms the equations dening the normal dy-namics are decoupled and dened as NNM and FDNF3


Oscillations

5552 NNM

Free-damped Oscillations



pppRpR2p + CppppR

2pRp = 0

5553 FDNF3

Systems under excitation

Rp =Sp minus 2bpppSpFp minus cpppFpRp (58)

Sp =minus ω2pRp minus 2ξpωpRp

minus (hpppp +Apppp)R3p minusBp

pppRpR2p minus CppppR2

pRp

+ Fp minus 2βpppSpFp minus 2γpppFpRp

It is shown that when F is zero (58) becomes (57) In other words (57) is aspecic case of (58) or (58) is a generalization of (57) (58) shows that trans-forming variables expressed in X minus Y coordinates into R minus S coordinates will addcoupling terms in the excitation vector F ( 2βpppSpFp minus 2γpppFpRp terms) The NFapproximation decouples the state-variables but adds some coupling terms in theexcitation vector components

5554 Reconstructing the Results for the Original System

Multi-scale calculating methods can be used to compute an approximate analyticalsolution of (57) and (58) From Rp Sp variables Xp Yp can be reconstructed usingtransformations (56) The eciency of FDNF3 is validated by a case-study basedon interconnected VSCs and compared to the classical linear analysis tool LNM

556 CaseStudy interconnected VSCs under Excitation

5561 Test System

Modern girds are more and more composed of renewable-energy-based power gen-erators that can be interconnected to the grid through long transmission lines [88]This weak grid conguration can lead to nonlinear oscillations [22 89] To illus-trate this particular case a test case composed of two interconnected VSCs to atransmission grid as been chosen as shown in Fig 57V SC1 and V SC2 are interconnected by a short connection line having a reactanceX12 and are both connected to the transmission grid by long transmission lineshaving reactances X1 X2To study the nonlinear interactions under disturbances or variable references orloads a detailed mathematical model is rsly made which consists of the physicalstructure as well as the current voltage and power loops To design the power


Figure 57 Test System Composed of interconnected VSCs

loop since the conventional PLL controller can cause synchronization problemswhen facing connections using long transmission lines [66] the Virtual SynchronousMachine (VSM) control strategy is adopted [29]For the chosen test case the equivalent switching frequency of VSCs is set as 1700Hz

and the control loops are tuned with time constants as Tcurrent = 5ms Tvoltage =

60ms Tpower = 147s Large Volume Capacitor is installed to avoid the voltagecollapseThe electromagnetic model of one VSC is a set of 13th order equations [29] andthe order of the overall system model can be higher than 26 Considering that thecurrent and voltage loops are linear and that the time constants of the three loopsrespect Tpower Tvoltage Tcurrent the voltage V1 and V2 at the Point of CommonCouplings (PCC) of VSCs are assumed sinusoidal with constant amplitudes andpower ows are represented by dynamics in power angles Thus a reduced model tostudy the nonlinear interactions of the system presented in Fig 57 can be describedas (54) A 4th order model is then chosen to govern the nonlinear dynamics of thesystem

M1d2δ1dt2

+D1dδ1dt +

V1VgX1

sin δ1 + V1V2X12


M2d2δ2dt2

+D2dδ2dt +

V2VgX2

sin δ2 + V1V2X12


(59)

In (53) δ1 and δ2 represent the power angles X1 and X2 are the reactances of thetransmission lines which connect the two VSCs to the transmission grid X12 is thereactance of the line interconnecting the two VSCs V1 V2 Vg are the RMS voltagesat the PCC of the VSCs and at the grid Finally P lowast1 and P lowast2 are the active power


Oscillations

0 05 1 15 2 2505

055

06

065

07

075

08

085

09

t(s)

δ1(rad)

Dynamics

EXLNMFDNF3

0 05 1 15 2 2502

025

03

035

04

045

05

055

06

065

t(s)

δ2(rad)

Dynamics

EXLNMFDNF3

Figure 58 System Dynamics under Step Excitation

references which are set according to the expected operating points

5562 Selected Case Study

A case is selected to test the proposed method Step Excitation in the PowerReference V SC1 and V SC2 respectively transfer an amount of power as 068pu

and 028pu It is supposed that there is an increase in the wind energy supplywhich implies a step in the power reference of 02pu for both VSCs The systemexperiences then oscillations due to excitationThe time-domain simulation is done by using a EMT software to assess the perfor-mance of the proposed method

5563 Original Results

The dynamics in the power angles of the VSCs δ1 and δ2 are shown in Fig 58The step excitation in P lowast1 = P lowast equal to 02pu happens at the time t = 05Fig 59 shows that as proposed by the classical linear analysis the system dynamiccan be decoupled into 2 simpler ones Ex asymp Sub1 + Sub2 making it possible toidentify that Sub1 has a lower frequency and Sub2 has a much higher frequencyAs a generalization of NNM FDNF3 has the property of decomposing a N -dimensional nonlinear system into a nonlinear sum of nonlinear 1-dimensional non-linear systems analogous to linear analysis that decompose a N -dimensional linearsystem into a linear sum of 1-dimensional linear systems In this way the complexpower dynamics can be reviewed as the combination simpler onesIt is shown in Fig 58 that FDNF3 can describe the system dynamic responsemuch more accurately than the linear analysis LNM especially the overshoot whichcorresponds to the limit cycles [55] or the stability bound of systems nonlineardynamics [5]Also as shown in Fig 510 even both VSCs are excited only one nonlinear modeis excited in R minus S coordinates Therefore inversely by controlling R1 R2 we canmake the decoupling control of V SC1 and V SC2 possible even if they work in the


0 05 1 15 2minus05

0

05

1

15

t(simulation time)

δ1(rad)

Mode Decomposition

EXMode 1Mode 2

0 05 1 15 2minus05

0

05

1

15

t(simulation time)

δ2(rad)

Figure 59 System Dynamics Decomposed by NNM

0 05 1 15 2 25minus01

0

01

02

03

04

05

06

t(s)

NonlinearModalState

Variables

Excitation of the Modes

R1R2

Figure 510 Decomposition the Modes under Excitation R1R2


Oscillations

nonlinear region It could be one way to overcome the diculty on how to controlVSCs independently when they work in nonlinear regions

557 Signicance of the Proposed Method

Transforming the original system dynamic into its normal dynamics it gives a clearpicture on how the excitation inuences the system dynamics Compared to theclassical linear analysis LNM and the NNM the advantages of FDNF3 approachcan be summarized as

1 Describing more exactly the power system dynamics under disturbances or un-der excitation by decomposition of the complex system dynamics into simplerones

2 Making possible to independently control interconnected systems even if thereare working in highly nonlinear regions

In this PhD work to the authors knowledge it is the rst time that Normal Formtheory is applied to study the power system dynamics under excitation (ie withvariable references or loads) The proposed method is validated by comparing resultswith time-domain simulations based on an EMT software For the sake of simplicityonly the results under the case N = 2 are presented but the possibility of workingwith large-scale problems will be investigated by the methodology suggested in thisdissertationAlthough the chosen test case is composed of inteconnected VSCs since the sys-tem model is the same as groups of generators working in parallel the proposedmethodology can be applied for studying the modal oscillations between classicalgenerators working in parallelThe proposed method only shows its ability to describe the power system responseby decomposing the complex dynamics Further applications may be found in modalstructural analysis under excitation or stability analysis This lays the foundationof the future work that

5571 Analysis of Dynamical System under Excitation

The proposed FDNF3 gives the normal forms of the dynamical system under ex-citation It makes possible to do stability analysis when there is a change in theinput (such as an increase of load or power generation) Also when measuring thesystems oscillations with experiments it is important to impose the correct forcePreviously the nonlinear analysis under excitation is conducted by imposing theforce on the free-oscillation Normal Forms



pppRpR2p + CppppR

2pRp = F

Comparing (510) with (58) (510) fails to take into account the couplings inthe F For the experimental measurements this will cause large error since other

56 Step II Implementing Nonlinear Modal Control by EMR+IBC193

modes than p may also excited imposing Fp ie the energy is not only injected intomode p and the invariance is broken

5572 Signicance in the Control of Dynamical System

The decomposition of nonlinear system makes it possible to control the two VSCsindependently in the nonlinear R minus S coordinates to reduce the oscillations in thesystem dynamics However this is a open-loop control To increase the resolutionand compensate for the uncertainties closed-loop control should be implemented

56 Step II Implementing Nonlinear Modal Control by

EMR+IBC

Presented in the previous section the nonlinear transformation can approximatelydecouple the system dynamics into nonlinear modes that can be independently con-trolled by imposing the nonlinear modal force The next step is to implement thecontrol strategy which can be realized by EMR+IBC

561 The Control Schema Implemented by Inversion-Based Con-trol Strategy

If we represent the decoupling process of the system dynamic model by EnergeticMacroscopic Representation (EMR) we can then develop the control schema byinversion-based control (IBC) strategyThe topology of the nonlinear modal control is shown as Fig 511 considering thenonlinearities up to 3rd order The linear transformation decouples the linear trans-formations and the nonlinear transformation weakens the linear coupling rendering

Figure 511 The Control Schema of the Nonlinear Modal Control Implemented byInversion-Based Control Strategy


Oscillations

As observed from the Fig 511 by decomposing the nonlinear system into 1 di-mensional subsystems the nonlinear modal force needed to excite the desired modecan be calculated Then the force should be imposed on the system is a nonlinearcombination of the nonlinear modal force obtained using the Normal Form trans-formation

562 Primitive Results

The exploration of nonlinear modal control is conducted on a gantry system asshown in Fig 512

Figure 512 An Example of Gantry System

which can be simplied as Fig 513For such a system closed loop control strategy is proposed as shown in Fig 514 [90]

However the nonlinear modal control implemented in this way doesnt ensure a goodperformance even for nonlinear modal control up to 2nd order where the normalform is a linear system controlled by PID controller More investigations should beconductedThe bottle neck may lay in


Figure 513 The Simplied System of the Gantry System



Oscillations

bull the inverse nonlinear transformation The way to do the inverse nonlineartransformation has been attached in the Appendix B for the future researchesbull the validity region of normal form transformationsbull the way to implement the PID controller for the decoupled subsystems

57 Exploration in the Control Form of Normal Forms

If NF can simplify the analysis by transforming the variables into the normal formcoordinates is it possible to simplify the control by the same principleThis question gives birth to the Normal Form control for which Poincareacute didntgive out any theory or mathematical expressions

571 A Literature Review

It is well-known that a state space description of a controllable linear system can betransformed to controllable or controller form by a linear change of state variablesIn the sense of being transformed into a simpler form it would be referred to asLinear Controller Normal Form [91]If a nonlinear control system admits a controller normal form it can be transformedinto a linear system by a change of coordinates and feedback Therefore the designof a locally stabilizing state feedback control low is a straightforward task In such acase we say the system is feedback linearizable On the other hand most nonlinearsystems do not admit a controller normal form under change of coordinates and in-vertible state feedback The nonlinear generalization of the linear controller normalforms which can be referred to as Nonlinear Controller Normal Form (NCNF) hasbeen systematically studied since 1980s by Krener [91] This approach was extendedto control systems in continuous-time by Kang and Krener ( [92] see also [93] fora survey) and Tall and Respondek ( [94] see [95] for a survey) and by Monaco etall [96] and Hamzi [97] et al in discretetime However this approach mainly focuseson single input system NCNF for multi-input system is awaited to be developed tocater for the studied case in this thesisOn another side the emphasis is on the reduction of the number of monomials inthe Taylor expansion one of the main reasons for the success of normal forms liesin the fact that it allows to analyze a dynamical system based on a simpler formand a simpler form doesnt necessarily mean to remove the maximum number ofterms in the Taylor series expansion This observation led to introduce the so-called inner-product normal forms in [92] They are based on properly choosingan inner product that allows to simplify the computations This inner-product willcharacterize the space overwhich one performs the Taylor series expansionThe methods reviewed brought several mathematical achievements while their ap-plicability are limited mainly in system having controllable linearization and inadding the feedback linearization nonlinear control

58 Conclusion 197

572 A Nouvel Proposed Methodology to Deduce the NormalForm of Control System

In fact all those methods referred to in the above focus on the generalization oflinear controller normal form to the nonlinear domain the objective is to rendera linear normal form by the normal form transformation to aid in the feedbacklinearization control However as shown in Chapter 2 Chapter 3 and Chapter 4 alinear normal form is not ensured for all the cases And taking into the 3rd ordernonlinear terms in the normal forms can suggest more information in the analysisof both the nonlinear modal interaction and nonlinear stabilityTherefore inspired by the normal form analysis another approach is proposed toderive the control forms of normal forms ie similar to the analysis we take thecontrol form

x = f(xu) (511)

the dimension of the system T isin Rn timesRm and u isin Rm representing the controlAnd by the cloased-loop control the u can be represented by

u = fc(x) (512)

Therefore the control system can be rewritten as

x = f(x fc(x)) (513)

58 Conclusion

In this chapter explorations from nonlinear analysis to nonlinear control are con-ducted Two approaches and three proposals are illustrated and investigated Dueto the time limit of the PhD the control strategy is not fully implemented yetHowever those proposal are very innovative and can inspire further investigationsIt lays the foundations for future researches Whats more to the authors knowl-edge analytical investigations of nonlinear system dynamics under excitation areconducted for the rst time on the power systems


Oscillations

Chapter 6

Conclusions and Perspectives

In this chapter conclusions on achievements of nonlinear analysis and

The assumptions of methods of normal forms are discussed and investigated The

classes of the

Unfortunately the author doesnt have enough time to fully develop the methodology

of nonlinear modal control during the 3 years PhD period This PhD research may

kick the door for future researches or another couple of PhD dissertations

Keywords Nonlinear Analysis Nonlinear Modal Control Multiple InputSystem with Coupled Dynamics

199

200 Chapter 6 Conclusions and Perspectives

61 A Summary of the Achievements

This PhD research works on the very fundamental problems concerning nonlin-ear systems having multiple input with coupled dynamics Multiple input systemwith coupled dynamics almost cover all the modern industrial systems since high-performance industrial electromechanical systems are often composed of a collectionof interconnected subsystems working in collaboration These dynamical systemscan be found everywhere and contribute largely to our daily life to the nationaleconomy and global developmentHaving made some achievements in the nonlinear analysis to have a clear picture ofthe system dynamics several proposals of nonlinear control are investigated whichkicks the door for the future investigationsThe achievements can be summarized below and more detailed discussions are putin the following sectionsbull Practical (Application) study of the nonlinear modal interaction andthe stability assessment of interconnected power systems This solved along-time existing problem and explained some nonlinear phenomenonIn the previous chapters of this PhD dissertation methodologies to analyzethe stability of some types of multiple input systems with coupled dynamicsare proposed and validated with case-studies based on interconnected powersystems

bull Foundation for control2 innovative approaches are proposed investigated and discussed It opensthe way for further developments in both application and normal form theory

bull Theoretical (Methodology) A systematic methodology is proposed which isinnovative and general for nonlinear dynamical systems development of mathematically accurate normal forms up to 3rd orderfor analysis and control of nonlinear dynamical systems with coupleddynamics by both the PoincarAtilde ccopys approach and TouzAtilde ccopys ap-proach

nonlinear indexes are proposed to quantify the nonlinear properties

62 Conclusions of Analysis of Power System Dynamics

The applications of this PhD work are to study the rotor angle dynamics of inter-connected power systems when there is disturbance in the transmission line Themethodologies are established in Chapter 2 and are applied to study the interareaoscillations in Chapter 3 and assess the transient stability in Chapters 3 amp 4Some achievements are made in this PhD research and a transaction paper hassubmitted with revision and another transaction is going to be proposedSome conclusions are made in this chapter as follows

62 Conclusions of Analysis of Power System Dynamics 201

621 Analysis of Nonlinear Interarea Oscillation

Nowadays power systems are composed of a collection of interconnected subsystemsworking in collaboration to supply a common load such as groups of generators [1]or interconnected Voltage Sources Converters working in parallel [212298] One ofthe most important issue in large-scale interconnected power systems under stressis the oscillations in the power system dynamics [28]As the oscillations are essentially caused by the modal interactions between the sys-tem components they are called Modal Oscillations The complexity of analyzingsuch oscillations lays in the strong couplings between the components of the systemUnderstanding of the essence of such oscillations is necessary to help stabilizing andcontrolling power systemsTo cater for the above demands analytical analysis tools are developed in this PhDwork for

1 Dynamic performance analysis suggesting approximate analytical solutions2 Modal analysis describing and quantifying how the system components inter-

act with each other3 Nonlinear Stability analysis predicting the stability of the nonlinear system

by taking into account the nonlinear terms in the Taylor series

622 Transient Stability Assessment

Transient stability has become a major operating constraint of the interconnectedpower system The existing practical tools for transient stability analysis are eithertime-domain simulations or direct methods based on energy-function and Lyapunovdirect method the methods of Transient Energy Function (TEF) Though there arenumerous meticulous researches on TEF this method is dicult to implement withlarge systemIn this PhD research the stability is assessed by using nonlinear indexes based on3rd order Normal FormsThe Transient stability assessment can be done either by the vector eld approachor by the normal mode approachThe Poincareacutes approach (vector eld approach) using the nonlinear index SI hasthe applicability for all generator models whose dynamics can be formulated asx = f(x) Unlike the TEF method it is an easy-to-implement tool with systematicprocedures that doesnt need further discussions in the implementationDeveloped in this PhD work the Touzeacutes approach (normal mode approach) usingthe index PSB has been presented to be ecient on the classical models and alsopossible to be extended to work with more complex models Since it is rather easyto implement it can be expected to solve the TSA problem with bright perspectivesOne would like to ask a question

What are the dierences between the stability assessments made by the twoapproaches Are they essentially the same

In fact the SI is more accurate than PSB since SI takes into accunt both the


coupled nonlinearities and decoupled nonlinearities and PSB neglects the couplednonlinearities in the normal forms SI is based on method 3-3-3 PSB is based onmethod NNM which is somewhat equivalent to method 3-3-3(S)However for the applications PSB is more adept for multi-machine systems sinceit has a much less computational burden and much easier to be calculated

623 A Summary of the Procedures

For both methodologies the procedures of analyzing the power system can be sum-marized as the owchart in Fig 61To do the Normal Form analysis of electric power systems there are two stagesThe rst stage is to formulate the dierential equations containing polynomial non-linearities by perturbing the system around the stable equilibrium point (SEP) byTaylor series which is labelled as SEP Initialization once the perturbation modelis xed the Normal Form coecients are xedAfter the SEP Initialization the second stage is to calculate the initial conditionsin the Normal Form coordinates which is labelled as Disturbance InitializationIt depends on the disturbance which the system experiences

624 Issues Concerned with the Methods of Normal Forms inPractice

To make the methodologies practical for the readers (mathematicians engineers)the issues to implement the Normal Form analysis are discussed in Section 28and Section 36 such as the search for initial conditions computational burdenmodel dependence and etc The scalability problem and applicability to large-scalesystems are also discussed in Section 35 Also the help the readers to implementthe methodologies programs and simulation les are made available online Moredetails can be found in the appendixes

63 Conclusions of the Control Proposals

In this PhD dissertation it points out two approaches to implement the nonlinearcontrol in Chapter 5The rst is automatic tuning the controller parameters to obtain the desired non-linear dynamic performance quantied by the nonlinear indexes which is named asthe analysis based approachThe second is imposing the nonlinear modal force on the approximately decouplednonlinear modes which is named as the decoupling-basis approachDue to the time limit of the PhD those control proposals are not fully implementedyet however the presented investigations and discussions lay the foundation offuture researches

63 Conclusions of the Control Proposals 203

Form the DAE model

Eqs (1)(2)

Power Flow analysis by

solving Eq (2) to obtain SEP

Taylor series expansion

around SEP Eq (3)

Transform Eq (3) into

its Jordan form Eq (4)

Search the initial points

in the NF coordinates

Calculate the indexes

for modal interaction

and nonlinear stability

S

E

P

I

n

i

t

i

a

l

i

z

a

t

i

o

n

Disturbance Initialization

variabledisturbance

variableSEP

Do eigen analysis

of matrix A

Calculate the NF

transform coefficients

h2h3 c

End

Start

Rapid

Figure 61 The Flowchart of Applying Methods of Normal Forms in Analysis ofSystem Dynamics


64 Conclusions of the Existing and Proposed Method-

ologies

The most important achievements of this PhD research work is the proposedmethodologies The proposed methodologies in this PhD work has been success-fully applied to nonlinear analysis of interconnected power system However theapplicability of the proposed methodologies is not limited in electrical domain Aspresented in Chapter 2 the proposed method 3-3-3 and method NNM are generalmethodologies for study of dynamical systems and the extracted nonlinear proper-ties are universal in multiple input system with nonlinear coupled dynamicsHowever to use the proposed

641 Validity region

One important concern of methods of normal forms is the validity region From thenumerical results it can be concluded that the methods of normal forms are validin the entire working regions of the power systemFor other systems we can employ the methodology proposed in [51] to investigatethe validity region of the methods of normal formsIt should be noted that if the amplitudes of the state-variables go beyond the validityregion then the information extracted from the normal forms is NONSENSE

642 Resonance

In this PhD work when deducing the normal forms internal resonances due to thecoincidences of the eigenvalue frequencies are not considered If there exists internalresonance other than the trivially resonant terms the normal form may include more2nd-order and 3rd-order terms and therefore the stability assessment will becomedicult However the quantication of the nonlinear modal interaction will notbecome more dicult Investigations should be done on the resonant cases

643 Possibility to do the Physical Experiments

Since the researches in power systems are mainly validated by numerical simulationsone question may arise

Is it possible to validate the proposed methodologies by physical

experimentsConducting physical experiments on interconnected power systems is much moreexpensive then conducting experiments on mechanical systems However since theelectromechanical oscillations can be measured from the rotor which may containnonlinear modal interactions It is possible to validate the conclusions drawn in thisPhD dissertation by physical experiments

65 Multiple Input Systems with Coupled Dynamics 205

65 Multiple Input Systems with Coupled Dynamics

In this PhD research dissertation we only focus on system with coupled dynamicson the state-variables However there can be other type of coupled dynamics

651 Classes of Multiple-input Systems with Coupled DynamicsStudied by the Proposed Methodologies

Two approaches are established which cover the studies of a large number of non-linear dynamical systems composed of interconnected subsystems working in collab-oration as far as their dynamics can be modelled by either (61) or (53)

6511 Class of System Studied by Poincareacutes Approach

x = f(xu) (61)

where x is the state-variables vector u is the systems inputs vector and f is adierentiable nonlinear vector eld Expanding this system in Taylor series arounda stable equilibrium point (SEP) u = uSEP x = xSEP one obtains

4x = H1(4x) +1

2H2(4x) +

1

3H3(4x) +O(4) (62)



] H3jklm =


]and O(4) are


6512 Class of System Studied by Touzeacutes Approach

Mq +Cq +Kq + fnl(q) = F (63)


652 System with Coupled Dynamics in the First-Order Dieren-tial of State-Variables of System

If there is a system with coupled dynamics in the rst-order dierential of state-variables of systems as modelled by Eq (64)

x = f(x x) (64)

or Eq (65)

Mq +Cnl(q) +Kq + fq = F (65)


we can also endeavour to study the system dynamics by methods of normal formsBy Poincareacutes approach one solution is to dene the auxiliary state variables asx = xs However this may lead to zero eigenvalues in the Jordan form If λj = 0trivial resonance may occur in second-order nonlinear terms since the conjugatepairs (λ2k λ2kminus1) satisfy ωj = ω2k +ω2kminus1 and therefore the second-order nonlinearterms must be kept in the normal forms Its normal forms are more complicated andthe stability assessment may be dierent than what we have proposed However thestudy of nonlinear modal interaction can employ the procedures and the nonlinearindexes will be in the similar formBy Touzeacutes approach we can still try to use the procedures proposed in previousChapters to decouple the system dynamics This may take another 3 years PhDwork However it deserves that Such type of coupled dynamics can be found inelectrical machine which are electromagnetically coupled and for a long time noanalytical investigation is available to study its nonlinear dynamics according tothe authors knowledge Solving such a system can overcome a great ever-existingproblem in electrical engineering

653 System with Coupled Dynamics in the Second-Order Dier-ential of State-Variables of System


x = f(x x) (66)

or Eq (67)

Mclq +C(q) +Kq + fq = F (67)

I hold the similar opinions as in the last section However investigations should beconducted to make a more rigorous conclusion

66 The End or a new Start

In this chapter achievements are summarized Also due to the limited time of thePhD and the limited talents of the PhD student there are still a lot things to do tocomplete the theory and enlarge the range of applicationsMore questions are asked then answered For the questions that are not answeredthis PhD dissertation has pointed out the possible way to answer them It is theend of this PhD dissertation but it is a new start of more interesting researches

List of Publications

1 T Tian and X Kestelyn and O Thomas and A Hiroyuki and A R MessinaAn Accurate Third-order Normal Form Approximation for Power SystemNonlinear AnalysisIEEE Transactions on Power Systems 2017

2 T Tian X Kestelyn and O Thomas Normal Form based Analytical Inves-tigation of Nonlinear Power System Dynamics under Excitation 2017 IEEE

Power and Energy Society General Meeting (PESGM) July 2017 pp 1-53 T Tian X Kestelyn G Denis X Guillaud and O Thomas Analytical

investigation of nonlinear interactions between voltage source converters in-terconnected to a transmission grid 2016 IEEE Power and Energy Society

General Meeting (PESGM) July 2016 pp 1-54 T Tian X Kestelyn and O Thomas Third-order based analytical investiga-

tion of nonlinear interactions between voltage source converters interconnectedto a transmission grid 2016 18th European Conference on Power Electronics

and Applications (EPE16 ECCE Europe) Sept 2016 pp 1-105 TTian XKestelyn et al Transient Stability Assessment Based on Nonlinear

Normal Mode [Article to be proposed]

207

208 Publications

Appendix A

Interconnected VSCs

This part of appendix is dedicated to the interconnected VSCsSection A1 illustrates how the reduced 4th order model is formulatedIn this part of Appendix the crucial les are presented to facilitate the readerscomprehension of the PhD dissertationAll the programs and les associated with interconnected VSCs can be found inhttpsdrivegooglecomdrivefolders0B2onGI-yt3FwRVZQbUFkcGl5enc

usp=sharing

A1 Reduced Model of the Interconnected VSCs

As stated in Chapter 1 nonlinear oscillations are predominant in stressed powersystem such as the weak grid with long transmission lineTherefore to test the proposed methodologies a renewable energy transmissionsystem is chosen where predominant nonlinear oscillations are involved It is ofpractical signicance to study such a case as renewable energy is our future sourceenvironmentally friendly and sustainable and has a very bright prospect EuropeanCommunity has proposed to reach 20 energy share in 2020 while the US proposed22 With 35 policies in Asia and Pacic countries rank as the most encouraging

A11 A Renewable Transmission System

Among all the renewable energy sources wind energy takes a large part in themarket A typical o-shore wind farm is shown as Fig A1 the generators aresituated in remote areas convert the mechanical energy into electricity integratedinto the great grid by voltage source converters (VSC) and transferred by longcablesTo improve the reliability the generators are interconnectedBecause whenthe cables are long it is likely to be broken that means the fault level is lowMoreover unlike the coral energywhich is always constant the uncertainty andvariability of wind and solar generation can pose challenges for grid operators Andthe reliability of both the generation and transmission of renewable energy shouldbe enforced One solution is to connect the dierent VSCs together The eectiveinteraction between the VSCs reduce peak and ll valley of the generation andensure the power stability even if one cable is cut-oThis topology eectively solves the transmission problem and enforces the reliabilityof power supply but it leads to nonlinear oscillations Because a) the long cablesconnecting the generators to the great grid has very high impedancemake the powersystem very weak called weak grid meaning that it is very sensitive to the dis-

209

210 Appendix A Interconnected VSCs

Figure A1 Renewable Distributed Energy Located in Remote Areas [99]

turbance in the power system b) the line connecting generators to each other has avery low impedance in the high voltage transmission system this will cause strongcoupling of the generators when there is some large disturbanceTo study the oscillations in power system a popular method is to linearizing thesystem around the equilibrium point and study its eigenmodes [1] That is assumethe system to exhibit linear oscillation under small disturbance In our case theconventional small disturbance method is not applicable any more Nonlinear modalanalysis should be done

A111 The Basic Unit

As the generator can be modeled as DCsource so the simplest unit of such a sys-tem can be modeled two grid-connectedvoltage source converters interconnectedto each other shown as Figure A2 AsVoltage Source Converters can be used tocontrol the voltage currents as well aspower ows this topology can representthe principle functions of the orignal large-scale system

Figure A2 The Basic Unit to Study theInter-oscillations in Weak Grid

A12 Methodology

The steps of this methodology can be summarized as below1 Model the System(Section A13) it contains three parts

A1 Reduced Model of the Interconnected VSCs 211

a) Establish the mathematical model of the studied system by EMR developinversion-based control strategy correspondingly use linear modal analysis totune linear controllersb) Reduce the model to focus on nonlinear behaviorc) Make the disturbance model to study the damped free vibration aroundthe equilibrium under large disturbance

2 Modal Analysis Do linear modal analysis then nonlinear modal analysisbased on nonlinear normal modes(NNM) to decouple the N-dimensional Equa-tion into 1-dimensional Equations (decouple the linear and nonlinear cou-pling)

3 Solve the 1-dimensional problem by numerical integration to obtain numericalresults or use perturbation method to obtain analytical results

A13 Modeling the System

A131 The Detailed Model

To start the case study a single classic VSC is modeled and controlledFig A3 [29] presents a classical physical conguration of grid-connected VSC andthe currents and voltages as well as power injected into the grid should be controlledThere are two crucial issues concerning the control of the grid-connected VSCFirstproblem is the low frequency resonances that can interact with the vector controlSecond problem is due to the PLL dynamics when the power converter is synchro-nized to a weak grid To handle the rst issue the controller parameters will betuned by iterative tuning based on linear modal analysis [80]To handle the secondissue the virtual synchronous machine power control will be adopted [29]

Figure A3 Grid-side-VSC Conguration [29]

According to Figure A3 the equations of the components can be written as Equation(A1)(A2)(A3)(A4) For simplicityall the varialbes and parameters are expressedin per unit


AC LoadSource

lgωbmiddot s middot iod = vod minus rg middot iod + ωg middot lg middot ioq minus vgd

lgωbmiddot s middot ioq = voq minus rg middot ioq minus ωg middot lg middot iod minus vgq

(A1)LC Filter Filter L

l1ωbmiddot s middot icd = vod minus r1icd + ωgl1icq minus vod

l1ωbmiddot s middot icq = voq minus r1icq minus ωgl1icd minus voq

(A2)

Filter C

c1ωbmiddot s middot vod = icd + ωg middot c1 middot voq minus iod

c1ωbmiddot s middot voq = icq minus ωg middot c1 middot vod minus ioq

(A3)Power Estimation(Neglecting the resistance rg of the grid impedance)

Pg =vgd middot voqlg middot ωg

Qg =vgd middot (voq minus vgd)

lg middot ωg(A4)

To model and decide the control structure Energetic Macroscopic Representa-tion(EMR) is used EMR is multi-physical graphical description based on the in-teraction principle (systemic) and the causality principle (energy) It vivies therelations between the subsystems as action and reaction decided by the inputs andoutputs owing between each other It classies subsystems into source accumula-tion conversion and distribution by their energetic functions [100] It uses blocksin dierent color and shapes to represent subsystems with dierent functions corre-spondinglyFrom the conguration and the equations we can see that the LC lter and theAC source are energetic accumulation systemsFor the inductance of LC lter theinputs are the voltages vo vc imposed by c1 and the converterand the output is icFor the capacitancethe inputs are the currentsic io injected from l1 and the gridFor the AC sourceload it is similar to the inductance of LC lter the inputs arethe voltages vg vo while the output is ioThe converter is a conversion blockof which the inputs are tunable The grid voltagesource and the DC-link Bus serve as the sources to the systemThus the representation of the physical system is accomplished shown as the Fig-ureA4 The output of the converter is vc and the variables to be controlled arevo io as well as the power injected into the grid There are dierent ways to estimatethe power leading to dierent inversion-based control strategies According to theinversion-based control theory [100] the reference voltagevcref to the converter isproduced by current loop via inversion the equation of inductance component of theLC lter the reference currentilowastc to the current loop can be calculated by inverting


Figure A4 EMR and Inversion Based Control Strategy

Figure A5 The cascaded control loops [29]

the equation of the capacitance component of the LC lter the power is controlledby estimator virtual synchronous machine (VSM) avoiding the unsynchronizationThen the cascaded control loops are decided shown in Figure A5 [29]To tune the control parameters the whole system is tuned by sensitivity analysisbased on linear modal analysiswhere the whole system is expressed in statespaceand controller parameters are tuned to obtain desired eigenvalues(corresponding tolinear modes) [29] [80]This section illustrates the analysis and control of a single VSC and it also servesas an example to the application of linear modal analysis However seen from thissection we can see the control structure is complex enough and the whole systemis 13th-order If we use such analytical model to study two VSCs interconnected toeach other the complexity will be at least twice and it is impossible to use sucha complex model to study the system One would like to reduce the model to just


focus on the nonlinearities

A132 The Reduced Model

One way to reduce the model is to reduce the voltage loop and current loop asV lowastod = Vod V

lowastod = Vod i

lowastod = iod i

lowastoq = ioq as the time constants of the current

loop voltage loop and power loop verify that Tpower Tvoltage Tcurrent This isdone by simplifying the loops one by one rst the current loop secondly the voltageloop The process to reduce the model is illustrated in Figure A6 where the loopsare the mathematics expression of iod vod is demonstrated to analyze the systemdynamicsOne should note that the process to reduce the model is not a simple deleting outthe loops and poles The eigenvalues of the current loop and voltage loop can becomplex the eigenvalues of the power loop are realIt is based on two important pre-conditions a)there are big gaps between the timeconstant of the three loops b)the voltage loop and the current loop are conservativeand linear c)the nonlinearities must be reserved After reducing the modelthesystem can be expressed as Equation (A5)

J1d2δ1dt2

+D1(ω1 minus ωg) +V1VgL1

sin δ1 + V1V2L12


J2d2δ1dt2


sin δ2 + V1V2L12


(A5)

Where J1 D1 J2 D2 are the controller parameters of the power loop δ1δ2 are thepower anglesV1 V2 are the voltages P lowast1 P

lowast2 are the references values of the power

ω1ω2 are angular frequencies of the two VSCs ωg is the angular frequency of thegrid and L1 L2 is the impedance of the cable connecting VSC to the grid and L12

is the impedance of the transmission line connecting the two VSCs to each otherL1 L2 is very high (07pu) leading a very big power angle of VSC to transfer theelectricity L12 is very low due to high voltage of the transmission system leadingto strong coupling between the two VSCs When there is a large disturbance onone VSC there will be drastic nonlinear oscillations between the two VSCs whichis veried by the simulation results

A14 The Disturbance Model

As stated before we would like to study the nonlinear oscillations of the studiedcase under large disturbance So we need a disturbance model This is done bymaking Taylors Series Expansion around the equilibrium point AssumeV1 = const V2 = const Vg = const ωg = const

The equilibrium point ω10 ω20 δ10 δ20 As ωg = ω10 = ω20 = const so ω1 = ω1 +

ω10 ω2 = ω2+ω20dω1dt =

d(ω1minusωg)dt dω2

dt =d(ω2minusωg)

dt θ1minusθg = δ1 θ2minusθg = δ2 θ1minusθ2 =

δ1 minus δ2 = δ12 = δ1 + δ10 minus δ2 + δ20 = δ1 minus δ2 + δ120dω1dt = d(ω1+ω10)

dt = dω1dt

dω2dt =

d(ω2+ω20)dt = dω2

dt dδ1dt = d(δ1+δ10)

dt = dδ1dt = ωbω1

dδ2dt = d(δ2+δ20)

dt = dδ2dt = ωbω2 And as


Figure A6 How the three-loops system can be reduced to one-loop


Table A1 The parameters of the Interconnected VSCs

J1 16 L1(pu) 07 V1 1

J2 16 L2(pu) 07 V2 1

D1 10 L12(pu) 001 Vg 1

D2 10 wb 314

the system operates at the equilibrium pointthere will be

P lowast1 =V1VgL1

sin δ10 +V1V2L12

sin δ120 Plowast2 =

V1VgL2

sin δ20 +V1V2L12

sin δ210

Therefore the system can be expressed as Equation(A6)

J1ωb

d2δ1dt2

+ D1ωb

dδ1dt +

V1VgL1

(sin(δ1 + δ10)minus sin δ10) + V1V2L12

(sin(δ1 minus δ2 + δ120)minus sin δ120) = 0

J2ωb

d2δ2dt2

+ D2ωb

dδ2dt +

V2VgL2



(A6)To simulate the system under a large disturbance and then damped to the equilib-rium point we need to expand the power in the expression of sin(δ + δ0)minus sin(δ0)

There are two methods to do the Taylors Series Expansion around the equilibriumδ0bull a)Just keep the linear term

sin(δ + δ0)minus sin δ0 = cos δ0δ

bull b)Keep the quadratic and cubic termssin(δ + δ0)minus sin δ0 = cos δ0δ minus sin δ0

2 δ2 minus cos δ06 δ3

In our study we keep the nonlinearities to the cubic order In the previous re-searches concerning power systemonly quadratic terms are considered The type ofnonlinearities(quadratic or cubic) depends on the initial state δ10 δ20 Soby settingdierent values for δ10 δ20 we can study the impacts of the nonlinearities on thesystem dynamics The results of the disturbance model is put in next section formodal analysis

A2 Parameters and Programs

A21 Parameters of Interconnected VSCs

A22 Initial File for Interconnected VSCs

delta10=input(Type in the stable equibrium point delta10 )

delta20=input(Type in the stable equilibrium point delta20 )

disturbance1=input(Type in the disturbance around the equilibrium point disturbance1 )

A2 Parameters and Programs 217


J1=0216D1=20J2=0216D2=20w_b=100314

g1=J1w_b

g2=J2w_b

d1=D1w_b

d2=D2w_b

syms g1 g2 d1 d2

syms w_b delta10 delta20 P1 P2

delta120=delta10-delta20

delta210=-delta120

L1=07L2=07L12=001wg=1w_b=502314

AC sourceload parameters

V1=1V2=1Vg=1w10=1w20=1

k1=V1VgL1k2=V2VgL2k12=V1V2L12

compensation of springs efforts

Natural frequencies

M=[g1 00 g2]

K=[k1cos(delta10)+k12cos(delta120) -k12cos(delta120)

-k12cos(delta120) k2cos(delta20)+k12cos(delta210)]

D=[d1 00 d2]

modal matrix and eigenvalues matrix

[TrE]=eig(inv(M)K)

natural frequencies matrix

OMEGA=[sqrt(E(11)) sqrt(E(22))]

Normalisation of T

Tr=Trdet(Tr)

leg=sqrt(sum(Tr^2))

Tr=Trleg(ones(12))

normalization

[ Tn]=normalization(T)

modal matrices

Mm=TrMTr

Dm=TrDTr

Km=TrKTr

T11=Tr(11)

T12=Tr(12)

T21=Tr(21)

T22=Tr(22)

division by Mm


Dm_div=inv(Mm)Dm

OMEGA=double(OMEGA)

for i=12

xi(i)=05Dm_div(ii)OMEGA(i)

end

TEQ(111)=k1(-sin(delta10)2)+k12(-sin(delta120)2)

TEQ(112)=k12sin(delta120)

TEQ(122)=k12(-sin(delta120)2)




TEC(1111)=k1(-cos(delta10)6)+k12(-cos(delta120)6)

TEC(1112)=k12(cos(delta120)2)

TEC(1122)=k12(-cos(delta120)2)






Auxilliary tensors of quadratic and cubic coefficients

Gg=zeros(2222)

Gg(111)=TEQ(111)T11^2+TEQ(112)T11T21+TEQ(122)T21^2

Gg(112)=TEQ(111)2T11T12+TEQ(112)(T11T22+ T12T21)+TEQ(122)2T21T22


Hg(1111)=TEC(1111)T11^3+TEC(1112)T21T11^2+

TEC(1122)T11T21^2+TEC(1222)T21^3

Hg(1112)=TEC(1111)3T11^2T12+TEC(1112)(T22T11^2+2T21T11T12)+

TEC(1122)(T12T21^2+2T11T21T22)+TEC(1222)3T21^2T22

Hg(1122)=TEC(1111)3T11T12^2+TEC(1112)(2T22T11T12+T21T12^2)+

TEC(1122)(2T12T21T22+T11T22^2)+TEC(1222)3T21T22^2

Hg(1222)=TEC(1111)T12^3+TEC(1112)T22T12^2+

TEC(1122)T12T22^2+TEC(1222)T22^3




Hg(2111)=TEC(2111)T11^3+TEC(2112)T21T11^2+TEC(2122)T11T21^2+

TEC(2222)T21^3


TEC(2122)(T12T21^2+2T11T21T22)+TEC(2222)3T21^2T22


TEC(2122)(2T12T21T22+T11T22^2)+TEC(2222)3T21T22^2



TEC(2222)T22^3

G tensor of quadratic coefficients

H tensor of cubic coefficients

G=zeros(222)

H=zeros(2222)

G(111)=(T11Gg(111)T21Gg(211))Mm(11)

G(112)=(T11Gg(112)T21Gg(212))Mm(11)

G(122)=(T11Gg(122)T21Gg(222))Mm(11)

G(211)=(T12Gg(211)T22Gg(211))Mm(22)

G(212)=(T12Gg(212)T22Gg(212))Mm(22)

G(222)=(T12Gg(222)T22Gg(222))Mm(22)

for p=12

for k=12

for j=k2

G(pkj)=(Tr(1p)Gg(1kj)+Tr(2p)Gg(2kj))Mm(pp)

++j

end

++k

end

++p

end

for p=12

for i=12

for k=i2

for j=k2

H(pikj)=(Tr(1p)Hg(1ikj)+Tr(2p)Hg(2ikj))Mm(pp)

++j

end

++k

end

++i

end

++p

end

H=zeros(2222)

c1=[G(111) G(112) G(122) H(1111) H(1112) H(1122) H(1222)]

c2=[G(211) G(212) G(222) H(2111) H(2112) H(2122) H(2222)]

[ABCALPHABETAGAMMAAABBCCRSTULAMBDAMUNUZETA]=forcedcoef(OMEGAGHxi)

sim(Damped_free_vibration_interconnectedVSCs)


A23 M-le to Calculate the Coef

File calculating the coefs of the NNM considering a linear damping of the

system

callingsystem_parametersm OMEGA xi G H

calculating ABCALPHABETAGAMMAAABBCCRSTULAMBDAMUNUZETA

Created the 0705 by Ghofrane BH using coefdirect_dampc_XK_forceem

function [ABCALPHABETAGAMMAAABBCCRSTULAMBDAMUNUZETA]=

forcedcoef(OMEGAGHxi)

N=length(OMEGA)

correction=1

quadratic coefs

A=zeros(NNN)

B=zeros(NNN)

C=zeros(NNN)

ALPHA=zeros(NNN)

BETA=zeros(NNN)

GAMMA=zeros(NNN)

for p=1N

for i=1N

A(pii) = -(64OMEGA(i)^4xi(i)^4+8OMEGA(i)^4-

96OMEGA(i)^3xi(i)^3xi(p)OMEGA(p)-8OMEGA(i)^3xi(p)OMEGA(p)xi(i)

+20xi(i)^2OMEGA(i)^2OMEGA(p)^2+32OMEGA(i)^2xi(i)^2xi(p)^2OMEGA(p)^2-

6OMEGA(i)^2OMEGA(p)^2+8xi(p)^2OMEGA(p)^2OMEGA(i)^2-12xi(p)OMEGA(p)^3

xi(i)OMEGA(i)+OMEGA(p)^4)G(pii)(64OMEGA(i)^6xi(i)^2-32OMEGA(i)^4xi(i)^2

OMEGA(p)^2+64xi(i)^4OMEGA(i)^4OMEGA(p)^2+20xi(i)^2OMEGA(i)^2OMEGA(p)^4+

16OMEGA(i)^2OMEGA(p)^4xi(p)^2+16OMEGA(i)^4OMEGA(p)^2-8OMEGA(i)^2OMEGA(p)^4

+OMEGA(p)^6-128OMEGA(i)^5xi(i)^3xi(p)OMEGA(p)+192OMEGA(i)^4xi(i)^2

xi(p)^2OMEGA(p)^2-96xi(i)^3OMEGA(i)^3OMEGA(p)^3xi(p)-12xi(i)OMEGA(i)

OMEGA(p)^5xi(p)+32xi(i)^2OMEGA(i)^2OMEGA(p)^4xi(p)^2-64OMEGA(i)^5

xi(p)OMEGA(p)xi(i)-64OMEGA(i)^3xi(p)^3OMEGA(p)^3xi(i))

B(pii) = -2G(pii)(4OMEGA(i)^2+8xi(i)^2OMEGA(i)^2-OMEGA(p)^2-12

xi(p)OMEGA(p)xi(i)OMEGA(i)+4xi(p)^2OMEGA(p)^2)(64OMEGA(i)^6xi(i)^2-

32OMEGA(i)^4xi(i)^2OMEGA(p)^2+64xi(i)^4OMEGA(i)^4OMEGA(p)^2+20

xi(i)^2OMEGA(i)^2OMEGA(p)^4+16OMEGA(i)^2OMEGA(p)^4xi(p)^2+16

OMEGA(i)^4OMEGA(p)^2-8OMEGA(i)^2OMEGA(p)^4+OMEGA(p)^6-128OMEGA(i)^5

xi(i)^3xi(p)OMEGA(p)+192OMEGA(i)^4xi(i)^2xi(p)^2OMEGA(p)^2-96xi(i)^3

OMEGA(i)^3OMEGA(p)^3xi(p)-12xi(i)OMEGA(i)OMEGA(p)^5xi(p)+32xi(i)^2

OMEGA(i)^2OMEGA(p)^4xi(p)^2-64OMEGA(i)^5xi(p)OMEGA(p)xi(i)-64


OMEGA(i)^3xi(p)^3OMEGA(p)^3xi(i))

C(pii) = -4G(pii)(4OMEGA(i)^3xi(i)+16xi(i)^3OMEGA(i)^3-24xi(i)^2

OMEGA(i)^2xi(p)OMEGA(p)+xi(i)OMEGA(i)OMEGA(p)^2-xi(p)OMEGA(p)^3+

8xi(p)^2OMEGA(p)^2xi(i)OMEGA(i))(64OMEGA(i)^6xi(i)^2-32OMEGA(i)^4

xi(i)^2OMEGA(p)^2+64xi(i)^4OMEGA(i)^4OMEGA(p)^2+20xi(i)^2OMEGA(i)^2

OMEGA(p)^4+16OMEGA(i)^2OMEGA(p)^4xi(p)^2+16OMEGA(i)^4OMEGA(p)^2-

8OMEGA(i)^2OMEGA(p)^4+OMEGA(p)^6-128OMEGA(i)^5xi(i)^3xi(p)OMEGA(p)+

192OMEGA(i)^4xi(i)^2xi(p)^2OMEGA(p)^2-96xi(i)^3OMEGA(i)^3OMEGA(p)^3

xi(p)-12xi(i)OMEGA(i)OMEGA(p)^5xi(p)+32xi(i)^2OMEGA(i)^2OMEGA(p)^4

xi(p)^2-64OMEGA(i)^5xi(p)OMEGA(p)xi(i)-64OMEGA(i)^3xi(p)^3OMEGA(p)^3

xi(i))

GAMMA(pii)=2(A(pii) - OMEGA(i)^2B(pii)) -2xi(i)OMEGA(i)C(pii)

ALPHA(pii) = -OMEGA(i)^2C(pii)

BETA(pii) = -4xi(i)OMEGA(i)B(pii) + C(pii)

end

end

for p=1N

for i=1N

for j=i+1N

D=(OMEGA(i) + OMEGA(j) - OMEGA(p))(OMEGA(i) + OMEGA(j) + OMEGA(p))

(OMEGA(i)- OMEGA(j) + OMEGA(p))(OMEGA(i) - OMEGA(j) - OMEGA(p))

if abs(D) lt 1e-12

A(pij)=0

B(pij)=0

C(pij)=0

C(pji)=0

ALPHA(pij)=0

BETA(pij)=0

GAMMA(pij)=0

GAMMA(pji)=0

else

A2=[OMEGA(j)^2+OMEGA(i)^2-OMEGA(p)^2 -2OMEGA(j)^2OMEGA(i)^2 -2OMEGA(j)^3

xi(j)+2xi(p)OMEGA(p)OMEGA(j)^2 -2OMEGA(i)^3xi(i)+2xi(p)OMEGA(p)

OMEGA(i)^2 -2 -4xi(i)^2OMEGA(i)^2-8xi(i)OMEGA(i)xi(j)OMEGA(j)-4

xi(j)^2OMEGA(j)^2+OMEGA(i)^2+OMEGA(j)^2-OMEGA(p)^2+4xi(p)OMEGA(p)

xi(i)OMEGA(i)+4xi(p)OMEGA(p)xi(j)OMEGA(j) 2xi(i)OMEGA(i)+

4xi(j)OMEGA(j)-2xi(p)OMEGA(p) 4xi(i)OMEGA(i)+2xi(j)OMEGA(j)-2

xi(p)OMEGA(p) 2xi(j)OMEGA(j)-2xi(p)OMEGA(p) -2OMEGA(i)^3xi(i)-


4OMEGA(i)^2xi(j)OMEGA(j)+2xi(p)OMEGA(p)OMEGA(i)^2 OMEGA(j)^2+OMEGA(i)^2-

4xi(j)^2OMEGA(j)^2-OMEGA(p)^2+4xi(p)OMEGA(p)xi(j)OMEGA(j) 2OMEGA(i)^2

2xi(i)OMEGA(i)-2xi(p)OMEGA(p) -4OMEGA(j)^2xi(i)OMEGA(i)-2OMEGA(j)^3

xi(j)+2xi(p)OMEGA(p)OMEGA(j)^2 2OMEGA(j)^2 OMEGA(i)^2+OMEGA(j)^2-

4xi(i)^2OMEGA(i)^2-OMEGA(p)^2+4xi(p)OMEGA(p)xi(i)OMEGA(i)]

B2=[G(pij) 0 0 0]

solabcij=inv(A2)B2

solabcij=A2B2

A(pij)=solabcij(1)

B(pij)=solabcij(2)

C(pij)=solabcij(3)

C(pji)=solabcij(4)

ALPHA(pij) = -C(pij)OMEGA(j)^2 - C(pji)OMEGA(i)^2

BETA(pij) = -2(xi(i)OMEGA(i) + xi(j)OMEGA(j))B(pij)+ C(pij) + C(pji)

GAMMA(pij) = A(pij) - OMEGA(i)^2B(pij) - 2xi(j)OMEGA(j)C(pij)

GAMMA(pji) = A(pij) - OMEGA(j)^2B(pij) - 2xi(i)OMEGA(i)C(pji)

end

end

end

end

Calcul intermediaire AA BB et CC

AA=zeros(NNNN)

BB=zeros(NNNN)

CC=zeros(NNNN)

AA2=zeros(NNNN)

BB2=zeros(NNNN)

CC2=zeros(NNNN)

for p=1N

for i=1N

for j=1N

for k=1N

V1=squeeze(G(piiN))

