PhD thesis

Modeling and Control of VSC-HVDC

Links Connected to Weak AC Systems

Lidong Zhang

ROYAL INSTITUTE OF TECHNOLOGY

SCHOOL OF ELECTRICAL ENGINEERING

ELECTRICAL MACHINES AND POWER ELECTRONICS

Stockholm 2010

Submitted to the School of Electrical Engineering in partial fulfillment of the

requirements for the degree of Doctor of Philosophy.

Stockholm 2010

TRITA–EE 2010:022

ISSN 1653-5146

ISRN KTH-EE–10/22–SE

ISBN 978-91-7415-640-9

This document was prepared using LATEX.

To Yibin, Karin, Vivianne

and my parents

and my sister Lixia

Abstract

For high-voltage direct-current (HVDC) transmission, the strength of the ac system is

important for normal operation. An ac system can be considered as weak either because its

impedance is high or its inertia is low. A typical high-impedance system is when an HVDC

link is terminated at a weak point of a large ac system where the short-circuit capacity

of the ac system is low. Low-inertia systems are considered to have limited number of

rotating machines, or no machines at all. Examples of such applications can be found

when an HVDC link is powering an island system, or if it is connected to a windfarm.

One of the advantages of applying a voltage-source converter (VSC) based HVDC system

is its potential to be connected to very weak ac systems where the conventional line-

commutated converter (LCC) based HVDC system has difficulties.

In this thesis, the modeling and control issues for VSC-HVDC links connected

to weak ac systems are investigated. In order to fully utilize the potential of the VSC-

HVDC system for weak-ac-system connections, a novel control method, i.e., power-

synchronization control, is proposed. By using power-synchronization control, the VSC

resembles the dynamic behavior of a synchronous machine. Several additional functions,

such as high-pass current control, current limitation, etc. are proposed to deal with various

practical issues during operation.

For modeling of ac/dc systems, the Jacobian transfer matrix is proposed as a uni-

fied modeling approach. With the ac Jacobian transfer matrix concept, a synchronous ac

system is viewed upon as one multivariable feedback system. In the thesis, it is shown

that the transmission zeros and poles of the Jacobian transfer matrix are closely related to

several power-system stability phenomena. The similar modeling concept is extended to

model a dc system with multiple VSCs. It is mathematically proven that the dc system is

an inherently unstable process, which requires feedback controllers to be stabilized.

For VSC-HVDC links using power-synchronization control, the short-circuit ratio

(SCR) of the ac system is no longer a limiting factor, but rather the load angles. The right-

half plane (RHP) transmission zero of the ac Jacobian transfer matrix moves closer to the

origin with larger load angles, which imposes a fundamental limitation on the achievable

bandwidth of the VSC. As an example, it is shown that a VSC-HVDC link using power-

synchronization control enables a power transmission of 0.86 p.u. from a system with an

SCR of 1.2 to a system with an SCR of 1.0. For low-inertia system connections, simulation

studies show that power-synchronization control is flexible for various operation modes

v

related to island operation and handles the mode shifts seamlessly.

Keywords: Control, modeling, multivariable feedback control, HVDC, power systems,

stability, subsynchronous torsional interaction, voltage-source converter, weak ac

systems.

vi

Acknowledgements

First of all, my deepest gratitude goes to my supervisors, Prof. Hans-Peter Nee and

Prof. Lennart Harnefors. It is an honor and a pleasure for me to have Prof. Hans-Peter

Nee as my supervisor. His patience and support helped me to go through the hardest

moments of the research work. It is also a privilege for me to be a student of Prof. Lennart

Harnefors. I am grateful for his generosity to share with me his deep understanding on

scientific work. Without his guidance, this project cannot reach the same level as it is

today.

This work has been carried out within Elektra Project 30630 and has been funded by

Energimyndigheten, ELFORSK, ABB Power Systems, ABB Corporate Research, Ban-

verket. The financial funding is greatly acknowledged.

My acknowledgements also go to the members of the steering group: Gunnar As-

plund (ABB Power Systems), Pablo Rey (ABB Power Systems), Hongbo Jiang (Banver-

ket), Torbjorn Thiringer (Chalmers University of Technology). During the last two and

half years, I had many inspiring discussions with the steering group members. Their fruit-

ful comments and inputs have greatly improved the quality of the research. Especially, I

would like to thank Gunnar Asplund, who was the chairman of the group before his retire-

ment from ABB Power Systems. Gunnar Asplund initiated the project and gave valuable

suggestions at the beginning of the project.

I would like also to thank my supervisor, Prof. Math Bollen, during my Licentiate

study at Chalmers. Prof. Math Bollen brought me into the scientific world. I received

endless support from him during my study at Chalmers and after graduation.

To my colleagues at ABB, I am grateful for all the supports I have received during

this period. In particular, I would like to thank Ying-Jiang Hafner, Magnus Ohrstrom,

Cuiqing Du, and Rolf Ottersten for interesting discussions as well as many helps with

thesis writing. Ying-Jiang Hafner carefully reviewed the manuscript of the thesis and

gave important suggestions. I would like to give a special thank to Pablo Rey, my group

manager at ABB, for allowing me to be absent from the group for the Ph.D study.

At KTH, I would like to thank all the colleagues in the Electrical Machines and

Power Electronics department. In particular, I would like to thank Prof. Chandur Sadaran-

gani for reviewing the manuscript of the thesis. I am also grateful to Hailian Xie for her

help with thesis writing, to Peter Lonn for his computer support, to Eva Pettersson and

Brigitt Hogberg for their help with the administrative work.

vii

Many thanks to my parents and my parents-in-law for their love and support. My

mother-in-law, Prof. Renmu He, is a renowned professor in power systems in China. I

received many helps from her in my professional life as well as my family life for the past

years. Her valuable suggestions during her stay in Sweden shed light on my research and

influenced the content of this thesis. I would like also to thank my sister and nephew for

their love and encouragement for all the time.

Last but not least, I would like to thank my beloved wife and daughters. Yibin,

thank you so much for your endless love, support and understanding. Thank you, Karin

and Vivianne, for the joys you have brought to my life.

Lidong Zhang

Stockholm, Sweden

April 2010

viii

Contents

Abstract v

Acknowledgements vii

Contents ix

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Project objectives and outline of the thesis . . . . . . . . . . . . . . . . . 3

1.3 Scientific contributions of the thesis . . . . . . . . . . . . . . . . . . . . 4

1.4 List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 High-Voltage Direct-Current Transmission 9

2.1 DC versus AC transmission . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 HVDC transmission using line-commutated current-source converters . . 11

2.3 HVDC transmission using forced-commutated voltage-source converters . 14

3 Control Methods for VSC-HVDC Systems 21

3.1 Power-angle control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Vector current control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3 Power-synchronization control . . . . . . . . . . . . . . . . . . . . . . . 30

3.3.1 Power-synchronization mechanism in ac systems . . . . . . . . . 31

3.3.2 Power-synchronization control of grid-connected VSCs . . . . . . 32

3.3.3 Bumpless-transfer and anti-windup schemes . . . . . . . . . . . . 35

3.3.4 Negative-sequence current control . . . . . . . . . . . . . . . . . 41

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 Dynamic Modeling of AC/DC Systems 47

4.1 Jacobian transfer matrix for ac-system modeling . . . . . . . . . . . . . . 47

4.1.1 Power-system stability and dynamic modeling . . . . . . . . . . 47

4.1.2 Feedback-control view of power systems . . . . . . . . . . . . . 50

4.2 Grid-connected VSCs using power-synchronization control . . . . . . . . 52

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Contents

4.2.1 Impedance-source neglecting the ac capacitor at the filter bus . . . 52

4.2.2 Impedance-source including the ac capacitor at the filter bus . . . 63

4.2.3 AC-source feeding from a series-compensated ac line . . . . . . . 70

4.3 Grid-connected VSCs using vector current control . . . . . . . . . . . . . 74

4.4 Jacobian transfer matrix for dc-system modeling . . . . . . . . . . . . . . 81

4.5 Summary of the properties of the Jacobian transfer matrix . . . . . . . . . 87

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5 Control of VSC-HVDC Links Connected to High-Impedance AC Systems 89

5.1 General aspects of high-impedance ac systems . . . . . . . . . . . . . . . 89

5.2 Comparison of power-synchronization control and vector current control . 91

5.3 Multivariable feedback designs . . . . . . . . . . . . . . . . . . . . . . . 100

5.3.1 Internal model control . . . . . . . . . . . . . . . . . . . . . . . 102

5.3.2 H∞ control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.3.3 Performance and robustness comparison . . . . . . . . . . . . . . 111

5.4 Direct-voltage control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.4.1 Controller design . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.4.2 DC-capacitance requirement . . . . . . . . . . . . . . . . . . . . 119

5.5 Interconnection of two very weak ac systems . . . . . . . . . . . . . . . 122

5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6 Control of VSC-HVDC Links Connected to Low-Inertia AC Systems 131

6.1 General aspects of low-inertia ac systems . . . . . . . . . . . . . . . . . 131

6.2 Controller design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.2.1 Frequency droop control . . . . . . . . . . . . . . . . . . . . . . 133

6.2.2 Alternating-voltage droop control . . . . . . . . . . . . . . . . . 134

6.3 Dynamic modeling and linear analysis of a typical island system . . . . . 135

6.3.1 Jacobian transfer matrix . . . . . . . . . . . . . . . . . . . . . . 136

6.3.2 Integrated linear model . . . . . . . . . . . . . . . . . . . . . . . 147

6.3.3 Linear analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

6.4 Simulation studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

6.5 Jacobian transfer matrix for other input devices . . . . . . . . . . . . . . 155

6.5.1 Synchronous generator . . . . . . . . . . . . . . . . . . . . . . . 155

6.5.2 Induction motor . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

6.6 Subsynchronous characteristics . . . . . . . . . . . . . . . . . . . . . . . 162

6.6.1 Frequency-scanning method . . . . . . . . . . . . . . . . . . . . 164

6.6.2 Large ac-system connection . . . . . . . . . . . . . . . . . . . . 165

6.6.3 Island operation . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

6.6.4 Summary of the subsynchronous characteristics . . . . . . . . . . 177

6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

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Contents

7 Conclusions and Future Work 181

7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

7.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

References 185

A Fundamentals of the Phasor and the Space-Vector Theory 197

A.1 Fundamentals of the phasor theory . . . . . . . . . . . . . . . . . . . . . 197

A.2 Fundamentals of the space-vector theory . . . . . . . . . . . . . . . . . . 198

A.3 Implementation of αβ and dq transformations . . . . . . . . . . . . . . . 200

A.3.1 abc-αβ transformation . . . . . . . . . . . . . . . . . . . . . . . 200

A.3.2 αβ-dq transformation . . . . . . . . . . . . . . . . . . . . . . . . 201

B Jacobian Transfer Matrix 203

B.1 Derivation of the transfer functions in Table 4.1 . . . . . . . . . . . . . . 203

B.1.1 Transfer function JPθ (s) . . . . . . . . . . . . . . . . . . . . . . 203

B.1.2 Transfer function JQθ (s) . . . . . . . . . . . . . . . . . . . . . . 205

B.1.3 Transfer function JUf θ (s) . . . . . . . . . . . . . . . . . . . . . 206

B.1.4 Transfer function JPV (s) . . . . . . . . . . . . . . . . . . . . . . 207

B.1.5 Transfer function JQV (s) . . . . . . . . . . . . . . . . . . . . . . 207

B.1.6 Transfer function JUf V (s) . . . . . . . . . . . . . . . . . . . . . 208

B.2 Proof of the instability of the dc Jacobian transfer matrix . . . . . . . . . 208

C Technical Data of the Test System 211

C.1 The VSC-HVDC link . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

C.2 The synchronous generator . . . . . . . . . . . . . . . . . . . . . . . . . 212

C.3 The induction motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

D List of Symbols and Abbreviations 215

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Contents

xii

Chapter 1

Introduction

This chapter describes the background of the thesis. The aim and the outline, as well as

the major scientific contributions of the thesis are presented. Finally, a list of publications

is given.

1.1 Background

In 1954, the first commercial high-voltage direct-current (HVDC) link between mainland

Sweden to Gotland island was commissioned. Since then, the accumulated installed power

of HVDC transmission worldwide has increased steadily, and recently a dramatic increase

in volume has been initiated. So far, most of the HVDC systems installed worldwide are

line-commutated converter (LCC) systems using thyristor valves. However, with grad-

ually reduced losses and costs, the recently developed voltage-source converter (VSC)

technology has shown to be more advantageous in many aspects [1–4].

The conventional line-commutated HVDC technology has an inherent weakness,

i.e., the commutation of the converter valves is dependent on the stiffness of the alter-

nating voltage. The converter cannot work properly if the connected ac system is weak.

Substantial research has been performed in this field [5–9]. The most outstanding con-

tribution on this subject is [5], which recommends to use short-circuit ratio (SCR) as a

description of the strength of the ac system relative to the power rating of the HVDC

link. Both [8] and [9] conclude that, for ac systems with an SCR lower than 1.5, syn-

chronous condensers have to be installed to increase the short-circuit capacity of the ac

system. However, synchronous condensers can substantially increase the investment and

maintenance costs of an HVDC project.

In contrast to the conventional LCC-HVDC system, the VSC-HVDC system is

based on self-commutated pulse-width modulation (PWM) technology, i.e., a VSC can

produce its own voltage waveform independent of the ac system. Thus, a VSC-HVDC

system has the potential to be connected to very weak ac systems. However, with the

traditional vector current control the potential of the VSC is not fully utilized [10–12],

1

Chapter 1. Introduction

e.g., Ref. [11] shows that the maximum power that a VSC-HVDC link using vector cur-

rent control can transmit to the ac system with SCR = 1.0 is 0.4 p.u. Ref. [12] shows

that the inner-current controller of vector current control may interact with low-frequency

resonances that are typically present in weak ac systems. In addition, the phase-locked

loop (PLL) dynamics of vector current control might also have a negative impact on the

performance of VSC-HVDC links for weak ac-system connections [10, 11, 13]. The poor

performance of vector current control for weak-ac-system connections has become an

obstacle for VSC-HVDC transmission to be applied in more challenging ac-system con-

ditions.

The application of high power-electronic devices, such as HVDC systems and

FACTS devices also imposes new challenges for power-system stability analysis and dy-

namic modeling. For classical power-system stability analysis, the phasor theory is the

major mathematical tool. With the phasor approach, the electromagnetic transients of

the ac network are neglected. This is a practical solution for conventional power sys-

tems where the electromagnetic transients have negligible effects on the stability issue of

concern. However, for high power-electronic devices, such a simplification is not accept-

able. The dynamic frequency range of high power-electronic devices is much higher than

that of the conventional power-system components. In this frequency, the phasor theory

cannot properly reflect the dynamic interaction between the ac system and the power-

electronic devices on the one hand, and between different power-electronic devices on

the other hand. For example, it has been shown by [14] that the conclusions drawn by

the phasor-based quasi-static analysis might not always agree with the results obtained by

time simulations with electromagnetic-transient programs.

The space-vector theory is based on instantaneous values, and therefore it is able to

represent the electromagnetic transients of the ac network [15]. Traditionally, the space-

vector theory is mainly applied for analyzing electrical machines and control of power-

electronic devices [15–17]. Several methodologies for dynamic modeling of three-phase

systems based on the space-vector theory have been been proposed. In [18], the complex

transfer functions are applied for analyzing three-phase ac machines. In [19], a three-

phase linear current controller is analyzed in the frequency domain based on the space-

vector approach. In [20], the space-vector theory is applied for modeling of three-phase

dynamic systems using the transfer matrix concept. For subsynchronous torsional inter-

action (SSTI) analysis, the ac network is normally required to be modeled by the space-

vector approach to take into account the electromagnetic transients [21, 22]. In recent

years, the space-vector theory has also been applied to study the dynamic interactions be-

tween high power-electronic devices. In [12], the dynamic interaction between an LCC-

HVDC link and a VSC-STATCOM in the frequency domain is analyzed based on the

space-vector theory.

While the space-vector theory has been applied successfully for analyzing high-

frequency stability phenomena in power systems, the theoretical work to connect the

2

1.2. Project objectives and outline of the thesis

high-frequency stability to the classical power-system stability defined by the phasor ap-

proach is missing in the literature. Such a connection is, however, necessary for HVDC

systems and FACTS applications. On the one hand, the dynamic frequency range of such

devices is high. On the other hand, the ratings of those devices are often high enough

to have a significant impact on most of the classical power-system stability phenomena,

such as angle stability and voltage stability. In the foreseeable future, the number of such

devices in power systems is expected to increase considerably. Thus, there is a need for a

unified modeling approach to address both the high-frequency and low-frequency stability

phenomena.

1.2 Project objectives and outline of the thesis

The objectives of the project are:

1. Develop a new control method for VSC-HVDC links connected to weak ac sys-

tems.

2. Develop a unified approach for dynamic modeling of ac/dc systems.

3. Investigate various modeling and control issues for VSC-HVDC links connected

to high-impedance ac systems.

4. Investigate various modeling and control issues for VSC-HVDC links connected

to low-inertia ac systems.

The project is conducted by both theoretical analysis and time simulations. The outline of

the thesis is:

Chapter 2 A short introduction of various technologies for HVDC transmission is

given.

Chapter 3 Two existing control methods for VSC-HVDC systems, i.e., power-angle

control and vector current control are described. A novel control method, i.e., power-

synchronization control, is proposed to solve the problem for VSC-HVDC links

connected to weak ac systems.

Chapter 4 A unified dynamic modeling approach, i.e., the Jacobian transfer matrix,

is proposed for modeling of ac/dc systems. Grid-connected VSCs using power-

synchronization control and vector current control are modeled by the proposed

concept.

Chapter 5 The control issues for VSC-HVDC links connected to high-impedance ac

systems are investigated. The dynamic performance of a VSC-HVDC link using

3


power-synchronization control and vector current control are compared. Two mul-

tivariable feedback-control designs, i.e., internal model control (IMC) and H∞ con-

trol are investigated. A direct-voltage controller is proposed. A control structure for

interconnection of two very weak ac systems is proposed.

Chapter 6 Power-synchronization control is applied to VSC-HVDC links connected to

low-inertia ac systems. A frequency droop controller and a voltage droop controller

are proposed. A linear model of a typical island system is developed for tuning the

control parameters of the VSC-HVDC link. The subsynchronous characteristics of

a VSC-HVDC converter are analyzed for both the large ac-system connection and

island operation.

Chapter 7 Summarizes the thesis and provides suggestions for future work.

1.3 Scientific contributions of the thesis

The main contributions of the thesis are:

• A novel control method for grid-connected VSCs, i.e., power-synchronization con-

trol, is proposed. The VSC using power-synchronization control basically resem-

bles the dynamic behavior of a synchronous machine. A group of additional con-

trol functions, such as high-pass current control, current limitation function, anti-

windup schemes, etc. are proposed to deal with various practical issues during op-

eration.

• A novel modeling concept, i.e., the Jacobian transfer matrix, is proposed as a uni-

fied dynamic modeling technique for ac/dc systems. With the proposed concept, a

synchronous power system is viewed upon as a multivariable feedback control sys-

tem. The proposed concept is intended to be a unified framework for analyzing both

the low-frequency and high-frequency stability phenomena in power systems.

• The theoretical connections between the stability defined by the Jacobian trans-

fer matrix concept and the classical power-system stability defined by the phasor

approach are analyzed. It is discovered that the transmission zeros of the Jacobian

transfer matrix have a close relationship with angle and voltage stability in power

systems.

• A similar modeling concept, i.e., the dc Jacobian matrix, is proposed for modeling

of dc systems. By using a π-link dc model, it is mathematically proven that the

dc system (constructed by VSCs) is an inherently unstable process, where the dc

resistance gives a destabilizing effect.

4

1.3. Scientific contributions of the thesis

• Grid-connected VSCs using power-synchronization control and vector current con-

trol are modeled by the Jacobian transfer matrix concept. The transfer functions are

validated with frequency-scanning results from PSCAD/EMTDC.

• The Jacobian transfer matrix concept is also applied for modeling of two conven-

tional power components, i.e., the synchronous generator and the induction motor.

It is discovered that the transmission zeros of the Jacobian transfer matrix are useful

for interpreting some classical concepts, such as the synchronizing torque for the

synchronous generator and the pull-out slip for the induction motor, from a feed-

back control point of view. The transfer functions of the Jacobian transfer matrices

are also validated with frequency-scanning results from PSCAD/EMTDC.

• The performance of power-synchronization control and vector current control are

compared for VSC-HVDC links connected to weak ac systems, where it is con-

cluded that power-synchronization control is more suitable for weak-ac-system con-

nections.

• Two multivariable feedback-control design methods, i.e., IMC and H∞ control are

investigated for VSC-HVDC links connected to high-impedance ac systems. The

performance and robustness of various control designs are compared and discussed.

• A two-degree-of-freedom direct-voltage controller for VSC-HVDC system is pro-

posed where a prefilter is applied to remove the overshoot of the direct voltage. A

notch filter is proposed to reduce the dc-resonance peak.

• The requirement of dc capacitance for VSC-HVDC links connected to weak ac

systems is derived.

• A control structure for VSC-HVDC links interconnecting two very weak ac sys-

tems is proposed. The linear model is validated with time simulations from PSCAD/-

EMTDC for each major design step.

• A frequency droop controller and an alternating-voltage droop controller are pro-

posed for VSC-HVDC links connected to low-inertia systems.

• A complete linear model is developed for a typical island system which includes

a synchronous generator, an induction motor, a VSC-HVDC link and some RLC

loads. The root-locus technique is applied to tune the control parameters of the

VSC-HVDC link.

• Simulation studies are performed to demonstrate the flexibility of power-synchroni-

zation control for various operation modes related to island operation.

• The subsynchronous characteristics of a VSC-HVDC converter using power-synch-

ronization control are analyzed using the frequency-scanning method.

5


1.4 List of publications

- The doctoral thesis has resulted in the following publications:

I L. Zhang and H.-P. Nee, “Multivariable feedback design of VSC-HVDC connected

to weak ac systems”, in PowerTech 2009, Bucharest, Romania, 2009.

II L. Zhang, L. Harnefors and H.-P. Nee, “Power-synchronization control of grid-

connected voltage-source converters”, IEEE Trans. Power Systems, vol. 25, no. 2,

pp. 809-820, May 2010.

III L. Zhang, L. Harnefors and H.-P. Nee, “Modeling and control of VSC-HVDC

links connected to island systems” accepted for publication at IEEE Power and

Energy Society General Meeting, 2010, Minneapolis, USA.

IV L. Zhang, L. Harnefors and H.-P. Nee, “Interconnection of two very weak ac

systems by VSC-HVDC links using power-synchronization control”, accepted for

publication in IEEE Trans. Power Systems.

V L. Zhang, H.-P. Nee and L. Harnefors, “Analysis of stability limitations of a

VSC-HVDC link using power-synchronization control”, submitted to IEEE Trans.

Power Systems.

- The author has co-authored the following publication during the course of the Ph.D

study:

VI L. Harnefors, L. Zhang and M. Bongiorno, “Frequency-domain passivity-based

current controller design”, IET Power Electron., vol. 1, no. 4, pp. 455-465, 2008.

- During the course of the licentiate study, the author has authored and co-authored the

following publications:

VII L. Zhang and M. H. J. Bollen, “A method for characterizing unbalanced voltage

dips with symmetrical components”, IEEE Power Engineering Letter, pp. 50-52,

July 1998.

VIII L. Zhang and M. H. J. Bollen, “A method for characterization of three-phase un-

balanced dips from recorded voltage waveshapes”, in International Telecommuni-

cation Energy Conference, Copenhagen, Danmark, 1999.

IX L. Zhang and M. H. J. Bollen, “Characteristics of voltage dips in power systems”,

IEEE Trans. Power Delivery, vol. 15, no. 2, pp. 827-832, April 2000.

X L. Zhang, “Three-phase unbalance of voltage dips”, Licentiate thesis, Techni-

cal Report no. 322L, ISBN 91-7197-855-0, Chalmers University of Technology,

Goteborg, Sweden, 1999.

6

1.4. List of publications

XI M. H. J. Bollen and L. Zhang, “Analysis of voltage tolerance of ac adjustable-speed

drives for three-phase balanced and unbalanced sags”, IEEE Trans. Ind. Applicat.,

vol. 36, no. 3, pp. 904-910, May/June 2000.

XII M. H. J. Bollen, J. Svensson, and L. Zhang, “Testing of grid-connected power con-

verters for the effects of short circuits in the grid”, in European Power Electronics

Conference, Lausanne, Switzerland, 1999.

- In addition, during the working time in ABB, the author has authored and co-authored

the following publications that are relevant to the subjects of the doctoral and licentiate

studies:

XIII L. Zhang and L. Dofnas, “A novel method to mitigate commutation failures in

HVDC systems”, in International Conference on Power System Technology, Kun-

ming, China, 2002.

XIV L. Zhang, L. Harnefors and P. Rey, “Power system reliability and transfer capa-

bility improvement by VSC-HVDC (HVDC Light)”, in Cigre Regional meeting,

Tallin, Estonia, 2007.

XV M. H. J. Bollen and L. Zhang, “Different methods for classification of three-phase

unbalanced voltage dips due to faults”, Electric Power Systems Research, vol. 66,

no. 1, pp. 59-69, July 2003.

7


8

Chapter 2

High-Voltage Direct-Current

Transmission

This chapter presents general aspects of HVDC transmission. Two major HVDC tech-

nologies, i.e., HVDC transmission using line-commutated current-source converters and

HVDC transmission using forced-commutated voltage-source converters are described.

2.1 DC versus AC transmission

The history of electric power systems began with direct-current (dc) transmission. In

1882, Thomas Edison built the first power system with dc transmission with a low voltage

level. However, dc transmission was quickly replaced by three-phase alternating current

(ac) transmission because of several advantages of the latter. The most prominent advan-

tage of ac transmission is that power can be transformed to different voltage levels. By

using transformers, long-distance power transmission becomes possible. In addition, cir-

cuit breakers for alternating current can take advantage of the natural current zeros that

occur twice per cycle, and ac motors are cheaper and more robust than dc motors.

In spite of the principal use of ac transmission in power systems, the interests on dc

transmission still remain [23]. In 1954, the first commercial HVDC link between main-

land Sweden to Gotland island was commissioned. Since then, the accumulated installed

power of HVDC transmission systems worldwide has increased steadily, and recently a

dramatic increase in volume has been initiated. Given the extra costs and losses related

to the converter stations, HVDC transmission is justified by some particular conditions

where the dc technology is the most feasible or may be the only solution:

• Power transmission via cables. Due to their physical structures, cables have much

higher capacitance than overhead lines. The capacitive current in cables created by

the alternating voltage makes ac power transmission over long distance impossible.

Even for a moderate length (50km), the losses created by the capacitive current can

9

Chapter 2. High-Voltage Direct-Current Transmission

be so high that reactive compensating equipment has to be installed in the middle

of the cable [24]. However, installation of reactive compensating equipment is ex-

pensive and not always practical, e.g., with submarine cable transmission under sea.

On the other hand, if the power is transmitted by direct currents, there will be no

losses related to capacitive currents. Therefore, for long-distance submarine cable

transmission, HVDC transmission is the only feasible technical solution.

• Bulk-power transmission over long-distance. Interestingly, given the fact that ac

won the “battle of currents” due to its possibility to transmit power over long dis-

tance [25], HVDC transmission wins the battle back after a century. To transmit

the same amount of power, dc transmission needs fewer power lines than ac trans-

mission. Accordingly, the costs and losses of the converter stations get balanced by

savings on the overhead lines where the break-even distance is around 400 km to

700 km depending on the land conditions and project specifications [26]. Besides,

dc transmission does not have the stability limitation related to ac transmission over

long distance.

• Unsynchronized ac-system connection. AC transmission is only possible if the

two interconnected ac systems have the same nominal frequency and operate syn-

chronously, but dc transmission does not have such requirements. Many back-to-

back HVDC links have been built for such purposes.

Besides the above essential arguments, there are additional benefits by having embedded

HVDC links in ac systems:

• Power-system stability improvement. One of the major features of the HVDC tech-

nology is its capability to manipulate large amount of power in a very short time,

which can often be utilized to improve the stability of the ac system. One example is

the improvement of transient stability by running up or running back the dc power

for emergency power supports [4, 27]. Another example is that HVDC system can

be used to damp low-frequency oscillations in ac systems by having an auxiliary

damping controller [28].

• Firewall function. Large interconnected ac systems have many well-known advan-

tages, e.g., the possibility to use larger and more economical power plants, reduction

of reserve capacity in the systems, utilization of the most efficient energy resources,

as well as achieving an increase in system reliability [25,29]. However, larger inter-

connected ac systems also increase the system complexity from the operation point

of view. One of the consequence of such complexity is the large blackouts in Amer-

ica and Europe [30]. In this aspect, HVDC links have the “firewall” function in

preventing cascaded ac-system outages spreading from one system to another [31].

10

2.2. HVDC transmission using line-commutated current-source converters

-

-

-

+

+

+

aU

N

bU

cU

L

L

L

aI

bI

cI

Ldci

dcu

V1

V6

V5

V2V4

V3

Fig. 2.1 Graetz bridge for LCC-HVDC system.

2.2 HVDC transmission using line-commutated current-

source converters

The converter technology used for HVDC transmission in the early days was based on

mercury valves. The major problem with mercury-arc technology was ark-back fault

which destroyed the rectifying property of the converter valve and consequently triggered

other problems [23]. In the late 1960s, the thyristor valve technology was developed that

overcame the problems of mercury-arc technology. Converters based on either mercury

valves or thyristor valves are called line-commutated converters (LCCs), or current-source

converters (CSCs). The basic module of an LCC is the three-phase full-wave bridge cir-

cuit shown in Fig. 2.1. This topology is known as the Graetz bridge. Although there are

several alternative configurations possible, the Graetz bridge has been universally used

for LCC-HVDC converters as it provides better utilization of the converter transformer

and a lower voltage across the valve when not conducting [32].

The Graetz bridge can be used for transmitting power in two directions, i.e., the

rectifier mode and the inverter mode. This is achieved by applying different firing angles

on the valves. If the firing angle is lower than 90◦, the direct current is flowing from the

positive terminal of the dc circuit, thus the power is following from the ac side to the dc

side; If the firing angle is higher than 90◦, the direct voltage changes polarity, thus the

direct current is flowing from the negative terminal of the dc circuit. The power is then

flowing from the dc side to the ac side. An HVDC link is essentially constructed by two

Graetz bridges, which are interconnected on the dc sides. The interconnection could be

an overhead line, a cable, or a back-to-back connection.

The application of LCC-HVDC technology has been very successful and the instal-

lations of LCC-HVDC links are expected to grow at least in the near future. However, the

LCC technology suffers from several inherent weaknesses.

11


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

-1

0

1

ud

c (

p.u

.)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04-2

0

2

I v (

p.u

.)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

-1

0

1

Uv (

p.u

.)

time (sec)

V5 V1 V3 V1 V3

V4 V6 V2 V4 V6 V2

Uab Ubc Uca Ucb Uac Ubc Uca Ucb Uba Uac Uab Uba

Uab Uac Uac Uab Uab

Fig. 2.2 Commutation failures of an LCC-HVDC inverter. Upper plot: direct voltage. Middle plot:

valve currents. Lower plot: valve voltage.

One problem is that the LCC always consumes reactive power, either in recti-

fier mode or in inverter mode. Depending on the firing angles, the reactive power con-

sumption of an LCC-HVDC converter station is approximately 50 − 60% of the ac-

tive power. The reactive-power consumption requires compensation by connecting large

ac filters/capacitors at the converter stations. For a common LCC-HVDC link, the fil-

ters/capacitors not only increase the costs, but also occupy large amounts of space of

the converter stations. Besides, large filters/capacitors also contribute to the temporary

overvoltage (TOV) and low-order harmonic resonance problems of the HVDC link when

connects to a weak ac system [5].

Another well-known problem of the LCC-HVDC system is the occurrence of com-

mutation failures at the inverter station typically caused by disturbances in the ac system.

Either depressed voltage magnitude or phase-angle shift of the alternating voltage may

reduce the extinction volt-time area of the inverter valve [33, 34]. If the extinction angle

of the inverter valve is smaller than 5 − 6◦, the previously conducted valve will regain

current, which will end up with a commutation failure. Fig. 2.2 shows plots of a typical

commutation failure. A disturbance on Uab appears during the commutation between V1

and V3. Because V1 does not get the reverse voltage that is needed to switch off the cur-

rent, V1 continues to be conducted and the valve current of V3 goes down to zero again.

When the next commutation occurs between V2 and V4, V1 and V4 conduct at the same

time. From Fig. 2.2 it can be observed that the commutation failure, in fact, creates a short

circuit on the dc side, which essentially temporarily stops the power transmission.

Commutation failures are common phenomena of LCC-HVDC systems. A single

12

2.2. HVDC transmission using line-commutated current-source converters

commutation failure generally does no harm to either the converter valves or the ac sys-

tem. However, a number of repeated commutation failures may force the HVDC link to

trip [35].

While the above two problems can be mitigated by relatively easy measures, the

third problem is more fundamental, which can become a limiting factor for LCC-HVDC

applications. For LCCs, the successful commutation of the alternating current from one

valve to the next relies on the stiffness of the alternating voltage, i.e., the network strength

of the ac system. If the ac system has low short-circuit capacity relative to the power

rating of the HVDC link, i.e., low SCR, more problematic interactions between the ac

and the dc systems are expected. Besides, the SCR of the ac system also imposes an

upper limitation on the HVDC power transmission, which is often described by the well-

known maximum power curve (MPC) [5]. As mentioned before, an LCC-HVDC link

normally requires reactive-power compensation by connecting larger ac filters/capacitors

at the converter stations. These ac filters/capacitors create additional problems in weak ac

systems as described below.

One such problem is the aforementioned TOV issue. In case of a sudden change in

the active power, or the blocking of converter, the large filters/capacitors at the converter

station together with the high inductance of the ac system cause a temporary overvolt-

age before the protection system disconnects the filters/capacitors. The magnitude of the

overvoltage is directly related to the strength of the ac system. Ref. [5] gives the following

estimation of the fundamental components of TOV (TOVfc) regarding the SCR of the ac

system:

• SCR > 3: TOVfc lower than 1.25 p.u.

• 2 < SCR < 3: TOVfc higher than 1.25 p.u. but lower than 1.4 p.u.

• SCR < 2: TOVfc higher than 1.4 p.u.

TOVs can influence the design and costs of the dc stations. TOVs can also lead to satura-

tion of the converter transformer or transformers close to the dc station.

Another problem with weak-ac-system connections is the low-order harmonic res-

onance. The high inductance of the ac system and the large filters/capacitors of the HVDC

link create a resonance with low frequency fres, which can be estimated approximately by

fres ≈ f1

√2 · SCR (2.1)

where f1 is the nominal frequency of the ac system. That is to say, the resonance frequency

tends to be lower for weak ac systems. Generally speaking, the lower the resonance fre-

quency, the greater the risk for harmful interaction with the converter control system.

An improved topology of the LCC-HVDC system to overcome part of the above

mentioned problems is the capacitor-commutated converter (CCC)-HVDC technology,

13


dci

dcu

-

-

-

+

+

+

aU

N

bU

cU

L

L

L

aI

bI

cI

Fig. 2.3 Two-level voltage-source converter.

where ac capacitors are inserted in series between the valves and converter transform-

ers [36]. The series-connected capacitors not only supply the reactive power consumed

by the valves, it also improve the dynamic performance of the HVDC system. However,

the major drawback of the CCC concept is that the series capacitors increase the insula-

tion costs of the valves. Thus, the CCC-HVDC technology has been so far only applied

to back-to-back HVDC links, where the voltage level of the valves is much lower.

2.3 HVDC transmission using forced-commutated voltage-

source converters

Voltage-source converters are a new converter technology for HVDC transmission [2].

The first commercial VSC-HVDC (HVDC Light) link with a rating of 50 MW was com-

missioned in 1999 in Gotland island of Sweden, close to the world’s first LCC-HVDC

link.

Voltage-source converters (VSCs) utilize self-commutating switches, e.g., gate turn

off thyristors (GTOs) or insulated-gate bipolar transistors (IGBTs), which can be turned

on or off freely. This is in contrast to the LCC where the thyristor valve can only be

turned off by reversed line voltages. Therefore, a VSC can produce its own sinusoidal

voltage waveform using pulse-width modulation (PWM) technology independent of the

ac system.

Many different topologies have been proposed for VSCs. However, for HVDC ap-

plications, they have been so far limited to three major types: two-level converter, three-

level converter, and modular multilevel converter (M2C) [37–39].

Fig.2.3 shows a two-level grid-connected VSC. The two-level bridge is the simplest

topology that can be used in order to build up a three-phase forced-commutated VSC

bridge. The bridge consists of six valves and each valve consists of a switching device

14

2.3. HVDC transmission using forced-commutated voltage-source converters

dci

dcu

-

-

-

+

+

+

aU

N

bU

cU

L

L

L

aI

bI

cI

Fig. 2.4 Three-level neutral-point-clamped voltage-source converter.

and an anti-parallel diode. For an HVDC link, two VSCs are interconnected on the dc

side. For high-voltage applications, series connection of switching devices is necessary.

The operation principle of the two-level bridge is simple. Each phase of the VSC can be

connected either to the positive dc terminal, or the negative dc terminal. By adjusting the

width of pulses, the reference voltage can be reproduced, as shown in the upper plot of

Fig. 2.6. After filtering by phase reactors and shunt filters, this series of voltage pulses

resembles the voltage waveform of the reference voltage.

The three-level VSC shown in Fig. 2.4 is also called neutral-point-clamped (NPC)

converter. The key components that distinguish this topology from the two-level converter

are the two clamping diodes in each phase. These two diodes clamp the switch voltage to

half of the dc voltage. Thus, each phase of the VSC can switch to three different voltage

levels, i.e., the positive dc terminal, the negative dc terminal and the mid-point. Con-

sequently, voltage pulses produced by a three-level VSC match closer to the reference

voltage. Therefore, the three-level NPC converter has less harmonic content as shown in

the middle plot of Fig. 2.6. Additionally, the three-level NPC converter has lower switch-

ing losses. Compared to two-level VSCs, three-level NPC VSCs require more diodes for

neutral-point clamping. However, the total number of switching components does not

necessarily have to be higher. The reason for this is that, for HVDC applications, a valve

consists of many series-connected switches. In the two-level case a valve has to with-

stand twice as high voltage than in the three-level case. Accordingly, the total number of

15


A

B

A

B

A

BA

B

A

B...

-

+aU

A

B

A

B

A

B

...

dcu

(a)

(b)

Positive arm

Negative arm

V1

V2

Fig. 2.5 Modular multilevel voltage-source converter. (a) One M2C module. (b) One phase topol-

ogy.

switches is approximately equal.

The NPC concept can be extended to higher number of voltage levels, which can

result in further improved harmonic reduction and lower switching losses [40]. However,

for high-voltage converter applications, the neutral-clamped diodes complicate the insu-

lation and cooling design of the converter valve. Therefore, NPC concepts with a number

of voltage levels higher than three has never been considered for HVDC applications [37].

The recently proposed modular multilevel converter (M2C) concept [39, 41–43]

attracts significant interests for high-voltage converter applications. Fig. 2.5 shows the

M2C topology for one phase. Compared to the above two topologies, one major feature

of the M2C is that no common capacitor is connected at the dc side. Instead, the dc

capacitors are distributed into each module, while the converter is built up by cascade-

connected modules.

Fig. 2.5(a) shows an M2C module. Each M2C module consists of two valves which

can be switched in three different ways:

• V2 is turned on and V1 is turned off, the capacitor is inserted into the circuit from

A to B. The module contributes with voltage to the phase voltage. The capacitor is

charged if the current is from A to B, and discharged otherwise.

• V1 is turned on and V2 is turned off, the capacitor is by passed. The module does

not contribute with voltage to the phase voltage.

• Both V1 and V2 are turned off, the module is blocked.

16


0 0.01 0.02 0.03 0.04 0.05 0.06

-100

0

100

UL

1 (

kV

)

0 0.01 0.02 0.03 0.04 0.05 0.06

-100

0

100

UL

1 (

kV

)

0 0.01 0.02 0.03 0.04 0.05 0.06

-100

0

100

time (sec)

UL

1 (

kV

)

Fig. 2.6 Pulse-width modulation for different converter topologies. Upper plot: two-level con-

verter. Middle plot: three-level converter. Lower plot: M2C with five modules .

The M2C concept is especially attractive for high-voltage applications, since the con-

verter can be easily scaled up by inserting additional modules in each arm. If consider-

able amounts of modules are cascaded (approximately 100 modules would be common

for HVDC applications), each module theoretically only needs to switch on and off once

per period, which greatly reduces the switching losses of the valves. However, prelimi-

nary investigation indicates that slightly higher switching frequencies are necessary. The

lower plot of Fig. 2.6 shows the voltage waveform produced by a five-module (five for

each arm) M2C. With only five modules, the waveform already resembles much better

the sinusoidal voltage reference than the other two topologies. With M2C, the harmonic

content of the voltage produced by the VSC is so low that additional filtering equipment

is almost unnecessary.

An additional benefit of the M2C is that the control system has an extra freedom in

dealing with faults at the dc side. The dc capacitors are not necessarily discharged during

faults. Thus, the fault recovery can be faster [39].

Compared to the other two topologies, the major drawback of the M2C topology is

that the required switching components are doubled since only one of the valves of each

module contributes to the phase voltage when the module is inserted in. In addition, the

design and control of the M2C are generally more complex at least than the two-level con-

verter. However, since the switching frequency of the M2C can be kept very low switches

with higher blocking voltages may be used, which in turn limits the increase in number

of switches. On the other hand, the reduction of switching losses and savings on filtering

equipment of the M2C may eventually justify its application for HVDC transmission.

17


5.0

0.1

5.0 0.1

Reactive power

Over-voltage

limitation

Under-voltage

limitation

Converter current

limitation

Active power

Fig. 2.7 PQ diagram for a typical VSC-HVDC converter.

No matter what converter topology is used, the VSC can always be treated as an

ideal voltage source where the control system has the freedom to specify the magnitude,

phase, and frequency of the produced sinusoidal voltage waveform. However, for control

design and stability analysis, it is important to take into account the limitation of the

converter in terms of active and reactive power transfer capability.

One such limit is the converter-current limitation, which is imposed by the current

carrying capability of the VSC valves. Since both the active power and the reactive power

contribute to the current flowing through the valves, this limitation is manifested as a

circle in a PQ diagram. Accordingly, if the converter is intended to support the ac system

with reactive-power supply/consumption, the maximum active power has to be limited to

make sure that the valve current is within the limit.

Another limitation which determines the reactive-power capability of the VSC is

the over/under voltage magnitude of the VSC (modulation index limitation). The over-

voltage limitation is imposed by the direct-voltage level of the VSC. The under-voltage

limit, however, is limited by the main-circuit design and the active-power transfer capa-

bility, which requires a minimum voltage magnitude to transmit the active power. In this

respect, the tap-changer of the converter transformer can play an important role to extend

the reactive-power limitation of the VSC. This could be an argument to have converter

transformers in VSC-HVDC systems. Fig. 2.7 shows the PQ diagram with the above

mentioned limitations for a typical VSC-HVDC converter [44].

VSC-HVDC technology overcomes most of the weaknesses of the LCC-HVDC

technology. In addition, it supports the ac system with reactive-power supply/consumption.

18


Similar to an LCC-HVDC system, a VSC-HVDC system can quickly run up or run back

the active power for ac system emergency-power support, but it can also instantly reverse

the active power [4].