V2=squeeze(A(iNjk))

V3=squeeze(G(p1ii))

V4=squeeze(A(1ijk))

AA(pijk)=V1V2 + V3V4

V5=squeeze(G(piiN))

V6=squeeze(B(iNjk))

V7=squeeze(G(p1ii))


V8=squeeze(B(1ijk))

BB(pijk)=V5V6 + V7V8

V9=squeeze(G(piiN))

V10=squeeze(C(iNjk))

V11=squeeze(G(p1ii))

V12=squeeze(C(1ijk))

CC(pijk)=V9V10 + V11V12

end

end

end

end

cubic coefs

R=zeros(NNNN)

S=zeros(NNNN)

T=zeros(NNNN)

U=zeros(NNNN)

LAMBDA=zeros(NNNN)

MU=zeros(NNNN)

NU=zeros(NNNN)

ZETA=zeros(NNNN)

treated in different loops to express all the particular cases

for p=1N

for i=1N

if i~=p

D1=(OMEGA(p)^2-OMEGA(i)^2)(OMEGA(p)^2-9OMEGA(i)^2)

if abs(D1) lt 1e-12

R(piii)=0

S(piii)=0

T(piii)=0

U(piii)=0

LAMBDA(piii)=0

MU(piii)=0

NU(piii)=0

ZETA(piii)=0

else

A3iii=[-OMEGA(p)^2+3OMEGA(i)^2 0 2xi(p)OMEGA(p)OMEGA(i)^2-

2OMEGA(i)^3xi(i) -2OMEGA(i)^4 0 -OMEGA(p)^2+12xi(p)OMEGA(p)


xi(i)OMEGA(i)+ 3OMEGA(i)^2-36xi(i)^2OMEGA(i)^2 -2 -2xi(p)OMEGA(p)+

10xi(i)OMEGA(i) -6xi(p)OMEGA(p)+6xi(i)OMEGA(i) -6OMEGA(i)^4

-OMEGA(p)^2+4xi(p)OMEGA(p)xi(i)OMEGA(i)+7OMEGA(i)^2-4xi(i)^2OMEGA(i)^2

4xi(p)OMEGA(p)OMEGA(i)^2-12OMEGA(i)^3xi(i) -6 6xi(p)OMEGA(p)

OMEGA(i)^2-30OMEGA(i)^3xi(i) -4xi(p)OMEGA(p)+12xi(i)OMEGA(i)

-OMEGA(p)^2+8xi(p)OMEGA(p)xi(i)OMEGA(i)+7OMEGA(i)^2-16xi(i)^2OMEGA(i)^2]

B3iii=[H(piii)+AA(piii) 0 CC(piii) BB(piii)]

sol3iii=inv(A3iii)B3iii

sol3iii=A3iiiB3iii

R(piii)=sol3iii(1)

S(piii)=sol3iii(2)

T(piii)=sol3iii(3)

U(piii)=sol3iii(4)

LAMBDA(piii) = -OMEGA(i)^2T(piii)

MU(piii) = U(piii) -6xi(i)OMEGA(i)S(piii)

NU(piii)=3R(piii)-2xi(i)OMEGA(i)T(piii)-2OMEGA(i)^2U(piii)

ZETA(piii)= -3OMEGA(i)^2S(piii)+2T(piii)-4xi(i)OMEGA(i)U(piii)

end

end

end

end

for p=1N

for i=1N-1

if i~=p

for j=i+1N

D2=(OMEGA(p)+OMEGA(i)-2OMEGA(j))(OMEGA(p)+OMEGA(i)+2OMEGA(j))

(-OMEGA(p)+OMEGA(i)+2OMEGA(j))(-OMEGA(p)+OMEGA(i)-2OMEGA(j))

if ((abs(D2) lt 1e-12) | (abs(OMEGA(i)^2-OMEGA(p)^2)lt1e-12))

R(pijj) = 0

S(pijj) = 0

T(pijj) = 0

T(pjij) = 0

U(pijj) = 0

U(pjij) = 0

LAMBDA(pijj)=0

MU(pijj) = 0

NU(pijj) =0

NU(pjij) =0

ZETA(pijj) = 0

b 21n132

ZETA(pjij) = 0

else


Aip=[-OMEGA(p)^2+2OMEGA(j)^2+OMEGA(i)^2 02xi(p)OMEGA(p)OMEGA(i)^2-

2OMEGA(i)^3xi(i) 2xi(p)OMEGA(p)OMEGA(j)^2-2OMEGA(j)^3xi(j)

-2OMEGA(j)^4 -2OMEGA(j)^2OMEGA(i)^2 0 -OMEGA(p)^2+4xi(p)

OMEGA(p)xi(i)OMEGA(i)+8xi(p)OMEGA(p)xi(j)OMEGA(j)+2OMEGA(j)^2+

OMEGA(i)^2-4xi(i)^2OMEGA(i)^2-16xi(i)OMEGA(i)xi(j)OMEGA(j)-

16xi(j)^2OMEGA(j)^2 -2 -2 -2xi(p)OMEGA(p)+8xi(j)OMEGA(j)+2

xi(i)OMEGA(i) -2xi(p)OMEGA(p)+4xi(i)OMEGA(i)+6xi(j)OMEGA(j)

-2xi(p)OMEGA(p)+2xi(i)OMEGA(i) -2OMEGA(j)^4 -OMEGA(p)^2+4xi(p)

OMEGA(p)xi(i)OMEGA(i)+OMEGA(i)^2+2OMEGA(j)^2-4xi(i)^2OMEGA(i)^2

2OMEGA(j)^2 0 2xi(p)OMEGA(p)OMEGA(j)^2-4OMEGA(j)^2xi(i)OMEGA(i)-

2OMEGA(j)^3xi(j) -4xi(p)OMEGA(p)+4xi(j)OMEGA(j) -4OMEGA(j)^2

OMEGA(i)^2 4OMEGA(i)^2 -OMEGA(p)^2+4xi(p)OMEGA(p)xi(j)OMEGA(j)+

4OMEGA(j)^2-4xi(j)^2OMEGA(j)^2+OMEGA(i)^2 4xi(p)OMEGA(p)

OMEGA(j)^2-12OMEGA(j)^3xi(j) 2xi(p)OMEGA(p)OMEGA(i)^2-4OMEGA(i)^2

xi(j)OMEGA(j)-2OMEGA(i)^3xi(i) -2 2xi(p)OMEGA(p)OMEGA(i)^2-

2OMEGA(i)^3xi(i)-8OMEGA(i)^2xi(j)OMEGA(j) 0 -2xi(p)OMEGA(p)+

6xi(j)OMEGA(j) -OMEGA(p)^2+8xi(p)OMEGA(p)xi(j)OMEGA(j)+OMEGA(i)^2+

2OMEGA(j)^2-16xi(j)^2OMEGA(j)^2 2OMEGA(i)^2 -4 -12OMEGA(j)^3

xi(j)+4xi(p)OMEGA(p)OMEGA(j)^2-8OMEGA(j)^2xi(i)OMEGA(i) 8xi(i)

OMEGA(i)+4xi(j)OMEGA(j)-4xi(p)OMEGA(p) 4xi(j)OMEGA(j)-2xi(p)

OMEGA(p)+2xi(i)OMEGA(i) 4OMEGA(j)^2 -OMEGA(p)^2-4xi(j)^2

OMEGA(j)^2+OMEGA(i)^2+4OMEGA(j)^2-8xi(i)OMEGA(i)xi(j)OMEGA(j)-

4xi(i)^2OMEGA(i)^2+4xi(p)OMEGA(p)xi(i)OMEGA(i)+4xi(p)OMEGA(p)

xi(j)OMEGA(j)]

Bip=[H(pijj)+AA(pijj)+AA(pjij) 0 CC(pjji) CC(pjij)+

CC(pijj) BB(pijj) BB(pjij)]

solrstuip=inv(Aip)Bip

solrstuip=AipBip

R(pijj) = solrstuip(1)

S(pijj) = solrstuip(2)

T(pijj) = solrstuip(3)

T(pjij) = solrstuip(4)

U(pijj) = solrstuip(5)

U(pjij) = solrstuip(6)

LAMBDA(pijj) = -OMEGA(j)^2T(pjij)-OMEGA(i)^2T(pijj)

ZETA(pijj) = -OMEGA(i)^2S(pijj)+T(pjij)-4xi(j)OMEGA(j)U(pijj)

NU(pijj) = R(pijj)-OMEGA(j)^2U(pjij)-2xi(i)OMEGA(i)T(pijj)

MU(pijj) = U(pjij)+U(pijj)-2xi(i)OMEGA(i)S(pijj)-

4xi(j)OMEGA(j)S(pijj)

NU(pjij) = 2R(pijj)-2xi(j)OMEGA(j)T(pjij)-OMEGA(i)^2U(pjij)-

2OMEGA(j)^2U(pijj)


ZETA(pjij) = -2OMEGA(j)^2S(pijj)+2T(pijj)+T(pjij)-

2xi(i)OMEGA(i)U(pjij)-2xi(j)OMEGA(j)U(pjij)

end

end

end

end

end

for p=1N

for i=1N-1

for j=(i+1)N

if j~=p

D3= (OMEGA(p)+2OMEGA(i)-OMEGA(j))(OMEGA(p)+2OMEGA(i)+OMEGA(j))

(-OMEGA(p)+2OMEGA(i)+OMEGA(j))(-OMEGA(p)+2OMEGA(i)-OMEGA(j))

if ((abs(D3) lt 1e-12) | (abs(OMEGA(p)^2 -OMEGA(j)^2)lt1e-12))

R(piij) =0

S(piij) =0

T(pjii) =0

T(piij) =0

U(pjii) =0

U(piij) =0

LAMBDA(piij)=0

NU(pjii) =0

NU(piij) =0

MU(piij) =0

ZETA(pjii) =0

ZETA(piij) =0

else

Ajpp=[-OMEGA(p)^2+OMEGA(j)^2+2OMEGA(i)^2 0 2xi(p)OMEGA(p)OMEGA(i)^2-


-2OMEGA(j)^2OMEGA(i)^2 -2OMEGA(i)^4 0 -OMEGA(p)^2+8xi(p)OMEGA(p)

xi(i)OMEGA(i)+4xi(p)OMEGA(p)xi(j)OMEGA(j)+OMEGA(j)^2+2OMEGA(i)^2-

16xi(i)^2OMEGA(i)^2-16xi(i)OMEGA(i)xi(j)OMEGA(j)-4xi(j)^2OMEGA(j)^2

-2 -2 -2xi(p)OMEGA(p)+6xi(i)OMEGA(i)+4xi(j)OMEGA(j)

-2xi(p)OMEGA(p)+8xi(i)OMEGA(i)+2xi(j)OMEGA(j)

-4xi(p)OMEGA(p)+4xi(i)OMEGA(i) -4OMEGA(j)^2OMEGA(i)^2 -OMEGA(p)^2+

4xi(p)OMEGA(p)xi(i)OMEGA(i)+4OMEGA(i)^2-4xi(i)^2OMEGA(i)^2+OMEGA(j)^2

4OMEGA(j)^2 2xi(p)OMEGA(p)OMEGA(j)^2-4OMEGA(j)^2xi(i)OMEGA(i)-

2OMEGA(j)^3xi(j) 4xi(p)OMEGA(p)OMEGA(i)^2-12OMEGA(i)^3xi(i)

-2xi(p)OMEGA(p)+2xi(j)OMEGA(j) -2OMEGA(i)^4 2OMEGA(i)^2 -OMEGA(p)^2+

4xi(p)OMEGA(p)xi(j)OMEGA(j)+OMEGA(j)^2-4xi(j)^2OMEGA(j)^2+2OMEGA(i)^2

2xi(p)OMEGA(p)OMEGA(i)^2-4OMEGA(i)^2xi(j)OMEGA(j)-2OMEGA(i)^3xi(i) 0

-4 -12OMEGA(i)^3xi(i)+4xi(p)OMEGA(p)OMEGA(i)^2-8OMEGA(i)^2xi(j)OMEGA(j)

4xi(i)OMEGA(i)+2xi(j)OMEGA(j)-2xi(p)OMEGA(p) 8xi(j)OMEGA(j)+4xi(i)OMEGA(i)


-4xi(p)OMEGA(p) -OMEGA(p)^2+OMEGA(j)^2+4OMEGA(i)^2-4xi(j)^2OMEGA(j)^2-


xi(j)OMEGA(j)-8xi(i)OMEGA(i)xi(j)OMEGA(j) 4OMEGA(i)^2 -2

2xi(p)OMEGA(p)OMEGA(j)^2-8OMEGA(j)^2xi(i)OMEGA(i)-2OMEGA(j)^3xi(j)

-2xi(p)OMEGA(p)+6xi(i)OMEGA(i) 0 2OMEGA(j)^2 -OMEGA(p)^2+8xi(p)

OMEGA(p)xi(i)OMEGA(i)+OMEGA(j)^2+2OMEGA(i)^2-16xi(i)^2OMEGA(i)^2]

Bjpp=[H(piij)+AA(piij)+AA(pjii) 0 CC(pjii)+CC(piji) CC(piij)

BB(piij) BB(pjii)]

solrstujpp=inv(Ajpp)Bjpp

solrstujpp=AjppBjpp

R(piij) =solrstujpp(1)

S(piij) =solrstujpp(2)

T(piij) =solrstujpp(3)

T(pjii) =solrstujpp(4)

U(piij) =solrstujpp(5)

U(pjii) =solrstujpp(6)

NU(pjii) = R(piij)-2xi(j)OMEGA(j)T(pjii)-OMEGA(i)^2U(piij)

ZETA(pjii) = -OMEGA(j)^2S(piij)+T(piij)-4xi(i)OMEGA(i)U(pjii)

LAMBDA(piij) = -OMEGA(j)^2T(pjii)-OMEGA(i)^2T(piij)

MU(piij) = U(pjii)+U(piij)-4xi(i)OMEGA(i)S(piij)-

2xi(j)OMEGA(j)S(piij)

NU(piij) = 2R(piij)-2xi(i)OMEGA(i)T(piij)-2OMEGA(i)^2U(pjii)-

OMEGA(j)^2U(piij)

ZETA(piij) = -2OMEGA(i)^2S(piij)+2T(pjii)+T(piij)-

2xi(i)OMEGA(i)U(piij)-2xi(j)OMEGA(j)U(piij)

end

end

end

end

end

for p=1N

for i=1N-2

for j=(i+1)N-1

for k=(j+1)N

D4=(OMEGA(p)+OMEGA(j)+OMEGA(k)-OMEGA(i))(-OMEGA(p)+OMEGA(j)-OMEGA(k)+

OMEGA(i))(OMEGA(p)+OMEGA(j)+OMEGA(k)+OMEGA(i))(-OMEGA(p)+OMEGA(j)-

OMEGA(k)-OMEGA(i))(OMEGA(p)+OMEGA(j)-OMEGA(k)-OMEGA(i))(-OMEGA(p)+OMEGA(j)+

OMEGA(k)+OMEGA(i))(OMEGA(p)+OMEGA(j)-OMEGA(k)+OMEGA(i))(-OMEGA(p)+OMEGA(j)+

OMEGA(k)-OMEGA(i))

if abs(D4) lt 1e-12

U(pijk) =0


U(pjik) =0

U(pkij) =0

R(pijk) =0

MU(pijk) =0

NU(pkij) =0

NU(pjik) =0

NU(pijk) =0

else

Alastijkp=[-OMEGA(p)^2+OMEGA(k)^2+OMEGA(j)^2+OMEGA(i)^2 0 2xi(p)

OMEGA(p)OMEGA(i)^2-2OMEGA(i)^3xi(i) 2xi(p)OMEGA(p)OMEGA(j)^2-

2OMEGA(j)^3xi(j) 2xi(p)OMEGA(p)OMEGA(k)^2-2OMEGA(k)^3xi(k)

-2OMEGA(k)^2OMEGA(j)^2 -2OMEGA(k)^2OMEGA(i)^2-2OMEGA(j)^2OMEGA(i)^2

0 OMEGA(i)^2+ OMEGA(j)^2-OMEGA(p)^2+OMEGA(k)^2+4xi(p)OMEGA(p)

xi(i)OMEGA(i)+4xi(p)OMEGA(p)xi(j)OMEGA(j)+4xi(p)OMEGA(p)xi(k)

OMEGA(k)-4xi(i)^2OMEGA(i)^2-4xi(j)^2OMEGA(j)^2-4xi(k)^2OMEGA(k)^2-

8xi(i)OMEGA(i)xi(j)OMEGA(j)-8xi(i)OMEGA(i)xi(k)OMEGA(k)-8xi(j)

OMEGA(j)xi(k)OMEGA(k) -2 -2 -2 4xi(j)OMEGA(j)+4xi(k)OMEGA(k)-

2xi(p)OMEGA(p)+2xi(i)OMEGA(i) 4xi(i)OMEGA(i)+4xi(k)OMEGA(k)-

2xi(p)OMEGA(p)+2xi(j)OMEGA(j) 4xi(i)OMEGA(i)+4xi(j)OMEGA(j)-

2xi(p)OMEGA(p)+2xi(k)OMEGA(k) -2xi(p)OMEGA(p)+2xi(i)OMEGA(i)

-2OMEGA(k)^2OMEGA(j)^2 -OMEGA(p)^2+4xi(p)OMEGA(p)xi(i)OMEGA(i)+

OMEGA(i)^2-4xi(i)^2OMEGA(i)^2+OMEGA(j)^2+OMEGA(k)^2 2OMEGA(j)^2

2OMEGA(k)^2 0 2xi(p)OMEGA(p)OMEGA(k)^2-4xi(i)OMEGA(i)OMEGA(k)^2-

2OMEGA(k)^3xi(k) 2xi(p)OMEGA(p)OMEGA(j)^2-4xi(i)OMEGA(i)

OMEGA(j)^2-2OMEGA(j)^3xi(j) -2xi(p)OMEGA(p)+2xi(j)OMEGA(j)

-2OMEGA(k)^2OMEGA(i)^2 2OMEGA(i)^2 -OMEGA(p)^2+4xi(p)OMEGA(p)

xi(j)OMEGA(j)+OMEGA(j)^2-4xi(j)^2OMEGA(j)^2+OMEGA(i)^2+OMEGA(k)^2

2OMEGA(k)^2 2xi(p)OMEGA(p)OMEGA(k)^2-4xi(j)OMEGA(j)OMEGA(k)^2-

2OMEGA(k)^3xi(k) 0 2xi(p)OMEGA(p)OMEGA(i)^2-4xi(j)OMEGA(j)

OMEGA(i)^2-2OMEGA(i)^3xi(i) -2xi(p)OMEGA(p)+2xi(k)OMEGA(k)

-2OMEGA(j)^2OMEGA(i)^2 2OMEGA(i)^2 2OMEGA(j)^2 -OMEGA(p)^2+4

xi(p)OMEGA(p)xi(k)OMEGA(k)+OMEGA(k)^2-4xi(k)^2OMEGA(k)^2+

OMEGA(i)^2+OMEGA(j)^2 2xi(p)OMEGA(p)OMEGA(j)^2-4xi(k)OMEGA(k)

OMEGA(j)^2-2OMEGA(j)^3xi(j) 2xi(p)OMEGA(p)OMEGA(i)^2-4xi(k)

OMEGA(k)OMEGA(i)^2-2OMEGA(i)^3xi(i) 0 -2 2xi(p)OMEGA(p)

OMEGA(i)^2-2OMEGA(i)^3xi(i)-4xi(j)OMEGA(j)OMEGA(i)^2-4xi(k)

OMEGA(k)OMEGA(i)^2 0 -2xi(p)OMEGA(p)+4xi(j)OMEGA(j)+2xi(k)

OMEGA(k) -2xi(p)OMEGA(p)+4xi(k)OMEGA(k)+2xi(j)OMEGA(j) -OMEGA(p)^2+

4xi(p)OMEGA(p)xi(j)OMEGA(j)+4xi(p)OMEGA(p)xi(k)OMEGA(k)+OMEGA(i)^2+

OMEGA(j)^2+OMEGA(k)^2-4xi(j)^2OMEGA(j)^2-8xi(j)OMEGA(j)xi(k)OMEGA(k)-

4xi(k)^2OMEGA(k)^2 2OMEGA(i)^2 2OMEGA(i)^2 -2 2xi(p)OMEGA(p)

OMEGA(j)^2-4xi(i)OMEGA(i)OMEGA(j)^2-2OMEGA(j)^3xi(j)-4xi(k)

OMEGA(k)OMEGA(j)^2 -2xi(p)OMEGA(p)+4xi(i)OMEGA(i)+2xi(k)OMEGA(k)

0 -2xi(p)OMEGA(p)+4xi(k)OMEGA(k)+2xi(i)OMEGA(i) 2OMEGA(j)^2


-OMEGA(p)^2+4xi(p)OMEGA(p)xi(i)OMEGA(i)+4xi(p)OMEGA(p)xi(k)

OMEGA(k)+OMEGA(j)^2+OMEGA(i)^2+OMEGA(k)^2-4xi(i)^2OMEGA(i)^2-8xi(i)

OMEGA(i)xi(k)OMEGA(k)-4xi(k)^2OMEGA(k)^2 2OMEGA(j)^2 -2

2xi(p)OMEGA(p)OMEGA(k)^2-4xi(i)OMEGA(i)OMEGA(k)^2-4xi(j)

OMEGA(j)OMEGA(k)^2-2OMEGA(k)^3xi(k) -2xi(p)OMEGA(p)+4xi(i)OMEGA(i)+

2xi(j)OMEGA(j) -2xi(p)OMEGA(p)+4xi(j)OMEGA(j)+2xi(i)OMEGA(i) 0

2OMEGA(k)^2 2OMEGA(k)^2 -OMEGA(p)^2+4xi(p)OMEGA(p)xi(i)OMEGA(i)+

4xi(p)OMEGA(p)xi(j)OMEGA(j)+OMEGA(k)^2+OMEGA(i)^2+OMEGA(j)^2-

4xi(i)^2OMEGA(i)^2-8xi(i)OMEGA(i)xi(j)OMEGA(j)-4xi(j)^2OMEGA(j)^2]

Blastijkp=[H(pijk)+AA(pijk)+AA(pkij)+AA(pjik) 0

CC(pkji)+CC(pjki) CC(pkij)+CC(pikj)

CC(pjik)+CC(pijk) BB(pijk) BB(pjik) BB(pkij)]

Am1=inv(Alastijkp)

solrstu=AlastijkpBlastijkp

R(pijk)=solrstu(1)

S(pijk)=solrstu(2)

T(pijk)=solrstu(3)

T(pjik)=solrstu(4)

T(pkij)=solrstu(5)

U(pijk)=solrstu(6)

U(pjik)=solrstu(7)

U(pkij)=solrstu(8)

NU(pkij)= R(pijk)-2xi(k)OMEGA(k)T(pkij)-OMEGA(i)^2U(pjik)-

OMEGA(j)^2U(pijk)

ZETA(pkij)= -OMEGA(k)^2S(pijk)+T(pjik)+T(pijk)-

2(xi(i)OMEGA(i)+xi(j)OMEGA(j))U(pkij)

LAMBDA(pijk)=-OMEGA(k)^2T(pkij)-OMEGA(j)^2T(pjik)-

OMEGA(i)^2T(pijk)

MU(pijk)=U(pkij)+U(pjik)+U(pijk)-2(xi(i)OMEGA(i)+xi(j)

OMEGA(j)+xi(k)OMEGA(k))S(pijk)

NU(pjik)=R(pijk)-2xi(j)OMEGA(j)T(pjik)-OMEGA(i)^2U(pkij)-

OMEGA(k)^2U(pijk)

NU(pijk)=R(pijk)-2xi(i)OMEGA(i)T(pijk)-OMEGA(j)^2U(pkij)-

OMEGA(k)^2U(pjik)

ZETA(pjik)=-OMEGA(j)^2S(pijk)+T(pkij)+T(pijk)-2(xi(i)OMEGA(i)+

xi(k)OMEGA(k))U(pjik)


ZETA(pijk)=-OMEGA(i)^2S(pijk)+T(pkij)+T(pjik)-2(xi(j)OMEGA(j)+

xi(k)OMEGA(k))U(pijk)

end

end

end

end

end

disp(end)

A3 Simulation Files

The simulation les can be found in httpsdrivegooglecomopenid=

0B2onGI-yt3FwRVZQbUFkcGl5enc with1 Damped_free_vibration_interconnectedVSCsmdl to study the free oscilla-

tions2 Forced_damped_vibration_interconnectedVSCsmdl to study the forced

oscillations

Appendix B

Inverse Nonlinear Transformationon Real-time

This part is dedicated to illustrate the real-time algorithm of inverting nonlineartransformation

B1 Theoretical Formulation

A nonlinear transformation is proposed to obtain the Normal Forms of the systemof equations which is proposed to render the simplest Normal Forms and reads

Xp = Rp +

Nsumi=1

Nsumjgei

(apijRiRj + bpijSiSj) +

Nsumi=1

Nsumj=1

cpijRiSj

+

Nsumi=1

Nsumjgei

Nsumkgej

(rpijkRiRjRk + spijkSiSjSk)

+

Nsumi=1

Nsumj=1

Nsumkgej

(tpijkSiRjRk + upijkRiSjSk)

Yp = Sp +

Nsumi=1

Nsumjgei

(αpijRiRj + βp

ijSiSj) +

Nsumi=1

Nsumj=1

γpijRiSj

+

Nsumi=1

Nsumjgei

Nsumkgej

(λpijkRiRjRk + microp

ijkSiSjSk)

+

Nsumi=1

Nsumj=1

Nsumkgej

(νpijkSiRjRk + ζpijkRiSjSk)

The diculty to implement in inverting the matrix is the nonlinearities Howeverin real-time in each step the Rp and Sp increase slightly Thus the real-time (RT)values can be approximated with great accuracy as XRT

p = Xinitp + 4Xp Y

RTp =

Y initp +4Yp RRTp = Rinitp +4Rp SRTp = Sinitp +4Sp Keeping terms up to degree4 and neglect terms with higher degrees Eq (B1) leads a linearization of thenonlinear transformation

231

232 Appendix B Inverse Nonlinear Transformation on Real-time

B2 Linearization of the Nonlinear Transformation

The linearization of the nonlinear transformation can be done as (B1)

4Xp = 4Rp +Nsumi=1

PR(1)pi4Ri +Nsumi=1

PS(1)pi4Si

Nsumi=1

PR(R2)pi4Ri +Nsumi=1

PS(S2)pi4Si

Nsumi=1

PR(S2)pi4Ri +

Nsumi=1

PS(R2)pi4Si

4Yp = 4Sp +Nsumi=1

QR(1)pi4Ri +Nsumi=1

QS(1)pi4Si

Nsumi=1

QR(R2)pi4Ri +Nsumi=1

QS(S2)pi4Si

Nsumi=1

QR(S2)pi4Ri +

Nsumi=1

QS(R2)pi4Si

with

PR(1)pi =isum

j=1

apjiRj +Nsumjgei

apijRj +Nsumj=1

cpijSj (B1)

PS(1)pi =isum

j=1

bpjiSj +Nsumjgei

bpijSj +Nsumj=1

cpjiRj

PR(R2)pi =

isumj=1

isumkgej

rpkjiRjRk +

isumj=1

Nsumkgei

rpjikRjRk +

Nsumjgei

Nsumkgej

rpijkRjRk

PS(S2)pi =

isumj=1

isumkgej

spkjiSjSk +

isumj=1

Nsumkgei

spjikSjSk +

Nsumjgei

Nsumkgej

spijkSjSk

PR(S2)pi =Nsumj=1

isumk=1

tpjkiSjRk +Nsumj=1

Nsumkgei

tpjikSjRk +Nsumj=1

Nsumkgej

upijkSjSk

PS(R2)pi =

Nsumj=1

isumk=1

upjkiRjSk +

Nsumj=1

Nsumkgei

upjikRjSk +

Nsumj=1

Nsumkgej

tpijkRjRk

B2 Linearization of the Nonlinear Transformation 233

QR(1)pi =isum

j=1

αpjiRj +Nsumjgei

αpijRj +Nsumj=1

γpijSj (B2)

QS(1)pi =

isumj=1

βpjiSj +

Nsumjgei

βpijSj +

Nsumj=1

γpjiRj

QR(R2)pi =

isumj=1

jsumkgej

λpkjiRjRk +

isumj=1

Nsumkgei

λpjikRjRk +

Nsumjgei

Nsumkgej

λpijkRjRk

QS(S2)pi =isum

j=1

isumkgej

micropkjiSjSk +isum

j=1

Nsumkgei

micropjikSjSk +Nsumjgei

Nsumkgej

micropijkSjSk

QR(S2)pi =Nsumj=1

isumk=1

νpjkiSjRk +Nsumj=1

Nsumkgei

νpjikSjRk +Nsumj=1

Nsumkgej

ζpijkSjSk

QS(R2)pi =

Nsumj=1

isumk=1

ζpjkiRjSk +

Nsumj=1

Nsumkgei

ζpjikRjSk +

Nsumj=1

Nsumkgej

νpijkRjRk

Let

XR = PR(1) + PR(R2) + PR(S2) XS = PS(1) + PS(R2) + PS(S2)

Y R = QR(1) +QR(R2) +QR(S2) Y S = QS(1) +QS(R2) +QS(S2)

(B3)

Then we obtain 4X

4Y

=

I +XR XS

Y R Y S

4R

4S

(B4)

and 4R

4S

=

I +XR XS

Y R Y S

minus1 4X

4Y

=

[NLRT

]minus1

4X

4Y

(B5)

The initial values can be searched o-line which will need dozens of steps And afterthat the change in the variable 4R4S can be calculated from 4X4Y at real-time And after the calculation matrix NLRT will be updated for the calculationat next sampling time


Appendix C

Kundurs 4 Machine 2 AreaSystem

C1 Parameters of Kundurs Two Area Four Machine

System

Table C1 Generator Data in PU on Machine Base

Parameter Value Parameter Value

Ra 00025 xl 0002

xd 180 τprimed0 80

xq 170 τprimeq0 040

xprimed 03 τ

primeprimed0 00

xprimeq 03 τ

primeprimed0 00

xprimeprimed 00 MVAbase 900

xprimeprimeq 00 H 65

Table C2 Power Flow Data for Case 1 and Case 2 in Steady State after the faultis cleared Load

Load Voltage Zshunt

Bus (pu230kv)

B5 1029 6 1975 P = 66584MW Q = minus35082Mvar

B6 0987 6 minus 4737 P = 144338MW Q = minus35002Mvar

235

236 Appendix C Kundurs 4 Machine 2 Area System

Table C3 Power Flow Data for Case 1 and Case 2 in Steady State after the faultis clearedGenerator

Generator Bus Terminal Voltage P Q

Type (pu20kv) (MW) (Mvar)

G1 PV 1006 1402 74750 4994

G2 swingbus 10006 00 40650 14074

G3 PV 1006 minus 5817 53925 11399

G4 PV 1006 minus 6611 52500 14314

Table C4 Mechanical Damping and Exciter Gain of Generators for case 1 case 2and case 3

GeneratorMechanical Damping Exciter Gain

Case 1 Case 2 Case 3 Case 1 Case 2 Case 3

Gen 1 15 35 35 150 150 240

Gen 2 15 35 35 150 150 240

Gen 3 11 2 2 150 150 240

Gen 4 10 2 2 150 150 240

C2 Programs and Simulation Files

This part of appendix is dedicated to the interconnected VSCsSection A1 illustrates how the reduced 4th order model is formulatedIn this part of Appendix the crucial les are presented to facilitate the readerscomprehension of the PhD dissertationAll the programs and les associated with interconnected VSCs can be found inhttpsdrivegooglecomopenid=0B2onGI-yt3FwdnBZQnJ6SUQ1UHM

C21 SEP Initialization

--file Reduce27thOrderEquationm --

Created by Tian TIAN PhD in EE 070916

This is a symboic program of IEEE 4 machine system composed of

C2 Programs and Simulation Files 237

the 4th order generator model+ 3rd oder Exciter model with AVR

clear all

close all

Load the system

The admittance matrix

clear all

This program is used to calculate the interconnected power system with

a number of Gn generators

Define the system

Gn=4 This is the 4 generator case

omega_s=120pi

delta_Syn_1=18477 delta_Syn_2=12081 delta_Syn_3=078541 delta_Syn_4=053684

omega_Syn_1=1omega_Syn_2=1omega_Syn_3=1 omega_Syn_4=1

elq_Syn_1=08652 elq_Syn_2=10067 elq_Syn_3=088605elq_Syn_4=091746

eld_Syn_1=06477 eld_Syn_2=033037 eld_Syn_3=05998 eld_Syn_4=052695

vm_Exc_1=103 vr3_Exc_1=0 vf_Exc_1=18658




Yt=[39604-395821i -39604+396040i 00000+00000i 00000+00000i

-39604+396040i 103914-424178i 00000+00000i 04578+41671i

00000+00000i 00000+00000i 39604-395821i -39604+396040i

00000+00000i 04578+41671i -39604+396040i 182758-431388i]

Gt=real(Yt)

Bt=imag(Yt)

The admittance matrix of the network reduced to the

internal bus of the generators

Parameters for mechanical part of generators

H=[65656565] Inertial constant

D=[353522] Mechanical damping

Parameters for electrical part of generators in steady state


Ra=ones(41)0 Xd=ones(41)180 Xq=ones(41)170

Stator resist ance Equivalent reactance in the d axis

Equivalent reactance in the q axis

Parameters for electromagentic transient process

Tld0=ones(41)80 Transient time constant in the d axis

Tlq0=ones(41)040 Transient time constant in the q axis

Xld=ones(41)03 Equivalent transient reactance in the d axis

Xlq=ones(41)03 Equivalent transient reactance in the q axis

Parameters for the exciters equipped with AVR(automatic voltage regulator)

K_A=[150150150150] Exciter gain

T_e=ones(41)001

T_1=ones(41)10

T_2=ones(41)1

T_r=ones(41)001 Delay for sampling of Vt

Define the variables for each generator

$delta_i$ $omega_i$ $eq$edEfdXE1XE2

deltaG = sym(deltaG [Gn-1 1])

delta=[deltaG0]+delta_Syn_4ones(Gn1)

omega= sym(omega [Gn 1])

elq= sym(elq [Gn 1])

eld= sym(eld [Gn 1])

Vf= sym(Vf [Gn 1])

Vm= sym(Vm [Gn 1])

Vr= sym(Vr [Gn 1])

Yr=zeros(2Gn2Gn)

for l=1Gn

for m=1Gn

Yr(2l-12m-1)=Gt(lm)

Yr(2l-12m)=-Bt(lm)

Yr(2l2m-1)=Bt(lm)

Yr(2l2m)=Gt(lm)

end

end


Transformation matrices

T1 from the $dq_i$ reference frame to the common reference frame

T2 from the bus terminal to the generator transient axes

T1 and T2 are defined in Chapter 3

for l=1Gn

T1(2l-12l-1)=sin(delta(l)) T1(2l-12l)=cos(delta(l))

T1(2l2l-1)=-cos(delta(l)) T1(2l2l)=sin(delta(l))

T2(2l-12l-1)=Ra(l) T2(2l-12l)=-Xlq(l)

T2(2l2l-1)=Xld(l) T2(2l2l)=Ra(l)

end

Y=inv(inv(Yr)+T29)

for l=1Gn

for m=1Gn

G1(lm)=Y(2l-12m-1)

B1(lm)=-Y(2l-12m)

B1(lm)=Y(2l2m-1)

G1(1m)=Y(2l2m)

end

end

Ydelta=inv(T1)YT1

for j=1Gn

eldq(2j-1)=eld(j)

eldq(2j)=elq(j)

end

Idq=Ydeltaeldq

Auxiliary variable--idiq

for j=1Gn

id(j)=Idq(2j-1)

iq(j)=Idq(2j)

end

for j=1Gn

Vt(j)=sqrt((elq(j)-Xld(j)id(j)9)^2+(eld(j)+Xlq(j)iq(j)9)^2)

Pe(j)=eld(j)id(j)+elq(j)iq(j)

elx(j)=sin(delta(j))eld(j)+cos(delta(j))elq(j)

ely(j)=-cos(delta(j))eld(j)+sin(delta(j))elq(j)

end

for j=1Gn


elxy(2j-11)=elx(j)

elxy(2j1)=ely(j)

end

Ixy=Yelxy

Initial conditions of the physical state variables

vf_Syn_1=18658 pm_Syn_1=74911 p_Syn_1=74911 q_Syn_1=071811




Pm=[pm_Syn_1pm_Syn_2pm_Syn_3pm_Syn_4]

Pe0=[p_Syn_1p_Syn_2p_Syn_3p_Syn_4]

Qe0=[q_Syn_1q_Syn_2q_Syn_3q_Syn_4]

Vref=zeros(Gn1)

Vref_Exc_1=103

Vref_Exc_2=101

Vref_Exc_3=103

Vref_Exc_4=101

Vref=[Vref_Exc_1Vref_Exc_2Vref_Exc_3Vref_Exc_4]

delta0=[delta_Syn_1-delta_Syn_4delta_Syn_2-delta_Syn_4delta_Syn_3-delta_Syn_4]

omega0=[omega_Syn_1omega_Syn_2omega_Syn_3omega_Syn_4]

elq0=[elq_Syn_1elq_Syn_2elq_Syn_3elq_Syn_4]

eld0=[eld_Syn_1eld_Syn_2eld_Syn_3eld_Syn_4]

Vm0=[vm_Exc_1vm_Exc_2vm_Exc_3vm_Exc_4]

Vr0=[vr3_Exc_1vr3_Exc_2vr3_Exc_3vr3_Exc_4]

Vf0=[vf_Exc_1vf_Exc_2vf_Exc_3vf_Exc_4]

for j=1Gn-1

ddelta_Gn(j)=omega_s(omega(j)-omega(Gn))

domega(j)=1(2H(j))(Pm(j)9-Pe(j)9-D(j)(omega(j)-1))

delq(j)=1Tld0(j)(Vf(j)-(Xd(j)-Xld(j))id(j)9-elq(j))

deld(j)=1Tlq0(j)((Xq(j)-Xlq(j))9iq(j)-eld(j))

dVm(j)=(Vt(j)-Vm(j))T_r(j)

dVr(j)=(K_A(j)(1-T_1(j)T_2(j))(Vref(j)-Vm(j))-Vr(j))T_2(j)

dVf(j)=(Vr(j)+K_A(j)T_1(j)T_2(j)(Vref(j)-Vm(j))+Vf0(j)-Vf(j))T_e(j)

end

for j=Gn








end

for j=1Gn-1

X27(7j-67j)=[deltaG(j)omega(j)elq(j)eld(j)Vm(j)Vr(j)Vf(j)]

F27(7j-67j)=[ddelta_Gn(j)domega(j)delq(j)deld(j)dVm(j)dVr(j)dVf(j)]

end

for j=Gn

X27(7j-67j-1)=[omega(j)elq(j)eld(j)Vm(j)Vr(j)Vf(j)]

F27(7j-67j-1)=[domega(j)delq(j)deld(j)dVm(j)dVr(j)dVf(j)]

end

---- Input of the operating point obtained from the power flow analysis

run by the software PSAT ----------

for j=1Gn-1

x027(7j-67j)=[delta0(j)omega0(j)elq0(j)eld0(j)Vm0(j)Vr0(j)Vf0(j)]

end

for j=Gn

x027(7j-67j-1)=[omega0(j)elq0(j)eld0(j)Vm0(j)Vr0(j)Vf0(j)]

end

x027=x027

Fv_R=vpa(F27)

FvS_R=simplify(Fv_R)

TS_R=taylor(FvS_RX27x027Order4)

TStest=subs(TSXX+x0)

x0test=zeros(281)

[F1t_RF2t_RF3t_R]=EqNl2Matrix(TS_RX27x027)

[F1tF2tF3t]=EqNl2Matrix(F27X27x027)

EIV_R=eig(F1t_R)

[TuLambdaF2N_RF3N_R]=NLcoefJordan(F1t_RF2t_RF3t_R)

C211 Doing the Taylors Expansion Series

----function [F1F2F3]=EqNl2Matrix(FXx0)----

Created by Tian TIAN PhD in EE ENSAM Lille 2017


function [F1F2F3]=EqNl2Matrix(FXx0)

This functions is used to do the taylors expansion of

a set of nonlinear functions of multi-variables

X=[x1x2xn] around the Equilibrium Point x0=[x10x20xn0]

ie

F(x)=F(x0)+F1(X-x0)+F2(X-x0)^2+F3(X-x0)^3+terms higher than 3rd-order

Input the nonlinear equations--F the variable vector--X the equilibrium point-- x0

Output first-order derivative matrix--F1

second derivative matrix-- F2 (05Hessin matrix)

third-order derivative matrix-- F3(16 H3 matrix)

In fact this function can be extended to calculate the Taylors series

coefficients matrix to order higher than 3 such as n+1th order

What is needed is to derivate the matrix n-th order derivative matrix

repetitively until to the expected order

ie H(n+1)(pqrn)=jacobian(H(n)(pqrn)X)

TS3=taylor(FXx0Order3)

L=length(X) Get the length of the matrix

J = jacobian(FX) Jocobian matrix obtaining the first-order derivative

N1 = double(subs(JXx0)) Obtain the matrix F1 at the equilibirum point x0

for p=1L

H2(p) = hessian(F(p)X)

end

for p=1L

for k=1L

H3(pk)=hessian(J(pk)X)

end

end

N2=double(subs(H2Xx0))


F1=N1

F2=05N2 Return the 05 Hessian Matrix

F3=16N3 Return the 16 Tensor Matrix

C212 Calculating the Nonlinear Matrices in Jordan Form


function [TuLambdaF2F3]=NLcoefJordan(AN2N3)


This function is used to change the nonlinear matrix of

the original system dynamics in the jordan coordinate

ie form dotx=Ax+N2(x)+N3(x)

to doty=lambda y+F2(y)+F3(y)

L=length(A)get the dimension of the system of equations

[TuLambda]=eig(A)

Tu=Tu(det(Tu))^(127)

invTu=inv(Tu)

Calculate F2

F21=zeros(LLL)

F2=zeros(LLL)

for p=1L

F2(p)=invTu(Tusqueeze(N2(p))Tu)

F21(p)=Tusqueeze(N2(p))Tu

end

for p=1L

for j=1L

for k=1L

F2(pjk)=invTu(p)F21(jk)

end

end

end

Calculate F3

F31=zeros(LLLL)

F3=zeros(LLLL)

for j=1L

for p=1L

for q=1L

for r=1L

for l=1L

for m=1L

for n=1L

F31(jpqr)=F31(jpqr)+N3(jlmn)Tu(lp)Tu(mq)Tu(nr)

end

end

end

end

end

end


end

for j=1L

for p=1L

for q=1L

for r=1L

F3(jpqr)=invTu(j)F31(pqr)

end

end

end

end

C3 Calculation of Normal Form Coecients

C31 Calculating the NF Coecients h2jkl of Method 2-2-1


function h2=classic_coef(EIVCG)

File calculating the coefs of the normal from transformation (VF)

N=length(EIV)

quadratic coefs

Cla=zeros(NNN)

for p=1N

for j=1N

for k=1N

Cla(pjk) = CG(pjk)(EIV(j) + EIV(k)-EIV(p))

end

end

end

disp(end to find the coefficients )

end

subsectionFinding the Oscillatory Modes

beginverbatim


function ModeNumber=OsciModeNO(EIV)

This function is to get the numerical order of oscillatory modes from

the eigenvalue vector

ModeNumber=[]

This vector is used to record the numerical order

C3 Calculation of Normal Form Coecients 245

of the oscillatory modes

m=0

ModeSize=length(EIV)

for j=1ModeSize

if abs(imag(EIV(j)))gt1e-3

m=m+1

ModeNumber(m)=j

end

end

C32 Calculation the NF Coecients of Method 3-3-3

function [h2 h3 C3 F2H2 c3]=NF_coef333R(EIVF2F3ModeNumber)


L=length(EIV)

M=length(ModeNumber)

MN=ModeNumber

N1718=0

xdot=zeros(L1)

c3=zeros(LL)

term_3=zeros(L1)

F2H2=zeros(LLLL)

F2c=zeros(LL)

C3=zeros(LLLL)

h2=zeros(LLL)

h3=zeros(LLLL)

for p=1M

for l=1M

for m=1M

for n=1M

h2(MN(p)MN(l)MN(m))= F2(MN(p)MN(l)MN(m))(EIV(MN(l))+EIV(MN(m))-EIV(MN(p)))

end

end

end

end

for p=1L

for l=1L

for k=1L


F2c(pl)=F2c(pl)+F2(plk)+F2(pkl)

end

for m=1L

for n=1L

F2H2(plmn)=F2c(pl)h2(pmn)

C3(plmn)=F2H2(plmn)+F3(plmn)

end

end

end

end

for i=1(M2)

for j=1(M2)

if j~=i

c3(MN(2i)MN(2j))=C3(MN(2i)MN(2i)MN(2j)MN(2j-1))+

C3(MN(2i)MN(2j)MN(2i)MN(2j-1))+

C3(MN(2i)MN(2j)MN(2j-1)MN(2i))+

C3(MN(2i)MN(2i)MN(2j-1)MN(2j))+

C3(MN(2i)MN(2j-1)MN(2i)MN(2j))+

C3(MN(2i)MN(2j-1)MN(2j)MN(2i))

c3(MN(2i-1)MN(2j))=C3(MN(2i-1)MN(2i)-1MN(2j)MN(2j-1))+

C3(MN(2i-1)MN(2j)MN(2i-1)MN(2j-1))+

C3(MN(2i-1)MN(2j)MN(2j-1)MN(2i-1))+

C3(MN(2i-1)MN(2i-1)MN(2j-1)MN(2j))+

C3(MN(2i-1)MN(2j-1)MN(2i-1)MN(2j))+

C3(MN(2i-1)MN(2j-1)MN(2j)MN(2i-1))

else

c3(MN(2i)MN(2i))=C3(MN(2i)MN(2i)MN(2i)MN(2i-1))+

C3(MN(2i)MN(2i)MN(2i-1)MN(2i))+

C3(MN(2i)MN(2i-1)MN(2i)MN(2i))

c3(MN(2i-1)MN(2i))=C3(MN(2i-1)MN(2i-1)MN(2i)MN(2i-1))+

C3(MN(2i-1)MN(2i-1)MN(2i-1)MN(2i))+

C3(MN(2i-1)MN(2i)MN(2i-1)MN(2i-1))

xdot(2i-1)=conj(x(2i))

end

end

end

for p=1M

for l=1M

for m=1M

for n=1M


h3(MN(p)MN(l)MN(m)MN(n))= C3(MN(p)MN(l)MN(m)MN(n))

(EIV(MN(l))+EIV(MN(m))+EIV(MN(n))-EIV(MN(p)))

end

end

end

end

for k=1(M2)

for l=1(M2)

for m=1(M2)

for n=1(M2)

if l~=k

h3(MN(2k)MN(2k)MN(2l)MN(2l-1))=0h3(MN(2k)MN(2l)MN(2k)MN(2l-1))=0

h3(MN(2k)MN(2l)MN(2l-1)MN(2k))=0h3(MN(2k)MN(2k)MN(2l-1)MN(2l))=0

h3(MN(2k)MN(2l-1)MN(2k)MN(2l))=0h3(MN(2k)MN(2l-1)MN(2l)MN(2k))=0

h3(MN(2k-1)MN(2k-1)MN(2l)MN(2l-1))=0

h3(MN(2k-1)MN(2l)MN(2k-1)MN(2l-1))=0

h3(MN(2k-1)MN(2l)MN(2l-1)MN(2k-1))=0

h3(MN(2k-1)MN(2k-1)MN(2l-1)MN(2l))=0

h3(MN(2k-1)MN(2l-1)MN(2k-1)MN(2l))=0

h3(MN(2k-1)MN(2l-1)MN(2l)MN(2k-1))=0

else

h3(MN(2k)MN(2k)MN(2k)MN(2k-1))=0

h3(MN(2k)MN(2k)MN(2k-1)MN(2k))=0

h3(MN(2k)MN(2k-1)MN(2k)MN(2k))=0

h3(MN(2k-1)MN(2k-1)MN(2k)MN(2k-1))=0

h3(MN(2k-1)MN(2k)MN(2k-1)MN(2k-1))=0

h3(MN(2k-1)MN(2k-1)MN(2k-1)MN(2k))=0

end

end

end

end

end

disp(-------------------------End to find the coefficients for method 3-3-3------------------------)

end

C33 Calculation of Nonlinear Indexes MI3

--file calculationnonlinearindexesm--


Created by Tian TIAN 30012017

This file is used to calculate the files for h2 h3 in the proposed 3-3-3 method

function [h2h3C3F2H2c3]=NonlinearIndex(EIVF2F3ModeNumber)


L=length(EIV_R)The number of modes

M=length(ModeNumber)The number of oscillatory modes

MN=ModeNumber

M=L2

EIV=EIV_R

w=imag(EIV)

sigma=real(EIV)

F2=F2N_R

F3=F3N_R

quadratic coefs

h3=h3test_R

for j=1L

for p=1L

h3(jppp)=h3(jppp)

for q=(p+1)L

h3(jpqq)=h3(jpqq)+h3(jqpq)+h3(jqqp)

for r=(q+1)L

h3(jpqr)=h3(jpqr)+h3(jprq)+h3(jrpq)+h3(jqpr)+

h3(jqrp)+h3(jrqp)

end

end

end

end

for j=1L

for p=1L

for q=1L

for r=1L

MI333(jpqr)=abs(h3(jpqr)z0333(p)z0333(q)z0333(r))abs(z0333(j))

end

end

end

end

MaxMI333=zeros(277)

for j=1L

[Ch3Ih3] = max(MI333(j))

[Ih3_1Ih3_2Ih3_3] = ind2sub([27 27 27]Ih3)

MaxMI333(j)=[jIh3_1Ih3_2Ih3_3EIV_R(j)(EIV_R(Ih3_1)+EIV_R(Ih3_2)+EIV_R(Ih3_3))


MI3(jIh3_1Ih3_2Ih3_3)]

end

disp(-----------End to calculate the nonlinear indexes for method 3-3-3

------------------------)

C34 Normal Dynamics

Solving the Normal Dynamics the solution of the normal forms will be obtainedwhich contains the properties of the system in the normal form coordinates andwhose reconstruction by the NF transformation can approximate the original systemdynamics

C341 Normal Dynamics for method 3-2-3S and method 3-3-3


function xdot=NF_333(txCEc3)

L=length(CE)

xdot=zeros(L1)

term_3=zeros(L1)

Cam=zeros(42)

F2H2=zeros(LLLL)

F2c=zeros(LL)

C3=zeros(LLLL)

for i=1(L2)

for j=1(L2)

term_3(2i)=term_3(2i)+c3(2ij)x(2i)x(2j)x(2j-1)

term_3(2i-1)=term_3(2i-1)+c3(2i-1j)x(2i-1)x(2j)x(2j-1)


end

end

for p=1L

xdot(p)=CE(pp)x(p)+term_3(p)

end

end

C342 Normal Dynamics for method 3-3-1 and 2-2-1

function xdot=NF_1(txCE)


L=length(CE)

xdot=zeros(L1)

for p=1L

xdot(p)=CE(pp)x(p)

end

end

C35 Search for Initial Conditions

C351 Newton Raphsons Method [67]

function [x resnorm F exitflag output jacob] = newtonraphson(funx0options)