Since the direct voltage of a VSC-HVDC system varies in a much smaller range

than that of a LCC-HVDC system, extruded cables can be used for VSC-HVDC systems.

The extruded cable reduces the cable cost and the construction cost. The latter makes

long-distance land-cable transmission possible [2].

Besides the above features, the most essential one is that a VSC-HVDC system

has an unlimited connection capability with ac systems, i.e., with properly designed con-

trol systems, VSC-HVDC system has the potential to be connected to any kind of ac

system with any number of links. This outstanding property will eventually bring the dc-

transmission technology to ever broader application fields!

19


20

Chapter 3

Control Methods for VSC-HVDC

Systems

This chapter describes various control methods used for VSC-HVDC systems. In Sec-

tion 3.1 and Section 3.2 two existing control methods, i.e., power-angle control and

vector current control are described. A novel control method, i.e., the so-called power-

synchronization control, is introduced in Section 3.3. The major results of this chapter are

summarized in Section 3.4. Some results of this chapter are included in [45].

3.1 Power-angle control

Power-angle control is also called voltage-angle control. It is perhaps the most straightfor-

ward controller for grid-connected VSCs [46–48]. The principle of power-angle control

is based on the following well-known equations

P =U1U2 sin θ

X

Q =U2

1 − U1U2 cos θ

X(3.1)

where P and Q are the active and reactive powers between two electrical nodes in ac

systems with voltage magnitudes U1 and U2. The quantities θ and X are the phase-angle

difference and line reactance between the two nodes. From (3.1) it follows that the active

power is mainly related to the phase angle θ, while the reactive power is more related to

the voltage-magnitude difference. These mathematical relationships are the foundation of

power-angle control, i.e., the active power is controlled by the phase angle of the VSC

voltage, while the reactive power or filter-bus voltage is controlled by the magnitude of

the VSC voltage.

Fig. 3.1 shows the main-circuit and control block diagram of a VSC-HVDC con-

verter using power-angle control. Lc is the inductance of the phase reactor, and Ln is the

21

Chapter 3. Control Methods for VSC-HVDC Systems

vθ

nL cL

-+refPP

QP,

E

VSC

ref

av

ref

bvref

cv

+

-fu

fC

APC

PLL

RPC/AVC-+refQQ V∆

tω

ci

v

+

-

Voltage

reference

control

-+

fU

refU

Fig. 3.1 Main-circuit and control block diagram of a VSC-HVDC converter using power-angle

control.

inductance of the ac system. Cf is the ac capacitor connected at the filter bus. The bold

letter symbols E, uf , and v represent the voltage vectors of the ac source, the filter bus,

and the VSC respectively. P and Q are the active power and reactive power from the VSC

to the ac system. The quantity ic is the current vector of the phase reactor.

To produce three-phase alternating voltages, the VSC needs three variables: magni-

tude, phase angle and frequency. With power-angle control, these three variables are given

by three different controllers, i.e., the reactive-power controller (RPC) or the alternating-

voltage controller (AVC), the active-power controller (APC), and the phase-locked loop

(PLL). The above controllers are briefly described in below:

• Reactive-power controller. The reactive power to/from the VSC is controlled by

the magnitude of the VSC voltage. A proportional-integral (PI) controller can be

applied, e.g.,

∆V =

(KQ

p +KQ

i

s

)[Qref −Q] . (3.2)

where the output ∆V gives the change in magnitude of the VSC reference voltage.

• Alternating-voltage controller. Alternatively, the VSC-HVDC converter controls

the filter-bus voltage instead of the reactive power. The output of the AVC is the

same as that of the RPC. A PI controller can be applied, e.g.,

∆V =

(KU

p +KU

i

s

)[Uref − Uf ]. (3.3)

• Active-power controller. The active power to/from the VSC is controlled by the

phase angle of the VSC voltage. A proportional-integral (PI) controller can be ap-

22

3.1. Power-angle control

plied, i.e.,

θv =

(KP

p +KP

i

s

)[Pref − P ]. (3.4)

where the output θv gives the change in phase angle of the VSC reference voltage.

• Phase-locked loop. The function of the PLL is to synchronize the VSC to the ac

system. Below a description of a PLL design suitable for power-angle control is

given.

If ω1 is the angular frequency of the ac system, and ω is the angular frequency of the

VSC, a PLL controller has the objective to follow the phase angle of the filter-bus

voltage by minimizing

e = (ω1 − ω)t. (3.5)

If the error e in (3.5) is sufficiently small, (3.5) can be approximated by

e ≈ sin(ω1t− ωt)

= sinω1t cosωt− cosω1t sinωt (3.6)

where sinω1t and cosω1t can be obtained by αβ transformation of the filter-bus

voltage as shown below.

The phase quantities of the filter-bus voltage can be defined as

ufa = Uf0 cos(ω1t)

ufb = Uf0 cos(ω1t− 120◦)

ufc = Uf0 cos(ω1t− 240◦). (3.7)

The corresponding real and imaginary parts of the vector uf in the stationary αβ

reference frame (see Appendix A) can be written as

ufα = Uf0 cosω1t, ufβ = Uf0 sinω1t. (3.8)

Substituting (3.8) into (3.6), yields

e ≈ ufβ

Uf0cosωt− ufα

Uf0sinωt. (3.9)

A PI controller can be used to minimize the error e, i.e.,

θPLL =

(KPLL

p +KPLL

i

s

)e. (3.10)

Fig. 3.2 shows the control block diagram of the PLL. The angle change θPLL is

added to a reference frequency signal ωreft.

23


abc

αβ

trefω

fau

fbu

fcu

COS

SIN

0

1

fU

0

1

fU

s

KK i

p

PLLPLL +

tω

αfu

βfu

+

- PLLθ

+

+

Fig. 3.2 PLL for power-angle control.

• Voltage-reference control. By having ∆V , θv and ωt, the three-phase reference

voltages of the VSC can be formulated as:

vrefa = (V0 + ∆V ) cos(ωt+ θv)

vrefb = (V0 + ∆V ) cos(ωt+ θv − 120◦)

vrefc = (V0 + ∆V ) cos(ωt+ θv − 240◦) (3.11)

where V0 is a nominal voltage reference, e.g., V0 = 1.0 p.u.

As shown in this section, the design and implementation of power-angle control

is simple and straightforward. However, power-angle control practically has never been

applied to any real VSC-HVDC system, since it suffers from two fundamental problems:

1. The control system has no general means to damp the various resonances in the

ac system. Therefore, the bandwidth of the controller is very much limited by the

resonances in the ac system, especially the one at the grid frequency [48].

Fig. 3.3 shows a plot of active-power and reactive-power step responses with power-

angle control. The resonance at the grid frequency can be easily observed. Although

the resonance can be damped out by applying notch filters in the active-power and

reactive-power controllers, or canceled by some model-based control designs [49],

the effects of such measures are doubtful since the ac system is a highly uncertain

process where not all of the resonance frequencies are known.

2. The control system does not have the capability to limit the valve current of the

converter. This is a serious problem, as the converters of a VSC-HVDC link usually

do not have over-current capability. It is very important for the control system to

limit the valve current to prevent the converters from being blocked (tripped) at

disturbances.

24

3.2. Vector current control

0 0.5 1 1.5 2-0.05

0

0.05

0.1

0.15

time (sec)

Pre

f, P

(p

.u.)

Pref

P

0 0.5 1 1.5 2-0.05

0

0.05

0.1

0.15

time (sec)

Qre

f, Q

(p

.u.)

Qref

Q

Fig. 3.3 Step response of active power (upper plot) and reactive power (lower plot) with power-

angle control. Observe the resonance at the grid frequency.

3.2 Vector current control

Vector current control of VSCs has initially been applied to variable-speed drives, where

the VSC is connected to an ac motor [50,51]. By utilizing the dq decoupling technique, the

control system is able to control the torque and flux independently. Once field orientation

is obtained the ac motor can be controlled using principles quite similar to those of dc-

motor control. Vector current control is generally considered as a substantial step in ac-

motor control [52].

The application of vector current control on grid-connected VSCs is often consid-

ered as a dual problem of drive control [17,53]. The basic principle is to control the active

power and the reactive power independently through an inner-current control loop [54,55].

As shown in Fig. 3.4, the essence of vector current control is that the control system

creates a converter dq frame, where a PLL is applied to make sure that the d-axis of the

converter dq frame is always aligned with the filter-bus voltage in order to synchronize

the VSC to the ac system. A simple implementation of a current controller is achieved by

the proportional-type control law

vcref = αcLc (iref − icc) + jω1Lci

cc +HLP(s)uc

f (3.12)

where αc is the desired closed-loop bandwidth of the inner-current controller, iref is the

converter current reference, and vcref is the voltage reference of the VSC. The superscript

c denotes the converter dq frame. The term jω1Lcicc is used to remove the so-called cross-

coupling. The function HLP(s) is a low-pass filter to improve the disturbance rejection

25


Grid-q

Grid-d

fu

v

E

VSC-d

VSC-q

Fig. 3.4 Converter dq frame of vector current control.

capability of the current controller. HLP(s) has the following expression

HLP(s) =αf

s + αf(3.13)

where αf is typically chosen with a bandwidth (40 − 100 rad/s) [13].

By applying the control law in (3.12), vector current control is claimed to be able to

control the active power and the reactive power independently by the d and q components

of the current reference iref . However, this is a conditional correct conclusion, which is

only valid if the filter-bus voltage is sufficiently stiff such that its dynamics are negligible.

A brief analysis is given below.

In a stationary frame, the dynamic equation of the phase reactor of the VSC can be

described by Kirchhoff’s voltage law as

Lcdis

dt= vs − us

f (3.14)

where the superscript s denotes the stationary reference frame. If it is assumed that

ω = ω1 (3.15)

at all times, i.e., the angular frequency ω of the converter dq frame equals the angular

frequency ω1 of the grid, the following relations are established

usf = uc

fejω1t, isc = icce

jω1t, vs = vcejω1t. (3.16)

Substituting (3.16) into (3.14) yields the dynamic equation in the converter dq frame

Lcdiccdt

= vc − ucf − jω1Lci

cc. (3.17)

If the switching-time delay is neglected and it is assumed that |vcref | does not exceed the

maximum voltage modulus, then vc = vcref . Substituting (3.12) into (3.17) yields

Lcdiccdt

= αcLc(iref − icc) −s

s + αfuc

f . (3.18)

26


By writing (3.18) in component form and applying Laplace transform (s = d/dt), yields

iccd =αc

s+ αcirefd − s

Lc(s+ αc)(s+ αf )uc

fd

iccq =αc

s+ αcirefq − s

Lc(s+ αc)(s+ αf )uc

fq. (3.19)

The above design approach for the inner-current controller is often referred to as internal-

model control (IMC) design [51], since the bandwidth of the inner-current control is

explicitly specified in the control parameters. Another common design approach is the

deadbeat-current control design [56, 57], which can only be realized by digital imple-

mentations. Generally speaking, if the bandwidth αc of IMC is chosen sufficiently high,

IMC and deadbeat-current control give similar results. For either of the control design,

the control bandwidth is basically limited by the switching frequency of the PWM and

the sampling period of the computer. Moreover, both methods rely on a good knowledge

of the value of Lc.

The following analysis will establish the relations between the active/reactive power

and the current references of the inner-current control. Assuming per unit quantities, the

instantaneous active power and reactive power from the VSC to the filter bus are given by

P = Re{uc

f(icc)

∗}, Q = Im

{uc

f (icc)

∗}. (3.20)

Linearizing (3.20) yields the following expressions

∆P =

[iccd0

iccq0

]T [∆uc

fd

∆ucfq

]+

[uc

fd0

ucfq0

]T [∆iccd

∆iccq

]

∆Q =

[icd0

−icq0

]T [∆uc

fq

∆ucfd

]+

[uc

fd0

ucfq0

]T [ −∆iccq

∆iccd

](3.21)

where the subscript 0 denotes the operating-point value. In the converter dq frame, in the

steady state, the q component of the filter-bus voltage equals zero and the d component

equals the voltage magnitude, i.e.,

ucfd0 = Uf0, u

cfq0 = 0. (3.22)

If the dynamics of the filter-bus voltage are neglected, it follows that

∆ucfd = ∆uc

fq = 0. (3.23)

By substituting (3.22) and (3.23) into (3.21), the expressions of ∆P and ∆Q can be

simplified as

∆P = Uf0∆iccd, ∆Q = Uf0∆i

ccq. (3.24)

27


Linearizing (3.19) and further substituting it into (3.24) [with the condition in (3.23)]

yields the following relationship in transfer matrix form

[∆P

∆Q

]=

[Uf0

αc

s+αc0

0 Uf0αc

s+αc

]

︸︷︷︸J(s)

[∆irefd

∆irefq

]. (3.25)

The transfer matrix J(s) is called Jacobian transfer matrix in this thesis, which is a gen-

eral concept for dynamic modeling of ac/dc systems that is to be introduced in Chapter 4.

Eq. (3.25) shows that the Jacobian transfer matrix J(s) is diagonal, i.e., ∆P is

only related to ∆irefd while ∆Q is only related to ∆irefq , and no cross-coupling between

the two loops exists. However, (3.25) is derived based on the assumptions of (3.15) and

(3.23). Both of the assumptions are related to the stiffness of the filter-bus voltage. If the

ac system is strong enough, i.e., Ln ≪ Lc, the dynamics of the filter-bus voltage can be

neglected. However, if the ac system is weak, the assumptions in (3.15) and (3.23) no

longer hold. Therefore, the weaker the ac system, the higher the off-diagonal elements in

J(s), i.e., the more interactions between the active-power and the reactive-power control.

Consequently, to analyze the stability of vector current control for VSC-HVDC

links connected to weak ac systems, the dynamics of the filter-bus voltage have to be

considered. That is, the Jacobian transfer matrix J(s) in (3.25) should take into account

the grid inductance Ln and the dynamics of the PLL. Such a model will be developed in

Chapter 4. An in-depth analysis of the difficulty with vector current control for weak-ac-

system connections will be given in Chapter 5.

Fig. 3.5 shows the main-circuit and control block diagram of a VSC-HVDC con-

verter using vector current control. The active-power controller and the reactive-power or

the alternating-voltage controller of vector current control can be designed in a similar

way as power-angle control but with irefd and irefq as outputs. However, the PLL can be de-

signed in a more concise way by utilizing the concept of the converter dq frame , i.e., a PI

controller is applied to minimize the q component of the filter-bus voltage in the converter

dq frame

θPLL =

(KPLL

p +KPLL

i

s

)Im{uc

f}. (3.26)

In this way, the VSC is synchronized to the ac system. Fig. 3.6 shows the control block

diagram of the PLL for vector current control. With vector current control, the voltage

reference of the VSC is formulated by vrefd , vref

q and ωt. This is essentially the same as

power-angle control where the reference of the VSC voltage is formulated by the magni-

tude, the phase angle and the frequency. In the former case the rectangular form is used

while the latter uses polar form. The mathematical expressions of the dq-αβ and αβ−abcblocks in Fig. 3.5 are given in Appendix A.3.

For vector current control, given sufficiently high bandwidth, the dq components of

the converter current always follow the corresponding current references. Consequently,

28


nLcL

-+refPP

QP,

E

+

-fu

fC

APC

PLL

RPC/AVC-+refQQ

tω

ci

v

+

-

Inner-

current

controller

dqαβ abc

αβ

fuci

-+refU

fU

ref

αv

ref

βv

ref

dv

ref

qvref

di

ref

qi

VSC

ref

av

ref

bvref

cv

Fig. 3.5 Main-circuit and control block diagram of a VSC-HVDC converter using vector current

control.

++

s

KK i

p

PLLPLL +

c

fu}{Im c

fuPLLθ tω

trefω

Fig. 3.6 PLL for vector current control.

by limiting the modulus of the current references, the valve current of the converter is

limited. The simulation results in Fig. 3.7 show the fault ride-through capability of a VSC-

HVDC link using vector current control. A three-phase ac-system fault with 0.2 s duration

is applied at 0.1 s close to the filter bus. The modulus of the current reference [|irefc | =√(irefd )2 + (irefq )2] reaches the current limit Imax immediately after the fault occurrence.

The control system automatically limits the converter current. After the fault is detected,

the control system reduces the fault current to half of the maximum load current (or any

other desired values to minimize the short-circuit current contribution to the ac system)

except a very short current spike at the fault inception. In VSC applications, regardless

of the control principle, the converter always tries to protect itself from excessive over

currents. This fast protection is often implemented as a low-level hardware system, and

its objective is to protect the converter in cases where the higher levels of control fail.

Since the current spike in Fig. 3.7 is so short ( < 1.6 pu in magnitude and < 5 ms in

duration ), it neither does any harm to the converter valve, nor contributes much to the

short-circuit current to the ac system.

Besides the fault-current limitation capability, the current control also has a damp-

ing effect on resonances in the ac system. Therefore, vector current control overcomes

the two fundamental problems of power-angle control. In practice, vector current control

29


0 0.2 0.4 0.6 0.8 10

0.5

1

P (

p.u

.)

0 0.2 0.4 0.6 0.8 1

0

0.5

1

i re

f

d, i r

ef

q (

p.u

.)

0 0.2 0.4 0.6 0.8 10

1

2

time (sec)

| i c

| (p

.u.)

i ref

d

i ref

q

Fig. 3.7 Fault ride-through capability of vector current control. Upper plot: active power from the

VSC. Middle plot: dq components of the current reference. Lower plot: modulus of the

converter current.

has been successfully applied to a number of commercial VSC-HVDC links. However, a

major drawback of vector current control is its poor performance for VSC-HVDC links

connected to weak ac systems, which becomes an obstacle for VSC-HVDC transmission

to be applied in more challenging ac-system conditions.

To overcome the problem of vector current control with weak-ac-system connec-

tions, a novel control method, i.e., power-synchronization control, is proposed in the next

section. In some sense, power-synchronization control might be viewed upon as a combi-

nation of power-angle control and vector current control.

3.3 Power-synchronization control

With the two control methods described in the previous sections, a PLL is used to synchro-

nize the VSC with the ac system. This has since long been believed to be a pre-condition

for any grid-connected VSC. However, there is, in fact, an internal synchronization mech-

anism in ac systems that the VSC can utilize to synchronize with the ac system. In this

section, a control method based on such type of synchronization is proposed. The major

goal of the proposed control method is to overcome the problem of vector current control

with weak-ac-system connections.

30

3.3. Power-synchronization control

1SM 2SMX

222

2 mem TT

dt

dJ −=

ω11

11 em

m TTdt

dJ −=

ω

Fig. 3.8 Synchronization mechanism between SMs in an ac system.

3.3.1 Power-synchronization mechanism in ac systems

In this sub-section, the power-synchronization mechanism between synchronous machines

(SMs) in ac systems is described. The mechanism is illustrated by a simple system con-

sisting of two interconnected SMs as shown in Fig. 3.8. SM1 operates as a generator and

SM2 operates as a motor. The reactance X is the sum of the reactances of the SMs and

that of the line interconnecting the two SMs. All resistances and other damping effects

are disregarded.

Initially, it is assumed that the two SMs operate at steady state, as described by the

phasor diagram in Fig. 3.9(a). The phasorsE1 andE2 represent the line-to-line equivalents

of the inner emfs of the two SMs respectively. These emfs are assumed to be constant at

all times (even during transients). The electric power transmitted from SM1 to SM2 is

given by

P =E1E2 sin θ

X(3.27)

where θ is the electrical angle separating the two emfs E1 and E2. The mechanical torque

Tm1 of SM1 is now increased by a certain amount for a short duration and then brought

back to its initial value. As a consequence of the temporary increase of Tm1, the me-

chanical angle of the rotor of SM1 advances, as predicted by the generator-mode swing

equation

J1dωm1

dt= Tm1 − Te1 (3.28)

where J1 is the total inertia of the shaft-system of SM1 , ωm1 is the rotor speed, and Te1 is

the electromagnetic torque of of SM1. Since the emf of a synchronous machine is tightly

connected to the rotor position, the advance of the mechanical angle of the rotor of SM1

inevitably causes an advance of the phase of the emf of SM1, as indicated by the phasor

E′

1 in Fig. 3.9(b). The initial position of E1 is shown as a dashed line in Fig. 3.9(b). Due

to the phase advancement indicated by E′

1, the phase difference between the emfs of the

two SMs is increased. According to (3.28), this translates into an increase of the electric

power transmitted from SM1 to SM2 . This increase in power is equivalent to an increase

in the electromagnetic torque Te2 of SM2 . Assuming that SM2 has a constant load torque

Tm2, the rotor of SM2 starts to accelerate as dictated by

J2dωm2

dt= Te2 − Tm2 (3.29)

31


'1E1E 2E

θ2E

(a) (b)

Fig. 3.9 Phasor diagrams describing power synchronization.

where J2 is the total inertia of the shaft-system of SM2 , and ωm2 is the mechanical angular

velocity of SM2 . As the rotor of SM2 starts to accelerate, the same thing occurs with the

phase of E2, as indicated by the arc-shaped arrow in Fig. 3.9(b). The acceleration of the

phasor E2 results in a reduction of the phase difference between the emfs of the two

SMs. After a transient, which in reality involves a certain amount of damping, the phase

difference between the emfs of the two SMs is brought back to its initial value (as the

transmitted electric power), and the system is again at steady state.

The synchronization mechanism described above is known to all power system spe-

cialists, i.e., the synchronization process is achieved by means of a transient power trans-

fer. The same kind of synchronization also appears in large systems of interconnected

synchronous machines.

Due to the fact that the synchronous machines can maintain operation in various ac-

network conditions while the vector-current-controlled VSCs are prone to fail, it makes

sense to suggest a control method based on a synchronization process where the electric

power is the communicating medium. In the next sub-section, a controller based on power

synchronization is proposed.

3.3.2 Power-synchronization control of grid-connected VSCs

From the discussion in the preceding sub-section it is known that the SMs in an ac system

maintain synchronism by means of power synchronization, i.e., a transient power trans-

fer. This power transfer involves a current which is determined by the interconnecting

network. Generally, this current is unknown. If power synchronization should be used to

control a VSC, therefore, it cannot be combined with a vector current controller, which

requires a known current reference. As will be shown below, the active power output from

the VSC is instead controlled directly by the power-synchronization loop and the reactive

power (or alternating voltage) is controlled by the magnitude of the VSC voltage. Conse-

quently, an inner current loop is not necessary from a power and voltage control point of

view.

However, as it was mentioned in Section 3.2, besides power and voltage control,

the current controller is also important in

1. Providing damping effects to poorly-damped resonances in ac systems.

2. Limiting the valve current of the converter during severe ac-system faults.

32


To make the essence of power-synchronization control easier to be captured, the ini-

tial proposal of the control law only considers the damping issue, while the fault-current-

limitation issue is proposed as a modification of the initial design.

Accordingly, an initial control design based on the power-synchronization law is

proposed as

• Power-synchronization loop (PSL). The control law is given by

θv =kp

s(Pref − P ). (3.30)

where θv supplies the synchronization input to the VSC, i.e., ωt = ωreft + θv. The

power-synchronization loop is essentially an emulation of the swing equation, how-

ever, not an exact copy. Since the mechanical angular velocity ωm is the derivative

of the angular position, (3.28) represents a double integration when going from

torque (or electric power) to angular position. This double integration, inherently,

yields a poor phase margin even with considerable damping. Therefore, the pro-

posed power-synchronization law in (3.30) employs only a single integration.

• Alternating-voltage control (AVC). The control law is given by

∆V =ku

s(Uref − Uf ). (3.31)

where ∆V gives the change in magnitude of the VSC reference voltage. The AVC

can also be viewed as an emulation of the exciter control of a synchronous ma-

chine. A normal exciter control of a synchronous machine is of proportional type.

However, it is found to be more beneficial to have integral process for the VSC to

suppress high-frequency disturbances. If there are other voltage-controlling devices

connected close to the filter bus, a load compensation should be applied to avoid

voltage hunting. This issue will be discussed in Chapter 6.

• Reactive-power control (RPC). When operating against a weak ac system, the

VSC-HVDC converter should preferably be operated in AVC mode to give the ac

system the best possible voltage support. In case reactive-power control is required,

the output of this controller should be added to the alternating-voltage reference,

and the added amount should be limited. The PI-type controller proposed in (3.2)

can be used for the RPC but with voltage reference change ∆Uref as output.

• Voltage-vector control law. The control law of the voltage vector of the VSC is

proposed as

vcref = (V0 + ∆V ) −HHP (s) icc (3.32)

where V0 is the nominal value, e.g., V0 = 1.0 p.u., and ∆V is given by the AVC.

HHP (s) is a high-pass filter for damping purpose, which is expressed by

HHP (s) =kvs

s+ αv

(3.33)

33


where αv should be chosen low enough to cover all the possible resonances in the

ac system. Typically αv should be chosen between 30 rad/s and 50 rad/s to also

cover the subsynchronous resonance in ac systems. The gain kv determines the level

of the damping effect with typical values between 0.2 p.u. and 0.6 p.u. The effects

of HHP (s) for resonance damping will be further analyzed in Chapter 4.

In the following, a current limitation scheme is proposed as a modification of the

initial design. The principle is to have the control system in current-limitation mode au-

tomatically once the converter current is above the limit.

In case that the converter current is above the maximum limit Imax, the desired

control law of the VSC is the inner-current control law of vector current control in (3.12).

However, instead of giving a constant current reference to (3.12), the value of iref in (3.12)

is given by

iref =1

αcLc

[(V0 + ∆V ) −HHP (s) icc −HLP(s)uc

f − jω1Lcicc

]+ icc. (3.34)

The current reference in (3.34) is designed in such a way that the control law in (3.12)

becomes (3.32) in normal operation. This can be easily verified by substituting (3.34)

into (3.12). However, the current reference iref in (3.34) gives an indication of the actual

converter current. During ac-system faults, current limitation is automatically achieved by

limiting the modulus of iref to the maximum current limit Imax. A brief analysis of this is

given below.

The dynamics of the converter current in the converter dq frame can be described

by

Lcdiccdt

= vc − ucf − jω1Lci

cc. (3.35)

Assuming vc = vcref , substituting (3.12) into (3.35) yields,

Lcdic

dt= αcLc (iref − icc) −

s

s+ αf

ucf . (3.36)

By setting the time derivative and the Laplace operator s to zero, it is found that

iref = icc. (3.37)

That is, the current reference is identical to the actual converter current in the steady state.

In other words, by limiting the modulus of the current reference, the converter current is

limited.

Fig. 3.10 shows the overview of power-synchronization control. The “Current refer-

ence control” block corresponds to the control law described by (3.34), while the “Current

controller” block corresponds to the control law described by (3.12).

34


dqαβ abc

αβ

vθ

nLcL

-+refPP

ciQP,

E

VSC

+

-fu

-+

fU

V∆

v

+

-

fC

Current

controller

Current-

reference

control

ref

di

ref

qi

ref

dv

ref

qv

trefω

tω+

+s

kp

s

ku

PSL

AVCRPC

+

-

+refQ

Q

ref

αv

ref

βv

refU

refU∆

ref

av

ref

bvref

cv

Fig. 3.10 Main-circuit and control block diagram of a VSC-HVDC converter using power-

synchronization control.

3.3.3 Bumpless-transfer and anti-windup schemes

In some situations, power synchronization cannot be applied, and a back-up PLL has to

be used instead. Those situations are:

1. Before the VSC is de-blocked, the back-up PLL provides the synchronization input

to the VSC. After the converter is de-blocked, the PLL is replaced by the PSL.

This procedure is similar to the auto-synchronization process used for synchronous

machines before they are connected to the grid.

2. During severe ac-system faults, the control system has to limit the converter cur-

rent. Thus, the PSL cannot be applied.

According to the control law given by (3.32), the converter dq frame created by the PSL

has the d-axis aligned with the voltage vector v of the VSC. However, the dq frame created

by the PLL has the d-axis aligned with the voltage vector uf of the filter bus. As shown in

Fig. 3.11, the two dq frames differ by an angle θc. Consequently, the dq frame created by

the PLL should be corrected to minimize the “bump” of transfer when the synchronization

input of the VSC is switched from the PSL to the PLL. The angle θc can be derived from

the power-angle equation in (3.27) with the p.u. expresseion

θc = arcsin(PXc

UfV) ≈ arcsin(PXc) (3.38)

35


where Xc is the reactance of the phase reactor. Uf and V are the magnitudes of the filter-

bus voltage and the VSC voltage, which are close to 1.0 p.u. in normal operation. Fig. 3.12

shows the block diagram of the backup PLL, which is essentially the same as the PLL for

vector current control in Fig. 3.6 but with angle correction.

However, if the synchronization input of the VSC is switched to the PLL, the PSL

still controls the angle output θv. At the time when the synchronization input is switched

back from the PLL to the PSL, θv might not necessarily be the same as θPLL. This most

likely happens during severe ac-system faults, where the PSL tends to increase θv rapidly.

In Fig. 3.12, a bumpless-transfer scheme is proposed to handle the above situation. Once

there is an error between θv and θv , the scheme integrates the error and feeds back a term

to cancel the integral action of the PSL. From the scheme, the following relationship is

established

θv =kp

s + γp(Pref − P ) +

γp

s+ γpθv. (3.39)

If γp is sufficiently large, (3.39) yields θv ≈ θv . In other words, once the synchronization

input of the VSC is switched from the PSL to the PLL, the bumpless-transfer scheme will

make sure that the output of the PSL tracks the output of the PLL to avoid the transfer

“bump” when the synchronization input is switched back to the PSL. 1

In the following, the anti-windup issue of the alternating-voltage controller is dis-

cussed. With power-synchronization control, the alternating-voltage controller controls

the d component of the voltage reference vrefd [cf. (3.32)], which is essentially the magni-

tude of the VSC voltage. However, vrefd might be limited by either of the two fundamental

limitations of the VSC:

1. Converter-current limitation. This mainly happens during ac-system faults as it has

been described above.

2. Modulation-index limitation. Occasionally, the magnitude of vrefd is above the max-

imum value of voltage that the VSC can produce. Thus, the control has to limit the

magnitude of vrefd to prevent over-modulating the valve. Modulation-index limi-

tation tends to be reached more often with weak-ac-system connections, where a

relatively higher VSC voltage magnitude is necessary to keep the filter-bus voltage

to be nominal. Usually, the tap-changer of the converter transformer is used to pre-

vent modulation-index limitation. However, it can still occur with transient voltage

swings, or if the tap-changer of the converter transformer has reached the limit of

the tapping range.

The principle of the anti-windup scheme of the alternating-voltage control is basically the

same as the bumpless-transfer scheme of the PSL, as shown in Fig. 3.13. Whatever the

1Interestingly, this is essentially the same problem as transient stability phenomena in ac systems where

synchronous machines accelerate the speed of their rotors during ac-system faults. Thus, the well-known

equal-area criterion might be viewed as an integrator-windup problem. Unfortunately, no bumpless-transfer

or anti-windup scheme can ever be designed for a real machine.

36


Grid-d

fu

v

E

PLL-d

PSL-d

PLL-q

PSL-q

Grid-q

cθ

Angle

correction

Fig. 3.11 Converter dq frames created by the PSL and the PLL. The two dq frames differ by the

angle θc.

vθ

-

+refP

P

tω++

s

kp

'

PLLθ

+-

+

+

vθ

s

pγ

trefω

++PLLθ

cθ

s

KK i

p

PLLPLL +

c

fu}{Im c

fu

PSL

Backup-PLL

Angle

correction

Fig. 3.12 Bumpless-transfer scheme for switching the synchronization input of the VSC.

37


ref

dv

-

+refU

fU

s

ku

+-

+

+

c

cdisHV )(HP0 −

V∆refˆdv+

+

s

Eγ

AVC

Currrent

limitation

Modulation

index

limitation

Fig. 3.13 Anti-windup scheme for the alternating-voltage controller.

reason of the voltage limitation, the anti-windup scheme integrates the error of vrefd and

vrefd , and feeds back a term to cancel the integrator of the alternating-voltage controller. By

choosing γE large enough, the output vrefd tracks the limited d component reference vref

d .

Once the voltage limitation lifts, the alternating-voltage control gets a smooth re-start.

In the following, some simulation results from PSCAD/EMTDC are shown to dem-

onstrate the performance of a VSC-HVDC link using power-synchronization control. The

VSC-HVDC link is connected to a very weak ac system with SCR = 1.0. The converter

station at the other side of the VSC-HVDC link controls the direct voltage, and the con-

verter is assumed to be connected to a strong system. It is also assumed that the bandwidth

of the direct-voltage controller at the other station is high enough such that the variation

of the dc-link voltage has negligible effects on the dynamics of the converter connected

to the weak ac system. The issue of interconnection of two very weak ac systems will be

discussed in Chapter 5, where the direct-voltage control plays a central role.

Fig. 3.14 shows the converter deblocking and power ramping-up process. In the

lower plot, it is shown that the VSC uses the PLL as its synchronization input before the

converter is deblocked. The output of the PSL, θv, is forced to be equal to the output of

the PLL, θ′

PLL . After the converter is deblocked at 0.1 s, the PSL takes over the synchro-

nization input of the VSC, and ramps up the active power to 0.86 p.u. The PLL tracks

the PSL as the power is ramping up, and is exactly identical to the PSL when the system

reaches the steady state.

In Fig. 3.15, a three-phase ac-system fault with duration 0.2 s is applied close to

the filter bus. By detecting the ac-system fault (by current limitation or magnitude drop of

the filter-bus voltage), the control system switches the synchronization input of the VSC

to the backup-PLL. In the middle plot, it is shown that the PSL initially quickly increases

the angle output θv, but after the back-up PLL takes over the synchronization input of the

VSC, the bumpless transfer scheme re-directs the output of the PSL to follow the output

of the PLL. Once the ac-system fault is cleared, the PSL takes over the synchronization

input and brings the power back to the pre-fault level. The PLL again tracks the PSL and

38


0 0.2 0.4 0.6 0.8 1

-1

0

1

Uf (

p.u

.)

0 0.2 0.4 0.6 0.8 1

0

0.5

1P

(p

.u.)

0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5

time (sec)

θv,

θ´ P

LL (

rad

)

PSL

PLL

Fig. 3.14 Converter deblocking and power ramping up of the VSC-HVDC link using power-

synchronization control. Upper plot: three-phase filter-bus voltages. Middle plot: active

power of the VSC-HVDC link. Lower plot: outputs of the PSL and the PLL.

equals the PSL in the steady state.

The lower plot of Fig. 3.15 demonstrates the anti-windup scheme of the alternating-

voltage controller. During the fault period, the d component of the VSC voltage reference

vrefd is limited by the current controller. Similar to the bumpless-transfer scheme of the

PSL, the anti-windup scheme of the alternating-voltage controller also re-directs the volt-

age output of the AVC to follow the limited voltage reference. After the fault is cleared,

the AVC takes over the voltage control.

Fig. 3.16 shows the actions of the current controller during the fault. After the three-

phase ac-system fault is applied at 0.1 s, the modulus of the current reference [|iref | =√(irefd )2 + (irefq )2] reaches the current limit Imax. The control system seamlessly switches

to the control law (3.12), where the limited irefd and irefq become the inputs to the current

controller. Only a short current spike is observed on the valve current |ic| at the fault

occurrence stage, which usually does no harm to the converter valve. After detecting the

faults, the current controller reduces the valve current to half of the maximum load current

Imax or any other desired values to minimize the short-circuit current contribution to the

ac system. After the fault is cleared at 0.3 s, the current limitation lifts, and the voltage-

vector control law is back to (3.32).

As shown by the time simulation in this section, power-synchronization control,

in fact, uses vector current control and the PLL during severe ac-system faults, since

the power-synchronization law is not applicable. A question is naturally raised: if vector

39


0 0.2 0.4 0.6 0.8 1

-1

0

1

Uf (

p.u

.)

0 0.2 0.4 0.6 0.8 1

0

1

2

3

θv,

θ´ P

LL (

rad

)

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

time (sec)

vre

f

d (

p.u

.)

PSL PLL

Controller vref

d

Limited vref

d

Fig. 3.15 Bumpless-transfer and anti-windup schemes of power-synchronization control during

a three-phase ac-system fault. Upper plot: three-phase filter-bus voltages. Middle plot:

outputs of the PSL and the PLL. Lower plot: d component of the VSC voltage reference.

0 0.2 0.4 0.6 0.8 1

0

0.5

1

P (

p.u

.)

0 0.2 0.4 0.6 0.8 1-1

0

1

i re

f

d, i r

ef

q (

p.u

.)

0 0.2 0.4 0.6 0.8 10

1

2

time (sec)

| i c

| (p

.u.)

i ref

d

i ref

q

Fig. 3.16 Current limitation of power-synchronization control during a three-phase ac-system

fault. Upper plot: active power of the VSC-HVDC link. Middle plot: dq components

of the current reference. Lower plot: modulus of the converter current.

40


current control does not work well with weak-ac-system connections, could instability

occur during ac-system faults?

As will be shown in Chapter 5, the stability of the current controller deteriorates

with weak-ac-system connections but to an acceptable level. What is not acceptable is

using current references to control the active power and the alternating voltage. The justi-

fication of using power-synchronization control is that it is meaningless to just control the

current during normal operation. However, it is enough during severe ac-system faults.

As shown in Fig. 3.16, vector current control works fairly stable during ac-system faults

given the fact that the ac system only has SCR = 1.0.

VSCs using power-synchronization control basically emulate the operation of a

synchronous machine. Therefore, it contributes with short-circuit capacity to the ac sys-

tem at the PCC. In order to obtain a simple estimate of the impact of the VSC on the

short-circuit capacity of ac system, the effects of the alternating-voltage control and the

ac filter can be disregarded. Doing so, the short-circuit capacity of the ac system including

the VSC at the PCC can be expressed as

Sac =U2

f0(Ln + Lc)

ω1LnLc

. (3.40)

However, VSCs do not necessarily increase short-circuit currents to the ac system during

ac-system faults thanks to the current limitation function.

3.3.4 Negative-sequence current control

In reality, there is often phase unbalance in the three-phase voltages of the ac system. The

phase unbalance may appear in the steady state due to unevenly connected loads in the

ac system, but more often caused by unbalanced voltage dips [34]. The zero-sequence

voltage in the ac system cannot pass through the converter transformer where Y −∆ type

connection is commonly used, but the negative-sequence voltage can create unbalanced

converter currents, which may overload the converter and produce double-grid-frequency

ripples on the direct voltage [58]. In this section, a negative-sequence current controller is

proposed to mitigate the phase-unbalance problem.

Fig. 3.17 shows the control block diagram of the proposed negative-sequence cur-

rent controller, which is principally the same as the positive-sequence current controller in

Fig. 3.5 and Fig. 3.10 but with negative-sequence variables. The outputs of the negative-

sequence current controller vrefa−, vref

b− and vrefc− are directly added to the outputs of the

positive-sequence current controller vrefa , vref

b and vrefc . Sometimes, the two current con-

trollers together are referred to as dual vector current controller [59, 60]. The minus sign

in the subscripts of the variables in Fig. 3.17 represents the negative-sequence quantities.

However, the dq frame of the negative-sequence current controller rotates in the opposite

direction to the converter dq frame. Therefore, −ωt is used as the synchronization input.

The proposed control law of the negative-sequence current controller is expressed as

41


dqαβ abc

αβ

ref

−avref

−bvref

−cv

Inner-

current

controller

ref

−di

ref

−qi

ref

−dv

ref

−qv

tω−

−fu−ci

ref

−αv

ref

−βv

Fig. 3.17 Negative-sequence current controller.

vref−

= αc−Lc

(iref−

− icc−)

+ jω1Lcicc− + uc

f− (3.41)

where αc− is the desired bandwidth of the negative-sequence current controller. Eq. (3.41)

is essentially the same expression as (3.12) but with negative-sequence variables. The

negative-sequence current reference iref−

is usually set to zero if there is no other special

purpose.

Thus, the principle of the negative-sequence current controller is to create phase

unbalance on the VSC voltages so as to minimize the negative-sequence current flowing

through the converter. There are different ways to obtain negative-sequence variables. The

following is a common approach [61].

In a stationary reference frame, if the zero-sequence component is disregarded, a

complex vector ys(t) can be expressed as

ys(t) = Y+ejωt

︸︷︷︸ys

+(t)

+ Y−e−jωt

︸︷︷︸ys−

(t)

(3.42)

where Y+ is the magnitude of the positive-sequence vector ys+(t), while Y− is the mag-

nitude of the negative-sequence vector ys−(t). The quantity ω is the angular speed of the

rotating vector ys+(t). It follows from (3.42) that a voltage vector delayed with a quarter

period (Tp/4) and multiplied by j is

jys(t− Tp/4) = j[Y+ejω(t−Tp/4) + Y−e

−jω(t−Tp/4)] (3.43)

or

jys(t− Tp/4) = Y+ejωt − Y−e

−jωt. (3.44)

Accordingly, the negative sequence vector ys−(t) can be obtained from (3.42) and (3.44)

as

ys−(t) = Y−e

jωt =1

2[ys(t) − jys(t− Tp/4)]. (3.45)

Fig. 3.18 demonstrates the effect of the negative-sequence current controller by time

simulations in PSCAD/EMTDC. At 0.1 s, a negative-sequence voltage source with 10%

magnitude of the positive-sequence voltage is added on the ac source E, which makes the

42

3.4. Summary

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

-1

0

1

Uf (

p.u

.)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

-0.2

-0.1

0

0.1v

re

f

d-

, v

re

f

q-

(p

.u.)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

0.4

0.6

0.8

time (sec)

| i c

| (p

.u.)

v ref

q-

v ref

d-

Fig. 3.18 Effects of negative-current control in the steady state. Upper plot: three-phase filter-bus

voltages. Middle plot: dq components of the negative-sequence VSC voltage reference.

Lower plot: modulus of the converter current.

converter current unsymmetrical. The lower plot in Fig. 3.18 shows the modulus of the

converter-current vector in the converter frame that oscillates with double-grid frequency

caused by the negative-sequence component. If the operating point of the VSC is close to

the maximum valve-current limit, this phase unbalance might overload the converter. At

0.3 s, the negative-sequence current controller is activated, which successfully removes

the negative-sequence current component. Fig. 3.19 and Fig. 3.20 show the effect of the

negative-sequence current controller during a single-line-to-ground fault in the ac system.

With the negative-current controller activated, as shown in Fig. 3.20, the unbalance of

valve currents is much reduced.

It should be noted that, since unbalanced VSC voltage need to be produced in order

to reduce the negative-sequence current, the magnitudes of some phases of the VSC volt-

age might go above the modulation-index limitation. This is the reason why there is still

remaining negative-sequence component on the converter current during the unbalanced

fault even with the negative-sequence current controller applied in Fig. 3.20.

3.4 Summary

In this chapter, two existing control methods, i.e., power-angle control and vector current

control, for VSC-HVDC systems are reviewed, and a novel control method, i.e., power-

synchronization control, is proposed. Power-angle control is simple and straightforward

to implement. However, power-angle control has no general means to damp resonances in

43


0 0.2 0.4 0.6 0.8 1

-1

0

1

Uf (

p.u

.)

0 0.2 0.4 0.6 0.8 1-1

0

1

i re

f

d, i r

ef

q (

p.u

.)

0 0.2 0.4 0.6 0.8 10

1

2

time (sec)

| i c

| (p

.u.)

i ref

di ref

q

Fig. 3.19 Single-line-to-ground fault without the negative-sequence current controller applied.

Upper plot: three-phase filter-bus voltages. Middle plot: dq components of the current

reference. Lower plot: modulus of the converter current.

0 0.2 0.4 0.6 0.8 1

-1

0

1

Uf (

p.u

.)

0 0.2 0.4 0.6 0.8 1-1

0

1

i re

f

d, i r

ef

q (

p.u

.)

0 0.2 0.4 0.6 0.8 10

1

2

time (sec)

| i c

| (p

.u.)

i ref

di ref

q

Fig. 3.20 Single-line-to-ground fault with the negative-sequence current controller applied.

44

3.4. Summary

the ac system and the converter may get over current during ac-system faults. Vector cur-

rent control is the most popular control method used for grid-connected VSCs. However,

for HVDC applications, vector current control has poor performance for weak-ac-system

connections. VSCs using power-synchronization control basically resemble the dynamic

behavior of a synchronous machine. Several additional functions, such as high-pass cur-

rent control, current limitation, etc. are proposed to deal with various practical issues

during operation.

45


46

Chapter 4

Dynamic Modeling of AC/DC Systems

This chapter discusses the dynamic modeling issue for ac/dc systems. In Section 4.1, a

new concept, i.e., the so-called Jacobian transfer matrix, is introduced for modeling of ac

systems. By the proposed modeling concept, an synchronous ac system is viewed upon as

one multivariable feedback control system where the Jacobian transfer matrix is defined

as the controlled process. In Section 4.2 and Section 4.3, the Jacobian transfer matrix con-

cept is applied to model grid-connected VSCs using power-synchronization control and

vector current control. In Section 4.4, the Jacobian transfer matrix concept is extended to

modeling of dc systems. The properties of the Jacobian transfer matrix are summarized in

Section 4.5. The major results of this chapter are summarized in Section 4.6. Some results

of this chapter are included in [62, 63].

4.1 Jacobian transfer matrix for ac-system modeling

In this section, a new concept for dynamic modeling of ac systems is described. In Sec-

tion 4.1.1, the existing concepts for power-system stability and dynamic modeling are re-

viewed. In Section 4.1.2, the Jacobian transfer matrix is introduced as a unified dynamic-

modeling concept for ac systems.

4.1.1 Power-system stability and dynamic modeling

Power-system stability has been a major subject for power engineering for many years.

The definitions of power-system stability and dynamic-modeling techniques have contin-

uously been extended with the evolvement of power systems [32, 64–66].

In the early days, the stability problem was one of maintaining synchronous oper-

ation. Thus, transient stability was the most classical stability problem of concern. Tran-

sient stability is defined as the ability of a power system to maintain synchronism when

subjected to severe transient disturbance [67]. Transient stability is basically an issue of

energy balance of the rotors of synchronous machines, where the equal-area criterion is

47

Chapter 4. Dynamic Modeling of AC/DC Systems

often used to describe the mechanism.

Small-signal stability is another stability phenomenon related to synchronous ma-

chines. In today’s power systems, small-signal stability is largely a problem of insufficient

damping of oscillations [32]. The fast excitation control has been the traditional culprit

for reduced or even negative damping torque [68]. Insufficient damping may also be a

consequence of large power systems interconnected by weak ties [69]. The power-system

stabilizer (PSS) are generally considered as an easy and effective solution for small-signal

stability issues, at least for the local-mode oscillations [70]. HVDC systems and flexible

ac transmission systems (FACTS) have also been shown to be effective in damping inter-

area oscillations [4, 28, 71]. For small-signal stability, modal analysis based on linearized

ac-system models is the most popular tool for analyzing the phenomenon and evaluating

various damping solutions [32].

The above two forms of stability are mainly related to the dynamic behavior of

synchronous machines. Consequently, they are often referred to as “rotor-angle stability”.

Voltage stability was identified as a problem in the 1970s. Voltage stability is de-

fined as the ability of a power system to maintain acceptable voltage at all buses in the

system under normal operating conditions and after being subjected to a disturbance [67].

Voltage stability is often caused by lack of reactive-power support in heavy loaded ar-

eas [66]. In contrast to rotor-angle stability, voltage stability is more related to the dy-

namic behavior of loads and their interactions with the ac system. Dynamic load modeling

is generally believed to be of central importance for voltage-stability study [72, 73].

For voltage-stability analysis, an interesting methodology was proposed, i.e., the

Jacobian matrix. The Jacobian matrix was originally applied for solving power equations

using the Newton-Raphson algorithm in power-flow programs. It has been found that the

singularities of the Jacobian matrix have close relationships with voltage-stability prob-

lems. For instance, it can easily be shown that the critical points on the well-known P −Vcurves exactly correspond to the operation conditions when the Jacobian matrix becomes

singular. Based on the Jacobian matrix, a modal analysis technique that is similar to the

one used for small-signal analysis was developed [74]. However, the applications of the

Jacobian matrix and other related methods in voltage-stability analysis have been ques-

tioned for their mathematical foundations [75]. Voltage instability, like any other kind

of instability, is a dynamic phenomenon. A rigorous analysis, therefore, would require a

dynamic system formulation and the application of appropriate dynamic criteria, but the

Jacobian matrix is merely a static matrix that describes the power-flow deviations related

to voltage and angle variations in the system at a certain operating point.

One argument to support the Jacobian matrix and the related methods is that the Ja-

cobian matrix might be viewed as a dynamic description of the ac system if the frequency

range of the stability of concern is “quasi-static” [76]. As it will be shown later by the

Jacobian transfer matrix modeling technique introduced by this thesis, such an explana-

tion does make sense. However, the question is, if the Jacobian matrix can be viewed as

48

4.1. Jacobian transfer matrix for ac-system modeling

a dynamic description of the ac system that is valid in the “quasi-static” frequency range,

then what is the dynamic description of the ac system that is valid in the whole frequency

range, and how are they related mathematically ?

The subsynchronous resonance (SSR) problem appeared also in the 1970s. The first

SSR problem was experienced in 1970 resulting in the failure of a turbine-generator shaft

at the Mohave plant in southern California, USA [32]. In the beginning, it was believed

that SSR occurs only in series-compensated transmission systems, where the long shaft of

the rotor of thermal units may interact with the series capacitor at a frequency lower than

the fundamental frequency. However, it was later discovered that similar interactions can

also occur with an HVDC converter [77]. Therefore, subsynchronous torsional interaction

(SSTI) is suggested to be a more general definition [78].

For rotor-angle and voltage stability, the phasor theory is the major mathematical

tool for dynamic modeling and analysis. With the phasor approach, electromagnetic tran-

sients of the ac network are neglected. Such a simplification is acceptable for rotor-angle

and voltage stability but not for SSTI, since the dynamic frequency range of SSTI is

much higher than rotor-angle and voltage stability. For analyzing SSTI, the electromag-

netic transients of the ac network have to be properly represented. Thus, the space-vector

theory is commonly applied to formulate the dynamic model of the ac network.

There are predominantly two methods for SSTI analysis: eigenvalue analysis and

frequency scanning [22, 79, 80]. Eigenvalue analysis is a rigorous tool for power-system

modeling, since the whole power system is modeled as one single state-space representa-

tion. The computation of eigenvalues and eigenvectors is an excellent method of providing

crucial information about the nature of the power system [21]. The major drawback of the

eigenvalue analysis is its complexity once the order of the system is high. The frequency-

scanning method is based on the complex-torque concept [81]. The risk of SSTI can thus

be evaluated by plotting the damping-torque curve of the generator in the frequency do-

main. Although frequency scanning is a frequency-domain concept, the damping-torque

curve is also possibly obtained from time-simulation programs by perturbing the rotor

speed with a small sweeping-frequency sinusoidal signal. As the aforementioned reason,

the time-simulation program used for frequency scanning should be of electromagnetic-

transient type, such as EMTP or PSCAD/EMTDC.

In the foreseeable future, the main challenges for power-system stability analysis

and dynamic modeling are imposed by high power-electronic devices, such as HVDC

systems and FACTS if the number of such devices is considerably increased in power

systems. On the one hand, the ratings of those devices are often high enough to have a

significant impact on all the above-mentioned stability phenomena. On the other hand,

their dynamic frequency range are much higher than the traditional power-system com-

ponents. They might interact with power systems and with each other in a more complex

way.

From the above brief review of power-system stability and the related dynamic-

49


1K

5K

3K

4K

Jacobian transfer

matrix )(sJ

Induction

motor

FACTSHVDC

system

Constant

power load

Generator

2K

Fig. 4.1 An ac network connected to various input devices.

modeling techniques, it can be concluded:

1. Due to historical and practical reasons, power-system stability has traditionally

been defined mainly based on physical phenomena, but the mathematical relation-

ships between different forms of stability are not rigourously clarified. The dynamic

modeling techniques are also versatile for various forms of stability.

2. For any dynamic system, the stability of the system is usually closely related to

feedback control. However, power-system stability has rarely been interpreted from

the feedback-control point of view.

3. High power-electronic devices might interact with the ac system and each other

in a more complex way than conventional power components. New modeling and

analysis techniques are required to meet the challenges in the future.

In the next sub-section, the Jacobian transfer matrix is introduced as a unified ac-system

dynamic modeling technique to address these issues.

4.1.2 Feedback-control view of power systems

The fundamental idea of the Jacobian transfer matrix modeling is that a synchronous

power system is modeled as one multivariable feedback-control system, where the feed-

back controllers and the controlled process of the power system are explicitly defined.

As shown in Fig. 4.1, for a normal power system, there are various power components

50

4.1. Jacobian transfer matrix for ac-system modeling

+-

yr)(sK )(sJ

Fig. 4.2 Feedback-control view of a power system.

connected to it, such as generators, induction motors, HVDC systems, FACTS devices,

and loads, etc. The controllers, as will be defined later, are pulled out from those power

components and form a controller transfer matrix as

K(s) =

K1(s)

K2(s)

. . .

Kn(s)

and the rest of the power system, i.e., the ac network and the electrical parts of those

power components form another transfer matrix J(s). A synchronous power system thus

is modeled by the feedback-control system shown in Fig. 4.2. The transfer matrix J(s) is

called Jacobian transfer matrix in this thesis, i.e., the Jacobian transfer matrix is defined

as the controlled process of a power system.1 With the proposed modeling concept, the

stability of a power system is uniquely defined as the stability of the closed-loop system

formed by the controller transfer matrix K(s) and the Jacobian transfer matrix J(s). Some

terminologies related to the proposed modeling concept are defined in below:

Input device: Any power component connected to a power system which has a feedback-

control property. The aforementioned synchronous generators, induction motors,

HVDC systems, and FACTS are all treated as input devices. An input device con-

sists of two parts: the electrical part and the controller.

AC network: The passive power components in a power system, such as transmission

lines, transformers, line inductors, shunt capacitors, resistive-inductive-capacitive

(RLC) loads, etc.

Controller transfer matrix: A transfer matrix formed by all the controllers from each in-

dividual input device. The controller transfer matrix is usually a diagonal or block-

diagonal transfer matrix unless there is cross controls between the input devices.

The controller as defined in this thesis is a generalized terminology. For example,

the rotors of synchronous generators and induction motors are treated as controllers

even though physically they are not intentionally implemented controllers as such.

1The name “Jacobian ” will be justified in the following sections.

51


refU

-

+

refP P∆vθ∆

fU∆

+-

)(sJ

s

kp

PSL

s

ku

AVC

Jacobian

transfer

matrix

0V

V∆

Fig. 4.3 Closed-loop system for grid-connected VSCs using power-synchronization control

Jacobian transfer matrix: A transfer matrix represents the ac network and the electri-

cal parts of the input devices. One exception is vector control of grid-connected

VSCs, which is also formed into the Jacobian transfer matrix for mathematical con-

venience even though they are not electrical.

In the next few sections, the Jacobian transfer matrix concept will be applied for mod-

eling of grid-connected VSCs connected to some simplified ac-network configurations,

where the VSC is the only input device of the ac system. In Chapter 6, some other input

devices, such as synchronous generators and induction motors, are also modeled to create

a complete linear model for an island system with several input devices.

4.2 Grid-connected VSCs using power-synchronization con-

trol

In this section, the Jacobian transfer matrix is developed for grid-connected VSCs using

power-synchronization control. Fig. 4.3 shows the closed-loop system for grid-connected

VSCs using power-synchronization control, i.e., the Jacobian transfer matrix is the con-

trolled process which represents the ac circuit of the VSC and the ac network.

4.2.1 Impedance-source neglecting the ac capacitor at the filter bus

Fig. 4.4 shows the main circuit of a VSC connected to an ac network represented by a

simple impedance source. Lc andRc are the inductance and resistance of the phase reactor

of the VSC, and Ln and Rn are the inductance and resistance of the ac system. Cf is the

ac capacitor connected at the filter bus. The bold letter symbols E, uf , and v represent the

voltage vectors of the ac source, the filter bus, and the VSC respectively. E0, Uf0, and V0

are their corresponding voltage magnitudes. The ac source, which is a constant-frequency

stiff voltage source, is used as the voltage reference, and the phase angles of uf and v are

52

4.2. Grid-connected VSCs using power-synchronization control

+

-

++

-

VSC

-

nL cL

Ef

u

QP,

vfC

nRcR

00 vV θ∠0

0 0∠E

cini

00 ufU θ∠

Fig. 4.4 Main circuit of a VSC connected to an impedance ac source.

R LX 1ω= I

QP,

0∠= EE θ∠= VV

Fig. 4.5 Equivalent circuit by neglecting the ac capacitor for angle-stability analysis.

θu0 and θv0 respectively. P and Q are the active and reactive powers from the VSC to the

ac system. The quantity ic is the current vector of the phase reactor, and in is the current

vector to the ac source. At the dc side, the VSC is assumed to connect to a stiff direct-

voltage source. The ac capacitor at the filter bus of the VSC is neglected in the analysis in

this subsection. Such a simplification is useful for explaining some basic concepts of the

proposed modeling technique in an analytical way. Especially, it is easier to demonstrate

how angle and voltage stability defined in the classical power-system theory can be cast

into the Jacobian transfer matrix modeling frame work.

Stability analysis by the phasor theory

In the following, the stability of the small power system is analyzed by the classical phasor

theory. Fig. 4.5 shows the equivalent circuit of Fig. 4.4 by neglecting the ac capacitor,

where E and V represent the phasor of the ac source and the VSC. R and X are the

equivalent resistance and reactance between the ac source and the VSC. Neglecting the

resistance, the angle stability of the equivalent system is given by the well-known power-

angle equation

P =EV

Xsin θ. (4.1)

From (4.1) it follows that the power that can be transmitted to/from the VSC is limited

by the fact that −1 ≤ sin θ ≤ 1. Fig. 4.6 shows the power-angle curve with different line

reactances.

For voltage-stability analysis, suppose that the VSC controls the reactive power to

the ac system. Neglecting the resistance, the reactive power Q can be expressed by the

53


0 50 100 150 2000

0.5

1

1.5

2

2.5

3

3.5

4

X = 1 p.u.

X = 0.6 p.u.

X = 0.3 p.u.

θ (deg.)

P (

p.u

.)

Fig. 4.6 Power-angle curves with different line reactance.

following expression:

Q =V 2 −EV cos θ

X. (4.2)

By eliminating the angle θ in (4.1) and (4.2), the following equation is derived:

(V 2)2 −

(2QX + E2

)V 2 +X2

(P 2 +Q2

)= 0. (4.3)

To get real solutions of V 2 in (4.3), the required condition is

(2QX + E2

)2 − 4X2(P 2 +Q2

)≥ 0. (4.4)

If (4.1) and (4.2) are substituted back into (4.4), as

V ≥ E

2 cos θ. (4.5)

If the condition (4.5) is fulfilled, two solutions of V in (4.3) are obtained

V =

√E2

2+QX ±

√E4

4−X2P 2 +XE2Q. (4.6)

The relation in (4.6) is plotted in Fig. 4.7 where

tanφ = − Q

|P | . (4.7)

These plots are commonly called P − V curves, or “nose curves”. Besides the P − V

54


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

tanφ= 1.5

tanφ= 0.4

tanφ= 0

tanφ= -0.2

|P|X/E2

V/E

tanφ= -0.5

Fig. 4.7 P − V curves with different tan φ of the load.

curve, another way to determine the voltage-stability limit is by the singularity of the

power-flow Jacobian matrix [74]. Jacobian matrix J is expressed as

∆P

∆Q

=

[JPθ JPV

JQθ JQV

]

︸︷︷︸J

∆θv

∆VV

(4.8)

where ∆P , ∆Q, ∆θv, and ∆V are the incremental changes in the active power, the reac-

tive power, the phase angle, and the magnitude of the bus respectively. JPθ, JPV, JQθ, JQV

are the partial derivatives of the power-flow equations with respect to the voltage angle

and magnitude of the bus.

For the simple ac system in Fig. 4.5, the Jacobian matrix J at the VSC bus can be

derived analytically from the active-power and reactive-power expressions (4.1) and (4.2).

Accordingly,

JPθ =∂P

∂θ=EV cos θ

X

JPV =∂P

∂VV =

EV sin θ

X

JQθ =∂Q

∂θ=EV sin θ

X

JQV =∂Q

∂VV =

2V 2 − EV cos θ

X. (4.9)

55


The singularity of the Jacobian matrix J is satisfied if its determinant equals zero, i.e.,

det [J] = JPθJQV − JPVJQθ = 0 (4.10)

giving

V =E

2 cos θ. (4.11)

Comparing (4.11) to (4.5), it is easily found that the operating conditions where the Ja-

cobian matrix becomes singular exactly correspond to the “critical voltage” points on the

P − V curves in Fig. 4.7.

Stability analysis by the Jacobian transfer matrix approach

In the next, the ac system and the VSC in Fig. 4.4 is modeled by the proposed Jacobian

transfer matrix concept. As mentioned before, VSCs have two feasible operation modes,

i.e., alternating-voltage control and reactive-power control. For the two operation modes,

the process models can be written in the following input-output form:

∆P

∆Uf

=

[JPθ (s) JPV (s)

JUf θ (s) JUf V (s)

]

︸︷︷︸JPU(s)

∆θv

∆VV0

∆P

∆Q

=

[JPθ (s) JPV (s)

JQθ (s) JQV (s)

]

︸︷︷︸JPQ(s)

∆θv

∆VV0

(4.12)

where the definitions of ∆P , ∆Q, ∆θv , and ∆V , in fact, are the same as previously

defined for the power-flow Jacobian matrix (4.8), and ∆Uf is the incremental change of

the voltage magnitude of the filter bus. The transfer matrices JPU (s) and JPQ (s) are the

Jacobian transfer matrices for the two operation modes. JPQ (s) has the exact input and

output variables as the Jacobian matrix in (4.8). However, a fundamental difference is that

Jacobian transfer matrix is a dynamic description of the ac system in all frequency range,

while Jacobian matrix is only valid in the “quasi-static” frequency range.

Appendix B.1 gives the detailed procedure for deriving the transfer functions in

(4.12) by applying the space-vector theory. The six transfer functions of JPU (s) and

JPQ (s) generally have the following form

Jxx (s) =a0s

2 + a1s+ a2

(sL+R)2 + (ω1L)2 (4.13)

56


where the a-coefficients are summarized in Table 4.1. The k parameters are defined as

k1 = V0Uf0 cos (θv0 − θu0)

k2 = V0Uf0 sin (θv0 − θu0)

k3 = E0V0 cos θv0

k4 = E0V0 sin θv0. (4.14)

Table 4.1 Coefficients of the transfer functions in the Jacobian transfer matrices.

a0Lω1

(k3 − k1)

JPθ (s) a1Rω1

(k3 − k1) − Lk2

a2 ω1Lk3 − Rk4

a0Lω1

(k4 − k2)

JQθ (s) a1Rω1

(k4 − k2) − Lk1

a2 ω1L (k4 − 2k2) +R (k3 − 2k1)

a0 −LLnk2/Uf0

JUf θ (s) a1 −RLnk2/Uf0

a2 −ω21LLnk2/Uf0 − RLnk1/Uf0

a0Lω1

(k4 − k2)

JPV (s) a1Rω1

(k4 − k2) + Lk1

a2 ω1Lk4 +Rk3

a0Lω1

(k1 − k3)

JQV (s) a1Rω1

(k1 − k3) − Lk2

a2 ω1L (2k1 − k3) +R (k4 − 2k2)

a0 LLnk1/Uf0

JUf V (s) a1 RLnk1/Uf0

a2 ω21LLnk1/Uf0 −RLnk2/Uf0

It should be noted that the output variables of the transfer functions derived for

the Jacobian transfer matrices are the active power, the reactive power and the voltage

magnitude at the filter bus. To compare with the results from the phasor analysis, it is

necessary to substitute Uf0 = V0 and θu0 = θv0 into the coefficients in Table 4.1, i.e.,

the condition for the phasor analysis is considered as a special case that the filter bus is

identical to the VSC bus.

57


By substituting Uf0 = V0 and θu0 = θv0 into the transfer functions of JPQ (s) , and

further assuming R = 0 and s = 0, the four transfer functions JPθ (s) , JPV (s) , JQθ (s),

and JQV (s), are identical to the four elements of the Jacobian matrix J defined in (4.8).

In other words, the power-flow Jacobian matrix is the static form of the Jacobian transfer

matrix JPQ (s) that is valid in the “quasi-static” frequency range.

Eq. (4.13) shows that all the six transfer functions have a pair of resonant (complex)

poles

p1,2 = −RL

± jω1. (4.15)

and two zeros. The resonant poles will be further discussed in the next subsection. In

this subsection, the transmission zeros of JPU(s) and JPQ(s) are analyzed, since their

locations have close relationships with angle and voltage stability defined by the phasor

approach.

The computation of the transmission zeros for a general multi-input multi-output

(MIMO) process is a complex issue [82]. However, for a square matrix, the transmission

zeros can be obtained simply by its determinant, i.e., the transmission zeros of JPU (s)

are the values of s = z that satisfy

det [JPU (s)] = JPθ (s) JUf V (s) − JPV (s) JUf θ (s) = 0. (4.16)

If the resistance R is neglected, the solutions to (4.16) are given by

z1,2 = ±ω1

√E0 cos θu0

Uf0 −E0 cos θu0. (4.17)

Graphically, the locations of the transmission zeros of JPU (s) can be divided by the fol-

lowing borders:

• The border where JPU (s) gets zeros at the origin. This is equivalent to

E0 cos θu0 = 0 (4.18)

giving θu0 = ±90◦.

• The border where JPU (s) gets zeros at infinity. This is equivalent to

Uf0 − E0 cos θu0 = 0 (4.19)

giving

θu0 = ± arccos

(Uf0

E0

). (4.20)

• The border where JPU (s) gets zeros at ±ω1. This is equivalent to

E0 cos θu0

Uf0 − E0 cos θu0= 1 (4.21)

58


0 0.5 1 1.5 2-100

-80

-60

-40

-20

0

20

40

60

80

100

Uf0

/E0

θu

0 (

de

g.) Infinity border

Origin border

Origin border

±ω1 border

Fig. 4.8 Locations of the transmission zeros of JPU (s) .

giving

θv0 = ± arccos

(Uf0

2E0

). (4.22)

The ±ω1 border gives an idea about how much the zeros limit the achievable band-

width of the control system, even though it is not a “real” border.

From Fig. 4.8, it is easily observed that the transmission zeros of JPU (s) depend very

much on the operating points. Both Uf0/E0 and θu0 affect their locations. However, within

moderate voltage levels, e.g., 0.8 ≤ Uf0/E0 ≤ 1.2, they are affected mainly by θu0. The

tendency is that, with higher θu0, the zeros get closer to the origin border. In control theory,

a process which has right-half plane (RHP) zeros is called non-minimum-phase system.

From a feedback-control perspective, the RHP zero of the process causes an additional

time delay, which imposes a fundamental limitation on the achievable bandwidth of the

control loop [82], i.e., the closed-loop system cannot achieve a higher bandwidth than

the location of the RHP zero. When the RHP zero reaches the origin, i.e., θu0 = ±90◦,

it causes 180◦ phase shift even in the steady state, which means that tight control at low

frequencies is not possible [82]. If θu0 is replaced by θv0 in (4.18), it is easily found

that the operating conditions where the transmission zeros of the Jacobian transfer matrix

JPU (s) reach the origin exactly correspond to the angle-stability limit defined by the

phasor approach.

From the above analysis, it can be clearly seen that the ac system is, in fact, a

non-minimum-phase system, where the angle-stability limitation is a consequence of the

non-minimum-phase effect. How can this phenomenon be interpreted physically? A brief

59


Real

Imaginary

fU '

fU

V

E

ILjω

'ILjω'V

θ∆

Instant

I

'I Slow

Fig. 4.9 Non-minimum-phase phenomena hidden by the phasor diagram.

analysis of the physical mechanism is given below.

Fig. 4.9 shows the phasor diagram of the ac system, where the resistance is ne-

glected. Now, suppose that the voltage vector of the VSC rotates with the phase angle

∆θv and its magnitude is kept constant during the angle rotation. The current magnitude

|I| naturally increases to |I ′| in response to such angular rotation, but the voltage magni-

tude along the ac line drops instead, e.g., |U ′

f | < |Uf |. This can be easily observed from

the phasor diagram but it can also be proven by the transfer function JUf θ (s) in (B.28).

However, the phasor diagram hides one important fact, i.e., the voltage change is abrupt,

but the current does not increase instantly due to the inductance of the ac line.

Similar to JPU (s), the transmission zeros of JPQ (s) are the values of s = z that

satisfy

det [JPQ (s)] = JPθ (s) JQV (s) − JPV (s) JQθ (s) = 0. (4.23)

If the resistance R is neglected, the solutions to (4.23) are given by

z1,2 = ±ω1

√2E0Uf0 cos θu0 −E2

0

E20 + U2

f0 − 2E0Uf0 cos θu0. (4.24)

The locations of the transmission zeros of JPQ (s) can also be divided by the following

borders:

• The border where JPQ (s) gets transmission zeros at the origin. This is equivalent

to

2E0Uf0 cos θu0 − E20 = 0 (4.25)

giving

cos θu0 =E0

2Uf0. (4.26)

60


• The border where JPQ (s) gets transmission zeros at ±ω1. This is equivalent to

2E0Uf0 cos θu0 − E20

E20 + U2

f0 − 2E0Uf0 cos θu0

= 1 (4.27)

giving

θu0 = ± arccos

(E0

2Uf0+Uf0

4E0

). (4.28)

The ±ω1 border gives an idea about how much the transmission zeros limit the

achievable bandwidth of the VSC controller.

From Fig. 4.10, it is easily observed that the locations of the transmission zeros of

JPQ (s) also depend on the operating points, in the same way as for JPU (s). But besides

the dependency on the phase angle, the locations of the transmission zeros of JPQ (s) also

strongly depend on the voltage magnitude of the filter bus. The tendency is that the zeros

are closer to the origin border with higher θu0 and lower Uf0/E0.

For the special condition where the filter bus is identical to the VSC bus, the trans-

mission zeros of JPQ (s) become

s = ±ω1

√2E0V0 cos θv0 −E2

0

E20 + V 2

0 − 2E0V0 cos θv0. (4.29)

Accordingly, the origin border of JPQ (s) becomes

cos θv0 =E0

2V0. (4.30)

Comparing (4.30) with (4.5) and (4.11), it is easily found that the operating condi-

tions where the transmission zeros of JPQ (s) reach the origin exactly correspond to the

voltage-stability limit defined by the phasor approach.

By comparing Fig. 4.8 with Fig. 4.10, it can be found that within moderate voltage

levels, the transmission zeros of JPQ (s) are closer to the origin than the transmission

zeros of JPU (s), which implies that the achievable bandwidth of the VSC is higher in

alternating-voltage control mode than in reactive-power control mode.

To interpret the physical mechanism of voltage stability, the phasor diagram in

Fig. 4.9 can still be used. Different from in alternating-voltage control mode, in reactive-

power control mode, the voltage magnitude of the VSC is no longer held constant during

the phase-angle rotation. On the contrary, in order to keep the reactive power constant, the

VSC decreases the voltage magnitude following the phase-angle rotation. This behavior

obviously makes the non-minimum-phase effect worse, and the effect also becomes more

dependent on the VSC voltage magnitude.

Remark 1: The above analysis clearly shows the close relationships of the trans-

mission zeros of the Jacobian transfer matrices with angle and voltage stability in power

61


0 0.5 1 1.5 2 2.5 3 3.5

-80

-60

-40

-20

0

20

40

60

80

Uf0

/E0

θu

0 (

de

g.)

Origin border

±ω1 border

Fig. 4.10 Locations of the transmission zeros of JPQ (s).

systems. It also explains why the singularity of the power-flow Jacobian matrix can be

used as a criterion to determine voltage instability in power systems. But, strictly speak-

ing, the only correct criterion for power-system instability is the poles of the closed-loop

system. A Jacobian transfer matrix having transmission zeros at the origin only indicates

that the process is impossible to be tightly controlled at low frequencies. Instability, how-

ever, can only be caused by the feedback control.

Remark 2: For the particular case in this section, the physical reason to the non-

minimum phase phenomena are explained. However, non-minimum phenomena cannot

be generalized to the whole power system as a reason to angle and voltage stability, since

zeros are input dependent. For instance, in Chapter 6, it will be shown that the Jacobian

transfer matrix for induction motors only has a left-half plane (LHP) real zero moving

towards the origin as the slip of the induction motor increases.

Remark 3: For the particular case in this section, it has been shown that the power-

flow Jacobian matrix is the static form of the Jacobian transfer matrix. However, this

cannot be generalized either. One reason is that the Jacobian transfer matrix includes

the ac network as well as the electrical parts of the input devices, while the power-flow

Jacobian matrix normally only includes the ac network. Another reason is that, as it will

be shown in later chapters, the outputs of the controllers of different input devices are

versatile, while the power-flow Jacobian matrix only has voltage magnitude and phase

angle as inputs.

Remark 4: Based on the analysis in this section, it can be concluded that reactive-

power control basically is not a suitable operation mode for VSC-HVDC links connected

62


to weak ac systems. Therefore, in the later sections, only the transfer matrix JPU (s) will

be derived, and with a short name J (s).

4.2.2 Impedance-source including the ac capacitor at the filter bus

In this section, the Jacobian transfer matrix will be derived for the main circuit shown

in Fig. 4.4 by including the ac capacitor at the filter bus. If the high-pass current control

HHP(s) is disregarded, the voltage-vector control law of the VSC in (3.32) becomes

vcref = V0 + ∆V. (4.31)

In a synchronous grid dq reference frame with the d axis chosen aligned with the ac source

E, if the switching-time delay is neglected and it is assumed that |vcref | does not exceed

the maximum voltage modulus, then

v = vcrefe

jθv = (V0 + ∆V )ejθv . (4.32)

With the voltage vector v expressed by (4.32), the dynamic equations of the main circuit

in Fig. 4.4 can be written as

Lcdicdt

= (V0 + ∆V )ejθv − uf − Rcic − jω1Lcic (4.33)

Cfduf

dt= ic − in − jω1Cfuf (4.34)

Lndindt

= uf − E −Rnin − jω1Lnin (4.35)

and in dq-component form

Lcdicddt

= (V0 + ∆V ) cos θv − ufd − Rcicd + ω1Lcicq

Lcdicqdt

= (V0 + ∆V ) sin θv − ufq − Rcicq − ω1Lcicd

Cfdufd

dt= icd − ind + ω1Cfufq

Cfdufq

dt= icq − inq − ω1Cfufd

Lndind

dt= ufd − E0 −Rnind + ω1Lninq

Lndinq

dt= ufq −Rninq − ω1Lnind. (4.36)

The output variables are the active power P and the voltage magnitudeUf at the filter bus.

In per unit form these two quantities are defined as

P = Re {uf i∗

n} , Uf =√u2

fd + u2fq. (4.37)

63


The state-space model can be obtained by linearizing (4.36) and (4.37), which yields the

following form

d

dtx = Ax +Bu

y = Cx +Du (4.38)

where

A =

−Rc

Lcω1 − 1

Lc0 0 0

−ω1 −Rc

Lc0 − 1

Lc0 0

1Cf

0 0 ω1 − 1Cf

0

0 1Cf

−ω1 0 0 − 1Cf

0 0 1Ln

0 −Rn

Lnω1

0 0 0 1Ln

−ω1 −Rn

Ln

B =

[−V0 sin θv0

Lc

V0 cos θv0

Lc0 0 0 0

V0 cos θv0

Lc

V0 sin θv0

Lc0 0 0 0

]T

C =

[0 0 ind0 inq0 ufd0 ufq0

0 0ufd0

Uf0

ufq0

Uf00 0

]

D =

[0 0

0 0

], u =

[∆θv

∆VV0

], y =

[∆P

∆Uf

]

x =[

∆icd ∆icq ∆ufd ∆ufq ∆ind ∆inq

]T. (4.39)

The state-space representation (4.38) can also be written in input-output transfer matrix

form

y =[C (sI −A)−1B +D

]u (4.40)

which yields

∆P

∆Uf

=

[JPθ (s) JPV (s)

JUf θ (s) JUf V (s)

]

︸︷︷︸J(s)

∆θ

∆VV0

. (4.41)

Due to the order of the system, the analytical expressions of the transfer functions of the

Jacobian transfer matrix J (s) in (4.41) are difficult to obtain. However, the analytical

expressions of the transmission zeros and poles are possible to be derived from the state-

space representation.

64


The transmission zeros of J (s) can be derived by the QZ method suggested in [83,

84]. A polynomial system matrix P (s) is defined as

P (s) =

[A− sI B

C D

]. (4.42)

The transmission zeros are then the values s = z for which P (s) loses rank, resulting in

zero output for some non-zero input values, i.e., the zeros are found by the values s = z

that satisfy

det[P (s)] = 0. (4.43)

By neglecting the resistances Rc and Rn, the solutions to (4.43) are found to be exactly

identical to those of (4.17). In other words, the ac capacitor at the filter bus has no influ-

ence on the transmission zeros of the Jacobian transfer matrix!

As will be explained below, this somewhat surprising result is reasonable. In Sec-

tion 4.2.1, it was shown that the transmission zeros of the Jacobian transfer matrix have

a close relationship with angle stability of the ac system. As long as the filter-bus volt-

age is controlled, the angle stability of the ac system is indeed only determined by the

phase angle between the ac source and the filter bus, and independent of the ac capaci-

tor. Of course, in practice, the ac capacitor will affect how much reactive power the VSC

needs to supply in order to keep the filter-bus voltage constant, but this will not affect the

stability unless the converter is forced into voltage limitation.

The poles of J (s) are the eigenvalues of the A matrix. If the resistances Rc and Rn

are neglected also in this case, the poles can be solved analytically with the expressions

p1,2 = ±jω1

p3,4 = ±j(√

1

LnCf

+1

LcCf

− ω1

)

p5,6 = ±j(√

1

LnCf

+1

LcCf

+ ω1

). (4.44)

The two pairs of poles p3,4 and p5,6 are related to the ac capacitor connected at the filter

bus. However, the pole pair p1,2 at the grid frequency is not related to any ac capacitor. In

Fig. 3.3, the resonance at the grid frequency has already been observed during the power

step response of the VSC-HVDC link using power-angle control.

In contrast to the transmission zeros, the poles of the Jacobian transfer matrix are

independent of the operating points, but are usually very poorly damped due to the low

resistance in transmission systems. For grid-connected VSCs, the control system has to

provide damping to these poles to achieve a reasonable bandwidth.

In the following, it is demonstrated how these resonant poles are damped by the

high-pass current control HHP (s) proposed for power-synchronization control in Chap-

ter 3.

65


By including HHP (s), the voltage vector v of the VSC is expressed as

v = vcrefe

jθv

= [(V0 + ∆V ) −HHP(s)icc]ejθv

= (V0 + ∆V )ejθv −HHP(s)ic. (4.45)

Thus, the dynamic equation in (4.33) is modified as

Lcdicdt

= (V0 + ∆V )ejθv −HHP (s) ic − uf − Rcic − jω1Lcic. (4.46)

To eliminate the Laplace transform variable s in HHP (s) which in (4.46) should be in-

terpreted as s = d/dt, a new state variable ρc needs to be introduced. With the new state

variable ρc, the dynamic equation of the phase reactor in (4.46) is expressed as

Lcdicdt

= Lcρc + (V0 + ∆V )ejθv − (Rc + kv + αvLc) ic − uf − jω1Lcic

Lcdρc

dt= αv

[(V0 + ∆V )ejθv −Rcic − uf − jω1Lcic

]. (4.47)

Replacing the dynamic equations of the phase-reactor in (4.33) by (4.47) and following

the same procedure, the Jacobian transfer matrix J (s) includingHHP (s) can be obtained.

For mathematical convenience, HHP (s) has been treated as a part of the Jacobian

transfer matrix, i.e., the controlled process, even though it physically belongs to the con-

trol system. Fig. 4.11 shows the pole-zero map of J (s) with a variation of the gain kv of

the high pass-current control. The system parameters of the Jacobian transfer matrix are

given in Table 4.2. As shown in Fig. 4.11, the effect of HHP (s) is to shift the resonant

poles of J (s) towards the left-half plane (LHP). This damping effect is general for any

resonances in the ac system since HHP (s) basically emulates the dynamic behavior of a

physical resistor.

From Fig. 4.11 it can also be observed that the two transmission zeros are not

affected by HHP (s). This property, in fact, can also be analytically verified by applying

the QZ method for zero calculation.

Figs. 4.12-4.15 show the Bode plots of the four transfer functions JPθ(s), JPV(s),

JUf θ(s), and JUf V(s) overlapped with plots produced by frequency-scanning results from

PSCAD/EMTDC (dashed lines). The transfer functions with and without HHP (s) are

placed side by side in the figures. It clearly shows the damping effects of HHP (s) to all

the resonant poles for all the transfer functions.

The resonant poles p3,4 and p5,6 caused by the ac capacitor at the filter bus appear

in all the four transfer functions. However, it seems that the resonant pole pair p1,2 at the

grid frequency only affects JPθ(s) and JPV(s), but not JUf θ(s) and JUf V(s). The reason is

that, by neglecting the resistances Rn and Rc, JUf θ(s) and JUf V(s) get exactly pole-zero

cancelation at the grid frequency.

66


ω (rad/sec)

ω (

rad

/se

c)

-2000 -1500 -1000 -500 0 500 1000 1500 2000

-2000

-1500

-1000

-500

0

500

1000

1500

2000

z1

z2

p5

p3

p6

p4

p1

p2

Fig. 4.11 Damping effects of HHP (s) on the resonant poles of J (s). Variations of the gain kv

from 0.0 p.u. to 0.6 p.u.

Table 4.2 Parameters of the Jacobian transfer matrix for a grid-connected VSC using power-

synchronization control connected to an impedance-source. Per unit based on 350 MVAand 195 kV.

Parameters Value

Rc 0.01 p.u.

ω1Lc 0.2 p.u.

Main-circuit parameters ω1Cf 0.17 p.u.

Rn 0.01 p.u.

ω1Ln 1.0 p.u.

E0 1.0 p.u.

Initial conditions V0 1.0 p.u.

θu0 (θv0) 30◦ (35.8◦)

kv 0.45 p.u.

High-pass current control αv 40 rad/s

67


101

102

103

10-2

100

102

|JP

θ(j

ω)|

101

102

103

-800

-600

-400

-200

0

ω (rad/sec)

AR

G J

Pθ(j

ω)

(deg.)

(a) Without HHP (s).

101

102

103

10-2

100

102

|JP

θ(j

ω)|

101

102

103

-800

-600

-400

-200

0

ω (rad/sec)

AR

G J

Pθ(j

ω)

(deg.)

(b) With HHP (s).

Fig. 4.12 Bode plots of JPθ(s) (solid: linear models, dashed: frequency-scanning results from

PSCAD/EMTDC).

101

102

103

10-2

100

102

|JP

V(j

ω)|

101

102

103

-300

-200

-100

0

100

ω (rad/sec)

AR

G J

PV(j

ω)

(deg.)


101

102

103

10-2

100

102

|JP

V(j

ω)|

101

102

103

-300

-200

-100

0

100

ω (rad/sec)

AR

G J

PV(j

ω)

(deg.)

(b) With HHP (s).

Fig. 4.13 Bode plots of JPV(s) (solid: linear models, dashed: frequency-scanning results from

PSCAD/EMTDC).

68


101

102

103

10-2

100

102

|JU

fθ(j

ω)|

101

102

103

-600

-400

-200

ω (rad/sec)

AR

G J

Ufθ

(jω

) (d

eg.)


101

102

103

10-2

100

102

|JU

fθ(j

ω)|

101

102

103

-600

-400

-200

ω (rad/sec)

AR

G J

Ufθ

(jω

) (d

eg.)

(b) With HHP (s).

Fig. 4.14 Bode plots of JUf θ(s) (solid: linear models, dashed: frequency-scanning results from

PSCAD/EMTDC).

101

102

103

10-2

100

102

|JU

fV(j

ω)|

101

102

103

-300

-200

-100

0

ω (rad/sec)

AR

G J

UfV

(jω

) (d

eg.)


101

102

103

10-2

100

102

|JU

fV(j

ω)|

101

102

103

-300

-200

-100

0

ω (rad/sec)

AR

G J

UfV

(jω

) (d

eg.)

(b) With HHP (s).

Fig. 4.15 Bode plots of JUf V(s) (solid: linear models, dashed: frequency-scanning results from

PSCAD/EMTDC).

69


+

-

++

--

nLcL

Efu

QP,

v

nRcR

00 vV θ∠0

0 0∠E

ci

00 ufU θ∠

VSC

nC

ni+-

nu

Fig. 4.16 AC-source feeding from a series-compensated ac line.

This property can also be confirmed by the analytical expressions of JUf θ(s) and

JUf V(s) in Table 4.1 for the network configuration without considering the ac capacitor

at the filter bus. Substituting R = 0 into the transfer functions it is found that

JUf θ (s) = −Ln

LV0 sin (θv0 − θu0)

JUf V (s) =Ln

LV0 cos (θv0 − θu0) (4.48)

i.e., the poles are canceled by the zeros.

4.2.3 AC-source feeding from a series-compensated ac line

Fig. 4.16 shows a network configuration with an ac source feeding from a series-compen-

sated transmission line, where Cn is the capacitance of the series ac capacitor, while un

is the voltage vector across the series ac capacitor. The ac capacitor at the filter bus of

the VSC is neglected in this configuration in order to obtain analytical expressions of the

transmission zeros and poles of the Jacobian transfer matrix.

In a synchronous dq reference frame where the d axis is chosen aligned with the ac

source, the dynamic equations of the main circuit in Fig. 4.16 can be expressed as

Ldi

dt= v − un −E − Ri − jω1Li

Cndun

dt= i − jω1Cnun

(4.49)

where L = Lc + Ln, R = Rc + Rn, and i = ic = in. In component form, (4.49) can be

70


written as

Ldiddt

= vd − und − E0 − Rid + ω1Liq

Ldiqdt

= vq − unq −Riq − ω1Lid

Cndund

dt= id + ω1Cnunq

Cndunq

dt= iq − ω1Cnund.

(4.50)

Since the ac capacitor at the filter bus is neglected, the filter-bus voltage vector uf is

neither a state variable nor an input. Such a variable has to be represented by other states

and/or inputs. For this network configuration, uf can be solved from the following two

equations

Lndi

dt= uf − un − Rni − jω1Lni (4.51)

Lcdi

dt= v − uf − Rci − jω1Lci. (4.52)

By dividing (4.51) with Ln and (4.52) with Lc, and further subtracting (4.52) from (4.51),

it is possible to express uf as

uf =Ln

Lv +Rxi +

Lc

Lun. (4.53)

where Rx = (RnLc − RcLn)/L.