NEWTONRAPHSON Solve set of non-linear equations using Newton-Raphson method

[X RESNORM F EXITFLAG OUTPUT JACOB] = NEWTONRAPHSON(FUN X0 OPTIONS)

FUN is a function handle that returns a vector of residuals equations F

and takes a vector x as its only argument When the equations are

solved by x then F(x) == zeros(size(F() 1))

Optionally FUN may return the Jacobian Jij = dFidxj as an additional

output The Jacobian must have the same number of rows as F and the same

number of columns as x The columns of the Jacobians correspond to ddxj and

the rows correspond to dFid

EG J23 = dF2dx3 is the 2nd row ad 3rd column

If FUN only returns one output then J is estimated using a center

difference approximation

Jij = dFidxj = (Fi(xj + dx) - Fi(xj - dx))2dx

NOTE If the Jacobian is not square the system is either over or under

constrained

X0 is a vector of initial guesses

OPTIONS is a structure of solver options created using OPTIMSET

EG options = optimset(TolX 0001)

The following options can be set

OPTIONSTOLFUN is the maximum tolerance of the norm of the residuals

[1e-6]


OPTIONSTOLX is the minimum tolerance of the relative maximum stepsize

[1e-6]

OPTIONSMAXITER is the maximum number of iterations before giving up

[100]

OPTIONSDISPLAY sets the level of display off iter

[iter]

X is the solution that solves the set of equations within the given tolerance

RESNORM is norm(F) and F is F(X) EXITFLAG is an integer that corresponds to

the output conditions OUTPUT is a structure containing the number of

iterations the final stepsize and exitflag message and JACOB is the J(X)

See also OPTIMSET OPTIMGET FMINSEARCH FZERO FMINBND FSOLVE LSQNONLIN

References

httpenwikipediaorgwikiNewtons_method

httpenwikipediaorgwikiNewtons_method_in_optimization

97 Globally Convergent Methods for Nonlinear Systems of Equations 383

Numerical Recipes in C Second Edition (1992)

httpwwwnrbookcomabookcpdfphp

Version 05

allow sparse matrices replace cond() with condest()

check if Jstar has NaN or Inf return NaN or Inf for cond() and return

exitflag -1 matrix is singular

fix bug max iteration detection and exitflag reporting typos

Version 04

allow lsq curve fitting type problems IE non-square matrices

exit if J is singular or if dx is NaN or Inf

Version 03

Display RCOND each step

Replace nargout checking in funwrapper with ducktypin

Remove Ftyp and F scaling bc F(typx)-gt0 amp FFtyp-gtInf

User Numerical Recipies minimum Newton step backtracking line search

with alpha = 1e-4 min_lambda = 01 and max_lambda = 05

Output messages exitflag and min relative step

Version 02

Remove òptionsFinDiffRelStep` and òptionsTypicalX` since not in MATLAB

Set `dx = eps^(13)` in `jacobian` function

Remove òptions` argument from `funwrapper` amp `jacobian` functions

since no longer needed

Set typx = x0 typx(x0==0) = 1 use initial guess as typx if 0 use 1

Replace `feval` with èvalf` since `feval` is builtin


initialize

There are no argument checks

x0 = x0() needs to be a column vector

set default options

oldopts = optimset(

TolX 1e-8 TolFun 1e-5 MaxIter 2000 Display iter)

if narginlt3

options = oldopts use defaults

else

options = optimset(oldopts options) update default with user options

end

FUN = (x)funwrapper(fun x) wrap FUN so it always returns J

get options

TOLX = optimget(options TolX) relative max step tolerance

TOLFUN = optimget(options TolFun) function tolerance

MAXITER = optimget(options MaxIter) max number of iterations

DISPLAY = strcmpi(iter optimget(options Display)) display iterations

TYPX = max(abs(x0) 1) x scaling value remove zeros

ALPHA = 1e-4 criteria for decrease

MIN_LAMBDA = 01 min lambda

MAX_LAMBDA = 05 max lambda

set scaling values

TODO let user set weights

weight = ones(numel(FUN(x0))1)

J0 = weight(1TYPX) Jacobian scaling matrix

set display

if DISPLAY

fprintf(n10s 10s 10s 10s 10s 12sn Niter resnorm stepnorm

lambda rcond convergence)

for n = 167fprintf(-)endfprintf(n)

fmtstr = 10d 104g 104g 104g 104g 124gn

printout = (n r s l rc c)fprintf(fmtstr n r s l rc c)

end

check initial guess

x = x0 initial guess

[F J] = FUN(x) evaluate initial guess

Jstar = JJ0 scale Jacobian

if any(isnan(Jstar())) || any(isinf(Jstar()))

exitflag = -1 matrix may be singular

else

exitflag = 1 normal exit

end

if issparse(Jstar)

rc = 1condest(Jstar)


else

if any(isnan(Jstar()))

rc = NaN

elseif any(isinf(Jstar()))

rc = Inf

else

rc = 1cond(Jstar) reciprocal condition

end

end

resnorm = norm(F) calculate norm of the residuals

dx = zeros(size(x0))convergence = Inf dummy values

solver

Niter = 0 start counter

lambda = 1 backtracking

if DISPLAYprintout(Niter resnorm norm(dx) lambda rc convergence)end

while (resnormgtTOLFUN || lambdalt1) ampamp exitflaggt=0 ampamp Niterlt=MAXITER

if lambda==1

Newton-Raphson solver

Niter = Niter+1 increment counter

dx_star = -JstarF calculate Newton step

NOTE use isnan(f) || isinf(f) instead of STPMAX

dx = dx_starTYPX rescale x

g = FJstar gradient of resnorm

slope = gdx_star slope of gradient

fold = FF objective function

xold = x initial value

lambda_min = TOLXmax(abs(dx)max(abs(xold) 1))

end

if lambdaltlambda_min

exitflag = 2 x is too close to XOLD

break

elseif any(isnan(dx)) || any(isinf(dx))


break

end

x = xold+dxlambda next guess

[F J] = FUN(x) evaluate next residuals

Jstar = JJ0 scale next Jacobian

f = FF new objective function

check for convergence

lambda1 = lambda save previous lambda

if fgtfold+ALPHAlambdaslope

if lambda==1

lambda = -slope2(f-fold-slope) calculate lambda


else

A = 1(lambda1 - lambda2)

B = [1lambda1^2-1lambda2^2-lambda2lambda1^2lambda1lambda2^2]

C = [f-fold-lambda1slopef2-fold-lambda2slope]

coeff = num2cell(ABC)

[ab] = coeff

if a==0

lambda = -slope2b

else

discriminant = b^2 - 3aslope

if discriminantlt0

lambda = MAX_LAMBDAlambda1

elseif blt=0

lambda = (-b+sqrt(discriminant))3a

else

lambda = -slope(b+sqrt(discriminant))

end

end

lambda = min(lambdaMAX_LAMBDAlambda1) minimum step length

end

elseif isnan(f) || isinf(f)

limit undefined evaluation or overflow


else

lambda = 1 fraction of Newton step

end

if lambdalt1

lambda2 = lambda1f2 = f save 2nd most previous value

lambda = max(lambdaMIN_LAMBDAlambda1) minimum step length

continue

end

display

resnorm0 = resnorm old resnorm

resnorm = norm(F) calculate new resnorm

convergence = log(resnorm0resnorm) calculate convergence rate

stepnorm = norm(dx) norm of the step



break

end

if issparse(Jstar)


else



end

if DISPLAYprintout(Niter resnorm stepnorm lambda1 rc convergence)end

end

output

outputiterations = Niter final number of iterations

outputstepsize = dx final stepsize

outputlambda = lambda final lambda

if Nitergt=MAXITER

exitflag = 0

outputmessage = Number of iterations exceeded OPTIONSMAXITER

elseif exitflag==2

outputmessage = May have converged but X is too close to XOLD

elseif exitflag==-1

outputmessage = Matrix may be singular Step was NaN or Inf

else

outputmessage = Normal exit

end

jacob = J

end

function [F J] = funwrapper(fun x)

if nargoutlt2 use finite differences to estimate J

try

[F J] = fun(x)

catch

F = fun(x)

J = jacobian(fun x) evaluate center diff if no Jacobian

end

F = F() needs to be a column vector

end

function J = jacobian(fun x)

estimate J

dx = eps^(13) finite difference delta

nx = numel(x) degrees of freedom

nf = numel(fun(x)) number of functions

J = zeros(nfnx) matrix of zeros

for n = 1nx

create a vector of deltas change delta_n by dx

delta = zeros(nx 1) delta(n) = delta(n)+dx

dF = fun(x+delta)-fun(x-delta) delta F

J( n) = dF()dx2 derivatives dFd_n

end

end


C352 Mismatch function for Method 3-3-3 and Method 3-3-1

Created by Tian TIAN PhD in EE ENSAM Lille

function f=fzinit3(zh2h3y0)

Reconstruction the system dynamics(SD) from the normal dynamics(ND)

From the z coordinate to the x coordinate

dynamics=linear system+second order dynamics+non-resonant 3rd order dynamics+

resonant 3rd order dynamics

from z to y y=z+h2(z)+h3(z)+c3(z) for y to x x=Uy

The input is the solution in the time-series

the time length of data(discretation)tt

N=length(tt)

L=length(y0)

ynl=zeros(L1)

ynl3=zeros(L1)

H3=zeros(L1)

for l=1L

for p=1L

for q=1L

for r=1L

H3(l)=H3(l)+h3(1pqr)z(p)z(q)z(r)

end

end

end

end

for l=1L

ynl3(l)=z(l)+zsqueeze(h2(l))z+H3(l)

end

f=ynl3-y0

C353 Mismatch Equation for method 2-2-1 and 3-2-3S


function f=fzinit2(zh2y0)








N=length(tt)


L=length(y0)

ynl=zeros(L1)

ynl3=zeros(L1)

for l=1L

ynl3(l)=z(l)+zsqueeze(h2(l))z

end

f=ynl3-y0

C4 Simulation les


0B2onGI-yt3FwdnBZQnJ6SUQ1UHM withbull power_PSS3_unbalancedslx to assess the performance of the pre-sented methods by numerical step-by-step integration based on the demopower_PSS


Appendix D

New England New York 16Machine 5 Area System

This part of appendix is dedicated to New England New York 16 Machine 5 AreaSystemIn this part of Appendix the crucial les are presented to facilitate the readerscomprehension of the PhD dissertationAll the programs and les associated with interconnected VSCs can be found inhttpsdrivegooglecomopenid=0B2onGI-yt3FwdnBZQnJ6SUQ1UHM

D1 Programs

Initial File of New England New York 16 Machine 5 Area System --Init_16m_NFanalysism --

This is the initi alization and modal analysis file for

the 68-Bus Benchmark system with 16-machines and

86-lines It requires following files to successfully run

1 calcm

2 chq_limm

3 data16m_benchmarkm

4 form_jacm

5 loadflowm

6 y_sparsem

7 Benchmark_IEEE_standardmdl

Version 33

Authors Abhinav Kumar Singh Bikash C Pal

Affiliation Imperial College London

Date December 2013

Largely modified by Tian TIAN PhD in EE ENSAM Lille 2017

to obtain the nonlinear matrices in Jordan Form in preparation

for the Normal Form Analysis

clear all

clc

MVA_Base=1000

f=600

deg_rad = pi1800 degree to radian

259

260 Appendix D New England New York 16 Machine 5 Area System

rad_deg = 1800pi radian to degree

j=sqrt(-1)

Load Data

data16m_benchmark

N_Machine=size(mac_con1)

N_Bus=size(bus1)

N_Line=size(line1)

Loading Data Ends

Run Load Flow

tol = 1e-12iter_max = 50 vmin = 05 vmax = 15 acc = 10

disply=yflag = 2

svolt = bus(2) stheta = bus(3)deg_rad

bus_type = round(bus(10))

swing_index=find(bus_type==1)

[bus_solline_solline_flowY1ytpschrg] =

loadflow(buslinetoliter_maxaccdisplyflag)

clc

display(Running)

Load Flow End

Initialize Machine Variables

BM=mac_con(3)MVA_Baseuniform base conversion matrix

xls=mac_con(4)BM

Ra=mac_con(5)BM

xd=mac_con(6)BM

xdd=mac_con(7)BM

xddd=mac_con(8)BM

Td0d=mac_con(9)

Td0dd=mac_con(10)

xq=mac_con(11)BM

xqd=mac_con(12)BM

xqdd=mac_con(13)BM

Tq0d=mac_con(14)

Tq0dd=mac_con(15)

H=mac_con(16)BM

D=mac_con(17)BM

M=2H

wB=2pif

Tc=001ones(N_Machine1)

Vs=00ones(N_Machine1)

if xddd~=0 okltBDSCIgt

Zg= Ra + jxdddZg for sub-transient model

D1 Programs 261

else

Zg= Ra + jxdd

end

Yg=1Zg

Machine Variables initialization ends

AVR Initialization

Tr=ones(N_Machine1)

KA=zeros(N_Machine1)

Kp=zeros(N_Machine1)

Ki=zeros(N_Machine1)

Kd=zeros(N_Machine1)

Td=ones(N_Machine1)

Ka=ones(N_Machine1)

Kad=zeros(N_Machine1)for DC4B Efd0 initialization

Ta=ones(N_Machine1)

Ke=ones(N_Machine1)

Aex=zeros(N_Machine1)

Bex=zeros(N_Machine1)

Te=ones(N_Machine1)

Kf=zeros(N_Machine1)

Tf=ones(N_Machine1)

Efdmin=zeros(N_Machine1)

Efdmax=zeros(N_Machine1)

Efdmin_dc=zeros(N_Machine1)

Efdmax_dc=zeros(N_Machine1)

len_exc=size(exc_con)

Vref_Manual=ones(N_Machine1)

for i=11len_exc(1)

Exc_m_indx=exc_con(i2)present machine index

Vref_Manual(Exc_m_indx)=0

if(exc_con(i3)~=0)

Tr(Exc_m_indx)=exc_con(i3)

end

if(exc_con(i5)~=0)

Ta(Exc_m_indx)=exc_con(i5)

end

if(exc_con(i1)==1)

Kp(Exc_m_indx)=exc_con(i16)

Kd(Exc_m_indx)=exc_con(i17)

Ki(Exc_m_indx)=exc_con(i18)

Td(Exc_m_indx)=exc_con(i19)

Ke(Exc_m_indx)=exc_con(i8)

Te(Exc_m_indx)=exc_con(i9)


Kf(Exc_m_indx)=exc_con(i14)

Tf(Exc_m_indx)=exc_con(i15)

Bex(Exc_m_indx)=log(exc_con(i11)exc_con(i13))(exc_con(i10)-exc_con(i12))

Aex(Exc_m_indx)=exc_con(i11)exp(-Bex(Exc_m_indx)exc_con(i10))

Ka(Exc_m_indx)=exc_con(i4)

Kad(Exc_m_indx)=1

Efdmin_dc(Exc_m_indx)=exc_con(i7)

Efdmax_dc(Exc_m_indx)=exc_con(i6)

else

KA(Exc_m_indx)=exc_con(i4)

Efdmin(Exc_m_indx)=exc_con(i7)

Efdmax(Exc_m_indx)=exc_con(i6)

end

end

AVR initialization ends

PSS initialization

Ks=zeros(N_Machine1)

Tw=ones(N_Machine1)

T11=ones(N_Machine1)






Vs_max=zeros(N_Machine1)

Vs_min=zeros(N_Machine1)

len_pss=size(pss_con)

for i=11len_pss(1)

Pss_m_indx=pss_con(i2)present machine index

Ks(Pss_m_indx)=pss_con(i3)pssgain

Tw(Pss_m_indx)=pss_con(i4)washout time constant

T11(Pss_m_indx)=pss_con(i5)first lead time constant

T12(Pss_m_indx)=pss_con(i6)first lag time constant

T21(Pss_m_indx)=pss_con(i7)second lead time constant

T22(Pss_m_indx)=pss_con(i8)second lag time constant

T31(Pss_m_indx)=pss_con(i9)third lead time constant

T32(Pss_m_indx)=pss_con(i10)third lag time constant

Vs_max(Pss_m_indx)=pss_con(i11)maximum output limit

Vs_min(Pss_m_indx)=pss_con(i12)minimum output limit

end

PSS initialization ends

Initialize Network Variables

D1 Programs 263

Y=full(Y1)sparse to full matrix

MM=zeros(N_BusN_Machinedouble)

Multiplying matrix to convert Ig into vector of correct length

for i=11N_Machine

MM(mac_con(i2)mac_con(i1))=1

end

YG=MMYg

YG=diag(YG)

V=bus_sol(2)

theta=bus_sol(3)pi180

YL=diag((bus(6)-jbus(7))V^2)Constant Impedence Load model

Y_Aug=Y+YL+YG

Z=inv(Y_Aug) It=

Y_Aug_dash=Y_Aug

Network Variables initialization ends

Initial Conditions

Vg=MMVthg=MMtheta

P=MMbus_sol(4)Q=MMbus_sol(5)

mp=2

mq=2

kp=bus_sol(6)V^mp

kq=bus_sol(7)V^mq

V0=V(cos(theta)+ jsin(theta))

y_dash=y

from_bus = line(1)

to_bus = line(2)

MW_s = V0(from_bus)conj((V0(from_bus) - tpsV0(to_bus))y_dash

+ V0(from_bus)(jchrg2))(tpsconj(tps))

P_s = real(MW_s) active power sent out by from_bus

Q_s = imag(MW_s)

voltage = Vg(cos(thg) + jsin(thg))

current = conj((P+jQ)voltage)

Eq0 = voltage + (Ra+jxq)current

id0 = -abs(current) (sin(angle(Eq0) - angle(current)))

iq0 = abs(current) cos(angle(Eq0) - angle(current))

vd0 = -abs(voltage) (sin(angle(Eq0) - angle(voltage)))

vq0 = abs(voltage) cos(angle(Eq0) - angle(voltage))

Efd0 = abs(Eq0) - (xd-xq)id0

Eq_dash0 = Efd0 + (xd - xdd) id0

Ed_dash0 = -(xq-xqd) iq0

Psi1d0=Eq_dash0+(xdd-xls)id0

Psi2q0=-Ed_dash0+(xqd-xls)iq0

Edc_dash0=(xddd-xqdd)iq0


Te0 = Eq_dash0iq0(xddd-xls)(xdd-xls) + Ed_dash0id0(xqdd-xls)

(xqd-xls)+(xddd-xqdd)id0iq0 - Psi2q0id0(xqd-xqdd)(xqd-xls) +

Psi1d0iq0(xdd-xddd)(xdd-xls)

delta0 = angle(Eq0)

IG0 = (Yg(vq0+jvd0)+ (iq0+jid0))exp(jdelta0) The sum of currents

IQ0 = real((iq0+jid0)exp(jdelta0))

ID0 = imag((iq0+jid0)exp(jdelta0))

VDQ=MM(Y_Aug_dash(MMIG0))

VD0=imag(VDQ)

VQ0=real(VDQ)

Vref=Efd0

V_Ka0=zeros(N_Machine1)

V_Ki0=zeros(N_Machine1)

for i=11len_exc(1)

Exc_m_indx=exc_con(i2)

if(exc_con(i1)==1)

V_Ka0(Exc_m_indx)=

Efd0(Exc_m_indx)(Ke(Exc_m_indx)+Aex(Exc_m_indx)exp(Efd0(Exc_m_indx)Bex(Exc_m_indx)))

V_Ki0(Exc_m_indx)=V_Ka0(Exc_m_indx)Ka(Exc_m_indx)

Vref(Exc_m_indx)=Vg(Exc_m_indx)

else

Vref(Exc_m_indx)=(Efd0(Exc_m_indx)KA(Exc_m_indx))+Vg(Exc_m_indx)

end

end

Tm0=Te0

Pm0=Tm0

Sm0=00ones(N_Machine1)

Gn=16

delta = sym(delta [Gn 1])


Sm= sym(Sm [Gn 1])

Eq_dash=sym(Eq_dash[N_Machine 1])

Ed_dash=sym(Ed_dash[N_Machine 1])


Edc_dash=sym(Edc_dash[N_Machine 1])

Psi1d=sym(Psi1d[N_Machine 1])

Psi2q=sym(Psi2q[N_Machine 1])

D1 Programs 265

Efd=sym(Efd[N_Machine 1])

PSS1=sym(PSS1[N_Machine 1])




V_r=sym(V_f[N_Machine 1])

V_Ki=sym(V_Ki[N_Machine 1])

V_Kd=sym(V_Kd[N_Machine 1])

V_a=sym(V_a[N_Machine 1])

V_f=sym(V_f[N_Machine 1])

Efdd2=sym(Efdd2[N_Machine 1])

ig=(Eq_dash(xddd-xls)(xdd-xls)+Psi1d(xdd-xddd)(xdd-xls)+

1i(Ed_dash(xqdd-xls)(xqd-xls)-

Psi2q(xqd-xqdd)(xqd-xls)+Edc_dash))Yg

Vdq=MMZ(MM(igexp(jdelta)))exp(-jdelta)

Vq=real(Vdq)

Vd=imag(Vdq)

idq=(Eq_dash(xddd-xls)(xdd-xls)+Psi1d(xdd-xddd)(xdd-xls)+

1i(Ed_dash(xqdd-xls)(xqd-xls)-Psi2q(xqd-xqdd)(xqd-xls)+Edc_dash)-(Vq+1iVd))Yg

iq=real(idq)

id=imag(idq)

TeG=Eq_dashiq(xddd-xls)(xdd-xls)+Ed_dashid(xqdd-xls)(xqd-xls)+

(xddd-xqdd)idiq-Psi2qid(xqd-xqdd)(xqd-xls)+Psi1diq(xdd-xddd)(xdd-xls)

Differential Equations of the Electromechanical Part

ddelta=wBSm

dSm=1(2H)(Tm0-TeG-DSm)

dEd_dash=1Tq0d(-Ed_dash+(xq-xqd)(-iq+(xqd-xqdd)(xqd-xls)^2((xqd-xls)id-

Ed_dash-Psi2q)))

dEdc_dash=1Tc(iq(xddd-xqdd)-Edc_dash)

dEq_dash=1Td0d(Efd-Eq_dash+(xd-xdd)(id+(xdd-xddd)(xdd-xls)^2


(Psi1d-(xdd-xls)id-Eq_dash)))

dPsi1d=1Td0dd(Eq_dash+(xdd-xls)id-Psi1d)

dPsi2q=1Tq0dd(-Ed_dash+(xqd-xls)iq-Psi2q)

ddelta_Gn(Gn)=[]

Differential Equations of the PSS

KPSS1=(T11-T12)T12 KPSS3=(T21-T22)T22 KPSS6=(T31-T32)T32

PSS_in1=KsSm-PSS1

PSS_in2=PSS_in1+(PSS_in1-PSS2)KPSS1



Vs=PSS_in3+(PSS_in3-PSS4)KPSS6

N_PSS=[] This vector records the numer of machines on which the PSSs are equipped

for i=11len_pss(1)


dPSS1(Pss_m_indx)=1Tw(Pss_m_indx)(Sm(Pss_m_indx)Ks(Pss_m_indx)-PSS1(Pss_m_indx))

dPSS2(Pss_m_indx)=1T12(Pss_m_indx)(PSS_in1(Pss_m_indx)-PSS2(Pss_m_indx))


dPSS3(i)=1T22(i)(((KsSm-PSS1)T11T12-PSS2(i))(T11(i)-T12(i))T12(i)-PSS3(i))


N_PSS=[N_PSS i]

end

Vt=sqrt(Vd^2+Vq^2)

Differential Equations of the Exciter

N_ST1A=[] This vector record the number of machines equipped with exciter type ST1A

N_DC4B=[] Tihs vector record the number of machines equipped with exciter type DC4B

for i=11len_exc(1)


if(exc_con(i1)==1) The differential equations for exciter type DC4B

DC4B_in1(Exc_m_indx)=Vs(Exc_m_indx)+Vref(Exc_m_indx)-V_r(Exc_m_indx)-

Kf(Exc_m_indx)Tf(Exc_m_indx)(Efd(Exc_m_indx)-V_f(Exc_m_indx))

DC4B_in2(Exc_m_indx)=DC4B_in1(Exc_m_indx)Kp(Exc_m_indx)+V_Ki(Exc_m_indx)+

DC4B_in1(Exc_m_indx)Kd(Exc_m_indx)Td(Exc_m_indx)-V_Kd(Exc_m_indx)

dEfd(Exc_m_indx)=1Te(Exc_m_indx)(V_a(Exc_m_indx)-(Ke(Exc_m_indx)Efd(Exc_m_indx)+

Efd(Exc_m_indx)Aex(Exc_m_indx)exp(Bex(Exc_m_indx)Efd(Exc_m_indx))))

D1 Programs 267

dV_r(Exc_m_indx)=1Tr(Exc_m_indx)(Vt(Exc_m_indx)-V_r(Exc_m_indx))

dV_Ki(Exc_m_indx)=DC4B_in1(Exc_m_indx)Ki(Exc_m_indx )

dV_Kd(Exc_m_indx)=(DC4B_in1(Exc_m_indx)Kd(Exc_m_indx)Td(Exc_m_indx)-

V_Kd(Exc_m_indx))1Td(Exc_m_indx)

dV_a(Exc_m_indx)=(DC4B_in2(Exc_m_indx)Ka(Exc_m_indx)-

V_a(Exc_m_indx))1Ta(Exc_m_indx)

dV_f(Exc_m_indx)=(Efd(Exc_m_indx)-V_f(Exc_m_indx))1Tf(Exc_m_indx)

N_DC4B=[N_DC4B Exc_m_indx]

else

dEfdd2(Exc_m_indx)=(Vt(Exc_m_indx)-Efdd2(Exc_m_indx))1Tr(Exc_m_indx)

Efd(Exc_m_indx)=KA(Exc_m_indx)(Vs(Exc_m_indx)+Vref(Exc_m_indx)-Efdd2(Exc_m_indx))

N_ST1A=[N_ST1A Exc_m_indx]

end

end

V_r0=Vg V_f0=Efd0Kad V_Kd0=zeros(N_Machine1)

Efdd20=VgPSS10=zeros(N_Machine1) PSS20=zeros(N_Machine1)

PSS30=zeros(N_Machine1) PSS40=zeros(N_Machine1)

Cla2=classic_coef(EIVCGCH)

F=[ddeltadSmdEd_dashdEdc_dashdEq_dashdPsi1ddPsi2qdEfd(N_DC4B)

dV_r(N_DC4B)dV_Ki(N_DC4B)dV_Kd(N_DC4B)dV_a(N_DC4B)dV_f(N_DC4B)

dEfdd2(N_ST1A)dPSS1(N_PSS)dPSS2(N_PSS)dPSS3(N_PSS)dPSS4(N_PSS)]

X=[deltaSmEd_dashEdc_dashEq_dashPsi1dPsi2qEfd(N_DC4B)V_r(N_DC4B)

V_Ki(N_DC4B)V_Kd(N_DC4B)V_a(N_DC4B)V_f(N_DC4B)Efdd2(N_ST1A)PSS1(N_PSS)

PSS2(N_PSS)PSS3(N_PSS)PSS4(N_PSS)]

x0=[delta0Sm0Ed_dash0Edc_dash0Eq_dash0Psi1d0Psi2q0Efd0(N_DC4B)

V_r0(N_DC4B)V_Ki0(N_DC4B)V_Kd0(N_DC4B)V_Ka0(N_DC4B)V_f0(N_DC4B)

Efdd20(N_ST1A)PSS10(N_PSS)PSS20(N_PSS)PSS30(N_PSS)PSS40(N_PSS)]

EIV=eig(F1t)

[TuLambdaF2NF3N]=NLcoefJordan(F1tF2tF3t)

damping_ratioP=-real(EIV)abs(EIV)

N_State=size(EIV1)

for i=11N_State

if(abs(EIV(i))lt=1e-10)


damping_ratioP(i)=1

end

end

[DrPIdxP]=sort(damping_ratioP100) value calculated by the m-file

--clacm--

function [delPdelQPQconv_flag] =

calc(VangYPgQgPlQlsw_bnog_bnotol)

Syntax [delPdelQPQconv_flag] =


Purpose calculates power mismatch and checks convergence

also determines the values of P and Q based on the

supplied values of voltage magnitude and angle

Version 20 eliminates do loop

Input nbus - total number of buses

bus_type - load_bus(3) gen_bus(2) swing_bus(1)

V - magnitude of bus voltage

ang - angle(rad) of bus voltage

Y - admittance matrix

Pg - real power of generation

Qg - reactive power of generation

Pl - real power of load

Ql - reactive power of load

sw_bno - a vector having zeros at all swing_bus locations ones otherwise

g_bno - a vector having zeros at all generator bus locations ones otherwise

tol - a tolerance of computational error

Output delP - real power mismatch

delQ - reactive power mismatch

P - calculated real power

Q - calculated reactive power

conv_flag - 0 converged

1 not yet converged

See also

Calls

Called By loadflow

(c) Copyright 1991 Joe H Chow - All Rights Reserved

History (in reverse chronological order)

D1 Programs 269

Version 20

Author Graham Rogers

Date July 1994

Version 10

Author Kwok W Cheung Joe H Chow

Date March 1991

jay = sqrt(-1)

swing_bus = 1

gen_bus = 2

load_bus = 3

voltage in rectangular coordinate

V_rect = Vexp(jayang)

bus current injection

cur_inj = YV_rect

power output based on voltages

S = V_rectconj(cur_inj)

P = real(S) Q = imag(S)

delP = Pg - Pl - P

delQ = Qg - Ql - Q

zero out mismatches on swing bus and generation bus

delP=delPsw_bno

delQ=delQsw_bno

delQ=delQg_bno

total mismatch

[pmisip]=max(abs(delP))

[qmisiq]=max(abs(delQ))

mism = pmis+qmis

if mism gt tol

conv_flag = 1

else

conv_flag = 0

end

return

--chq_limm--

function f = chq_lim(qg_maxqg_min)

Syntax

f = chq_lim(qg_maxqg_min)

function for detecting generator vars outside limit

sets Qg to zero if limit exceded sets Ql to negative of limit

sets bus_type to 3 and recalculates ang_red and volt_red


changes generator bus_type to type 3

recalculates the generator index

inputs qg_max and qg_min are the last two clumns of the bus matrix

outputsf is set to zero if no limit reached or to 1 if a limit is reached

Version 11


Date May 1997

Purpose Addition of var limit index

Version 10


Date October 1996

(c) copyright Joe Chow 1996

global Qg bus_type g_bno PQV_no PQ_no ang_red volt_red

global Ql

global gen_chg_idx

gen_chg_idx indicates those generators changed to PQ buses

gen_cgq_idx = ones(n of bus1) if no gen at vars limits

= 0 at the corresponding bus if generator at var limit

f = 0

lim_flag = 0 indicates whether limit has been reached

gen_idx = find(bus_type ==2)

qg_max_idx = find(Qg(gen_idx)gtqg_max(gen_idx))

qg_min_idx = find(Qg(gen_idx)ltqg_min(gen_idx))

if ~isempty(qg_max_idx)

some q excedes maximum

set Qg to zero

Qg(gen_idx(qg_max_idx)) = zeros(length(qg_max_idx)1)

modify Ql

Ql(gen_idx(qg_max_idx)) = Ql(gen_idx(qg_max_idx))

- qg_max(gen_idx(qg_max_idx))

modify bus_type to PQ bus

bus_type(gen_idx(qg_max_idx)) = 3ones(length(qg_max_idx)1)

gen_chg_idx(gen_idx(qg_max_idx)) = zeros(length(qg_max_idx)1)

lim_flag = 1

end

if ~isempty(qg_min_idx)

some q less than minimum

set Qg to zero

Qg(gen_idx(qg_min_idx)) = zeros(length(qg_min_idx)1)

modify Ql

Ql(gen_idx(qg_min_idx)) = Ql(gen_idx(qg_min_idx))

- qg_min(gen_idx(qg_min_idx))

D1 Programs 271


bus_type(gen_idx(qg_min_idx)) = 3ones(length(qg_min_idx)1)

gen_chg_idx(gen_idx(qg_min_idx)) = zeros(length(qg_min_idx)1)

lim_flag = 1

end

if lim_flag == 1

recalculate g_bno

nbus = length(bus_type)

g_bno = ones(nbus1)

bus_zeros=zeros(nbus1)

bus_index=(11nbus)

PQV_no=find(bus_type gt=2)

PQ_no=find(bus_type==3)

gen_index=find(bus_type==2)

g_bno(gen_index)=bus_zeros(gen_index)

construct sparse angle reduction matrix

il = length(PQV_no)

ii = (11il)

ang_red = sparse(iiPQV_noones(il1)ilnbus)

construct sparse voltage reduction matrix

il = length(PQ_no)

ii = (11il)

volt_red = sparse(iiPQ_noones(il1)ilnbus)

end

f = lim_flag

return

--Benchmark_IEEE_standardm--

This is the data file for the 68-Bus Benchmark system with 16-machines

and 86-lines The data is partially taken from the book Robust Control

in Power Systems by B Pal and B Chaudhuri with some of the parameters

modified to account for a more realistic model

Version 33



Date December 2013

BUS DATA STARTS

bus data format

bus number voltage(pu) angle(degree) p_gen(pu) q_gen(pu)

p_load(pu) q_load(pu)G-shunt (pu) B shunt (pu) bus_type

bus_type - 1 swing bus

- 2 generator bus (PV bus)

- 3 load bus (PQ bus)

system_base_mva = 1000


bus = [

01 1045 000 250 000 000 000 000 000 2 999 -999

02 098 000 545 000 000 000 000 000 2 999 -999

03 0983 000 650 000 000 000 000 000 2 999 -999

04 0997 000 632 000 000 000 000 000 2 999 -999

05 1011 000 505 000 000 000 000 000 2 999 -999

06 1050 000 700 000 000 000 000 000 2 999 -999

07 1063 000 560 000 000 000 000 000 2 999 -999

08 103 000 540 000 000 000 000 000 2 999 -999

09 1025 000 800 000 000 000 000 000 2 999 -999

10 1010 000 500 000 000 000 000 000 2 999 -999

11 1000 000 10000 000 000 000 000 000 2 999 -999

12 10156 000 1350 000 000 000 000 000 2 999 -999

13 1011 000 3591 000 000 000 000 000 2 999 -999

14 100 000 1785 000 000 000 000 000 2 999 -999

15 1000 000 1000 000 000 000 000 000 2 999 -999

16 1000 000 4000 000 000 000 000 000 1 0 0

17 100 000 000 000 6000 300 000 000 3 0 0

18 100 000 000 000 2470 123 000 000 3 0 0

19 100 000 000 000 000 000 000 000 3 0 0

20 100 000 000 000 6800 103 000 000 3 0 0

21 100 000 000 000 2740 115 000 000 3 0 0

22 100 000 000 000 000 000 000 000 3 0 0

23 100 000 000 000 2480 085 000 000 3 0 0

24 100 000 000 000 309 -092 000 000 3 0 0

25 100 000 000 000 224 047 000 000 3 0 0

26 100 000 000 000 139 017 000 000 3 0 0

27 100 000 000 000 2810 076 000 000 3 0 0

28 100 000 000 000 2060 028 000 000 3 0 0

29 100 000 000 000 2840 027 000 000 3 0 0

30 100 000 000 000 000 000 000 000 3 0 0

31 100 000 000 000 000 000 000 000 3 0 0

32 100 000 000 000 000 000 000 000 3 0 0

33 100 000 000 000 112 000 000 000 3 0 0

34 100 000 000 000 000 000 000 000 3 0 0

35 100 000 000 000 000 000 000 000 3 0 0

36 100 000 000 000 102 -01946 000 000 3 0 0

37 100 000 000 000 000 000 000 000 3 0 0

38 100 000 000 000 000 000 000 000 3 0 0

39 100 000 000 000 267 0126 000 000 3 0 0

40 100 000 000 000 06563 02353 000 000 3 0 0

41 100 000 000 000 1000 250 000 000 3 0 0

42 100 000 000 000 1150 250 000 000 3 0 0

43 100 000 000 000 000 000 000 000 3 0 0

D1 Programs 273

44 100 000 000 000 26755 00484 000 000 3 0 0

45 100 000 000 000 208 021 000 000 3 0 0

46 100 000 000 000 1507 0285 000 000 3 0 0

47 100 000 000 000 20312 03259 000 000 3 0 0

48 100 000 000 000 24120 0022 000 000 3 0 0

49 100 000 000 000 16400 029 000 000 3 0 0

50 100 000 000 000 100 -147 000 000 3 0 0

51 100 000 000 000 337 -122 000 000 3 0 0

52 100 000 000 000 158 030 000 000 3 0 0

53 100 000 000 000 2527 11856 000 000 3 0 0

54 100 000 000 000 000 000 000 000 3 0 0

55 100 000 000 000 322 002 000 000 3 0 0

56 100 000 000 000 200 0736 000 000 3 0 0

57 100 000 000 000 000 000 000 000 3 0 0

58 100 000 000 000 000 000 000 000 3 0 0

59 100 000 000 000 234 084 000 000 3 0 0

60 100 000 000 000 2088 0708 000 000 3 0 0

61 100 000 000 000 104 125 000 000 3 0 0

62 100 000 000 000 000 000 000 000 3 0 0

63 100 000 000 000 000 000 000 000 3 0 0

64 100 000 000 000 009 088 000 000 3 0 0

65 100 000 000 000 000 000 000 000 3 0 0

66 100 000 000 000 000 000 000 000 3 0 0

67 100 000 000 000 3200 15300 000 000 3 0 0

68 100 000 000 000 3290 032 000 000 3 0 0

]

BUS DATA ENDS

LINE DATA STARTS

line =[

01 54 0 00181 0 10250 0

02 58 0 00250 0 10700 0

03 62 0 00200 0 10700 0

04 19 00007 00142 0 10700 0

05 20 00009 00180 0 10090 0

06 22 0 00143 0 10250 0

07 23 00005 00272 0 0 0

08 25 00006 00232 0 10250 0

09 29 00008 00156 0 10250 0

10 31 0 00260 0 10400 0

11 32 0 00130 0 10400 0

12 36 0 00075 0 10400 0

13 17 0 00033 0 10400 0

14 41 0 00015 0 10000 0


15 42 0 00015 0 10000 0

16 18 0 00030 0 10000 0

17 36 00005 00045 03200 0 0

18 49 00076 01141 11600 0 0

18 50 00012 00288 20600 0 0

19 68 00016 00195 03040 0 0

20 19 00007 00138 0 10600 0

21 68 00008 00135 02548 0 0

22 21 00008 00140 02565 0 0

23 22 00006 00096 01846 0 0

24 23 00022 00350 03610 0 0

24 68 00003 00059 00680 0 0

25 54 00070 00086 01460 0 0

26 25 00032 00323 05310 0 0

27 37 00013 00173 03216 0 0

27 26 00014 00147 02396 0 0

28 26 00043 00474 07802 0 0

29 26 00057 00625 10290 0 0

29 28 00014 00151 02490 0 0

30 53 00008 00074 04800 0 0

30 61 000095 000915 05800 0 0

31 30 00013 00187 03330 0 0

31 53 00016 00163 02500 0 0

32 30 00024 00288 04880 0 0

33 32 00008 00099 01680 0 0

34 33 00011 00157 02020 0 0

34 35 00001 00074 0 09460 0

36 34 00033 00111 14500 0 0

36 61 00011 00098 06800 0 0

37 68 00007 00089 01342 0 0

38 31 00011 00147 02470 0 0

38 33 00036 00444 06930 0 0

40 41 00060 00840 31500 0 0

40 48 00020 00220 12800 0 0

41 42 00040 00600 22500 0 0

42 18 00040 00600 22500 0 0

43 17 00005 00276 0 0 0

44 39 0 00411 0 0 0

44 43 00001 00011 0 0 0

45 35 00007 00175 13900 0 0

45 39 0 00839 0 0 0

45 44 00025 00730 0 0 0

46 38 00022 00284 04300 0 0

D1 Programs 275

47 53 00013 00188 13100 0 0

48 47 000125 00134 08000 0 0

49 46 00018 00274 02700 0 0

51 45 00004 00105 07200 0 0

51 50 00009 00221 16200 0 0

52 37 00007 00082 01319 0 0

52 55 00011 00133 02138 0 0

54 53 00035 00411 06987 0 0

55 54 00013 00151 02572 0 0

56 55 00013 00213 02214 0 0

57 56 00008 00128 01342 0 0

58 57 00002 00026 00434 0 0

59 58 00006 00092 01130 0 0

60 57 00008 00112 01476 0 0

60 59 00004 00046 00780 0 0

61 60 00023 00363 03804 0 0

63 58 00007 00082 01389 0 0

63 62 00004 00043 00729 0 0

63 64 00016 00435 0 10600 0

65 62 00004 00043 00729 0 0

65 64 00016 00435 0 10600 0

66 56 00008 00129 01382 0 0

66 65 00009 00101 01723 0 0

67 66 00018 00217 03660 0 0

68 67 00009 00094 01710 0 0

27 53 00320 03200 04100 0 0

]

LINE DATA ENDS

MACHINE DATA STARTS

Machine data format

1 machine number

2 bus number

3 base mva

4 leakage reactance x_l(pu)

5 resistance r_a(pu)

6 d-axis sychronous reactance x_d(pu)

7 d-axis transient reactance x_d(pu)

8 d-axis subtransient reactance x_d(pu)

9 d-axis open-circuit time constant T_do(sec)

10 d-axis open-circuit subtransient time constant

T_do(sec)


11 q-axis sychronous reactance x_q(pu)

12 q-axis transient reactance x_q(pu)

13 q-axis subtransient reactance x_q(pu)

14 q-axis open-circuit time constant T_qo(sec)

15 q-axis open circuit subtransient time constant

T_qo(sec)

16 inertia constant H(sec)

17 damping coefficient d_o(pu)

18 dampling coefficient d_1(pu)

19 bus number

20 saturation factor S(10)


note all the following machines use subtransient reactance model

mac_con = [

01 01 100 00125 00 01 0031 0025 102 005 0069 00416667 0025 15

0035 42 0 0 01 0 0

02 02 100 0035 00 0295 00697 005 656 005 0282 00933333 005 15

0035 302 0 0 02 0 0

03 03 100 00304 00 02495 00531 0045 57 005 0237 00714286 0045 15

0035 358 0 0 03 0 0

04 04 100 00295 00 0262 00436 0035 569 005 0258 00585714 0035 15

0035 286 0 0 04 0 0

05 05 100 0027 00 033 0066 005 54 005 031 00883333 005 044

0035 26 0 0 05 0 0

06 06 100 00224 00 0254 005 004 73 005 0241 00675000 004 04

0035 348 0 0 06 0 0

07 07 100 00322 00 0295 0049 004 566 005 0292 00666667 004 15

0035 264 0 0 07 0 0

08 08 100 0028 00 029 0057 0045 67 005 0280 00766667 0045 041

0035 243 0 0 08 0 0

09 09 100 00298 00 02106 0057 0045 479 005 0205 00766667 0045 196

0035 345 0 0 09 0 0

10 10 100 00199 00 0169 00457 004 937 005 0115 00615385 004 1

0035 310 0 0 10 0 0

D1 Programs 277

11 11 100 00103 00 0128 0018 0012 41 005 0123 00241176 0012 15 0035 282 0 0 11 0 0

12 12 100 0022 00 0101 0031 0025 74 005 0095 00420000 0025 15 0035 923 0 0 12 0 0

13 13 200 00030 00 00296 00055 0004 59 005 00286 00074000 0004 15 0035 2480 0 0 13 0 0

14 14 100 00017 00 0018 000285 00023 41 005 00173 00037931 00023 15 0035 3000 0 0 14 0 0

15 15 100 00017 00 0018 000285 00023 41 005 00173 00037931 00023 15 0035 3000 0 0 15 0 0

16 16 200 00041 00 00356 00071 00055 78 005 00334 00095000 00055 15 0035 2250 0 0 16 0 0

]

MACHINE DATA ENDS

EXCITER DATA STARTS

Description of Exciter data starts

exciter data DC4BST1A model

1 - exciter type (1 for DC4B 0 for ST1A)

2 - machine number

3 - input filter time constant T_R

4 - voltage regulator gain K_A

5 - voltage regulator time constant T_A

6 - maximum voltage regulator output V_Rmax

7 - minimum voltage regulator output V_Rmin

8 - exciter constant K_E

9 - exciter time constant T_E

10 - E_1

11 - S(E_1)

12 - E_2

13 - S(E_2)

14 - stabilizer gain K_F

15 - stabilizer time constant T_F

16 - K_P

17 - K_I

18 - K_D

19 - T_D

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

exc_con = [

1 1 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

1 2 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01


1 3 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

1 4 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

1 5 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

1 6 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

1 7 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

1 8 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

0 9 001 200 000 50 -50 00 0 0 0 0 0 0 0 0 0 0

0

1 10 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

1 11 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

1 12 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

]

EXCITER DATA ENDS

PSS DATA STARTS

1-S No

2-present machine index

3-pssgain

4-washout time constant

5-first lead time constant

6-first lag time constant

7-second lead time constant

8-second lag time constant

9-third lead time constant

10-third lag time constant

11-maximum output limit

12-minimum output limit

pss_con = [

1 1 12 10 015 004 015 004 015 004 02 -005

2 2 20 15 015 004 015 004 015 004 02 -005

3 3 20 15 015 004 015 004 015 004 02 -005

4 4 20 15 015 004 015 004 015 004 02 -005

5 5 12 10 015 004 015 004 015 004 02 -005

6 6 12 10 015 004 015 004 015 004 02 -005

7 7 20 15 015 004 015 004 015 004 02 -005

D1 Programs 279

8 8 20 15 015 004 015 004 015 004 02 -005

9 9 12 10 015 004 015 004 015 004 02 -005

9 9 15 10 009 002 009 002 1 1 02 -005

10 10 20 15 015 004 015 004 015 004 02 -005

11 11 20 15 015 004 015 004 015 004 02 -005

12 12 20 15 009 004 015 004 015 004 02 -005

]

PSS DATA ENDS

--file form_jacm--

function [Jac11Jac12Jac21Jac22]=form_jac(VangYang_redvolt_red)

Syntax [Jac] = form_jac(VangYang_redvolt_red)

[Jac11Jac12Jac21Jac22] = form_jac(VangY

ang_redvolt_red)

Purpose form the Jacobian matrix using sparse matrix techniques

Input V - magnitude of bus voltage



ang_red - matrix to eliminate swing bus voltage magnitude and angle

entries

volt_red - matrix to eliminate generator bus voltage magnitude

entries

Output Jac - jacobian matrix

Jac11Jac12Jac21Jac22 - submatrices of

jacobian matrix

See also

Calls

Called By vsdemo loadflow

(c) Copyright 1991-1996 Joe H Chow - All Rights Reserved


Version 20


Date March 1994

Purpose eliminated do loops to improve speed

Version 10


Date March 1991


jay = sqrt(-1)

exp_ang = exp(jayang)

Voltage rectangular coordinates

V_rect = Vexp_ang

CV_rect=conj(V_rect)

Y_con = conj(Y)

vector of conjugate currents

i_c=Y_conCV_rect

complex power vector

S=V_recti_c

S=sparse(diag(S))

Vdia=sparse(diag(V_rect))

CVdia=conj(Vdia)

Vmag=sparse(diag(abs(V)))

S1=VdiaY_conCVdia

t1=((S+S1)Vmag)volt_red

t2=(S-S1)ang_red

J11=-ang_redimag(t2)

J12=ang_redreal(t1)

J21=volt_redreal(t2)

J22=volt_redimag(t1)

if nargout gt 3

Jac11 = J11 clear J11




else

Jac11 = [J11 J12

J21 J22]

end

--loadflowm--

function [bus_solline_solline_flowYytpschrg] =

loadflow(buslinetoliter_maxaccdisplayflag)

Syntax [bus_solline_solline_flow] =


81297

Purpose solve the load-flow equations of power systems

modified to eliminate do loops and improve the use

sparse matices

Input bus - bus data

D1 Programs 281

line - line data

tol - tolerance for convergence

iter_max - maximum number of iterations

acc - acceleration factor

display - y generate load-flow study report

else no load-flow study report

flag - 1 form new Jacobian every iteration

2 form new Jacobian every other

iteration

Output bus_sol - bus solution (see report for the

solution format)

line_sol - modified line matrix

line_flow - line flow solution (see report)

See also

Algorithm Newton-Raphson method using the polar form of

the equations for P(real power) and Q(reactive power)

Calls Y_sparse calc form_jac chq_lim



Modification to correct generator var error on output

Graham Rogers November 1997

Version 21


Date October 1996

Purpose To add generator var limits and on-load tap changers

Version 20


Date March 1994

Version 10

Authors Kwok W Cheung Joe H Chow

Date March 1991

global bus_int


global Q Ql

global gen_chg_idx


global ac_line n_dcl

tt = clock start the total time clock

jay = sqrt(-1)

load_bus = 3

gen_bus = 2

swing_bus = 1

if exist(flag) == 0

flag = 1

end

lf_flag = 1

set solution defaults

if isempty(tol)tol = 1e-9end

if isempty(iter_max)iter_max = 30end

if isempty(acc)acc = 10 end

if isempty(display)display = nend

if flag lt1 || flag gt 2

error(LOADFLOW flag not recognized)

end

[nline nlc] = size(line) number of lines and no of line cols

[nbus ncol] = size(bus) number of buses and number of col

set defaults

bus data defaults

if ncollt15

set generator var limits

if ncollt12

bus(11) = 9999ones(nbus1)

bus(12) = -9999ones(nbus1)

end

if ncollt13bus(13) = ones(nbus1)end



volt_min = bus(15)

volt_max = bus(14)

else

volt_min = bus(15)

volt_max = bus(14)

end

no_vmin_idx = find(volt_min==0)

if ~isempty(no_vmin_idx)

volt_min(no_vmin_idx) = 05ones(length(no_vmin_idx)1)

end

no_vmax_idx = find(volt_max==0)

if ~isempty(no_vmax_idx)

D1 Programs 283

volt_max(no_vmax_idx) = 15ones(length(no_vmax_idx)1)

end

no_mxv = find(bus(11)==0)

no_mnv = find(bus(12)==0)

if ~isempty(no_mxv)bus(no_mxv11)=9999ones(length(no_mxv)1)end

if ~isempty(no_mnv)bus(no_mnv12) = -9999ones(length(no_mnv)1)end

no_vrate = find(bus(13)==0)

if ~isempty(no_vrate)bus(no_vrate13) = ones(length(no_vrate)1)end

tap_it = 0

tap_it_max = 10

no_taps = 0

line data defaults sets all tap ranges to zero - this fixes taps

if nlc lt 10

line(710) = zeros(nline4)

no_taps = 1

disable tap changing

end

outer loop for on-load tap changers

mm_chk=1

while (tap_itlttap_it_maxampampmm_chk)

tap_it = tap_it+1

build admittance matrix Y

[YnSWnPVnPQSB] = y_sparse(busline)

process bus data

bus_no = bus(1)

V = bus(2)

ang = bus(3)pi180

Pg = bus(4)

Qg = bus(5)

Pl = bus(6)

Ql = bus(7)

Gb = bus(8)