Following the same procedure as the previous section, the state-space model in

the form of (4.38) can be obtained by linearizing (4.50) and (4.37) with ∆ufd and ∆ufq

derived from the linearized component form of (4.53). The corresponding matrices and

vectors in (4.38) are expressed as

A =

−RL

ω1 − 1L

0

−ω1 −RL

0 − 1L

1Cn

0 0 ω1

0 1Cn

−ω1 0

, B =

[−V0 sin θv0

Lc

V0 cos θv0

Lc0 0

V0 cos θv0

Lc

V0 sin θv0

Lc0 0

]T

C =

id0Rx + ufd0 iq0Rx + ufq0

id0Lc

L

iq0Lc

L

ufd0Rx

Uf0

ufq0Rx

Uf0

ufd0Lc

Uf0L

ufq0Lc

Uf0L

D =

[ LnV0

L(iq0 cos θv0 − id0 sin θv0)

LnV0

L(id0 cos θv0 + iq0 sin θv0)

LnV0

LUf0(ufq0 cos θv0 − ufd0 sin θv0)

LnV0

LUf0(ufd0 cos θv0 + ufq0 sin θv0)

]

x =[

∆id ∆iq ∆und ∆unq

]T. (4.54)

71


The definitions of u and y are the same as in (4.39). Similarly, the Jacobian transfer matrix

J(s) can be obtained from the state-space representation by applying (4.40).

The poles of J(s) are the eigenvalues of the A matrix. These are given by

p1,2 = − R

2L± 1

2

√√√√R2

L2− 4ω2

1 −4

LCn

+ 4

√4ω2

1

LCn

− R2ω21

L2

p3,4 = − R

2L± 1

2

√√√√R2

L2− 4ω2

1 −4

LCn

− 4

√4ω2

1

LCn

− R2ω21

L2. (4.55)

If the resistance R is neglected, (4.55) can be simplified as

p1,2 = ±j(ω1 −

1√LCn

)

p3,4 = ±j(ω1 +

1√LCn

). (4.56)

For series compensation, the reactance of the series capacitor is always smaller than the

reactance of the ac line. Consequently, 1/√LCn is always smaller than ω1. Thus, (4.56)

shows that J(s) has two pairs of poles, one pair (p1,2) in the subsynchronous frequency

range, and the other pair (p3,4) in the supersynchronous frequency range. IfCn = ∞, these

two pairs of poles correspond to the pole pair at the grid frequency for the impedance-

source configuration in the previous subsection. The smaller the ac capacitance, i.e., the

higher degree that the ac line is compensated, the further the pole pairs are separated.

The pole pair p1,2 in the subsynchronous frequency range is usually troublesome.

Besides the SSR problem with the rotor shaft of the thermal plant as mentioned before,

they can also create problems for power-electronic devices connected in the vicinity since

the subsynchronous frequency range is also where the control systems of most power-

electronic devices are active.

If the resistances Rc and Rn are neglected, the transmission zeros of J(s) can be

derived analytically by applying the QZ method, which gives

z1,2 = ±

√Uf0(ω

21LnCn − 1) +

√b

2LnCn(Uf0 −E0 cos θu0)−(ω2

1 +1

LnCn

)

z3,4 = ±

√Uf0(ω

21LnCn − 1) −

√b

2LnCn(Uf0 −E0 cos θu0)−(ω2

1 +1

LnCn

)(4.57)

where the term b is expressed as

b = 8E0 cos θu0

[ω2

1LnCn(2E0 cos θu0 − 3Uf0) − Uf0

]+ U2

f0(ω21LnCn + 3)2. (4.58)

Fig. 4.17 shows the loci of the two pairs of zeros as the load angle θu0 is increased. For

the Jacobian transfer matrix, the main-circuit parameters are chosen as given by Table 4.2

72


-600 -400 -200 0 200 400 600

-600

-400

-200

0

200

400

600

p1

p2

p3

p4

ω (rad/sec)

ω (

rad

/se

c)

z1

z2

z3

z4

θu0

=32°

θu0

=41°

θu0

=54°

θu0

=32°

θu0

=41°

θu0

=54°

Fig. 4.17 Loci of the transmission zeros with increased load angles.

but with a 60% degree of series compensation, i.e., the capacitance of the series capacitor

is chosen as ω1Cn = 1.667 p.u. The zero pair z1,2 is on the real axis, while z3,4 is on the

imaginary axis. With increased load angles, the resonant zero pair z3,4 moves from p1,2

towards p3,4. The real zero pair z1,2 moves towards the origin. If θu0 = 90◦ is substituted

into the analytical expression of z1,2 in (4.57), it precisely gives z1,2 = 0. It can also be

analytically verified that, if Cn = ∞ is substituted into (4.57), then the mathematical

expression of z1,2 is identical to (4.17), while z3,4 = ∞.

In contrast to the resonant poles, for which HHP (s) provides significant damping,

HHP (s) has no effect on the resonant zero pair z3,4, which might negatively affect the

phase margin of the control system around the subsynchronous frequency range. Care

must be taken if the bandwidth of the control system of the VSC is intended to be higher

than the frequency of z3,4. Fig. 4.18 shows the Bode plot of the open-loop transfer function

of the power-synchronization loop, which shows the effect of HHP (s). The open-loop

transfer function of the power-synchronization loop is expressed as

HPSL (s) = JPθ (s)kp

s. (4.59)

During the process of describing the modeling of VSCs using power-synchronization

control with the concept of Jacobian transfer matrix, simplified ac-network configurations

have been used. This has been done in order to develop a basic understanding of the Ja-

cobian transfer matrix modeling concept, as well as to understand the physical meanings

73


101

102

103

10-4

10-2

100

102

|HP

SL(j

ω)|

Without HHP

(s)

With HHP

(s)

101

102

103

-300

-200

-100

0

arg

HP

SL(j

ω)

(de

g.)

ω (rad/sec)

Fig. 4.18 Bode plot of the open-loop transfer function HPSL (s), kp = 130 rad/s. Initial condi-

tions: θu0 = 24.9◦ (θv0 = 37◦).

of the transmission zeros and poles of the Jacobian transfer matrix in an analytical way.

Within the investigated ac-network configurations so far, a constant-frequency ac

voltage source with a voltage vector E is assumed. The d axis of the grid dq frame has

been chosen aligned with the voltage vector for analysis. Such voltage sources, of course,

do not exist in the real system. However, if the VSC is connected to a large ac system,

assuming such an equivalent network configuration is acceptable. In case that the VSC is

connected to an island system, a common ac-network R − I frame needs to be defined

which is not related to any voltage sources. Such issues will be discussed in Chapter 6.

4.3 Grid-connected VSCs using vector current control

In Chapter 3, the Jacobian transfer matrix was developed for a grid-connected VSC using

vector current control. However, the model was based on the assumption that the filter-bus

voltage is stiff, whereas such assumption is not valid if the VSC is connected to a weak ac

system. In this section, the Jacobian transfer matrix is developed taking into account the

dynamics of the filter bus, as well as the dynamics of the PLL. The methodology is inde-

pendent of ac-network configurations, but for simplicity the impedance-source network

configuration shown in Fig. 4.4 is used for model development.

Fig. 4.19 shows the small-signal block diagram of vector current control, where the

Jacobian transfer matrix is defined as the transfer matrix which has the current reference

∆irefd and ∆irefq as inputs, and the active power ∆P and the the filter-bus voltage ∆Uf as

74

4.3. Grid-connected VSCs using vector current control

P∆

fU∆

cdi∆

cqi∆

fdu∆

fqu∆

ref

dv∆cvdref∆

c

fdu∆

c

fqu∆

c

cdi∆

c

cqi∆

ref

di∆ ref

qi∆

fdu∆

fqu∆

cdi∆

cqi∆

c

fdu∆

c

cdi∆

c

cqi∆

PLLθ∆

fdu∆

fqu∆

c

fqu∆

)(snJ

)(PLL sG

vM

uM

iM

)(cc sG

)(sJ

ref

qv∆c

vqref∆

PLLθ∆

PLLθ∆

PLLθ∆

Jacobian transfer matrix

Fig. 4.19 Definition of the Jacobian transfer matrix for a grid-connected VSC using vector current

control.

outputs.2

By defining the Jacobian transfer matrix, Fig. 4.20 shows the closed-loop sys-

tem of a grid-connected VSC using vector current control, where the minus sign in the

alternating-voltage control is due to the direction of the phase-reactor current reference.

In the following, the mathematical expressions of the three transfer matrices Jn(s),

GPLL(s), Gcc(s), and the three real matrices Mv , Mu, and Mi shown in Fig. 4.19 are

described, where the superscript c on the variables represents the converter dq frame.

1. Network Jacobian transfer matrix Jn(s). This is the ac-network Jacobian trans-

fer matrix which has the dq components of the VSC voltage ∆vd and ∆vq as the

inputs, and the active power ∆P , the filter-bus voltage magnitude ∆Uf , dq com-

ponents of the phase-reactor current ∆icd and ∆icq, and the dq components of the

filter-bus voltage ∆ufd and ∆ufq as the outputs. To distinguish it from the final

Jacobian transfer matrix for vector current control, the subscript n is used. Jn(s)

2The procedure of model development is the same if reactive power ∆Q is used as output instead of

∆Uf , but the alternating-voltage control mode is focused on as it is the preferred operation mode for weak-

ac-system connections.

75


refU

-

+

refP P∆ref

di∆

fU∆

+-

)(sJ)(

s

KK

U

iU

p +−

APC

AVC

ref

qi∆

s

KK

P

iP

p +Jacobian

transfer

matrix

Fig. 4.20 Closed-loop system for VSCs using vector current control.

is similar to the Jacobian transfer matrix developed for power-synchronization con-

trol in Section 4.2.2, but differs by the inputs and the outputs. Those matrices in the

state-space representation that are different from (4.39) are

B =

[1 0 0 0 0 0

0 1 0 0 0 0

]T

, C =

0 0 ind0 inq0 ufd0 ufq0

0 0ufd0

Uf0

ufq0

Uf00 0

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

u =[

∆vd ∆vq

]T, y =

[∆P ∆Uf ∆icd ∆icq ∆ufd ∆ufq

]T.

(4.60)

The network Jacobian transfer matrix Jn(s) thus can be obtained from (4.40), i.e.,

y = Jn(s)u. (4.61)

2. Phase-locked loop (PLL) transfer matrix GPLL(s). As described in Chapter 3, the

control law of the PLL is given by

θPLL =

(KPLL

p +KPLL

i

s

)

︸︷︷︸FPLL(s)

Im{ucf} = FPLL(s)uc

fq. (4.62)

The filter-bus voltage ucf in the converter dq frame is related to uf in the grid dq

frame by

ucf = ufe

−jθPLL (4.63)

76


or in component form

ucfd + juc

fq = (ufd + jufq)(cos θPLL − j sin θPLL) (4.64)

which yields

ucfq = ufq cos θPLL − ufd sin θPLL. (4.65)

Substituting (4.65) into (4.62), yields

θPLL = FPLL(s)(ufq cos θPLL − ufd sin θPLL) (4.66)

which is further linearized as

∆θPLL =−FPLL(s) sin θu0

1 + FPLL(s)(ufd0 cos θu0 + ufq0 sin θu0)︸︷︷︸APLL(s)

∆ufd

+FPLL(s) cos θu0

1 + FPLL(s)(ufd0 cos θu0 + ufq0 sin θu0)︸︷︷︸BPLL(s)

∆ufq (4.67)

where θu0 is the angle output of the PLL in the steady state, which corresponds

to the phase angle of the filter-bus voltage in the grid dq frame. Eq. (4.67) can be

further expressed in transfer matrix form as

∆θPLL =[APLL(s) BPLL(s)

]︸︷︷︸

GPLL(s)

[∆ufd

∆ufq

]. (4.68)

3. Frame-transformation matrices Mv, Mu, and Mi. The VSC voltage reference vref ,

the filter-bus voltage uf , and the phase-reactor current ic, are related in the converter

dq frame and the grid dq frame as

vref = vcrefe

jθPLL, ucf = ufe

−jθPLL, icc = ice−jθPLL. (4.69)

By writing (4.69) in component form and applying linearization, (4.69) can be ex-

77


pressed in input-output matrix form as

[∆vref

d

∆vrefq

]=

[cos θu0 − sin θu0 (−vc

d0 sin θu0 − vcq0 cos θu0)

sin θu0 cos θu0 (vcd0 cos θu0 − vc

q0 sin θu0)

]

︸︷︷︸Mv

∆vcdref

∆vcqref

∆θPLL

[∆uc

fd

∆ucfq

]=

[cos θu0 sin θu0 (−ufd0 sin θu0 + ufq0 cos θu0)

− sin θu0 cos θu0 (−ufd0 cos θu0 − ufq0 sin θu0)

]

︸︷︷︸Mu

∆ufd

∆ufq

∆θPLL

[∆iccd

∆iccq

]=

[cos θu0 sin θu0 (−icd0 sin θu0 + icq0 cos θu0)

− sin θu0 cos θu0 (−icd0 cos θu0 − icq0 sin θu0)

]

︸︷︷︸Mi

∆icd

∆icq

∆θPLL

.

(4.70)

4. Current-control transfer matrix Gcc(s). The current control law is given by (3.12).

After writing (3.12) in component form and applying linearization, the current con-

troller can be expressed in input-output transfer matrix form as

[∆vc

dref

∆vcqref

]=

[αcLc 0 −αcLc −ω1Lc HLP(s) 0

0 αcLc ω1Lc −αcLc 0 HLP(s)

]

︸︷︷︸Gcc(s)

∆irefd

∆irefq

∆iccd

∆iccq

∆ucfd

∆ucfq

.

(4.71)

If the switching-time delay of the converter is neglected and it is assumed that |vref | does

not exceed the maximum voltage modulus, then v = vref . The above derived transfer/real

matrices can be interconnected by the block diagram shown in Fig. 4.19. Accordingly, the

Jacobian transfer matrix for vector current control is obtained as

∆P

∆Uf

=

[JPId (s) JPIq (s)

JUf Id (s) JUf Iq (s)

]

︸︷︷︸J(s)

∆irefd

∆irefq

. (4.72)

78


ω (rad/sec)

ω (

rad

/se

c)

-2000 -1500 -1000 -500 0 500 1000 1500 2000

-2000

-1500

-1000

-500

0

500

1000

1500

2000

z1

z2

p5

p3

p1

p2

p4

p6

Fig. 4.21 Damping effects of the current controller on the resonant poles of J (s) for vector current

control. Variations of αc from 1200 rad/s to 2500 rad/s.

Table 4.3 Control parameters of the Jacobian transfer matrix for a grid-connected VSC using

vector current control connected to an impedance source.

Parameters Value

Inner-current control αc 2500 rad/s

αf 80 rad/s

PLL KPLLp 20

KPLLi 20

Similar to the HHP(s) function of power-synchronization control, the current con-

troller of vector current control also provides damping to the resonant poles of the ac

network, as shown in Fig. 4.21. The main-circuit parameters and initial conditions of the

Jacobian transfer matrix for vector current control are chosen the same as those given

in Table 4.2 for power-synchronization control, and the control parameters of the inner

current control and the PLL are given in Table 4.3.

The higher the bandwidth of the current controller, the more damping it adds. For a

modern transistor PWM converter with a switching frequency of 1−2 kHz, the bandwidth

of the current controller can be chosen as αc = 1000− 2500 rad/s [51], which is enough

for damping purpose.

When analyzing the transmission zeros of the Jacobian transfer matrix, an inter-

79


101

102

103

10-1

100

|JP

I d

(jω

)|

101

102

103

-200

-100

0

ω (rad/sec)

AR

G J

PI d

(jω

) (d

eg.)

(a) JPId (s).

101

102

103

10-2

100

|JP

I q

(jω

)|

101

102

103

-600

-400

-200

ω (rad/sec)

AR

G J

PI q

(jω

) (d

eg.)

(b) JPIq (s).

101

102

103

10-1

100

|JU

fI d

(jω

)|

101

102

103

-600

-400

-200

ω (rad/sec)

AR

G J

UfI d

(jω

) (d

eg.)

(c) JUf Id (s).

101

102

103

10-2

100

|JU

fI q

(jω

)|

101

102

103

-600

-400

-200

ω (rad/sec)

AR

G J

UfI q

(jω

) (d

eg.)

(d) JUf Iq (s).

Fig. 4.22 Bode plots of the transfer functions of the Jacobian transfer matrix for vec-

tor current control (solid: linear models, dashed: frequency-scanning results from

PSCAD/EMTDC). Parameters and initial conditions are given in Table 4.2 and Ta-

ble 4.3.

esting observation can be made. With the same main-circuit parameters and initial con-

ditions, by comparing Fig. 4.21 with Fig. 4.11, it can be easily found that the locations

of the transmission zeros of the Jacobian transfer matrix for vector current control are

exactly identical to those for power-synchronization control! Thus, from the fundamen-

tal limitation point of view, i.e., the bandwidth limitation by the RHP zero, there is no

difference between vector current control and power-synchronization control.

Fig. 4.22 shows the Bode plots of the transfer functions of the Jacobian transfer

matrix with overlapped frequency-scanning plots from PSCAD/EMTDC. The two plots

generally show good agreement for all the transfer functions.

80

4.4. Jacobian transfer matrix for dc-system modeling

VSC1

VSC3

dc1P

dc2P

dc4P

VSC2

dc3P

dc5P

VSC4

VSC5

)(dc sG

ac2P

ac1P

ac4P

ac3P

ac5P

DC-Jacobian

transfer matrix

Fig. 4.23 A dc network connected to multiple VSCs.

+- y

r

)(ac sK)(sJ

)(dc sGdcac PP ∆=∆−

dcu∆

-

ref

dcu

+

+

)(dc sK

AC-Jacobian

transfer matrix

DC-Jacobian

transfer matrix

Fig. 4.24 Feedback-control view of the ac/dc combined system.

4.4 Jacobian transfer matrix for dc-system modeling

The concept of Jacobian transfer matrix for dc-system modeling is similar to ac-system

modeling. As shown in Fig. 4.23, a dc system is also treated as a multivariable feedback

control system where the Jacobian transfer matrix Gdc(s) is defined as the controlled

process. In this thesis, only VSCs are considered as input devices to the dc system. The

Jacobian transfer matrix of a dc system thus represents the passive components in the dc

system. The input variables to Gdc(s) are the dc powers from the VSCs, while the output

variables from Gdc(s) are the direct voltages at each VSC terminal.

Fig. 4.24 shows the feedback-control view of the ac/dc combined system, where

Kac(s) is the controller transfer matrix of the ac system while Kdc(s) is the controller

transfer matrix of the dc system, i.e., the direct-voltage controllers. In other words, the

dc Jacobian transfer matrix Gdc(s) is treated as an additional controlled process of the

81


dcR dcL

dc1C dc2C

dc1P dc2P

+

-

+

-

dc2u

dci

dc1u

ac1P ac2P

dcC

dc1P dc2P

+

-

dcu

ac1P ac2P

(a) (b)

VSC1 VSC2 VSC1 VSC2

Fig. 4.25 DC-link representation for a two terminal VSC-HVDC link. (a) Single dc capacitor. (b)

π-link.

closed-loop system of the combined ac/dc system. The ac Jacobian transfer matrix J(s)

and the dc Jacobian transfer matrix Gdc(s) are connected by the equivalence of the ac-

power vector Pac and the dc-power vector Pdc, i.e.,

Pac = −Pdc (4.73)

where the minus sign is due to the definition of the power directions. By using the equiv-

alence in (4.73), it should be pointed out that the losses of the converter valves can intro-

duce an error, which will be discussed later.

In the following, a dc link connected to two VSCs, i.e., a two-terminal VSC-HVDC

link, is modeled by the dc Jacobian transfer matrix modeling concept. Fig. 4.25(a) shows

an dc-link circuit that is represented by a single dc capacitor. The capacitor bank at the

dc link is an energy storage. The time derivative of the stored energy must equal the sum

of the instantaneous power infeed from the two converters (neglecting the losses). The

direct-voltage dynamics can thus be written as

1

2Cdc

d (u2dc)

dt= Pdc1 + Pdc2 (4.74)

where Cdc is the dc-link capacitance and udc is the direct voltage. Pdc1 and Pdc2 are the

instantaneous powers from VSC1 and VSC2. If the direct-voltage controller were to oper-

ate directly on the error urefdc − udc, the closed-loop dynamics would be dependent on the

operating point udc0. This inconvenience is avoided by selecting the direct-voltage con-

troller operating instead on the error(uref

dc

)2−u2dc as suggested in [13,85]. Consequently,

the dc-link dynamics can be written in the linearized form

∆u2dc =

2

sCdc︸︷︷︸Gdc(s)

(∆Pdc1 + ∆Pdc2) . (4.75)

If the dc transmission line is a long overhead line, then the resistance and the inductance

of the dc line have to be taken into account. Fig. 4.25(b) shows a dc link represented by

82


a π-link model. Based on Kirchhoff’s voltage and current laws, the dynamic equations of

the dc circuit can be written as

Cdc1dudc1

dt=Pdc1

udc1− idc

Ldcdidc

dt= udc1 − udc2 − Rdcidc

Cdc2dudc2

dt= idc +

Pdc2

udc2(4.76)

where Cdc1 and Cdc2 are lumped capacitances at the two VSC stations. The quantities

udc1 and udc2 are the direct voltages. Pdc1 and Pdc2 are the instantaneous powers from

VSC1 and VSC2. Rdc and Ldc are the resistance and the inductance of the dc line. The

quantity idc is the direct current of the dc line. If the linearized deviation variables ∆Pdc1

and ∆Pdc2 of Pdc1 and Pdc2 are chosen as the inputs, and ∆u2dc1 and ∆u2

dc2 of udc1 and

udc2 are chosen as the outputs, a state-space model can be obtained by linearization of

(4.76)

d

dtx = Ax +Bu

y = Cx +Du (4.77)

where

A =

− Pdc10

u2dc10

Cdc1− 1

Cdc10

1Ldc

−Rdc

Ldc− 1

Ldc

0 1Cdc2

− Pdc20

u2dc20

Cdc2

, B =

[1

udc10Cdc10 0

0 0 1udc20Cdc2

]T

C =

[2udc10 0 0

0 0 2udc20

], D =

[0 0

0 0

], x =

[∆udc1 ∆idc ∆udc2

]T

u =[

∆Pdc1 ∆Pdc2

]T, y =

[∆u2

dc1 ∆u2dc2

]T. (4.78)

The state-space representation in (4.77) can also be written in input-output transfer matrix

form by (4.40) which yields

∆u2dc1

∆u2dc2

=

[Gdc11 (s) Gdc12 (s)

Gdc21 (s) Gdc22 (s)

]

︸︷︷︸Gdc(s)

∆Pdc1

∆Pdc2

. (4.79)

The poles of Gdc(s) can be solved analytically for the operating point where Pdc10 =

Pdc20 = 0 with the expression

p1 = 0 (4.80)

p2,3 = − Rdc

2Ldc± j

√R2

dc

4L2dc

− Cdc1 + Cdc2

LdcCdc1Cdc2.

83


Table 4.4 Parameters of the dc Jacobian transfer matrix for the π-link model. Per unit based on

350 MW and 150 kV.

Parameters Value

Rdc 0.04 p.u.

Main-circuit parameters Ldc 0.0025 p.u.

Cdc1 0.0077 p.u.

Cdc2 0.0077 p.u.

Udc10 1.0 p.u.

Initial conditions Pdc10 0.0 p.u.

As shown in Fig. 4.26, the poles of Gdc(s) are dependent on the operating point.

With the increase of loading, the frequencies of the two complex poles p2,3 are reduced,

and the real pole p1 at the origin moves into the right-half plane. The main-circuit param-

eters and initial conditions of the dc Jacobian transfer matrix for the π-link are given in

Table 4.4. In control theory, the RHP pole of the process imposes a fundamental lower

limit on the bandwidth of the controller, i.e., the closed-loop system of the direct-voltage

control has to achieve a bandwidth that is higher than the location of the RHP pole of

Gdc(s) to stabilize the process. Recalling also the upper limit of bandwidth imposed by

the RHP transmission zero of the ac Jacobian transfer matrix [cf. (4.17)], it is generally

more complicated to operate grid-connected VSCs at high load angles.

The instability of Gdc(s) is to do with the resistance of the dc link. With dc powers

as the inputs, as the way how VSCs work to a dc link, the dc resistance gives a destabi-

lizing effect. The analytical solutions of the poles of Gdc(s) with other operating point

than Pdc10 = Pdc20 = 0 are difficult to obtain. However, Appendix B.2 gives a rigorous

mathematical proof of the instability of Gdc(s) for other operating points than Pdc10 = 0

or Pdc20 = 0. The mathematical proof also shows the role of the dc resistance to the in-

stability of Gdc(s). Fig. 4.27 shows the root-loci of Gdc(s) by varying the dc-resistance

values. With increased dc resistance, the resonant pole pair p2,3 becomes more damped,

but p0 moves towards the right-half plane. It should be noted that the destabilizing effect

of the dc resistance only becomes apparent if the dc-transmission line is sufficiently long,

e.g., HVDC transmission over long-distance overhead lines.

By applying the QZ method, the transmission zero of Gdc(s) for the π-link model

is derived with a surprisingly simple expression

z = −Rdc

Ldc. (4.81)

In contrast to the zeros of the ac Jacobian transfer matrix, the location of the zero of the

dc Jacobian transfer matrix is independent of the operating points.

84


-30 -20 -10 0 10 20 30-400

-300

-200

-100

0

100

200

300

400

z

ω (rad/sec)

ω (

rad

/se

c)

p2

p1

p3

Fig. 4.26 Root-loci of Gdc(s) regarding variations of Pdc10 from 0.0 p.u. to 1.0 p.u.

-200 -150 -100 -50 0 50 100 150 200-400

-300

-200

-100

0

100

200

300

400

ω (rad/sec)

ω (

rad

/se

c)

p2

p1

p3

z

Fig. 4.27 Root-loci of Gdc(s) regarding variations of Rdc from 0.0 p.u. to 0.24 p.u. Initial con-

ditions: Pdc10 = 1.0 p.u., udc10 = 1.0 p.u.

85


)(sJ

dc1VSC1 PP ∆=∆−vθ∆

fU∆ dc2P∆

2

dc1u∆

2

dc2u∆)(dc sG

)(u sJdθ

0V

V∆

Fig. 4.28 Definition of the transfer function Judθ(s).

To verify the proposed modeling concept with frequency-scanning results from time

simulations, a transfer function Judθ(s) is defined as

Judθ(s) = JPθ(s)Gdc11(s) (4.82)

which is shown graphically in Fig. 4.28. It should be noted that the active power derived

previously for the ac Jacobian transfer matrix is the power from the filter bus, which

is somewhat different from the active power flowing from the VSC due to the energy

stored in the phase reactor. Therefore, the ac power ∆Pac should be obtained from the

linearization of

Pac = PVSC = Re {vi∗c} . (4.83)

In Fig. 4.28, the active power flowing from the VSC is denoted as PVSC to distinguish it

from the active power from the filter bus to the ac system.

Fig. 4.29 shows the Bode plots of Judθ(s) overlapped with the frequency-scanning

results from PSCAD/EMTDC for the two types of dc-link representations respectively,

where the impedance-source system shown in Fig. 4.4 is used as the ac-network configu-

ration. Besides, the high-pass current control HHP(s) has been applied in the ac Jacobian

transfer matrix.

Some discrepancies can be observed between the plots of the linear models and the

frequency-scanning results, which are mainly due to the lack of valve-loss representation

of the linear models. The most noticeable one is the resonance peak of the π-link model

in Fig. 4.29(b). The frequency-scanning results show that the valve losses have a damping

effect on the resonance peak while the linear model tends to overestimate the impact of

the dc resonance. In addition, the slopes of the magnitudes of the linear models in both

Fig. 4.29(a) and Fig. 4.29(b) are generally steeper than those from the frequency-scanning

results, which can be considered as the resistive effects of the valve losses. To properly

represent the losses of the converter valves, the linear model should take into account the

topology of the converter as well as the applied PWM technique, which is of nonlinear

nature. These issues certainly require further investigations in the future research.

86

4.5. Summary of the properties of the Jacobian transfer matrix

101

102

103

10-5

100

105

|JU

dθ(j

ω)|

101

102

103

-800

-600

-400

-200

0

ω (rad/sec)

AR

G J

Udθ(j

ω)

(deg.)

(a) Single dc capacitor model.

101

102

103

10-5

100

105

|JU

dθ(j

ω)|

101

102

103

-800

-600

-400

-200

0

ω (rad/sec)

AR

G J

Udθ(j

ω)

(deg.)

(b) π-link model.

Fig. 4.29 Bode plots of Judθ(s) (solid: linear models, dashed: frequency-scanning results from

PSCAD/EMTDC).

4.5 Summary of the properties of the Jacobian transfer

matrix

In this chapter, the concept of Jacobian transfer matrix is proposed for dynamic mod-

eling of ac/dc systems. By investigating some simplified ac/dc network configurations

connected to VSCs, several properties of the ac/dc Jacobian transfer matrix can be sum-

marized below:

1. The complex poles of the ac/dc Jacobian transfer matrix generally reflect the res-

onances in the ac/dc network. The poles of the ac Jacobian transfer matrix are in-

dependent of the operating point, while the poles of the dc Jacobian transfer matrix

are dependent on the operating point except in the special case when the dc network

is represented by a single dc capacitor.

2. The ac Jacobian transfer matrix is an inherently stable process, although the poles

might be poorly damped and the zeros can move to the origin. Therefore, instability

of an ac system can only be caused by feedback control. On the other hand, the

dc Jacobian transfer matrix is an inherently unstable process or has a pole at the

origin. Therefore, a dc system has to be stabilized by feedback control, i.e., the

direct-voltage control.

3. The zeros (on the real axis) of the ac Jacobian transfer matrix are dependent on the

operating point. By moving to the origin, the zeros impose a fundamental limitation

on power transmission in the ac network. This limitation has manifested as angle-

stability and voltage-stability phenomena in the classical power-system stability

theory. If the zero of the ac Jacobian transfer matrix appears on the right-half plane,

87


it also imposes an upper limitation on the achievable bandwidth of the input device,

i.e., the bandwidth of the power controller of the input device should be lower than

the location of the RHP zero. On the other hand, the dc Jacobian transfer matrix

does not have such operating-point-dependent zero. The above difference of ac and

dc systems explains the fact that power can be transmitted in the dc network up to

the thermal limit but is often limited in the ac network, especially for long-distance

power transmissions.

4. The pole on the right-half plane of the dc Jacobian transfer matrix imposes a fun-

damental lower limitation on the bandwidth of the direct-voltage controller, i.e., the

bandwidth of the direct-voltage controller should be higher than the location of the

RHP pole to stabilize the process.

It should be pointed out that the above conclusions about the dc system are only valid

for the dc system defined in this thesis, i.e., the VSCs are the only input devices to the

dc network, and dc powers are the inputs while direct voltages are the outputs of the dc

Jacobian transfer matrix. For example, for dc systems connected to LCCs, the dynamic

modeling is certainly different. Such scenarios are, however, out of the scope of this thesis.

In this chapter, the ac Jacobian transfer matrix has only considered the VSC as the

input device. However, the concept of the ac Jacobian transfer matrix is meant for any

type of input device in ac systems. In Chapter 6, the Jacobian transfer matrices for two

other input devices, i.e., synchronous generators and induction motors, are developed.

4.6 Summary

In this chapter, the concept of Jacobian transfer matrix is proposed for dynamic model-

ing of ac/dc systems. The fundamental idea of the ac Jacobian transfer matrix modeling

is that a synchronous power system is modeled as one multivariable feedback-control

system, where the feedback controllers and the controlled process of the power system

are explicitly defined. The concept has been applied to model grid-connected VSCs us-

ing power-synchronization control and vector current control for several simplified ac-

network configurations. The modeling concept is also extended to model dc systems con-

structed by multiple VSCs. By theoretical analysis, it is found that the poles and zeros

of the Jacobian transfer matrix give useful information about the properties of the ac/dc

system.

88

Chapter 5

Control of VSC-HVDC Links

Connected to High-Impedance AC

Systems

In this chapter, the control of VSC-HVDC links connected to high-impedance weak ac

systems is investigated. In Section 5.1, general aspects of high-impedance ac systems

are described. In Section 5.2, the performance of vector current control and power-

synchronization control for weak-ac-system connections are compared based on the Ja-

cobian transfer matrix modeling concept. In Section 5.3, multivariable feedback designs

of power-synchronization control are investigated by two design approaches, i.e., inter-

nal model control and H∞ control. In Section 5.4, a direct-voltage controller is proposed

for power-synchronization control. The dc-capacitance requirement for VSC-HVDC links

connected to weak ac systems is discussed. Finally, a control structure for interconnecting

two very weak ac systems is proposed in Section 5.5. The major results of this chapter are

summarized in Section 5.6. Some results of this chapter are included in [63, 86].

5.1 General aspects of high-impedance ac systems

As it was mentioned in Chapter 2, there is an inherent weakness with the conventional

LCC-HVDC system, i.e., the commutations of the thyristor valves are dependent on the

stiffness of the alternating voltage. The converter cannot work properly if the connected

ac system is weak. Substantial research has been performed in this field [5, 6, 8]. The

most outstanding contribution on this subject is [5]. According to [5], an ac system can be

considered as weak from two aspects: 1. ac-system impedance is high. 2. ac-system inertia

is low. Either of the two system conditions may become an obstacle for conventional

LCC-HVDC applications.

If an HVDC link is terminated at the weak point of a large ac system, i.e., the

89

Chapter 5. Control of VSC-HVDC Links Connected to High-Impedance AC Systems

equivalent impedance of the ac system is high, the alternating voltage at the filter bus

will become sensitive to power variations of the HVDC link. This difficulty is usually

measured by the short-circuit ratio (SCR), which is a ratio of the ac-system short-circuit

capacity vs. the rated power of the HVDC link. If the ac system is represented as an

impedance source as shown in Fig. 4.4, then the SCR is directly related to the ac-system

inductance Ln. According to [5], SCR is defined as

SCR =Sac

PdcN

(5.1)

where Sac is the short-circuit capacity of the ac system at the filter bus, while PdcN is the

rated dc power of the HVDC link. The short-circuit capacity of the ac system Sac can be

expressed as

Sac =U2

f0

Z≈

U2f0

ω1Ln(5.2)

where ω1 is the angular frequency of the ac system and Z is the equivalent impedance of

the ac system. To further simplify the expression of SCR in (5.1), the filter-bus voltage

is assumed to be identical to the base value, i.e, Uf0 = UacN, and the rated power of the

HVDC link PdcN is used as the base power of the ac system, i.e., SacN = PdcN. If ω1Ln is

expressed in per unit, it follows from (5.1) and (5.2) that SCR can be expressed as

SCR =1

ω1Ln

. (5.3)

The following is a definition of the strength of an ac system based on the classification

of [5]:

• Strong system, if the SCR of the ac system is greater than 3.0.

• Weak system, if the SCR of the ac system is between 2.0 and 3.0.

• Very weak system, if the SCR of the ac system is lower than 2.0.

One of the driving forces to develop the VSC-HVDC technology is to overcome the

weak-ac-system connection problem of the conventional LCC-HVDC system. By apply-

ing VSC techniques, the two notorious weak-ac-system related problems for conventional

LCC-HVDC systems, i.e., the transient over voltage (TOV) and the low-order harmonic

resonance, are no longer big issues. For VSC-HVDC systems, large ac capacitors are

not needed for reactive-power compensation. Moreover, as shown in Chapter 4, for both

vector current control and power-synchronization control, the control system can easily

provide damping to resonances in ac systems at any frequency 1.

However, weak-ac-system connections still represent more challenging operation

conditions for VSC-HVDC systems due to the following reasons:

1However, as it is shown by [12], the problem of low-harmonics resonance might not be trivial for vector

current control, but it is not a problem for power-synchronization control thanks to the high-pass current

control function.

90

5.2. Comparison of power-synchronization control and vector current control

1. The filter-bus voltage is more sensitive to power variations of the VSC-HVDC link

in weak ac systems. Thus, a weak-ac-system connection requires that the control

system of the VSC-HVDC link must be less dependent on the stiffness of the filter-

bus voltage.

2. The SCR of the ac system imposes a theoretical limitation on the maximum power

that the VSC-HVDC system is possible to transmit to or from the ac system. This

limitation has traditionally been analyzed by the maximum power curve (MPC) for

conventional LCC-HVDC systems [5, 87]. As it was discussed in Chapter 4, such

power limitations are basic characteristics of the ac system that are related to the

operating-point-dependent zeros of the ac Jacobian transfer matrix. Moreover, for

a VSC-HVDC link, the RHP zero of the ac Jacobian transfer matrix imposes a

fundamental limitation on the achievable bandwidth of the control system.

3. The off-diagonal elements of the Jacobian transfer matrix are larger for weak-ac-

system connections, especially with high loadings. This problem is in common for

vector current control and power-synchronization control, but it is more serious for

vector current control. High off-diagonal elements imply more interactions between

the active-power control and the alternating-voltage control.

4. Model uncertainties, either due to variations of ac-network configurations or op-

erating points, are generally higher for weak-ac-system connections. The control

system of the VSC-HVDC link is required to be robust for model variations.

In the following sections, the above issues related to high-impedance ac systems will be

addressed. The low-inertia ac-system connection issue will be discussed in Chapter 6.

5.2 Comparison of power-synchronization control and vec-

tor current control

In this section, the dynamic performance of power-synchronization control and vector

current control are compared for VSC-HVDC links connected to high-impedance weak

ac systems. For simplicity, the impedance-source system shown in Fig. 4.4 is assumed

as the ac-network configuration. Except for the quantities that are varied, the parameters

of the network configuration, initial conditions, and control systems, are based on those

defined in Table 4.2 and Table 4.3. The comparisons are performed by analyzing the char-

acteristics of the Jacobian transfer matrices in the frequency domain and by verifications

using time simulations.


matrix for vector current control regarding the variations of the SCRs of the ac system.

The VSC-HVDC converter is assumed to operate with zero loading P = 0.0 p.u., i.e.,

91


static synchronous compensator (STATCOM) mode. For simplicity, only the magnitudes

of the transfer functions are plotted.

As it was mentioned before, one typical feature of high-impedance ac systems is

that the filter-bus voltage is more sensitive to power variations of the VSC-HVDC link.

This effect is reflected in transfer functions JUf Id (s) and JUf Iq (s) where their magnitudes

are getting larger with decreased SCRs of the ac system. Meanwhile, the magnitudes of

the off-diagonal transfer functions, increase with lower SCRs, which imply difficulties

for the outer-loop controllers, i.e., more interactions between the active-power control

loop and the alternating-voltage control loop are expected. However, even more severe

difficulties are experienced with increased loadings of the VSC-HVDC link that are shown

below.

Fig. 5.2 shows the Bode plots of the transfer functions of the Jacobian transfer ma-

trix regarding the variations of the loadings of a VSC-HVDC link connected to an ac

system with SCR = 1.0. It is easily observed that the magnitudes of the off-diagonal

transfer functions increase dramatically with increased loadings, while the magnitude of

JPId (s) decreases. Beyond 50% of the VSC-HVDC loading (the dashed curves), the mag-

nitudes of the off-diagonal transfer functions are greater than those of the diagonal transfer

functions. In control theory, such a process is called ill-conditioned process which implies

particular difficulties for the outer-loop controllers [82]. Of all the transfer functions, the

variation of JUf Id (s) plays the most critical role. From Fig. 4.22c in Chapter 4, it can be

found that JUf Id (s) has −180◦ phase shift at low frequencies which means that the in-

crease of irefd always results in a decreased filter-bus voltage Uf , or vise versa.2 This effect

is greatly amplified with higher-loading conditions for a VSC-HVDC converter connected

to a weak ac system. This is also the reason to that the magnitude of JPId (s) decreases

with high-loading conditions as will be shown by the time-simulation results in below.

Figs. 5.3-5.5 show the time-simulation results from PSCAD/EMTDC by applying a

0.1 p.u. step to irefd at 0.1 s. The pre-conditions of the three figures correspond to the three

operating points of the Bode plots in Fig. 5.2 respectively. The time-simulation results

agree with the conclusion drawn from the frequency-domain analysis. For example, the

time-simulation results in Fig. 5.5 show that, with 70% loading of the VSC-HVDC link,

the 0.1 p.u. step of irefd does not give any increase of the active-power output due to the

decreased filter-bus voltage Uf , although the converter current iccd still follows the current

reference irefd . The same conclusion can be drawn by the frequency-domain analysis.

In the following, similar frequency-domain analysis and time simulations are per-

formed for power-synchronization control. The network configurations and operating

points are chosen identical as those for the analysis of vector current control.


matrix for power-synchronization control regarding variations of SCRs of the ac system.

2JPIq (s) and JUf Iq (s) also have −180◦ phase shift at low frequencies, but they are in the same direction

for irefq .

93


0 0.1 0.2 0.3 0.4 0.50

0.05

0.1ire

f

d, ic c

d (

p.u

.)

iref

d

ic

cd

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

P (

p.u

.)

0 0.1 0.2 0.3 0.4 0.50.8

1

time (sec)

Uf (

p.u

.)

Fig. 5.3 Step response of id for vector current control at P = 0.0 p.u. with SCR = 1.0.

0 0.1 0.2 0.3 0.4 0.50.5

0.55

0.6

0.65

iref

d, ic c

d (

p.u

.)

iref

d

ic

cd

0 0.1 0.2 0.3 0.4 0.50.5

0.55

0.6

0.65

P (

p.u

.)

0 0.1 0.2 0.3 0.4 0.50.8

1

time (sec)

Uf (

p.u

.)


94


0 0.1 0.2 0.3 0.4 0.50.7

0.75

0.8

iref

d, ic c

d (

p.u

.)

iref

d

ic

cd

0 0.1 0.2 0.3 0.4 0.50.65

0.7

0.75

0.8

P (

p.u

.)

0 0.1 0.2 0.3 0.4 0.50.8

1

time (sec)

Uf (

p.u

.)


100

102

104

10-2

100

ω (rad/sec)

|JP

θ(j

ω)|

100

102

104

10-2

100

ω (rad/sec)

|JP

V(j

ω)|

100

102

104

10-2

100

ω (rad/sec)

|JU

fθ(j

ω)|

100

102

104

10-2

100

ω (rad/sec)

|JU

fV(j

ω)|

Fig. 5.6 Bode plots of the transfer functions of the Jacobian transfer matrix for power-synchro-

nization control with P = 0.0 p.u. (solid: SCR = 5.0, dashed: SCR = 2.0, dotted:

SCR = 1.0).

95


It is interesting to note that the magnitudes of the off-diagonal transfer functions decrease

slightly with decreased SCRs of the ac system. However, the variations of the magnitudes

of JPθ(s) are larger than JPId (s) in Fig. 5.1. Similar to vector current control, Fig. 5.7

shows that the magnitudes of the off-diagonal transfer functions increase with higher

loadings of the VSC-HVDC system for power-synchronization control, which means that

the outer-loop controllers of power-synchronization control also experience difficulties

with increased loadings of the VSC-HVDC system. However, the magnitude of JUf θ(s)

is much lower than JUf Id (s) with the same operating conditions. Besides, the magnitude

of JPθ(s) does not decrease as dramatically as JPId (s) with increased loadings of the

VSC-HVDC system.

The above difference between JUf θ(s) and JUf Id (s) is the key factor that affects

the dynamic performance of power-synchronization control and vector current control

for VSC-HVDC links connected to weak ac systems. With power-synchronization con-

trol, the active power is controlled by the phase angle of the VSC voltage, meanwhile

the magnitude of the VSC voltage is held constant. Such a control strategy minimizes

dynamic variations of the filter-bus voltage. However, with vector current control, the

inner-current control loop only controls the currents, while control of the filter-bus volt-

age is left to the outer-loop controller. Accordingly, dynamic variations of the filter-bus

voltage are much higher for vector current control than for power-synchronization con-

trol. Such dynamic variations of the filter-bus voltage create substantial difficulties for the

outer-loop controllers.