Bb = bus(9)


qg_max = bus(11)

qg_min = bus(12)

sw_bno=ones(nbus1)

g_bno=sw_bno

set up index for Jacobian calculation

form PQV_no and PQ_no




sw_bno(swing_index)=bus_zeros(swing_index)





sw_bno is a vector having ones everywhere but the swing bus locations

g_bno is a vector having ones everywhere but the generator bus locations


il = length(PQV_no)

ii = (11il)



il = length(PQ_no)

ii = (11il)


iter = 0 initialize iteration counter

calculate the power mismatch and check convergence

[delPdelQPQconv_flag] =


st = clock start the iteration time clock

start iteration process for main Newton_Raphson solution

while (conv_flag == 1 ampamp iter lt iter_max)

iter = iter + 1

Form the Jacobean matrix

clear Jac

Jac=form_jac(VangYang_redvolt_red)

reduced real and reactive power mismatch vectors

red_delP = ang_reddelP

red_delQ = volt_reddelQ

clear delP delQ

solve for voltage magnitude and phase angle increments

temp = Jac[red_delP red_delQ]

expand solution vectors to all buses

delAng = ang_redtemp(1length(PQV_no))

delV = volt_redtemp(length(PQV_no)+1length(PQV_no)+length(PQ_no))

update voltage magnitude and phase angle

V = V + accdelV

D1 Programs 285

V = max(Vvolt_min) voltage higher than minimum

V = min(Vvolt_max) voltage lower than maximum

ang = ang + accdelAng




check if Qg is outside limits


Qg(gen_index) = Q(gen_index) + Ql(gen_index)

lim_flag = chq_lim(qg_maxqg_min)

if lim_flag == 1

disp(Qg at var limit)

end

end

if iter == iter_max

imstr = int2str(iter_max)

disp([inner ac load flow failed to converge after imstr iterations])

tistr = int2str(tap_it)

disp([at tap iteration number tistr])

else

disp(inner load flow iterations)

disp(iter)

end

if no_taps == 0

lftap

else

mm_chk = 0

end

end

if tap_it gt= tap_it_max

titstr = int2str(tap_it_max)

disp([tap iteration failed to converge aftertitstr iterations])

else

disp( tap iterations )

disp(tap_it)

end

ste = clock end the iteration time clock

vmx_idx = find(V==volt_max)

vmn_idx = find(V==volt_min)

if ~isempty(vmx_idx)

disp(voltages at)

bus(vmx_idx1)

disp(are at the max limit)

end


if ~isempty(vmn_idx)

disp(voltages at)

bus(vmn_idx1)

disp(are at the min limit)

end


load_index = find(bus_type==3)

Pg(gen_index) = P(gen_index) + Pl(gen_index)


gend_idx = find((bus(10)==2)amp(bus_type~=2))

if ~isempty(gend_idx)

disp(the following generators are at their var limits)

disp( bus Qg)

disp([bus(gend_idx1) Q(gend_idx)])

Qlg = Ql(gend_idx)-bus(gend_idx7) the generator var part of the load

Qg(gend_idx)=Qg(gend_idx)-Qlg restore the generator vars

Ql(gend_idx)=bus(gend_idx7) restore the original load vars

end

Pl(load_index) = Pg(load_index) - P(load_index)

Ql(load_index) = Qg(load_index) - Q(load_index)

Pg(SB) = P(SB) + Pl(SB) Qg(SB) = Q(SB) + Ql(SB)

VV = Vexp(jayang) solution voltage

calculate the line flows and power losses

tap_index = find(abs(line(6))gt0)

tap_ratio = ones(nline1)

tap_ratio(tap_index)=line(tap_index6)

phase_shift(1) = line(7)

tps = tap_ratioexp(jayphase_shiftpi180)

from_bus = line(1)

from_int = bus_int(round(from_bus))

to_bus = line(2)

to_int = bus_int(round(to_bus))

r = line(3)

rx = line(4)

chrg = line(5)

z = r + jayrx

y = ones(nline1)z

MW_s = VV(from_int)conj((VV(from_int) - tpsVV(to_int))y

+ VV(from_int)(jaychrg2))(tpsconj(tps))


to to_bus

D1 Programs 287

Q_s = imag(MW_s) reactive power sent out by

from_bus to to_bus

MW_r = VV(to_int)conj((VV(to_int)

- VV(from_int)tps)y

+ VV(to_int)(jaychrg2))

P_r = real(MW_r) active power received by to_bus

from from_bus

Q_r = imag(MW_r) reactive power received by

to_bus from from_bus

iline = (11nline)

line_ffrom = [iline from_bus to_bus P_s Q_s]

line_fto = [iline to_bus from_bus P_r Q_r]

keyboard

P_loss = sum(P_s) + sum(P_r)

Q_loss = sum(Q_s) + sum(Q_r)

bus_sol=[bus_no V ang180pi Pg Qg Pl Ql Gb Bb

bus_type qg_max qg_min bus(13) volt_max volt_min]

line_sol = line

line_flow(1nline ) =[iline from_bus to_bus P_s Q_s]

line_flow(1+nline2nline) = [iline to_bus from_bus P_r Q_r]

Give warning of non-convergence

if conv_flag == 1

disp(ac load flow failed to converge)

error(stop)

end

display results

if display == y

clc

disp( LOAD-FLOW STUDY)

disp( REPORT OF POWER FLOW CALCULATIONS )

disp( )

disp(date)

fprintf(SWING BUS BUS g n SB)

fprintf(NUMBER OF ITERATIONS g n iter)

fprintf(SOLUTION TIME g secnetime(stest))

fprintf(TOTAL TIME g secnetime(clocktt))

fprintf(TOTAL REAL POWER LOSSES gnP_loss)

fprintf(TOTAL REACTIVE POWER LOSSES gnnQ_loss)

if conv_flag == 0

disp( GENERATION LOAD)

disp( BUS VOLTS ANGLE REAL REACTIVE REAL REACTIVE )

disp(bus_sol(17))


disp( LINE FLOWS )

disp( LINE FROM BUS TO BUS REAL REACTIVE )

disp(line_ffrom)

disp(line_fto)

end

end

if iter gt iter_max

disp(Note Solution did not converge in g iterationsn iter_max)

lf_flag = 0

end

return

--file y_sparsem--

function [YnSWnPVnPQSB] = y_sparse(busline)

Syntax [YnSWnPVnPQSB] = y_sparse(busline)

Purpose build sparse admittance matrix Y from the line data


line - line data

Output Y - admittance matrix

nSW - total number of swing buses

nPV - total number generator buses

nPQ - total number of load buses

SB - internal bus numbers of swing bus

See also

Calls

Called By loadflow form_j calc

(c) Copyright 1994-1996 Joe Chow - All Rights Reserved


Version 20


Date April 1994

Version 10


D1 Programs 289

Date March 1991

global bus_int

jay = sqrt(-1)

swing_bus = 1

gen_bus = 2

load_bus = 3

nline = length(line(1)) number of lines

nbus = length(bus(1)) number of buses

r=zeros(nline1)

rx=zeros(nline1)

chrg=zeros(nline1)

z=zeros(nline1)

y=zeros(nline1)

Y = sparse(110nbusnbus)

set up internal bus numbers for second indexing of buses

busmax = max(bus(1))

bus_int = zeros(busmax1)

ibus = (1nbus)

bus_int(round(bus(1))) = ibus

process line data and build admittance matrix Y

r = line(3)

rx = line(4)

chrg =jaysparse(diag( 05line(5)))

z = r + jayrx line impedance

y = sparse(diag(ones(nline1)z))

determine connection matrices including tap changers and phase shifters

from_bus = round(line(1))

from_int = bus_int(from_bus)

to_bus = round(line(2))

to_int = bus_int(to_bus)


tap=ones(nline1)

tap(tap_index)=1 line(tap_index6)

phase_shift = line(7)

tap = tapexp(-jayphase_shiftpi180)


sparse matrix formulation

iline = [11nline]

C_from = sparse(from_intilinetapnbusnlinenline)

C_to = sparse(to_intilineones(nline1)nbusnlinenline)

C_line = C_from - C_to

form Y matrix from primative line ys and connection matrices

Y=C_fromchrgC_from + C_tochrgC_to

Y = Y + C_lineyC_line

Gb = bus(8) bus conductance

Bb = bus(9) bus susceptance

add diagonal shunt admittances

Y = Y + sparse(ibusibusGb+jayBbnbusnbus)

if nargout gt 1

count buses of different types

nSW = 0

nPV = 0

nPQ = 0

bus_type=round(bus(10))

load_index=find(bus_type==3)


SB=find(bus_type==1)

nSW=length(SB)

nPV=length(gen_index)

nPQ=length(load_index)

end

return

D2 Simulation Files


0B2onGI-yt3FwcTBQWFgwQTYwRU0 withbull Benchmark_IEEE_standardmdl to study the system dynamics by step-by-step numerical integration

and Init_MultiMachine13faultm is the initial le to simulate the case when thereis a fault near Generator 13

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multiple-input systems with coupled

dynamics by the method of Normal

Forms



Analyse et controcircle de systegravemes non lineacuteaires agrave entreacutees multiples et agrave dynamiques coupleacutees par la meacutethode des formes normales

Application aux interactions modales non lineacuteaires et eacutevaluation de la stabiliteacute des

reacuteseaux eacutelectriques

RESUME Les systegravemes composeacutes drsquoune somme de sous-systegravemes interconnecteacutes offrent les avantages majeurs de flexibiliteacute drsquoorganisation et de redondance synonyme

de fiabiliteacute accrue Une des plus belles reacutealisations baseacutee sur ce concept reacuteside dans les reacuteseaux eacutelectriques qui sont reconnus agrave ce jour comme la plus grande et la plus complexe des structures existantes jamais deacuteveloppeacutees par lrsquohomme

Les pheacutenomegravenes de plus en plus non lineacuteaires rencontreacutes dans lrsquoeacutetude des

nouveaux reacuteseaux eacutelectriques amegravenent au deacuteveloppement de nouveaux outils permettant lrsquoeacutetude des interactions entre les diffeacuterents eacuteleacutements qui les composent Parmi les outils drsquoanalyse existants ce meacutemoire preacutesente le deacuteveloppement et

lrsquoapplication de la theacuteorie des Formes Normales agrave lrsquoeacutetude des interactions preacutesentes dans un reacuteseau eacutelectrique Les objectifs speacutecifiques de cette thegravese concernent le deacuteveloppement de la meacutethode des Formes Normales jusqursquoagrave lrsquoordre 3 lrsquoapplication de

cette meacutethode agrave lrsquoeacutetude des oscillations preacutesentes dans des reacuteseaux tests et lrsquoapport

de la meacutethode deacuteveloppeacutee dans lrsquoeacutetude de la stabiliteacute des reacuteseaux

Mots cleacutes Systegravemes interconnecteacutes oscillations non-lineacuteaires theacuteorie des formes normales

Analysis and control of nonlinear multiple input systems with coupled dynamics by the method of normal forms

Application to nonlinear modal interactions and transient stability assessment of


ABSTRACT Systems composed with a sum of interconnected sub-systems offer the advantages of a better flexibility and redundancy for an increased reliability One of the largest and biggest system based on this concept ever devised by man is the interconnected power system

Phenomena encountered in the newest interconnected power systems are more and more nonlinear and the development of new tools for their study is od major concern Among the existing tools this PhD work presents the development and the application of the Normal Form theory to the study of the interactions existing on an interconnected power system The specific objectives of this PhD work are the development of the Normal Form theory up to the third order the application of this method to study power system interarea oscillations and the gain of the developed method for the study of stability of power systems Keywords interconnected systems nonlinear oscillations normal form theory

List of Figures

List of Tables


Introductions

General Context

Multiple Input System with Coupled Dynamics

Background and Collaborations of this PhD Research Work

What has been achieved in this PhD research

Introduction to the Analysis and Control of Dynamics of Interconnected Power Systems

Why interconnecting power systems

Why do we study the interconnected power system

The Questions to be Solved

The Principle of Electricity Generation

The Focus on the Rotor Angle Dynamics

Assumptions to Study the Rotor Angle Dynamics in this PhD dissertation

A Brief Review of the Conventional Small-singal (Linear Modal) Analysis and its Limitations

Small-Signal Stability

Participation Factor and Linear Modal Interaction

Linear analysis fails to study the system dynamics when it exhibits Nonlinear Dynamical Behavior

How many nonlinear terms should be considered

Linear Analysis Fails to Predict the Nonlinear Modal Interaction

Reiteration of the Problems

Need for Nonlinear Analysis Tools

Perspective of Nonlinear Analysis

The Difficulties in Developing a Nonlinear Analysis Tool

A Solution Methods of Normal Forms

Several Remarkable Works on Methods of Normal Forms

Why choosing normal form methods

A Brief Introduction to Methods of Normal Forms

Resonant Terms

How can the methods of normal forms answer the questions

Classification of the Methods of Normal Forms

Current Developments by Methods of Normal Forms Small-Signal Analysis taking into account nonlinearities

Inclusion of 2nd Order Terms and Application in Nonlinear Modal Interaction

Inclusion of 3rd Order Terms

What is needed

Objective and Fulfilment of this PhD Dissertation

Objective and the Task

A Brief Introduction to the Achievements of PhD Work with A Simple Example

An example of Two Machines Connected to the Infinite Bus

The proposed method 3-3-3

Theoretical Improvements of Method 3-3-3 Compared to existing methods

Performance of Method 3-3-3 in Approximating the System Dynamics Compared to the Existing Methods

The Proposed Method NNM

Applications of the Proposed Methods

Scientific Contribution of this PhD Research

Contribution to Small Signal Analysis

Contribution to Transient Stability Assessment

Contribution to Nonlinear Control

Contribution to the Nonlinear Dynamical Systems

A Summary of the Scientific Contributions

The Format of the Dissertation

The Methods of Normal Forms and the Proposed Methodologies

General Introduction to NF Methods

Normal Form Theory and Basic Definitions

Damping and Applicability of Normal Form Theory

Illustrative Examples

General Derivations of Normal Forms for System Containing Polynomial Nonlinearities

Introduction

Linear Transformation

General Normal Forms for Equations Containing Nonlinearities in Multiple Degrees

Summary analysis of the different methods reviewed and proposed in the work

General Procedures of Applying the Method of Normal Forms in Power System

Derivations of the Normal Forms of Vector Fields for System of First Order Equations

Class of First-Order Equation that can be studied by Normal Form methods

Step1Simplifying the linear terms

Step 2 Simplifying the nonlinear terms

Analysis of the Normal Dynamics using z variables

Other Works Considering Third Order Terms ndash Summary of all the Normal Form Methods

Interpretation the Normal Form Methods by Reconstruction of Normal Dynamics into System Dynamics

The significance to do 3-3-3

Another Way to Derive the Normal Forms

Comparison to the Modal Series Methods

Nonlinear Indexes Based on the 3rd order Normal Forms of Vector Fields

3rd Order Modal Interaction Index

Stability Index

Nonlinear Modal Persistence Index

Stability Assessment

Derivations of Normal Forms of Nonlinear Normal Modes for System of Second-Order Differential Equations

Class of Second-Order Equations that can be studied by Methods of Normal Forms


Normal Modes and Nonlinear Normal Modes

Second-Order Normal Form Transformation

Third-order Normal Form Transformation

Eliminating the Maximum Nonlinear Terms

Invariance Property

One-Step Transformation of Coordinates from XY to RS

Comparison with Shaw-Pierre Method where Nonlinear Normal Modes are Formed

Possible Extension of the Normal Mode Approach

A Summary of the Existing and Proposed MNFs Based on Normal Mode Approaches

Nonlinear Indexes Based on Nonlinear Normal Modes

Amplitude Dependent Frequency-Shift of the Oscillatory Modes

Stability Bound

Analysis of Nonlinear Interaction Based on Closed-form Results

The Performance of Method NNM Tested by Case Study

Interconnected VSCs

Analysis based on EMT Simulations

Coupling Effects and Nonlinearity

Mode Decomposition and Reconstruction

Amplitude-dependent Frequency-shift

Analytical Investigation on the Parameters

Closed Remarks

Issues Concerned with the Methods of Normal Forms in Practice

Searching for Initial Condition in the Normal Form Coordinates

The Scalability Problem in Applying Methods of Normal Forms on Very Large-Scale System

The Differential Algebraic Equation Formalism

Comparisons on the two Approaches

Similarity and Difference in the Formalism

Applicability Range and Model Dependence

Computational Burden

Possible Extension of the Methods

Conclusions and Originality

Application of Method 3-3-3 Nonlinear Modal Interaction and Stability Analysis of Interconnected Power Systems

Introduction

Need for inclusion of higher-order terms

A Brief Review of the existing Normal Forms of Vector Fields and the Method 3-3-3

Modeling of the System Dynamics

Method 2-2-1

Method 3-2-3S

Method 3-3-1

Method 3-3-3

Chosen Test Benches and Power System Modeling

Assessing the Proposed Methodology on IEEE Standard Test system 1 Kundurs 4 machine 2 area System

Modelling of Mutli-Machine Multi-Area Interconnected Power System

Network Connectivity and Nonlinear Matrix

Programming Procedures for Normal Form Methods

Cases Selected and Small Signal Analysis

Case 1-2 Benefits of using 3rd order approximation for modal interactions and stability analysis

Nonlinear Analysis Based on Normal Forms

Applicability to Larger Networks with More Complex Power System Models

Modelling of the System of Equations

Case-study and Results

Factors Influencing Normal Form Analysis


Strong Resonance and Model Dependence

Conclusion of the Reviewed and Proposed Methods

Significance of the Proposed Research

Future Works and Possible Applications of Method 3-3-3

Application of Method NNM Transient Stability Assessment

Introduction

Study of Transient Stability Assessment and Transfer Limit

Existing Practical Tools

Advantages of the Presented Methods

Challenges for the Presented Methods

A Brief Review of the Proposed Tool Method NNM

Technical Contribution of the Method NNM

Structure of this Chapter

Network Reduction Modelling of the Power System

Modelling of SMIB System

Modeling of Multi-Machine System

Taylor Series Around the Equilibrium Point

Reducing the State-Variables to Avoid the Zero Eigenvalue

Case-studies

SMIB Test System

Multi-Machine Multi-Area Case IEEE 4 Machine Test System

Conclusion and Future Work

Significance of this Research

On-going Work

Originality and Perspective

From Analysis to Control and Explorations on Forced Oscillations

From Nonlinear Analysis to Nonlinear Control

Two Approaches to Propose the Control Strategies

Approach I Analysis Based Control Strategy

Approach II the Nonlinear Modal Control

State-of-Art The Applied Nonlinear Control

The Limitations

Decoupling Basis Control

A Linear Example of Independent Modal Space Control Strategy

Nonlinear Modal Control

How to do the nonlinear modal control

Step I Analytical Investigation of Nonlinear Power System Dynamics under Excitation

Limitations and Challenges to Study the System Dynamics under Excitation

Originality

Theoretical Formulation

Linear Transformation Modal Expansion

Normal Form Transformation in Modal Coordinates

CasendashStudy interconnected VSCs under Excitation

Significance of the Proposed Method

Step II Implementing Nonlinear Modal Control by EMR+IBC

The Control Schema Implemented by Inversion-Based Control Strategy

Primitive Results

Exploration in the Control Form of Normal Forms

A Literature Review

A Nouvel Proposed Methodology to Deduce the Normal Form of Control System

Conclusion


A Summary of the Achievements

Conclusions of Analysis of Power System Dynamics

Analysis of Nonlinear Interarea Oscillation

Transient Stability Assessment

A Summary of the Procedures


Conclusions of the Control Proposals

Conclusions of the Existing and Proposed Methodologies

Validity region

Resonance

Possibility to do the Physical Experiments

Multiple Input Systems with Coupled Dynamics

Classes of Multiple-input Systems with Coupled Dynamics Studied by the Proposed Methodologies

System with Coupled Dynamics in the First-Order Differential of State-Variables of System

System with Coupled Dynamics in the Second-Order Differential of State-Variables of System

The End or a new Start


Interconnected VSCs

Reduced Model of the Interconnected VSCs

A Renewable Transmission System

Methodology

Modeling the System

The Disturbance Model

Parameters and Programs

Parameters of Interconnected VSCs

Initial File for Interconnected VSCs

M-file to Calculate the Coef

Simulation Files

Inverse Nonlinear Transformation on Real-time


Linearization of the Nonlinear Transformation

Kundurs 4 Machine 2 Area System

Parameters of Kundurs Two Area Four Machine System

Programs and Simulation Files

SEP Initialization

Calculation of Normal Form Coefficients

Calculating the NF Coefficients h2jkl of Method 2-2-1

Calculation the NF Coefficients of Method 3-3-3

Calculation of Nonlinear Indexes MI3

Normal Dynamics

Search for Initial Conditions

Simulation files

New England New York 16 Machine 5 Area System

Programs

Simulation Files

Bibliography

Ndeg 2009 ENAM XXXX

Arts et Meacutetiers ParisTech - Campus de Lille L2EP LSIS

2016-ENAM-00XX

Eacutecole doctorale ndeg 432 Sciences des Meacutetiers de lrsquoingeacutenieur


Tian TIAN


Analysis and control of nonlinear multiple input sy stems with coupled dynamics by the method of normal forms

Doctorat ParisTech

T H Egrave S E pour obtenir le grade de docteur deacutelivreacute par

lrsquoEacutecole Nationale Supeacuterieure dArts et Meacutetiers

Speacutecialiteacute ldquo geacutenie eacutelectrique rdquo

Directeur de thegravese Xavier KESTELYN Co-encadrement de la thegravese Olivier THOMAS

T

H

Egrave

S

E

Jury Mme Manuela SECHILARIU Professeur AVENUES EA 7284 UTC Compiegravegne Preacutesidente M Claude LAMARQUE Professeur LTDS UMR 5513 ENTPE Vaux-en-Velin Rapporteur M Bogdan MARINESCU Professeur LS2N UMR 6004 Eacutecole Centrale de Nantes Rapporteur M Olivier THOMAS Professeur LSIS UMR 7296 Arts et Meacutetiers ParisTech Examinateur M Xavier KESTELYN Professeur L2EP EA 2697 Arts et Meacutetiers ParisTech Examinateur

2


Doctorat ParisTech





TianTIAN







Jury


2








ii

iii

Acknowledgements






iv






Leonardo da Vinci

vi


















vii

viii

Contents

List of Figures xix

List of Tables xxi












ix

x Contents

















Contents xi












xii Contents















Contents xiii











xiv Contents


















Contents xv







Bibliography 291

xvi Contents

List of Figures













xvii

















List of Figures xix





for δep = π12 169


for δep = π12 169







xx List of Figures

List of Tables


methods 52


NNM 113










2 and case 3 236

xxi

xxii List of Tables









1









3






















5





x = f(xu) (1)

g(xu) = 0



x0 = f(x0u0) = 0 (2)



4x = A4x+O(2) (3)




4y = Λ4y (4)







pki = UkiVik (5)




7























9











11



Problegraveme



























13








x = f(x) (6)


Systems




M q +Dq +Kq + fnl(q) = 0 (7)






doctorat






15







L2EP et LSIS















17























bull


Chapter 1

Introductions












19


11 General Context





















1122 L2EP and EMR




























































δ = ωsω


(11)





Innite Bus

Great Grid

G1

Pe

jXe

minusrarr

E


t(s)0 5 10 15 20 25

δ(degree)

30

40

50

60Predisturbance



















Example Case 1




0 2 4 6 8 10 12

δ1(degree)

40

60

80

100

120

140

t(s)0 2 4 6 8 10 12

δ2(degree)

0

20

40

60

80

100

Exact

Linear




x = f(x) (12)











pki = ukivik (15)












ζωnStates


primeq2 E

primeq3 E

primeq4
















t(s)0 05 1 15 2 25

δ(rad)

-1

0

1

2





Pe =EprimeV0X

sin δ (16)








minus sin δsep2










Pow

erf(δ)

02

05

08

1

Linear Analysis

2nd Order Analysis

3rd Order Analysis

Regionmaybeunstable



Example Case 2



0 5 10 15 20 25 30

δ1(degree)

40

70

100

130140

t(s)0 5 10 15 20 25 30

δ2(degree)

0

30

60

90100

ExactLinear


0 036 072 108 144 180 212 252Hz

101

102

Mag

(

of F

unda

men

tal)



2nd OrderFrequency

3rd OrderFrequency

































x = f(xu) (18)


4x = A4x+ f2(4x2) + f3(4x3) (19)




















































Perturbationmodel





3)





161 Resonant Terms



mode j








zj = λjzj +Nsum

ωkωl

NsumisinR2

gj2klzkzl +Nsum

ωpωq ωr

NsumisinR3

gj3pqrzpzqzr (111)


j3







Fn hn gn S or




nonlinearities









2 2 1











3 3 1









3 2 3 S



173 What is needed











G1 δ1

G2 δ2

jX12jXe

ZZPe

Innite Bus

E


Pe1 = V1V2X12




Pe2 = V1V2X12




(113)



4x = A4x+ a2x2 + a3x

3 (114)


yj = λjyj +4sum

k=1

4suml=1

ykyl +4sump=1

4sumq=1

4sumr=1

ypyqyr (115)




3)






(116)


3 2 3 S







2) + hj3(z3)


+gj3zjz3z4

















0 05 1 15 2minus05

0

05

1

15

t(simulation time)

δ1(rad)

Mode Decomposition

EXMode 1Mode 2

0 05 1 15 2minus05

0

05

1

15

t(simulation time)

δ2(rad)






























Chapter 2





























59


Methodologies













x = y + h2(y)

= z + h3(z) + h2(z + h3(z)) (23)

= [z + h2(z)] + h3(z) +Dh2(z)h3(z)

= z + h2(z) + h3(z) + (O)(4)

2112 Normal Form




2113 Resonant Terms












Methodologies








2131 Linear System


x = Ax (26)


y = Λy (27)


yj = λjyj (28)





X = sX + a2X2 + a3X


apXp (29)



X = Y + α2Y2 (210)



(1 + 2α2Y )Y = sY + (sα2 + a2)Y2 +O(Y 3) (211)





Y = sY + (a2 minus sα2)Y2 +O(Y 3) (213)


α2 =a2s

(214)


Y = sY +O(Y 3) (215)



Methodologies




221 Introduction


u = Au+ εF2(u) + ε2F3(u) + (216)
























Methodologies

and






ε0 y = Λy (224)





+

ksuml=2


where termssumk


















Methodologies















x = f(xu) (228)




4x = H1(4x) +1

2H2(4x) +

1

3H3(4x) +O(4) (229)



] H3jklm =


]and O(4) are






yj = λjyj +Nsumk=1

Nsuml=1

F2jklykyl +Nsump=1

Nsumq=1

Nsumr=1



sumNi=1 vji

[UThj3U

] F3jpqr = 1

6

sumNi=1 vji

sumNk=1

sumNk=1

sumNm=1H3ikLNMu

lpumq u

Nr vji




y = Λy + F (y) (232)




y = Λy + F2(y) (233)


Methodologies


yj = λjyj +Nsumk=1

Nsuml=1

F2jklykyl (234)


y = z + h2(z) (235)


yj = zj +Nsumk=1

Nsuml=1

h2jklzkzl (236)


z + ˙h2(z) = Λz + Λh2(z) + F2(z + h2(z)) (237)





z = Λz (order 1)










h2jkl =F2jkl





z = Λz +DF2(z)h2(z) +O(4) (241)

= Λz +O(3)

= Λz



yj = λjyj +

Nsumk=1

Nsuml=1

F2jklykyl +

Nsump=1

Nsumq=1

Nsumr=1

F3jpqrypyqyr (242)


z + ˙h2(z) = Λz + Λh2(z) + F2(z + h2(z) (243)



+HF2(z)(h2(z))2 + F3(z) +DF3zh2(z))


+DF2(z)h2(z) + F3(z) +O(4))


z = Λz (order 1)




(order 4)



z = Λz +DF2(z)h2(z) + F3(z) +O(4) (244)


Methodologies



zj =λjzj (245)

+

Nsump=1

Nsumq=1

Nsumr=1

(Nsuml=1


)zpzqzr

=λjzj +Nsump=1

Nsumq=1

Nsumr=1

Cjpqrzpzqzr

where Cjpqr =sumN




z = w + h3(w) (246)



(247)




+O(5))(Λw + Λh3(w) +DF2(w)h2(w) + F3(w) +O(5))


w = Λw (order 1)



(order 5)


w =Λw +DF2(w)h2(w) (248)





wj =λjwj

+

Nsump=1

Nsumq=1

Nsumr=1


+O(5) (249)


h3jpqr =F3jpqr +




wj = λjwj +

Msuml=1




Methodologies


cj=2k2l =C2k


(2l)(2k)(2lminus1)



cj=2k2k =C2k


(2kminus1)(2k)(2k) (253)












zj =wj +

Nsump=1

Nsumq=1

Nsumr=1










h3jpqr =F3jpqr



wj = λjwj +O(3) (258)



2361 2-2-1 and 3-2-3S


y = z + h2(z) (259)



+

Nsumk=1

Nsuml=1


+O(3) (260)

For 3-2-3S


+

Nsumk=1

Nsuml=1


+O(3) (261)


Methodologies

2362 3-3-3 and 3-3-1


y =Λw + h2(z) (262)

Λw + h3(w) + h2(w + h3(w))

=Λw + h3(w) + h2(w)



y = w + h2(w) + h3(w) (263)

For 3-3-1


+Nsumk=1

Nsuml=1


+Nsump=1

Nsumq=1

Nsumr=1


(264)

For 3-3-3


+

Nsumk=1

Nsuml=1


+

Nsump=1

Nsumq=1

Nsumr=1


(265)












(z + h2(z)) = z +Dh2(z)z = z +Dh2(z)Λz (266)





Methodologies







f1j = λjf1j (268)

f2j = λjf2j +

Nsumk=1

Nsuml=1

Cjklf1kf

1l (269)

f3j = λjf3j +

Nsumk=1

Nsuml=1

Cjkl(f1kf

2l + f2kf

1l ) +

Nsumk=1

Nsuml=1

Nsumk=1

Djpqrf

1p f

1q f

1r (270)


f1j (t) = y0j eλjt (271)


f2j = λjf2j +

Nsumk=1

Nsuml=1

Cjkly0ky

0l e

(k+l)t (272)


f2j (s1 s2) =

Nsumk=1

Nsuml=1

Cjkl1


1l (s2) (273)



f2j =Nsum

k=1

Nsuml=1


=Nsum

k=1

Nsuml=1

Cjkly0ky

0l S

jkl(t)

where

Sjkl(t) =1





k=1

Nsuml=1

Cjkl1


1l (s3) + f1k (s1)f

2l (s2 + s3))+

(275)Nsum

k=1

Nsuml=1

Nsumr=1

Djpqr

1


1q (s2)f

1r (s3)


f3j (s1 s2 s3) =

Nsumk=1

Nsuml=1

Cjkl1


[Nsum

k=1

Nsuml=1

Cjkl1


1l (s2)

]f1l (s3)

(276)

+ f1k (s1)

[Nsum

k=1

Nsuml=1

Cjkl1


1l (s3)

]

+Nsum

k=1

Nsuml=1

Nsumr=1

Djpqr

1


1q (s2)f

1r (s3)




Methodologies















zj = wj +Nsump=1

Nsumq=1

Nsumr=1

h3jpqrwpwqwr (277)







MI3jpqr =|h3jpqrw0

pw0qw

0r |

|w0j |

(278)


242 Stability Index


wj = λjwj +

Msuml=1


= (λj +Msuml=1

cj2l|w2l|2)wj

(280)



Since

λj +Msuml=1



SIIj2l)) (282)


Methodologies










j2l times Tr



SIj = σj +Msuml=1




(285)






Equations




Mq +Dq +Kq + fnl(q) = 0 (286)


fpnl(q) =Nsum

i=1jgeif2pijqiqj +

Nsumi=1jgeikgej

f3pijkqiqjqk (287)


Methodologies







+Nsumi=1

Nsumjgei

gpijXiXj +Nsumi=1

Nsumjgei

Nsumkgej

hpijkXiXjXk = 0



[gp] =

Nsumi=1



hpijk =

Nsumi=1

Φminus1pi (

NsumP=1

NsumQ=1

NsumR=1













Nsumi=1

Nsumjgei

gpijXiXj = 0 (293)


Xp = Yp (294)


Nsumjgei

gpijXiXj (295)


X = U + ε(a(U2) + b(V 2) + c(UV ))

Y = V + ε(α(U2) + β(V 2) + γ(UV ))

(296)


Methodologies


Xp = Up +Nsumi=1

Nsumjgei


Nsumj=1

cijUiVj

Yp = Vp +Nsumi=1

Nsumjgei


Nsumj=1

γijUiVj (297)


pij c

pij α

pij β

pij γ

pij



Uj = Vj +O(U2i V

2i ) (298)


i V2i ) (299)



3i ) (2100)


pij + bpji)UiVj



3i )


pij + βpji)UiVj



D(a+ b+ c) = α+ β + γ (2101)

D(α+ β + γ) + 2ξωD(a+ b+ c) (2102)




Dc(UV )V = α (U2)

Dc(UV )U = β (V 2)

(2103)






ing (2103)(2104)apij b

pij c

pij α

pij β

pij γ

pij



Up + 2ξpωpUp + ω2pUp +O(3) = 0 (2105)

with

O =Nsumi=1

Nsumjge1

Nsumkgej


+Nsumi=1

Nsumjgei

Nsumkgej

BpijkUiUjUk +

Nsumi=1

Nsumjgei

Nsumkgej

CpijkUiUjUk



Apijk =

Nsumlgei

gpilaljk +

sumllei

gplialjk (2107)

Bpijk =

Nsumlgei

gpilbljk +

sumllei

gplibljk (2108)

Cpijk =

Nsumlgei

gpilcljk +

sumllei

gplicljk (2109)


Methodologies




U = R+ ε2(r(R3) + s(S3) + t(R2 S) + u(RS2))


(2110)



D(r + s+ t+ u) = λ+ micro+ ν + ζ (2111)



To degree ε2




DuR = micro (S3)

(2113)





(2114)









foralli le p



(2115)






ωp = ωi + ωj ωp = 2ωi (2116)





Methodologies





X + ω2X + αX3 = 0 (2119)









1minus ξ2p (2121)









pppRpR2p + CppppR

2pRp



P(2) =

Nsumjgep


2j +Bp


](2124)

+sumilep


2i +Bp


]

Q(2) =

Nsumjgep

(BpjpjRjRj + CpjjpR

2j ) +

Nsumilep

(BpiipRiRi + CpiipR

2i ) (2125)




Methodologies


pppp C

pppp)




pppRpR2p + CppppR

2pRp = 0




Xp = Rp +

Nsumi=1

Nsumjgei


Nsumi=1

Nsumj=1

cijRiSj

+Nsumi=1

Nsumjgei

Nsumkgej


+

Nsumi=1

Nsumjgei

Nsumkgej



Yp = Sp +Nsumi=1

Nsumjgei


2sumj=1

γijRiSj

+

Nsumi=1

Nsumjgei

Nsumkgej


+Nsumi=1

Nsumjgei

Nsumkgej






y

=

micro 0

0 minus1

x

y

+

xy + cx3

bx2

(2128)


microc = 0 xc = x xs = y (2129)



y = bx2 + (|x micro|3) (2130)


Methodologies


x = microx+ (b+ c)x3 (2131)



micro 0

0 minus1








Modes












p S0p [18] And R0

p S0p

satises X0p = R0

p +R(3)(R0i S

0i ) Y 0

p = S0p + S(3)(R0

i S0i )


ωnlp = ωp

(1 +


pppp

8ω2p

a2

)(2133)


262 Stability Bound



3p


2p

]Rp (2134)



2p ge 0 (2135)


Methodologies

with

Rp le

radicminus

ω2p

hpppp +Apppp(2136)



2p + rppppR

3p (2137)


Knl =RpXp

=1







NIp =

radicsumNk=1

sumNj=1


∣∣∣2|Xp0|

(2140)





pBpppp

16πωp(2141)



Study





Methodologies



J1d2δ1dt2


sin δ1 + V1V2L12


J2d2δ1dt2


sin δ2 + V1V2L12


(2142)





R1 + 2ξ1ω1R1 + ω21R1 + (h1111 +A1

111)R31 +B1

111R1R12

+ C1111R

21R1

+R1((A1212 +A1

122 + h1122)R22 +B1

122R22

+ (C1122 + C1

212)R2R2) + R1(B1212R2R2 + C1

221R22) = 0

R2 + 2ξ2ω2R2 + ω22R2 + (h2222 +A2

222)R32 +B2

222R2R22

+ C2222R

22R2

+R2((A2112 +A2

211 + h2112)R22 +B2

211R12

+ (C2211 + C2

121)R1R1)) + R2(B2112R1R2 + C2

112R22) = 0

(2143)



R1 + 2ξ1ω1R1 + ω21R1 + (h1

111 +A1111)R

31 +B1

111R1R12+ C1

111R21R1 +R2P

(2)1 (Ri Ri) + R2Q

(2)1 (Ri Ri) = 0

R2 + 2ξ2ω2R2 + ω22R2 + (h2

222 +A2222)R

32 +B2

222R2R22+ C2

222R22R2 +R1P

(2)2 (Ri Ri) + R1Q

(2)2 (Ri Ri) = 0

(2144)


R1 + 2ξ1ω1R1 + ω21R1 + (h1111 +A1

111)R31 +B1

111R1R12

+ C1111R

21R1 = 0

(2145)


R1 + 2ξ1ω1R1 + ω21R1 + (h1111 +A1

111)R31 +B1

111R1R12

+ C1111R

21R1 = 0

R2 + 2ξ2ω2R2 + ω22R2 + (h2222 +A2

222)R32 +B2

222R2R22

+ C2222R

22R2 = 0

(2146)



Methodologies




0 02 04 06 08 10

05

1

15

δ1(rad)

EXNNMLNM

0 02 04 06 08 1minus05

0

05

1

15

t(s)

δ2(rad)

EXNNMLNM

0 005 01 015 020

05

1

15

δ1(rad)

EXNNMLNM

0 005 01 015 02minus05

0

05

1

15

t(s)

δ2(rad)

EXNNMLNM






0 02 04 06 08 1minus02

minus01

0

01

02

Errorin

δ1(rad)

Error

FNNMNNMNM2

0 02 04 06 08 1minus02

minus01

0

01

02

Errorin

δ2(rad)








Methodologies

0 05 1 15 2minus05

0

05

1

15

t(simulation time)

δ1(rad)

Mode Decomposition

EXMode 1Mode 2

0 05 1 15 2minus05

0

05

1

15

t(simulation time)

δ2(rad)








0 02 04 06 08 1

005

1105

11

115

Disturbance in δ1

Mode Frequency

Disturbance in δ2

ω1(rad)

0 02 04 06 08 1

005

1137

1375

138

Disturbance in δ1

Disturbance in δ2

ω2(rad)


01 03 07 09 110

005

01

015

02

025

03

035

04

045

05


NI

Mode1Mode2

01 03 07 09 110

05

1

15

2

25

3

35


FS

Mode1Mode2


0001 0005 001 005 010

005

01

015

02

025

03

035


NI

Mode1Mode2

0001 0005 001 005 010

1

2

3

4

5

6

7

8

9

10


FS

Mode1Mode2



Methodologies

0 005 01 015 02minus01

0

01

02

δ1(rad)

Dynamics

0 005 01 015 02minus02

minus01

0

01

02

t(s)

δ2(rad)

EXNNMLNM

(a) X1 = X2 = 01

0 005 01 015 0205

1

15

2

δ1(rad)

Dynamics

EXNNMLNM

0 005 01 015 020

05

1

15

t(s)

δ2(rad)

(b) X1 = X2 = 11

0 02 04 06 08 1minus5

0

5x 10

minus4

Errorin

δ1(rad)

Error

0 02 04 06 08 1minus5

0

5x 10

minus4

Errorin

δ2(rad)

NNMLNM

0 02 04 06 08 1

minus05

0

05

Errorin

δ1(rad)

Error

0 02 04 06 08 1

minus05

0

05

Errorin

δ2(rad)

NNMLNM



0 005 01 015 020

05

1

15

δ1(rad)

Dynamics

EXNNMLNM

0 005 01 015 02minus1

0

1

2

t(s)

δ2(rad)

(a) X12 = 0001

0 005 01 015 020

05

1

15

δ1(rad)

Dynamics

EXNNMLNM

0 005 01 015 02minus1

0

1

2

t(s)

δ2(rad)

(b) X12 = 01

0 02 04 06 08 1minus05

0

05

Errorin

δ1(rad)

Error

NNMLNM

0 02 04 06 08 1minus05

0

05

Errorin

δ2(rad)

0 02 04 06 08 1minus01

0

01

Errorin

δ1(rad)

Error

NNMLNM

0 02 04 06 08 1minus01

0

01

Errorin

δ2(rad)



Methodologies


277 Closed Remarks



Forms in Practice




f(z) = z minus y0 + h(z) = 0 (2147)









fj(w) = wj +Nsumk=1

Nsuml=1


Nsumq=1

Nsumr




fj(w(s)) = w

(s)j +

Nsumk=1

Nsuml=1

h2jklh2jklw(s)k w

(s)l +

Nsump=1

Nsumq=1

Nsumr


q w(s)r = 0


Methodologies


[A(ws)] =

[partf

partw

]w=w(s)




w(s+1) = w(s) + ρ4w(s)



The algorithm ends




fp = Rp +Nsumi=1

Nsumjgei

apijRiRj +Nsumi=1

Nsumjgei

Nsumkgej











Methodologies




X1

Y1

Xp

Yp

XN

˙YN

=

0 1



0

0 1


0

0


0 1


X1

Y1

Xp

Yp

XN

YN

+

g(X1) + h(X1)

0

g(Xp) + h(Xp)

0

g(XN) + h(XN)

0

(2149)





sumNk=1


p=1

sumNq=1











sumNi=1 vji

[UThj3U

]

F3jpqr = 16

sumNi=1 vji

sumNk=1

sumNk=1

sumNm=1H3ikLNMu

lpumq u

Nr





Methodologies

















Methodologies

Chapter 3



















115



31 Introduction






31 Introduction 117






Powerf(δ)

02

04

06

08

1

12

14


Exact

Linear

Tayl(2)

Tayl(3)

Regionmaybeunstable

3rd Order Analysis

2nd Order Analysis

Linear Analysis









x = f(xu) (31)


4x = H1(4x) +1

2H2(4x) +

1

3H3(4x) +O(4) (32)



] H3jklm =


]and O(4) are






yj = λjyj +

Nsumk=1

Nsuml=1

F2jklykyl +

Nsump=1

Nsumq=1

Nsumr=1



sumNi=1 vji

[UThj3U

] F3jpqr = 1

6

sumNi=1 vji

sumNk=1

sumNk=1

sumNm=1H3iklmu

lpumq u

nr vji is






322 Method 2-2-1



with h2jkl =F2jkl


323 Method 3-2-3S



then

y = z + h2(z)rArr

zj = λjwj +

sumMl=1 c



(37)


324 Method 3-3-1





325 Method 3-3-3



wj = λjwj +

sumMl=1 c














tem









di

qi

x

y

dk

qk

δiδk

ωs





δi = ωsωk

ωi = 12Hi


˙eprimeqi = 1

Tprimed0i


primeqi)

˙eprimedi = 1

Tprimeq0i


primedi)

(310)





4idqminusxy =

sin δi cos δi


4ixyminusdq =


cos δi sin δi

(311)



primed0T

primeq0


primed0 are the


primeq






11+sTR

times 1+sTC1+sTB

KA1+sTA

VT

+

Vref

minusXE1 XE2 Efd



˙Efdi = KAiXE2iTAi

minus 1TAi

Efdi

˙XE1i = minus 1TRi

XE1i + 1TRi

VTi

˙XE2i = 1TBi


TBiXE2i minus TCi

TBiTRiVT i + 1

TBiVrefi

(312)


VT i =radic


primeqiiqi)







primed e


]






eprimexieprimeyi

= 4idqminusxy

eprimedieprimeqi

It = Y

[eprimex1 e


primexn e

primeyn



primeyiiyi

(315)

where eprimeqi e


primexi e

primeyi are









I1

I2

Im

Im+1

Im+n

=






V1

V2

Vm

Vm+1

Vm+n

(317)




In

=

Ymm Ymn

Ynm Ynn

Vm

Vtn

(318)



]Vtn (319)



](320)



Ytijej =Nsumj=1


Nsumj=1

[(Gtijexi minusBt



Yr =

Gt11 minusBt11

Bt11 Gt11


Gt1n minusBt1n

Bt1n Gt1n

Gtn1 minusBt

n1

Btn1 Gtn1


Gtnn minusBtnn

Btnn Gtnn

(321)




primen

] an admittance




primexn e

primeyn

](322)



primexn e

primeyn




Since edieqi

=

eprimedieprimeqi


idiiqi

=

eprimedieprimeqi

minus Ziidiiqi

(323)


4idqminusxy

edieqi

= 4idqminusxy

eprimedieprimeqi

minus4idqminusxyZk

idiiqi

(324)


=

eprimexieprimeyi


xyminusdq

ixiiyi

(325)



dqminusxy

] T2 =


ex1

ey1

ex2

ey2

exn

eyn

=

eprimex1

eprimey1

eprimex2

eprimey2

eprimexn

eprimeyn

minus T1T2Tminus11

ix1

iy1

ix2

iy2

ixn

iyn

=rArr Y minus1r

ix1

iy1

ix2

iy2

ixn

iyn

= Y minus1

ix1

iy1

ix2

iy2

ixn

iyn

minus T1T2Tminus11

ix1

iy1

ix2

iy2

ixn

iyn

(326)




If Xprimeq = X

primed then 4i






(328)



prime=[

eprimed1 e


primedn e

primeqn

]can be obtained as


]T= Tminus11 Y T1

[eprimed1 e


primedi e


]T(329)




ωi = 12Hi


eprimeqi = 1

Tprimed0i


primeqi)

eprimedi = 1

Tprimeq0i


primedi)

˙Efdi = KAiXE2iTAi

minus 1TAi

Efdi

˙XE1i = minus 1TRi

XE1i + 1TRi

VT i

˙XE2i = 1TBi


TBiXE2i minus TCi

TBiTRiVT i + 1

TBiVrefi

(331)




δi

ωi

eprimeqi

eprimedi

˙Efdi

˙XE1i

˙XE2i

=

0 ωs 0 0 0 0 0

0 minus Di2Hi

0 0 0 0 0

0 0 minus 1

Tprimed0i

0 1

Tprimed0i

0 0

0 0 0 minus 1

Tprimeq0i

0 0 0

0 0 0 0 minus 1TAi

0 KAiTAi


0

0 0 0 0 0 1TBi


TBi

δi

ωi

eprimeqi

eprimedi

Efdi

XE1i

XE2i

+

0


2Hi


Tprimed0i

idi

XqiminusXprimeqi

Tprimeq0i

iqi

0

VTiTRi

minus TCiTBiTRi

VT i

+

0

Pmi2Hi

0

0

0

0

VrefiTBi

(332)


As [Idq] = Yδ

[eprimedq



xi =

δi

ωi

eprimeqi

eprimedi

Efdi

XE1i

XE2i

Agi =

0 ωs 0 0 0 0 0

0 minus Di2Hi

0 0 0 0 0

0 0 minus 1

Tprimed0i

0 0 0 0

0 0 0 minus 1

Tprimeq0i

0 0 0

0 0 0 0 minus 1TAi

0 KAiTAi


0

0 0 0 0 0 1TBi


TBi

Ni(X) =

0


2Hi


Tprimed0i

idi

XqiminusXprimeqi

Tprimeq0i

iqi

0

VTiTRi

minus TCiTBiTRi

VT i

Ui =

0

Pmi2Hi

0

0

0

0

VrefiTBi

(333)


x1

x2

xn

=




x1

x2

xn

+

N1(x)

N2(x)

Nn(x)

+

U1

U2

Un

(334)

x = Agx+N(x) + U (335)




4x = Ag4x+N1(4x) +1

2N2(4x) +

1

3N3(4x) (336)


H1 = Ag +N1 H2 = N2 H3 = N3 (337)

























States


primeq2 E

primeq3 E

primeq4










States


primeq2 E

primeq3 E

primeq4










0 5 10 15 20 25 30

δ1(degree)

40

70

100

130140

t(s)0 5 10 15 20 25 30

δ2(degree)

0

30

60

90100

Exact 3-2-3S 2-2-1 3-3-1 3-3-3 Linear



Hz0 036 072 108 144 180 212 252

Mag

(

of F

unda

men

tal)

101

102

Exact3-2-3S2-2-13-3-13-3-3Linear


0 5 10 15 20 25 30

δ1(degree)

-100

0

100

180

t(s)0 5 10 15 20 25 30

δ2(degree)

-100

-50

0

50

100

150180200

Exact 3-2-3S 3-3-3 2-2-1 3-3-1 Linear





t(s)0 10 20 30

(pu)

085

09

095

1

105

11

115Eprime

q1 Eprime

q2 Eprime

q3 Eprime

q4

Eq1

Eq2

Eq3

Eq4

t(s)0 5 10 15 20 25 30

(pu)

-55Vf

min

-4

-3

-2

-1

0

1

2

3

4

Vfmax


Vmex2

Vfex2



primeq2 E

primeq3 E









j y0 z0221 z0331 z0333

5 514 + j293 389 + j238 388 + j238 03105 + j03263





15 032 + j010 004minus 00012i 005minus j00012 0072 + j005

17 087 + j252 082 + j216 084 + j210 170 + j131i



(171717) 0046minus j0118 0029 + j0034 0017minus j0019


01220 00068 + j0008 00560 033














7 048 + j096 minus621 + j883 096

9 00117 + j017 minus1837 + j628 098


13 0074minus j025 minus178 + j673 085

15 089 + j0532 minus196 + j651 086

17 012minus j0678 minus0405 + j228 033











5 minus5354minus j10977 033 -176682











17 029 + j131 11801 034











Tprimeq0i


primeqi)minusIqi +

(XprimeqiminusX

primeprimeqi)




Tprimed0i

dEprimeqi


primedi)Idi +

(XprimediminusX

primeprimedi)



primeqi)

Tprimeprimed0i

dψ1didt = E

primeqi + (X


prime1di)

Tprimeprimeq0i

dψ2qi


prime2qi)


primeprimedi minusX


primedci

(338)




(XprimeqiminusXlsi)

+ EprimeqiIqi


(XprimediminusXlsi)


primeprimeqi) +


primeprimedi)

(XprimediminusXlsi)


primeprimeqi)

(XprimeqiminusXlsi)


primeqi


(XprimediminusXlsi)

+ψprime1di

(XprimediminusX

primeprimedi)

(XprimediminusXlsi)

minusVqi+j[Eprimedi


(XprimeqiminusXlsi)

minus

ψprime2qi

(XprimeqiminusX

primeprimeqi)

(XprimeqiminusXlsi)


qi

di

x

y

qk

dk

δiδk

ωs






TedEfd





s + sKdTd+1)



(339)







(340)




Vss = KpsssTw

1 + sTw

(1 + sT11)

(1 + sT12)

(1 + sT21)

(1 + sT22)

(1 + sT31)

(1 + sT32)Sm (341)









1 -0438 0404 δ(13) 1 ω(13) 0741 ω(15) 0556 ω(14) 0524

2 0937 0526 δ(14) 1 ω(16) 0738 ω(14) 05 δ(13) 0114

3 -3855 061 δ(13) 1 ω(13) 083 δ(12) 0137 ω(6) 0136

4 3321 0779 δ(15) 1 ω(15) 0755 ω(14) 0305 δ(14) 0149

5 0256 0998 δ(2) 1 ω(2) 0992 δ(3) 0913 ω(3) 0905

6 3032 1073 δ(12) 1 ω(12) 0985 ω(13) 0193 δ(13) 0179

7 -1803 1093 δ(9) 1 ω(9) 0996 δ(1) 0337 ω(1) 0333

8 3716 1158 δ(5) 1 ω(5) 1 ω(6) 0959 δ(6) 0958

9 3588 1185 ω(2) 1 δ(2) 1 δ(3) 0928 ω(3) 0928

10 0762 1217 δ(10) 1 ω(10) 0991 ω(9) 0426 δ(9) 042

11 1347 126 ω(1) 1 δ(1) 0996 δ(10) 0761 ω(10) 0756

12 6487 1471 δ(8) 1 ω(8) 1 ω(1) 0435 δ(1) 0435

13 7033 1487 δ(4) 1 ω(4) 1 ω(5) 0483 δ(5) 0483

14 6799 1503 δ(7) 1 ω(7) 1 ω(6) 0557 δ(6) 0557

15 3904 1753 δ(11) 1 ω(11) 0993 Ψ1d(11) 0056 ω(10) 0033






1 33537 0314 ω(15) 1 ω(16) 0982 δ(7) 0884 δ(4) 0857

2 3621 052 δ(14) 1 ω(16) 0645 ω(14) 0521 δ(15) 0149

3 9625 0591 δ(13) 1 ω(13) 083 ω(16) 0111 ω(12) 0096

4 3381 0779 δ(15) 1 ω(15) 0755 ω(14) 0304 δ(14) 0148

5 27136 0972 ω(3) 1 δ(3) 0962 ω(2) 0924 δ(2) 0849

6 18566 108 ω(12) 1 δ(12) 0931 Eprimeq(12) 0335 PSS4(11) 0196

7 23607 0939 δ(9) 1 ω(9) 0684 PSS3(9) 0231 PSS2(9) 0231

8 30111 1078 δ(5) 1 ω(5) 0928 ω(6) 0909 δ(6) 0883

9 28306 1136 ω(2) 1 δ(2) 0961 δ(3) 0798 ω(3) 0777

10 13426 1278 ω(1) 1 δ(1) 0969 Eprimeq(1) 0135 ω(8) 0111

11 1883 1188 δ(10) 1 ω(10) 0994 Eprimeq(10) 0324 PSS3(10) 019

12 32062 1292 δ(8) 1 ω(8) 0935 Eprimeq(8) 061 PSS2(8) 0422

13 39542 1288 δ(4) 1 ω(4) 0888 Eprimeq(4) 0706 PSS4(4) 0537

14 33119 1367 δ(7) 1 ω(7) 0909 δ(6) 0653 ω(6) 0612

15 2387 5075 ω(11) 1 Eprimeq(11) 0916 Ψ1d(11) 0641 PSS4(10) 0464



Frequency (Hz)0 02 04 06 08 10 12 14 16 18 2

Mag

( o

f Fun

dam

enta

l)

100

101

102G1G12G13G14G15



Mode 1 2 3 4


frequency (Hz) 07788 06073 05226 03929










Frequency (Hz)0 02 04 06 08 10 12 14 16 18 20

Mag

(

of F

unda

men

tal)

100

101

102G6G9G13G14G15





3-3-3 12 1294e-12 17 1954e-6

2-2-13-2-3 3 1799e-11 6 1838e-06

3-3-1 7 1187e-09 11 7786e-09

literature












real(cj=2k2k )

= 0




















Chapter 4
















151


Assessment

41 Introduction



fer Limit


δ = ω (41)



















Assessment







δ = ωsω (42)




t(s)0 5 10 15 20 25 30 35 40 45 50

δ(rad)

-1

-05

0

05

1

15




Assessment


with

Pmax =EprimeqE0



t(s)0 05 1 15 2 25 3 35 4

δ(rad)

-3

-2

-1

0

1

2

3

Exact Linear SEP



















Assessment



























Assessment















o-line2 Step-by-

step inte-gration



1 Mostly-Numerical


















Mq +Dq +Kq + fnl(q) = 0 (44)


fpnl(q) =

Nsumi=1jgei

f2pijqiqj +

Nsumi=1jgeikgej

f3pijkqiqjqk (45)


4241 LNM



Assessment



+

Nsumi=1

Nsumjgei

gpijXiXj +

Nsumi=1

Nsumjgei

Nsumkgej

hpijkXiXjXk = 0




4242 NM2


Up + 2ξpωpUp + ω2pUp +O(3) = 0 (48)

4243 NNM




pppRpR2p + CppppR

2pRp = 0







with

Rp le

radicminus

ω2p

hpppp +Apppp(411)



Knl =RpXp

=1












δ = ωsω (413)


with

Pmax =EprimeqV0Xe




Assessment




2Hi


Dωi

ωbδi (416)

where

Pei = E2iGgii +

Nsumk=1k 6=i









(418)



=

sumN

k=1k 6=i


] j = i


] j 6= i

(419)



2part2Peipartδiδj

reads

Ki2Sii = minus

sumNk=1k 6=i


]Ki

2Sij = EiEj


] j 6= i

Ki2Sjj = minusEiEj


] j 6= i

(420)



it reads

Ki3Siii = 1

6

sumNk=1k 6=i


]Ki

3Siij = minus12EiEj


] j 6= i

Ki3Sijj = 1

2EiEj


] j 6= i

Ki3Sjjj = minus1

6EiEj


](421)




2Hi

ωbδi minus

Dwi


2Hn

ωbδn minus

Dwn



2H



2H

ωb4δN +

D



Assessment





](426)



reads

Kprimei2Sii = Ki


]Kprimei2Sij = Ki

2Sijj 6= i

Ki2Sjj = Ki

2Sjj + EnEj


]j 6= i

(427)



it reads

Kprimei3Siii = Ki


]Kprimei3Siij = Ki

3Siij j 6= i

Kprimei3Sijj = Ki

3Sijj j 6= i

Kprimei3Sjjj = Ki

3Sjjj + 16EnEj


](428)

44 Case-studies






44 Case-studies 167

t(s)0 05 1 15 2 25 3 35 4

δ(rad)

1

12

14

16

EX LNM NM2 NNM SEP


t(s)0 02 04 06 08 1 12 14 16 18 2

δ(rad)

-3

-2

-1

0

1

2

3

EX LNM NM2 NNM SEP





Assessment




PSBminusEABEAB




44 Case-studies 169

t(s)0 05 1 15 2 25 3

δ(rad)

-4

-2

0

2

4

6

EX LNM NM2 NNM SEP


12

t(s)0 05 1 15 2 25 3

δ(rad)

-4

-2

0

2

4



12



Assessment








452 On-going Work



0 5 10 15 20

δ14(rad)

0

05

1

15

2

25

t(s)0 5 10 15 20

δ24(rad)

-05

0

05

1

15

2EX LNM NM2 NNM SEP


0 5 10 15 20

δ14(rad)

0

1

2

3

EX LNM NM2 NNM SEP

t(s)0 5 10 15 20

δ24(rad)

0

1

2



Assessment

0 5 10 15 20

δ14(rad)

-2

0

2

4

EX LNM NM2 NNM SEP

t(s)0 5 10 15 20

δ24(rad)

-2

0

2

4


0 5 10 15 20

δ14(rad)

-2

0

2

4

EX LNM NM2 NNM SEP

t(s)0 5 10 15 20

δ24(rad)

-2

0

2

4






Assessment

Chapter 5


Oscillations


























175


Oscillations













Oscillations









Oscillations












Oscillations




542 The Limitations











Oscillations






M q +Dq +Kq = F (51)



ΦTj Fj

Mj(52)













Mq +Cq +Kq + fnl(q) = 0 (53)


Oscillations









552 Originality





Mq +Cq +Kq + fnl(q) = F (54)





+

Nsumi=1

Nsumjgei

gpijXiXj +

Nsumi=1

Nsumjgei

Nsumkgej

hpijkXiXjXk = Fp











Oscillations

5552 NNM




pppRpR2p + CppppR

2pRp = 0

5553 FDNF3






pRp






5561 Test System







M1d2δ1dt2

+D1dδ1dt +

V1VgX1

sin δ1 + V1V2X12


M2d2δ2dt2

+D2dδ2dt +

V2VgX2

sin δ2 + V1V2X12


(59)



Oscillations

0 05 1 15 2 2505

055

06

065

07

075

08

085

09

t(s)

δ1(rad)

Dynamics

EXLNMFDNF3

0 05 1 15 2 2502

025

03

035

04

045

05

055

06

065

t(s)

δ2(rad)

Dynamics

EXLNMFDNF3









0 05 1 15 2minus05

0

05

1

15

t(simulation time)

δ1(rad)

Mode Decomposition

EXMode 1Mode 2

0 05 1 15 2minus05

0

05

1

15

t(simulation time)

δ2(rad)


0 05 1 15 2 25minus01

0

01

02

03

04

05

06

t(s)

NonlinearModalState

Variables


R1R2



Oscillations











pppRpR2p + CppppR

2pRp = F







EMR+IBC






Oscillations











Oscillations






58 Conclusion 197



x = f(xu) (511)


u = fc(x) (512)


x = f(x fc(x)) (513)

58 Conclusion



Oscillations

Chapter 6




classes of the





199




























Form the DAE model

Eqs (1)(2)




around SEP Eq (3)








S

E

P

I

n

i

t

i

a

l

i

z

a

t

i

o

n


variabledisturbance

variableSEP

Do eigen analysis

of matrix A

Calculate the NF


h2h3 c

End

Start

Rapid




ologies


641 Validity region


642 Resonance












x = f(xu) (61)


4x = H1(4x) +1

2H2(4x) +

1

3H3(4x) +O(4) (62)



] H3jklm =


]and O(4) are



Mq +Cq +Kq + fnl(q) = F (63)




x = f(x x) (64)

or Eq (65)

Mq +Cnl(q) +Kq + fq = F (65)





x = f(x x) (66)

or Eq (67)

Mclq +C(q) +Kq + fq = F (67)













207

208 Publications

Appendix A

Interconnected VSCs


usp=sharing





209




A111 The Basic Unit



A12 Methodology












AC LoadSource






(A2)

Filter C






lg middot ωg(A4)










lowastod = Vod i

lowastod = iod i



J1d2δ1dt2


sin δ1 + V1V2L12


J2d2δ1dt2


sin δ2 + V1V2L12


(A5)








ω10 ω2 = ω2+ω20dω1dt =


dt =d(ω2minusωg)



dt = dω1dt

dω2dt =










J1 16 L1(pu) 07 V1 1

J2 16 L2(pu) 07 V2 1

D1 10 L12(pu) 001 Vg 1

D2 10 wb 314


P lowast1 =V1VgL1

sin δ10 +V1V2L12


V1VgL2

sin δ20 +V1V2L12

sin δ210


J1ωb

d2δ1dt2

+ D1ωb

dδ1dt +

V1VgL1



J2ωb

d2δ2dt2

+ D2ωb

dδ2dt +

V2VgL2

















J1=0216D1=20J2=0216D2=20w_b=100314

g1=J1w_b

g2=J2w_b

d1=D1w_b

d2=D2w_b

syms g1 g2 d1 d2



delta210=-delta120

L1=07L2=07L12=001wg=1w_b=502314


V1=1V2=1Vg=1w10=1w20=1



Natural frequencies

M=[g1 00 g2]



D=[d1 00 d2]


[TrE]=eig(inv(M)K)



Normalisation of T

Tr=Trdet(Tr)

leg=sqrt(sum(Tr^2))

Tr=Trleg(ones(12))

normalization


modal matrices

Mm=TrMTr

Dm=TrDTr

Km=TrKTr

T11=Tr(11)

T12=Tr(12)

T21=Tr(21)

T22=Tr(22)

division by Mm


Dm_div=inv(Mm)Dm

OMEGA=double(OMEGA)

for i=12


end
















Gg=zeros(2222)




Hg(1111)=TEC(1111)T11^3+TEC(1112)T21T11^2+

TEC(1122)T11T21^2+TEC(1222)T21^3


TEC(1122)(T12T21^2+2T11T21T22)+TEC(1222)3T21^2T22


TEC(1122)(2T12T21T22+T11T22^2)+TEC(1222)3T21T22^2

Hg(1222)=TEC(1111)T12^3+TEC(1112)T22T12^2+

TEC(1122)T12T22^2+TEC(1222)T22^3





TEC(2222)T21^3


TEC(2122)(T12T21^2+2T11T21T22)+TEC(2222)3T21^2T22


TEC(2122)(2T12T21T22+T11T22^2)+TEC(2222)3T21T22^2



TEC(2222)T22^3



G=zeros(222)

H=zeros(2222)

G(111)=(T11Gg(111)T21Gg(211))Mm(11)

G(112)=(T11Gg(112)T21Gg(212))Mm(11)

G(122)=(T11Gg(122)T21Gg(222))Mm(11)

G(211)=(T12Gg(211)T22Gg(211))Mm(22)

G(212)=(T12Gg(212)T22Gg(212))Mm(22)

G(222)=(T12Gg(222)T22Gg(222))Mm(22)

for p=12

for k=12

for j=k2


++j

end

++k

end

++p

end

for p=12

for i=12

for k=i2

for j=k2


++j

end

++k

end

++i

end

++p

end

H=zeros(2222)

c1=[G(111) G(112) G(122) H(1111) H(1112) H(1122) H(1222)]

c2=[G(211) G(212) G(222) H(2111) H(2112) H(2122) H(2222)]






system






N=length(OMEGA)

correction=1

quadratic coefs

A=zeros(NNN)

B=zeros(NNN)

C=zeros(NNN)

ALPHA=zeros(NNN)

BETA=zeros(NNN)

GAMMA=zeros(NNN)

for p=1N

for i=1N































xi(i))




end

end

for p=1N

for i=1N

for j=i+1N



if abs(D) lt 1e-12

A(pij)=0

B(pij)=0

C(pij)=0

C(pji)=0

ALPHA(pij)=0

BETA(pij)=0

GAMMA(pij)=0

GAMMA(pji)=0

else














B2=[G(pij) 0 0 0]

solabcij=inv(A2)B2

solabcij=A2B2

A(pij)=solabcij(1)

B(pij)=solabcij(2)

C(pij)=solabcij(3)

C(pji)=solabcij(4)





end

end

end

end


AA=zeros(NNNN)

BB=zeros(NNNN)

CC=zeros(NNNN)

AA2=zeros(NNNN)

BB2=zeros(NNNN)

CC2=zeros(NNNN)

for p=1N

for i=1N

for j=1N

for k=1N

V1=squeeze(G(piiN))

V2=squeeze(A(iNjk))

V3=squeeze(G(p1ii))

V4=squeeze(A(1ijk))


V5=squeeze(G(piiN))

V6=squeeze(B(iNjk))

V7=squeeze(G(p1ii))


V8=squeeze(B(1ijk))


V9=squeeze(G(piiN))





end

end

end

end

cubic coefs

R=zeros(NNNN)

S=zeros(NNNN)

T=zeros(NNNN)

U=zeros(NNNN)

LAMBDA=zeros(NNNN)

MU=zeros(NNNN)

NU=zeros(NNNN)

ZETA=zeros(NNNN)


for p=1N

for i=1N

if i~=p


if abs(D1) lt 1e-12

R(piii)=0

S(piii)=0

T(piii)=0

U(piii)=0

LAMBDA(piii)=0

MU(piii)=0

NU(piii)=0

ZETA(piii)=0

else












sol3iii=A3iiiB3iii

R(piii)=sol3iii(1)

S(piii)=sol3iii(2)

T(piii)=sol3iii(3)

U(piii)=sol3iii(4)





end

end

end

end

for p=1N

for i=1N-1

if i~=p

for j=i+1N




R(pijj) = 0

S(pijj) = 0

T(pijj) = 0

T(pjij) = 0

U(pijj) = 0

U(pjij) = 0

LAMBDA(pijj)=0

MU(pijj) = 0

NU(pijj) =0

NU(pjij) =0

ZETA(pijj) = 0

b 21n132

ZETA(pjij) = 0

else

























xi(j)OMEGA(j)]




solrstuip=AipBip













2OMEGA(j)^2U(pijj)




end

end

end

end

end

for p=1N

for i=1N-1

for j=(i+1)N

if j~=p




R(piij) =0

S(piij) =0

T(pjii) =0

T(piij) =0

U(pjii) =0

U(piij) =0

LAMBDA(piij)=0

NU(pjii) =0

NU(piij) =0

MU(piij) =0

ZETA(pjii) =0

ZETA(piij) =0

else

























BB(piij) BB(pjii)]


solrstujpp=AjppBjpp













OMEGA(j)^2U(piij)



end

end

end

end

end

for p=1N

for i=1N-2

for j=(i+1)N-1

for k=(j+1)N





OMEGA(k)-OMEGA(i))

if abs(D4) lt 1e-12

U(pijk) =0


U(pjik) =0

U(pkij) =0

R(pijk) =0

MU(pijk) =0

NU(pkij) =0

NU(pjik) =0

NU(pijk) =0

else


















































Am1=inv(Alastijkp)


R(pijk)=solrstu(1)

S(pijk)=solrstu(2)

T(pijk)=solrstu(3)

T(pjik)=solrstu(4)

T(pkij)=solrstu(5)

U(pijk)=solrstu(6)

U(pjik)=solrstu(7)

U(pkij)=solrstu(8)


OMEGA(j)^2U(pijk)




OMEGA(i)^2T(pijk)




OMEGA(k)^2U(pijk)


OMEGA(k)^2U(pjik)






end

end

end

end

end

disp(end)

A3 Simulation Files




oscillations

Appendix B





Xp = Rp +

Nsumi=1

Nsumjgei


Nsumi=1

Nsumj=1

cpijRiSj

+

Nsumi=1

Nsumjgei

Nsumkgej


+

Nsumi=1

Nsumj=1

Nsumkgej


Yp = Sp +

Nsumi=1

Nsumjgei

(αpijRiRj + βp

ijSiSj) +

Nsumi=1

Nsumj=1

γpijRiSj

+

Nsumi=1

Nsumjgei

Nsumkgej


ijkSiSjSk)

+

Nsumi=1

Nsumj=1

Nsumkgej



p = Xinitp + 4Xp Y

RTp =


231




4Xp = 4Rp +Nsumi=1

PR(1)pi4Ri +Nsumi=1

PS(1)pi4Si

Nsumi=1


PS(S2)pi4Si

Nsumi=1

PR(S2)pi4Ri +

Nsumi=1

PS(R2)pi4Si

4Yp = 4Sp +Nsumi=1

QR(1)pi4Ri +Nsumi=1

QS(1)pi4Si

Nsumi=1


QS(S2)pi4Si

Nsumi=1

QR(S2)pi4Ri +

Nsumi=1

QS(R2)pi4Si

with

PR(1)pi =isum

j=1

apjiRj +Nsumjgei

apijRj +Nsumj=1

cpijSj (B1)

PS(1)pi =isum

j=1

bpjiSj +Nsumjgei

bpijSj +Nsumj=1

cpjiRj

PR(R2)pi =

isumj=1

isumkgej

rpkjiRjRk +

isumj=1

Nsumkgei

rpjikRjRk +

Nsumjgei

Nsumkgej

rpijkRjRk

PS(S2)pi =

isumj=1

isumkgej

spkjiSjSk +

isumj=1

Nsumkgei

spjikSjSk +

Nsumjgei

Nsumkgej

spijkSjSk

PR(S2)pi =Nsumj=1

isumk=1

tpjkiSjRk +Nsumj=1

Nsumkgei

tpjikSjRk +Nsumj=1

Nsumkgej

upijkSjSk

PS(R2)pi =

Nsumj=1

isumk=1

upjkiRjSk +

Nsumj=1

Nsumkgei

upjikRjSk +

Nsumj=1

Nsumkgej

tpijkRjRk


QR(1)pi =isum

j=1

αpjiRj +Nsumjgei

αpijRj +Nsumj=1

γpijSj (B2)

QS(1)pi =

isumj=1

βpjiSj +

Nsumjgei

βpijSj +

Nsumj=1

γpjiRj

QR(R2)pi =

isumj=1

jsumkgej

λpkjiRjRk +

isumj=1

Nsumkgei

λpjikRjRk +

Nsumjgei

Nsumkgej

λpijkRjRk

QS(S2)pi =isum

j=1

isumkgej

micropkjiSjSk +isum

j=1

Nsumkgei


Nsumkgej

micropijkSjSk

QR(S2)pi =Nsumj=1

isumk=1

νpjkiSjRk +Nsumj=1

Nsumkgei

νpjikSjRk +Nsumj=1

Nsumkgej

ζpijkSjSk

QS(R2)pi =

Nsumj=1

isumk=1

ζpjkiRjSk +

Nsumj=1

Nsumkgei

ζpjikRjSk +

Nsumj=1

Nsumkgej

νpijkRjRk

Let



(B3)

Then we obtain 4X

4Y

=

I +XR XS

Y R Y S

4R

4S

(B4)

and 4R

4S

=

I +XR XS

Y R Y S

minus1 4X

4Y

=

[NLRT

]minus1

4X

4Y

(B5)



Appendix C



System



Ra 00025 xl 0002

xd 180 τprimed0 80


xprimed 03 τ

primeprimed0 00

xprimeq 03 τ

primeprimed0 00




Load Voltage Zshunt

Bus (pu230kv)

B5 1029 6 1975 P = 66584MW Q = minus35082Mvar


235





G1 PV 1006 1402 74750 4994

G2 swingbus 10006 00 40650 14074

G3 PV 1006 minus 5817 53925 11399

G4 PV 1006 minus 6611 52500 14314




Gen 1 15 35 35 150 150 240

Gen 2 15 35 35 150 150 240

Gen 3 11 2 2 150 150 240

Gen 4 10 2 2 150 150 240









clear all

close all

Load the system


clear all



Define the system


omega_s=120pi









Yt=[39604-395821i -39604+396040i 00000+00000i 00000+00000i

-39604+396040i 103914-424178i 00000+00000i 04578+41671i

00000+00000i 00000+00000i 39604-395821i -39604+396040i

00000+00000i 04578+41671i -39604+396040i 182758-431388i]

Gt=real(Yt)

Bt=imag(Yt)


















T_e=ones(41)001

T_1=ones(41)10

T_2=ones(41)1









Vf= sym(Vf [Gn 1])

Vm= sym(Vm [Gn 1])

Vr= sym(Vr [Gn 1])

Yr=zeros(2Gn2Gn)

for l=1Gn

for m=1Gn

Yr(2l-12m-1)=Gt(lm)

Yr(2l-12m)=-Bt(lm)

Yr(2l2m-1)=Bt(lm)

Yr(2l2m)=Gt(lm)

end

end






for l=1Gn



T2(2l-12l-1)=Ra(l) T2(2l-12l)=-Xlq(l)


end

Y=inv(inv(Yr)+T29)

for l=1Gn

for m=1Gn

G1(lm)=Y(2l-12m-1)

B1(lm)=-Y(2l-12m)

B1(lm)=Y(2l2m-1)

G1(1m)=Y(2l2m)

end

end

Ydelta=inv(T1)YT1

for j=1Gn

eldq(2j-1)=eld(j)

eldq(2j)=elq(j)

end

Idq=Ydeltaeldq


for j=1Gn

id(j)=Idq(2j-1)

iq(j)=Idq(2j)

end

for j=1Gn





end

for j=1Gn


elxy(2j-11)=elx(j)

elxy(2j1)=ely(j)

end

Ixy=Yelxy









Vref=zeros(Gn1)

Vref_Exc_1=103

Vref_Exc_2=101

Vref_Exc_3=103

Vref_Exc_4=101









for j=1Gn-1








end

for j=Gn








end

for j=1Gn-1



end

for j=Gn



end



for j=1Gn-1


end

for j=Gn


end

x027=x027

Fv_R=vpa(F27)




x0test=zeros(281)



EIV_R=eig(F1t_R)










ie















for p=1L


end

for p=1L

for k=1L


end

end



F1=N1












[TuLambda]=eig(A)


invTu=inv(Tu)

Calculate F2

F21=zeros(LLL)

F2=zeros(LLL)

for p=1L



end

for p=1L

for j=1L

for k=1L


end

end

end

Calculate F3

F31=zeros(LLLL)

F3=zeros(LLLL)

for j=1L

for p=1L

for q=1L

for r=1L

for l=1L

for m=1L

for n=1L


end

end

end

end

end

end


end

for j=1L

for p=1L

for q=1L

for r=1L


end

end

end

end






N=length(EIV)

quadratic coefs

Cla=zeros(NNN)

for p=1N

for j=1N

for k=1N


end

end

end


end


beginverbatim





ModeNumber=[]




m=0


for j=1ModeSize


m=m+1

ModeNumber(m)=j

end

end




L=length(EIV)


MN=ModeNumber

N1718=0

xdot=zeros(L1)

c3=zeros(LL)

term_3=zeros(L1)

F2H2=zeros(LLLL)

F2c=zeros(LL)

C3=zeros(LLLL)

h2=zeros(LLL)

h3=zeros(LLLL)

for p=1M

for l=1M

for m=1M

for n=1M


end

end

end

end

for p=1L

for l=1L

for k=1L



end

for m=1L

for n=1L



end

end

end

end

for i=1(M2)

for j=1(M2)

if j~=i








C3(MN(2i-1)MN(2j)MN(2i-1)MN(2j-1))+

C3(MN(2i-1)MN(2j)MN(2j-1)MN(2i-1))+

C3(MN(2i-1)MN(2i-1)MN(2j-1)MN(2j))+

C3(MN(2i-1)MN(2j-1)MN(2i-1)MN(2j))+

C3(MN(2i-1)MN(2j-1)MN(2j)MN(2i-1))

else





C3(MN(2i-1)MN(2i-1)MN(2i-1)MN(2i))+

C3(MN(2i-1)MN(2i)MN(2i-1)MN(2i-1))


end

end

end

for p=1M

for l=1M

for m=1M

for n=1M




end

end

end

end

for k=1(M2)

for l=1(M2)

for m=1(M2)

for n=1(M2)

if l~=k




h3(MN(2k-1)MN(2k-1)MN(2l)MN(2l-1))=0

h3(MN(2k-1)MN(2l)MN(2k-1)MN(2l-1))=0

h3(MN(2k-1)MN(2l)MN(2l-1)MN(2k-1))=0

h3(MN(2k-1)MN(2k-1)MN(2l-1)MN(2l))=0

h3(MN(2k-1)MN(2l-1)MN(2k-1)MN(2l))=0

h3(MN(2k-1)MN(2l-1)MN(2l)MN(2k-1))=0

else




h3(MN(2k-1)MN(2k-1)MN(2k)MN(2k-1))=0

h3(MN(2k-1)MN(2k)MN(2k-1)MN(2k-1))=0

h3(MN(2k-1)MN(2k-1)MN(2k-1)MN(2k))=0

end

end

end

end

end


end










MN=ModeNumber

M=L2

EIV=EIV_R

w=imag(EIV)

sigma=real(EIV)

F2=F2N_R

F3=F3N_R

quadratic coefs

h3=h3test_R

for j=1L

for p=1L

h3(jppp)=h3(jppp)

for q=(p+1)L


for r=(q+1)L


h3(jqrp)+h3(jrqp)

end

end

end

end

for j=1L

for p=1L

for q=1L

for r=1L


end

end

end

end

MaxMI333=zeros(277)

for j=1L


[Ih3_1Ih3_2Ih3_3] = ind2sub([27 27 27]Ih3)




end


------------------------)

C34 Normal Dynamics





L=length(CE)

xdot=zeros(L1)

term_3=zeros(L1)

Cam=zeros(42)

F2H2=zeros(LLLL)

F2c=zeros(LL)

C3=zeros(LLLL)

for i=1(L2)

for j=1(L2)




end

end

for p=1L


end

end




L=length(CE)

xdot=zeros(L1)

for p=1L

xdot(p)=CE(pp)x(p)

end

end


















constrained






[1e-6]



[1e-6]


[100]


[iter]






References






Version 05





Version 04



Version 03







Version 02








initialize



set default options

oldopts = optimset(


if narginlt3


else


end


get options









set scaling values




set display

if DISPLAY




fmtstr = 10d 104g 104g 104g 104g 124gn


end

check initial guess






else


end

if issparse(Jstar)



else


rc = NaN


rc = Inf

else


end

end



solver





if lambda==1











end



break



break

end








if lambda==1



else





[ab] = coeff

if a==0

lambda = -slope2b

else


if discriminantlt0


elseif blt=0


else


end

end


end




else


end

if lambdalt1



continue

end

display







break

end

if issparse(Jstar)


else



end


end

output




if Nitergt=MAXITER

exitflag = 0


elseif exitflag==2


elseif exitflag==-1


else


end

jacob = J

end



try

[F J] = fun(x)

catch

F = fun(x)


end


end


estimate J





for n = 1nx





end

end












N=length(tt)

L=length(y0)

ynl=zeros(L1)

ynl3=zeros(L1)

H3=zeros(L1)

for l=1L

for p=1L

for q=1L

for r=1L


end

end

end

end

for l=1L


end

f=ynl3-y0











N=length(tt)


L=length(y0)

ynl=zeros(L1)

ynl3=zeros(L1)

for l=1L


end

f=ynl3-y0

C4 Simulation les




Appendix D



D1 Programs





1 calcm

2 chq_limm


4 form_jacm

5 loadflowm

6 y_sparsem


Version 33



Date December 2013




clear all

clc

MVA_Base=1000

f=600


259



j=sqrt(-1)

Load Data

data16m_benchmark


N_Bus=size(bus1)

N_Line=size(line1)

Loading Data Ends

Run Load Flow


disply=yflag = 2






clc

display(Running)

Load Flow End



xls=mac_con(4)BM

Ra=mac_con(5)BM

xd=mac_con(6)BM

xdd=mac_con(7)BM

xddd=mac_con(8)BM

Td0d=mac_con(9)

Td0dd=mac_con(10)

xq=mac_con(11)BM

xqd=mac_con(12)BM

xqdd=mac_con(13)BM

Tq0d=mac_con(14)

Tq0dd=mac_con(15)

H=mac_con(16)BM

D=mac_con(17)BM

M=2H

wB=2pif





D1 Programs 261

else

Zg= Ra + jxdd

end

Yg=1Zg


AVR Initialization

Tr=ones(N_Machine1)





Td=ones(N_Machine1)

Ka=ones(N_Machine1)


Ta=ones(N_Machine1)

Ke=ones(N_Machine1)



Te=ones(N_Machine1)


Tf=ones(N_Machine1)







for i=11len_exc(1)



if(exc_con(i3)~=0)


end

if(exc_con(i5)~=0)


end

if(exc_con(i1)==1)













Kad(Exc_m_indx)=1



else




end

end


PSS initialization


Tw=ones(N_Machine1)










for i=11len_pss(1)












end



D1 Programs 263




for i=11N_Machine


end

YG=MMYg

YG=diag(YG)

V=bus_sol(2)



Y_Aug=Y+YL+YG

Z=inv(Y_Aug) It=

Y_Aug_dash=Y_Aug


Initial Conditions

Vg=MMVthg=MMtheta


mp=2

mq=2

kp=bus_sol(6)V^mp

kq=bus_sol(7)V^mq


y_dash=y

from_bus = line(1)

to_bus = line(2)




Q_s = imag(MW_s)


















delta0 = angle(Eq0)





VD0=imag(VDQ)

VQ0=real(VDQ)

Vref=Efd0



for i=11len_exc(1)


if(exc_con(i1)==1)

V_Ka0(Exc_m_indx)=




else


end

end

Tm0=Te0

Pm0=Tm0


Gn=16



Sm= sym(Sm [Gn 1])







D1 Programs 265
















Vq=real(Vdq)

Vd=imag(Vdq)



iq=real(idq)

id=imag(idq)




ddelta=wBSm



Ed_dash-Psi2q)))







ddelta_Gn(Gn)=[]



PSS_in1=KsSm-PSS1






for i=11len_pss(1)







N_PSS=[N_PSS i]

end

Vt=sqrt(Vd^2+Vq^2)




for i=11len_exc(1)









D1 Programs 267









else




end

end














EIV=eig(F1t)



N_State=size(EIV1)

for i=11N_State



damping_ratioP(i)=1

end

end


--clacm--


























1 not yet converged

See also

Calls

Called By loadflow



D1 Programs 269

Version 20


Date July 1994

Version 10


Date March 1991

jay = sqrt(-1)

swing_bus = 1

gen_bus = 2

load_bus = 3




cur_inj = YV_rect




delP = Pg - Pl - P

delQ = Qg - Ql - Q


delP=delPsw_bno

delQ=delQsw_bno

delQ=delQg_bno

total mismatch



mism = pmis+qmis

if mism gt tol

conv_flag = 1

else

conv_flag = 0

end

return

--chq_limm--


Syntax










Version 11


Date May 1997


Version 10


Date October 1996



global Ql

global gen_chg_idx




f = 0







set Qg to zero


modify Ql






lim_flag = 1

end



set Qg to zero


modify Ql



D1 Programs 271




lim_flag = 1

end

if lim_flag == 1

recalculate g_bno


g_bno = ones(nbus1)


bus_index=(11nbus)






il = length(PQV_no)

ii = (11il)



il = length(PQ_no)

ii = (11il)


end

f = lim_flag

return






Version 33



Date December 2013

BUS DATA STARTS

bus data format








bus = [

01 1045 000 250 000 000 000 000 000 2 999 -999

02 098 000 545 000 000 000 000 000 2 999 -999

03 0983 000 650 000 000 000 000 000 2 999 -999

04 0997 000 632 000 000 000 000 000 2 999 -999

05 1011 000 505 000 000 000 000 000 2 999 -999

06 1050 000 700 000 000 000 000 000 2 999 -999

07 1063 000 560 000 000 000 000 000 2 999 -999

08 103 000 540 000 000 000 000 000 2 999 -999

09 1025 000 800 000 000 000 000 000 2 999 -999

10 1010 000 500 000 000 000 000 000 2 999 -999

11 1000 000 10000 000 000 000 000 000 2 999 -999

12 10156 000 1350 000 000 000 000 000 2 999 -999

13 1011 000 3591 000 000 000 000 000 2 999 -999

14 100 000 1785 000 000 000 000 000 2 999 -999

15 1000 000 1000 000 000 000 000 000 2 999 -999

16 1000 000 4000 000 000 000 000 000 1 0 0

17 100 000 000 000 6000 300 000 000 3 0 0

18 100 000 000 000 2470 123 000 000 3 0 0

19 100 000 000 000 000 000 000 000 3 0 0

20 100 000 000 000 6800 103 000 000 3 0 0

21 100 000 000 000 2740 115 000 000 3 0 0

22 100 000 000 000 000 000 000 000 3 0 0

23 100 000 000 000 2480 085 000 000 3 0 0

24 100 000 000 000 309 -092 000 000 3 0 0

25 100 000 000 000 224 047 000 000 3 0 0

26 100 000 000 000 139 017 000 000 3 0 0

27 100 000 000 000 2810 076 000 000 3 0 0

28 100 000 000 000 2060 028 000 000 3 0 0

29 100 000 000 000 2840 027 000 000 3 0 0

30 100 000 000 000 000 000 000 000 3 0 0

31 100 000 000 000 000 000 000 000 3 0 0

32 100 000 000 000 000 000 000 000 3 0 0

33 100 000 000 000 112 000 000 000 3 0 0

34 100 000 000 000 000 000 000 000 3 0 0

35 100 000 000 000 000 000 000 000 3 0 0

36 100 000 000 000 102 -01946 000 000 3 0 0

37 100 000 000 000 000 000 000 000 3 0 0

38 100 000 000 000 000 000 000 000 3 0 0

39 100 000 000 000 267 0126 000 000 3 0 0

40 100 000 000 000 06563 02353 000 000 3 0 0

41 100 000 000 000 1000 250 000 000 3 0 0

42 100 000 000 000 1150 250 000 000 3 0 0

43 100 000 000 000 000 000 000 000 3 0 0

D1 Programs 273

44 100 000 000 000 26755 00484 000 000 3 0 0

45 100 000 000 000 208 021 000 000 3 0 0

46 100 000 000 000 1507 0285 000 000 3 0 0

47 100 000 000 000 20312 03259 000 000 3 0 0

48 100 000 000 000 24120 0022 000 000 3 0 0

49 100 000 000 000 16400 029 000 000 3 0 0

50 100 000 000 000 100 -147 000 000 3 0 0

51 100 000 000 000 337 -122 000 000 3 0 0

52 100 000 000 000 158 030 000 000 3 0 0

53 100 000 000 000 2527 11856 000 000 3 0 0

54 100 000 000 000 000 000 000 000 3 0 0

55 100 000 000 000 322 002 000 000 3 0 0

56 100 000 000 000 200 0736 000 000 3 0 0

57 100 000 000 000 000 000 000 000 3 0 0

58 100 000 000 000 000 000 000 000 3 0 0

59 100 000 000 000 234 084 000 000 3 0 0

60 100 000 000 000 2088 0708 000 000 3 0 0

61 100 000 000 000 104 125 000 000 3 0 0

62 100 000 000 000 000 000 000 000 3 0 0

63 100 000 000 000 000 000 000 000 3 0 0

64 100 000 000 000 009 088 000 000 3 0 0

65 100 000 000 000 000 000 000 000 3 0 0

66 100 000 000 000 000 000 000 000 3 0 0

67 100 000 000 000 3200 15300 000 000 3 0 0

68 100 000 000 000 3290 032 000 000 3 0 0

]

BUS DATA ENDS

LINE DATA STARTS

line =[

01 54 0 00181 0 10250 0

02 58 0 00250 0 10700 0

03 62 0 00200 0 10700 0

04 19 00007 00142 0 10700 0

05 20 00009 00180 0 10090 0

06 22 0 00143 0 10250 0

07 23 00005 00272 0 0 0

08 25 00006 00232 0 10250 0

09 29 00008 00156 0 10250 0

10 31 0 00260 0 10400 0

11 32 0 00130 0 10400 0

12 36 0 00075 0 10400 0

13 17 0 00033 0 10400 0

14 41 0 00015 0 10000 0


15 42 0 00015 0 10000 0

16 18 0 00030 0 10000 0

17 36 00005 00045 03200 0 0

18 49 00076 01141 11600 0 0

18 50 00012 00288 20600 0 0

19 68 00016 00195 03040 0 0

20 19 00007 00138 0 10600 0

21 68 00008 00135 02548 0 0

22 21 00008 00140 02565 0 0

23 22 00006 00096 01846 0 0

24 23 00022 00350 03610 0 0

24 68 00003 00059 00680 0 0

25 54 00070 00086 01460 0 0

26 25 00032 00323 05310 0 0

27 37 00013 00173 03216 0 0

27 26 00014 00147 02396 0 0

28 26 00043 00474 07802 0 0

29 26 00057 00625 10290 0 0

29 28 00014 00151 02490 0 0

30 53 00008 00074 04800 0 0

30 61 000095 000915 05800 0 0

31 30 00013 00187 03330 0 0

31 53 00016 00163 02500 0 0

32 30 00024 00288 04880 0 0

33 32 00008 00099 01680 0 0

34 33 00011 00157 02020 0 0

34 35 00001 00074 0 09460 0

36 34 00033 00111 14500 0 0

36 61 00011 00098 06800 0 0

37 68 00007 00089 01342 0 0

38 31 00011 00147 02470 0 0

38 33 00036 00444 06930 0 0

40 41 00060 00840 31500 0 0

40 48 00020 00220 12800 0 0

41 42 00040 00600 22500 0 0

42 18 00040 00600 22500 0 0

43 17 00005 00276 0 0 0

44 39 0 00411 0 0 0

44 43 00001 00011 0 0 0

45 35 00007 00175 13900 0 0

45 39 0 00839 0 0 0

45 44 00025 00730 0 0 0

46 38 00022 00284 04300 0 0

D1 Programs 275

47 53 00013 00188 13100 0 0

48 47 000125 00134 08000 0 0

49 46 00018 00274 02700 0 0

51 45 00004 00105 07200 0 0

51 50 00009 00221 16200 0 0

52 37 00007 00082 01319 0 0

52 55 00011 00133 02138 0 0

54 53 00035 00411 06987 0 0

55 54 00013 00151 02572 0 0

56 55 00013 00213 02214 0 0

57 56 00008 00128 01342 0 0

58 57 00002 00026 00434 0 0

59 58 00006 00092 01130 0 0

60 57 00008 00112 01476 0 0

60 59 00004 00046 00780 0 0

61 60 00023 00363 03804 0 0

63 58 00007 00082 01389 0 0

63 62 00004 00043 00729 0 0

63 64 00016 00435 0 10600 0

65 62 00004 00043 00729 0 0

65 64 00016 00435 0 10600 0

66 56 00008 00129 01382 0 0

66 65 00009 00101 01723 0 0

67 66 00018 00217 03660 0 0

68 67 00009 00094 01710 0 0

27 53 00320 03200 04100 0 0

]

LINE DATA ENDS

MACHINE DATA STARTS

Machine data format

1 machine number

2 bus number

3 base mva








T_do(sec)







T_qo(sec)




19 bus number




mac_con = [

01 01 100 00125 00 01 0031 0025 102 005 0069 00416667 0025 15

0035 42 0 0 01 0 0

02 02 100 0035 00 0295 00697 005 656 005 0282 00933333 005 15

0035 302 0 0 02 0 0

03 03 100 00304 00 02495 00531 0045 57 005 0237 00714286 0045 15

0035 358 0 0 03 0 0

04 04 100 00295 00 0262 00436 0035 569 005 0258 00585714 0035 15

0035 286 0 0 04 0 0

05 05 100 0027 00 033 0066 005 54 005 031 00883333 005 044

0035 26 0 0 05 0 0

06 06 100 00224 00 0254 005 004 73 005 0241 00675000 004 04

0035 348 0 0 06 0 0

07 07 100 00322 00 0295 0049 004 566 005 0292 00666667 004 15

0035 264 0 0 07 0 0

08 08 100 0028 00 029 0057 0045 67 005 0280 00766667 0045 041

0035 243 0 0 08 0 0

09 09 100 00298 00 02106 0057 0045 479 005 0205 00766667 0045 196

0035 345 0 0 09 0 0

10 10 100 00199 00 0169 00457 004 937 005 0115 00615385 004 1

0035 310 0 0 10 0 0

D1 Programs 277

11 11 100 00103 00 0128 0018 0012 41 005 0123 00241176 0012 15 0035 282 0 0 11 0 0

12 12 100 0022 00 0101 0031 0025 74 005 0095 00420000 0025 15 0035 923 0 0 12 0 0

13 13 200 00030 00 00296 00055 0004 59 005 00286 00074000 0004 15 0035 2480 0 0 13 0 0

14 14 100 00017 00 0018 000285 00023 41 005 00173 00037931 00023 15 0035 3000 0 0 14 0 0

15 15 100 00017 00 0018 000285 00023 41 005 00173 00037931 00023 15 0035 3000 0 0 15 0 0

16 16 200 00041 00 00356 00071 00055 78 005 00334 00095000 00055 15 0035 2250 0 0 16 0 0

]

MACHINE DATA ENDS

EXCITER DATA STARTS




2 - machine number








10 - E_1

11 - S(E_1)

12 - E_2

13 - S(E_2)



16 - K_P

17 - K_I

18 - K_D

19 - T_D

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

exc_con = [

1 1 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

1 2 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01


1 3 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

1 4 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

1 5 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

1 6 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

1 7 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

1 8 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

0 9 001 200 000 50 -50 00 0 0 0 0 0 0 0 0 0 0

0

1 10 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

1 11 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

1 12 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

]

EXCITER DATA ENDS

PSS DATA STARTS

1-S No


3-pssgain










pss_con = [

1 1 12 10 015 004 015 004 015 004 02 -005

2 2 20 15 015 004 015 004 015 004 02 -005

3 3 20 15 015 004 015 004 015 004 02 -005

4 4 20 15 015 004 015 004 015 004 02 -005

5 5 12 10 015 004 015 004 015 004 02 -005

6 6 12 10 015 004 015 004 015 004 02 -005

7 7 20 15 015 004 015 004 015 004 02 -005

D1 Programs 279

8 8 20 15 015 004 015 004 015 004 02 -005

9 9 12 10 015 004 015 004 015 004 02 -005

9 9 15 10 009 002 009 002 1 1 02 -005

10 10 20 15 015 004 015 004 015 004 02 -005

11 11 20 15 015 004 015 004 015 004 02 -005

12 12 20 15 009 004 015 004 015 004 02 -005

]

PSS DATA ENDS

--file form_jacm--




ang_redvolt_red)






entries


entries



jacobian matrix

See also

Calls




Version 20


Date March 1994


Version 10


Date March 1991


jay = sqrt(-1)



V_rect = Vexp_ang


Y_con = conj(Y)


i_c=Y_conCV_rect


S=V_recti_c

S=sparse(diag(S))


CVdia=conj(Vdia)


S1=VdiaY_conCVdia


t2=(S-S1)ang_red


J12=ang_redreal(t1)



if nargout gt 3





else

Jac11 = [J11 J12

J21 J22]

end

--loadflowm--





81297



sparse matices


D1 Programs 281

line - line data








iteration


solution format)



See also








Version 21


Date October 1996


Version 20


Date March 1994

Version 10


Date March 1991

global bus_int


global Q Ql

global gen_chg_idx




jay = sqrt(-1)

load_bus = 3

gen_bus = 2

swing_bus = 1

if exist(flag) == 0

flag = 1

end

lf_flag = 1








end



set defaults

bus data defaults

if ncollt15


if ncollt12



end




volt_min = bus(15)

volt_max = bus(14)

else

volt_min = bus(15)

volt_max = bus(14)

end




end



D1 Programs 283


end







tap_it = 0

tap_it_max = 10

no_taps = 0


if nlc lt 10


no_taps = 1


end


mm_chk=1


tap_it = tap_it+1



process bus data

bus_no = bus(1)

V = bus(2)

ang = bus(3)pi180

Pg = bus(4)

Qg = bus(5)

Pl = bus(6)

Ql = bus(7)

Gb = bus(8)

Bb = bus(9)


qg_max = bus(11)

qg_min = bus(12)

sw_bno=ones(nbus1)

g_bno=sw_bno














il = length(PQV_no)

ii = (11il)



il = length(PQ_no)

ii = (11il)









iter = iter + 1


clear Jac





clear delP delQ







V = V + accdelV

D1 Programs 285











if lim_flag == 1


end

end

if iter == iter_max





else


disp(iter)

end

if no_taps == 0

lftap

else

mm_chk = 0

end

end




else


disp(tap_it)

end





disp(voltages at)

bus(vmx_idx1)


end



disp(voltages at)

bus(vmn_idx1)


end








disp( bus Qg)





end











from_bus = line(1)


to_bus = line(2)


r = line(3)

rx = line(4)

chrg = line(5)

z = r + jayrx

y = ones(nline1)z




to to_bus

D1 Programs 287


from_bus to to_bus


- VV(from_int)tps)y



from from_bus



iline = (11nline)



keyboard





line_sol = line




if conv_flag == 1


error(stop)

end

display results

if display == y

clc



disp( )

disp(date)







if conv_flag == 0



disp(bus_sol(17))


disp( LINE FLOWS )


disp(line_ffrom)

disp(line_fto)

end

end

if iter gt iter_max


lf_flag = 0

end

return

--file y_sparsem--





line - line data






See also

Calls




Version 20


Date April 1994

Version 10


D1 Programs 289

Date March 1991

global bus_int

jay = sqrt(-1)

swing_bus = 1

gen_bus = 2

load_bus = 3



r=zeros(nline1)

rx=zeros(nline1)

chrg=zeros(nline1)

z=zeros(nline1)

y=zeros(nline1)





ibus = (1nbus)



r = line(3)

rx = line(4)










tap=ones(nline1)






iline = [11nline]











if nargout gt 1


nSW = 0

nPV = 0

nPQ = 0





nSW=length(SB)



end

return

D2 Simulation Files




Bibliography



















291

292 Bibliography


















Bibliography 293





















294 Bibliography


















Bibliography 295






















296 Bibliography


















Bibliography 297




















298 Bibliography












Forms



















List of Figures

List of Tables


Introductions

General Context

























Resonant Terms






What is needed























Introduction

















Stability Index










Invariance Property







Stability Bound



Interconnected VSCs






Closed Remarks












Introduction




Method 2-2-1

Method 3-2-3S

Method 3-3-1

Method 3-3-3



















Introduction













Case-studies

SMIB Test System




On-going Work








The Limitations







Originality








Primitive Results


A Literature Review


Conclusion










Validity region

Resonance








Interconnected VSCs



Methodology

Modeling the System






Simulation Files







SEP Initialization





Normal Dynamics


Simulation files


Programs

Simulation Files

Bibliography

2


Doctorat ParisTech





TianTIAN







Jury


2








ii

iii

Acknowledgements






iv






Leonardo da Vinci

vi


















vii

viii

Contents

List of Figures xix

List of Tables xxi












ix

x Contents

















Contents xi












xii Contents















Contents xiii











xiv Contents


















Contents xv







Bibliography 291

xvi Contents

List of Figures













xvii

















List of Figures xix





for δep = π12 169


for δep = π12 169







xx List of Figures

List of Tables


methods 52


NNM 113










2 and case 3 236

xxi

xxii List of Tables









1









3






















5





x = f(xu) (1)

g(xu) = 0



x0 = f(x0u0) = 0 (2)



4x = A4x+O(2) (3)




4y = Λ4y (4)







pki = UkiVik (5)




7























9











11



Problegraveme



























13








x = f(x) (6)


Systems




M q +Dq +Kq + fnl(q) = 0 (7)






doctorat






15







L2EP et LSIS















17























bull


Chapter 1

Introductions












19


11 General Context





















1122 L2EP and EMR




























































δ = ωsω


(11)





Innite Bus

Great Grid

G1

Pe

jXe

minusrarr

E


t(s)0 5 10 15 20 25

δ(degree)

30

40

50

60Predisturbance



















Example Case 1




0 2 4 6 8 10 12

δ1(degree)

40

60

80

100

120

140

t(s)0 2 4 6 8 10 12

δ2(degree)

0

20

40

60

80

100

Exact

Linear




x = f(x) (12)











pki = ukivik (15)












ζωnStates


primeq2 E

primeq3 E

primeq4
















t(s)0 05 1 15 2 25

δ(rad)

-1

0

1

2





Pe =EprimeV0X

sin δ (16)








minus sin δsep2










Pow

erf(δ)

02

05

08

1

Linear Analysis

2nd Order Analysis

3rd Order Analysis

Regionmaybeunstable



Example Case 2



0 5 10 15 20 25 30

δ1(degree)

40

70

100

130140

t(s)0 5 10 15 20 25 30

δ2(degree)

0

30

60

90100

ExactLinear


0 036 072 108 144 180 212 252Hz

101

102

Mag

(

of F

unda

men

tal)



2nd OrderFrequency

3rd OrderFrequency

































x = f(xu) (18)


4x = A4x+ f2(4x2) + f3(4x3) (19)




















































Perturbationmodel





3)





161 Resonant Terms



mode j








zj = λjzj +Nsum

ωkωl

NsumisinR2

gj2klzkzl +Nsum

ωpωq ωr

NsumisinR3

gj3pqrzpzqzr (111)


j3







Fn hn gn S or




nonlinearities









2 2 1











3 3 1









3 2 3 S



173 What is needed











G1 δ1

G2 δ2

jX12jXe

ZZPe

Innite Bus

E


Pe1 = V1V2X12




Pe2 = V1V2X12




(113)