Figs. 5.8-5.10 show the simulation results from PSCAD/EMTDC when applying

a 0.1 rad step to θv for power-synchronization control. Comparing with the simulation

results with vector current control, the variations of the filter-bus voltage are much smaller

for power-synchronization control in response to the same power changes. Besides, the

dynamic responses of the open-loop transfer functions for power-synchronization control

are more stable than those for vector current control since the Jacobian transfer matrix

for vector current control includes more feedback loops, especially the PLL introduces a

delay. The difference of the dynamic response can also be explained by the Bode plots of

the transfer functions, where the magnitudes are “flatter” at low frequencies for power-

synchronization control than for vector current control.

Generally speaking, with power-synchronization control, the control system can

maintain stable operation with load angle up to θu0 = 60◦, i.e., approximately 86% of the

rated power, for VSC-HVDC links connected to ac systems with SCR = 1.0, whereas

it is very difficult for vector current control to operate even with 50% loading. Ref. [45]

gives more simulation results for such comparisons.

However, if the connected ac system is strong, vector current control seems to be

more advantageous than power-synchronization control. Figs. 5.11-5.12 show the Bode

plots of the transfer functions of the Jacobian transfer matrix for vector current control

and power-synchronization control with SCR variations from 3 to 10. In Fig. 5.11 vector

96


100

102

104

10-2

100

ω (rad/sec)

|JP

θ(j

ω)|

100

102

104

10-2

100

ω (rad/sec)

|JP

V(j

ω)|

100

102

104

10-2

100

ω (rad/sec)

|JU

fθ(j

ω)|

100

102

104

10-2

100

ω (rad/sec)

|JU

fV(j

ω)|


nization control with SCR =1.0 p.u. (solid: P = 0.0 p.u., dashed: P = 0.5 p.u., dotted:

P = 0.7 p.u.).

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

θv (

rad

)

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

P (

p.u

.)

0 0.1 0.2 0.3 0.4 0.50.8

1

time (sec)

Uf (

p.u

.)

Fig. 5.8 Step response of θv for power-synchronization control at P = 0.0 p.u. with SCR = 1.0.

97


0 0.1 0.2 0.3 0.4 0.50.6

0.65

0.7

θv (

rad

)

0 0.1 0.2 0.3 0.4 0.50.5

0.55

0.6

0.65

P (

p.u

.)

0 0.1 0.2 0.3 0.4 0.50.8

1

time (sec)

Uf (

p.u

.)


0 0.1 0.2 0.3 0.4 0.50.9

0.95

1

1.05

θv (

rad

)

0 0.1 0.2 0.3 0.4 0.50.65

0.7

0.75

0.8

P (

p.u

.)

0 0.1 0.2 0.3 0.4 0.50.8

1

time (sec)

Uf (

p.u

.)


98


current control controls the reactive power instead of controlling the alternating voltage,

which is acceptable for strong ac-system connections.

By comparing Fig. 5.11 and Fig. 5.12, it can be seen that the magnitude of the off-

diagonal elements of the Jacobian transfer matrix for vector current control are generally

less than those for power-synchronization control, and more importantly, the magnitudes

of the diagonal elements of the Jacobian transfer matrix for vector current control are

much less sensitive to SCR variations than those for power-synchronization control. In

other words, vector current control is more robust than power-synchronization control if

the connected ac system is guaranteed to be strong.

Fig. 5.13 shows the time-simulation results of vector current control from PSCAD/-

EMTDC by applying 0.1 p.u. steps with irefd and irefq respectively where the connected ac

system has an SCR of 10. Vector current control shows excellent reference tracking and

decoupling capabilities.

It should be noted that the Jacobian transfer matrices include various control func-

tions for both power-synchronization control and vector current control. The parameters

of those control functions can also affect the above frequency-domain analysis and time-

simulation results. However, the selection of control parameters will not affect the ma-

jor conclusions drawn in this section since the difference between vector current control

and power-synchronization control is more related to their fundamental control strategies.

Therefore, such an analysis is not given in the thesis but typical values for those control

parameters are selected.

Among the two control methods compared in this section, power-synchronization

control is the most suitable control system for VSC-HVDC links connected to weak ac

systems. Consequently, only power-synchronization control will be applied for the re-

maining work of the thesis.

5.3 Multivariable feedback designs

As shown in Section 5.2, the magnitudes of the off-diagonal transfer functions of the Ja-

cobian transfer matrix become larger for weak-ac-system connections. To achieve high

control bandwidth and to decouple the interactions between the active-power control loop

and the alternating-voltage control loop, multivariable feedback control design should be

applied. Fig. 5.14 shows the general control block diagram for multivariable feedback de-

signs, i.e., a controller transfer matrix K(s) with four controllers K11(s), K12(s), K21(s),

and K22(s) is applied for active-power and alternating-voltage control. In this section,

two multivariable feedback design approaches are investigated: Internal model control

(IMC) and H∞ control. For simplicity, the impedance-source system shown in Fig. 4.4 is

assumed as the ac-network configuration.

100

5.3. Multivariable feedback designs

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

iref

d, ic c

d (

p.u

.)

iref

d

ic

cd

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

P (

p.u

.)

0 0.1 0.2 0.3 0.4 0.5-0.05

0

0.05

0.1

iref

q, ic c

q (

p.u

.)

iref

q

ic

cq

0 0.1 0.2 0.3 0.4 0.5

-0.1

-0.05

0

0.05

time (sec)

Q (

p.u

.)

Fig. 5.13 Step response of irefd and irefq for vector current control at P = 0.0 p.u. with ac-system

SCR = 10.0.

refU

-+

refPP∆vθ∆

fU∆

+-

++

(s)11K

(s)21K

(s)12K

(s)22K

++

)(sJ

AC-Jacobian

transfer

matrix0V

V∆

Fig. 5.14 Multlivariable feedback control of grid-connected VSCs using power-synchronization

control

101


+-

(s)K

)(~

sJ

+-

y(s)J

y~

r u

Fig. 5.15 Block diagram of the internal model control system.

5.3.1 Internal model control

The concept of IMC design is based on the idea that the transfer functions of the closed-

loop system can be directly shaped by explicitly including a process model in the con-

troller [88]. Fig. 5.15 shows the the block diagram of the IMC design, where r = [Pref Uref ]T ,

u = [∆θv ∆V/V0]T , and y = [P Uf ]

T . From Fig. 5.15, it can be easily observed that, if

the process J (s) is stable and the process model J (s) is equal to J (s), the whole system

is internally stable if and only if K (s) is stable. Thus, if K (s) is chosen as

K (s) = J−1 (s)F (s) (5.4)

the output response would be

y = J (s) J−1 (s)F (s) r = F (s) r. (5.5)

Therefore, the bandwidth of the closed-loop system would be solely determined by the

filter F (s), which is often defined as a first-order filter with the desired bandwidth. How-

ever, for VSC-HVDC applications, the above philosophy cannot work directly, since the

Jacobian transfer matrix J (s) contains an RHP transmission zero as shown in Chapter 4.

A direct inverse of J (s) would end up with an unstable controller.

In such a situation, a factorization technique needs to be applied [89]. Consequently,

the process model can be factorized into two parts

J (s) = Jp (s) Jn (s) (5.6)

where Jn (s) contains the invertible elements and Jp (s) contains the non-invertible ele-

ments.

An easy factorization method is to place the RHP transmission zero at the diagonal

part of Jp (s) as

Jp (s) =

[ ZRHP−sZRHP+s

0

0 ZRHP−sZRHP+s

](5.7)

where ZRHP is the RHP transmission zero of J (s). The invertible matrix Jn (s) can be

solved by (5.6) and (5.7). A low-pass filter matrix F (s) is chosen to specify the closed-

102


+-

++

(s)K

)(~

sJ

y(s)J

r

(s)'K

u

Fig. 5.16 IMC in classical feedback-control structure.

loop bandwidth

F (s) =

[ αp

s+αp0

0 αu

s+αu

](5.8)

where αp and αu are the desired bandwidths of the active-power controller and alternating-

voltage controller. In specifying the bandwidth αp and αu, the RHP transmission zero of

J (s) has to be considered. As a rule of thumb, the bandwidth of the closed-loop system

should be chosen at least lower than half of the location of the RHP zero [82].

Thus, the controller is designed as

K (s) = J−1n (s)F (s) . (5.9)

If the model is assumed to be perfect, i.e., J (s) J−1n (s) = Jp (s), then the output response

would be

y = Jp (s)F (s) r. (5.10)

Because both Jp (s) and F (s) are diagonal, the resulting closed-loop response is also

diagonal.

The IMC controller can easily be formulated into the classical feedback-control

structure, as shown by the block diagram in Fig. 5.16. The controller K′

(s), therefore,

can be written as

K′

(s) = K (s) [I − J (s)K (s)]−1 (5.11)

where I represents the identity matrix. Substituting (5.6) and (5.9) into (5.11) yields

K′

(s) = J−1n (s)F (s) [I − Jp (s)F (s)]−1. (5.12)

Due to its feedback-control nature, IMC is able to compensate for load disturbances and

model uncertainties. However, for controller design, a “nominal” model needs to be de-

fined. The “nominal” model should be chosen in the “center” of the possible process

models such that the model uncertainties are minimized within all the possible ac-network

configurations and operating conditions.

103


For VSC-HVDC applications, this is not a simple task, since there are a few factors

that affect the model uncertainties of the Jacobian transfer matrix. The following is just a

simple rule to give an initial guidance. The nominal model has to be re-adjusted based on

the results of the robustness tests as shown later.

For power-synchronization control, the power-angle relationship has the dominant

role in the controller design. From Fig. 5.6, it can be observed that the magnitude of

JPθ(s) varies with the SCR of the ac system. The uncertainty of JPθ(s) should be given

higher priority in the controller design since it affects the effective gain of the active-power

controller. If the ac system is simplified as an impedance source and the ac capacitor at

the filter bus is neglected, it is known from Table 4.1 that the static magnitude of JPθ(s) is

inversely proportional to the inductance L, which is the sum of the ac-system inductance

Ln and the phase-reactor inductance Lc. If the short-circuit ratios of the weakest and

strongest ac-system scenarios are SCRmin and SCRmax respectively, it is reasonable to

choose the short-circuit ratio SCRnom of the nominal ac system such that

1

ω1Lc + 1SCRnom

=1

2

(1

ω1Lc + 1SCRmin

+1

ω1Lc + 1SCRmax

). (5.13)

The solution to (5.13) is found to be

SCRnom =2ω1LcSCRminSCRmax + (SCRmin + SCRmax)

2 + ω1Lc(SCRmin + SCRmax). (5.14)

If there is no other preference, the operating point of the nominal model should be chosen

with the full loading of the VSC-HVDC link.

Since the process model is included in the IMC controller, the order of the con-

troller is high if the ac-system configuration is complex. Thus, a simple process model

should be chosen as the nominal model to reduce the order of the controller. For this par-

ticular application, the ac capacitor at the filter bus is neglected in the nominal model, but

included in the robustness tests as a model uncertainty.

Fig. 5.17 shows the step response of the active-power and alternating-voltage of

the closed-loop system of the nominal model. The parameters and initial conditions of the

nominal model are given in Table 5.1. As shown by the figure, the IMC controller success-

fully decouples the cross-coupling between the two control channels. However, the test

result shows that the controller has poor robustness with power-direction uncertainty. The

reason is that the transfer function JPV(s) is power-direction dependent (cf. Table 4.1).

Since the power direction is a known information for a VSC-HVDC link, it is feasible to

design two different controllers for the rectifier and the inverter operation modes respec-

tively.

Fig. 5.18 and Fig. 5.19 show the robustness-test results of the IMC controllers re-

garding variations of the SCRs of the ac system and the power levels for rectifier and

inverter operation respectively. With an SCR = 1.0 p.u., the maximum loading is chosen

as P = 0.86 p.u. instead of P = 1.0 p.u.

104


0 0.1 0.2 0.3

0

0.5

1

From: Pref

To: P

time (sec)

P (

p.u

.)

0 0.1 0.2 0.3

0

0.5

1

From: Uref

To: P

time (sec)

P (

p.u

.)

0 0.1 0.2 0.3

0

0.5

1

From: Pref

To: Uf

time (sec)

Uf (

p.u

.)

0 0.1 0.2 0.3

0

0.5

1

From: Uref

To: Uf

time (sec)

Uf (

p.u

.)

Fig. 5.17 Active-power and alternating-voltage step responses of the IMC controller with the

nominal model. Control parameters: αp = 100 rad/sec, αu = 100 rad/sec.

Table 5.1 Parameters of the nominal model used for IMC and H∞ control design. Per unit based

on 350 MVA and 195 kV.

Parameters Values

Rc 0.01 p.u.

ω1Lc 0.2 p.u.

Main-circuit parameters ω1Cf 0.17 p.u.

Rn 0.01 p.u.

ω1Ln 0.667 p.u.

E0 1.0 p.u.

Initial conditions V0 1.08 p.u.

θu0 (θv0) 40.6◦ (51◦)

kv 0.45 p.u.

High-pass current control αv 40 rad/s

105


0 0.1 0.2 0.3

-0.5

0

0.5

1

1.5

From: Pref

To: P

time (sec)

P (

p.u

.)

0 0.1 0.2 0.3

-0.5

0

0.5

1

1.5

From: Uref

To: P

time (sec)

P (

p.u

.)

0 0.1 0.2 0.3

0

0.5

1

From: Pref

To: Uf

time (sec)

Uf (

p.u

.)

0 0.1 0.2 0.3

0

0.5

1

From: Uref

To: Uf

time (sec)

Uf (

p.u

.)

Fig. 5.18 Robustness tests by active-power and alternating-voltage step responses with the IMC

controller for the VSC-HVDC converter operating in inverter mode. Model variations:

SCR = 1, 2, 5. P = 0.0, 0.5, 1.0 p.u.

0 0.1 0.2 0.3

-0.5

0

0.5

1

1.5

From: Pref

To: P

time (sec)

P (

p.u

.)

0 0.1 0.2 0.3

-0.5

0

0.5

1

1.5

From: Uref

To: P

time (sec)

P (

p.u

.)

0 0.1 0.2 0.3

0

0.5

1

From: Pref

To: Uf

time (sec)

Uf (

p.u

.)

0 0.1 0.2 0.3

0

0.5

1

From: Uref

To: Uf

time (sec)

Uf (

p.u

.)

Fig. 5.19 Robustness tests by active-power and alternating-voltage step responses with the IMC

controller for the VSC-HVDC converter operating in rectifier mode. Model variations:

SCR = 1, 2, 5. P = 0.0,−0.5,−1.0 p.u.

106


+-

(s)K

++ y

(s)Jr u

( )sA

�( )sM

�I +

Fig. 5.20 Additive and multiplicative uncertainties.

For the real system, the measuring signals are usually low-pass filtered, and the

controller computation and PWM switching can also introduce a delay. Thus, a transfer

matrix with the expression

∆J (s) =

[1

1+τse−Tds 0

0 11+τs

e−Tds

](5.15)

is cascade-connected to the Jacobian transfer matrix J (s) for the robustness tests. The

parameters of ∆J (s) are chosen as τ = 3 ms and Td = 0.5 ms.

The design of the IMC controller is an iterative process. If the robustness is not

satisfied, the controller has to be detuned, i.e., choosing lower values of αp and αu, to meet

the robustness requirement. In the next subsection, another multivariable feedback design

approach, i.e., H∞ control, is introduced. With H∞ control, the robustness requirement

on the controller can be explicitly specified in the initial controller design stage.

5.3.2 H∞ control

H∞ control is another methodology for multivariable controller design. The main fea-

ture of H∞ control is its explicit way in dealing with model uncertainties, i.e., an H∞

controller can achieve closed-loop stability with satisfactory performance under process

variations as well as in the presence of other uncertainties such as disturbances and errors

in the sensors [82, 90].

Fig. 5.20 shows a standard multivariable feedback-control block diagram includ-

ing process uncertainties. ∆A (s) and ∆M (s) represent the additive and multiplicative

uncertainties respectively. The transfer matrices S(s), R(s) and T(s) are defined as

S (s) = [I + J (s)K (s)]−1

R (s) = K (s) [I + J (s)K (s)]−1

T (s) = J (s)K (s) [I + J (s)K (s)]−1 (5.16)

where S(s) and T(s) are known as sensitivity and complementary sensitivity functions

respectively. The matrix R(s) does not have a name yet.

107


The maximum singular value of the sensitivity function σ [S (jω)] determines the

disturbance attenuation, because S (s) is, in fact, the closed-loop transfer function from

load disturbances to process output. Thus, a disturbance attenuation performance specifi-

cation can be written as

σ [S (jω)] ≤ |W−11 (jω) |. (5.17)

Allowing the weighting function |W−11 (jω) | to depend on frequency ω enables to specify

a different attenuation factor for each frequency ω. With H∞ control, the performance can

be achieved by finding the controller K (s) through solving the problem

||W1 (jω)S (jω) ||∞ < γ (5.18)

where ∞ denotes the infinite norm. The value γ is a constant which indicates the accuracy

to which the optimal loop matches the desired loop shape. W1(s) is a diagonal transfer

matrix that has W1(s) as the diagonal elements.

The maximum singular value of the complementary sensitivity function σ [T (jω)]

is used to measure the stability margins with respect to ∆M , as shown in Fig. 5.20.

Assuming the additive uncertainty ∆A (jω) = 0, taking σ [∆M (jω)] to be the

definition of the size of ∆M (jω), the size of the smallest multiplicative destabilizing

uncertainty ∆M (jω) is

σ [∆M (jω)] =1

σ [T (jω)]. (5.19)

The smaller is σ [T (jω)], the greater will be the size of the smallest destabilizing multi-

plicative perturbation, and hence the greater will be the stability margin.

A similar result is available for relating the stability margin with respect to the

additive plant perturbations

σ [∆A (jω)] =1

σ [R (jω)]. (5.20)

As a consequence of (5.19) and (5.20), the stability margins of control systems are speci-

fied via singular-value inequalities such as

σ [R (jω)] ≤ |W−12 (jω) |

σ [T (jω)] ≤ |W−13 (jω) |. (5.21)

It is common practice to lump the effects of all plant uncertainties into a single ficti-

tious multiplicative perturbation ∆M , i.e., allowing the weighting function |W−13 (jω) |

to depend on frequency ω to specify a different attenuation factor for each frequency ω.

The stability margin can be achieved by finding the controller K (s) through solving the

problem

||W3 (jω)T (jω) ||∞ < γ (5.22)

where W3(s) is also a diagonal transfer matrix that has W3(s) as the diagonal elements.

In order to guarantee closed-loop stability and at the same time to achieve desired control

108


performance under process uncertainties, the objectives of performance and robust stabil-

ity can be simultaneously achieved by finding a controller K (s) that satisfies both (5.18)

and (5.22). The solution to the optimal problem is often called H∞ controller based on

mixed performance and robustness objectives. The numerical methods for solving H∞

optimization problem are usually complex. Thus, commercial softwares, such as MAT-

LAB, are commonly used to ease such tasks. The major work for the designer is then to

specify W1 (jω) and W3 (jω) to meet the design requirement of the control performance

and robustness.

Following the definition in (5.16), the sensitivity function S (s) and complementary

sensitivity function T (s) have the relation

S (s) + T (s) = I (5.23)

i.e., σ [S (jω)] and σ [T (jω)] cannot both be small at the same frequency. The relationship

between S (s) and T (s) reflects the inherent conflict between control performance and

robustness. Usually, this conflict can be resolved by requiring σ [S (jω)] to be small at low

frequencies, and σ [T (jω)] to be small at high frequencies, due to the fact that the control

performance is more important in the low-frequency range, while measurement noise and

process uncertainties are more often of high-frequency nature.3

With H∞ design, the closed-loop system performance is basically defined by the

weighting functions W1 (jω) and W3 (jω). Optimization algorithms are used to synthe-

size the controller with the bandwidth between the crossover frequency of W1 (jω) and

W3 (jω). Similar to IMC design, the RHP transmission zero of the Jacobian transfer ma-

trix has to be considered in specifying the bandwidth of the closed-loop system, i.e., the

desired bandwidth should be lower than at least half of the location of the RHP zero. The

basic principle for selecting weighting functions are to give W1 (jω) a low-pass property

and W3 (jω) a high-pass property. It is necessary to ensure that the crossover frequency

for the Bode plot of W1 (jω) is below the crossover frequency of W3 (jω), such that there

is a gap for the desired loop shape to pass. Fig. 5.21 shows the Bode plots of W1 (jω) and

W3 (jω), which have the following transfer functions

W1 (s) =sM

+ ωS

s+ ASωS

W3 (s) =s+ ωT

M

AT s+ ωT(5.24)

where M represents the desired bounds on ||S(jω)||∞ and ||T(jω)||∞, AS and AT are

the desired disturbance attenuation inside bandwidth for S(jω) and T(jω), and ωS and

ωT are the crossover frequencies of W1(jω) and W3(jω) respectively.

Fig. 5.22 shows the active-power and alternating-voltage step responses of the lin-

ear closed-loop system of the nominal model where the nominal model is also chosen

3Unfortunately, this is not fully true for VSC-HVDC applications, since the variations of SCRs of the ac

system and power levels of the VSC-HVDC link create process uncertainties in the low-frequency range.

109


100

101

102

103

104

10-1

100

101

102

|W1 (

jω)|

, |W

3 (

jω)|

ω (rad/sec)

|W1 (jω)| |W

3 (jω)|

Fig. 5.21 Bode plots (magnitude) of wighting functions W1 (jω) and W3 (jω). Parameters: M =2.0, AS = 0.0005, AT = 0.03, ωS = 70 rad/sec, ωT = 130 rad/sec [solid: weighting

function |W1 (jω) |, dashed: weighting function |W3 (jω) |].

0 0.1 0.2 0.3

0

0.5

1

From: Pref

To: P

time (sec)

P (

p.u

.)

0 0.1 0.2 0.3

0

0.5

1

From: Uref

To: P

time (sec)

P (

p.u

.)

0 0.1 0.2 0.3

0

0.5

1

From: Pref

To: Uf

time (sec)

Uf (

p.u

.)

0 0.1 0.2 0.3

0

0.5

1

From: Uref

To: Uf

time (sec)

Uf (

p.u

.)

Fig. 5.22 Active-power and alternating-voltage step responses of the H∞ controller with the nom-

inal model.

110


as defined in Table 5.1. Similar to the IMC controller, the H∞ controller also success-

fully decouples the cross-coupling between the two control channels. However, different

from IMC, where a transparent algorithm was used, H∞ control achieved the decou-

pling through optimization. Consequently, a slight cross-coupling can still be observed in

Fig. 5.22. The parameters of the weighting functionsW1 (s) and W3 (s) have been chosen

in such a way that the H∞ controller has the same response time as the IMC controller in

order to compare the results from the two controllers.

Fig. 5.23 and Fig. 5.24 show the robustness-test results of the H∞ controller re-

garding variations of SCRs of the ac system and power levels of the VSC-HVDC link for

rectifier and inverter operations respectively.

In these two sub-sections, two multivariable feedback design approaches, i.e., IMC

and H∞ control, have been applied for a VSC-HVDC link connected to a weak ac sys-

tem. Time simulations are performed with the linear models to verify the robustness of

the controllers regarding variations of SCRs of the ac system and power levels of the

VSC-HVDC link. From the time-simulation results, no significant difference is observed

between the two controllers. In the next subsection, a comparison in the frequency domain

is given.

5.3.3 Performance and robustness comparison

To show the advantages of the multivariable feedback designs proposed in the previous

subsections, the diagonal controller proposed in Chapter 3 is used as a reference design,

i.e.,

K (s) =

[ kp

s0

0 ku

s

]. (5.25)

Fig. 5.25 shows the active-power and alternating-voltage step responses of the VSC-

HVDC converter using the diagonal controller. The nominal model is chosen to be the

same as that is used for the IMC and the H∞ control design, i.e., with the parameters

given in Table 5.1. The control parameters kp and ku are tuned so as to give similar re-

sponse times as the other two controllers.

Fig. 5.26 and Fig. 5.27 show the robustness tests for the VSC-HVDC converter

operating in rectifier mode and inverter mode respectively. Generally speaking, the time-

domain responses of the diagonal controller are more oscillatory than those of the other

two controllers since the diagonal controller disregards the cross-coupling between the

two control loops.

Fig. 5.28 shows the comparison of the above three controllers by plotting the maxi-

mum singular values of their sensitivity functions σ [S (jω)] together with the magnitude

of the weighting function W−11 (s) that was used to synthesize the H∞ controller. The

values of σ [S (jω)] for the IMC and H∞ controllers are almost identical, which is in

111


0 0.1 0.2 0.3

-0.5

0

0.5

1

1.5

From: Pref

To: P

time (sec)

P (

p.u

.)

0 0.1 0.2 0.3

-0.5

0

0.5

1

1.5

From: Uref

To: P

time (sec)

P (

p.u

.)

0 0.1 0.2 0.3

0

0.5

1

From: Pref

To: Uf

time (sec)

Uf (

p.u

.)

0 0.1 0.2 0.3

0

0.5

1

From: Uref

To: Uf

time (sec)

Uf (

p.u

.)

Fig. 5.23 Robustness tests by active-power and alternating-voltage step responses with the H∞

controller for the VSC-HVDC converter operating in inverter mode. Model variations:

SCR = 1, 2, 5. P = 0.0, 0.5, 1.0 p.u.

0 0.1 0.2 0.3

-0.5

0

0.5

1

1.5

From: Pref

To: P

time (sec)

P (

p.u

.)

0 0.1 0.2 0.3

-0.5

0

0.5

1

1.5

From: Uref

To: P

time (sec)

P (

p.u

.)

0 0.1 0.2 0.3

0

0.5

1

From: Pref

To: Uf

time (sec)

Uf (

p.u

.)

0 0.1 0.2 0.3

0

0.5

1

From: Uref

To: Uf

time (sec)

Uf (

p.u

.)

Fig. 5.24 Robustness tests by active-power and alternating-voltage step responses with the H∞

controller for the VSC-HVDC converter operating in rectifier mode. Model variations:

SCR = 1, 2, 5. P = 0.0,−0.5,−1.0 p.u.

112


0 0.1 0.2 0.3

0

0.5

1

From: Pref

To: P

time (sec)

P (

p.u

.)

0 0.1 0.2 0.3

0

0.5

1

From: Uref

To: P

time (sec)

P (

p.u

.)0 0.1 0.2 0.3

0

0.5

1

From: Pref

To: Uf

time (sec)

Uf (

p.u

.)

0 0.1 0.2 0.3

0

0.5

1

From: Uref

To: Uf

time (sec)

Uf (

p.u

.)

Fig. 5.25 Active-power and alternating-voltage step responses of the diagonal controller with the

nominal model. Control parameters: kp = 50 rad/s, ku = 60.

0 0.2 0.4 0.6

-0.5

0

0.5

1

1.5

From: Pref

To: P

time (sec)

P (

p.u

.)

0 0.2 0.4 0.6

-0.5

0

0.5

1

1.5

From: Uref

To: P

time (sec)

P (

p.u

.)

0 0.2 0.4 0.6

0

0.5

1

From: Pref

To: Uf

time (sec)

Uf (

p.u

.)

0 0.2 0.4 0.6

0

0.5

1

From: Uref

To: Uf

time (sec)

Uf (

p.u

.)

Fig. 5.26 Robustness tests of the diagonal controller for the VSC-HVDC converter operating in

rectifier mode. Model variations: SCR = 1, 2, 5. P = 0.0,−0.5,−1.0 p.u.

113


0 0.2 0.4 0.6

-0.5

0

0.5

1

1.5

From: Pref

To: P

time (sec)

P (

p.u

.)

0 0.2 0.4 0.6

-0.5

0

0.5

1

1.5

From: Uref

To: P

time (sec)

P (

p.u

.)

0 0.2 0.4 0.6

0

0.5

1

From: Pref

To: Uf

time (sec)

Uf (

p.u

.)

0 0.2 0.4 0.6

0

0.5

1

From: Uref

To: Uf

time (sec)

Uf (

p.u

.)

Fig. 5.27 Robustness tests of the diagonal controller for the VSC-HVDC converter operating in

inverter mode. Model variations: SCR = 1, 2, 5. P = 0.0, 0.5, 1.0 p.u.

agreement with the step responses shown in Fig. 5.17 and Fig. 5.22. However, the value

of σ [S (jω)] of the diagonal controller is not bounded by W−1 (s), which implies that

the performance of the diagonal controller is worse than the performance requirement

specified by W−1 (s), especially with the peak at around ω = 80 rad/s. This also gener-

ally agrees with the time-domain performance. The step response shown in Fig. 5.25 for

the diagonal controller, is more oscillatory than the step responses shown in Fig. 5.17 and

Fig. 5.22 for the IMC and H∞ controllers. The oscillation mode of the diagonal controller

is at around ω = 80 rad/s.

Fig. 5.29 shows the comparison of the above three controllers by plotting the max-

imum singular values of their complementary sensitivity functions σ [T (jω)] together

with the magnitude of the weighting function W−13 (s) that was used to synthesize the

H∞ controller. The values of σ [T (jω)] for the IMC and H∞ controllers are almost iden-

tical, but the IMC controller, in fact, shows even slightly lower values at frequencies above

ω = 400 rad/s. Of course, this difference cannot be reflected by the robustness tests in

the time domain since a τ = 3 ms (ω = 333.3 rad/s) low-pass filter has been applied in

the test model [cf. (5.15)]. The frequency-domain comparison confirms the similarity of

robustness of the two controllers by the time-domain tests. In Fig. 5.29, it shows that the

value of σ [T (jω)] of the diagonal controller is not bounded by the weighting function

W−13 (s), which implies that the robustness of the diagonal controller is worse than the

robustness requirement specified by W−13 (s), mainly at the peak around ω = 80 rad/s.

By comparing Fig. 5.28 with Fig. 5.29, both σ [S (jω)] and σ [T (jω)] for the diagonal

controller have a high peak around ω = 80 rad/s, which seem to be contradictory to the

114


100

101

102

103

104

10-2

10-1

100

ω (rad/sec)

σ [S

(jω

)]

H∞

controller

IMC controller

Diagonal controller |W-1

1(jω)|

Fig. 5.28 Maximum singular values of the sensitivity functions for nominal performance compar-

ison [solid: H∞ controller, dashed: IMC controller, dotted: diagonal controller, thick

line: weighting function |W−11 (jω) |].

100

101

102

103

104

10-2

10-1

100

ω (rad/sec)

σ [ T

(jω

)] H∞

controller

IMC controller

Diagonal controller|W

-1

3(jω)|

Fig. 5.29 Maximum singular values of the complementary sensitivity functions for robustness

comparison [solid: H∞ controller, dashed: IMC controller, dotted: diagonal controller,

thick line: weighting function |W−13 (jω) |].

115


constraint in (5.23). However, since S(s) and T(s) are transfer matrices, they also differ

by the phases of the transfer functions. Thus, the real meaning of the constraint in (5.23)

is that σ [S (jω)] and σ [T (jω)] cannot both be small at the same frequency, but they can

certainly both be large at the same frequency. In fact, it can be mathematically proven

that σ [S (jω)] and σ [T (jω)] differ at most by 1 [91], i.e., a very high σ [S (jω)] always

implies a very high σ [T (jω)]. In practice, for a controller that yields very poor perfor-

mance with the nominal model, it certainly implies that the controller would also have

poor robustness for model variations!

Remark 1: It is remarkable that the IMC controller solved by the simple algorithm

achieves similar performance and robustness (or even slightly better robustness in the

high-frequency range) as the more advanced optimization-based H∞ control. Generally

speaking, IMC design might yield poor controllers for processes that have poorly damped

poles, since IMC simply inverses the process while the poorly damped poles, in fact,

still remain in the system. However, with power-synchronization control, this problem is

avoided, since the high-pass current control function HHP(s) has shifted all the resonant

poles of the Jacobian transfer matrix towards the left-half plane (cf. Fig. 4.11). On the

other hand, compared to H∞ control, the order of the controller solved by IMC is lower.

With the nominal model used in this section, IMC design yields a controller only of 4th

order while H∞ design yields a controller of 8th order.

Remark 2: The sensitivity function S(s) is a good indicator of the performance

of the closed-loop system. However, it is only valid for the performance of the nominal

model. Meanwhile, the bounded complementary sensitivity function T(s) only ensures

robust stability, i.e., the closed-loop system is only stable within the specified range of

model uncertainties. Thus, these two functions cannot be used as indicators of the per-

formance of the worst-case scenario, i.e., robust performance. To synthesize controllers

that fulfil robust performance, the structured singular value, i.e., the µ, needs to be ap-

plied [92]. However, controllers synthesized by µ can be conservative for this application

since the variations of SCRs of the ac system and power levels of the VSC-HVDC link are

real-number uncertainties, while µ is only tight for complex-number perturbations [93],

e.g., model uncertainties in the high-frequency range. On the other hand, in this section, it

has been demonstrated that the robustness tests performed in the time domain provides a

good overview of the performance of the closed-loop system with all the possible scenar-

ios. Accordingly, the frequency-domain approach and the time-domain approach should

be used as complementary tools for controller design and evaluation.

Remark 3: In this section, the controllers are designed and tested based on the sim-

ple impedance-source representation of the ac system. In reality, the topology of the ac

system is, of course, much more complex. However, given the same SCR and the same

operating point of the VSC-HVDC converter, such variations with the process model ap-

pear mainly in the high-frequency range. These types of uncertainties can easily be dealt

116

5.4. Direct-voltage control

with by feedback control, especially with the high-pass current control function applied.

Therefore, for VSC-HVDC applications, the major challenges for robustness of the con-

troller are the variations of SCRs of the ac system and power levels of the VSC-HVDC

converter. One exception is that, if there is a series-compensated ac line feeding into the

converter station, the series-compensated ac line should be included in the nominal model

for the controller design instead of being treated as a model uncertainty, since the char-

acteristics of a series-compensated ac line can be special as shown in Chapter 4. Another

situation is that, if there are other large input devices in the vicinity of the converter sta-

tions and their dynamic interactions with the VSC-HVDC link are of concern, then an

equivalent system is necessary to be developed to properly reflect the topology of the ac

system.

5.4 Direct-voltage control

For VSC-HVDC applications, at least one of the converter stations has to control the direct

voltage, while the other converter station controls the active power. The active power is

thus automatically balanced between the two converter stations. In this section, various

aspects of direct-voltage control are discussed.

5.4.1 Controller design

The block diagram of the direct-voltage controller is shown in Fig. 5.30, where the multi-

variable power-synchronization controller is used as an inner loop. For the direct-voltage

controller, a PI controller is proposed with the control law

Pref = −(Kpd +

Kid

s

)

︸︷︷︸Fdc(s)

[(uref

dc )2 − u2dc

]. (5.26)

If the power-synchronization controller is assumed to be sufficiently fast, i.e., Pref =

PVSC, and the dc link is assumed to be a single capacitor, i.e., Gdc(s) = 2/(sCdc)

[cf. (4.75)], then the closed-loop system is expressed as

u2dc =

2(Kpds +Kid)

s2Cdc + 2Kpds+ 2Kid(uref

dc )2. (5.27)

The control parameters are selected as

Kpd = αdCdc and Kid = α2dCdc/2 (5.28)

which places two real poles of the closed-loop system at s = −αd. It is generally difficult

to achieve reference tracking and disturbance reduction by a single feedback controller.

117


2ref

dc1)(u

ref

1fU

-+

ref

1P 1P∆

(s)dc1F

0V

V∆

vθ∆

1fU∆+-

+-

++

VSC2P∆−VSC1P∆−

(s)11K

(s)21K

(s)12K

(s)22K

++

(s)rK

2

dc1u∆ 2

dc2u∆

)(sJ

)(dc sG

DC-Jacobian

transfer

matrix

AC-Jacobian

transfer

matrix

Power-synchronization

control inner loop

Fig. 5.30 Control block diagram of the direct-voltage controller.

The solution is to use a two degree-of-freedom controller, where the reference (urefdc )2

is connected to a prefilter to improve the reference tracking while the controller Fdc(s)

is only tuned for disturbance reduction. A convenient practical choice of prefilter is the

lead-lag network

Kr(s) =1 + sT1

1 + sT2

(5.29)

where T1 > T2 is chosen to speed up the response and T1 < T2 is chosen to slow down

the response. The effect of the prefilter is demonstrated in Fig. 5.31, which shows that the

overshoot of the step response of the direct-voltage controller is removed by the prefilter.

As it has been shown in Chapter 4, for long overhead lines, the dc-line inductance

and the dc capacitors may create a resonance peak in the low-frequency range. If the

resonance appears within the bandwidth where the direct-voltage controller is active, an

effective way to mitigate its impact is to add a notch filter. A notch filter commonly has

the expression

Fn(s) =s2 + 2ξ1ωns+ ω2

n

s2 + 2ξ2ωns+ ω2n

(5.30)

where the three adjustable parameters are ξ1, ξ2, and ωn. The ratio of ξ2/ξ1 sets the depth

118


0 0.05 0.1 0.15 0.2

0

0.5

1

time (sec)

ud

c (

p.u

.)

Fig. 5.31 Step response of the direct voltage. The overshoot of the direct voltage is removed by a

prefilter (solid: without prefilter, dashed: with prefilter). Direct-voltage controller: αd =40 rad/sec. Prefilter: T1 = 0.075 sec, T2 = 0.0925 sec.

of the notch, and ωn is the resonance frequency. Fig. 5.32 shows the effect of the notch

filter by the open-loop transfer function of the direct-voltage control, which has the ex-

pression

Hdc(s) = −Fdc(s)Fn(s)Gdc11(s). (5.31)

where the parameters of Gdc11(s), i.e., Gdc(s) are given in Table 4.4. It should be noted

that the notch filter may adversely affect the phase margin of the direct-voltage controller,

as shown in Fig. 5.32.

5.4.2 DC-capacitance requirement

In Section 5.4.1, it has been shown that, by explicitly including the dc capacitance in the

parameters of the direct-voltage controller, one can freely place the poles of the closed-

loop system. This implies that, as long as its size is known, the dc-capacitance does not

affect the linear stability of the direct-voltage control. However, for disturbance reduction,

the dc capacitance has an important role. In this section, the dc-capacitance requirement

for weak-ac-system connections is discussed. The applied method follows the procedure

introduced by [85]. To simplify the analysis, the dc link is assumed to be the single

dc-capacitor model, i.e., Gdc(s) = 2/(sCdc). If the power-synchronization loop has a

bandwidth that is considerably higher than that of the direct-voltage control, the transfer

function from the load disturbance to the error signal becomes

Gpe (s) =2

sCdc

1 + αdCdc

(s+

αd2

s

)2

sCdc

=2s

Cdc (s + αd)2 . (5.32)

If the worst scenario is considered, i.e., the active-power output to the other converter

station is changed stepwise from the maximum active power Pm to 0, e.g., the converter

is suddenly blocked. The step response of the error signal edc (t) in the time domain

119


101

102

103

10-2

100

102

|Hd

c(j

ω)|

101

102

103

-300

-200

-100

0

100

arg

Hd

c(j

ω)

(de

g.)

ω (rad/sec)

Fig. 5.32 Bode plots of Hdc(s) to show the effect of the notch filter for reducing the dc resonance

peak (solid: without notch filter, dashed: with notch filter). Direct-voltage controller:

αd = 40 rad/sec. Notch filter: ωn = 322 rad/sec, ξ1 = 0.2, ξ2 = 0.8.

becomes

edc (t) = L−1

{Gpe (s)

Pm

s

}

= L−1

{2Pm

Cdc (s+ αd)2

}=

2Pm

Cdcte−

αdt . (5.33)

The time derivative of edc (t) is

dedc

dt=

2Pm

Cdc(1 − αdt) e

−αdt (5.34)

which has a local maximum for t = 1/αd. By substituting this into (5.33), the maximum

error is found to be

edc,max =2Pm

αdCdce−1. (5.35)

By considering udc,max the maximum direct-voltage allowed, then edc,max = u2dc,max −

(urefdc )2, the required dc capacitance is

Cdc >2Pme

−1

[u2dc,max − (uref

dc )2]αd

. (5.36)

A common expression for dc capacitance is using its time constant Tdc, which is defined

as

Tdc =Cdc0U

2dcN

2PdcN

=Cdc

2(5.37)

120


where UdcN is the rated (base) direct voltage (kV) and PdcN is the rated (base) power

(MW) of the VSC-HVDC link, and Cdc0 is the dc capacitance (µF). Substituting (5.37)

into (5.36) yields

Tdc >Pme

−1

[u2dc,max − (uref

dc )2]αd

. (5.38)

It should be noted, for the dc-capacitance requirement, only per unit values of the di-

rect voltage and the dc power are applicable for (5.38), while either per unit values or

real values can be applied in (5.36). For weak-ac-system connections, the VSC-HVDC

link needs to operate with large load angles, where the RHP transmission zero of the ac

Jacobian transfer matrix J(s) moves closer to the origin. The RHP zero limits the band-

width of the power-synchronization controller, and eventually limits the bandwidth of the

direct-voltage control. A rule of thumb is that the bandwidth of the power-synchronization

controller should be lower than half of the location of the RHP zero [82]. If the voltages

of the ac source and the filter bus are assumed to be nominal, i.e., E0 ≈ 1.0 p.u. and

Uf0 ≈ 1.0 p.u., the location of the RHP zero can be simplified from (4.17) as

s = ±ω1

√E0 cos θu0

Uf0 −E0 cos θu0≈ ±ω1

√cos θu0

1 − cos θu0. (5.39)

Based on the definition of SCR in (5.3), the well-known power-angle equation between

the ac source and the filter bus can be expressed as

P =E0Uf0

ω1Lnsin θu0 ≈ SCR sin θu0. (5.40)

If it is further assumed that the direct-voltage control is four times slower than the power-

synchronization loop, then αd can be solved by (5.39) and (5.40). Accordingly,

αd <1

8ω1

√cos θu0

1 − cos θu0=

1

8ω1

√√√√√

√1 − ( Pm

SCR)2

1 −√

1 − ( Pm

SCR)2. (5.41)

Another issue that needs to be taken into account is that, if the bandwidth of the direct-

voltage controller is chosen four times slower than the power-synchronization loop, the

inner loop will affect the maximum voltage variation. Fig. 5.33 shows the ratio of band-

width reduction by the inner loop, which is approximated by a first-order filter with band-

width αp. Considering this effect, (5.38) is adjusted as

Tdc >1.3Pme

−1

[u2dc,max − (uref

dc )2]αd

. (5.42)

Given the maximum loading Pm and maximum allowed direct voltage udc,max, the rela-

tionship between dc-capacitance requirement and the SCR of the ac system can be es-

tablished by (5.41) and (5.42). If the worst scenario is considered, the maximum loading

121


1 2 3 4 5 6 7 8 9 101

1.5

2

2.5

αp/α

d

e d

c,d

ma

x/e

° dc,m

ax

Fig. 5.33 Bandwidth reduction by the inner loop.

1 1.2 1.4 1.6 1.8 25

10

15

20

25

30

Td

c (

ms)

SCR

Fig. 5.34 DC-capacitance requirements for weak-ac-system connections with urefdc = 1.0 p.u.

(solid: udc,max = 1.2 p.u., dashed: udc,max = 1.3 p.u., dotted: udc,max = 1.4 p.u.).

should be chosen as Pm = 1.0 p.u. However, as it was mentioned before, for very weak-

ac-system connections, it is recommended that the load angle shall not be above 60◦ to

maintain a reasonable stability margin. For instance, if the SCR of the ac system is 1.0,

then the maximum loading is Pm = SCR sin θu0 = 0.86 p.u. Fig. 5.34 shows the plots

of dc-capacitance requirements for ac systems with SCR ≤ 2.0 with different allowed

udc,max.