4x = A4x+ a2x2 + a3x

3 (114)


yj = λjyj +4sum

k=1

4suml=1

ykyl +4sump=1

4sumq=1

4sumr=1

ypyqyr (115)




3)






(116)


3 2 3 S







2) + hj3(z3)


+gj3zjz3z4

















0 05 1 15 2minus05

0

05

1

15

t(simulation time)

δ1(rad)

Mode Decomposition

EXMode 1Mode 2

0 05 1 15 2minus05

0

05

1

15

t(simulation time)

δ2(rad)






























Chapter 2





























59


Methodologies













x = y + h2(y)

= z + h3(z) + h2(z + h3(z)) (23)

= [z + h2(z)] + h3(z) +Dh2(z)h3(z)

= z + h2(z) + h3(z) + (O)(4)

2112 Normal Form




2113 Resonant Terms












Methodologies








2131 Linear System


x = Ax (26)


y = Λy (27)


yj = λjyj (28)





X = sX + a2X2 + a3X


apXp (29)



X = Y + α2Y2 (210)



(1 + 2α2Y )Y = sY + (sα2 + a2)Y2 +O(Y 3) (211)





Y = sY + (a2 minus sα2)Y2 +O(Y 3) (213)


α2 =a2s

(214)


Y = sY +O(Y 3) (215)



Methodologies




221 Introduction


u = Au+ εF2(u) + ε2F3(u) + (216)
























Methodologies

and






ε0 y = Λy (224)





+

ksuml=2


where termssumk


















Methodologies















x = f(xu) (228)




4x = H1(4x) +1

2H2(4x) +

1

3H3(4x) +O(4) (229)



] H3jklm =


]and O(4) are






yj = λjyj +Nsumk=1

Nsuml=1

F2jklykyl +Nsump=1

Nsumq=1

Nsumr=1



sumNi=1 vji

[UThj3U

] F3jpqr = 1

6

sumNi=1 vji

sumNk=1

sumNk=1

sumNm=1H3ikLNMu

lpumq u

Nr vji




y = Λy + F (y) (232)




y = Λy + F2(y) (233)


Methodologies


yj = λjyj +Nsumk=1

Nsuml=1

F2jklykyl (234)


y = z + h2(z) (235)


yj = zj +Nsumk=1

Nsuml=1

h2jklzkzl (236)


z + ˙h2(z) = Λz + Λh2(z) + F2(z + h2(z)) (237)





z = Λz (order 1)










h2jkl =F2jkl





z = Λz +DF2(z)h2(z) +O(4) (241)

= Λz +O(3)

= Λz



yj = λjyj +

Nsumk=1

Nsuml=1

F2jklykyl +

Nsump=1

Nsumq=1

Nsumr=1

F3jpqrypyqyr (242)


z + ˙h2(z) = Λz + Λh2(z) + F2(z + h2(z) (243)



+HF2(z)(h2(z))2 + F3(z) +DF3zh2(z))


+DF2(z)h2(z) + F3(z) +O(4))


z = Λz (order 1)




(order 4)



z = Λz +DF2(z)h2(z) + F3(z) +O(4) (244)


Methodologies



zj =λjzj (245)

+

Nsump=1

Nsumq=1

Nsumr=1

(Nsuml=1


)zpzqzr

=λjzj +Nsump=1

Nsumq=1

Nsumr=1

Cjpqrzpzqzr

where Cjpqr =sumN




z = w + h3(w) (246)



(247)




+O(5))(Λw + Λh3(w) +DF2(w)h2(w) + F3(w) +O(5))


w = Λw (order 1)



(order 5)


w =Λw +DF2(w)h2(w) (248)





wj =λjwj

+

Nsump=1

Nsumq=1

Nsumr=1


+O(5) (249)


h3jpqr =F3jpqr +




wj = λjwj +

Msuml=1




Methodologies


cj=2k2l =C2k


(2l)(2k)(2lminus1)



cj=2k2k =C2k


(2kminus1)(2k)(2k) (253)












zj =wj +

Nsump=1

Nsumq=1

Nsumr=1










h3jpqr =F3jpqr



wj = λjwj +O(3) (258)



2361 2-2-1 and 3-2-3S


y = z + h2(z) (259)



+

Nsumk=1

Nsuml=1


+O(3) (260)

For 3-2-3S


+

Nsumk=1

Nsuml=1


+O(3) (261)


Methodologies

2362 3-3-3 and 3-3-1


y =Λw + h2(z) (262)

Λw + h3(w) + h2(w + h3(w))

=Λw + h3(w) + h2(w)



y = w + h2(w) + h3(w) (263)

For 3-3-1


+Nsumk=1

Nsuml=1


+Nsump=1

Nsumq=1

Nsumr=1


(264)

For 3-3-3


+

Nsumk=1

Nsuml=1


+

Nsump=1

Nsumq=1

Nsumr=1


(265)












(z + h2(z)) = z +Dh2(z)z = z +Dh2(z)Λz (266)





Methodologies







f1j = λjf1j (268)

f2j = λjf2j +

Nsumk=1

Nsuml=1

Cjklf1kf

1l (269)

f3j = λjf3j +

Nsumk=1

Nsuml=1

Cjkl(f1kf

2l + f2kf

1l ) +

Nsumk=1

Nsuml=1

Nsumk=1

Djpqrf

1p f

1q f

1r (270)


f1j (t) = y0j eλjt (271)


f2j = λjf2j +

Nsumk=1

Nsuml=1

Cjkly0ky

0l e

(k+l)t (272)


f2j (s1 s2) =

Nsumk=1

Nsuml=1

Cjkl1


1l (s2) (273)



f2j =Nsum

k=1

Nsuml=1


=Nsum

k=1

Nsuml=1

Cjkly0ky

0l S

jkl(t)

where

Sjkl(t) =1





k=1

Nsuml=1

Cjkl1


1l (s3) + f1k (s1)f

2l (s2 + s3))+

(275)Nsum

k=1

Nsuml=1

Nsumr=1

Djpqr

1


1q (s2)f

1r (s3)


f3j (s1 s2 s3) =

Nsumk=1

Nsuml=1

Cjkl1


[Nsum

k=1

Nsuml=1

Cjkl1


1l (s2)

]f1l (s3)

(276)

+ f1k (s1)

[Nsum

k=1

Nsuml=1

Cjkl1


1l (s3)

]

+Nsum

k=1

Nsuml=1

Nsumr=1

Djpqr

1


1q (s2)f

1r (s3)




Methodologies















zj = wj +Nsump=1

Nsumq=1

Nsumr=1

h3jpqrwpwqwr (277)







MI3jpqr =|h3jpqrw0

pw0qw

0r |

|w0j |

(278)


242 Stability Index


wj = λjwj +

Msuml=1


= (λj +Msuml=1

cj2l|w2l|2)wj

(280)



Since

λj +Msuml=1



SIIj2l)) (282)


Methodologies










j2l times Tr



SIj = σj +Msuml=1




(285)






Equations




Mq +Dq +Kq + fnl(q) = 0 (286)


fpnl(q) =Nsum

i=1jgeif2pijqiqj +

Nsumi=1jgeikgej

f3pijkqiqjqk (287)


Methodologies







+Nsumi=1

Nsumjgei

gpijXiXj +Nsumi=1

Nsumjgei

Nsumkgej

hpijkXiXjXk = 0



[gp] =

Nsumi=1



hpijk =

Nsumi=1

Φminus1pi (

NsumP=1

NsumQ=1

NsumR=1













Nsumi=1

Nsumjgei

gpijXiXj = 0 (293)


Xp = Yp (294)


Nsumjgei

gpijXiXj (295)


X = U + ε(a(U2) + b(V 2) + c(UV ))

Y = V + ε(α(U2) + β(V 2) + γ(UV ))

(296)


Methodologies


Xp = Up +Nsumi=1

Nsumjgei


Nsumj=1

cijUiVj

Yp = Vp +Nsumi=1

Nsumjgei


Nsumj=1

γijUiVj (297)


pij c

pij α

pij β

pij γ

pij



Uj = Vj +O(U2i V

2i ) (298)


i V2i ) (299)



3i ) (2100)


pij + bpji)UiVj



3i )


pij + βpji)UiVj



D(a+ b+ c) = α+ β + γ (2101)

D(α+ β + γ) + 2ξωD(a+ b+ c) (2102)




Dc(UV )V = α (U2)

Dc(UV )U = β (V 2)

(2103)






ing (2103)(2104)apij b

pij c

pij α

pij β

pij γ

pij



Up + 2ξpωpUp + ω2pUp +O(3) = 0 (2105)

with

O =Nsumi=1

Nsumjge1

Nsumkgej


+Nsumi=1

Nsumjgei

Nsumkgej

BpijkUiUjUk +

Nsumi=1

Nsumjgei

Nsumkgej

CpijkUiUjUk



Apijk =

Nsumlgei

gpilaljk +

sumllei

gplialjk (2107)

Bpijk =

Nsumlgei

gpilbljk +

sumllei

gplibljk (2108)

Cpijk =

Nsumlgei

gpilcljk +

sumllei

gplicljk (2109)


Methodologies




U = R+ ε2(r(R3) + s(S3) + t(R2 S) + u(RS2))


(2110)



D(r + s+ t+ u) = λ+ micro+ ν + ζ (2111)



To degree ε2




DuR = micro (S3)

(2113)





(2114)









foralli le p



(2115)






ωp = ωi + ωj ωp = 2ωi (2116)





Methodologies





X + ω2X + αX3 = 0 (2119)









1minus ξ2p (2121)









pppRpR2p + CppppR

2pRp



P(2) =

Nsumjgep


2j +Bp


](2124)

+sumilep


2i +Bp


]

Q(2) =

Nsumjgep

(BpjpjRjRj + CpjjpR

2j ) +

Nsumilep

(BpiipRiRi + CpiipR

2i ) (2125)




Methodologies


pppp C

pppp)




pppRpR2p + CppppR

2pRp = 0




Xp = Rp +

Nsumi=1

Nsumjgei


Nsumi=1

Nsumj=1

cijRiSj

+Nsumi=1

Nsumjgei

Nsumkgej


+

Nsumi=1

Nsumjgei

Nsumkgej



Yp = Sp +Nsumi=1

Nsumjgei


2sumj=1

γijRiSj

+

Nsumi=1

Nsumjgei

Nsumkgej


+Nsumi=1

Nsumjgei

Nsumkgej






y

=

micro 0

0 minus1

x

y

+

xy + cx3

bx2

(2128)


microc = 0 xc = x xs = y (2129)



y = bx2 + (|x micro|3) (2130)


Methodologies


x = microx+ (b+ c)x3 (2131)



micro 0

0 minus1








Modes












p S0p [18] And R0

p S0p

satises X0p = R0

p +R(3)(R0i S

0i ) Y 0

p = S0p + S(3)(R0

i S0i )


ωnlp = ωp

(1 +


pppp

8ω2p

a2

)(2133)


262 Stability Bound



3p


2p

]Rp (2134)



2p ge 0 (2135)


Methodologies

with

Rp le

radicminus

ω2p

hpppp +Apppp(2136)



2p + rppppR

3p (2137)


Knl =RpXp

=1







NIp =

radicsumNk=1

sumNj=1


∣∣∣2|Xp0|

(2140)





pBpppp

16πωp(2141)



Study





Methodologies



J1d2δ1dt2


sin δ1 + V1V2L12


J2d2δ1dt2


sin δ2 + V1V2L12


(2142)





R1 + 2ξ1ω1R1 + ω21R1 + (h1111 +A1

111)R31 +B1

111R1R12

+ C1111R

21R1

+R1((A1212 +A1

122 + h1122)R22 +B1

122R22

+ (C1122 + C1

212)R2R2) + R1(B1212R2R2 + C1

221R22) = 0

R2 + 2ξ2ω2R2 + ω22R2 + (h2222 +A2

222)R32 +B2

222R2R22

+ C2222R

22R2

+R2((A2112 +A2

211 + h2112)R22 +B2

211R12

+ (C2211 + C2

121)R1R1)) + R2(B2112R1R2 + C2

112R22) = 0

(2143)



R1 + 2ξ1ω1R1 + ω21R1 + (h1

111 +A1111)R

31 +B1

111R1R12+ C1

111R21R1 +R2P

(2)1 (Ri Ri) + R2Q

(2)1 (Ri Ri) = 0

R2 + 2ξ2ω2R2 + ω22R2 + (h2

222 +A2222)R

32 +B2

222R2R22+ C2

222R22R2 +R1P

(2)2 (Ri Ri) + R1Q

(2)2 (Ri Ri) = 0

(2144)


R1 + 2ξ1ω1R1 + ω21R1 + (h1111 +A1

111)R31 +B1

111R1R12

+ C1111R

21R1 = 0

(2145)


R1 + 2ξ1ω1R1 + ω21R1 + (h1111 +A1

111)R31 +B1

111R1R12

+ C1111R

21R1 = 0

R2 + 2ξ2ω2R2 + ω22R2 + (h2222 +A2

222)R32 +B2

222R2R22

+ C2222R

22R2 = 0

(2146)



Methodologies




0 02 04 06 08 10

05

1

15

δ1(rad)

EXNNMLNM

0 02 04 06 08 1minus05

0

05

1

15

t(s)

δ2(rad)

EXNNMLNM

0 005 01 015 020

05

1

15

δ1(rad)

EXNNMLNM

0 005 01 015 02minus05

0

05

1

15

t(s)

δ2(rad)

EXNNMLNM






0 02 04 06 08 1minus02

minus01

0

01

02

Errorin

δ1(rad)

Error

FNNMNNMNM2

0 02 04 06 08 1minus02

minus01

0

01

02

Errorin

δ2(rad)








Methodologies

0 05 1 15 2minus05

0

05

1

15

t(simulation time)

δ1(rad)

Mode Decomposition

EXMode 1Mode 2

0 05 1 15 2minus05

0

05

1

15

t(simulation time)

δ2(rad)








0 02 04 06 08 1

005

1105

11

115

Disturbance in δ1

Mode Frequency

Disturbance in δ2

ω1(rad)

0 02 04 06 08 1

005

1137

1375

138

Disturbance in δ1

Disturbance in δ2

ω2(rad)


01 03 07 09 110

005

01

015

02

025

03

035

04

045

05


NI

Mode1Mode2

01 03 07 09 110

05

1

15

2

25

3

35


FS

Mode1Mode2


0001 0005 001 005 010

005

01

015

02

025

03

035


NI

Mode1Mode2

0001 0005 001 005 010

1

2

3

4

5

6

7

8

9

10


FS

Mode1Mode2



Methodologies

0 005 01 015 02minus01

0

01

02

δ1(rad)

Dynamics

0 005 01 015 02minus02

minus01

0

01

02

t(s)

δ2(rad)

EXNNMLNM

(a) X1 = X2 = 01

0 005 01 015 0205

1

15

2

δ1(rad)

Dynamics

EXNNMLNM

0 005 01 015 020

05

1

15

t(s)

δ2(rad)

(b) X1 = X2 = 11

0 02 04 06 08 1minus5

0

5x 10

minus4

Errorin

δ1(rad)

Error

0 02 04 06 08 1minus5

0

5x 10

minus4

Errorin

δ2(rad)

NNMLNM

0 02 04 06 08 1

minus05

0

05

Errorin

δ1(rad)

Error

0 02 04 06 08 1

minus05

0

05

Errorin

δ2(rad)

NNMLNM



0 005 01 015 020

05

1

15

δ1(rad)

Dynamics

EXNNMLNM

0 005 01 015 02minus1

0

1

2

t(s)

δ2(rad)

(a) X12 = 0001

0 005 01 015 020

05

1

15

δ1(rad)

Dynamics

EXNNMLNM

0 005 01 015 02minus1

0

1

2

t(s)

δ2(rad)

(b) X12 = 01

0 02 04 06 08 1minus05

0

05

Errorin

δ1(rad)

Error

NNMLNM

0 02 04 06 08 1minus05

0

05

Errorin

δ2(rad)

0 02 04 06 08 1minus01

0

01

Errorin

δ1(rad)

Error

NNMLNM

0 02 04 06 08 1minus01

0

01

Errorin

δ2(rad)



Methodologies


277 Closed Remarks



Forms in Practice




f(z) = z minus y0 + h(z) = 0 (2147)









fj(w) = wj +Nsumk=1

Nsuml=1


Nsumq=1

Nsumr




fj(w(s)) = w

(s)j +

Nsumk=1

Nsuml=1

h2jklh2jklw(s)k w

(s)l +

Nsump=1

Nsumq=1

Nsumr


q w(s)r = 0


Methodologies


[A(ws)] =

[partf

partw

]w=w(s)




w(s+1) = w(s) + ρ4w(s)



The algorithm ends




fp = Rp +Nsumi=1

Nsumjgei

apijRiRj +Nsumi=1

Nsumjgei

Nsumkgej











Methodologies




X1

Y1

Xp

Yp

XN

˙YN

=

0 1



0

0 1


0

0


0 1


X1

Y1

Xp

Yp

XN

YN

+

g(X1) + h(X1)

0

g(Xp) + h(Xp)

0

g(XN) + h(XN)

0

(2149)





sumNk=1


p=1

sumNq=1











sumNi=1 vji

[UThj3U

]

F3jpqr = 16

sumNi=1 vji

sumNk=1

sumNk=1

sumNm=1H3ikLNMu

lpumq u

Nr





Methodologies

















Methodologies

Chapter 3



















115



31 Introduction






31 Introduction 117






Powerf(δ)

02

04

06

08

1

12

14


Exact

Linear

Tayl(2)

Tayl(3)

Regionmaybeunstable

3rd Order Analysis

2nd Order Analysis

Linear Analysis









x = f(xu) (31)


4x = H1(4x) +1

2H2(4x) +

1

3H3(4x) +O(4) (32)



] H3jklm =


]and O(4) are






yj = λjyj +

Nsumk=1

Nsuml=1

F2jklykyl +

Nsump=1

Nsumq=1

Nsumr=1



sumNi=1 vji

[UThj3U

] F3jpqr = 1

6

sumNi=1 vji

sumNk=1

sumNk=1

sumNm=1H3iklmu

lpumq u

nr vji is






322 Method 2-2-1



with h2jkl =F2jkl


323 Method 3-2-3S



then

y = z + h2(z)rArr

zj = λjwj +

sumMl=1 c



(37)


324 Method 3-3-1





325 Method 3-3-3



wj = λjwj +

sumMl=1 c














tem









di

qi

x

y

dk

qk

δiδk

ωs





δi = ωsωk

ωi = 12Hi


˙eprimeqi = 1

Tprimed0i


primeqi)

˙eprimedi = 1

Tprimeq0i


primedi)

(310)





4idqminusxy =

sin δi cos δi


4ixyminusdq =


cos δi sin δi

(311)



primed0T

primeq0


primed0 are the


primeq






11+sTR

times 1+sTC1+sTB

KA1+sTA

VT

+

Vref

minusXE1 XE2 Efd



˙Efdi = KAiXE2iTAi

minus 1TAi

Efdi

˙XE1i = minus 1TRi

XE1i + 1TRi

VTi

˙XE2i = 1TBi


TBiXE2i minus TCi

TBiTRiVT i + 1

TBiVrefi

(312)


VT i =radic


primeqiiqi)







primed e


]






eprimexieprimeyi

= 4idqminusxy

eprimedieprimeqi

It = Y

[eprimex1 e


primexn e

primeyn



primeyiiyi

(315)

where eprimeqi e


primexi e

primeyi are









I1

I2

Im

Im+1

Im+n

=






V1

V2

Vm

Vm+1

Vm+n

(317)




In

=

Ymm Ymn

Ynm Ynn

Vm

Vtn

(318)



]Vtn (319)



](320)



Ytijej =Nsumj=1


Nsumj=1

[(Gtijexi minusBt



Yr =

Gt11 minusBt11

Bt11 Gt11


Gt1n minusBt1n

Bt1n Gt1n

Gtn1 minusBt

n1

Btn1 Gtn1


Gtnn minusBtnn

Btnn Gtnn

(321)




primen

] an admittance




primexn e

primeyn

](322)



primexn e

primeyn




Since edieqi

=

eprimedieprimeqi


idiiqi

=

eprimedieprimeqi

minus Ziidiiqi

(323)


4idqminusxy

edieqi

= 4idqminusxy

eprimedieprimeqi

minus4idqminusxyZk

idiiqi

(324)


=

eprimexieprimeyi


xyminusdq

ixiiyi

(325)



dqminusxy

] T2 =


ex1

ey1

ex2

ey2

exn

eyn

=

eprimex1

eprimey1

eprimex2

eprimey2

eprimexn

eprimeyn

minus T1T2Tminus11

ix1

iy1

ix2

iy2

ixn

iyn

=rArr Y minus1r

ix1

iy1

ix2

iy2

ixn

iyn

= Y minus1

ix1

iy1

ix2

iy2

ixn

iyn

minus T1T2Tminus11

ix1

iy1

ix2

iy2

ixn

iyn

(326)




If Xprimeq = X

primed then 4i






(328)



prime=[

eprimed1 e


primedn e

primeqn

]can be obtained as


]T= Tminus11 Y T1

[eprimed1 e


primedi e


]T(329)




ωi = 12Hi


eprimeqi = 1

Tprimed0i


primeqi)

eprimedi = 1

Tprimeq0i


primedi)

˙Efdi = KAiXE2iTAi

minus 1TAi

Efdi

˙XE1i = minus 1TRi

XE1i + 1TRi

VT i

˙XE2i = 1TBi


TBiXE2i minus TCi

TBiTRiVT i + 1

TBiVrefi

(331)




δi

ωi

eprimeqi

eprimedi

˙Efdi

˙XE1i

˙XE2i

=

0 ωs 0 0 0 0 0

0 minus Di2Hi

0 0 0 0 0

0 0 minus 1

Tprimed0i

0 1

Tprimed0i

0 0

0 0 0 minus 1

Tprimeq0i

0 0 0

0 0 0 0 minus 1TAi

0 KAiTAi


0

0 0 0 0 0 1TBi


TBi

δi

ωi

eprimeqi

eprimedi

Efdi

XE1i

XE2i

+

0


2Hi


Tprimed0i

idi

XqiminusXprimeqi

Tprimeq0i

iqi

0

VTiTRi

minus TCiTBiTRi

VT i

+

0

Pmi2Hi

0

0

0

0

VrefiTBi

(332)


As [Idq] = Yδ

[eprimedq



xi =

δi

ωi

eprimeqi

eprimedi

Efdi

XE1i

XE2i

Agi =

0 ωs 0 0 0 0 0

0 minus Di2Hi

0 0 0 0 0

0 0 minus 1

Tprimed0i

0 0 0 0

0 0 0 minus 1

Tprimeq0i

0 0 0

0 0 0 0 minus 1TAi

0 KAiTAi


0

0 0 0 0 0 1TBi


TBi

Ni(X) =

0


2Hi


Tprimed0i

idi

XqiminusXprimeqi

Tprimeq0i

iqi

0

VTiTRi

minus TCiTBiTRi

VT i

Ui =

0

Pmi2Hi

0

0

0

0

VrefiTBi

(333)


x1

x2

xn

=




x1

x2

xn

+

N1(x)

N2(x)

Nn(x)

+

U1

U2

Un

(334)

x = Agx+N(x) + U (335)




4x = Ag4x+N1(4x) +1

2N2(4x) +

1

3N3(4x) (336)


H1 = Ag +N1 H2 = N2 H3 = N3 (337)

























States


primeq2 E

primeq3 E

primeq4










States


primeq2 E

primeq3 E

primeq4










0 5 10 15 20 25 30

δ1(degree)

40

70

100

130140

t(s)0 5 10 15 20 25 30

δ2(degree)

0

30

60

90100

Exact 3-2-3S 2-2-1 3-3-1 3-3-3 Linear



Hz0 036 072 108 144 180 212 252

Mag

(

of F

unda

men

tal)

101

102

Exact3-2-3S2-2-13-3-13-3-3Linear


0 5 10 15 20 25 30

δ1(degree)

-100

0

100

180

t(s)0 5 10 15 20 25 30

δ2(degree)

-100

-50

0

50

100

150180200

Exact 3-2-3S 3-3-3 2-2-1 3-3-1 Linear





t(s)0 10 20 30

(pu)

085

09

095

1

105

11

115Eprime

q1 Eprime

q2 Eprime

q3 Eprime

q4

Eq1

Eq2

Eq3

Eq4

t(s)0 5 10 15 20 25 30

(pu)

-55Vf

min

-4

-3

-2

-1

0

1

2

3

4

Vfmax


Vmex2

Vfex2



primeq2 E

primeq3 E









j y0 z0221 z0331 z0333

5 514 + j293 389 + j238 388 + j238 03105 + j03263





15 032 + j010 004minus 00012i 005minus j00012 0072 + j005

17 087 + j252 082 + j216 084 + j210 170 + j131i



(171717) 0046minus j0118 0029 + j0034 0017minus j0019


01220 00068 + j0008 00560 033














7 048 + j096 minus621 + j883 096

9 00117 + j017 minus1837 + j628 098


13 0074minus j025 minus178 + j673 085

15 089 + j0532 minus196 + j651 086

17 012minus j0678 minus0405 + j228 033











5 minus5354minus j10977 033 -176682











17 029 + j131 11801 034











Tprimeq0i


primeqi)minusIqi +

(XprimeqiminusX

primeprimeqi)




Tprimed0i

dEprimeqi


primedi)Idi +

(XprimediminusX

primeprimedi)



primeqi)

Tprimeprimed0i

dψ1didt = E

primeqi + (X


prime1di)

Tprimeprimeq0i

dψ2qi


prime2qi)


primeprimedi minusX


primedci

(338)




(XprimeqiminusXlsi)

+ EprimeqiIqi


(XprimediminusXlsi)


primeprimeqi) +


primeprimedi)

(XprimediminusXlsi)


primeprimeqi)

(XprimeqiminusXlsi)


primeqi


(XprimediminusXlsi)

+ψprime1di

(XprimediminusX

primeprimedi)

(XprimediminusXlsi)

minusVqi+j[Eprimedi


(XprimeqiminusXlsi)

minus

ψprime2qi

(XprimeqiminusX

primeprimeqi)

(XprimeqiminusXlsi)


qi

di

x

y

qk

dk

δiδk

ωs






TedEfd





s + sKdTd+1)



(339)







(340)




Vss = KpsssTw

1 + sTw

(1 + sT11)

(1 + sT12)

(1 + sT21)

(1 + sT22)

(1 + sT31)

(1 + sT32)Sm (341)









1 -0438 0404 δ(13) 1 ω(13) 0741 ω(15) 0556 ω(14) 0524

2 0937 0526 δ(14) 1 ω(16) 0738 ω(14) 05 δ(13) 0114

3 -3855 061 δ(13) 1 ω(13) 083 δ(12) 0137 ω(6) 0136

4 3321 0779 δ(15) 1 ω(15) 0755 ω(14) 0305 δ(14) 0149

5 0256 0998 δ(2) 1 ω(2) 0992 δ(3) 0913 ω(3) 0905

6 3032 1073 δ(12) 1 ω(12) 0985 ω(13) 0193 δ(13) 0179

7 -1803 1093 δ(9) 1 ω(9) 0996 δ(1) 0337 ω(1) 0333

8 3716 1158 δ(5) 1 ω(5) 1 ω(6) 0959 δ(6) 0958

9 3588 1185 ω(2) 1 δ(2) 1 δ(3) 0928 ω(3) 0928

10 0762 1217 δ(10) 1 ω(10) 0991 ω(9) 0426 δ(9) 042

11 1347 126 ω(1) 1 δ(1) 0996 δ(10) 0761 ω(10) 0756

12 6487 1471 δ(8) 1 ω(8) 1 ω(1) 0435 δ(1) 0435

13 7033 1487 δ(4) 1 ω(4) 1 ω(5) 0483 δ(5) 0483

14 6799 1503 δ(7) 1 ω(7) 1 ω(6) 0557 δ(6) 0557

15 3904 1753 δ(11) 1 ω(11) 0993 Ψ1d(11) 0056 ω(10) 0033






1 33537 0314 ω(15) 1 ω(16) 0982 δ(7) 0884 δ(4) 0857

2 3621 052 δ(14) 1 ω(16) 0645 ω(14) 0521 δ(15) 0149

3 9625 0591 δ(13) 1 ω(13) 083 ω(16) 0111 ω(12) 0096

4 3381 0779 δ(15) 1 ω(15) 0755 ω(14) 0304 δ(14) 0148

5 27136 0972 ω(3) 1 δ(3) 0962 ω(2) 0924 δ(2) 0849

6 18566 108 ω(12) 1 δ(12) 0931 Eprimeq(12) 0335 PSS4(11) 0196

7 23607 0939 δ(9) 1 ω(9) 0684 PSS3(9) 0231 PSS2(9) 0231

8 30111 1078 δ(5) 1 ω(5) 0928 ω(6) 0909 δ(6) 0883

9 28306 1136 ω(2) 1 δ(2) 0961 δ(3) 0798 ω(3) 0777

10 13426 1278 ω(1) 1 δ(1) 0969 Eprimeq(1) 0135 ω(8) 0111

11 1883 1188 δ(10) 1 ω(10) 0994 Eprimeq(10) 0324 PSS3(10) 019

12 32062 1292 δ(8) 1 ω(8) 0935 Eprimeq(8) 061 PSS2(8) 0422

13 39542 1288 δ(4) 1 ω(4) 0888 Eprimeq(4) 0706 PSS4(4) 0537

14 33119 1367 δ(7) 1 ω(7) 0909 δ(6) 0653 ω(6) 0612

15 2387 5075 ω(11) 1 Eprimeq(11) 0916 Ψ1d(11) 0641 PSS4(10) 0464



Frequency (Hz)0 02 04 06 08 10 12 14 16 18 2

Mag

( o

f Fun

dam

enta

l)

100

101

102G1G12G13G14G15



Mode 1 2 3 4


frequency (Hz) 07788 06073 05226 03929










Frequency (Hz)0 02 04 06 08 10 12 14 16 18 20

Mag

(

of F

unda

men

tal)

100

101

102G6G9G13G14G15





3-3-3 12 1294e-12 17 1954e-6

2-2-13-2-3 3 1799e-11 6 1838e-06

3-3-1 7 1187e-09 11 7786e-09

literature












real(cj=2k2k )

= 0




















Chapter 4
















151


Assessment

41 Introduction



fer Limit


δ = ω (41)



















Assessment







δ = ωsω (42)




t(s)0 5 10 15 20 25 30 35 40 45 50

δ(rad)

-1

-05

0

05

1

15




Assessment


with

Pmax =EprimeqE0



t(s)0 05 1 15 2 25 3 35 4

δ(rad)

-3

-2

-1

0

1

2

3

Exact Linear SEP



















Assessment



























Assessment















o-line2 Step-by-

step inte-gration



1 Mostly-Numerical


















Mq +Dq +Kq + fnl(q) = 0 (44)


fpnl(q) =

Nsumi=1jgei

f2pijqiqj +

Nsumi=1jgeikgej

f3pijkqiqjqk (45)


4241 LNM



Assessment



+

Nsumi=1

Nsumjgei

gpijXiXj +

Nsumi=1

Nsumjgei

Nsumkgej

hpijkXiXjXk = 0




4242 NM2


Up + 2ξpωpUp + ω2pUp +O(3) = 0 (48)

4243 NNM




pppRpR2p + CppppR

2pRp = 0







with

Rp le

radicminus

ω2p

hpppp +Apppp(411)



Knl =RpXp

=1












δ = ωsω (413)


with

Pmax =EprimeqV0Xe




Assessment




2Hi


Dωi

ωbδi (416)

where

Pei = E2iGgii +

Nsumk=1k 6=i









(418)



=

sumN

k=1k 6=i


] j = i


] j 6= i

(419)



2part2Peipartδiδj

reads

Ki2Sii = minus

sumNk=1k 6=i


]Ki

2Sij = EiEj


] j 6= i

Ki2Sjj = minusEiEj


] j 6= i

(420)



it reads

Ki3Siii = 1

6

sumNk=1k 6=i


]Ki

3Siij = minus12EiEj


] j 6= i

Ki3Sijj = 1

2EiEj


] j 6= i

Ki3Sjjj = minus1

6EiEj


](421)




2Hi

ωbδi minus

Dwi


2Hn

ωbδn minus

Dwn



2H



2H

ωb4δN +

D



Assessment





](426)



reads

Kprimei2Sii = Ki


]Kprimei2Sij = Ki

2Sijj 6= i

Ki2Sjj = Ki

2Sjj + EnEj


]j 6= i

(427)



it reads

Kprimei3Siii = Ki


]Kprimei3Siij = Ki

3Siij j 6= i

Kprimei3Sijj = Ki

3Sijj j 6= i

Kprimei3Sjjj = Ki

3Sjjj + 16EnEj


](428)

44 Case-studies






44 Case-studies 167

t(s)0 05 1 15 2 25 3 35 4

δ(rad)

1

12

14

16

EX LNM NM2 NNM SEP


t(s)0 02 04 06 08 1 12 14 16 18 2

δ(rad)

-3

-2

-1

0

1

2

3

EX LNM NM2 NNM SEP





Assessment




PSBminusEABEAB




44 Case-studies 169

t(s)0 05 1 15 2 25 3

δ(rad)

-4

-2

0

2

4

6

EX LNM NM2 NNM SEP


12

t(s)0 05 1 15 2 25 3

δ(rad)

-4

-2

0

2

4



12



Assessment








452 On-going Work



0 5 10 15 20

δ14(rad)

0

05

1

15

2

25

t(s)0 5 10 15 20

δ24(rad)

-05

0

05

1

15

2EX LNM NM2 NNM SEP


0 5 10 15 20

δ14(rad)

0

1

2

3

EX LNM NM2 NNM SEP

t(s)0 5 10 15 20

δ24(rad)

0

1

2



Assessment

0 5 10 15 20

δ14(rad)

-2

0

2

4

EX LNM NM2 NNM SEP

t(s)0 5 10 15 20

δ24(rad)

-2

0

2

4


0 5 10 15 20

δ14(rad)

-2

0

2

4

EX LNM NM2 NNM SEP

t(s)0 5 10 15 20

δ24(rad)

-2

0

2

4






Assessment

Chapter 5


Oscillations


























175


Oscillations













Oscillations









Oscillations












Oscillations




542 The Limitations











Oscillations






M q +Dq +Kq = F (51)



ΦTj Fj

Mj(52)













Mq +Cq +Kq + fnl(q) = 0 (53)


Oscillations









552 Originality





Mq +Cq +Kq + fnl(q) = F (54)





+

Nsumi=1

Nsumjgei

gpijXiXj +

Nsumi=1

Nsumjgei

Nsumkgej

hpijkXiXjXk = Fp











Oscillations

5552 NNM




pppRpR2p + CppppR

2pRp = 0

5553 FDNF3






pRp






5561 Test System







M1d2δ1dt2

+D1dδ1dt +

V1VgX1

sin δ1 + V1V2X12


M2d2δ2dt2

+D2dδ2dt +

V2VgX2

sin δ2 + V1V2X12


(59)



Oscillations

0 05 1 15 2 2505

055

06

065

07

075

08

085

09

t(s)

δ1(rad)

Dynamics

EXLNMFDNF3

0 05 1 15 2 2502

025

03

035

04

045

05

055

06

065

t(s)

δ2(rad)

Dynamics

EXLNMFDNF3









0 05 1 15 2minus05

0

05

1

15

t(simulation time)

δ1(rad)

Mode Decomposition

EXMode 1Mode 2

0 05 1 15 2minus05

0

05

1

15

t(simulation time)

δ2(rad)


0 05 1 15 2 25minus01

0

01

02

03

04

05

06

t(s)

NonlinearModalState

Variables


R1R2



Oscillations











pppRpR2p + CppppR

2pRp = F







EMR+IBC






Oscillations











Oscillations






58 Conclusion 197



x = f(xu) (511)


u = fc(x) (512)


x = f(x fc(x)) (513)

58 Conclusion



Oscillations

Chapter 6




classes of the





199




























Form the DAE model

Eqs (1)(2)




around SEP Eq (3)








S

E

P

I

n

i

t

i

a

l

i

z

a

t

i

o

n


variabledisturbance

variableSEP

Do eigen analysis

of matrix A

Calculate the NF


h2h3 c

End

Start

Rapid




ologies


641 Validity region


642 Resonance












x = f(xu) (61)


4x = H1(4x) +1

2H2(4x) +

1

3H3(4x) +O(4) (62)



] H3jklm =


]and O(4) are



Mq +Cq +Kq + fnl(q) = F (63)




x = f(x x) (64)

or Eq (65)

Mq +Cnl(q) +Kq + fq = F (65)





x = f(x x) (66)

or Eq (67)

Mclq +C(q) +Kq + fq = F (67)













207

208 Publications

Appendix A

Interconnected VSCs


usp=sharing





209




A111 The Basic Unit



A12 Methodology












AC LoadSource






(A2)

Filter C






lg middot ωg(A4)










lowastod = Vod i

lowastod = iod i



J1d2δ1dt2


sin δ1 + V1V2L12


J2d2δ1dt2


sin δ2 + V1V2L12


(A5)








ω10 ω2 = ω2+ω20dω1dt =


dt =d(ω2minusωg)



dt = dω1dt

dω2dt =










J1 16 L1(pu) 07 V1 1

J2 16 L2(pu) 07 V2 1

D1 10 L12(pu) 001 Vg 1

D2 10 wb 314


P lowast1 =V1VgL1

sin δ10 +V1V2L12


V1VgL2

sin δ20 +V1V2L12

sin δ210


J1ωb

d2δ1dt2

+ D1ωb

dδ1dt +

V1VgL1



J2ωb

d2δ2dt2

+ D2ωb

dδ2dt +

V2VgL2

















J1=0216D1=20J2=0216D2=20w_b=100314

g1=J1w_b

g2=J2w_b

d1=D1w_b

d2=D2w_b

syms g1 g2 d1 d2



delta210=-delta120

L1=07L2=07L12=001wg=1w_b=502314


V1=1V2=1Vg=1w10=1w20=1



Natural frequencies

M=[g1 00 g2]



D=[d1 00 d2]


[TrE]=eig(inv(M)K)



Normalisation of T

Tr=Trdet(Tr)

leg=sqrt(sum(Tr^2))

Tr=Trleg(ones(12))

normalization


modal matrices

Mm=TrMTr

Dm=TrDTr

Km=TrKTr

T11=Tr(11)

T12=Tr(12)

T21=Tr(21)

T22=Tr(22)

division by Mm


Dm_div=inv(Mm)Dm

OMEGA=double(OMEGA)

for i=12


end
















Gg=zeros(2222)




Hg(1111)=TEC(1111)T11^3+TEC(1112)T21T11^2+

TEC(1122)T11T21^2+TEC(1222)T21^3


TEC(1122)(T12T21^2+2T11T21T22)+TEC(1222)3T21^2T22


TEC(1122)(2T12T21T22+T11T22^2)+TEC(1222)3T21T22^2

Hg(1222)=TEC(1111)T12^3+TEC(1112)T22T12^2+

TEC(1122)T12T22^2+TEC(1222)T22^3





TEC(2222)T21^3


TEC(2122)(T12T21^2+2T11T21T22)+TEC(2222)3T21^2T22


TEC(2122)(2T12T21T22+T11T22^2)+TEC(2222)3T21T22^2



TEC(2222)T22^3



G=zeros(222)

H=zeros(2222)

G(111)=(T11Gg(111)T21Gg(211))Mm(11)

G(112)=(T11Gg(112)T21Gg(212))Mm(11)

G(122)=(T11Gg(122)T21Gg(222))Mm(11)

G(211)=(T12Gg(211)T22Gg(211))Mm(22)

G(212)=(T12Gg(212)T22Gg(212))Mm(22)

G(222)=(T12Gg(222)T22Gg(222))Mm(22)

for p=12

for k=12

for j=k2


++j

end

++k

end

++p

end

for p=12

for i=12

for k=i2

for j=k2


++j

end

++k

end

++i

end

++p

end

H=zeros(2222)

c1=[G(111) G(112) G(122) H(1111) H(1112) H(1122) H(1222)]

c2=[G(211) G(212) G(222) H(2111) H(2112) H(2122) H(2222)]






system






N=length(OMEGA)

correction=1

quadratic coefs

A=zeros(NNN)

B=zeros(NNN)

C=zeros(NNN)

ALPHA=zeros(NNN)

BETA=zeros(NNN)

GAMMA=zeros(NNN)

for p=1N

for i=1N































xi(i))




end

end

for p=1N

for i=1N

for j=i+1N



if abs(D) lt 1e-12

A(pij)=0

B(pij)=0

C(pij)=0

C(pji)=0

ALPHA(pij)=0

BETA(pij)=0

GAMMA(pij)=0

GAMMA(pji)=0

else














B2=[G(pij) 0 0 0]

solabcij=inv(A2)B2

solabcij=A2B2

A(pij)=solabcij(1)

B(pij)=solabcij(2)

C(pij)=solabcij(3)

C(pji)=solabcij(4)





end

end

end

end


AA=zeros(NNNN)

BB=zeros(NNNN)

CC=zeros(NNNN)

AA2=zeros(NNNN)

BB2=zeros(NNNN)

CC2=zeros(NNNN)

for p=1N

for i=1N

for j=1N

for k=1N

V1=squeeze(G(piiN))

V2=squeeze(A(iNjk))

V3=squeeze(G(p1ii))

V4=squeeze(A(1ijk))


V5=squeeze(G(piiN))

V6=squeeze(B(iNjk))

V7=squeeze(G(p1ii))


V8=squeeze(B(1ijk))


V9=squeeze(G(piiN))





end

end

end

end

cubic coefs

R=zeros(NNNN)

S=zeros(NNNN)

T=zeros(NNNN)

U=zeros(NNNN)

LAMBDA=zeros(NNNN)

MU=zeros(NNNN)

NU=zeros(NNNN)

ZETA=zeros(NNNN)


for p=1N

for i=1N

if i~=p


if abs(D1) lt 1e-12

R(piii)=0

S(piii)=0

T(piii)=0

U(piii)=0

LAMBDA(piii)=0

MU(piii)=0

NU(piii)=0

ZETA(piii)=0

else












sol3iii=A3iiiB3iii

R(piii)=sol3iii(1)

S(piii)=sol3iii(2)

T(piii)=sol3iii(3)

U(piii)=sol3iii(4)





end

end

end

end

for p=1N

for i=1N-1

if i~=p

for j=i+1N




R(pijj) = 0

S(pijj) = 0

T(pijj) = 0

T(pjij) = 0

U(pijj) = 0

U(pjij) = 0

LAMBDA(pijj)=0

MU(pijj) = 0

NU(pijj) =0

NU(pjij) =0

ZETA(pijj) = 0

b 21n132

ZETA(pjij) = 0

else

























xi(j)OMEGA(j)]




solrstuip=AipBip













2OMEGA(j)^2U(pijj)




end

end

end

end

end

for p=1N

for i=1N-1

for j=(i+1)N

if j~=p




R(piij) =0

S(piij) =0

T(pjii) =0

T(piij) =0

U(pjii) =0

U(piij) =0

LAMBDA(piij)=0

NU(pjii) =0

NU(piij) =0

MU(piij) =0

ZETA(pjii) =0

ZETA(piij) =0

else

























BB(piij) BB(pjii)]


solrstujpp=AjppBjpp













OMEGA(j)^2U(piij)



end

end

end

end

end

for p=1N

for i=1N-2

for j=(i+1)N-1

for k=(j+1)N





OMEGA(k)-OMEGA(i))

if abs(D4) lt 1e-12

U(pijk) =0


U(pjik) =0

U(pkij) =0

R(pijk) =0

MU(pijk) =0

NU(pkij) =0

NU(pjik) =0

NU(pijk) =0

else


















































Am1=inv(Alastijkp)


R(pijk)=solrstu(1)

S(pijk)=solrstu(2)

T(pijk)=solrstu(3)

T(pjik)=solrstu(4)

T(pkij)=solrstu(5)

U(pijk)=solrstu(6)

U(pjik)=solrstu(7)

U(pkij)=solrstu(8)


OMEGA(j)^2U(pijk)




OMEGA(i)^2T(pijk)




OMEGA(k)^2U(pijk)


OMEGA(k)^2U(pjik)






end

end

end

end

end

disp(end)

A3 Simulation Files




oscillations

Appendix B





Xp = Rp +

Nsumi=1

Nsumjgei


Nsumi=1

Nsumj=1

cpijRiSj

+

Nsumi=1

Nsumjgei

Nsumkgej


+

Nsumi=1

Nsumj=1

Nsumkgej


Yp = Sp +

Nsumi=1

Nsumjgei

(αpijRiRj + βp

ijSiSj) +

Nsumi=1

Nsumj=1

γpijRiSj

+

Nsumi=1

Nsumjgei

Nsumkgej


ijkSiSjSk)

+

Nsumi=1

Nsumj=1

Nsumkgej



p = Xinitp + 4Xp Y

RTp =


231




4Xp = 4Rp +Nsumi=1

PR(1)pi4Ri +Nsumi=1

PS(1)pi4Si

Nsumi=1


PS(S2)pi4Si

Nsumi=1

PR(S2)pi4Ri +

Nsumi=1

PS(R2)pi4Si

4Yp = 4Sp +Nsumi=1

QR(1)pi4Ri +Nsumi=1

QS(1)pi4Si

Nsumi=1


QS(S2)pi4Si

Nsumi=1

QR(S2)pi4Ri +

Nsumi=1

QS(R2)pi4Si

with

PR(1)pi =isum

j=1

apjiRj +Nsumjgei

apijRj +Nsumj=1

cpijSj (B1)

PS(1)pi =isum

j=1

bpjiSj +Nsumjgei

bpijSj +Nsumj=1

cpjiRj

PR(R2)pi =

isumj=1

isumkgej

rpkjiRjRk +

isumj=1

Nsumkgei

rpjikRjRk +

Nsumjgei

Nsumkgej

rpijkRjRk

PS(S2)pi =

isumj=1

isumkgej

spkjiSjSk +

isumj=1

Nsumkgei

spjikSjSk +

Nsumjgei

Nsumkgej

spijkSjSk

PR(S2)pi =Nsumj=1

isumk=1

tpjkiSjRk +Nsumj=1

Nsumkgei

tpjikSjRk +Nsumj=1

Nsumkgej

upijkSjSk

PS(R2)pi =

Nsumj=1

isumk=1

upjkiRjSk +

Nsumj=1

Nsumkgei

upjikRjSk +

Nsumj=1

Nsumkgej

tpijkRjRk


QR(1)pi =isum

j=1

αpjiRj +Nsumjgei

αpijRj +Nsumj=1

γpijSj (B2)

QS(1)pi =

isumj=1

βpjiSj +

Nsumjgei

βpijSj +

Nsumj=1

γpjiRj

QR(R2)pi =

isumj=1

jsumkgej

λpkjiRjRk +

isumj=1

Nsumkgei

λpjikRjRk +

Nsumjgei

Nsumkgej

λpijkRjRk

QS(S2)pi =isum

j=1

isumkgej

micropkjiSjSk +isum

j=1

Nsumkgei


Nsumkgej

micropijkSjSk

QR(S2)pi =Nsumj=1

isumk=1

νpjkiSjRk +Nsumj=1

Nsumkgei

νpjikSjRk +Nsumj=1

Nsumkgej

ζpijkSjSk

QS(R2)pi =

Nsumj=1

isumk=1

ζpjkiRjSk +

Nsumj=1

Nsumkgei

ζpjikRjSk +

Nsumj=1

Nsumkgej

νpijkRjRk

Let



(B3)

Then we obtain 4X

4Y

=

I +XR XS

Y R Y S

4R

4S

(B4)

and 4R

4S

=

I +XR XS

Y R Y S

minus1 4X

4Y

=

[NLRT

]minus1

4X

4Y

(B5)



Appendix C



System



Ra 00025 xl 0002

xd 180 τprimed0 80


xprimed 03 τ

primeprimed0 00

xprimeq 03 τ

primeprimed0 00




Load Voltage Zshunt

Bus (pu230kv)

B5 1029 6 1975 P = 66584MW Q = minus35082Mvar


235





G1 PV 1006 1402 74750 4994

G2 swingbus 10006 00 40650 14074

G3 PV 1006 minus 5817 53925 11399

G4 PV 1006 minus 6611 52500 14314




Gen 1 15 35 35 150 150 240

Gen 2 15 35 35 150 150 240

Gen 3 11 2 2 150 150 240

Gen 4 10 2 2 150 150 240









clear all

close all

Load the system


clear all



Define the system


omega_s=120pi









Yt=[39604-395821i -39604+396040i 00000+00000i 00000+00000i

-39604+396040i 103914-424178i 00000+00000i 04578+41671i

00000+00000i 00000+00000i 39604-395821i -39604+396040i

00000+00000i 04578+41671i -39604+396040i 182758-431388i]

Gt=real(Yt)

Bt=imag(Yt)


















T_e=ones(41)001

T_1=ones(41)10

T_2=ones(41)1









Vf= sym(Vf [Gn 1])

Vm= sym(Vm [Gn 1])

Vr= sym(Vr [Gn 1])

Yr=zeros(2Gn2Gn)

for l=1Gn

for m=1Gn

Yr(2l-12m-1)=Gt(lm)

Yr(2l-12m)=-Bt(lm)

Yr(2l2m-1)=Bt(lm)

Yr(2l2m)=Gt(lm)

end

end






for l=1Gn



T2(2l-12l-1)=Ra(l) T2(2l-12l)=-Xlq(l)


end

Y=inv(inv(Yr)+T29)

for l=1Gn

for m=1Gn

G1(lm)=Y(2l-12m-1)

B1(lm)=-Y(2l-12m)

B1(lm)=Y(2l2m-1)

G1(1m)=Y(2l2m)

end

end

Ydelta=inv(T1)YT1

for j=1Gn

eldq(2j-1)=eld(j)

eldq(2j)=elq(j)

end

Idq=Ydeltaeldq


for j=1Gn

id(j)=Idq(2j-1)

iq(j)=Idq(2j)

end

for j=1Gn





end

for j=1Gn


elxy(2j-11)=elx(j)

elxy(2j1)=ely(j)

end

Ixy=Yelxy









Vref=zeros(Gn1)

Vref_Exc_1=103

Vref_Exc_2=101

Vref_Exc_3=103

Vref_Exc_4=101









for j=1Gn-1








end

for j=Gn








end

for j=1Gn-1



end

for j=Gn



end



for j=1Gn-1


end

for j=Gn


end

x027=x027

Fv_R=vpa(F27)




x0test=zeros(281)



EIV_R=eig(F1t_R)










ie















for p=1L


end

for p=1L

for k=1L


end

end



F1=N1












[TuLambda]=eig(A)


invTu=inv(Tu)

Calculate F2

F21=zeros(LLL)

F2=zeros(LLL)

for p=1L



end

for p=1L

for j=1L

for k=1L


end

end

end

Calculate F3

F31=zeros(LLLL)

F3=zeros(LLLL)

for j=1L

for p=1L

for q=1L

for r=1L

for l=1L

for m=1L

for n=1L


end

end

end

end

end

end


end

for j=1L

for p=1L

for q=1L

for r=1L


end

end

end

end






N=length(EIV)

quadratic coefs

Cla=zeros(NNN)

for p=1N

for j=1N

for k=1N


end

end

end


end


beginverbatim





ModeNumber=[]




m=0


for j=1ModeSize


m=m+1

ModeNumber(m)=j

end

end




L=length(EIV)


MN=ModeNumber

N1718=0

xdot=zeros(L1)

c3=zeros(LL)

term_3=zeros(L1)

F2H2=zeros(LLLL)

F2c=zeros(LL)

C3=zeros(LLLL)

h2=zeros(LLL)

h3=zeros(LLLL)

for p=1M

for l=1M

for m=1M

for n=1M


end

end

end

end

for p=1L

for l=1L

for k=1L



end

for m=1L

for n=1L



end

end

end

end

for i=1(M2)

for j=1(M2)

if j~=i








C3(MN(2i-1)MN(2j)MN(2i-1)MN(2j-1))+

C3(MN(2i-1)MN(2j)MN(2j-1)MN(2i-1))+

C3(MN(2i-1)MN(2i-1)MN(2j-1)MN(2j))+

C3(MN(2i-1)MN(2j-1)MN(2i-1)MN(2j))+

C3(MN(2i-1)MN(2j-1)MN(2j)MN(2i-1))

else





C3(MN(2i-1)MN(2i-1)MN(2i-1)MN(2i))+

C3(MN(2i-1)MN(2i)MN(2i-1)MN(2i-1))


end

end

end

for p=1M

for l=1M

for m=1M

for n=1M




end

end

end

end

for k=1(M2)

for l=1(M2)

for m=1(M2)

for n=1(M2)

if l~=k




h3(MN(2k-1)MN(2k-1)MN(2l)MN(2l-1))=0

h3(MN(2k-1)MN(2l)MN(2k-1)MN(2l-1))=0

h3(MN(2k-1)MN(2l)MN(2l-1)MN(2k-1))=0

h3(MN(2k-1)MN(2k-1)MN(2l-1)MN(2l))=0

h3(MN(2k-1)MN(2l-1)MN(2k-1)MN(2l))=0

h3(MN(2k-1)MN(2l-1)MN(2l)MN(2k-1))=0

else




h3(MN(2k-1)MN(2k-1)MN(2k)MN(2k-1))=0

h3(MN(2k-1)MN(2k)MN(2k-1)MN(2k-1))=0

h3(MN(2k-1)MN(2k-1)MN(2k-1)MN(2k))=0

end

end

end

end

end


end










MN=ModeNumber

M=L2

EIV=EIV_R

w=imag(EIV)

sigma=real(EIV)

F2=F2N_R

F3=F3N_R

quadratic coefs

h3=h3test_R

for j=1L

for p=1L

h3(jppp)=h3(jppp)

for q=(p+1)L


for r=(q+1)L


h3(jqrp)+h3(jrqp)

end

end

end

end

for j=1L

for p=1L

for q=1L

for r=1L


end

end

end

end

MaxMI333=zeros(277)

for j=1L


[Ih3_1Ih3_2Ih3_3] = ind2sub([27 27 27]Ih3)




end


------------------------)

C34 Normal Dynamics





L=length(CE)

xdot=zeros(L1)

term_3=zeros(L1)

Cam=zeros(42)

F2H2=zeros(LLLL)

F2c=zeros(LL)

C3=zeros(LLLL)

for i=1(L2)

for j=1(L2)




end

end

for p=1L


end

end




L=length(CE)

xdot=zeros(L1)

for p=1L

xdot(p)=CE(pp)x(p)

end

end


















constrained






[1e-6]



[1e-6]


[100]


[iter]






References






Version 05





Version 04



Version 03







Version 02








initialize



set default options

oldopts = optimset(


if narginlt3


else


end


get options









set scaling values




set display

if DISPLAY




fmtstr = 10d 104g 104g 104g 104g 124gn


end

check initial guess






else


end

if issparse(Jstar)



else


rc = NaN


rc = Inf

else


end

end



solver





if lambda==1











end



break



break

end








if lambda==1



else





[ab] = coeff

if a==0

lambda = -slope2b

else


if discriminantlt0


elseif blt=0


else


end

end


end




else


end

if lambdalt1



continue

end

display







break

end

if issparse(Jstar)


else



end


end

output




if Nitergt=MAXITER

exitflag = 0


elseif exitflag==2


elseif exitflag==-1


else


end

jacob = J

end



try

[F J] = fun(x)

catch

F = fun(x)


end


end


estimate J





for n = 1nx





end

end












N=length(tt)

L=length(y0)

ynl=zeros(L1)

ynl3=zeros(L1)

H3=zeros(L1)

for l=1L

for p=1L

for q=1L

for r=1L


end

end

end

end

for l=1L


end

f=ynl3-y0











N=length(tt)


L=length(y0)

ynl=zeros(L1)

ynl3=zeros(L1)

for l=1L


end

f=ynl3-y0

C4 Simulation les




Appendix D



D1 Programs





1 calcm

2 chq_limm


4 form_jacm

5 loadflowm

6 y_sparsem


Version 33



Date December 2013




clear all

clc

MVA_Base=1000

f=600


259



j=sqrt(-1)

Load Data

data16m_benchmark


N_Bus=size(bus1)

N_Line=size(line1)

Loading Data Ends

Run Load Flow


disply=yflag = 2






clc

display(Running)

Load Flow End



xls=mac_con(4)BM

Ra=mac_con(5)BM

xd=mac_con(6)BM

xdd=mac_con(7)BM

xddd=mac_con(8)BM

Td0d=mac_con(9)

Td0dd=mac_con(10)

xq=mac_con(11)BM

xqd=mac_con(12)BM

xqdd=mac_con(13)BM

Tq0d=mac_con(14)

Tq0dd=mac_con(15)

H=mac_con(16)BM

D=mac_con(17)BM

M=2H

wB=2pif





D1 Programs 261

else

Zg= Ra + jxdd

end

Yg=1Zg


AVR Initialization

Tr=ones(N_Machine1)





Td=ones(N_Machine1)

Ka=ones(N_Machine1)


Ta=ones(N_Machine1)

Ke=ones(N_Machine1)



Te=ones(N_Machine1)


Tf=ones(N_Machine1)







for i=11len_exc(1)



if(exc_con(i3)~=0)


end

if(exc_con(i5)~=0)


end

if(exc_con(i1)==1)













Kad(Exc_m_indx)=1



else




end

end


PSS initialization


Tw=ones(N_Machine1)










for i=11len_pss(1)












end



D1 Programs 263




for i=11N_Machine


end

YG=MMYg

YG=diag(YG)

V=bus_sol(2)



Y_Aug=Y+YL+YG

Z=inv(Y_Aug) It=

Y_Aug_dash=Y_Aug


Initial Conditions

Vg=MMVthg=MMtheta


mp=2

mq=2

kp=bus_sol(6)V^mp

kq=bus_sol(7)V^mq


y_dash=y

from_bus = line(1)

to_bus = line(2)




Q_s = imag(MW_s)


















delta0 = angle(Eq0)





VD0=imag(VDQ)

VQ0=real(VDQ)

Vref=Efd0



for i=11len_exc(1)


if(exc_con(i1)==1)

V_Ka0(Exc_m_indx)=




else


end

end

Tm0=Te0

Pm0=Tm0


Gn=16



Sm= sym(Sm [Gn 1])







D1 Programs 265
















Vq=real(Vdq)

Vd=imag(Vdq)



iq=real(idq)

id=imag(idq)




ddelta=wBSm



Ed_dash-Psi2q)))







ddelta_Gn(Gn)=[]



PSS_in1=KsSm-PSS1






for i=11len_pss(1)







N_PSS=[N_PSS i]

end

Vt=sqrt(Vd^2+Vq^2)




for i=11len_exc(1)









D1 Programs 267









else




end

end














EIV=eig(F1t)



N_State=size(EIV1)

for i=11N_State



damping_ratioP(i)=1

end

end


--clacm--


























1 not yet converged

See also

Calls

Called By loadflow



D1 Programs 269

Version 20


Date July 1994

Version 10


Date March 1991

jay = sqrt(-1)

swing_bus = 1

gen_bus = 2

load_bus = 3




cur_inj = YV_rect




delP = Pg - Pl - P

delQ = Qg - Ql - Q


delP=delPsw_bno

delQ=delQsw_bno

delQ=delQg_bno

total mismatch



mism = pmis+qmis

if mism gt tol

conv_flag = 1

else

conv_flag = 0

end

return

--chq_limm--


Syntax










Version 11


Date May 1997


Version 10


Date October 1996



global Ql

global gen_chg_idx




f = 0







set Qg to zero


modify Ql






lim_flag = 1

end



set Qg to zero


modify Ql



D1 Programs 271




lim_flag = 1

end

if lim_flag == 1

recalculate g_bno


g_bno = ones(nbus1)


bus_index=(11nbus)






il = length(PQV_no)

ii = (11il)



il = length(PQ_no)

ii = (11il)


end

f = lim_flag

return






Version 33



Date December 2013

BUS DATA STARTS

bus data format








bus = [

01 1045 000 250 000 000 000 000 000 2 999 -999

02 098 000 545 000 000 000 000 000 2 999 -999

03 0983 000 650 000 000 000 000 000 2 999 -999

04 0997 000 632 000 000 000 000 000 2 999 -999

05 1011 000 505 000 000 000 000 000 2 999 -999

06 1050 000 700 000 000 000 000 000 2 999 -999

07 1063 000 560 000 000 000 000 000 2 999 -999

08 103 000 540 000 000 000 000 000 2 999 -999

09 1025 000 800 000 000 000 000 000 2 999 -999

10 1010 000 500 000 000 000 000 000 2 999 -999

11 1000 000 10000 000 000 000 000 000 2 999 -999

12 10156 000 1350 000 000 000 000 000 2 999 -999

13 1011 000 3591 000 000 000 000 000 2 999 -999

14 100 000 1785 000 000 000 000 000 2 999 -999

15 1000 000 1000 000 000 000 000 000 2 999 -999

16 1000 000 4000 000 000 000 000 000 1 0 0

17 100 000 000 000 6000 300 000 000 3 0 0

18 100 000 000 000 2470 123 000 000 3 0 0

19 100 000 000 000 000 000 000 000 3 0 0

20 100 000 000 000 6800 103 000 000 3 0 0

21 100 000 000 000 2740 115 000 000 3 0 0

22 100 000 000 000 000 000 000 000 3 0 0

23 100 000 000 000 2480 085 000 000 3 0 0

24 100 000 000 000 309 -092 000 000 3 0 0

25 100 000 000 000 224 047 000 000 3 0 0

26 100 000 000 000 139 017 000 000 3 0 0

27 100 000 000 000 2810 076 000 000 3 0 0

28 100 000 000 000 2060 028 000 000 3 0 0

29 100 000 000 000 2840 027 000 000 3 0 0

30 100 000 000 000 000 000 000 000 3 0 0

31 100 000 000 000 000 000 000 000 3 0 0

32 100 000 000 000 000 000 000 000 3 0 0

33 100 000 000 000 112 000 000 000 3 0 0

34 100 000 000 000 000 000 000 000 3 0 0

35 100 000 000 000 000 000 000 000 3 0 0

36 100 000 000 000 102 -01946 000 000 3 0 0

37 100 000 000 000 000 000 000 000 3 0 0

38 100 000 000 000 000 000 000 000 3 0 0

39 100 000 000 000 267 0126 000 000 3 0 0

40 100 000 000 000 06563 02353 000 000 3 0 0

41 100 000 000 000 1000 250 000 000 3 0 0

42 100 000 000 000 1150 250 000 000 3 0 0

43 100 000 000 000 000 000 000 000 3 0 0

D1 Programs 273

44 100 000 000 000 26755 00484 000 000 3 0 0

45 100 000 000 000 208 021 000 000 3 0 0

46 100 000 000 000 1507 0285 000 000 3 0 0

47 100 000 000 000 20312 03259 000 000 3 0 0

48 100 000 000 000 24120 0022 000 000 3 0 0

49 100 000 000 000 16400 029 000 000 3 0 0

50 100 000 000 000 100 -147 000 000 3 0 0

51 100 000 000 000 337 -122 000 000 3 0 0

52 100 000 000 000 158 030 000 000 3 0 0

53 100 000 000 000 2527 11856 000 000 3 0 0

54 100 000 000 000 000 000 000 000 3 0 0

55 100 000 000 000 322 002 000 000 3 0 0

56 100 000 000 000 200 0736 000 000 3 0 0

57 100 000 000 000 000 000 000 000 3 0 0

58 100 000 000 000 000 000 000 000 3 0 0

59 100 000 000 000 234 084 000 000 3 0 0

60 100 000 000 000 2088 0708 000 000 3 0 0

61 100 000 000 000 104 125 000 000 3 0 0

62 100 000 000 000 000 000 000 000 3 0 0

63 100 000 000 000 000 000 000 000 3 0 0

64 100 000 000 000 009 088 000 000 3 0 0

65 100 000 000 000 000 000 000 000 3 0 0

66 100 000 000 000 000 000 000 000 3 0 0

67 100 000 000 000 3200 15300 000 000 3 0 0

68 100 000 000 000 3290 032 000 000 3 0 0

]

BUS DATA ENDS

LINE DATA STARTS

line =[

01 54 0 00181 0 10250 0

02 58 0 00250 0 10700 0

03 62 0 00200 0 10700 0

04 19 00007 00142 0 10700 0

05 20 00009 00180 0 10090 0

06 22 0 00143 0 10250 0

07 23 00005 00272 0 0 0

08 25 00006 00232 0 10250 0

09 29 00008 00156 0 10250 0

10 31 0 00260 0 10400 0

11 32 0 00130 0 10400 0

12 36 0 00075 0 10400 0

13 17 0 00033 0 10400 0

14 41 0 00015 0 10000 0


15 42 0 00015 0 10000 0

16 18 0 00030 0 10000 0

17 36 00005 00045 03200 0 0

18 49 00076 01141 11600 0 0

18 50 00012 00288 20600 0 0

19 68 00016 00195 03040 0 0

20 19 00007 00138 0 10600 0

21 68 00008 00135 02548 0 0

22 21 00008 00140 02565 0 0

23 22 00006 00096 01846 0 0

24 23 00022 00350 03610 0 0

24 68 00003 00059 00680 0 0

25 54 00070 00086 01460 0 0

26 25 00032 00323 05310 0 0

27 37 00013 00173 03216 0 0

27 26 00014 00147 02396 0 0

28 26 00043 00474 07802 0 0

29 26 00057 00625 10290 0 0

29 28 00014 00151 02490 0 0

30 53 00008 00074 04800 0 0

30 61 000095 000915 05800 0 0

31 30 00013 00187 03330 0 0

31 53 00016 00163 02500 0 0

32 30 00024 00288 04880 0 0

33 32 00008 00099 01680 0 0

34 33 00011 00157 02020 0 0

34 35 00001 00074 0 09460 0

36 34 00033 00111 14500 0 0

36 61 00011 00098 06800 0 0

37 68 00007 00089 01342 0 0

38 31 00011 00147 02470 0 0

38 33 00036 00444 06930 0 0

40 41 00060 00840 31500 0 0

40 48 00020 00220 12800 0 0

41 42 00040 00600 22500 0 0

42 18 00040 00600 22500 0 0

43 17 00005 00276 0 0 0

44 39 0 00411 0 0 0

44 43 00001 00011 0 0 0

45 35 00007 00175 13900 0 0

45 39 0 00839 0 0 0

45 44 00025 00730 0 0 0

46 38 00022 00284 04300 0 0

D1 Programs 275

47 53 00013 00188 13100 0 0

48 47 000125 00134 08000 0 0

49 46 00018 00274 02700 0 0

51 45 00004 00105 07200 0 0

51 50 00009 00221 16200 0 0

52 37 00007 00082 01319 0 0

52 55 00011 00133 02138 0 0

54 53 00035 00411 06987 0 0

55 54 00013 00151 02572 0 0

56 55 00013 00213 02214 0 0

57 56 00008 00128 01342 0 0

58 57 00002 00026 00434 0 0

59 58 00006 00092 01130 0 0

60 57 00008 00112 01476 0 0

60 59 00004 00046 00780 0 0

61 60 00023 00363 03804 0 0

63 58 00007 00082 01389 0 0

63 62 00004 00043 00729 0 0

63 64 00016 00435 0 10600 0

65 62 00004 00043 00729 0 0

65 64 00016 00435 0 10600 0

66 56 00008 00129 01382 0 0

66 65 00009 00101 01723 0 0

67 66 00018 00217 03660 0 0

68 67 00009 00094 01710 0 0

27 53 00320 03200 04100 0 0

]

LINE DATA ENDS

MACHINE DATA STARTS

Machine data format

1 machine number

2 bus number

3 base mva








T_do(sec)







T_qo(sec)




19 bus number




mac_con = [

01 01 100 00125 00 01 0031 0025 102 005 0069 00416667 0025 15

0035 42 0 0 01 0 0

02 02 100 0035 00 0295 00697 005 656 005 0282 00933333 005 15

0035 302 0 0 02 0 0

03 03 100 00304 00 02495 00531 0045 57 005 0237 00714286 0045 15

0035 358 0 0 03 0 0

04 04 100 00295 00 0262 00436 0035 569 005 0258 00585714 0035 15

0035 286 0 0 04 0 0

05 05 100 0027 00 033 0066 005 54 005 031 00883333 005 044

0035 26 0 0 05 0 0

06 06 100 00224 00 0254 005 004 73 005 0241 00675000 004 04

0035 348 0 0 06 0 0

07 07 100 00322 00 0295 0049 004 566 005 0292 00666667 004 15

0035 264 0 0 07 0 0

08 08 100 0028 00 029 0057 0045 67 005 0280 00766667 0045 041

0035 243 0 0 08 0 0

09 09 100 00298 00 02106 0057 0045 479 005 0205 00766667 0045 196

0035 345 0 0 09 0 0

10 10 100 00199 00 0169 00457 004 937 005 0115 00615385 004 1

0035 310 0 0 10 0 0

D1 Programs 277

11 11 100 00103 00 0128 0018 0012 41 005 0123 00241176 0012 15 0035 282 0 0 11 0 0

12 12 100 0022 00 0101 0031 0025 74 005 0095 00420000 0025 15 0035 923 0 0 12 0 0

13 13 200 00030 00 00296 00055 0004 59 005 00286 00074000 0004 15 0035 2480 0 0 13 0 0

14 14 100 00017 00 0018 000285 00023 41 005 00173 00037931 00023 15 0035 3000 0 0 14 0 0

15 15 100 00017 00 0018 000285 00023 41 005 00173 00037931 00023 15 0035 3000 0 0 15 0 0

16 16 200 00041 00 00356 00071 00055 78 005 00334 00095000 00055 15 0035 2250 0 0 16 0 0

]

MACHINE DATA ENDS

EXCITER DATA STARTS




2 - machine number








10 - E_1

11 - S(E_1)

12 - E_2

13 - S(E_2)



16 - K_P

17 - K_I

18 - K_D

19 - T_D

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

exc_con = [

1 1 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

1 2 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01


1 3 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

1 4 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

1 5 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

1 6 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

1 7 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

1 8 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

0 9 001 200 000 50 -50 00 0 0 0 0 0 0 0 0 0 0

0

1 10 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

1 11 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

1 12 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

]

EXCITER DATA ENDS

PSS DATA STARTS

1-S No


3-pssgain










pss_con = [

1 1 12 10 015 004 015 004 015 004 02 -005

2 2 20 15 015 004 015 004 015 004 02 -005

3 3 20 15 015 004 015 004 015 004 02 -005

4 4 20 15 015 004 015 004 015 004 02 -005

5 5 12 10 015 004 015 004 015 004 02 -005

6 6 12 10 015 004 015 004 015 004 02 -005

7 7 20 15 015 004 015 004 015 004 02 -005

D1 Programs 279

8 8 20 15 015 004 015 004 015 004 02 -005

9 9 12 10 015 004 015 004 015 004 02 -005

9 9 15 10 009 002 009 002 1 1 02 -005

10 10 20 15 015 004 015 004 015 004 02 -005

11 11 20 15 015 004 015 004 015 004 02 -005

12 12 20 15 009 004 015 004 015 004 02 -005

]

PSS DATA ENDS

--file form_jacm--




ang_redvolt_red)






entries


entries



jacobian matrix

See also

Calls




Version 20


Date March 1994


Version 10


Date March 1991


jay = sqrt(-1)



V_rect = Vexp_ang


Y_con = conj(Y)


i_c=Y_conCV_rect


S=V_recti_c

S=sparse(diag(S))


CVdia=conj(Vdia)


S1=VdiaY_conCVdia


t2=(S-S1)ang_red


J12=ang_redreal(t1)



if nargout gt 3





else

Jac11 = [J11 J12

J21 J22]

end

--loadflowm--





81297



sparse matices


D1 Programs 281

line - line data








iteration


solution format)



See also








Version 21


Date October 1996


Version 20


Date March 1994

Version 10


Date March 1991

global bus_int


global Q Ql

global gen_chg_idx




jay = sqrt(-1)

load_bus = 3

gen_bus = 2

swing_bus = 1

if exist(flag) == 0

flag = 1

end

lf_flag = 1








end



set defaults

bus data defaults

if ncollt15


if ncollt12



end




volt_min = bus(15)

volt_max = bus(14)

else

volt_min = bus(15)

volt_max = bus(14)

end




end



D1 Programs 283


end







tap_it = 0

tap_it_max = 10

no_taps = 0


if nlc lt 10


no_taps = 1


end


mm_chk=1


tap_it = tap_it+1



process bus data

bus_no = bus(1)

V = bus(2)

ang = bus(3)pi180

Pg = bus(4)

Qg = bus(5)

Pl = bus(6)

Ql = bus(7)

Gb = bus(8)

Bb = bus(9)


qg_max = bus(11)

qg_min = bus(12)

sw_bno=ones(nbus1)

g_bno=sw_bno














il = length(PQV_no)

ii = (11il)



il = length(PQ_no)

ii = (11il)









iter = iter + 1


clear Jac





clear delP delQ







V = V + accdelV

D1 Programs 285











if lim_flag == 1


end

end

if iter == iter_max





else


disp(iter)

end

if no_taps == 0

lftap

else

mm_chk = 0

end

end




else


disp(tap_it)

end





disp(voltages at)

bus(vmx_idx1)


end



disp(voltages at)

bus(vmn_idx1)


end








disp( bus Qg)





end











from_bus = line(1)


to_bus = line(2)


r = line(3)

rx = line(4)

chrg = line(5)

z = r + jayrx

y = ones(nline1)z




to to_bus

D1 Programs 287


from_bus to to_bus


- VV(from_int)tps)y



from from_bus



iline = (11nline)



keyboard





line_sol = line




if conv_flag == 1


error(stop)

end

display results

if display == y

clc



disp( )

disp(date)







if conv_flag == 0



disp(bus_sol(17))


disp( LINE FLOWS )


disp(line_ffrom)

disp(line_fto)

end

end

if iter gt iter_max


lf_flag = 0

end

return

--file y_sparsem--





line - line data






See also

Calls




Version 20


Date April 1994

Version 10


D1 Programs 289

Date March 1991

global bus_int

jay = sqrt(-1)

swing_bus = 1

gen_bus = 2

load_bus = 3



r=zeros(nline1)

rx=zeros(nline1)

chrg=zeros(nline1)

z=zeros(nline1)

y=zeros(nline1)





ibus = (1nbus)



r = line(3)

rx = line(4)










tap=ones(nline1)






iline = [11nline]











if nargout gt 1


nSW = 0

nPV = 0

nPQ = 0





nSW=length(SB)



end

return

D2 Simulation Files




Bibliography



















291

292 Bibliography


















Bibliography 293





















294 Bibliography


















Bibliography 295






















296 Bibliography


















Bibliography 297




















298 Bibliography












Forms



















List of Figures

List of Tables


Introductions

General Context

























Resonant Terms






What is needed























Introduction

















Stability Index










Invariance Property







Stability Bound



Interconnected VSCs






Closed Remarks












Introduction




Method 2-2-1

Method 3-2-3S

Method 3-3-1

Method 3-3-3



















Introduction













Case-studies

SMIB Test System




On-going Work








The Limitations







Originality








Primitive Results


A Literature Review


Conclusion










Validity region

Resonance








Interconnected VSCs



Methodology

Modeling the System






Simulation Files







SEP Initialization





Normal Dynamics


Simulation files


Programs

Simulation Files

Bibliography


Doctorat ParisTech





TianTIAN







Jury


2








ii

iii

Acknowledgements






iv






Leonardo da Vinci

vi


















vii

viii

Contents

List of Figures xix

List of Tables xxi












ix

x Contents

















Contents xi












xii Contents















Contents xiii











xiv Contents


















Contents xv







Bibliography 291

xvi Contents

List of Figures













xvii

















List of Figures xix





for δep = π12 169


for δep = π12 169







xx List of Figures

List of Tables


methods 52


NNM 113










2 and case 3 236

xxi

xxii List of Tables









1









3






















5





x = f(xu) (1)

g(xu) = 0



x0 = f(x0u0) = 0 (2)



4x = A4x+O(2) (3)




4y = Λ4y (4)







pki = UkiVik (5)




7























9











11



Problegraveme



























13








x = f(x) (6)


Systems




M q +Dq +Kq + fnl(q) = 0 (7)






doctorat






15







L2EP et LSIS















17























bull


Chapter 1

Introductions












19


11 General Context





















1122 L2EP and EMR




























































δ = ωsω


(11)





Innite Bus

Great Grid

G1

Pe

jXe

minusrarr

E


t(s)0 5 10 15 20 25

δ(degree)

30

40

50

60Predisturbance



















Example Case 1




0 2 4 6 8 10 12

δ1(degree)

40

60

80

100

120

140

t(s)0 2 4 6 8 10 12

δ2(degree)

0

20

40

60

80

100

Exact

Linear




x = f(x) (12)











pki = ukivik (15)












ζωnStates


primeq2 E

primeq3 E

primeq4
















t(s)0 05 1 15 2 25

δ(rad)

-1

0

1

2





Pe =EprimeV0X

sin δ (16)








minus sin δsep2










Pow

erf(δ)

02

05

08

1

Linear Analysis

2nd Order Analysis

3rd Order Analysis

Regionmaybeunstable



Example Case 2



0 5 10 15 20 25 30

δ1(degree)

40

70

100

130140

t(s)0 5 10 15 20 25 30

δ2(degree)

0

30

60

90100

ExactLinear


0 036 072 108 144 180 212 252Hz

101

102

Mag

(

of F

unda

men

tal)



2nd OrderFrequency

3rd OrderFrequency

































x = f(xu) (18)


4x = A4x+ f2(4x2) + f3(4x3) (19)




















































Perturbationmodel





3)





161 Resonant Terms



mode j








zj = λjzj +Nsum

ωkωl

NsumisinR2

gj2klzkzl +Nsum

ωpωq ωr

NsumisinR3

gj3pqrzpzqzr (111)


j3







Fn hn gn S or




nonlinearities









2 2 1











3 3 1









3 2 3 S



173 What is needed











G1 δ1

G2 δ2

jX12jXe

ZZPe

Innite Bus

E


Pe1 = V1V2X12




Pe2 = V1V2X12




(113)



4x = A4x+ a2x2 + a3x

3 (114)


yj = λjyj +4sum

k=1

4suml=1

ykyl +4sump=1

4sumq=1

4sumr=1

ypyqyr (115)




3)






(116)


3 2 3 S







2) + hj3(z3)


+gj3zjz3z4

















0 05 1 15 2minus05

0

05

1

15

t(simulation time)

δ1(rad)

Mode Decomposition

EXMode 1Mode 2

0 05 1 15 2minus05

0

05

1

15

t(simulation time)

δ2(rad)






























Chapter 2





























59


Methodologies













x = y + h2(y)

= z + h3(z) + h2(z + h3(z)) (23)

= [z + h2(z)] + h3(z) +Dh2(z)h3(z)

= z + h2(z) + h3(z) + (O)(4)

2112 Normal Form




2113 Resonant Terms












Methodologies








2131 Linear System


x = Ax (26)


y = Λy (27)


yj = λjyj (28)





X = sX + a2X2 + a3X


apXp (29)



X = Y + α2Y2 (210)



(1 + 2α2Y )Y = sY + (sα2 + a2)Y2 +O(Y 3) (211)





Y = sY + (a2 minus sα2)Y2 +O(Y 3) (213)


α2 =a2s

(214)


Y = sY +O(Y 3) (215)



Methodologies




221 Introduction


u = Au+ εF2(u) + ε2F3(u) + (216)
























Methodologies

and






ε0 y = Λy (224)





+

ksuml=2


where termssumk


















Methodologies















x = f(xu) (228)




4x = H1(4x) +1

2H2(4x) +

1

3H3(4x) +O(4) (229)



] H3jklm =


]and O(4) are






yj = λjyj +Nsumk=1

Nsuml=1

F2jklykyl +Nsump=1

Nsumq=1

Nsumr=1



sumNi=1 vji

[UThj3U

] F3jpqr = 1

6

sumNi=1 vji

sumNk=1

sumNk=1

sumNm=1H3ikLNMu

lpumq u

Nr vji




y = Λy + F (y) (232)




y = Λy + F2(y) (233)


Methodologies


yj = λjyj +Nsumk=1

Nsuml=1

F2jklykyl (234)


y = z + h2(z) (235)


yj = zj +Nsumk=1

Nsuml=1

h2jklzkzl (236)


z + ˙h2(z) = Λz + Λh2(z) + F2(z + h2(z)) (237)





z = Λz (order 1)










h2jkl =F2jkl





z = Λz +DF2(z)h2(z) +O(4) (241)

= Λz +O(3)

= Λz



yj = λjyj +

Nsumk=1

Nsuml=1

F2jklykyl +

Nsump=1

Nsumq=1

Nsumr=1

F3jpqrypyqyr (242)


z + ˙h2(z) = Λz + Λh2(z) + F2(z + h2(z) (243)



+HF2(z)(h2(z))2 + F3(z) +DF3zh2(z))


+DF2(z)h2(z) + F3(z) +O(4))


z = Λz (order 1)




(order 4)



z = Λz +DF2(z)h2(z) + F3(z) +O(4) (244)


Methodologies



zj =λjzj (245)

+

Nsump=1

Nsumq=1

Nsumr=1

(Nsuml=1


)zpzqzr

=λjzj +Nsump=1

Nsumq=1

Nsumr=1

Cjpqrzpzqzr

where Cjpqr =sumN




z = w + h3(w) (246)



(247)




+O(5))(Λw + Λh3(w) +DF2(w)h2(w) + F3(w) +O(5))


w = Λw (order 1)



(order 5)


w =Λw +DF2(w)h2(w) (248)





wj =λjwj

+

Nsump=1

Nsumq=1

Nsumr=1


+O(5) (249)


h3jpqr =F3jpqr +




wj = λjwj +

Msuml=1




Methodologies


cj=2k2l =C2k


(2l)(2k)(2lminus1)



cj=2k2k =C2k


(2kminus1)(2k)(2k) (253)












zj =wj +

Nsump=1

Nsumq=1

Nsumr=1










h3jpqr =F3jpqr



wj = λjwj +O(3) (258)



2361 2-2-1 and 3-2-3S


y = z + h2(z) (259)



+

Nsumk=1

Nsuml=1


+O(3) (260)

For 3-2-3S


+

Nsumk=1

Nsuml=1


+O(3) (261)


Methodologies

2362 3-3-3 and 3-3-1


y =Λw + h2(z) (262)

Λw + h3(w) + h2(w + h3(w))

=Λw + h3(w) + h2(w)



y = w + h2(w) + h3(w) (263)

For 3-3-1


+Nsumk=1

Nsuml=1


+Nsump=1

Nsumq=1

Nsumr=1


(264)

For 3-3-3


+

Nsumk=1

Nsuml=1


+

Nsump=1

Nsumq=1

Nsumr=1


(265)












(z + h2(z)) = z +Dh2(z)z = z +Dh2(z)Λz (266)





Methodologies







f1j = λjf1j (268)

f2j = λjf2j +

Nsumk=1

Nsuml=1

Cjklf1kf

1l (269)

f3j = λjf3j +

Nsumk=1

Nsuml=1

Cjkl(f1kf

2l + f2kf

1l ) +

Nsumk=1

Nsuml=1

Nsumk=1

Djpqrf

1p f

1q f

1r (270)


f1j (t) = y0j eλjt (271)


f2j = λjf2j +

Nsumk=1

Nsuml=1

Cjkly0ky

0l e

(k+l)t (272)


f2j (s1 s2) =

Nsumk=1

Nsuml=1

Cjkl1


1l (s2) (273)



f2j =Nsum

k=1

Nsuml=1


=Nsum

k=1

Nsuml=1

Cjkly0ky

0l S

jkl(t)

where

Sjkl(t) =1





k=1

Nsuml=1

Cjkl1


1l (s3) + f1k (s1)f

2l (s2 + s3))+

(275)Nsum

k=1

Nsuml=1

Nsumr=1

Djpqr

1


1q (s2)f

1r (s3)


f3j (s1 s2 s3) =

Nsumk=1

Nsuml=1

Cjkl1


[Nsum

k=1

Nsuml=1

Cjkl1


1l (s2)

]f1l (s3)

(276)

+ f1k (s1)

[Nsum

k=1

Nsuml=1

Cjkl1


1l (s3)

]

+Nsum

k=1

Nsuml=1

Nsumr=1

Djpqr

1


1q (s2)f

1r (s3)




Methodologies















zj = wj +Nsump=1

Nsumq=1

Nsumr=1

h3jpqrwpwqwr (277)







MI3jpqr =|h3jpqrw0

pw0qw

0r |

|w0j |

(278)


242 Stability Index


wj = λjwj +

Msuml=1


= (λj +Msuml=1

cj2l|w2l|2)wj

(280)



Since

λj +Msuml=1



SIIj2l)) (282)


Methodologies










j2l times Tr



SIj = σj +Msuml=1




(285)






Equations




Mq +Dq +Kq + fnl(q) = 0 (286)


fpnl(q) =Nsum

i=1jgeif2pijqiqj +

Nsumi=1jgeikgej

f3pijkqiqjqk (287)


Methodologies







+Nsumi=1

Nsumjgei

gpijXiXj +Nsumi=1

Nsumjgei

Nsumkgej

hpijkXiXjXk = 0



[gp] =

Nsumi=1



hpijk =

Nsumi=1

Φminus1pi (

NsumP=1

NsumQ=1

NsumR=1













Nsumi=1

Nsumjgei

gpijXiXj = 0 (293)


Xp = Yp (294)


Nsumjgei

gpijXiXj (295)


X = U + ε(a(U2) + b(V 2) + c(UV ))

Y = V + ε(α(U2) + β(V 2) + γ(UV ))

(296)


Methodologies


Xp = Up +Nsumi=1

Nsumjgei


Nsumj=1

cijUiVj

Yp = Vp +Nsumi=1

Nsumjgei


Nsumj=1

γijUiVj (297)


pij c

pij α

pij β

pij γ

pij



Uj = Vj +O(U2i V

2i ) (298)


i V2i ) (299)



3i ) (2100)


pij + bpji)UiVj



3i )


pij + βpji)UiVj



D(a+ b+ c) = α+ β + γ (2101)

D(α+ β + γ) + 2ξωD(a+ b+ c) (2102)




Dc(UV )V = α (U2)

Dc(UV )U = β (V 2)

(2103)






ing (2103)(2104)apij b

pij c

pij α

pij β

pij γ

pij



Up + 2ξpωpUp + ω2pUp +O(3) = 0 (2105)

with

O =Nsumi=1

Nsumjge1

Nsumkgej


+Nsumi=1

Nsumjgei

Nsumkgej

BpijkUiUjUk +

Nsumi=1

Nsumjgei

Nsumkgej

CpijkUiUjUk



Apijk =

Nsumlgei

gpilaljk +

sumllei

gplialjk (2107)

Bpijk =

Nsumlgei

gpilbljk +

sumllei

gplibljk (2108)

Cpijk =

Nsumlgei

gpilcljk +

sumllei

gplicljk (2109)


Methodologies




U = R+ ε2(r(R3) + s(S3) + t(R2 S) + u(RS2))


(2110)



D(r + s+ t+ u) = λ+ micro+ ν + ζ (2111)



To degree ε2




DuR = micro (S3)

(2113)





(2114)









foralli le p



(2115)






ωp = ωi + ωj ωp = 2ωi (2116)





Methodologies





X + ω2X + αX3 = 0 (2119)









1minus ξ2p (2121)









pppRpR2p + CppppR

2pRp



P(2) =

Nsumjgep


2j +Bp


](2124)

+sumilep


2i +Bp


]

Q(2) =

Nsumjgep

(BpjpjRjRj + CpjjpR

2j ) +

Nsumilep

(BpiipRiRi + CpiipR

2i ) (2125)




Methodologies


pppp C

pppp)




pppRpR2p + CppppR

2pRp = 0




Xp = Rp +

Nsumi=1

Nsumjgei


Nsumi=1

Nsumj=1

cijRiSj

+Nsumi=1

Nsumjgei

Nsumkgej


+

Nsumi=1

Nsumjgei

Nsumkgej



Yp = Sp +Nsumi=1

Nsumjgei


2sumj=1

γijRiSj

+

Nsumi=1

Nsumjgei

Nsumkgej


+Nsumi=1

Nsumjgei

Nsumkgej






y

=

micro 0

0 minus1

x

y

+

xy + cx3

bx2

(2128)


microc = 0 xc = x xs = y (2129)



y = bx2 + (|x micro|3) (2130)


Methodologies


x = microx+ (b+ c)x3 (2131)



micro 0

0 minus1








Modes












p S0p [18] And R0

p S0p

satises X0p = R0

p +R(3)(R0i S

0i ) Y 0

p = S0p + S(3)(R0

i S0i )


ωnlp = ωp

(1 +


pppp

8ω2p

a2

)(2133)


262 Stability Bound



3p


2p

]Rp (2134)



2p ge 0 (2135)


Methodologies

with

Rp le

radicminus

ω2p

hpppp +Apppp(2136)



2p + rppppR

3p (2137)


Knl =RpXp

=1







NIp =

radicsumNk=1

sumNj=1


∣∣∣2|Xp0|

(2140)





pBpppp

16πωp(2141)



Study





Methodologies



J1d2δ1dt2


sin δ1 + V1V2L12


J2d2δ1dt2


sin δ2 + V1V2L12


(2142)





R1 + 2ξ1ω1R1 + ω21R1 + (h1111 +A1

111)R31 +B1

111R1R12

+ C1111R

21R1

+R1((A1212 +A1

122 + h1122)R22 +B1

122R22

+ (C1122 + C1

212)R2R2) + R1(B1212R2R2 + C1

221R22) = 0

R2 + 2ξ2ω2R2 + ω22R2 + (h2222 +A2

222)R32 +B2

222R2R22

+ C2222R

22R2

+R2((A2112 +A2

211 + h2112)R22 +B2

211R12

+ (C2211 + C2

121)R1R1)) + R2(B2112R1R2 + C2

112R22) = 0

(2143)



R1 + 2ξ1ω1R1 + ω21R1 + (h1

111 +A1111)R

31 +B1

111R1R12+ C1

111R21R1 +R2P

(2)1 (Ri Ri) + R2Q

(2)1 (Ri Ri) = 0

R2 + 2ξ2ω2R2 + ω22R2 + (h2

222 +A2222)R

32 +B2

222R2R22+ C2

222R22R2 +R1P

(2)2 (Ri Ri) + R1Q

(2)2 (Ri Ri) = 0

(2144)


R1 + 2ξ1ω1R1 + ω21R1 + (h1111 +A1

111)R31 +B1

111R1R12

+ C1111R

21R1 = 0

(2145)


R1 + 2ξ1ω1R1 + ω21R1 + (h1111 +A1

111)R31 +B1

111R1R12

+ C1111R

21R1 = 0

R2 + 2ξ2ω2R2 + ω22R2 + (h2222 +A2

222)R32 +B2

222R2R22

+ C2222R

22R2 = 0

(2146)



Methodologies




0 02 04 06 08 10

05

1

15

δ1(rad)

EXNNMLNM

0 02 04 06 08 1minus05

0

05

1

15

t(s)

δ2(rad)

EXNNMLNM

0 005 01 015 020

05

1

15

δ1(rad)

EXNNMLNM

0 005 01 015 02minus05

0

05

1

15

t(s)

δ2(rad)

EXNNMLNM






0 02 04 06 08 1minus02

minus01

0

01

02

Errorin

δ1(rad)

Error

FNNMNNMNM2

0 02 04 06 08 1minus02

minus01

0

01

02

Errorin

δ2(rad)








Methodologies

0 05 1 15 2minus05

0

05

1

15

t(simulation time)

δ1(rad)

Mode Decomposition

EXMode 1Mode 2

0 05 1 15 2minus05

0

05

1

15

t(simulation time)

δ2(rad)








0 02 04 06 08 1

005

1105

11

115

Disturbance in δ1

Mode Frequency

Disturbance in δ2

ω1(rad)

0 02 04 06 08 1

005

1137

1375

138

Disturbance in δ1

Disturbance in δ2

ω2(rad)


01 03 07 09 110

005

01

015

02

025

03

035

04

045

05


NI

Mode1Mode2

01 03 07 09 110

05

1

15

2

25

3

35


FS

Mode1Mode2


0001 0005 001 005 010

005

01

015

02

025

03

035


NI

Mode1Mode2

0001 0005 001 005 010

1

2

3

4

5

6

7

8

9

10


FS

Mode1Mode2



Methodologies

0 005 01 015 02minus01

0

01

02

δ1(rad)

Dynamics

0 005 01 015 02minus02

minus01

0

01

02

t(s)

δ2(rad)

EXNNMLNM

(a) X1 = X2 = 01

0 005 01 015 0205

1

15

2

δ1(rad)

Dynamics

EXNNMLNM

0 005 01 015 020

05

1

15

t(s)

δ2(rad)

(b) X1 = X2 = 11

0 02 04 06 08 1minus5

0

5x 10

minus4

Errorin

δ1(rad)

Error

0 02 04 06 08 1minus5

0

5x 10

minus4

Errorin

δ2(rad)

NNMLNM

0 02 04 06 08 1

minus05

0

05

Errorin

δ1(rad)

Error

0 02 04 06 08 1

minus05

0

05

Errorin

δ2(rad)

NNMLNM



0 005 01 015 020

05

1

15

δ1(rad)

Dynamics

EXNNMLNM

0 005 01 015 02minus1

0

1

2

t(s)

δ2(rad)

(a) X12 = 0001

0 005 01 015 020

05

1

15

δ1(rad)

Dynamics

EXNNMLNM

0 005 01 015 02minus1

0

1

2

t(s)

δ2(rad)

(b) X12 = 01

0 02 04 06 08 1minus05

0

05

Errorin

δ1(rad)

Error

NNMLNM

0 02 04 06 08 1minus05

0

05

Errorin

δ2(rad)

0 02 04 06 08 1minus01

0

01

Errorin

δ1(rad)

Error

NNMLNM

0 02 04 06 08 1minus01

0

01

Errorin

δ2(rad)



Methodologies


277 Closed Remarks



Forms in Practice




f(z) = z minus y0 + h(z) = 0 (2147)









fj(w) = wj +Nsumk=1

Nsuml=1


Nsumq=1

Nsumr




fj(w(s)) = w

(s)j +

Nsumk=1

Nsuml=1

h2jklh2jklw(s)k w

(s)l +

Nsump=1

Nsumq=1

Nsumr


q w(s)r = 0


Methodologies


[A(ws)] =

[partf

partw

]w=w(s)




w(s+1) = w(s) + ρ4w(s)



The algorithm ends




fp = Rp +Nsumi=1

Nsumjgei

apijRiRj +Nsumi=1

Nsumjgei

Nsumkgej











Methodologies




X1

Y1

Xp

Yp

XN

˙YN

=

0 1



0

0 1


0

0


0 1


X1

Y1

Xp

Yp

XN

YN

+

g(X1) + h(X1)

0

g(Xp) + h(Xp)

0

g(XN) + h(XN)

0

(2149)





sumNk=1


p=1

sumNq=1











sumNi=1 vji

[UThj3U

]

F3jpqr = 16

sumNi=1 vji

sumNk=1

sumNk=1

sumNm=1H3ikLNMu

lpumq u

Nr





Methodologies

















Methodologies

Chapter 3



















115



31 Introduction






31 Introduction 117






Powerf(δ)

02

04

06

08

1

12

14


Exact

Linear

Tayl(2)

Tayl(3)

Regionmaybeunstable

3rd Order Analysis

2nd Order Analysis

Linear Analysis









x = f(xu) (31)


4x = H1(4x) +1

2H2(4x) +

1

3H3(4x) +O(4) (32)



] H3jklm =


]and O(4) are






yj = λjyj +

Nsumk=1

Nsuml=1

F2jklykyl +

Nsump=1

Nsumq=1

Nsumr=1



sumNi=1 vji

[UThj3U

] F3jpqr = 1

6

sumNi=1 vji

sumNk=1

sumNk=1

sumNm=1H3iklmu

lpumq u

nr vji is






322 Method 2-2-1



with h2jkl =F2jkl


323 Method 3-2-3S



then

y = z + h2(z)rArr

zj = λjwj +

sumMl=1 c



(37)


324 Method 3-3-1





325 Method 3-3-3



wj = λjwj +

sumMl=1 c














tem









di

qi

x

y

dk

qk

δiδk

ωs





δi = ωsωk

ωi = 12Hi


˙eprimeqi = 1

Tprimed0i


primeqi)

˙eprimedi = 1

Tprimeq0i


primedi)

(310)





4idqminusxy =

sin δi cos δi


4ixyminusdq =


cos δi sin δi

(311)



primed0T

primeq0


primed0 are the


primeq






11+sTR

times 1+sTC1+sTB

KA1+sTA

VT

+

Vref

minusXE1 XE2 Efd



˙Efdi = KAiXE2iTAi

minus 1TAi

Efdi

˙XE1i = minus 1TRi

XE1i + 1TRi

VTi

˙XE2i = 1TBi


TBiXE2i minus TCi

TBiTRiVT i + 1

TBiVrefi

(312)


VT i =radic


primeqiiqi)







primed e


]






eprimexieprimeyi

= 4idqminusxy

eprimedieprimeqi

It = Y

[eprimex1 e


primexn e

primeyn



primeyiiyi

(315)

where eprimeqi e


primexi e

primeyi are









I1

I2

Im

Im+1

Im+n

=






V1

V2

Vm

Vm+1

Vm+n

(317)




In

=

Ymm Ymn

Ynm Ynn

Vm

Vtn

(318)



]Vtn (319)



](320)



Ytijej =Nsumj=1


Nsumj=1

[(Gtijexi minusBt



Yr =

Gt11 minusBt11

Bt11 Gt11


Gt1n minusBt1n

Bt1n Gt1n

Gtn1 minusBt

n1

Btn1 Gtn1


Gtnn minusBtnn

Btnn Gtnn

(321)




primen

] an admittance




primexn e

primeyn

](322)



primexn e

primeyn




Since edieqi

=

eprimedieprimeqi


idiiqi

=

eprimedieprimeqi

minus Ziidiiqi

(323)


4idqminusxy

edieqi

= 4idqminusxy

eprimedieprimeqi

minus4idqminusxyZk

idiiqi

(324)


=

eprimexieprimeyi


xyminusdq

ixiiyi

(325)



dqminusxy

] T2 =


ex1

ey1

ex2

ey2

exn

eyn

=

eprimex1

eprimey1

eprimex2

eprimey2

eprimexn

eprimeyn

minus T1T2Tminus11

ix1

iy1

ix2

iy2

ixn

iyn

=rArr Y minus1r

ix1

iy1

ix2

iy2

ixn

iyn

= Y minus1

ix1

iy1

ix2

iy2

ixn

iyn

minus T1T2Tminus11

ix1

iy1

ix2

iy2

ixn

iyn

(326)




If Xprimeq = X

primed then 4i






(328)



prime=[

eprimed1 e


primedn e

primeqn

]can be obtained as


]T= Tminus11 Y T1

[eprimed1 e


primedi e


]T(329)




ωi = 12Hi


eprimeqi = 1

Tprimed0i


primeqi)

eprimedi = 1

Tprimeq0i


primedi)

˙Efdi = KAiXE2iTAi

minus 1TAi

Efdi

˙XE1i = minus 1TRi

XE1i + 1TRi

VT i

˙XE2i = 1TBi


TBiXE2i minus TCi

TBiTRiVT i + 1

TBiVrefi

(331)




δi

ωi

eprimeqi

eprimedi

˙Efdi

˙XE1i

˙XE2i

=

0 ωs 0 0 0 0 0

0 minus Di2Hi

0 0 0 0 0

0 0 minus 1

Tprimed0i

0 1

Tprimed0i

0 0

0 0 0 minus 1

Tprimeq0i

0 0 0

0 0 0 0 minus 1TAi

0 KAiTAi


0

0 0 0 0 0 1TBi


TBi

δi

ωi

eprimeqi

eprimedi

Efdi

XE1i

XE2i

+

0


2Hi


Tprimed0i

idi

XqiminusXprimeqi

Tprimeq0i

iqi

0

VTiTRi

minus TCiTBiTRi

VT i

+

0

Pmi2Hi

0

0

0

0

VrefiTBi

(332)


As [Idq] = Yδ

[eprimedq



xi =

δi

ωi

eprimeqi

eprimedi

Efdi

XE1i

XE2i

Agi =

0 ωs 0 0 0 0 0

0 minus Di2Hi

0 0 0 0 0

0 0 minus 1

Tprimed0i

0 0 0 0

0 0 0 minus 1

Tprimeq0i

0 0 0

0 0 0 0 minus 1TAi

0 KAiTAi


0

0 0 0 0 0 1TBi


TBi

Ni(X) =

0


2Hi


Tprimed0i

idi

XqiminusXprimeqi

Tprimeq0i

iqi

0

VTiTRi

minus TCiTBiTRi

VT i

Ui =

0

Pmi2Hi

0

0

0

0

VrefiTBi

(333)


x1

x2

xn

=




x1

x2

xn

+

N1(x)

N2(x)

Nn(x)

+

U1

U2

Un

(334)

x = Agx+N(x) + U (335)




4x = Ag4x+N1(4x) +1

2N2(4x) +

1

3N3(4x) (336)


H1 = Ag +N1 H2 = N2 H3 = N3 (337)

























States


primeq2 E

primeq3 E

primeq4










States


primeq2 E

primeq3 E

primeq4










0 5 10 15 20 25 30

δ1(degree)

40

70

100

130140

t(s)0 5 10 15 20 25 30

δ2(degree)

0

30

60

90100

Exact 3-2-3S 2-2-1 3-3-1 3-3-3 Linear



Hz0 036 072 108 144 180 212 252

Mag

(

of F

unda

men

tal)

101

102

Exact3-2-3S2-2-13-3-13-3-3Linear


0 5 10 15 20 25 30

δ1(degree)

-100

0

100

180

t(s)0 5 10 15 20 25 30

δ2(degree)

-100

-50

0

50

100

150180200

Exact 3-2-3S 3-3-3 2-2-1 3-3-1 Linear





t(s)0 10 20 30

(pu)

085

09

095

1

105

11

115Eprime

q1 Eprime

q2 Eprime

q3 Eprime

q4

Eq1

Eq2

Eq3

Eq4

t(s)0 5 10 15 20 25 30

(pu)