5.5 Interconnection of two very weak ac systems

In this section, interconnection of two very weak ac systems by a VSC-HVDC link is

investigated by one design example and followed by linear-model verification and ac-fault

studies simulated by PSCAD/EMTDC. In the design example, a 350 MW VSC-HVDC

link is applied to interconnect two very weak ac systems, where the ac system connected

to the inverter side (station 1, power receiving end) has network strength SCR1 = 1.0,

while the ac system connected to the rectifier side (station 2, power sending end) has

network strength SCR2 = 1.2. The detailed parameters of the converter stations are given

in Appendix C.

From Fig. 5.30, it can be easily observed that, if one converter station controls

the direct voltage, while the other converter station controls the active power, the two

122

5.5. Interconnection of two very weak ac systems

2ref

dc1)(u

ref

1fU

ref

1P

(s)dc1F

+-

(s)r1K

VSC1P∆−

2ref

dc2 )(u

ref

2fU

ref

2P

(s)dc2F

+-

(s)r2K

VSC2P∆−

(s)pcF

ord

1P

1P∆

+-

+

2ref

dc1)(u∆

+

2

dc1u∆ 2

dc2u∆

)(dc sG

Power-

synchronization

inner loop

Power-

synchronization

inner loop

DC-Jacobian

transfer

matrix

Fig. 5.35 Control block diagram of a VSC-HVDC link interconnecting two weak ac systems.

converter stations are, in fact, linearly independent. This implies that the stability of one

converter station does not affect the stability of the other converter station. One might

consequently conclude that there is nothing more special for a VSC-HVDC link connected

to two weak systems than it is only connected to one weak system. This might be true

if only linear effects are considered. However, the real system is non-linear. For VSC-

HVDC operations, the direct voltage has to be carefully maintained around its nominal

value. For instance, a big direct-voltage drop might temporarily limit the capability of the

alternating-voltage controller and negatively affect the linear stability of the closed-loop

system.

The proposed control structure for weak-ac-system interconnection is shown in

Fig. 5.35, where the dashed block represents the power-synchronization control inner

loop in Fig. 5.30. The basic idea of the design is that both of the two converter stations

have direct-voltage controllers, while the active-power controlling station controls the

active power by adding an additional contribution to the reference of the direct-voltage

controller and its output shall be limited. With the proposed control structure the linear

independence between the rectifier station and the inverter station is lost. However, the

bandwidth of the direct-voltage controllers is much higher than the bandwidth of the ac-

system dynamics. This implies that the “firewall” effect of the VSC-HVDC link is still in

force.

In the following, the detailed procedure for controller design and parameter settings

are given. The linear model is compared to the simulation results from PSCAD/EMTDC

for each major design step.

1. Power-synchronization inner loop. Due to the very low SCR of the inverter ac

system, the maximum allowed power is Pm1 = 0.86 p.u., which corresponds to ap-

proximately θu10 = 60◦ load angle. The ac system at the rectifier side has a slightly

123


higher SCR, but considering the losses of the converter station and of the dc cable,

it has a maximum loading Pm2 = −0.91 p.u., which corresponds approximately to

a load angle of θu20 = −50◦.

The power-synchronization controller is used as the inner loop of the direct-voltage

controller as previously shown in Fig. 5.30. For this application, the IMC controller

investigated in Section 5.3 is applied to parameterize the power-synchronization

controller. According to the discussion in Section 5.4.2, the bandwidth of the power-

synchronization controller should be lower than half of the RHP-zero location of

the ac Jacobian transfer matrix. By using (5.39), the bandwidths of the active-

power controllers of power-synchronization control at the two converter stations

can be calculated as αp1 = 160 rad/s and αp2 = 210 rad/s. The bandwidths of the

alternating-voltage controllers, however, do not need to be very high. Thus, they

have been chosen as αu1 = αu2 = 50 rad/s.

2. Direct-voltage controller. As mentioned in Section 5.4.1, the bandwidth of the

direct-voltage controllers should be chosen at least four times slower than the active-

power controller of the power-synchronization control. Alternatively, the bandwidth

of the direct-voltage controllers can be calculated directly by substituting the SCR

and maximum loading into (5.41), which yieldsαd1 = 39 rad/s and αd2 = 53 rad/s.

However, the linear analysis shows that the direct-voltage controller at the rectifier

side (station 2) tends to have less phase margin for the same load angle than that at

the inverter (station 1) side. Therefore, the bandwidths of both of the direct-voltage

controllers have been chosen as αd1 = αd2 = 40 rad/s. Fig. 5.36 shows the step

response of the direct voltages at the two converter stations with the comparison

between the linear model and the nonlinear model in PSCAD/EMTDC.

The maximum allowed direct voltage is 1.3 p.u. Based on the curves of the dc-

capacitance requirement in Fig. 5.34, the dc capacitor has been chosen to make the

total dc capacitance Tdc = 15 ms. Fig. 5.37 shows the plots of disturbance reduc-

tion of the direct-voltage controllers at the two converter stations with the compar-

ison between the linear model and the nonlinear model in PSCAD/EMTDC. The

maximum direct-voltage peak in Fig. 5.37(b) is below the specification udc,max =

1.3 p.u. The discrepancies between the results of the linear models and the non-

linear simulations are due to that the results of the linear model are obtained at

one operating point, while the disturbance reduction in PSCAD/EMTDC involves

operating-point changes. The tests in PSCAD/EMTDC are performed by blocking

the converter at the other station with maximum loading.

3. Active-power controller. In principle, either of the VSC-HVDC converter stations

can be the power-controlling station. In this example, the inverter station (station 1)

controls the active power by a proportional-integral (PI) controller with the control

124


0 0.05 0.1 0.15 0.20.95

1

1.05

1.1

1.15

time (sec)

ud

c1 (

p.u

.)

(a)

0 0.05 0.1 0.15 0.20.95

1

1.05

1.1

1.15

time (sec)

ud

c2 (

p.u

.)

(b)

Fig. 5.36 Step response of the direct-voltage controllers (solid: nonlinear simulation, dashed: lin-

ear model). (a) Direct-voltage step at station 1. (b) Direct-voltage step at station 2.

0 0.1 0.2 0.3 0.4 0.5

0.6

0.8

1

time (sec)

ud

c1 (

p.u

.)

(a)

0 0.1 0.2 0.3 0.4 0.5

1

1.1

1.2

1.3

1.4

time (sec)

ud

c2 (

p.u

.)

(b)

Fig. 5.37 Disturbance reductions of the direct-voltage controllers with maximum load changes

(solid: nonlinear simulation, dashed: linear model). (a) Converter block at station 2. (b)

Converter block at station 1.

125


law

∆(urefd1 )2 = −Kpp(1 +

Kip

s)

︸︷︷︸Fpc(s)

[P ord

1 − P1

]. (5.43)

The parameters of the active-power controller are tuned by the root-locus technique

in two steps:

• Step 1: Start with tuning the proportional gain Kpp by applying P control, i.e.,

Fpc(s) = −Kpp.

• Step 2: The integral gain Kip is tuned by applying the full PI controller.

Fig. 5.38 shows the root-loci of the closed-loop system by applying the propor-

tional controller. By varying Kpp from 0.0 to 1.0, three dominant pole pairs are

affected. The pole pair p5,6 (p5 = −10.8 rad/s, p6 = −0.296 rad/s) is shifted to-

wards the left-half plane, which can be viewed as the stabilizing effect of feedback

control. However, both the pole pairs p1,2 (p1,2 = −63.7 ± j232 rad/s) and p3,4

(p3 = −37.1 rad/s, p4 = −50 rad/s) are shifted towards the right-half plane. The

frequency-domain analysis shows that p1,2 is related to the gain margin, while p3,4

is related to the phase margin of the active-power control. Kpp = 0.6 is chosen to

get a balance of stability and response time, which places the three pole pairs at

p1,2 = −53 ± j227 rad/s, p3,4 = −27 ± j24 rad/s, and p5,6 = −15.1 ± j3 rad/s.

Fig. 5.39 shows the root-loci of the closed-loop system by applying the PI-type

active-power controller, where p0 is a pole introduced by the integral controller. By

varying Kip from 0.0 to 60, p0 moves quickly towards the left-half plane. The pole

pair p1,2 is rather insensitive to the variation of Kip, and p5,6 move towards two left-

half plane zeros on the real axis. However, p3,4 is negatively affected. Frequency-

domain analysis also shows that the phase margin of the active-power control is

reduced by increased integral gain Kip. However, a larger integral gain is neces-

sary in reducing the power-recovery time after ac-system faults and minimizing the

steady-state error. Finally,Kip = 30 is chosen which places the three dominant pole

pairs at p1,2 = −54 ± j225 rad/s, p3,4 = −13.2 ± j29.4 rad/s, p5 = −28 rad/s,

and p6 = −10 rad/s.

Fig. 5.40 shows the step response of the active-power control at low and high power

levels respectively. The step response at the high power level corresponds to the

operating point applied for the root-locus tuning.

It should be noted that the output of the active-power controller should be limited

to avoid too large direct-voltage variations. In this example, the limitation of the

output of the active-power controller is chosen as ±0.25, which corresponds to

approximately +12% and −13% direct-voltage variations.

126


-80 -70 -60 -50 -40 -30 -20 -10 0 10-250

-200

-150

-100

-50

0

50

100

150

200

250

ω (rad/sec)

ω (

rad

/se

c)

p1

p2

p3

p4

p5

p6

Fig. 5.38 Root-loci of the dominant poles of the closed-loop system by applying P-type active-

power control. Kip = 0.0, variations of Kpp from 0 to 1.0.

-80 -70 -60 -50 -40 -30 -20 -10 0 10-250

-200

-150

-100

-50

0

50

100

150

200

250

ω (rad/sec)

ω (

rad

/se

c)

p1

p2

p3

p4

p0

Fig. 5.39 Root-loci of the dominant poles of the closed-loop system by applying PI-type active-

power control. Kpp = 0.6, variations of Kip from 0 to 60.

127


0 0.1 0.2 0.3 0.4 0.5

0.7

0.75

0.8

time (sec)

P1 (

p.u

.)

(a)

0 0.1 0.2 0.3 0.4 0.5

0

0.05

0.1

time (sec)

P1 (

p.u

.)

(b)

Fig. 5.40 Step response of the active-power controller at low and high power levels at converter

station 1 (solid: nonlinear simulation, dashed: linear model). Active-power controller:

Kpp = 0.6, Kip = 30. (a) Step from P1 = 0.7 p.u. to P1 = 0.8 p.u. (b) Step from

P1 = 0.0 p.u. to P1 = 0.1 p.u.

The fault ride-through capability of the VSC-HVDC link is tested in PSCAD/-

EMTDC by applying three-phase ac-system faults at both of the converter stations. The

VSC-HVDC link initially operates with the maximum loading, i.e., P1 = 0.86 p.u. The

VSC-HVDC link is supposed to ride through ac-system faults without relying on telecom-

munications between the two converter stations.

In Fig. 5.41, a three-phase ac fault with 0.2 sec duration is applied at the inverter

station (station 1), i.e., the power-controlling station in this example, at 0.1 sec. One con-

sequence of the ac fault is that the direct voltage increases to approximately 1.3 p.u. due

to the loss of power output. The rectifier station (station 2) brings down the direct voltage

to its nominal value after an initial overshooting. Another consequence of the ac fault is

the increase of the modulus of the valve current |ic1|. After detecting the fault, the current

limiter reduces the converter current to half of the maximum load current Imax (or any

other desired value) except a very short current spike at the fault inception.

Fig. 5.42 shows a three-phase ac fault with 0.2 sec duration applied at the rectifier

station, i.e., the direct-voltage controlling station in this example, at 0.1 sec. During the

ac-system fault, the power-controlling station controls the direct voltage to a lower voltage

level which is 13% less than the nominal value. The 13% is a result of the limitation of the

active-power controller. Since the ac fault is applied at the rectifier side, there is no over-

128


0 0.2 0.4 0.6 0.8 1

0

0.5

1

P1 (

p.u

.)

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5u

dc1 (

p.u

.)

0 0.2 0.4 0.6 0.8 10

1

2

time(sec)

| i c

1| (p

.u.)

Fig. 5.41 Fault ride-through capability of the VSC-HVDC link during a three-phase ac-system

fault at the inverter station (station 1). Upper plot: active power from the VSC-HVDC

link at station 1. Middle plot: direct voltage at station 1. Lower plot: valve current at

station 1.

0 0.2 0.4 0.6 0.8 1

0

0.5

1

P1 (

p.u

.)

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

ud

c1 (

p.u

.)

0 0.2 0.4 0.6 0.8 10

1

2

time (sec)

| i c

1| (p

.u.)

Fig. 5.42 Fault ride-through capability of the VSC-HVDC link during a three-phase ac-system

fault at the rectifier station (station 2). Upper plot: active power from the VSC-HVDC

link at station 1. Middle plot: direct voltage at station 1. Lower plot: valve current at

station 1.

129


current problem at the inverter station (station 1). A specific amount of active power flows

in the reverse direction during the fault period. However, the current limitation controller

at the rectifier (station 2) limits the fault current to half of the maximum load current.

5.6 Summary

In this chapter, the control and modeling issues for VSC-HVDC links connected to high-

impedance ac systems are investigated. Power-synchronization control and vector current

control are compared for VSC-HVDC links connected to weak ac systems. It is con-

cluded that, for weak-ac-system connections, the voltage variations at the filter bus are

much less for power-synchronization control than for vector current control. Thus, power-

synchronization control is the most suitable controller for VSC-HVDC links connected

to weak ac systems. Two multivariable feedback designs, i.e., internal model control and

H∞ control, are applied and compared with the simple diagonal controller. The multi-

variable feedback design show clear advantages in control performance and robustness.

The IMC controller is simple yet achieves similar results as the more advanced H∞ con-

troller. A two-degree-of-freedom direct-voltage control with a prefilter to reduce the over-

shoot is proposed. For VSC-HVDC links connected to weak ac systems, a higher value

of dc capacitance is required. A control structure for interconnecting two very weak ac

systems is proposed. As an example, it is shown that a VSC-HVDC link using power-

synchronization control enables a power transmission of 0.86 p.u. from a system with an

SCR of 1.2 to a system with an SCR of 1.0.

130

Chapter 6

Control of VSC-HVDC Links

Connected to Low-Inertia AC Systems

In this chapter, power-synchronization control is applied to control VSC-HVDC links con-

nected to low-inertia ac systems, or island systems. In Section 6.1, general aspects of

low-inertia ac systems are described. In Section 6.2, a frequency droop controller and an

alternating-voltage droop controller are proposed. In Section 6.3, a typical island system

which includes four common power components, i.e., a synchronous generator, an induc-

tion motor, some passive RLC loads, and a VSC-HVDC link is modeled by the Jacobian

transfer matrix modeling concept. In addition, the linear model is applied for control-

parameter tuning using root-locus techniques. In Section 6.4, simulation studies are per-

formed with PSCAD/EMTDC to demonstrate the flexibility of power-synchronization con-

trol for various network conditions. In Section 6.5, the Jacobian transfer matrices for a

synchronous generator and an induction motor are developed, and their characteristics

are analyzed. Finally, the subsynchronous characteristics of a VSC-HVDC converter are

analyzed by the frequency-scanning method in Section 6.6. The major results of this chap-

ter are summarized in Section 6.7. Some results of this chapter are included in [94].

6.1 General aspects of low-inertia ac systems

In this chapter, various aspects of a VSC-HVDC link connected to another type of weak

ac systems, i.e., low-inertia or island systems, are investigated. Low-inertia systems are

considered to have limited number of rotating machines in the system, or no rotating ma-

chine at all. Examples of such applications can be found when an HVDC link is powering

an island system, or if it is connected to a windfarm. It may also be the case when an

HVDC link initially is connected to a large ac system, but comes into island operation

due to tripping of critical ac lines to the large system, or black start after a blackout.

For the conventional LCC-HVDC system, the commutations of the converter valves

depend on the ability of the ac system to maintain the required voltage and frequency.

131

Chapter 6. Control of VSC-HVDC Links Connected to Low-Inertia AC Systems

Therefore, the conventional LCC-HVDC system has a requirement on the minimum iner-

tia of the connected ac system. Similar to the definition of SCR, it was proposed in [5] to

use the effective inertia constant Hdc as a measure of the relative rotational inertia, which

is defined as

Hdc =Hac

PdcN

(6.1)

where Hac (MW · s) is the total rotational inertia of the ac system and PdcN is the rated dc

power. For a conventional LCC-HVDC link connected to an island system, an effective

inertia constant, Hdc, of at least 2.0 s to 3.0 s is required for satisfactory operation [32].

Synchronous condensers (SCs) have been the traditional means to increase the in-

ertia of an island system [95]. Since SCs also increase the short-circuit capacity of the

ac system, the weak-ac-system problem of the conventional LCC-HVDC system, either

low SCR or low inertia, can generally be solved by installation of SCs. However, SCs can

substantially increase the investment and maintenance costs of an HVDC project.

In this respect, the VSC-HVDC technology has a clear advantage over the con-

ventional LCC-HVDC technology. Different from the thyristor valves, where relatively

stiff voltage and frequency of the ac system are pre-conditions for valve commutations,

the VSC can produce its own alternating-voltage waveform independent of the ac system.

Thus, a VSC-HVDC link can even be connected to a passive network with no other power

source at all [96, 97].

However, control of VSC-HVDC links connected to an island system is technically

different from connected to a large ac system for the following reasons:

1. Due to the relatively limited number of power generating units in a typical island

system, the VSC-HVDC link should participate in frequency control of the island

system instead of applying constant power control.

2. An island system with low inertia is often of small geographic extent and the gen-

erators and loads are found electrically close to the converter station. More interac-

tions are expected between the VSC-HVDC converter and other input devices, e.g.,

the risks of voltage-control hunting and SSTI with local synchronous generators are

higher.

3. In some situations, island operation is a consequence of ac-system failures by los-

ing critical ac lines to a large ac system. The control system of the VSC-HVDC link

is required to be able to handle the transition of operation modes seamlessly.

In the following sections, the above issues related to low-inertia ac systems are investi-

gated for a VSC-HVDC link using power-synchronization control.

132

6.2. Controller design

refP∆

f

f

sT

K

+1 s

kp

mT1

s

s+

vθ∆

ω∆

refP

P+

-

+refω

+ -

Island

system

Fig. 6.1 Frequency droop controller.

6.2 Controller design

In this chapter, it is assumed that the VSC-HVDC link is connected to a strong system

at the other converter station, which controls the direct voltage of the dc link constant

disregarding the power variations of the converter connected to the island system. Thus,

only the controllers of the converter station connected to the island system are discussed.

In Chapter 5, multivariable feedback designs have been applied for VSC-HVDC

links connected to high-impedance ac systems. However, such designs are not considered

necessary for low-inertia system connections, since the bandwidth requirement for the

control system is generally not as critical for island operation. Thus, to simplify the control

design discussions, the simple diagonal controller in (5.25) is applied. The tuning of the

control parameters kp and ku will be further discussed in Section 6.3 using the root-locus

techniques.

6.2.1 Frequency droop control

For a VSC-HVDC link connected to an island system, the VSC-HVDC converter shall

not have constant power control but rather participate in frequency control of the island

system. A frequency controller of the following form is proposed

∆Pref =

(Kf

1 + Tfs

)[ωref − ω] (6.2)

where ωref is the reference and ω is the measured value of the angular frequency. The

output of the frequency controller is added to the power reference of the active-power

controller, as shown in Fig. 6.1. In the figure, the frequency measurement is taken by

the derivative of the angle output of the active-power controller with a measuring time

constant Tm, typically in the range of 10 − 20 ms. Kf and Tf are the gain and time

constant of the frequency controller.

Such a design gives the VSC-HVDC link a frequency droop characteristic. If there

are any other power generating units in the island system, the load sharing is determined

133


SG

tgtg jXR +

tvtv jXR +

VSC

QP,

11,QP

22 ,QP

gi

vi

PCC

+

-ge

+

-fu

Fig. 6.2 Voltage droop control for parallel connected voltage-control units.

by the frequency droop of each power generating unit, which is defined as R = ∆ω/∆P .

For two power generating units with frequency droops R1 and R2, the following relation-

ship is established for the power outputs ∆P1 and ∆P2

∆P1

∆P2=R2

R1. (6.3)

Accordingly, the power generating unit with smaller frequency droop shares more loads

in the island system. The frequency droop of the VSC-HVDC link with the proposed

frequency controller in (6.2) can be expressed as

∆ω =1

1kp

+Kf︸︷︷︸

Rvsc

(Pref − P ). (6.4)

Thus, the frequency droop of the VSC-HVDC link depends on both Kf and kp. The time

constant Tf should be chosen similar to the time constant of the turbines of the local

generators in the island system.

6.2.2 Alternating-voltage droop control

If two or more alternating-voltage controlling units are connected to a common bus, the

alternating-voltage controls should be coordinated to avoid voltage hunting between the

units. Fig. 6.2 shows a VSC-HVDC link connected in parallel with a synchronous gen-

erator with step-up transformers. In Fig. 6.2, eg, uf are the terminal/filter-bus voltage

vectors of the generator, the VSC-HVDC converter. P and Q are the active power and

reactive power to the island system. P1 and Q1 are the active power and reactive power

from the synchronous generator, while P2 and Q2 are the active power and reactive power

from the VSC-HVDC link. Rtg and Xtg are the resistance and leakage reactance of the

step-up transformer of the synchronous generator, while Rtv and Xtv are the resistance

and leakage reactance of the converter transformer of the VSC-HVDC converter. The so-

lution to solve the voltage-control confliction between the synchronous generator and the

134

6.3. Dynamic modeling and linear analysis of a typical island system

Large ac

system

VSC

SG

IM

tgL

tmL

lL

lR lC

tvL

gi

mi

+

-ge

+

-me

+

-ve

+

-

pccuLi

vi

Passive

loadAC network

PCC

Fig. 6.3 A typical island system.

VSC-HVDC converter is using the so-called “load compensation”, which is a common

solution for synchronous generators terminated at the same bus [32]. That is, instead of

controlling the alternating voltage of the point-of-common-coupling (PCC) bus, the VSC-

HVDC converter and the synchronous generator both control voltages between their own

terminal/filter-bus voltages and the voltage of the PCC to give droop characteristics to

their alternating-voltage controls. With load compensation, the magnitudes of the result-

ing compensated voltages V gc of the synchronous generator and V v

c of the VSC-HVDC

converter are

V gc =|eg + kg(Rtg + jXtg)ig|V v

c =|uf + kc(Rtv + jXtv)iv| (6.5)

where the compensation ratios kg and kc are typically chosen between 50% and 80%. By

adjusting the compensating ratios, the reactive-power sharings between the VSC-HVDC

converter and the synchronous generator are re-distributed.

6.3 Dynamic modeling and linear analysis of a typical is-

land system

Fig. 6.3 shows an island system introduced by [98], which includes four common com-

ponents in a typical island system, i.e., a synchronous generator (SG), an induction motor

(IM), some passive RLC loads, and the VSC-HVDC link. The induction-motor loads are

135


included in the studied system since they are quite common in industrial systems [99].

The space vectors eg, em and ev are the terminal voltage vectors of the SG, the IM, and

the VSC-HVDC converter, while ig, im and iv are the corresponding injecting current

vectors. Ltg, Ltm, and Ltv are the leakage inductances of the step-up transformers of the

SG, the IM, and the converter transformer of the VSC-HVDC converter. Rl, Ll and Cl are

the resistance, inductance, and capacitance of the passive loads. iL is the current vector

flowing through Ll, and upcc is the voltage vector of the PCC. An ac line is connected to

a large ac system at the PCC bus. This connection is intentional added to demonstrate the

mode shift of the VSC-HVDC link in Section 6.4.

6.3.1 Jacobian transfer matrix

In Chapter 4, the concept of Jacobian transfer matrix was proposed as a unified dynamic

modeling technique for ac/dc systems. For the network configurations applied in Chap-

ter 4, the ac Jacobian transfer matrices were derived directly from the state-space repre-

sentation of the combined ac network and the main circuit of the VSC. Such a procedure

is practical if only one input device is connected to a simple ac network. However, for a

larger ac network connected to several input devices, it is necessary to develop the state-

space representation of the ac network and the electrical part of each individual input

device separately as several “modules”, while the ac Jacobian transfer matrix is derived

by connecting all the “modules” together. Such an approach not only eases the task for

model development, but more importantly, the developed model is easier to be modified

if the topology of the ac network and/or the input devices are changed.

In the following subsections, such an approach is applied for the model develop-

ment of the island system in Fig. 6.3. All the three input devices, i.e., the SG, the IM,

and the VSC-HVDC converter are modeled as current sources which inject currents to

the ac network, while the ac network supplies the terminal voltage to each input de-

vice [100, 101].

Modeling of the ac network

The ac network is defined as the electrical circuit inside the dashed box in Fig. 6.3. In

Chapter 4, a constant-frequency stiff voltage source E was assumed, while the d com-

ponent of the synchronous dq frame was chosen aligned with the real part of E. Such a

constant-frequency stiff voltage source cannot be assumed in an island system. However,

for model development this is not really a problem. In fact, any electrical node in the is-

land system can principally be chosen as a reference point, as long as the reference frame

is not “disturbed” by the dynamics of the system. For this particular application, the PCC

of the ac-network is chosen as the reference point, a common synchronous R − I refer-

ence frame is defined which has the R axis aligned with the real axis of the voltage vector

upcc. In the defined synchronous R − I reference frame, the dynamic equations of the ac

136


network can be expressed as

LldiLdt

= upcc − jω1LliL

Cldupcc

dt= ig + im + iv − iL − upcc

Rl− jω1Clupcc

eg = upcc + jω1Ltgig + Ltgdigdt

em = upcc + jω1Ltmim + Ltmdimdt

ev = upcc + jω1Ltviv + Ltvdivdt. (6.6)

By linearization and writing in component form, (6.6) can be expressed in state-space

form

dxn

dt= Anxn +Bnun

yn = Cnxn +Dnun +Dn1dun

dt(6.7)

where the matrices An and Bn are

An =

0 ω11Ll

0

−ω1 0 0 1Ll

− 1Cl

0 − 1ClRl

ω1

0 − 1Cl

−ω1 − 1ClRl

, Bn =

0 0 0 0 0 0

0 0 0 0 0 0

1Cl

0 1Cl

0 1Cl

0

0 1Cl

0 1Cl

0 1Cl

(6.8)

and the matrices Cn and Dn1 are

Cn =

0 0 0 0 0 0

0 0 0 0 0 0

1 0 1 0 1 0

0 1 0 1 0 1

T

, Dn1 =

Ltg 0 0 0 0 0

0 Ltg 0 0 0 0

0 0 Ltm 0 0 0

0 0 0 Ltm 0 0

0 0 0 0 Ltv 0

0 0 0 0 0 Ltv

(6.9)

137


aR q

g

r ψω+- lL

adL

dL1

dR1

sdidi1 fdi fdL

fdR

+

-fde

+

-

sde

aR+ - lL

aqL

qL1

qR1

sqiqi1

+

-

sqe

qL2

qR2

qi2

d

g

rψω

Fig. 6.4 Equivalent circuit of the synchronous generator.

and the matrix Dn is

Dn =

0 −ω1Ltg 0 0 0 0

ω1Ltg 0 0 0 0 0

0 0 0 −ω1Ltm 0 0

0 0 ω1Ltm 0 0 0

0 0 0 0 0 −ω1Ltv

0 0 0 0 ω1Ltv 0

(6.10)

and the inputs, outputs, and state variables are

un =[

∆igR ∆igI ∆imR ∆imI ∆ivR ∆ivI

]T

yn =[

∆egR ∆egI ∆emR ∆emI ∆evR ∆evI

]T

xn =[

∆iLR ∆iLI ∆upccR ∆upccI

]T. (6.11)

The subscript R represents the real component, while I represents the imaginary compo-

nent in the common ac-network R− I frame.

Modeling of the electrical part of the synchronous generator

Fig. 6.4 shows the equivalent circuit of a salient pole synchronous machine based on the

two-axis theory where the three-phase windings of the synchronous machine are repre-

138


sented by two orthogonal windings along the d and q axes [32]. In a dq frame chosen along

the rotor DQ axis, the dynamic equations of the stator and the rotor can be expressed as

esd =dψd

dt− ψqω

gr −Raisd

esq =dψq

dt+ ψdω

gr − Raisq

efd =dψfd

dt+Rfdifd

0 =dψ1d

dt+R1di1d

0 =dψ1q

dt+R1qi1q

0 =dψ2q

dt+R2qi2q (6.12)

where esd and esq are the stator voltage components in the d and q directions respectively.

The quantity efd is the field voltage. The quantities isd and isq are the stator current com-

ponents in the d and q directions respectively. The quantity ifd is the field current. The

quantities i1d, i1q , and i2q are the currents of the damping circuits (two damping circuits

in q axis). Ra is the stator resistance.R1d, R1q, and R2q are the resistances of the damping

circuits.Rfd is the resistance of the field circuit (d axis). ψd and ψq are the flux linkages of

the stator in the d and q directions. ψfd is the flux linkage of the field circuit. ψ1d, ψ1q , and

ψ2q are the rotor flux linkages. ωgr is the rotor angular speed. The flux linkages in (6.12)

are expressed as

ψd = −(Lad + Ll︸︷︷︸Lsd

)isd + Ladifd + Ladi1d

ψq = −(Laq + Ll︸︷︷︸Lsq

)isq + Laqi1q + Laqi2q

ψfd = −Ladisd + (Lad + Lfd︸︷︷︸Lfmd

)ifd + Ladi1d

ψ1d = −Ladisd + Ladifd + (Lad + L1d︸︷︷︸L1md

)i1d

ψ1q = −Laqisq + (Laq + L1q︸︷︷︸L1mq

)i1q + Laqi2q

ψ2q = −Laqisq + Laqi1q + (Laq + L2q︸︷︷︸L2mq

)i2q (6.13)

where Ll is the stator leakage inductance. Lad and Laq are the d and q axis mutual induc-

tances. Lfd is the field leakage inductance. L1d, L1q and L2q are the leakage inductances

of the damping circuits. By linearization, the dynamic equations of the synchronous gen-

139


erator can be expressed in state-space form

dxg

dt= −L−1R︸︷︷︸

Ag

xg + L−1Bu︸︷︷︸Bg

ug +[

L−1Bω L−1Bf

]︸︷︷︸

Bg1

ug1

yg =

[1 0 0 0 0 0

0 1 0 0 0 0

]

︸︷︷︸Cg

xg (6.14)

where the inductance matrix L is

L =

−Lsd 0 Lad Lad 0 0

0 −Lsq 0 0 Laq Laq

−Lad 0 Lfmd Lad 0 0

−Lad 0 Lad L1md 0 0

0 −Laq 0 0 L1mq Laq

0 −Laq 0 0 Laq L2mq

(6.15)

and the matrix R is

R =

−Ra ωgr0Lsq 0 0 −ωg

r0Laq −ωgr0Laq

−ωgr0Lsd −Ra ωg

r0Lad ωgr0Lad 0 0

0 0 Rfd 0 0 0

0 0 0 R1d 0 0

0 0 0 0 R1q 0

0 0 0 0 0 R2q

(6.16)

and the matrices Bu, Bω, and Bf are

Bu =

[1 0 0 0 0 0

0 1 0 0 0 0

]T

, Bf =[

0 0Rfd

ωgr0Lad

0 0 0]T

Bω =[−isq0Lsq (isd0Lsd − ifd0Lad) 0 0 0 0

]T(6.17)


ug =[

∆esd ∆esq

]T, ug1 =

[∆ωg

r ∆efd

]T

yg =[

∆isd ∆isq]T, xg =

[∆isd ∆isq ∆ifd ∆i1d ∆i1q ∆i2q

]T(6.18)

where the subscript 0 denotes the value of the operating point. In (6.18), the rotor speed

∆ωgr is connected to the transfer function of the rotor, while the field voltage ∆efd is

140


se

R

I

d

q

δ

sRe

sIe

sdesqe

1ω

g

rω

Fig. 6.5 AC-network R − I reference frame vs. synchronous-generator dq frame.

connected to the excitation control. The dynamic equations described by (6.12) is only

valid if the reference frame is chosen along the DQ axes of the rotor. In order to connect

the synchronous generator to the ac network, the state-space description in (6.14) has

to be transformed to the common ac-network R − I frame, as shown in Fig. 6.5. The

angle δ represents the electrical angle between the two reference frames. Accordingly, the

terminal voltage of the synchronous generator in the two reference frames are related as

edqs = eRI

s e−jδ (6.19)

in linearized component form

[∆esd

∆esq

]=

[cos δ0 sin δ0

− sin δ0 cos δ0

]

︸︷︷︸PE

[∆esR

∆esI

]+

[−esR0 sin δ0 + esI0 cos δ0

−esR0 cos δ0 − esI0 sin δ0

]

︸︷︷︸PE1

∆δ.

(6.20)

Similarly, the stator current of the synchronous generator in the two reference frames are

related as

iRIs = idq

s ejδ (6.21)

in linearized component form

[∆isR

∆isI

]=

[cos δ0 − sin δ0

sin δ0 cos δ0

]

︸︷︷︸PI

[∆isd

∆isq

]+

[−isd0 sin δ0 − isq0 cos δ0

isd0 cos δ0 − isq0 sin δ0

]

︸︷︷︸PI1

∆δ. (6.22)

Substituting (6.20) and (6.22) into (6.14), yields

dxg

dt= Agxg +B

′

gu′

g +B′

g1u′

g1

y′

g = C′

gxg +Dg1u′

g1 (6.23)

141


sR λsL

mL

si

-

+

-

se

rRλrL

ri

+

r

m

rj ψω

Fig. 6.6 Equivalent circuit of the induction motor.

where

u′

g =[

∆esR ∆esI

]T, y

′

g =[

∆isR ∆isI]T

u′

g1 =[

∆δ ∆ωgr ∆efd

]T, B

′

g = BgPE, B′

g1 =[BgPE1 Bg1

]

C′

g = PICg, Dg1 =[PI1 zeros (2, 2)

]. (6.24)

That is, the input and output variables have been transformed to the common ac-network

R− I frame. After reference-frame transformation, u′

g1 has one additional input variable

∆δ, which is connected to the rotor transfer function. It should be noted that the state

variables of (6.23) are still in the synchronous-generator dq frame.

Modeling of the electrical part of the induction motor

Fig. 6.6 shows the equivalent circuit of a single cage induction motor based on the two-

axis theory [32]. Different from the synchronous generator, the two axes of the induction

motor are symmetrical. Rs and Rr are the resistances of the stator and rotor respectively.

Lsλ and Lrλ are the inductances of the stator and rotor respectively. Lm is the mutual

inductance. The vectors is and ir are the stator and rotor current vectors respectively. ψr

is the rotor flux. The vector es is the stator voltage vector. The reference direction of the

stator current vector is chosen outwards to match the current direction defined in Fig. 6.3.

In a synchronous reference frame, the dynamic equations of the stator and rotor can be

expressed as

es =dψs

dt+ jψsω1 − Rsis

0 =dψr

dt+ jψr(ω1 − ωm

r ) +Rrir (6.25)

where ωmr is the rotor angular speed. ψs and ψr are the stator and rotor fluxes, which are

defined as

ψs = −Lsis + Lmir, ψr = Lrir − Lmis (6.26)

142


where Ls = Lsλ +Lm and Lr = Lrλ +Lm. The dynamic equations of the induction motor

thus can be written in state-space form

dxm

dt= −L−1R︸︷︷︸

Am

xm + L−1Bu︸︷︷︸Bm

um + L−1Bω︸︷︷︸Bm1

um1

ym =

[1 0 0 0

0 1 0 0

]

︸︷︷︸Cm

xm (6.27)

where the matrices L and R are

L =

−Ls 0 Lm 0

0 −Ls 0 Lm

−Lm 0 Lr 0

0 −Lm 0 Lr

R =

−Rs ω1Ls 0 −ω1Lm

−ω1Ls −Rs ω1Lm 0

0 (ω1 − ωmr0)Lm Rr −(ω1 − ωm

r0)Lr

−(ω1 − ωmr0)Lm 0 (ω1 − ωm

r0)Lr Rr

(6.28)

and the matrices Bu and Bω are

Bu =

[1 0 0 0

0 1 0 0

]T

Bω =[

0 0 (−Lmisq0 + Lrirq0) (Lmisd0 − Lrird0)]T

(6.29)


um =[

∆esd ∆esq

]T, um1 = ∆ωm

r

ym =[

∆isd ∆isq]T, xm =

[∆isd ∆isq ∆ird ∆irq

]T. (6.30)

The angular speed ∆ωmr of the induction motor is connected to the transfer function of the

rotor. For the induction motor, the common ac-networkR−I frame can be directly chosen

as the reference dq frame. Therefore, there is no need for reference-frame transformation.

Modeling of the electrical part of the VSC-HVDC converter

Fig. 6.7 shows the main circuit of the VSC-HVDC converter, where Lv represents a small

fictitious inductance for the convenience of model development. In a synchronous refer-

ence frame, by taking into account HHP (s), the dynamic equations of the main circuit of

143


+

-

fu fC

00 vV θ∠vL

vi

+

-

ve

cR cL

ci

v

+

-

VSC

Fig. 6.7 Main circuit of the VSC-HVDC converter.

the VSC-HVDC converter can be expressed as [cf. (4.45)]

Lcdicdt

= (V0 + ∆V )ejθv −HHP (s) ic − uf −Rcic − jω1Lcic

Cfduf

dt= ic − iv − jω1Cfuf

Lvdivdt

= uf − ev − jω1Lviv. (6.31)

The Laplace transform variable s of HHP (s) can be eliminated in the same way as shown

in (4.47). By linearization and writing in component form, the dynamic equation of the

VSC-HVDC converter can be written in state-space form

dxv

dt= Avxv +Bvuv +Bv1uv1

yv = Cvxv (6.32)

where the Av matrix is

Av =

−αv − Rc+kv

Lcω1 1 0 − 1

Lc0 0 0

−ω1 −αv − Rc+kv

Lc0 1 0 − 1

Lc0 0

−αvRc

Lcω1αv 0 0 −αv

Lc0 0 0

−ω1αv −αvRc

Lc0 0 0 −αv

Lc0 0

1Cf

0 0 0 0 ω1 − 1Cf

0

0 1Cf

0 0 −ω1 0 0 − 1Cf

0 0 0 0 1Lv

0 0 ω1

0 0 0 0 0 1Lv

−ω1 0

(6.33)

144


and the Bv, Bv1, and Cv matrices are

Bv =

[0 0 0 0 0 0 − 1

Lv0

0 0 0 0 0 0 0 − 1Lv

]T

Bv1 =

[−V0 sin θv0

Lc

V0 cos θv0

Lc−αvV0 sin θv0

Lc

αvV0 cos θv0

Lc0 0 0 0

V0 cos θv0

Lc

V0 sin θv0

Lc

αvV0 cos θv0

Lc

αvV0 sin θv0

Lc0 0 0 0

]T

Cv =

[0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 1

](6.34)


uv =[

∆evd ∆evq

]T, uv1 =

[∆θv

∆VV0

]T, yv =

[∆ivd ∆ivq

]T

xv =[

∆icd ∆icq ∆ρcd ∆ρcq ∆ufd ∆ufq ∆ivd ∆ivq

]T. (6.35)

The input ∆θv is connected to the active-power controller, while ∆V/V0 is connected to

the alternating-voltage controller. For the VSC-HVDC converter, the common ac-network

R− I frame can also be directly chosen as the synchronous dq frame to avoid reference-

frame transformation.

Jacobian transfer matrix of the island system

Before connecting to the ac network, the state-space representations of the electrical parts

of the three input devices, i.e., the synchronous generator, the induction motor and the

VSC-HVDC converter are lumped into one state-space model as

dxz

dt= Azxz +Bzuz +Bz1udz

yz = Czxz +Dz1uz1 (6.36)

where xz = [xTg xT

m xTv ]T , uz = [(u

′

g)T uT

m uTv ]T , yz = [(y

′

g)T yT

m yTv ]T , and uz1 =

[(u′

g1)T uT

m1 uTv1]

T . Az, Bz, Bz1, Cz, Dz1 are the block diagonal matrices composed of

the corresponding input-device matrices in (6.23), (6.27) and (6.32), i.e.,

Az =

Ag

Am

Av

, Bz =

B′

g

Bm

Bv

, Bz1 =

B′

g1

Bm1

Bv1

Cz =

C′

g

Cm

Cv

, Dz1 =

Dg1

zeros (1, 1)

zeros (2, 2)

.

145


Note that each input and output of the state-space model of the input devices in (6.36)

and the state-space model of the ac network in (6.7) are reciprocal, i.e., each output of

one system is the input of the other system. Consequently, the state-space model of the

combined electrical systems can be solved as

dxE

dt=

[N(Az +BzDnCz) NBzCn

BnCz An

]

︸︷︷︸AE

xE

+

[N(BzDnDz1 +Bz1)

BnDz1

]uE +

[NBzDn1Dz1

zeros (4, 6)

]duE

dt︸︷︷︸

BE ·uE

(6.37)

where N = (I − BzDn1Cz)−1, xE = [xT

z xTn ]T , and uE = uz1. In (6.37), the deriva-

tive of the input duE/dt appears on the right side of the state-space equation. However,

considering the structure of Dz1, the differentiation is, in fact, only effective on the rotor

angle ∆δ of the synchronous generator, which yields ∆ωgr = d∆δ/dt. Since ∆ωg

r belongs

to uE , (6.37) can still be fit in standard state-space form.

Assuming per unit quantities, the output variables of the electrical system can be

expressed as

T ge = Im {isconj(ψ)} = isqψd − isdψq

Eg =√e2sd + e2sq

Tme = Im {isconj(ψs)} = isqψsd − isdψsq

P = Re {ufconj(iv)} = ufdivd + ufqivq

Uf =√u2

fd + u2fq (6.38)

where T ge is the electrical torque of the synchronous generator. Eg is the terminal voltage

of the synchronous generator. Tme is the electrical torque of the induction motor. P is the

active power from the VSC-HVDC converter. Uf is the filter-bus voltage of the VSC-

HVDC converter. In (6.38), T ge , Tm

e , P and Uf are expressed by state variables, but Eg

is expressed by esd and esq, which are neither state variables nor inputs. Therefore, the

terminal voltages ( of all the input devices) are solved by (6.36) and (6.7), yields

uz =[Nt(DnCz +Dn1CzAz) NtCn

]xE

+Nt(DnDz1 +Dn1CzBz1)uE +NtDn1Dz1duE

dt(6.39)

where Nt = (I − Dn1CzBz)−1. By having all the output variables expressed by state

variables and inputs, the outputs of the electrical systems can be expressed in state-space

form after linearizing (6.38)

yE = CExE +DEuE . (6.40)

146


The state-space representation (6.37) and (6.40) can be further written in input-output

transfer matrix form

yE =[CE (sI − AE)−1BE +DE

]︸︷︷︸

J(s)

uE (6.41)

where the Jacobian transfer matrix J(s) is the linear description of the electrical part of

the island system. That is, J(s) is a 5x6 transfer matrix which has

uE =[

∆δ ∆ωgr ∆efd ∆ωm

r ∆θv∆VV0

]T(6.42)

as the inputs and

yE =[

∆T ge ∆Eg ∆Tm

e ∆P ∆Uf

]T(6.43)

as the outputs.