-55Vf

min

-4

-3

-2

-1

0

1

2

3

4

Vfmax


Vmex2

Vfex2



primeq2 E

primeq3 E









j y0 z0221 z0331 z0333

5 514 + j293 389 + j238 388 + j238 03105 + j03263





15 032 + j010 004minus 00012i 005minus j00012 0072 + j005

17 087 + j252 082 + j216 084 + j210 170 + j131i



(171717) 0046minus j0118 0029 + j0034 0017minus j0019


01220 00068 + j0008 00560 033














7 048 + j096 minus621 + j883 096

9 00117 + j017 minus1837 + j628 098


13 0074minus j025 minus178 + j673 085

15 089 + j0532 minus196 + j651 086

17 012minus j0678 minus0405 + j228 033











5 minus5354minus j10977 033 -176682











17 029 + j131 11801 034











Tprimeq0i


primeqi)minusIqi +

(XprimeqiminusX

primeprimeqi)




Tprimed0i

dEprimeqi


primedi)Idi +

(XprimediminusX

primeprimedi)



primeqi)

Tprimeprimed0i

dψ1didt = E

primeqi + (X


prime1di)

Tprimeprimeq0i

dψ2qi


prime2qi)


primeprimedi minusX


primedci

(338)




(XprimeqiminusXlsi)

+ EprimeqiIqi


(XprimediminusXlsi)


primeprimeqi) +


primeprimedi)

(XprimediminusXlsi)


primeprimeqi)

(XprimeqiminusXlsi)


primeqi


(XprimediminusXlsi)

+ψprime1di

(XprimediminusX

primeprimedi)

(XprimediminusXlsi)

minusVqi+j[Eprimedi


(XprimeqiminusXlsi)

minus

ψprime2qi

(XprimeqiminusX

primeprimeqi)

(XprimeqiminusXlsi)


qi

di

x

y

qk

dk

δiδk

ωs






TedEfd





s + sKdTd+1)



(339)







(340)




Vss = KpsssTw

1 + sTw

(1 + sT11)

(1 + sT12)

(1 + sT21)

(1 + sT22)

(1 + sT31)

(1 + sT32)Sm (341)









1 -0438 0404 δ(13) 1 ω(13) 0741 ω(15) 0556 ω(14) 0524

2 0937 0526 δ(14) 1 ω(16) 0738 ω(14) 05 δ(13) 0114

3 -3855 061 δ(13) 1 ω(13) 083 δ(12) 0137 ω(6) 0136

4 3321 0779 δ(15) 1 ω(15) 0755 ω(14) 0305 δ(14) 0149

5 0256 0998 δ(2) 1 ω(2) 0992 δ(3) 0913 ω(3) 0905

6 3032 1073 δ(12) 1 ω(12) 0985 ω(13) 0193 δ(13) 0179

7 -1803 1093 δ(9) 1 ω(9) 0996 δ(1) 0337 ω(1) 0333

8 3716 1158 δ(5) 1 ω(5) 1 ω(6) 0959 δ(6) 0958

9 3588 1185 ω(2) 1 δ(2) 1 δ(3) 0928 ω(3) 0928

10 0762 1217 δ(10) 1 ω(10) 0991 ω(9) 0426 δ(9) 042

11 1347 126 ω(1) 1 δ(1) 0996 δ(10) 0761 ω(10) 0756

12 6487 1471 δ(8) 1 ω(8) 1 ω(1) 0435 δ(1) 0435

13 7033 1487 δ(4) 1 ω(4) 1 ω(5) 0483 δ(5) 0483

14 6799 1503 δ(7) 1 ω(7) 1 ω(6) 0557 δ(6) 0557

15 3904 1753 δ(11) 1 ω(11) 0993 Ψ1d(11) 0056 ω(10) 0033






1 33537 0314 ω(15) 1 ω(16) 0982 δ(7) 0884 δ(4) 0857

2 3621 052 δ(14) 1 ω(16) 0645 ω(14) 0521 δ(15) 0149

3 9625 0591 δ(13) 1 ω(13) 083 ω(16) 0111 ω(12) 0096

4 3381 0779 δ(15) 1 ω(15) 0755 ω(14) 0304 δ(14) 0148

5 27136 0972 ω(3) 1 δ(3) 0962 ω(2) 0924 δ(2) 0849

6 18566 108 ω(12) 1 δ(12) 0931 Eprimeq(12) 0335 PSS4(11) 0196

7 23607 0939 δ(9) 1 ω(9) 0684 PSS3(9) 0231 PSS2(9) 0231

8 30111 1078 δ(5) 1 ω(5) 0928 ω(6) 0909 δ(6) 0883

9 28306 1136 ω(2) 1 δ(2) 0961 δ(3) 0798 ω(3) 0777

10 13426 1278 ω(1) 1 δ(1) 0969 Eprimeq(1) 0135 ω(8) 0111

11 1883 1188 δ(10) 1 ω(10) 0994 Eprimeq(10) 0324 PSS3(10) 019

12 32062 1292 δ(8) 1 ω(8) 0935 Eprimeq(8) 061 PSS2(8) 0422

13 39542 1288 δ(4) 1 ω(4) 0888 Eprimeq(4) 0706 PSS4(4) 0537

14 33119 1367 δ(7) 1 ω(7) 0909 δ(6) 0653 ω(6) 0612

15 2387 5075 ω(11) 1 Eprimeq(11) 0916 Ψ1d(11) 0641 PSS4(10) 0464



Frequency (Hz)0 02 04 06 08 10 12 14 16 18 2

Mag

( o

f Fun

dam

enta

l)

100

101

102G1G12G13G14G15



Mode 1 2 3 4


frequency (Hz) 07788 06073 05226 03929










Frequency (Hz)0 02 04 06 08 10 12 14 16 18 20

Mag

(

of F

unda

men

tal)

100

101

102G6G9G13G14G15





3-3-3 12 1294e-12 17 1954e-6

2-2-13-2-3 3 1799e-11 6 1838e-06

3-3-1 7 1187e-09 11 7786e-09

literature












real(cj=2k2k )

= 0




















Chapter 4
















151


Assessment

41 Introduction



fer Limit


δ = ω (41)



















Assessment







δ = ωsω (42)




t(s)0 5 10 15 20 25 30 35 40 45 50

δ(rad)

-1

-05

0

05

1

15




Assessment


with

Pmax =EprimeqE0



t(s)0 05 1 15 2 25 3 35 4

δ(rad)

-3

-2

-1

0

1

2

3

Exact Linear SEP



















Assessment



























Assessment















o-line2 Step-by-

step inte-gration



1 Mostly-Numerical


















Mq +Dq +Kq + fnl(q) = 0 (44)


fpnl(q) =

Nsumi=1jgei

f2pijqiqj +

Nsumi=1jgeikgej

f3pijkqiqjqk (45)


4241 LNM



Assessment



+

Nsumi=1

Nsumjgei

gpijXiXj +

Nsumi=1

Nsumjgei

Nsumkgej

hpijkXiXjXk = 0




4242 NM2


Up + 2ξpωpUp + ω2pUp +O(3) = 0 (48)

4243 NNM




pppRpR2p + CppppR

2pRp = 0







with

Rp le

radicminus

ω2p

hpppp +Apppp(411)



Knl =RpXp

=1












δ = ωsω (413)


with

Pmax =EprimeqV0Xe




Assessment




2Hi


Dωi

ωbδi (416)

where

Pei = E2iGgii +

Nsumk=1k 6=i









(418)



=

sumN

k=1k 6=i


] j = i


] j 6= i

(419)



2part2Peipartδiδj

reads

Ki2Sii = minus

sumNk=1k 6=i


]Ki

2Sij = EiEj


] j 6= i

Ki2Sjj = minusEiEj


] j 6= i

(420)



it reads

Ki3Siii = 1

6

sumNk=1k 6=i


]Ki

3Siij = minus12EiEj


] j 6= i

Ki3Sijj = 1

2EiEj


] j 6= i

Ki3Sjjj = minus1

6EiEj


](421)




2Hi

ωbδi minus

Dwi


2Hn

ωbδn minus

Dwn



2H



2H

ωb4δN +

D



Assessment





](426)



reads

Kprimei2Sii = Ki


]Kprimei2Sij = Ki

2Sijj 6= i

Ki2Sjj = Ki

2Sjj + EnEj


]j 6= i

(427)



it reads

Kprimei3Siii = Ki


]Kprimei3Siij = Ki

3Siij j 6= i

Kprimei3Sijj = Ki

3Sijj j 6= i

Kprimei3Sjjj = Ki

3Sjjj + 16EnEj


](428)

44 Case-studies






44 Case-studies 167

t(s)0 05 1 15 2 25 3 35 4

δ(rad)

1

12

14

16

EX LNM NM2 NNM SEP


t(s)0 02 04 06 08 1 12 14 16 18 2

δ(rad)

-3

-2

-1

0

1

2

3

EX LNM NM2 NNM SEP





Assessment




PSBminusEABEAB




44 Case-studies 169

t(s)0 05 1 15 2 25 3

δ(rad)

-4

-2

0

2

4

6

EX LNM NM2 NNM SEP


12

t(s)0 05 1 15 2 25 3

δ(rad)

-4

-2

0

2

4



12



Assessment








452 On-going Work



0 5 10 15 20

δ14(rad)

0

05

1

15

2

25

t(s)0 5 10 15 20

δ24(rad)

-05

0

05

1

15

2EX LNM NM2 NNM SEP


0 5 10 15 20

δ14(rad)

0

1

2

3

EX LNM NM2 NNM SEP

t(s)0 5 10 15 20

δ24(rad)

0

1

2



Assessment

0 5 10 15 20

δ14(rad)

-2

0

2

4

EX LNM NM2 NNM SEP

t(s)0 5 10 15 20

δ24(rad)

-2

0

2

4


0 5 10 15 20

δ14(rad)

-2

0

2

4

EX LNM NM2 NNM SEP

t(s)0 5 10 15 20

δ24(rad)

-2

0

2

4






Assessment

Chapter 5


Oscillations


























175


Oscillations













Oscillations









Oscillations












Oscillations




542 The Limitations











Oscillations






M q +Dq +Kq = F (51)



ΦTj Fj

Mj(52)













Mq +Cq +Kq + fnl(q) = 0 (53)


Oscillations









552 Originality





Mq +Cq +Kq + fnl(q) = F (54)





+

Nsumi=1

Nsumjgei

gpijXiXj +

Nsumi=1

Nsumjgei

Nsumkgej

hpijkXiXjXk = Fp











Oscillations

5552 NNM




pppRpR2p + CppppR

2pRp = 0

5553 FDNF3






pRp






5561 Test System







M1d2δ1dt2

+D1dδ1dt +

V1VgX1

sin δ1 + V1V2X12


M2d2δ2dt2

+D2dδ2dt +

V2VgX2

sin δ2 + V1V2X12


(59)



Oscillations

0 05 1 15 2 2505

055

06

065

07

075

08

085

09

t(s)

δ1(rad)

Dynamics

EXLNMFDNF3

0 05 1 15 2 2502

025

03

035

04

045

05

055

06

065

t(s)

δ2(rad)

Dynamics

EXLNMFDNF3









0 05 1 15 2minus05

0

05

1

15

t(simulation time)

δ1(rad)

Mode Decomposition

EXMode 1Mode 2

0 05 1 15 2minus05

0

05

1

15

t(simulation time)

δ2(rad)


0 05 1 15 2 25minus01

0

01

02

03

04

05

06

t(s)

NonlinearModalState

Variables


R1R2



Oscillations











pppRpR2p + CppppR

2pRp = F







EMR+IBC






Oscillations











Oscillations






58 Conclusion 197



x = f(xu) (511)


u = fc(x) (512)


x = f(x fc(x)) (513)

58 Conclusion



Oscillations

Chapter 6




classes of the





199




























Form the DAE model

Eqs (1)(2)




around SEP Eq (3)








S

E

P

I

n

i

t

i

a

l

i

z

a

t

i

o

n


variabledisturbance

variableSEP

Do eigen analysis

of matrix A

Calculate the NF


h2h3 c

End

Start

Rapid




ologies


641 Validity region


642 Resonance












x = f(xu) (61)


4x = H1(4x) +1

2H2(4x) +

1

3H3(4x) +O(4) (62)



] H3jklm =


]and O(4) are



Mq +Cq +Kq + fnl(q) = F (63)




x = f(x x) (64)

or Eq (65)

Mq +Cnl(q) +Kq + fq = F (65)





x = f(x x) (66)

or Eq (67)

Mclq +C(q) +Kq + fq = F (67)













207

208 Publications

Appendix A

Interconnected VSCs


usp=sharing





209




A111 The Basic Unit



A12 Methodology












AC LoadSource






(A2)

Filter C






lg middot ωg(A4)










lowastod = Vod i

lowastod = iod i



J1d2δ1dt2


sin δ1 + V1V2L12


J2d2δ1dt2


sin δ2 + V1V2L12


(A5)








ω10 ω2 = ω2+ω20dω1dt =


dt =d(ω2minusωg)



dt = dω1dt

dω2dt =










J1 16 L1(pu) 07 V1 1

J2 16 L2(pu) 07 V2 1

D1 10 L12(pu) 001 Vg 1

D2 10 wb 314


P lowast1 =V1VgL1

sin δ10 +V1V2L12


V1VgL2

sin δ20 +V1V2L12

sin δ210


J1ωb

d2δ1dt2

+ D1ωb

dδ1dt +

V1VgL1



J2ωb

d2δ2dt2

+ D2ωb

dδ2dt +

V2VgL2

















J1=0216D1=20J2=0216D2=20w_b=100314

g1=J1w_b

g2=J2w_b

d1=D1w_b

d2=D2w_b

syms g1 g2 d1 d2



delta210=-delta120

L1=07L2=07L12=001wg=1w_b=502314


V1=1V2=1Vg=1w10=1w20=1



Natural frequencies

M=[g1 00 g2]



D=[d1 00 d2]


[TrE]=eig(inv(M)K)



Normalisation of T

Tr=Trdet(Tr)

leg=sqrt(sum(Tr^2))

Tr=Trleg(ones(12))

normalization


modal matrices

Mm=TrMTr

Dm=TrDTr

Km=TrKTr

T11=Tr(11)

T12=Tr(12)

T21=Tr(21)

T22=Tr(22)

division by Mm


Dm_div=inv(Mm)Dm

OMEGA=double(OMEGA)

for i=12


end
















Gg=zeros(2222)




Hg(1111)=TEC(1111)T11^3+TEC(1112)T21T11^2+

TEC(1122)T11T21^2+TEC(1222)T21^3


TEC(1122)(T12T21^2+2T11T21T22)+TEC(1222)3T21^2T22


TEC(1122)(2T12T21T22+T11T22^2)+TEC(1222)3T21T22^2

Hg(1222)=TEC(1111)T12^3+TEC(1112)T22T12^2+

TEC(1122)T12T22^2+TEC(1222)T22^3





TEC(2222)T21^3


TEC(2122)(T12T21^2+2T11T21T22)+TEC(2222)3T21^2T22


TEC(2122)(2T12T21T22+T11T22^2)+TEC(2222)3T21T22^2



TEC(2222)T22^3



G=zeros(222)

H=zeros(2222)

G(111)=(T11Gg(111)T21Gg(211))Mm(11)

G(112)=(T11Gg(112)T21Gg(212))Mm(11)

G(122)=(T11Gg(122)T21Gg(222))Mm(11)

G(211)=(T12Gg(211)T22Gg(211))Mm(22)

G(212)=(T12Gg(212)T22Gg(212))Mm(22)

G(222)=(T12Gg(222)T22Gg(222))Mm(22)

for p=12

for k=12

for j=k2


++j

end

++k

end

++p

end

for p=12

for i=12

for k=i2

for j=k2


++j

end

++k

end

++i

end

++p

end

H=zeros(2222)

c1=[G(111) G(112) G(122) H(1111) H(1112) H(1122) H(1222)]

c2=[G(211) G(212) G(222) H(2111) H(2112) H(2122) H(2222)]






system






N=length(OMEGA)

correction=1

quadratic coefs

A=zeros(NNN)

B=zeros(NNN)

C=zeros(NNN)

ALPHA=zeros(NNN)

BETA=zeros(NNN)

GAMMA=zeros(NNN)

for p=1N

for i=1N































xi(i))




end

end

for p=1N

for i=1N

for j=i+1N



if abs(D) lt 1e-12

A(pij)=0

B(pij)=0

C(pij)=0

C(pji)=0

ALPHA(pij)=0

BETA(pij)=0

GAMMA(pij)=0

GAMMA(pji)=0

else














B2=[G(pij) 0 0 0]

solabcij=inv(A2)B2

solabcij=A2B2

A(pij)=solabcij(1)

B(pij)=solabcij(2)

C(pij)=solabcij(3)

C(pji)=solabcij(4)





end

end

end

end


AA=zeros(NNNN)

BB=zeros(NNNN)

CC=zeros(NNNN)

AA2=zeros(NNNN)

BB2=zeros(NNNN)

CC2=zeros(NNNN)

for p=1N

for i=1N

for j=1N

for k=1N

V1=squeeze(G(piiN))

V2=squeeze(A(iNjk))

V3=squeeze(G(p1ii))

V4=squeeze(A(1ijk))


V5=squeeze(G(piiN))

V6=squeeze(B(iNjk))

V7=squeeze(G(p1ii))


V8=squeeze(B(1ijk))


V9=squeeze(G(piiN))





end

end

end

end

cubic coefs

R=zeros(NNNN)

S=zeros(NNNN)

T=zeros(NNNN)

U=zeros(NNNN)

LAMBDA=zeros(NNNN)

MU=zeros(NNNN)

NU=zeros(NNNN)

ZETA=zeros(NNNN)


for p=1N

for i=1N

if i~=p


if abs(D1) lt 1e-12

R(piii)=0

S(piii)=0

T(piii)=0

U(piii)=0

LAMBDA(piii)=0

MU(piii)=0

NU(piii)=0

ZETA(piii)=0

else












sol3iii=A3iiiB3iii

R(piii)=sol3iii(1)

S(piii)=sol3iii(2)

T(piii)=sol3iii(3)

U(piii)=sol3iii(4)





end

end

end

end

for p=1N

for i=1N-1

if i~=p

for j=i+1N




R(pijj) = 0

S(pijj) = 0

T(pijj) = 0

T(pjij) = 0

U(pijj) = 0

U(pjij) = 0

LAMBDA(pijj)=0

MU(pijj) = 0

NU(pijj) =0

NU(pjij) =0

ZETA(pijj) = 0

b 21n132

ZETA(pjij) = 0

else

























xi(j)OMEGA(j)]




solrstuip=AipBip













2OMEGA(j)^2U(pijj)




end

end

end

end

end

for p=1N

for i=1N-1

for j=(i+1)N

if j~=p




R(piij) =0

S(piij) =0

T(pjii) =0

T(piij) =0

U(pjii) =0

U(piij) =0

LAMBDA(piij)=0

NU(pjii) =0

NU(piij) =0

MU(piij) =0

ZETA(pjii) =0

ZETA(piij) =0

else

























BB(piij) BB(pjii)]


solrstujpp=AjppBjpp













OMEGA(j)^2U(piij)



end

end

end

end

end

for p=1N

for i=1N-2

for j=(i+1)N-1

for k=(j+1)N





OMEGA(k)-OMEGA(i))

if abs(D4) lt 1e-12

U(pijk) =0


U(pjik) =0

U(pkij) =0

R(pijk) =0

MU(pijk) =0

NU(pkij) =0

NU(pjik) =0

NU(pijk) =0

else


















































Am1=inv(Alastijkp)


R(pijk)=solrstu(1)

S(pijk)=solrstu(2)

T(pijk)=solrstu(3)

T(pjik)=solrstu(4)

T(pkij)=solrstu(5)

U(pijk)=solrstu(6)

U(pjik)=solrstu(7)

U(pkij)=solrstu(8)


OMEGA(j)^2U(pijk)




OMEGA(i)^2T(pijk)




OMEGA(k)^2U(pijk)


OMEGA(k)^2U(pjik)






end

end

end

end

end

disp(end)

A3 Simulation Files




oscillations

Appendix B





Xp = Rp +

Nsumi=1

Nsumjgei


Nsumi=1

Nsumj=1

cpijRiSj

+

Nsumi=1

Nsumjgei

Nsumkgej


+

Nsumi=1

Nsumj=1

Nsumkgej


Yp = Sp +

Nsumi=1

Nsumjgei

(αpijRiRj + βp

ijSiSj) +

Nsumi=1

Nsumj=1

γpijRiSj

+

Nsumi=1

Nsumjgei

Nsumkgej


ijkSiSjSk)

+

Nsumi=1

Nsumj=1

Nsumkgej



p = Xinitp + 4Xp Y

RTp =


231




4Xp = 4Rp +Nsumi=1

PR(1)pi4Ri +Nsumi=1

PS(1)pi4Si

Nsumi=1


PS(S2)pi4Si

Nsumi=1

PR(S2)pi4Ri +

Nsumi=1

PS(R2)pi4Si

4Yp = 4Sp +Nsumi=1

QR(1)pi4Ri +Nsumi=1

QS(1)pi4Si

Nsumi=1


QS(S2)pi4Si

Nsumi=1

QR(S2)pi4Ri +

Nsumi=1

QS(R2)pi4Si

with

PR(1)pi =isum

j=1

apjiRj +Nsumjgei

apijRj +Nsumj=1

cpijSj (B1)

PS(1)pi =isum

j=1

bpjiSj +Nsumjgei

bpijSj +Nsumj=1

cpjiRj

PR(R2)pi =

isumj=1

isumkgej

rpkjiRjRk +

isumj=1

Nsumkgei

rpjikRjRk +

Nsumjgei

Nsumkgej

rpijkRjRk

PS(S2)pi =

isumj=1

isumkgej

spkjiSjSk +

isumj=1

Nsumkgei

spjikSjSk +

Nsumjgei

Nsumkgej

spijkSjSk

PR(S2)pi =Nsumj=1

isumk=1

tpjkiSjRk +Nsumj=1

Nsumkgei

tpjikSjRk +Nsumj=1

Nsumkgej

upijkSjSk

PS(R2)pi =

Nsumj=1

isumk=1

upjkiRjSk +

Nsumj=1

Nsumkgei

upjikRjSk +

Nsumj=1

Nsumkgej

tpijkRjRk


QR(1)pi =isum

j=1

αpjiRj +Nsumjgei

αpijRj +Nsumj=1

γpijSj (B2)

QS(1)pi =

isumj=1

βpjiSj +

Nsumjgei

βpijSj +

Nsumj=1

γpjiRj

QR(R2)pi =

isumj=1

jsumkgej

λpkjiRjRk +

isumj=1

Nsumkgei

λpjikRjRk +

Nsumjgei

Nsumkgej

λpijkRjRk

QS(S2)pi =isum

j=1

isumkgej

micropkjiSjSk +isum

j=1

Nsumkgei


Nsumkgej

micropijkSjSk

QR(S2)pi =Nsumj=1

isumk=1

νpjkiSjRk +Nsumj=1

Nsumkgei

νpjikSjRk +Nsumj=1

Nsumkgej

ζpijkSjSk

QS(R2)pi =

Nsumj=1

isumk=1

ζpjkiRjSk +

Nsumj=1

Nsumkgei

ζpjikRjSk +

Nsumj=1

Nsumkgej

νpijkRjRk

Let



(B3)

Then we obtain 4X

4Y

=

I +XR XS

Y R Y S

4R

4S

(B4)

and 4R

4S

=

I +XR XS

Y R Y S

minus1 4X

4Y

=

[NLRT

]minus1

4X

4Y

(B5)



Appendix C



System



Ra 00025 xl 0002

xd 180 τprimed0 80


xprimed 03 τ

primeprimed0 00

xprimeq 03 τ

primeprimed0 00




Load Voltage Zshunt

Bus (pu230kv)

B5 1029 6 1975 P = 66584MW Q = minus35082Mvar


235





G1 PV 1006 1402 74750 4994

G2 swingbus 10006 00 40650 14074

G3 PV 1006 minus 5817 53925 11399

G4 PV 1006 minus 6611 52500 14314




Gen 1 15 35 35 150 150 240

Gen 2 15 35 35 150 150 240

Gen 3 11 2 2 150 150 240

Gen 4 10 2 2 150 150 240









clear all

close all

Load the system


clear all



Define the system


omega_s=120pi









Yt=[39604-395821i -39604+396040i 00000+00000i 00000+00000i

-39604+396040i 103914-424178i 00000+00000i 04578+41671i

00000+00000i 00000+00000i 39604-395821i -39604+396040i

00000+00000i 04578+41671i -39604+396040i 182758-431388i]

Gt=real(Yt)

Bt=imag(Yt)


















T_e=ones(41)001

T_1=ones(41)10

T_2=ones(41)1









Vf= sym(Vf [Gn 1])

Vm= sym(Vm [Gn 1])

Vr= sym(Vr [Gn 1])

Yr=zeros(2Gn2Gn)

for l=1Gn

for m=1Gn

Yr(2l-12m-1)=Gt(lm)

Yr(2l-12m)=-Bt(lm)

Yr(2l2m-1)=Bt(lm)

Yr(2l2m)=Gt(lm)

end

end






for l=1Gn



T2(2l-12l-1)=Ra(l) T2(2l-12l)=-Xlq(l)


end

Y=inv(inv(Yr)+T29)

for l=1Gn

for m=1Gn

G1(lm)=Y(2l-12m-1)

B1(lm)=-Y(2l-12m)

B1(lm)=Y(2l2m-1)

G1(1m)=Y(2l2m)

end

end

Ydelta=inv(T1)YT1

for j=1Gn

eldq(2j-1)=eld(j)

eldq(2j)=elq(j)

end

Idq=Ydeltaeldq


for j=1Gn

id(j)=Idq(2j-1)

iq(j)=Idq(2j)

end

for j=1Gn





end

for j=1Gn


elxy(2j-11)=elx(j)

elxy(2j1)=ely(j)

end

Ixy=Yelxy









Vref=zeros(Gn1)

Vref_Exc_1=103

Vref_Exc_2=101

Vref_Exc_3=103

Vref_Exc_4=101









for j=1Gn-1








end

for j=Gn








end

for j=1Gn-1



end

for j=Gn



end



for j=1Gn-1


end

for j=Gn


end

x027=x027

Fv_R=vpa(F27)




x0test=zeros(281)



EIV_R=eig(F1t_R)










ie















for p=1L


end

for p=1L

for k=1L


end

end



F1=N1












[TuLambda]=eig(A)


invTu=inv(Tu)

Calculate F2

F21=zeros(LLL)

F2=zeros(LLL)

for p=1L



end

for p=1L

for j=1L

for k=1L


end

end

end

Calculate F3

F31=zeros(LLLL)

F3=zeros(LLLL)

for j=1L

for p=1L

for q=1L

for r=1L

for l=1L

for m=1L

for n=1L


end

end

end

end

end

end


end

for j=1L

for p=1L

for q=1L

for r=1L


end

end

end

end






N=length(EIV)

quadratic coefs

Cla=zeros(NNN)

for p=1N

for j=1N

for k=1N


end

end

end


end


beginverbatim





ModeNumber=[]




m=0


for j=1ModeSize


m=m+1

ModeNumber(m)=j

end

end




L=length(EIV)


MN=ModeNumber

N1718=0

xdot=zeros(L1)

c3=zeros(LL)

term_3=zeros(L1)

F2H2=zeros(LLLL)

F2c=zeros(LL)

C3=zeros(LLLL)

h2=zeros(LLL)

h3=zeros(LLLL)

for p=1M

for l=1M

for m=1M

for n=1M


end

end

end

end

for p=1L

for l=1L

for k=1L



end

for m=1L

for n=1L



end

end

end

end

for i=1(M2)

for j=1(M2)

if j~=i








C3(MN(2i-1)MN(2j)MN(2i-1)MN(2j-1))+

C3(MN(2i-1)MN(2j)MN(2j-1)MN(2i-1))+

C3(MN(2i-1)MN(2i-1)MN(2j-1)MN(2j))+

C3(MN(2i-1)MN(2j-1)MN(2i-1)MN(2j))+

C3(MN(2i-1)MN(2j-1)MN(2j)MN(2i-1))

else





C3(MN(2i-1)MN(2i-1)MN(2i-1)MN(2i))+

C3(MN(2i-1)MN(2i)MN(2i-1)MN(2i-1))


end

end

end

for p=1M

for l=1M

for m=1M

for n=1M




end

end

end

end

for k=1(M2)

for l=1(M2)

for m=1(M2)

for n=1(M2)

if l~=k




h3(MN(2k-1)MN(2k-1)MN(2l)MN(2l-1))=0

h3(MN(2k-1)MN(2l)MN(2k-1)MN(2l-1))=0

h3(MN(2k-1)MN(2l)MN(2l-1)MN(2k-1))=0

h3(MN(2k-1)MN(2k-1)MN(2l-1)MN(2l))=0

h3(MN(2k-1)MN(2l-1)MN(2k-1)MN(2l))=0

h3(MN(2k-1)MN(2l-1)MN(2l)MN(2k-1))=0

else




h3(MN(2k-1)MN(2k-1)MN(2k)MN(2k-1))=0

h3(MN(2k-1)MN(2k)MN(2k-1)MN(2k-1))=0

h3(MN(2k-1)MN(2k-1)MN(2k-1)MN(2k))=0

end

end

end

end

end


end










MN=ModeNumber

M=L2

EIV=EIV_R

w=imag(EIV)

sigma=real(EIV)

F2=F2N_R

F3=F3N_R

quadratic coefs

h3=h3test_R

for j=1L

for p=1L

h3(jppp)=h3(jppp)

for q=(p+1)L


for r=(q+1)L


h3(jqrp)+h3(jrqp)

end

end

end

end

for j=1L

for p=1L

for q=1L

for r=1L


end

end

end

end

MaxMI333=zeros(277)

for j=1L


[Ih3_1Ih3_2Ih3_3] = ind2sub([27 27 27]Ih3)




end


------------------------)

C34 Normal Dynamics





L=length(CE)

xdot=zeros(L1)

term_3=zeros(L1)

Cam=zeros(42)

F2H2=zeros(LLLL)

F2c=zeros(LL)

C3=zeros(LLLL)

for i=1(L2)

for j=1(L2)




end

end

for p=1L


end

end




L=length(CE)

xdot=zeros(L1)

for p=1L

xdot(p)=CE(pp)x(p)

end

end


















constrained






[1e-6]



[1e-6]


[100]


[iter]






References






Version 05





Version 04



Version 03







Version 02








initialize



set default options

oldopts = optimset(


if narginlt3


else


end


get options









set scaling values




set display

if DISPLAY




fmtstr = 10d 104g 104g 104g 104g 124gn


end

check initial guess






else


end

if issparse(Jstar)



else


rc = NaN


rc = Inf

else


end

end



solver





if lambda==1











end



break



break

end








if lambda==1



else





[ab] = coeff

if a==0

lambda = -slope2b

else


if discriminantlt0


elseif blt=0


else


end

end


end




else


end

if lambdalt1



continue

end

display







break

end

if issparse(Jstar)


else



end


end

output




if Nitergt=MAXITER

exitflag = 0


elseif exitflag==2


elseif exitflag==-1


else


end

jacob = J

end



try

[F J] = fun(x)

catch

F = fun(x)


end


end


estimate J





for n = 1nx





end

end












N=length(tt)

L=length(y0)

ynl=zeros(L1)

ynl3=zeros(L1)

H3=zeros(L1)

for l=1L

for p=1L

for q=1L

for r=1L


end

end

end

end

for l=1L


end

f=ynl3-y0











N=length(tt)


L=length(y0)

ynl=zeros(L1)

ynl3=zeros(L1)

for l=1L


end

f=ynl3-y0

C4 Simulation les




Appendix D



D1 Programs





1 calcm

2 chq_limm


4 form_jacm

5 loadflowm

6 y_sparsem


Version 33



Date December 2013




clear all

clc

MVA_Base=1000

f=600


259



j=sqrt(-1)

Load Data

data16m_benchmark


N_Bus=size(bus1)

N_Line=size(line1)

Loading Data Ends

Run Load Flow


disply=yflag = 2






clc

display(Running)

Load Flow End



xls=mac_con(4)BM

Ra=mac_con(5)BM

xd=mac_con(6)BM

xdd=mac_con(7)BM

xddd=mac_con(8)BM

Td0d=mac_con(9)

Td0dd=mac_con(10)

xq=mac_con(11)BM

xqd=mac_con(12)BM

xqdd=mac_con(13)BM

Tq0d=mac_con(14)

Tq0dd=mac_con(15)

H=mac_con(16)BM

D=mac_con(17)BM

M=2H

wB=2pif





D1 Programs 261

else

Zg= Ra + jxdd

end

Yg=1Zg


AVR Initialization

Tr=ones(N_Machine1)





Td=ones(N_Machine1)

Ka=ones(N_Machine1)


Ta=ones(N_Machine1)

Ke=ones(N_Machine1)



Te=ones(N_Machine1)


Tf=ones(N_Machine1)







for i=11len_exc(1)



if(exc_con(i3)~=0)


end

if(exc_con(i5)~=0)


end

if(exc_con(i1)==1)













Kad(Exc_m_indx)=1



else




end

end


PSS initialization


Tw=ones(N_Machine1)










for i=11len_pss(1)












end



D1 Programs 263




for i=11N_Machine


end

YG=MMYg

YG=diag(YG)

V=bus_sol(2)



Y_Aug=Y+YL+YG

Z=inv(Y_Aug) It=

Y_Aug_dash=Y_Aug


Initial Conditions

Vg=MMVthg=MMtheta


mp=2

mq=2

kp=bus_sol(6)V^mp

kq=bus_sol(7)V^mq


y_dash=y

from_bus = line(1)

to_bus = line(2)




Q_s = imag(MW_s)


















delta0 = angle(Eq0)





VD0=imag(VDQ)

VQ0=real(VDQ)

Vref=Efd0



for i=11len_exc(1)


if(exc_con(i1)==1)

V_Ka0(Exc_m_indx)=




else


end

end

Tm0=Te0

Pm0=Tm0


Gn=16



Sm= sym(Sm [Gn 1])







D1 Programs 265
















Vq=real(Vdq)

Vd=imag(Vdq)



iq=real(idq)

id=imag(idq)




ddelta=wBSm



Ed_dash-Psi2q)))







ddelta_Gn(Gn)=[]



PSS_in1=KsSm-PSS1






for i=11len_pss(1)







N_PSS=[N_PSS i]

end

Vt=sqrt(Vd^2+Vq^2)




for i=11len_exc(1)









D1 Programs 267









else




end

end














EIV=eig(F1t)



N_State=size(EIV1)

for i=11N_State



damping_ratioP(i)=1

end

end


--clacm--


























1 not yet converged

See also

Calls

Called By loadflow



D1 Programs 269

Version 20


Date July 1994

Version 10


Date March 1991

jay = sqrt(-1)

swing_bus = 1

gen_bus = 2

load_bus = 3




cur_inj = YV_rect




delP = Pg - Pl - P

delQ = Qg - Ql - Q


delP=delPsw_bno

delQ=delQsw_bno

delQ=delQg_bno

total mismatch



mism = pmis+qmis

if mism gt tol

conv_flag = 1

else

conv_flag = 0

end

return

--chq_limm--


Syntax










Version 11


Date May 1997


Version 10


Date October 1996



global Ql

global gen_chg_idx




f = 0







set Qg to zero


modify Ql






lim_flag = 1

end



set Qg to zero


modify Ql



D1 Programs 271




lim_flag = 1

end

if lim_flag == 1

recalculate g_bno


g_bno = ones(nbus1)


bus_index=(11nbus)






il = length(PQV_no)

ii = (11il)



il = length(PQ_no)

ii = (11il)


end

f = lim_flag

return






Version 33



Date December 2013

BUS DATA STARTS

bus data format








bus = [

01 1045 000 250 000 000 000 000 000 2 999 -999

02 098 000 545 000 000 000 000 000 2 999 -999

03 0983 000 650 000 000 000 000 000 2 999 -999

04 0997 000 632 000 000 000 000 000 2 999 -999

05 1011 000 505 000 000 000 000 000 2 999 -999

06 1050 000 700 000 000 000 000 000 2 999 -999

07 1063 000 560 000 000 000 000 000 2 999 -999

08 103 000 540 000 000 000 000 000 2 999 -999

09 1025 000 800 000 000 000 000 000 2 999 -999

10 1010 000 500 000 000 000 000 000 2 999 -999

11 1000 000 10000 000 000 000 000 000 2 999 -999

12 10156 000 1350 000 000 000 000 000 2 999 -999

13 1011 000 3591 000 000 000 000 000 2 999 -999

14 100 000 1785 000 000 000 000 000 2 999 -999

15 1000 000 1000 000 000 000 000 000 2 999 -999

16 1000 000 4000 000 000 000 000 000 1 0 0

17 100 000 000 000 6000 300 000 000 3 0 0

18 100 000 000 000 2470 123 000 000 3 0 0

19 100 000 000 000 000 000 000 000 3 0 0

20 100 000 000 000 6800 103 000 000 3 0 0

21 100 000 000 000 2740 115 000 000 3 0 0

22 100 000 000 000 000 000 000 000 3 0 0

23 100 000 000 000 2480 085 000 000 3 0 0

24 100 000 000 000 309 -092 000 000 3 0 0

25 100 000 000 000 224 047 000 000 3 0 0

26 100 000 000 000 139 017 000 000 3 0 0

27 100 000 000 000 2810 076 000 000 3 0 0

28 100 000 000 000 2060 028 000 000 3 0 0

29 100 000 000 000 2840 027 000 000 3 0 0

30 100 000 000 000 000 000 000 000 3 0 0

31 100 000 000 000 000 000 000 000 3 0 0

32 100 000 000 000 000 000 000 000 3 0 0

33 100 000 000 000 112 000 000 000 3 0 0

34 100 000 000 000 000 000 000 000 3 0 0

35 100 000 000 000 000 000 000 000 3 0 0

36 100 000 000 000 102 -01946 000 000 3 0 0

37 100 000 000 000 000 000 000 000 3 0 0

38 100 000 000 000 000 000 000 000 3 0 0

39 100 000 000 000 267 0126 000 000 3 0 0

40 100 000 000 000 06563 02353 000 000 3 0 0

41 100 000 000 000 1000 250 000 000 3 0 0

42 100 000 000 000 1150 250 000 000 3 0 0

43 100 000 000 000 000 000 000 000 3 0 0

D1 Programs 273

44 100 000 000 000 26755 00484 000 000 3 0 0

45 100 000 000 000 208 021 000 000 3 0 0

46 100 000 000 000 1507 0285 000 000 3 0 0

47 100 000 000 000 20312 03259 000 000 3 0 0

48 100 000 000 000 24120 0022 000 000 3 0 0

49 100 000 000 000 16400 029 000 000 3 0 0

50 100 000 000 000 100 -147 000 000 3 0 0

51 100 000 000 000 337 -122 000 000 3 0 0

52 100 000 000 000 158 030 000 000 3 0 0

53 100 000 000 000 2527 11856 000 000 3 0 0

54 100 000 000 000 000 000 000 000 3 0 0

55 100 000 000 000 322 002 000 000 3 0 0

56 100 000 000 000 200 0736 000 000 3 0 0

57 100 000 000 000 000 000 000 000 3 0 0

58 100 000 000 000 000 000 000 000 3 0 0

59 100 000 000 000 234 084 000 000 3 0 0

60 100 000 000 000 2088 0708 000 000 3 0 0

61 100 000 000 000 104 125 000 000 3 0 0

62 100 000 000 000 000 000 000 000 3 0 0

63 100 000 000 000 000 000 000 000 3 0 0

64 100 000 000 000 009 088 000 000 3 0 0

65 100 000 000 000 000 000 000 000 3 0 0

66 100 000 000 000 000 000 000 000 3 0 0

67 100 000 000 000 3200 15300 000 000 3 0 0

68 100 000 000 000 3290 032 000 000 3 0 0

]

BUS DATA ENDS

LINE DATA STARTS

line =[

01 54 0 00181 0 10250 0

02 58 0 00250 0 10700 0

03 62 0 00200 0 10700 0

04 19 00007 00142 0 10700 0

05 20 00009 00180 0 10090 0

06 22 0 00143 0 10250 0

07 23 00005 00272 0 0 0

08 25 00006 00232 0 10250 0

09 29 00008 00156 0 10250 0

10 31 0 00260 0 10400 0

11 32 0 00130 0 10400 0

12 36 0 00075 0 10400 0

13 17 0 00033 0 10400 0

14 41 0 00015 0 10000 0


15 42 0 00015 0 10000 0

16 18 0 00030 0 10000 0

17 36 00005 00045 03200 0 0

18 49 00076 01141 11600 0 0

18 50 00012 00288 20600 0 0

19 68 00016 00195 03040 0 0

20 19 00007 00138 0 10600 0

21 68 00008 00135 02548 0 0

22 21 00008 00140 02565 0 0

23 22 00006 00096 01846 0 0

24 23 00022 00350 03610 0 0

24 68 00003 00059 00680 0 0

25 54 00070 00086 01460 0 0

26 25 00032 00323 05310 0 0

27 37 00013 00173 03216 0 0

27 26 00014 00147 02396 0 0

28 26 00043 00474 07802 0 0

29 26 00057 00625 10290 0 0

29 28 00014 00151 02490 0 0

30 53 00008 00074 04800 0 0

30 61 000095 000915 05800 0 0

31 30 00013 00187 03330 0 0

31 53 00016 00163 02500 0 0

32 30 00024 00288 04880 0 0

33 32 00008 00099 01680 0 0

34 33 00011 00157 02020 0 0

34 35 00001 00074 0 09460 0

36 34 00033 00111 14500 0 0

36 61 00011 00098 06800 0 0

37 68 00007 00089 01342 0 0

38 31 00011 00147 02470 0 0

38 33 00036 00444 06930 0 0

40 41 00060 00840 31500 0 0

40 48 00020 00220 12800 0 0

41 42 00040 00600 22500 0 0

42 18 00040 00600 22500 0 0

43 17 00005 00276 0 0 0

44 39 0 00411 0 0 0

44 43 00001 00011 0 0 0

45 35 00007 00175 13900 0 0

45 39 0 00839 0 0 0

45 44 00025 00730 0 0 0

46 38 00022 00284 04300 0 0

D1 Programs 275

47 53 00013 00188 13100 0 0

48 47 000125 00134 08000 0 0

49 46 00018 00274 02700 0 0

51 45 00004 00105 07200 0 0

51 50 00009 00221 16200 0 0

52 37 00007 00082 01319 0 0

52 55 00011 00133 02138 0 0

54 53 00035 00411 06987 0 0

55 54 00013 00151 02572 0 0

56 55 00013 00213 02214 0 0

57 56 00008 00128 01342 0 0

58 57 00002 00026 00434 0 0

59 58 00006 00092 01130 0 0

60 57 00008 00112 01476 0 0

60 59 00004 00046 00780 0 0

61 60 00023 00363 03804 0 0

63 58 00007 00082 01389 0 0

63 62 00004 00043 00729 0 0

63 64 00016 00435 0 10600 0

65 62 00004 00043 00729 0 0

65 64 00016 00435 0 10600 0

66 56 00008 00129 01382 0 0

66 65 00009 00101 01723 0 0

67 66 00018 00217 03660 0 0

68 67 00009 00094 01710 0 0

27 53 00320 03200 04100 0 0

]

LINE DATA ENDS

MACHINE DATA STARTS

Machine data format

1 machine number

2 bus number

3 base mva








T_do(sec)







T_qo(sec)




19 bus number




mac_con = [

01 01 100 00125 00 01 0031 0025 102 005 0069 00416667 0025 15

0035 42 0 0 01 0 0

02 02 100 0035 00 0295 00697 005 656 005 0282 00933333 005 15

0035 302 0 0 02 0 0

03 03 100 00304 00 02495 00531 0045 57 005 0237 00714286 0045 15

0035 358 0 0 03 0 0

04 04 100 00295 00 0262 00436 0035 569 005 0258 00585714 0035 15

0035 286 0 0 04 0 0

05 05 100 0027 00 033 0066 005 54 005 031 00883333 005 044

0035 26 0 0 05 0 0

06 06 100 00224 00 0254 005 004 73 005 0241 00675000 004 04

0035 348 0 0 06 0 0

07 07 100 00322 00 0295 0049 004 566 005 0292 00666667 004 15

0035 264 0 0 07 0 0

08 08 100 0028 00 029 0057 0045 67 005 0280 00766667 0045 041

0035 243 0 0 08 0 0

09 09 100 00298 00 02106 0057 0045 479 005 0205 00766667 0045 196

0035 345 0 0 09 0 0

10 10 100 00199 00 0169 00457 004 937 005 0115 00615385 004 1

0035 310 0 0 10 0 0

D1 Programs 277

11 11 100 00103 00 0128 0018 0012 41 005 0123 00241176 0012 15 0035 282 0 0 11 0 0

12 12 100 0022 00 0101 0031 0025 74 005 0095 00420000 0025 15 0035 923 0 0 12 0 0

13 13 200 00030 00 00296 00055 0004 59 005 00286 00074000 0004 15 0035 2480 0 0 13 0 0

14 14 100 00017 00 0018 000285 00023 41 005 00173 00037931 00023 15 0035 3000 0 0 14 0 0

15 15 100 00017 00 0018 000285 00023 41 005 00173 00037931 00023 15 0035 3000 0 0 15 0 0

16 16 200 00041 00 00356 00071 00055 78 005 00334 00095000 00055 15 0035 2250 0 0 16 0 0

]

MACHINE DATA ENDS

EXCITER DATA STARTS




2 - machine number








10 - E_1

11 - S(E_1)

12 - E_2

13 - S(E_2)



16 - K_P

17 - K_I

18 - K_D

19 - T_D

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

exc_con = [

1 1 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

1 2 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01


1 3 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

1 4 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

1 5 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

1 6 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

1 7 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

1 8 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

0 9 001 200 000 50 -50 00 0 0 0 0 0 0 0 0 0 0

0

1 10 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

1 11 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

1 12 001 1 002 10 -10 10 785 39267 0070 52356 0910 0030 10 200 50 50

01

]

EXCITER DATA ENDS

PSS DATA STARTS

1-S No


3-pssgain










pss_con = [

1 1 12 10 015 004 015 004 015 004 02 -005

2 2 20 15 015 004 015 004 015 004 02 -005

3 3 20 15 015 004 015 004 015 004 02 -005

4 4 20 15 015 004 015 004 015 004 02 -005

5 5 12 10 015 004 015 004 015 004 02 -005

6 6 12 10 015 004 015 004 015 004 02 -005

7 7 20 15 015 004 015 004 015 004 02 -005

D1 Programs 279

8 8 20 15 015 004 015 004 015 004 02 -005

9 9 12 10 015 004 015 004 015 004 02 -005

9 9 15 10 009 002 009 002 1 1 02 -005

10 10 20 15 015 004 015 004 015 004 02 -005

11 11 20 15 015 004 015 004 015 004 02 -005

12 12 20 15 009 004 015 004 015 004 02 -005

]

PSS DATA ENDS

--file form_jacm--




ang_redvolt_red)






entries


entries



jacobian matrix

See also

Calls




Version 20


Date March 1994


Version 10


Date March 1991


jay = sqrt(-1)



V_rect = Vexp_ang


Y_con = conj(Y)


i_c=Y_conCV_rect


S=V_recti_c

S=sparse(diag(S))


CVdia=conj(Vdia)


S1=VdiaY_conCVdia


t2=(S-S1)ang_red


J12=ang_redreal(t1)



if nargout gt 3





else

Jac11 = [J11 J12

J21 J22]

end

--loadflowm--





81297



sparse matices


D1 Programs 281

line - line data








iteration


solution format)



See also








Version 21


Date October 1996


Version 20


Date March 1994

Version 10


Date March 1991

global bus_int


global Q Ql

global gen_chg_idx




jay = sqrt(-1)

load_bus = 3

gen_bus = 2

swing_bus = 1

if exist(flag) == 0

flag = 1

end

lf_flag = 1








end



set defaults

bus data defaults

if ncollt15


if ncollt12



end




volt_min = bus(15)

volt_max = bus(14)

else

volt_min = bus(15)

volt_max = bus(14)

end




end



D1 Programs 283


end







tap_it = 0

tap_it_max = 10

no_taps = 0


if nlc lt 10


no_taps = 1


end


mm_chk=1


tap_it = tap_it+1



process bus data

bus_no = bus(1)

V = bus(2)

ang = bus(3)pi180

Pg = bus(4)

Qg = bus(5)

Pl = bus(6)

Ql = bus(7)

Gb = bus(8)

Bb = bus(9)


qg_max = bus(11)

qg_min = bus(12)

sw_bno=ones(nbus1)

g_bno=sw_bno














il = length(PQV_no)

ii = (11il)



il = length(PQ_no)

ii = (11il)









iter = iter + 1


clear Jac





clear delP delQ







V = V + accdelV

D1 Programs 285











if lim_flag == 1


end

end

if iter == iter_max





else


disp(iter)

end

if no_taps == 0

lftap

else

mm_chk = 0

end

end




else


disp(tap_it)

end





disp(voltages at)

bus(vmx_idx1)


end



disp(voltages at)

bus(vmn_idx1)


end








disp( bus Qg)





end











from_bus = line(1)


to_bus = line(2)


r = line(3)

rx = line(4)

chrg = line(5)

z = r + jayrx

y = ones(nline1)z




to to_bus

D1 Programs 287


from_bus to to_bus


- VV(from_int)tps)y



from from_bus



iline = (11nline)



keyboard





line_sol = line




if conv_flag == 1


error(stop)

end

display results

if display == y

clc



disp( )

disp(date)







if conv_flag == 0



disp(bus_sol(17))


disp( LINE FLOWS )


disp(line_ffrom)

disp(line_fto)

end

end

if iter gt iter_max


lf_flag = 0

end

return

--file y_sparsem--





line - line data






See also

Calls




Version 20


Date April 1994

Version 10


D1 Programs 289

Date March 1991

global bus_int

jay = sqrt(-1)

swing_bus = 1

gen_bus = 2

load_bus = 3



r=zeros(nline1)

rx=zeros(nline1)

chrg=zeros(nline1)

z=zeros(nline1)

y=zeros(nline1)





ibus = (1nbus)



r = line(3)

rx = line(4)










tap=ones(nline1)






iline = [11nline]











if nargout gt 1


nSW = 0

nPV = 0

nPQ = 0





nSW=length(SB)



end

return

D2 Simulation Files




Bibliography



















291

292 Bibliography


















Bibliography 293





















294 Bibliography


















Bibliography 295






















296 Bibliography


















Bibliography 297




















298 Bibliography












Forms



















List of Figures

List of Tables


Introductions

General Context

























Resonant Terms






What is needed























Introduction

















Stability Index










Invariance Property







Stability Bound



Interconnected VSCs






Closed Remarks












Introduction




Method 2-2-1

Method 3-2-3S

Method 3-3-1

Method 3-3-3



















Introduction













Case-studies

SMIB Test System




On-going Work








The Limitations







Originality








Primitive Results


A Literature Review


Conclusion










Validity region

Resonance








Interconnected VSCs



Methodology

Modeling the System






Simulation Files







SEP Initialization





Normal Dynamics


Simulation files


Programs

Simulation Files

Bibliography

Tian Tian To cite this version - Accueil - TEL

Documents

Transcript of Tian Tian To cite this version - Accueil - TEL