6.3.2 Integrated linear model

Fig. 6.8 shows the integrated linear model of the island system. The central part is the

ac Jacobian transfer matrix J(s), i.e., the controlled process of the island system. The

non-electrical parts, i.e., the controllers, are described in the following:

Excitation system of the synchronous generator

The input signals of the exciter are the alternating-voltage reference Uref and the compen-

sated terminal voltage V gc . The output signal is the field voltage efd. KA and TA are the

gain and time constant of the exciter.

Rotor dynamics of the synchronous generator

Considering a single mass rotor, the dynamics of the rotor are determined by the equation

of motion:

d∆ωgr

dt=

1

2H(T g

m − T ge −KD∆ωg

r ). (6.44)

By applying the Laplace transform, the corresponding control law can be expressed as

∆ωgr =

1

2Hs+KD︸︷︷︸HRotor(s)

(T gm − T g

e ). (6.45)

147


Power-

synchronization

loop

AC-Jacobian

transfer matrix

J(s)

fde∆

gE∆

+

-

g

rω∆-

g

mT∆

+

g

eT∆

refU

-

-

refP

0V

V∆

vθ∆

+

-s

ku

P∆

fU∆

s

kp

A

A

T1

K

s+

GTs1

1

+

m

rω∆m

eT∆

m

mT

ω∆

f

f

sT1

K

+

R

1

ref

gE

CHs1

1

T+

Exciter

Rotor

GovernorTurbine

AC-voltage

control

Rotor

VSC-HVDC Synchronous generator

Induction motor

|| vtvcf

v

c jXkV iu +=

Load compensation

|| gtggg

g

c jXkV ie +=

Load compensation

refP

+

refP∆

+-

+

-

mT1

s

s+

s

1δ∆

Y∆

DK2Hs

1

+

DK2Hs

1

+

Frequency

controller

Frequency

measurement

Fig. 6.8 A complete linear model of the example island system.

148


Turbine and governing systems of the synchronous generator

In the thesis, a steam turbine model is assumed. A simplified steam turbine model can be

represented as

∆T gm =

1 + sFHPTRH

(1 + sTCH)(1 + sTRH)∆Y (6.46)

with the typical values FHP = 0.3, TRH = 7.0 s, TCH = 0.3 s [32]. For a non-reheat steam

turbine model [32], the transfer function is

∆T gm =

1

1 + sTCH∆Y. (6.47)

The steam flow in the turbine is controlled by the governing system. The governor senses

the frequency error and adjusts the steam into the turbine. The governor has the speed

deviation ∆ωgr and the load-reference set-point Pref as input variables. The output of the

governor is the steam gate ∆Y , which supplies the input to the steam turbine. The fre-

quency droop R determines the power-sharing proportion of the synchronous generator.

Rotor dynamics of the induction motor

The rotor dynamics of the induction motor are similar to the synchronous generator. The

input signals are the mechanical torque from the load and the electrical torque of the

motor. Similar to the synchronous generator, the rotor dynamics of the induction motor

can be expressed as

∆ωmr =

1

2Hs+KD(Tm

m − Tme ). (6.48)

It should be noted that the generator convention is applied in (6.48). In other words, if the

mechanical load is TL in motor operation, then it gives Tmm = −TL.

Active-power and alternating-voltage control of the VSC-HVDC link

The control laws are given by (5.25).

Frequency and alternating-voltage droop control of the VSC-HVDC link

The control laws are given by (6.2) and (6.5). The frequency measurement is obtained by

differentiating angle output of the active-power controller with a measuring time constant

Tm, typically in the range of 10− 20 ms. It should be noted that the load compensation of

the VSC-HVDC converter and the synchronous generator in Fig. 6.8 are in their nonlinear

form. It requires linearization if their effects are to be considered in the linear model.

The technical data of the VSC-HVDC link, the synchronous generator and the in-

duction motor are given in Appendix C. The rest parameters of the linear model for the

island system are given in Table 6.1.

149


Table 6.1 Parameters of the linear model for the island system. Loads: per unit based on 350 MVAand 400 kV. VSC-HVDC: per unit based on 350 MVA and 195 kV.

Parameters Values

Loads Rl, ω1Ll, ω1Cl 1.75 p.u., 16.8 p.u., 0.01 p.u.

KA, TA 200, 0.02 s

Synchronous generator R, TG, TCH, 0.067, 0.02 s, 0.3 s

kp, ku 100 rad/s, 60

kv, αv 0.3 p.u., 40 rad/s

VSC-HVDC Kf , Tf , Tm 3.72, 0.3 s, 0.02 s

6.3.3 Linear analysis

As it was discussed in Chapter 5, power-synchronization control of VSC-HVDC link is

fundamentally of MIMO nature. However, with the simple diagonal controller proposed

for island operation, the control parameters can be selected based on single-input single-

output (SISO) technique by tuning one loop at a time. Accordingly, the tuning process is

performed in two steps:

• Step 1: Starting with the integral gain of the alternating-voltage control ku = 0.0,

plot the root-loci of the dominant poles by varying the integral gain kp of the active-

power controller. Find the suitable parameter range of kp.

• Step 2: Using kp chosen at step 1, plot the root-loci of the dominant poles by varying

ku to find a suitable parameter range of ku.

Fig. 6.9 shows the root-loci of the dominant poles of the closed-loop system by

varying kp from 0 to 140 rad/s. Two pairs of dominant poles of the open-loop system are

affected by varying kp. The critical poles are the pole pair p1,2 (p1,2 = −127 ± j298),

which move towards the right-half plane with increased kp. The pole pair p1,2 imposes

an upper limitation on kp. Thus, kp should be chosen between 80 − 100 rad/s to get

a trade-off between dynamic-response time and stability margin. The pole pair at p3,4

(p3,4 = −36.4 ± j17.1) becomes more damped by increasing kp.

Fig. 6.10 shows the root-loci of the dominant poles of the closed-loop system by

varying ku with kp = 100 rad/s. The pole p0 is a pole at the origin which is introduced

by the integrator. By increasing ku from 0 to 120, p0 moves towards the left-half plane.

The two dominant pole pairs are also affected by variations of ku. The pole pair p3,4

moves towards the left-half plane, however, the pole pair at p1,2 slightly moves towards

the right-half plane instead. Therefore, the above tuning steps should be iterative to find a

compromised combination of kp and ku to place the two dominant pole pairs. Finally, the

150


-150 -100 -50 0

-300

-200

-100

0

100

200

300

ω (rad/sec)

ω (

rad

/se

c)

p1

p2

p3

p4

Fig. 6.9 Root-loci of the dominant poles of the island system. ku = 0.0, variations of kp from

0 rad/s to 140 rad/s.

-150 -100 -50 0

-300

-200

-100

0

100

200

300

ω (rad/sec)

ω (

rad

/se

c)

p1

p2

p3

p4

p0

Fig. 6.10 Root-loci of the dominant poles of the island system. kp = 100.0 rad/s, variations of

ku from 0 to 120.

151


control parameters are chosen with kp = 100 rad/s and ku = 60, which end up with the

dominant pole pairs at p1,2 = −77.8 ± j269 rad/s and p3,4 = −74.3 ± j56 rad/s.

6.4 Simulation studies

To demonstrate the dynamic performance of the VSC-HVDC link, the island system in

Fig. 6.3 is simulated with PSCAD/EMTDC. The other end of the VSC-HVDC link is

assumed to be connected to a strong ac system.

One of the features of power-synchronization control is its flexibility with differ-

ent network conditions, i.e., the proposed control structure can deal with various possible

operating conditions, as well as handle the transitions between the operation modes seam-

lessly. In this section, this feature of power-synchronization control is demonstrated.

Initially, the island system is connected to a large ac system with constant fre-

quency. In this operation mode, the VSC-HVDC link is possible to apply constant power

control. The initial condition is as following. The total loads in the island system is

320 MW whereas 120 MW from induction-motor loads (4 units) and 200 MW from

RLC passive loads. The synchronous generator supplies 100 MW, while the VSC-HVDC

link supplies 150 MW with constant power control. The rest of the loads are balanced by

the large system, i.e., 70 MW power transmitted to the island system.

Fig. 6.11 shows the simulation results when the line connection to the large ac

system is lost at 0.1 sec. By detecting this event (by transient frequency measurement

or the state of the ac breaker of the tripped ac line), the VSC-HVDC link enables the

frequency droop control. The power that is lost from the large ac system is shared by

the VSC-HVDC link and the local synchronous generator. The proportion of the power

sharing is determined by the frequency droops of the synchronous generator and the VSC-

HVDC link where it is equally shared in this case. After disconnection from the large

ac system, the frequency of the island system deviates slightly from the nominal value

due to the frequency droop characteristics of the VSC-HVDC link and the synchronous

generator.

Fig. 6.12 is a continuation simulation from Fig. 6.11. The local synchronous gener-

ator is also tripped at 0.1 sec (new time frame). The island system then becomes a passive

network with induction-motor loads and RLC passive loads. The VSC-HVDC link sup-

plies all the loads in the island system. Due to the frequency-droop characteristics of the

VSC-HVDC link, the frequency of the island system deviates much from the nominal

value. At 1.5 sec, the power order of the VSC-HVDC link is adjusted to 0.9 p.u. to match

the loads in the island system. Consequently, the frequency in the island system is back

to the nominal value.

Fig. 6.13 and Fig. 6.14 demonstrate the fault ride-through capability of the VSC-

HVDC link in two different operating conditions, i.e., large-ac-system connection and

island operation. With both of the operating conditions, the VSC-HVDC link manages

152

6.4. Simulation studies

0 0.5 1 1.5 2 2.5 3100

150

200

P d

c (

MW

)

0 0.5 1 1.5 2 2.5 350

100

150

200

P g

(M

W)

0 0.5 1 1.5 2 2.5 30.9

1

1.1

time (sec)

f 1 (

p.u

.)

Fig. 6.11 Trip of the ac line to the large ac system (entering island operation). Upper plot: ac-

tive power from the VSC-HVDC link. Middle plot: active power from the synchronous

generator. Lower plot: network frequency.

0 0.5 1 1.5 2 2.5 3100

200

300

400

P d

c (

MW

)

0 0.5 1 1.5 2 2.5 30

100

200

P g

(M

W)

0 0.5 1 1.5 2 2.5 30.8

0.9

1

1.1

time (sec)

f 1 (

p.u

.)

Fig. 6.12 Trip of the local synchronous generator (entering passive network operation). At 1.5 sec,

the power order of the VSC-HVDC link is adjusted to 0.9 p.u.

153


0 0.2 0.4 0.6 0.8 1

-1

0

1

U p

cc(p

.u.)

0 0.2 0.4 0.6 0.8 1

0

0.5

1

P d

c (

p.u

.)

0 0.2 0.4 0.6 0.8 10

1

2

time(sec)

| i c

| (p

.u.)

Fig. 6.13 Fault ride-through capability of the VSC-HVDC with large-ac-system connection. Up-

per plot: alternating voltage at the PCC. Middle plot: active power from the VSC-

HVDC. Lower plot: valve current of the VSC-HVDC.

0 0.2 0.4 0.6 0.8 1

-1

0

1

U p

cc(p

.u.)

0 0.2 0.4 0.6 0.8 1

0

0.5

1

P d

c (

p.u

.)

0 0.2 0.4 0.6 0.8 10

1

2

time(sec)

| i c

| (p

.u.)

Fig. 6.14 Fault ride-through capability of the VSC-HVDC in island operation.

154

6.5. Jacobian transfer matrix for other input devices

SG

gi+

-

ge

nL

+

-

E

0

0 0∠E00 ggE θ∠ nR

Fig. 6.15 A synchronous generator connected to an impedance ac source.

recovery from the severe three-phase ac-system faults applied at the PCC. After detect-

ing the faults, the current limiter reduces the fault currents to half of the maximum load

current Imax or any other desired values.

The above simulation studies demonstrate the flexibility of power-synchronization

control for various operating conditions. It should be noted that black start of the island

system is just the reverse procedure of the above. However, the synchronous generator

needs to be synchronized to the VSC-HVDC link before it can be re-connected to the

island system.

6.5 Jacobian transfer matrix for other input devices

In Chapter 4, the poles and transmission zeros of the ac Jacobian transfer matrices are ana-

lyzed for grid-connected VSCs connected to several simplified ac-network configurations.

It has been found that the poles and transmission zeros of the Jacobian transfer matrix are

informative in understanding the basic characteristics of the ac system. The major find-

ings are that, the resonant (complex) poles of the Jacobian transfer matrices generally

correspond to the resonances in the ac system (except the poles at the grid frequency),

while the operating-point-dependent zeros have a close relationship with angle stability

and voltage stability phenomena in ac systems. By moving to the origin, the zeros impose

a fundamental limitation on power transmission in the ac network. In addition, the RHP

zero of the Jacobian transfer matrix imposes a fundamental limitation on the achievable

bandwidth of the VSC.

In this section, the characteristics of the Jacobian transfer matrix are further an-

alyzed for two additional conventional input devices in power systems, i.e., the syn-

chronous generator and the induction motor. The goal of the analysis is to gain more

insights of the Jacobian transfer matrix modeling concept.

6.5.1 Synchronous generator

Fig. 6.15 shows a synchronous generator connected to an impedance ac source, which

is the classical one-machine infinite-bus (OMIB) system. Rn and Ln are the resistance

and inductance of the ac system. The bold letter symbols E and eg represent the volt-

age vectors of the ac source and the terminal voltage of the generator. E0 and Eg0 are

155


g

eT∆

gE∆ref

gE

-

+

g

mT g

rω∆+-

)(sJ

Rotor

Exciter

fde∆

DK2Hs

1

+

sA

A

T1

K

+

Jacobian

transfer

matrix

Fig. 6.16 Closed-loop system of the synchronous generator.

their corresponding voltage magnitudes. The ac source is used as the voltage reference,

and the phase angle of eg in the steady-state operating point is θg0. The quantity ig is

the current vector with the reference direction from the synchronous generator to the ac

system. The technical data of the synchronous generator are given in Appendix C (no

step-up transformer connected). The resistance and reactance of the ac system are chosen

as Rn = 0.05 p.u. and ω1Ln = 1.0 p.u. (per unit based on 150 MVA and 20 kV). The

reason for using such a long line with a comparably low voltage is to facilitate studies on

the zeros of the Jacobian transfer matrix.

By applying the Jacobian transfer matrix modeling concept, Fig. 6.16 shows the

closed-loop system of the synchronous generator connected to the ac system. For sim-

plicity, the governing system of the synchronous generator is neglected, i.e., a constant

mechanical torque is assumed. As it was shown in Fig. 6.8, the synchronous generator

has three input variables to the Jacobian transfer matrix, i.e, the rotor speed ∆ωgr , the ro-

tor angle ∆δ, and the field voltage ∆efd. This is rather inconvenient for analysis since

∆ωgr and ∆δ are not independent. Therefore, in Fig. 6.16, the boundary between the Ja-

cobian transfer matrix and the controllers are re-defined by only having ∆ωgr and ∆efd

as inputs, while the integrator between ∆ωgr and ∆δ is formed into the Jacobian transfer

matrix, i.e., the modified Jacobian transfer matrix has the expression

J(s) =

JTeω(s) +JTeδ(s)

s︸︷︷︸J′

Teω(s)

JTeEfd(s)

JEgω(s) +JEgδ(s)

s︸︷︷︸J′

Egω(s)

JEgEfd(s)

. (6.49)

Table 6.2 lists the poles and the transmission zeros of J(s) for the synchronous generator

at no-load operation, and Fig. 6.17 shows the pole-zero map of J(s) by varying the load

angle of the synchronous generator.

156


Table 6.2 Locations of poles and transmission zeros of the Jacobian transfer matrix J(s) for the

synchronous generator with θg0 = 0◦ (no load operation).

Poles rad/sec Zeros rad/sec

p1,2 −2.56 ± j314.1 z1,2 −0.002 ± j314.1

p3 −40.4 z3 −243.7

p4 −25.6 z4 −24.3

p5 −2.13 z5 −0.9

p6 −0.44 z6 INF

p7 0

-200 0 200 400 600 800 1000 1200

-300

-200

-100

0

100

200

300p

1

p2

z3

z4-z

5

p3-p

7

ω (rad/sec)

ω (

rad

/se

c)

z6

z1

z2

θg0

=17°

θg0

=27°

θg0

=45°

θg0

=90°

Fig. 6.17 Loci of the transmission zeros of the Jacobian transfer matrix for the synchronous gen-

erator with increased load angles.

157


It is interesting to note that, similar to that for the grid-connected VSCs, the Ja-

cobian transfer matrix for the synchronous generator also has a resonant pole pair at the

grid frequency (p1,2). However, a zero pair z1,2 appears very close to the locations of the

resonant pole pair. The zero pair z1,2 is affected only slightly by the load-angle variations,

i.e., the resonant poles at the grid frequency are canceled by the resonant zeros. Although

the pole-zero cancelation is basically not affected by operating-point variations, the bal-

ance can be broken by series-compensated ac lines. As it was shown in Section 4.2.3, a

series capacitor splits the resonant pole pair at the grid frequency into two other resonant

pole pairs, one at the subsynchronous frequency and the other at the supersynchronous

frequency [cf. (4.56)]. The SSR problem of the synchronous generator is related to the

pole pair at the subsynchronous frequency.

The integrating process between the rotor speed and the rotor angle has the ad-

vantage to reduce the peaks of various resonances in the ac system. However, the in-

tegrator also reduces the phase margin of the rotor loop. The small-signal stability or

low-frequency oscillations in power systems are largely due to the poor phase margin of

the rotor loop of the synchronous generators, which is worsened by fast excitation control.

It is believed by the author that the PSS solution can, in some sense, be viewed upon as

a multivariable feedback-control design which aims to decouple the interaction between

the rotor loop and the excitation control.

Similar to the VSC, the Jacobian transfer matrix for the synchronous generator also

has but one operating-point-dependent zero z6, which moves from infinity towards the

origin with increased load angles. By θg0 = 90◦, z6 is very close to the origin. However,

the RHP zero z6 usually does not create any problem for the synchronous generator, since

a synchronous generator normally operates with a load angle lower than 40◦ due to the

transient stability limitation [64]. The RHP zero z6 at the operating point θg0 = 40◦ will

not impose any bandwidth limitation on the synchronous generator that usually has a

bandwidth lower than 10 rad/s.

Fig. 6.18 shows the four transfer functions of the Jacobian transfer matrix, J′

Teω(s),

JTeEfd(s), J

′

Egω(s), and JEgEfd(s) overlapped with the frequency-scanning results from

PSCAD/EMTDC. The synchronous generator is assumed to operate with a load angle

θg0 = 30◦. Due to limitations of the applied frequency-scanning technique in PSCAD/-

EMTDC, only the results with frequencies higher than 6.28 rad/s (1 Hz) are shown.

The transfer functions of J′

Teω(s), JTeEfd(s) show the pole-zero cancelation phe-

nomena of the Jacobian transfer matrix at the grid frequency which are also confirmed by

the frequency-scanning results from PSCAD/EMTDC.

From the magnitude and phase plot of J′

Teω(s), it can be observed that the integrator

between the rotor speed and the rotor angle gives a negative slope to the magnitude, which

suppresses the high-frequency resonances. However, it also introduces a −90◦ phase shift

which limits the phase margin of the rotor loop.

158


10-2

100

102

10-5

100

105

|J´ T

eω(j

ω)|

10-2

100

102

-150

-100

-50

0

ω (rad/sec)

AR

G J

´ Teω(j

ω)

(deg.)

(a) J′

Teω(s).

10-2

100

102

10-5

100

|JT

eE

fd

(jω

)|

10-2

100

102

-200

-100

0

ω (rad/sec)

AR

G J

TeE

fd

(jω

) (d

eg.)

(b) JTeEfd(s).

10-2

100

102

10-5

100

105

|J´ E

gω(j

ω)|

10-2

100

102

-400

-300

-200

ω (rad/sec)

AR

G J

´ Egω(j

ω)

(deg.)

(c) J′

Egω(s).

10-2

100

102

10-5

100

|JE

gE

fd

(jω

)|

10-2

100

102

-200

-100

0

ω (rad/sec)

AR

G J

EgE

fd

(jω

) (d

eg.)

(d) JEgEfd(s).

Fig. 6.18 Bode plots of the transfer functions of the Jacobian transfer matrix for the syn-

chronous generator (solid: linear models, dashed: frequency-scanning results from

PSCAD/EMTDC).

6.5.2 Induction motor

Fig. 6.19 shows an induction motor connected to a constant-frequency stiff voltage source.

The quantity im is the stator-current vector of the induction motor. The reference direction

of the stator current vector is chosen outwards to be consistent with other input devices.

The technical data of the induction motor are given in Appendix C (no step-up transformer

connected).

By applying the Jacobian transfer matrix modeling concept, Fig. 6.20 shows the

closed-loop system of the induction motor connected to the ac system. The induction

motor only has the rotor speed ∆ωmr connected to the Jacobian transfer matrix, while the

electrical torque Tme is the output. Thus, the induction motor is modeled as a single-input-

159


mi

IM+

-

E

0

0 0∠E

Fig. 6.19 An induction motor connected to a stiff voltage source.

m

eT∆m

mT m

rω∆+

- )(sJRotor

DK2Hs

1

+

Jacobian

transfer matrix

Fig. 6.20 Closed-loop system of the induction motor.

single-output (SISO) feedback control system, where the rotor is treated as the controller,

similar to the synchronous generator. In practice, there are also rotor-speed or squared-

rotor-speed dependent contribution to the mechanical torque Tm, which can be modeled as

part of the controller. However, since the major subject of this section is about the Jacobian

transfer matrix, a constant mechanical torque is assumed for simplicity. Accordingly, the

Jacobian transfer matrix has the expression

J(s) = JTeω(s) (6.50)

i.e., the Jacobian transfer matrix is a 1x1 matrix with JTeω(s) as the only element. Ta-

ble 6.3 lists the poles and zeros of the Jacobian transfer matrix for the induction motor

at no-load operation. Fig. 6.21 shows the pole-zero map by increasing the loading of the

induction motor.

Table 6.3 Locations of poles and zeros of the Jacobian transfer matrix of the induction motor at

no-load operation (slip = 0.0).

Poles rad/sec Zeros rad/sec

p1,2 −9.4 ± j314.16 z1,2 −5.1 ± j314.1

p3 −13.3 z3 −13.3

p4 −13.3

Similar to the synchronous generator, the resonant pole pair p1,2 at the grid fre-

quency is also canceled by the resonant zero pair z1,2. This implies that the induction

motor might also suffer from SSR problems if a series-compensated ac line is connected

in the vicinity. Of course, it is exceptional rare for induction motors to have sufficiently

long rotor shafts to excite such oscillation modes.

160


-15 -10 -5 0 5

-300

-200

-100

0

100

200

300z

1

z2

p1

p2

ω (rad/sec)

ω (

rad

/se

c)

z3

p3

p4

slip=0 slip=0.02 slip=0.032 slip=0.042

Fig. 6.21 Locus of z3 with increased slips.

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0

0.5

1

1.5

2

2.5

3

1-slip

Tm e

(p

.u.)

z3=-13.3 rad/s

z3=-10.3 rad/s

z3=-5.6 rad/sz

3=0 rad/s

Fig. 6.22 Locus of z3 on the torque-slip curve of the induction motor .

161


10-1

100

101

102

103

10-4

10-2

100

|JT

eω(j

ω)|

10-1

100

101

102

103

-150

-100

-50

0

50

ω (rad/sec)

AR

G J

Teω(j

ω)

(de

g.)

Fig. 6.23 Bode plot of JTeω(s) (solid: linear model, dashed: frequency-scanning result from

PSCAD/EMTDC).

With slip = 0.0, two poles p3 and p4, as well as a real LHP zero z3 have exactly the

same location. By increasing the slip of the induction motor, p3 and p4 move slightly apart

along the imaginary-axis direction. However, z3 moves towards the origin, and it reaches

the origin with slip = 0.042.

Fig. 6.22 shows the well-known torque-slip curve of the induction motor. It is in-

teresting to note that z3 reaches the origin exactly at the pull-out slip of the induction

motor!

Fig. 6.23 shows the Bode plot of the transfer function JTeω(s) overlapped with the

frequency scanning result from PSCAD/EMTDC with frequencies higher than 6.28 rad/s.

The pole-zero cancelation phenomenon of the Jacobian transfer matrix at the grid fre-

quency can be observed. The frequency-scanning result from PSCAD/EMTDC confirms

the same phenomenon.

6.6 Subsynchronous characteristics

In the previous sections, the rotor of the synchronous generator was assumed to be made

up of a single mass. With such a representation of the rotor, the bandwidth of a syn-

chronous generator is usually in the frequency range of 0.2 to 2 Hz. In reality, however, the

rotor of a steam-turbine generator has a complex mechanical structure consisting of sev-

eral predominant masses connected by shafts of finite stiffness. Therefore, when the gen-

erator is perturbed, torsional oscillations occur between sections of the turbine-generator

162

6.6. Subsynchronous characteristics

nL

400 kV

20 kV

150 MVA

195 kV

VSC

350 MW

SG

nR

tgLtvL

+

-

E

350 MW

400 kV

20 kV 195 kV

VSCSG

lR

tgLtvL

150 MVA

(a) (b)

PCC PCC

Fig. 6.24 AC-network configurations for investigation of the subsynchronous characteristics of a

VSC-HVDC link. (a) Large ac-system connection. (b) Island operation.

rotor. These torsional oscillation modes, typically in the subsynchronous frequency range,

could interact with the electrical system in an adverse manner. There are predominantly

two types of subsynchronous torsional interactions (SSTIs) of concern:

1. Subsynchronous resonances with series-compensated transmission lines.

2. Torsional interactions with large power-electronic devices, such as HVDC systems

and FACTS devices.

In this section, the subsynchronous characteristics of a VSC-HVDC link located in the

vicinity of a steam-turbine synchronous generator are analyzed. The two ac-network con-

figurations to be investigated are shown in Fig. 6.24. With the ac-network configuration

shown in Fig. 6.24(a), a VSC-HVDC link and a synchronous generator are connected

through an ac transmission line to a constant-frequency stiff voltage source E, which

represents a large ac system. With the ac-network configuration shown in Fig. 6.24(b),

a VSC-HVDC link and a synchronous generator are connected in an island system. The

resistive load Rl is used to adjust the load flow in the island system.

163


-

+)(sGm

)(sGe

g

mT

g

eT∆

g

rω∆

Fig. 6.25 Closed-loop system formed by the linearized electrical and mechanical dynamics.

The technical data of the synchronous generator and the VSC-HVDC link are given

in Appendix C.

6.6.1 Frequency-scanning method

As it was mentioned in Chapter 4, there are predominantly two methods for SSTI analysis:

eigenvalue analysis and frequency scanning. The eigenvalue-analysis method is a rigor-

ous tool. However, it requires detailed representation of the mechanical system of the

synchronous generator. The frequency-scanning method is based on the complex-torque

theory [81]. With the closed-loop system shown in Fig. 6.25, the transfer function between

∆ωgr and ∆T g

e can be expressed as

Ge(jω) = De(jω) − jKe(jω) (6.51)

where De and Ke are defined as the damping torque and the synchronizing torque. With-

out the electrical system, the damping torque of the mechanical system is always positive

due to friction. Thus, ifDe(jω) > 0, unstable subsynchronous oscillations (SSOs) cannot

occur.

It is a common practice to use the electrical damping curve of the synchronous

generator to evaluate the subsynchronous characteristics of an HVDC link [102, 103]. If

the characteristics of the HVDC link are taken into consideration a modified electrical

damping curve D′

e(jω) is obtained. Now, if D′

e(jω) is larger than the original electrical

damping curve De(jω) without the HVDC link, i.e.,

D′

e(jω) > De(jω) (6.52)

the negative influence of the HVDC link on SSOs can be eliminated.

As a screening tool to evaluate the subsynchronous characteristics of an HVDC

link, the frequency-scanning method has the following merits:

1. There is no need to model the mechanical system of the synchronous generator.

2. It can be performed by both frequency-domain analysis and time-domain simula-

tions.

164


P∆

refU

-

+refPvθ∆

+-

s

k p

PSL

s

ku

AVC

fU∆ g

eT∆

g

rω∆

VSC1P∆−

VSC2P∆−

2

dc1u∆ 2

dc2u∆

)(dc sG

2ref

dc1)(u

)2

( dc

2

dcs

CC d

d

αα +−

+-

DVC

)(sJ

)(sGe

DC-Jacobian

transfer

matrix

AC-Jacobian

transfer

matrix

0V

V∆

Fig. 6.26 Jacobian transfer matrix formulation of the large ac-system connection configuration in

Fig. 6.24(a).

6.6.2 Large ac-system connection

System definition

Fig. 6.26 shows the Jacobian transfer matrix formulation of the large ac-system connec-

tion configuration in Fig. 6.24(a). The transfer function Ge(s) is defined as the transfer

function between ∆ωgr to ∆T g

e with all the feedback loops of the VSC-HVDC link closed

(inside the dashed box). In principle, Ge(s) should also take into account the effects of

the exciter and possibly the power-system stabilizer (PSS) of the synchronous generator.

However, since the goal of the investigation is to study the subsynchronous characteris-

tics of the VSC-HVDC link, where the difference between D′

e(jω) and De(jω) is of more

concern, the excitation system of the synchronous generator is neglected.

The VSC-HVDC converter has two possible control modes: active-power control

and direct-voltage control. As it was discussed in Chapter 5, if the direct voltage is con-

trolled by only one converter station, the two converter stations are linearly independent.

Consequently, in either of the operation modes, it is enough to have the control system

165


of one converter station modeled. Also for the sake of simplicity, the dc-link circuit is

assumed to be a single capacitor, i.e., Gdc(s) = 2/(sCdc). Since the dc capacitance Cdc is

explicitly included into the control parameters of the direct-voltage controller, the value

of Cdc has no impact on Ge(s).

Table 6.4 gives the base control parameters selected for the VSC-HVDC converter.

The controllers of the VSC-HVDC converter have been tuned relatively conservative to

operate the VSC-HVDC link for wider ac-network conditions. The impact of the control

parameters on the subsynchronous characteristics of the VSC-HVDC converter will be

further discussed after presenting the initial results based on the control parameters given

in Table 6.4.

Table 6.4 Control parameters of the VSC-HVDC converter with large ac-system connection. Per

unit based on 350 MVA and 195 kV.

Controller Parameters Values

Power-synchronization loop kp 60 rad/s

Alternating-voltage control ku 60

Direct-voltage control αd 40 rad/s

High-pass current control kv, αv 0.2 p.u., 40 rad/s

For analyzing SSTI between a synchronous generator and a nearby HVDC link, the

unit interaction factor (UIF) is commonly used [104, 105], which is defined as

UIF =MWdc

MVAg

(1 − SCg

SCtot

)2

(6.53)

where MWdc is the rating of the dc system, MVAg is the rating of the generator, SCtot is

the short-circuit capacity at the PCC including the synchronous generator unit, and SCg

is the short-circuit capacity at the PCC excluding the generator.

Based on the parameters of the synchronous generator and its step-up transformer

in Appendix C, the short-circuit capacity contribution from the synchronous generator at

the PCC is MVAg/(L′′

d + Ltg) = 545.3 MVA. By adjusting the parameters of the ac line

to the large system, ac-network configurations with different UIF values are obtained. As

shown by Table 6.5, the UIF value is directly related to the SCR of the ac system1, i.e., the

UIF values are higher with weak-ac-system connections. The resistances of the ac lines

are chosen such that the ac lines have 85◦ impedance angles.

In the following analysis, the synchronous generator is assumed to supply Pg =

0.1 p.u. power, while the VSC-HVDC link transmits 90% of the rated power, i.e., Pdc =

±0.9 p.u. for inverter and rectifier operation respectively.

1The synchronous generator is not included in the calculation of the SCR.

166


Model validation

Fig. 6.27 shows the model validation by comparing the frequency-scanning results from

the linear model and the time-domain simulation. The ac-network is chosen to have

UIF = 0.3. The VSC-HVDC converter operates as an inverter in power control mode. The

electrical damping curves are obtained by both the linear model and the time simulations

in PSCAD/EMTDC. The agreement between the linear model and the time simulation is

good.

Results from the linear analysis

In the following, the electrical damping curves of the synchronous generator are plotted

using the linear model to show the subsynchronous characteristics of the VSC-HVDC

converter for ac-network configurations with the four UIF values defined in Table. 6.5.

The VSC-HVDC converter operates in the following four modes:

1. Rectifier operation, power control mode.

2. Inverter operation, power control mode.

3. Rectifier operation, direct-voltage control mode.

4. Inverter operation, direct-voltage control mode.

Figs. 6.28-6.31 show the electrical damping curves of the synchronous generator affected

by the VSC-HVDC converter. The figures clearly show that the VSC-HVDC converter can

have a significant impact on the electrical damping of the generator, and the impact in-

creases with higher UIF values (weaker ac systems). In addition, the impact also depends

on the operation mode of the VSC-HVDC converter. The following can be observed from

the linear results:

1. In rectifier operation, power control mode, the impact of the VSC-HVDC con-

verter is positive in the entire subsynchronous frequency range. The reason could

be that the VSC-HVDC converter resembles the behavior of a resistive load in this

operation mode.

2. In inverter operation, power control mode, the impact of the VSC-HVDC con-

verter is positive in the lower subsynchronous frequency range. However, it gives

negative damping contributions in the higher subsynchronous frequency range for

approximately UIF > 0.1.

3. With direct-voltage control, for either rectifier or inverter operation, the VSC-

HVDC converter tends to give negative damping in the lower subsynchronous fre-

quency range for approximately UIF > 0.1. However, the negative contribution is

rarely above 10 Hz (62.8 rad/s) even with the highest UIF = 0.7.

167


Table 6.5 Parameters of the ac line for different UIF values. Per unit based on 350 MVA and

400 kV.

UIF Rn ω1Ln SCR

0.02 0.0056 p.u. 0.0655 p.u. 15.2

0.1 0.0147 p.u. 0.168 p.u. 5.9

0.3 0.0315 p.u. 0.3581 p.u. 2.8

0.7 0.0665 p.u. 0.7746 p.u. 1.3

0 50 100 150 200 250 300-1

0

1

2

3

4

5

ω (rad/sec)

De (

jω)

Linear modelTime simulation

(a) Without VSC-HVDC.

0 50 100 150 200 250 300-1

0

1

2

3

4

5

ω (rad/sec)

De (

jω)


(b) With VSC-HVDC.

Fig. 6.27 Model validation with UIF = 0.3 network configuration (solid: linear models, dashed:

time simulations). (a) Electrical damping curves of the generator (Pg = 0.1 p.u.) with-

out the VSC-HVDC link in the system. (b) Electrical damping curves of the generator

(Pg = 0.1 p.u.) with the VSC-HVDC link in the system (inverter operation, power

control mode, Pdc = 0.9 p.u.).

168


0 100 200 300-1

0

1

2

3

4

5

ω (rad/sec)

De (

jω)

Without HVDCWith HVDC

(a) UIF = 0.02.

0 100 200 300-1

0

1

2

3

4

5

ω (rad/sec)

De (

jω)


(b) UIF = 0.1.

0 100 200 300-1

0

1

2

3

4

5

ω (rad/sec)

De (

jω)


(c) UIF = 0.3.

0 100 200 300-1

0

1

2

3

4

5

ω (rad/sec)

De (

jω)


(d) UIF = 0.7.

Fig. 6.28 Electrical damping curves of the generator (Pg = 0.1 p.u.) affected by the VSC-HVDC

converter (rectifier operation, power control mode, Pdc = −0.9 p.u.).

0 100 200 300-1

0

1

2

3

4

5

ω (rad/sec)

De (

jω)


(a) UIF = 0.02.

0 100 200 300-1

0

1

2

3

4

5

ω (rad/sec)

De (

jω)


(b) UIF = 0.1.

0 100 200 300-1

0

1

2

3

4

5

ω (rad/sec)

De (

jω)


(c) UIF = 0.3.

0 100 200 300-1

0

1

2

3

4

5

ω (rad/sec)

De (

jω)


(d) UIF = 0.7.


converter (inverter operation, power control mode, Pdc = 0.9 p.u.).

169


0 100 200 300-1

0

1

2

3

4

5

ω (rad/sec)

De (

jω)


(a) UIF = 0.02.

0 100 200 300-1

0

1

2

3

4

5

ω (rad/sec)

De (

jω)


(b) UIF = 0.1.

0 100 200 300-1

0

1

2

3

4

5

ω (rad/sec)

De (

jω)


(c) UIF = 0.3.

0 100 200 300-1

0

1

2

3

4

5

ω (rad/sec)

De (

jω)


(d) UIF = 0.7.


converter (rectifier operation, direct-voltage control mode, Pdc = −0.9 p.u.).

0 100 200 300-1

0

1

2

3

4

5

ω (rad/sec)

De (

jω)


(a) UIF = 0.02.

0 100 200 300-1

0

1

2

3

4

5

ω (rad/sec)

De (

jω)


(b) UIF = 0.1.

0 100 200 300-1

0

1

2

3

4

5

ω (rad/sec)

De (

jω)


(c) UIF = 0.3.

0 100 200 300-2

0

2

4

ω (rad/sec)

De (

jω)


(d) UIF = 0.7.


converter (inverter operation, direct-voltage control mode, Pdc = 0.9 p.u.).

170


Impact of the control parameters of the VSC-HVDC converter

Besides the operation modes, the control parameters of the VSC-HVDC converter can

also have a strong impact on the electrical damping of the generator. The most critical

parameters are the gain kv of the high-pass current control HHP(s) and the gain kp of the

active-power control.

To evaluate the effect ofHHP(s) on the electrical damping, an ideal operation mode

of the VSC-HVDC converter is studied. The active-power control and the alternating-

voltage control loops are open, i.e., the VSC-HVDC converter operates as a constant-

frequency stiff voltage source. The results from the linear analysis in Fig. 6.32(a) show

that the high-pass current controlHHP(s), in fact, gives negative damping to the generator.

This seems to be contradictory to the role of HHP(s), which is introduced for the purpose

of damping in the first place.

To understand this phenomenon, the damping-torque concept of the synchronous

generator has to be interpreted by feedback-control theory. As shown by Fig. 6.26, the

electrical damping curve De is essentially the real part of the transfer function JTeω of

the Jacobian transfer matrix with the controllers of the VSC-HVDC converter closed.

Assuming constant mechanical torque input, the stability margin of the rotor loop of the

generator can be evaluated by its open-loop transfer function HGen, which is expressed as

HGen(s) = Ge(s)HRotor(s) (6.54)

where HRotor(s) is the transfer function of the rotor, which is expressed in (6.45) for a

single-mass rotor. The concept of damping torque (either electrical or mechanical) has

a close relationship with the stability margin of the rotor loop of the synchronous gen-

erator. The higher the damping torque, the more stability margin the synchronous gen-

erator has. However, such a stability margin is manifested by two aspects: phase margin

and gain margin. The reduction of either of the stability margins will result in reduced

damping torque. Based on this interpretation of the damping-torque concept, it would be

easier to understand the negative damping effect of HHP(s). As shown in Fig. 6.33(a),

with HHP(s) the gain of HGen(s) becomes steeper in the higher frequency range (ap-

proximately 15 − 250 rad/s), which is useful for suppressing other resonances in the ac

system. However, the reduction of gain HGen(s) also results in the reduction of the phase

of HGen. For this particular case, however, the phase margin of HGen(s) is more critical

for the stability of the generator. Consequently, HHP(s) yields negative damping to the

synchronous generator since it reduces the phase margin of the synchronous generator in

the subsynchronous frequency range. On the other hand, as shown in Fig. 6.33(b), a higher

gain kp of the power-synchronization control loop of the VSC-HVDC converter makes the

slope of HGen less steep in the higher frequency range (approximately 13 − 280 rad/s),

which substantially increases the phase margin ofHGen. This effect is reflected as positive

damping torque in Fig. 6.32(b).

171


0 50 100 150 200 250 300-2

-1

0

1

2

3

4

5

ω (rad/sec)

De (

jω)

kv=0.0

kv=0.2

kv=0.5

(a) kp = 0.0, ku = 0.0, variations of kv .

0 50 100 150 200 250 300-2

-1

0

1

2

3

4

5

ω (rad/sec)

De (

jω)

kp=0.0

kp=60

kp=100

(b) kv = 0.2 p.u., ku = 0.0, variations of kp.

Fig. 6.32 Electrical damping curves of the generator (Pg = 0.1 p.u.) affected by the con-

trol parameters of the VSC-HVDC converter (inverter operation, power control mode,

Pdc = 0.9 p.u.). UIF = 0.7.

101

102

10-4

10-2

100

|HG

en(j

ω)|

kv=0.0

kv=0.5

101

102

-200

-180

-160

-140

arg

HG

en(j

ω)

(deg.)

ω (rad/sec)

(a) kp = 0.0, ku = 0.0, variations of kv .

101

102

10-4

10-2

100

|HG

en(j

ω)|

kp=0.0

kp=100

101

102

-200

-180

-160

-140

arg

HG

en(j

ω)

(deg.)

ω (rad/sec)

(b) kv = 0.2 p.u., ku = 0.0, variations of kp.

Fig. 6.33 Phase margin of the generator (Pg = 0.1 p.u.) affected by the control parameters of

the VSC-HVDC converter (inverter operation, power control mode, Pdc = 0.9 p.u.).UIF = 0.7.

172


0 50 100 150 200 250 300-2

-1

0

1

2

3

4

5

ω (rad/sec)

De (

jω)

ku=0.0

ku=60

ku=100

(a) Rectifier operation, power control.

0 50 100 150 200 250 300-2

-1

0

1

2

3

4

5

ω (rad/sec)

De (

jω)

ku=0.0

ku=60

ku=100

(b) Inverter operation, power control.

Fig. 6.34 Electrical damping curves of the generator (Pg = 0.1 p.u.) affected by the alternating-

voltage control of the VSC-HVDC converter (power control mode, Pdc = ±0.9 p.u.).Variations of ku. UIF = 0.7.

The alternating-voltage controller of the VSC-HVDC converter also affects the

electrical damping of the synchronous generator. As shown in Fig. 6.34, the alternating-

voltage controller has the effect to bring the synchronous generator and the VSC-HVDC

converter electrically “closer” for both rectifier and inverter operation. Consequently, with

higher gain ku, the impact of the VSC-HVDC converter on the electrical damping curve

of the synchronous generator increases, either positive or negative.

6.6.3 Island operation

System definition

Fig. 6.35 shows the Jacobian transfer matrix formulation of the island system shown

in Fig. 6.24(b), where Ge(s) is the transfer function between ∆ωgr to ∆T g

e with all the

feedback-control loops of the VSC-HVDC converter closed (inside the dashed box). Ta-

ble 6.6 shows the base control parameters selected for the VSC-HVDC converter. In is-

land operation, more interactions between the synchronous generator and the VSC-HVDC

converter in the subsynchronous frequency range are expected. According to the defini-

tion in (6.53), the UIF with the island-operation configuration in Fig. 6.24(b) is

UIF =MWdc

MVAg= 2.33 (6.55)

simply because SCg = 0, i.e., the short-circuit power of the ac system is zero if the

synchronous generator is not included.

173


P∆

refU

-+refP

vθ∆

+-

s

k p

PSL

s

ku

AVC

fU∆

)(sJ

ω∆

f

f

sT1

K

+

Frequency

measurementrefP∆

mT1

s

s+

-

Frequency

controller

g

eT∆

g

rω∆

)(sGe

AC-Jacobian

transfer

matrix

0V

V∆

Fig. 6.35 Jacobian transfer matrix formulation of the island system shown in Fig. 6.24(b).

Table 6.6 Control parameters of the VSC-HVDC converter in island operation. Per unit based on

350 MVA and 195 kV.

Controller Parameters Values

Active-power control kp 60 rad/s

Alternating-voltage control ku 60

Frequency control Kf , Tf , Tm, 3.72, 0.3 s, 0.02 s

High-pass current control kv, αv 0.2 p.u., 40 rad/s

174


Rectifier operation

To operate the VSC-HVDC converter as a rectifier, the synchronous generator generates

85% of its rated power, i.e., Pg = 0.85 p.u. (150 MVA power base). The generated power

is then transmitted by the VSC-HVDC link to the ac system connected to the other con-

verter station, i.e., the VSC-HVDC converter operates with Pdc = −0.36 p.u. (350 MVA

power base). No load is connected in the island system, i.e., Rl = 0.0.

Fig. 6.36 shows the electrical damping curves of the generator. In Fig. 6.36(a),

the HVDC converter is represented by a constant-frequency stiff voltage source. Similar

to the large ac-system connection configuration, the VSC-HVDC converter operating in

rectifier mode in the island system contributes with only positive damping to the generator

in the entire subsynchronous frequency range. The damping effects are, however, much

higher due to the high UIF value. The frequency-scanning results from PSCAD/EMTDC

(dashed curve) confirm the same effect. The two plots produced by the linear model and

the time simulation match fairly well.

Inverter operation

To operate the VSC-HVDC converter as an inverter, the synchronous generator generates

10% of its rated power, i.e., Pg = 0.1 p.u., while the VSC-HVDC link operates with

Pdc = 0.9 p.u. Resistive loads (330 MW) are connected in the island system to consume

the power from the generator and the VSC-HVDC link.

Fig. 6.37 shows the electrical damping curves of the generator. In Fig. 6.37(a), the

HVDC converter is represented by a constant-frequency stiff voltage source. Compared

to Fig. 6.36(a), the electrical damping of the synchronous generator in Fig. 6.37(a) is

apparently higher, which is due to the damping effects contributed by the resistive loads

connected in the island system. As shown in Fig. 6.37(b), the influence on the damping

curve of the generator by the VSC-HVDC converter operating in inverter mode is also

similar to the large ac-system connection configuration but with a higher effect due to the

high UIF value. The VSC-HVDC converter contributes with positive damping in the lower

frequency range but with negative damping in the higher frequency range. The frequency-

scanning results from PSCAD/EMTDC (dashed curve) confirm the same effect. The two

plots in Fig. 6.37 produced by the linear model and the time simulation also match fairly

well.

The influence of the control parameters kp and kv of the VSC-HVDC converter

on the electrical damping is also similar to the large ac-system connection configuration.

As shown in Fig. 6.38, the high-pass current control contributes with negative damping,

while the active-power control contributes with positive damping. Therefore, from sub-

synchronous characteristics point of view, a high gain kp of the active-power control is

favorable even if it is not necessary from a power-control point of view.

The control parameters of the frequency controller of the VSC-HVDC converter

175


0 50 100 150 200 250 300-1

0

1

2

3

4

5

ω (rad/sec)

De (

jω)


(a) With constant-frequency stiff voltage source.

0 50 100 150 200 250 300-1

0

1

2

3

4

5

ω (rad/sec)

De (

jω)


(b) With VSC-HVDC converter.

Fig. 6.36 Electrical damping curves of the generator (Pg = 0.85 p.u.) affected by the VSC-

HVDC converter (rectifier operation, Pdc = −0.36 p.u.).

0 50 100 150 200 250 300-2

-1

0

1

2

3

4

5

ω (rad/sec)

De (

jω)


(a) With constant-frequency stiff voltage source.

0 50 100 150 200 250 300-2

-1

0

1

2

3

4

5

ω (rad/sec)

De (

jω)


(b) With VSC-HVDC converter.


converter [inverter operation, Pdc = 0.9 p.u., Rl = 1.06 p.u. (330 MW resistive

loads)].

176


0 50 100 150 200 250 300-2

-1

0

1

2

3

4

5

ω (rad/sec)

De (

jω)

kv=0.0

kv=0.2

kv=0.5

(a) kp = 0.0 rad/s, ku = 0.0, variation of kv .

0 50 100 150 200 250 300-2

-1

0

1

2

3

4

5

ω (rad/sec)

De (

jω)

kp=0.0

kp=60

kp=100

(b) kv = 0.2 p.u., ku = 0.0, variation of kp.

Fig. 6.38 Electrical damping curves of the generator (Pg = 0.1 p.u.) affected by the con-

trol parameters of the VSC-HVDC converter (inverter operation, power control mode,

Pdc = 0.9 p.u.).

might also affect its subsynchronous characteristics. However, since a large time constant

Tf is usually applied, such an influence mainly appears in the low-frequency range where

no risk of SSTI is expected. Therefore, no further discussion of the sensitivity of the

subsynchronous characteristics of the VSC-HVDC converter to those control parameters

is given in the thesis.

6.6.4 Summary of the subsynchronous characteristics

In this section, the subsynchronous characteristics of a VSC-HVDC converter using power-

synchronization control are investigated by the frequency-scanning method. Two types of

ac-network configurations, i.e., large ac-system connection and island operation, are in-

vestigated. The following conclusions are drawn for the large ac-system connection:

1. A VSC-HVDC converter can have considerable impact on the electrical damp-

ing curve of a synchronous generator for network configurations with UIF values

greater than 0.1. The impact increases with higher UIF values.

2. The VSC-HVDC converter might have different subsynchronous characteristics

depending on its operation mode. The safest mode is when the converter operates

as a rectifier in the power control mode. In this operation mode, the VSC contributes

with only positive damping in the entire subsynchronous frequency range, similar

to resistive loads. By operating as an inverter, the VSC contributes with positive

damping in the lower frequency range. However, it contributes with slightly nega-

tive damping in the higher frequency range. In the direct-voltage control mode, in

either rectifier or inverter operation, the VSC might contribute with negative damp-

177


ing in the lower frequency range. However, this frequency range is rarely above

62.8 rad/s (10 Hz) even with the highest UIF value (UIF = 0.7).

3. The control parameters of the VSC-HVDC converter might also affect the subsyn-

chronous characteristics. Surprisingly, the high-pass current control HHP(s), which

is introduced for the purpose of damping, negatively affects the electrical damping

of a synchronous generator. Further frequency-domain analysis shows that the rea-

son is that HHP(s) reduces the phase margin of the synchronous generator. On the

other hand, the active-power control of the VSC-HVDC converter provides positive

damping. In other words, the negative-damping frequency range in the inverter op-

eration mode can be reduced by selecting a higher gain of the active-power control.

The alternating-voltage control has the effect of shortening the electrical distance

between the VSC-HVDC converter and the synchronous generator. That is, with

higher gain of the alternating-voltage control, higher subsynchronous impact from

the VSC-HVDC converter is expected, either positive or negative.

In island operation, stronger interactions between the VSC-HVDC converter and the syn-

chronous generator in the subsynchronous frequency range are expected. The impact of

operation modes and control parameters on the subsynchronous characteristics of the

VSC-HVDC converter is similar to large ac-system connections, but with higher magni-

tudes. From an SSTI point of view, a higher gain of active-power control is more favorable

if the stability margin of the VSC-HVDC link can be ensured.

With the two types of investigated network configurations, the only possible risk

of SSTI is when the VSC-HVDC converter operates as an inverter with network con-

figurations having high UIF values, where the VSC-HVDC converter contributes with

negative damping in the higher subsynchronous frequency range. The negative-damping

frequency range, however, can be reduced by proper tuning of the control parameters.

Thus, a subsynchronous damping controller (SSDC) for a VSC-HVDC converter using

power-synchronization control is not considered necessary.

6.7 Summary

In this chapter, the control and modeling issues for VSC-HVDC links connected to low-

inertia ac systems are investigated. For VSC-HVDC links connected to low-inertia ac

systems, a frequency droop controller and an alternating-voltage droop controller are nor-

mally required to coordinate frequency and voltage control with local power/voltage con-

trolling units. The Jacobian transfer matrix modeling concept is applied to model a typical

island system which includes a synchronous generator, an induction motor, a VSC-HVDC

link and some RLC loads. The linear model has been used for tuning the control param-

eters of the VSC-HVDC link. The simulation studies show that power-synchronization

control is flexible for various operating modes related to island operation, while the mode

178

6.7. Summary

shifts are handled seamlessly. The subsynchronous characteristics of a VSC-HVDC link

using power-synchronization control are analyzed by the frequency-scanning method. A

VSC-HVDC converter might have different subsynchronous characteristics depending on

its operation mode. For some particular operation modes, a VSC-HVDC converter might

contribute with negative damping. However, the negative-damping frequency range can

be reduced by proper tuning of the control parameters.

179


180

Chapter 7

Conclusions and Future Work

This chapter presents the conclusions of the thesis and suggests future work.

7.1 Conclusions

PWM-based VSC-HVDC systems show many advantages compared to the thyristor-

based LCC-HVDC system. One prominent feature is that the VSC-HVDC system has

the potential to be connected to very weak ac systems where the LCC-HVDC system has

difficulties. In this thesis, the modeling and control issues for VSC-HVDC links connected

to weak ac systems are investigated.

In order to fully utilize the potential of the VSC-HVDC system for weak-ac-system

connections, a novel control method, i.e., power-synchronization control, for grid-con-

nected VSCs is proposed. A grid-connected VSC using power-synchronization control

basically resembles the dynamic behavior of a synchronous machine. However, due to

the technical requirements for a VSC-HVDC link and various limitations of VSC valves,

additional control functions are required to deal with various practical issues during oper-

ation. Such control functions include:

• A high-pass current control function to damp various resonances in ac systems.

• A current limitation function to ride through ac-system faults.

• A bumpless-transfer scheme for switching the synchronization input of the VSC,

and an anti-windup scheme for alternating-voltage control.

• A negative-sequence current controller to mitigate unbalance valve currents in the

steady state or during unbalanced ac-system faults.

By the comparison performed in the thesis, it is shown that power-synchronization control

is superior to the traditional vector current control for VSC-HVDC links connected to

weak ac systems.

181

Chapter 7. Conclusions and Future Work

The Jacobian transfer matrix is proposed as a unified modeling technique for dy-

namic modeling of ac/dc systems. With the ac Jacobian transfer matrix concept, a syn-

chronous ac system is viewed upon as one multivariable feedback-control system where

the feedback controllers and the controlled process, i.e., the Jacobian transfer matrix, are

explicitly defined. Thus, the stability of a power system is uniquely defined as the stability

of the closed-loop system formed by the controllers and the Jacobian transfer matrix.

One interesting finding is that the transmission zeros of the ac Jacobian transfer ma-

trix reflect several power-system stability phenomena. By using a simplified ac-network

configuration, the thesis shows that the transmission zeros of the ac Jacobian transfer

matrix have close relationships with angle and voltage stability defined by the classical

power-system theory. Two additional common power-system components, i.e., the syn-

chronous generator and induction motor, are also modeled by the Jacobian transfer matrix

concept. It is discovered that the transmission zeros of the ac Jacobian transfer matrices

are useful to interpret some classical power-system concepts, such as the synchronizing

torque for the synchronous generator and the pull-out slip for the induction motor, from a

feedback-control point of view.

The complex poles of the ac Jacobian transfer matrix represent the various reso-

nances in the ac system. The high-pass current control of power-synchronization control

and the inner current controller of vector current control are all shown to have general

damping effects on the resonances regardless of the frequency of the resonance.

The similar concept is extended to model dc systems constructed by multiple VSC

terminals. By using a π-link model, it is proven that the dc Jacobian transfer matrix is an

inherently unstable process, where the dc resistance has a destabilizing effect. Accord-

ingly, the feedback controller, i.e., the direct-voltage controller, has to be applied at least

at one of the converter stations to stabilize the dc system. Similar to the complex poles

of the ac Jacobian transfer matrix, the complex poles of the dc Jacobian transfer matrix

also represent various resonances in the dc system. Long overhead dc-transmission line

may create resonance peaks in the low frequency range where the control system of the

VSC-HVDC link is active. A notch filter in the direct-voltage control loop is proposed to

reduce the resonance peak.

For VSC-HVDC links using power-synchronization control, the SCR of the ac sys-

tem is no longer a critical limiting factor, i.e., VSC-HVDC links are feasible to be con-

nected to ac systems with any short-circuit capacity. In addition, the VSC-HVDC link

contributes with short-circuit capacity to the ac system at the PCC, however, without in-

creasing the short-circuit current thanks to its current limiting capability during ac-system

faults. The major consideration is the RHP zero of the ac Jacobian transfer matrix, which

moves closer to the origin with larger load angles. The RHP zero imposes a fundamental

limitation on the achievable bandwidth of the VSC, which implies that the VSC-HVDC

link shall not operate with a too large load angle for maintaining a reasonable stability

margin. For weak-ac-system connections, higher values of dc capacitance are also nec-

182

7.2. Future work

essary to reduce the direct-voltage variations. A control structure for interconnecting two

very weak ac systems is proposed in the thesis. This control structure enables a power

transmission of 0.86 p.u. from a system with an SCR of 1.2 to a system with an SCR of

1.0.

For VSC-HVDC links connected to low-inertia ac systems, a frequency droop con-

troller and an alternating-voltage droop controller are normally required to coordinate

frequency and voltage control with local power/voltage controlling units. In the thesis,

the Jacobian transfer matrix modeling concept is applied to model a typical island sys-

tem which includes a synchronous generator, an induction motor, a VSC-HVDC link and

some RLC loads. The linear model has been used for tuning the control parameters of

the VSC-HVDC link. The simulation studies show that power-synchronization control is

flexible for various operation modes related to island operation, while the mode shifts are

handled seamlessly.

The subsynchronous characteristics of a VSC-HVDC link using power-synchroni-

zation control are analyzed by the frequency-scanning method. Two ac-network config-

urations, i.e., large ac-system connection and island operation, are investigated. A VSC-

HVDC converter might have different subsynchronous characteristics depending on its

operation mode. For most of the operation modes, the VSC-HVDC converter contributes

with positive damping to a synchronous generator located in the vicinity. The only possi-

ble risk of SSTI is when the VSC-HVDC converter operates as an inverter in a high-UIF

system. The negative-damping frequency range, however, can be reduced by proper tuning

of the control parameters. Thus, a subsynchronous damping controller for a VSC-HVDC

link using power-synchronization control is not considered necessary.

7.2 Future work

The following is a list of possible future work:

• Power-synchronization control is a feasible control system for any grid-connected

VSC. Another possible application is control of STATCOMs with energy storage.

By using power-synchronization control, the STATCOM is able to operate with

various challenging operating conditions.

• By using power-synchronization control, the SCR of the ac system is no longer a

limiting factor for VSC-HVDC links connected to weak ac systems. However, the

variation of SCRs, i.e., model uncertainty, is still a challenge. Besides the robust

control methods proposed in the thesis, other possible solutions to deal with model

uncertainties are 1. adaptive control. 2. online detection of the SCR of the ac system.

• The losses of the VSC valves have been neglected in the linear models in the thesis.

However, it was shown by frequency-scanning results that the representation of the

183

Chapter 7. Conclusions and Future Work

valve losses in linear models has an importance if dc resonances are of concern.

To properly represent the valve losses, the linear model should take into account

the topology of the converter as well as the applied PWM technique, which are of

nonlinear nature.

• The structure proposed for interconnecting two very weak ac systems has a possi-

bility to be simplified. A possible simplification is to remove the power-synchroni-

zation control loop but to use the direct-voltage control to supply the synchroniza-

tion input to the VSC instead. Such a modification has the potential to improve the

response time of the VSC-HVDC link and reduce the dc-capacitance requirement.

• The modeling concept of the ac Jacobian transfer matrix is useful for studying the

interactions of high power-electronics devices located in the vicinity, e.g., multi-

infeed HVDC links, interactions between HVDC links and FACTS devices, etc.

• The modeling concept of the dc Jacobian transfer matrix is useful for studying

the potential future dc systems. In this thesis, only a π-link dc model with two

VSC terminals is analyzed. Larger dc systems with more VSC terminals should be

analyzed to gain a thorough understanding of the property of a dc system.

• In analyzing the subsynchronous characteristics of a VSC-HVDC converter, two

ac-network configurations, i.e., large ac-system connection and island operation,

are investigated. Another scenario to be investigated is when a series-compensated

ac line is connected in the vicinity.

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194

Appendices

195

Appendix A

Fundamentals of the Phasor and the

Space-Vector Theory

This appendix gives a brief review of the two different approaches for ac-circuit analysis,

i.e., the phasor approach and the space-vector approach in Section A.1 and Section A.2.

In the equations, phasors are denoted by capital letters with overline, while space vectors

are denoted by bold letters. In order to avoid confusions with scaling, per unit values are

assumed in the equations for both of the theory. In Section A.3, the implementations of αβ

and dq transformations for controller designs are described.

A.1 Fundamentals of the phasor theory

The phasor theory is the major mathematical tool for power systems. It has been used

widely in power-flow programs, short-circuit calculations, and stability analysis. In this

section, the fundamentals of the theory are briefly reviewed .

With the simplified three-phase ac system shown Fig. 4.5, the impedance Z of the

ac line is defined as

Z = R + jX (A.1)

where X = ω1L is defined as the reactance of the ac line, with ω1 = 2πf1, where f1 is

the fundamental frequency of the ac system.

In the phasor theory, Ohm’s law still holds for three-phase ac systems if they are

symmetrical, i.e., the current phasor I can be calculated as

I =V − E

Z=

V − E

R + jX. (A.2)

The phasor relations in (A.2) can also be conveniently visualized by the phasor diagram

as shown in Fig. A.1.

197

Chapter A. Fundamentals of the Phasor and the Space-Vector Theory

V

E

ILjω

IR

I

θ

Real

Imaginary

Fig. A.1 Phasor diagram of the electrical circuit in Fig. 4.5.

The complex power is defined as

S = P + jQ = V I∗

= (V cos θ + jV sin θ)

[V cos θ + jV sin θ −E

jX

]∗

=EV

Xsin θ + j

[V 2 − EV cos θ

X

](A.3)

where the resistance R is neglected in (A.3) for simplicity. From (A.3), the active and

reactive powers P and Q can be identified as

P =EV

Xsin θ (A.4)

Q =V 2 −EV cos θ

X. (A.5)

A.2 Fundamentals of the space-vector theory

In contrast to the phasor approach, where only fundamental frequency effects are treated,

the space-vector approach deals with instantaneous values in the electrical circuit. The

basic idea of the space-vector theory comes from the fact that one complex vector y can

be used to represent three-phase variables ya, yb, yc, if any zero-sequence components are

disregarded. As shown in Fig. A.2(a), the complex vector y is defined as

ys =2

3

(ya + ybe

j 2π3 + yce

j 4π3

)(A.6)

where superscript s denotes the stationary αβ reference frame.

At any instant, the rotating vector ys has two corresponding variables yα and yβ on

the αβ (i.e., real and imaginary) axes. To transform the rotating vector to a non-rotating

198

A.2. Fundamentals of the space-vector theory

α

β

αy

βy

a

b

c

y

α

β

dy

βy

y d t1ω

q

qy

αy

(a) (b)

Fig. A.2 Reference frame transformations. (a) From three-phase variables to a rotating vector in

the stationary αβ frame. (b) From the stationary αβ frame to the synchronous dq frame.

one, a rotating dq reference frame is introduced, as shown in Fig. A.2(b). The dq frame

rotates in the same direction as ys with a speed that equals the angular frequency ω1 of

the ac system. Consequently, the vector y in the dq frame is related to ys in the stationary

frame by

ys = ejω1ty. (A.7)

In the following, the simple ac system in Fig. 4.5 is analyzed by the space-vector ap-

proach. The equations for the dynamics of the ac system in Fig. 4.5 can be described by

Kirchhoff’s voltage law in the stationary frame as

Ldis

dt= vs − Es − Ris (A.8)

where vs, Es, and is denote the space vectors of the voltage of node V, the voltage of node

E, and the current of the ac line, respectively. To transform the electrical variables from

the stationary frame to the dq frame, the space αβ vectors are written as

Es = Eejω1t, is = iejω1t, vs = vejω1t. (A.9)

Substituting (A.9) into (A.8) yields the dynamic equation in the dq frame

Ldi

dt= v −E − Ri − jω1Li. (A.10)

By comparing (A.10) to (A.2), it can be observed that there is a relationship between

phasors and space vectors. If the same reference node is chosen, e.g., node E, the d and q

components of the space vectors correspond to the real and imaginary parts of the phasors

in the steady state (didt

= 0).

For the simple ac system in Fig. 4.5, ac capacitors are not included. However, it can

easily be shown that the same statement holds even with ac capacitors included.

199


Assuming per unit scaling [20], the instantaneous active power P and reactive

power Q are defined in the dq frame as

P = Re {vi∗} = vdid + vqiq (A.11)

Q = Im {vi∗} = vqid − vdiq. (A.12)

In the steady state, the active and reactive powers defined in (A.11) and (A.12) in the dq

frame equal those defined in (A.4) and (A.5) by the phasor approach.

A.3 Implementation of αβ and dq transformations

In the control block diagram of vector current control (Fig. 3.5) and power-synchronization

control (Fig. 3.10), αβ and dq transformations are applied for controller implementations.

In this section, the mathematical expressions for those blocks are described.

A.3.1 abc-αβ transformation

Following (A.6), the vector ys in the stationary frame is expressed as

ys = yα + jyβ =2

3

(ya + ybe

j 2π3 + yce

j 4π3

). (A.13)

Since the zero-sequence quantities in the three-phase variables are neglected, the follow-

ing relation is established

ya + yb + yc = 0. (A.14)

Based on (A.13) and (A.14), the αβ components can be derived from the three-phase

variables

yα = ya

yβ =1√3(yb − yc). (A.15)

Based on (A.15) and (A.14), the transformation from the αβ components to the three-

phase variables can also be derived

ya = yα

yb = −1

2yα +

√3

2yβ

yc = −1

2yα −

√3

2yβ. (A.16)

200

A.3. Implementation of αβ and dq transformations

A.3.2 αβ-dq transformation

According to (A.7), the vector y = yd + jyq in the synchronous dq frame is related to

ys = yα + jyβ in the stationary αβ frame by

y = e−jωtys (A.17)

where ωt is the synchronization variable (from PSL or PLL) of the VSC. Eq. (A.17) can

be written in component form as

yd = yα cosωt+ yβ sinωt

yq = −yα sinωt+ yβ cosωt. (A.18)

Similarly, the vector ys = yα + jyβ in the stationary αβ frame is related to y = yd + jyq

in the synchronous dq frame by

ys = ejωty (A.19)

which can also be written in component form as

yα = yd cosωt− yq sinωt

yβ = yd sinωt+ yq cosωt. (A.20)

201


202

Appendix B

Jacobian Transfer Matrix

In this appendix, some mathematical derivations and proofs related to Jacobian transfer

matrix are given. In Section B.1, the transfer functions of the ac Jacobian transfer matrix

for VSCs connected to an impedance source neglecting the ac capacitor at the filter bus

are derived. In Section B.2, a mathematical proof of the instability of the dc Jacobian

transfer matrix for a π-link dc circuit is given.

B.1 Derivation of the transfer functions in Table 4.1

B.1.1 Transfer function JPθ (s)

In a synchronous reference frame, by neglecting the ac capacitorCf , the dynamic equation

of the main circuit in Fig. 4.4 can be written in the dq frame as

Ldi

dt= v − E −Ri − jω1Li (B.1)

where R = Rn + Rc, L = Ln + Lc, and i = ic = in. Eq. (B.1) can also be written in

component form as

Ldiddt

= V cos θv − E0 − Rid + ω1Liq

Ldiqdt

= V sin θv − Riq − ω1Lid. (B.2)

Let the voltage magnitude of the VSC be kept constant, i.e., V = V0. If the operating

points in (B.2) are denoted with subscript 0 and the deviations around the operating points

are denoted with the prefix ∆, (B.2) can be linearized using

θv = θv0 + ∆θv, id = id0 + ∆id, iq = iq0 + ∆iq (B.3)

where cos θv and sin θv can be linearized as

sin (θv0 + ∆θv) ≈ sin θv0 + cos θv0∆θv

cos (θv0 + ∆θv) ≈ cos θv0 − sin θv0∆θv. (B.4)

203

Chapter B. Jacobian Transfer Matrix

Substituting (B.3) and (B.4) into (B.2), and keeping only the deviation parts yields the

linearized form of (B.2)

Ld∆iddt

= −V0 sin θv0∆θv − R∆id + ω1L∆iq

Ld∆iqdt

= V0 cos θv0∆θv −R∆iq − ω1L∆id. (B.5)

Moreover, by applying the Laplace transform to (B.5), the transfer functions of ∆id vs.

∆θv, and ∆iq vs. ∆θv are obtained

∆id = V0ω1L cos θv0 − (sL+R) sin θv0

(sL+R)2 + (ω1L)2 ∆θv

∆iq = V0ω1L sin θv0 + (sL+R) cos θv0

(sL+R)2 + (ω1L)2 ∆θv. (B.6)

According to the definition in (A.11), the instantaneous active power P from the filter bus

to the ac system is

P = Re {uf i∗} . (B.7)

Linearizing (B.7) yields the expression of the active-power deviation as

∆P = Re {i∗0∆uf + uf0∆i∗} (B.8)


∆P =

[id0

iq0

]T [∆ufd

∆ufq

]+

[ufd0

ufq0

]T [∆id

∆iq

]. (B.9)

The current vector i0 = id0 + jiq0 at the operation point can be derived by

i0 =Uf0e

jθu0 − E0

Rn + jω1Ln

(B.10)

which yields

id0 =Uf0 sin θu0

ω1Ln, iq0 =

E0 − Uf0 cos θu0

ω1Ln. (B.11)

The resistance Rn in (B.10) is neglected in (B.11) to simplify the expression. The voltage

vector of the filter bus at the operating point, uf0 = ufd0 + jufq0, can be expressed as

ufd0 = Uf0 cos θu0, ufq0 = Uf0 sin θu0. (B.12)

By linearization of the expression

Lndi

dt= uf − E −Rni − jω1Lni (B.13)

204

B.1. Derivation of the transfer functions in Table 4.1

and subdivision into d and q components, the voltage-deviation components ∆ufd and

∆ufq can be written as

∆ufd = sLg∆id − ω1Lg∆iq

∆ufq = sLg∆iq + ω1Lg∆id. (B.14)

The resistance Rn is also neglected in (B.14) to simplify the expression. By substituting

(B.6), (B.11), (B.12), and (B.14) into (B.9), the dynamic relation between ∆P and ∆θv

is found to be

∆P =a0s

2 + a1s+ a2

(sL+R)2 + (ω1L)2

︸︷︷︸JPθ(s)

∆θv (B.15)

where

a0 =L

ω1(k3 − k1)

a1 =R

ω1(k3 − k1) − Lk2

a2 = ω1Lk3 −Rk4. (B.16)

The k parameters used in (B.16) are defined as

k1 = V0Uf0 cos (θv0 − θu0)

k2 = V0Uf0 sin (θv0 − θu0)

k3 = E0V0 cos θv0

k4 = E0V0 sin θv0. (B.17)

B.1.2 Transfer function JQθ (s)

According to the definition in (A.12), the instantaneous reactive power Q from the filter

bus to the ac system is

Q = Im {uf i∗} . (B.18)

Linearizing (B.18) yields the expression of the reactive-power deviation. Thus,

∆Q = Im {i∗0∆uf + uf0∆i∗} (B.19)


∆Q =

[id0

−iq0

]T [∆ufq

∆ufd

]+

[ufd0

ufq0

]T [ −∆iq

∆id

]. (B.20)

205


By substituting (B.6), (B.11), (B.12), and (B.14) into (B.20), the dynamic relation be-

tween ∆Q and ∆θv is found to be

∆Q =a0s

2 + a1s+ a2

(sL+R)2 + (ω1L)2

︸︷︷︸JQθ(s)

∆θv (B.21)

where

a0 =L

ω1(k4 − k2)

a1 =R

ω1(k4 − k2) − Lk1

a2 = ω1L (k4 − 2k2) − R (k3 − 2k1) . (B.22)

B.1.3 Transfer function JUf θ (s)

The voltage magnitude at the filter bus can be expressed as

Uf =√u2

fd + u2fq (B.23)

which can be linearized by using

Uf = Uf0 + ∆Uf , ufd = ufd0 + ∆ufd, ufq = ufq0 + ∆ufq. (B.24)

Substituting (B.24) into (B.23) and only keeping the deviation parts yields the linearized

form of (B.23) as

∆Uf =ufd0

Uf0∆ufd +

ufq0

Uf0∆ufq. (B.25)

By substituting (B.6) into (B.14), and further substituting (B.14) and (B.12) into (B.25),

the dynamic relation between ∆Uf and ∆θv is found to be

∆Uf =a0s

2 + a1s+ a2

(sL+R)2 + (ω1L)2

︸︷︷︸JUf θ(s)

∆θv (B.26)

where

a0 = −LLnk2/Uf0

a1 = −RLnk2/Uf0

a2 = −ω21LLnk2/Uf0 − RLnk1/Uf0. (B.27)

If the resistance R in (B.27) is neglected, JUf θ (s) can be simplified as

JUf θ (s) = −Ln

LV0 sin (θv0 − θu0) . (B.28)

206

B.1. Derivation of the transfer functions in Table 4.1

B.1.4 Transfer function JPV (s)

To derive the transfer functions related to the voltage-magnitude deviation ∆V at the

VSC, let θv be kept constant, i.e., θv = θv0. Eq. (B.2) can be linearized using

V = V0 + ∆V, id = id0 + ∆id, iq = iq0 + ∆iq. (B.29)

Substituting (B.29) into (B.2), and keeping only the deviation parts yields the linearized

form of (B.2) as

Ld∆iddt

= ∆V cos θv0 − R∆id + ω1L∆iq

Ld∆iqdt

= ∆V sin θv0 −R∆iq − ω1L∆id. (B.30)

Moreover, by applying the Laplace transform to (B.30), the following transfer functions

are obtained for ∆id and ∆iq vs. ∆V

∆id =ω1L sin θv0 + (sL+R) cos θv0

(sL+R)2 + (ω1L)2 ∆V

∆iq =−ω1L cos θv0 + (sL+R) sin θv0

(sL+R)2 + (ω1L)2 ∆V. (B.31)

By substituting (B.11), (B.12), (B.14) and (B.31) into (B.9), the dynamic relation between

∆P and ∆V/V0 is found to be

∆P =a0s

2 + a1s+ a2

(sL+R)2 + (ω1L)2

︸︷︷︸JPV(s)

·∆VV0

(B.32)

where

a0 =L

ω1(k4 − k2)

a1 =R

ω1(k4 − k2) + Lk1

a2 = ω1Lk4 +Rk3. (B.33)

B.1.5 Transfer function JQV (s)

By substituting (B.11), (B.12), (B.14) and (B.31) into (B.20), the dynamic relation be-

tween ∆Q and ∆V/V0 is found to be

∆Q =a0s

2 + a1s + a2

(sL+R)2 + (ω1L)2

︸︷︷︸JQV(s)

·∆VV0

(B.34)

207


where

a0 =L

ω1(k1 − k3)

a1 =R

ω1(k1 − k3) − Lk2

a2 = ω1L (2k1 − k3) +R (k4 − 2k2) . (B.35)

B.1.6 Transfer function JUf V (s)

By substituting (B.31) into (B.14), and further substituting (B.14) and (B.12) into (B.25),

the dynamic relation between ∆Uf and ∆V/V0 is found to be

∆Uf =a0s

2 + a1s+ a2

(sL+R)2 + (ω1L)2

︸︷︷︸JUf V(s)

·∆VV0

(B.36)

where

a0 = LLnk1/Uf0

a1 = RLnk1/Uf0

a2 = ω21LLnk1/Uf0 − RLnk2/Uf0. (B.37)

If the resistance R in (B.37) is neglected, JUf V (s) can be simplified as

JUf V (s) =Ln

LV0 cos (θv0 − θu0) . (B.38)

B.2 Proof of the instability of the dc Jacobian transfer

matrix

In the following, a mathematical proof of the instability of the dc Jacobian transfer matrix

Gdc(s) for operating points that Pdc10 6= 0 and Pdc20 6= 0 is given. Gdc(s) is a linear

model of the dc π-link in Fig. 4.25(b).

The poles of Gdc(s) are the eigenvalues of the A matrix in (4.78), which has the

characteristic equation

det(λI − A) = λ3 + a1λ2 + a2λ

2 + a3 = 0 (B.39)

208

B.2. Proof of the instability of the dc Jacobian transfer matrix

with the coefficients

a1 =Pdc10

u2dc10Cdc1

+Pdc20

u2dc20Cdc2

+Rdc

Ldc

a2 =Pdc10Pdc20

u2dc10u

2dc20Cdc1Cdc2

+Pdc10Rdc

u2dc10LdcCdc1

+Pdc20Rdc

u2dc20LdcCdc2

+1

LdcCdc1

+1

LdcCdc2

a3 =Pdc10u

2dc20 + Pdc20u

2dc10 + Pdc10Pdc20Rdc

U2dc10u

2dc20LdcCdc1Cdc2

. (B.40)

To prove the instability of Gdc(s), it is sufficient that any one of the three coefficients in

(B.40) is negative. The following is a proof of

a3 < 0. (B.41)

It is obvious that the denominator of a3 in (B.40) is positive. Thus, (B.41) is identical to

Pdc10u2dc20 + Pdc20u

2dc10 + Pdc10Pdc20Rdc < 0. (B.42)

From the main circuit of the π-link model in Fig. 4.25(b), the following equality is estab-

lishedPdc10

udc10= idc0 = −Pdc20

udc20. (B.43)

For VSC-HVDC applications, it is apparent that udc10 and udc20 have the same polarity.

Consequently, it follows from (B.43) that

Pdc10

Pdc20< 0 (B.44)

or

Pdc10Pdc20 < 0. (B.45)

Based on the inequality in (B.45), by dividing Pd10Pd20 at both sides, (B.42) can be re-

written asu2

dc20

Pdc20+u2

dc10

Pdc10+Rdc > 0. (B.46)

Substituting

Pdc10 = udc10idc0

Pdc20 = −udc20idc0

udc20 = udc10 −Rdcidc0 (B.47)

into (B.46), yields

2Rdc > 0 (B.48)

which apparently holds. In other words, for other operating points when Pdc10 6= 0 and

Pdc20 6= 0, Gdc(s) is unstable as long as the dc-transmission line is not lossless.

209


210

Appendix C

Technical Data of the Test System

This appendix gives the technical data of the VSC-HVDC system, the synchronous gener-

ator, and the induction motor that are used in the thesis. A 50 Hz ac system is assumed.

C.1 The VSC-HVDC link

Table C.1 Technical data of the VSC-HVDC system. AC: per unit based on 350 MVA and 195 kV.

DC: per unit based on 350 MW and 150 kV.

Rated power PdcN 350 MW

Rated ac voltage (Line-to-line) UacN 195 kV

Nominal ac system frequency f1 50 Hz

Phase-reactor reactance ω1Lc 0.2 p.u.

Phase-reactor resistance Rc 0.01 p.u.

Converter topology two-level

PWM switching frequency fsw 1650 Hz

Maximum valve current Imax 1.08 p.u.

Rated direct voltage UdcN ±150 kV

DC capacitance Cdc 0.015 p.u.

Converter transformer rating 380 MVA

Converter transformer ratio 195 kV/400 kV

Transformer leakage inductance Ltv 12%

211

Chapter C. Technical Data of the Test System

C.2 The synchronous generator

Table C.2 Technical data of the synchronous generator. Per unit based on 150 MVA and 20 kV.

Rated power MVAg 150 MVA

Rated terminal voltage (Line-to-line) UacN 20 kV

Nominal ac system frequency f1 50 Hz

Stator resistance Ra 0.0045 p.u.

Stator leakage inductance Ll 0.13 p.u.

Synchronous inductance Ld, Lq 1.79 p.u., 1.71 p.u.

Transient inductance L′

d, L′

q 0.169 p.u., 0.228 p.u.

Subtransient inductance L′′

d , L′′

q 0.135 p.u., 0.2 p.u.

Transient time constant T′

d0, T′

q0 4.3 sec, 0.85 sec

Subtransient time constant T′′

d0, T′′

q0 0.032 sec, 0.05 sec

Inertia constant H 6.175 sec

Damping coefficient KD 0.005

Step-up transformer rating 150 MVA

Step-up transformer ratio 20 kV/400 kV

Transformer leakage inductance Ltg 14%

The parameters of the equivalent circuit can be calculated by the technical data given in

Table C.2:

1. The mutual inductances Lad and Laq are

Lad = Ld − Ll = 1.66 p.u.

Laq = Lq − Ll = 1.58 p.u. (C.1)

2. The leakage inductance of the field circuit Lfd is

Lfd =Lad(L

′

d − Ll)

Lad − (L′

d − Ll)= 0.04 p.u. (C.2)

212

C.3. The induction motor

3. The leakage inductances of the damping circuits L1d, L1q and L2q are

L1d =LadLfd(L

′′

d − Ll)

LadLfd − (Lad + Lfd)(L′′

d − Ll)= 0.0057 p.u.

L1q =Laq(L

′

q − Ll)

Laq − (L′

q − Ll)= 0.1045 p.u.

L2q =LaqL1q(L

′′

q − Ll)

LaqL1q − (Laq + L1q)(L′′

q − Ll)= 0.245 p.u. (C.3)

4. The resistance of the field circuit Rfd is

Rfd =Lad + Lfd

T′

d0ω1

= 0.0013 p.u. (C.4)

5. The resistances of the damping circuits R1d, R1q and R2q are

R1d =1

T′′

d0ω1

(L1d +

LadLfd

Lad + Lfd

)= 0.0044 p.u.

R1q =Laq + L1q

T′

q0ω1

= 0.0063 p.u.

R2q =1

T′′

q0ω1

(L2q +

LaqL1q

Laq + L1q

)= 0.0218 p.u. (C.5)

C.3 The induction motor

Table C.3 Technical data of the induction motor. Per unit based on 30 MVA and 12 kV.

Rated power MVAm 30 MVA

Rated terminal voltage (line-to-line) UacN 12 kV

Stator resistance Rs 0.0034 p.u.

Stator leakage inductance Lsλ 0.09 p.u.

Rotor resistance Rr 0.007 p.u.

Rotor leakage inductance Lrλ 0.08 p.u.

Magnetizing inductance Lm 1.9 p.u.

Inertia constant J = 2H 3.7267 sec

Damping coefficient KD 0.005

Step-up transformer rating 30 MVA

Step-up transformer ratio 12 kV/400 kV

Transformer leakage inductance Ltm 8%

213

Chapter C. Technical Data of the Test System

214

Appendix D

List of Symbols and Abbreviations

Symbols

A system matrix

AS desired attenuation for the sensitivity function

AT desired attenuation for the complementary sensitivity function

a coefficient

B input matrix

C output matrix

Cdc capacitance of the single dc capacitor dc-link representation

Cdc1, Cdc2 dc capacitances of the π-link representation

Cf capacitance of the ac capacitor connected to the filter-bus of the

VSC-HVDC system

Cn capacitance of a series capacitor

D feedthrough matrix

De damping torque

e control error signal

E voltage vector of a constant-frequency stiff voltage source

efd field voltage of the synchronous generator

eg terminal voltage vector of the synchronous generator

em terminal voltage vector of the induction motor

ev terminal voltage vector of the VSC-HVDC converter

f1 nominal frequency of the ac system

fsw PWM switching frequency

FPLL PLL controller

215

Chapter D. List of Symbols and Abbreviations

Fn notch filter for the direct-voltage controller

F low-pass filter transfer matrix

H inertia constant of the synchronous generator or the induction mo-

tor

Hac inertia of the ac system

Hdc relative rotational inertia of a dc link

HHP high-pass filter

HLP low-pass filter

I identity matrix

i current vector

j√−1

J, J Jacobian transfer matrix and transfer function

J process model of J

Jp non-invertible part of J

Jn invertible part of J

K, K controller transfer matrix and transfer function

kc load compensation ratio of the VSC

Ke synchronizing torque

ku gain of alternating-voltage control of the VSC

kv gain of high-pass current control of the VSC

kg load compensation ratio of the synchronous generator

kp gain of the power-synchronization loop of the VSC

Kr prefilter of direct-voltage control

L inductance

Lc phase-reactor inductance of the VSC

Ldc dc inductance of the π-link representation

P active power

PacN ac power base value

PdcN dc power base value

p pole

Q reactive power

R resistance

Rdc dc resistance of the π-link representation

r reference signal

216

S sensitivity function

s Laplace operator

Td time delay

Tdc time constant of the dc capacitor

Tp period time

Te electrical torque

Tm mechanical torque

T complementary sensitivity function

u input vector

UacN alternating-voltage base value

UdcN direct-voltage base value

uf filter-bus voltage vector

v VSC voltage vector

W1 sensitivity weighting function

W3 complementary sensitivity weighting function

X reactance

y output vector

z zero

ZRHP RHP zero

α bandwidth

δ load angle of the synchronous generator

γ parameter of the anti-windup scheme

L Laplace transform

µ structured singular value

ω1 fundamental angular frequency

ωr rotor angular frequency

ωn resonance frequency of Fn

ωS crossover frequency of W1

ωT crossover frequency of W3

φ power factor angle

ψ flux

ρc a state variable to represent HHP(s) in state-space form

σ singular value

217


θ phase angle

θu load angle of the VSC at the filter bus

θv load angle of the VSC at the VSC bus

τ time constant of a low-pass filter

ξ parameter of Fn

Subscripts

a,b,c phase quantities

ac alternating current quantity

c VSC quantity

d d component in the dq frame

dc direct current quantity

f filter-bus values of the VSC

g synchronous generator quantity

I imaginary component in the R− I frame

i integral control parameter

l load quantity

m induction motor quantity

max maximum value

min minimum value

n ac-system quantity

N base value

nom nominal value

p proportional control parameter

pcc PCC quantity

PLL PLL quantity

R real component in the R− I frame

ref reference value

s stator quantity

t transformer quantity

q q component in the dq frame

v VSC quantity

0 operating point value

218

∞ infinity norm

Superscripts

c quantity in the converter dq frame

g synchronous generator quantity

m induction motor quantity

P control parameter related to active-power control

PLL control parameter related to PLL

Q control parameter related to reactive-power control

ref reference value

s quantity in the synchronous αβ-frame

U control parameter related to alternating-voltage control

Abbreviations

ac alternating current

APC active-power control

AVC alternating-voltage control

CCC capacitor-commutated converter

conj conjugate

CSC current-source converter

dc direct current

det determinant

emf electromotive force

FACTS flexible ac transmission systems

GTO gate turn off

HP high pass

HVDC high-voltage direct-current

IGBT insulated-gate bipolar transistor

Im imaginary

IM induction motor

IMC internal model control

LCC line-commutated converter

LHP left-half plane

219


LP low pass

max maximum

min minimum

MIMO multi-input multi-output

MPC maximum power curve

M2C modular-multilevel converter

nom nominal

NPC neutral-point clamped

OMIB one-machine infinite-bus

PCC point-of-common-coupling

PI proportional-integral

PLL phase-locked loop

PSL power-synchronization loop

PSS power-system stabilizer

p.u. per unit

PWM pulse-width modulation

Re real

RHP right-half plane

RLC resistive-inductive-capacitive

RPC reactive-power control

SC synchronous condenser

SCR short-circuit ratio

SG synchronous generator

SISO single-input single-output

SM synchronous machine

SSDC subsynchronous damping controller

SSR subsynchronous resonance

SSTI subsynchronous torsional interaction

STATCOM static synchronous compensator

TOV transient over voltage

UIF unit interaction factor

VSC voltage-source converter

220

PhD thesis

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