Power System Protection

A Novel Approach for Tuning of Power

System Stabilizer Using Genetic Algorithm

A Thesis

Submitted for the Degree of

in the Faculty of Engineering

By

Ravindra Singh

Department of Electrical Engineering INDIAN INSTITUTE OF SCIENCE

Bangalore 560012, (INDIA)

July 2004

Abstract

The problem of dynamic stability of power system has challenged power system engineers

since over three decades now. In a generator, the electromechanical coupling between the

rotor and the rest of the system causes it to behave in a manner similar to a spring mass

damper system, which exhibits an oscillatory behaviour around the equilibrium state, follow-

ing any disturbance, such as sudden change in loads, change in transmission line parameters,

fluctuations in the output of turbine and faults etc. The use of fast acting high gain AVRs

and evolution of large interconnected power systems with transfer of bulk power across weak

transmission links have further aggravated the problem of low frequency oscillations. The

oscillations, which are typically in the frequency range of 0.2 to 3.0 Hz, might be excited by

the disturbances in the system or, in some cases, might even build up spontaneously. These

oscillations limit the power transmission capability of a network and, sometimes, even cause

a loss of synchronism and an eventual breakdown of the entire system.

The application of Power System Stabilizer (PSS) can help in damping out these oscilla-

tions and improve the system stability. The traditional and till date the most popular solu-

tion to this problem is application of conventional power system stabilizer (CPSS). However,

continual changes in the operating condition and network parameters result in corresponding

change in system dynamics. This constantly changing nature of power system makes the

design of CPSS a difficult task.

Adaptive control methods have been applied to overcome this problem with some degree of

success. However, the complications involved in implementing such controllers have restricted

their practical usage.

In recent years there has been a growing interest in robust stabilization and disturbance

attenuation problem. H control theory provides a powerful tool to deal with robust sta-

bilization and disturbance attenuation problem. However the standard H control theory

does not guarantee robust performance under the presence of all the uncertainties in the

power plants.

This thesis provides a method for designing fixed parameter controller for system to ensure

robustness under model uncertainties. Minimum performance required of PSS is decided a

priori and achieved over the entire range of operating conditions.

A new method has been proposed for tuning the parameters of a fixed gain power sys-

tem stabilizer. The stabilizer places the troublesome system modes in an acceptable region

in the complex plane and guarantees a robust performance over a wide range of operating

conditions. Robust D-stability is taken as primary specification for design. Conventional

lead/lag PSS structure is retained but its parameters are re-tuned using genetic algorithm

(GA) to obtain enhanced performance. The advantage of GA technique for tuning the PSS

parameters is that it is independent of the complexity of the performance index considered.

It suffices to specify an appropriate objective function and to place finite bounds on the op-

timized parameters. The efficacy of the proposed method has been tested on single machine

as well as multimachine systems. The proposed method of tuning the PSS is an attractive

alternative to conventional fixed gain stabilizer design as it retains the simplicity of the con-

ventional PSS and still guarantees a robust acceptable performance over a wide range of

operating and system condition.

The method suggested in this thesis can be used for designing robust power system sta-

bilizers for guaranteeing the required closed loop performance over a prespecified range of

operating and system conditions. The simplicity in design and implementation of the pro-

posed stabilizers makes them better suited for practical applications in real plants.

Acknowledgements

The completion and compilation of this thesis is the outcome of inspiring guidance of Dr.

Indraneel Sen. His keen interest in the progress of this work and patience to read through

my script are greatly acknowledged. I am thankful for his suggestions and discussions.

A special word of thank is due to Prof. K. R. Padiyar for his excellent teaching and who

influenced me to create a deep interest in the area of Power System Dynamics.

The help and cooperation of the chairman and staff of the Department of Electrical En-

gineering is gratefully acknowledged.

Perhaps words cannot express the gratitude I owe all my seniors like Mr. Anup Kumar

Singh, Mr. Maneesh Tewari, Mr. Nagesh Prabhu, Ms. Bijuna and Ms. Divya and friends

like Raghvendra Gupta, Raghvendra Pandey, Ashish, Ritwik, Amit and Vishal who helped

me in every way.

It, probably, goes without saying that I owe the biggest thank to my parents and family

members who have been a constant source of help and encouragement.

Finally, I thank everyone who have directly or indirectly helped me during the course of

this work.

Ravindra Singh

Contents

List of Tables iv

List of Figures v

1 Introduction 1

1.1 Low Frequency Oscillations in Power System . . . . . . . . . . . . . . . . . . 1

1.2 Fixed Parameter Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Conventional Stabilizers . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.2 Other Fixed Parameter Controllers . . . . . . . . . . . . . . . . . . . 4

1.2.3 The Drawbacks of Conventional Fixed Parameter Controllers . . . . . 4

1.3 Adaptive Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Fuzzy Logic Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Robust Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.6 Application of Genetic Algorithms to PSS Design . . . . . . . . . . . . . . . 9

1.7 Robust PSS design using Genetic Algorithms: the present approach . . . . . 10

1.8 Performance Requirements of Power System Damping Controllers . . . . . . 11

1.8.1 How Much Damping Do We Need? . . . . . . . . . . . . . . . . . . . 12

1.8.2 Performance Evaluation of a PSS . . . . . . . . . . . . . . . . . . . . 13

1.9 Scope of Present Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.10 Organization of Chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Mathematical Modelling of Power System 16

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 SMIB Model in Non-Linear Form . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.1 Rotor Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

i

Contents ii

2.2.2 Stator Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.3 Network Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Excitation System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4 PSS Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5 SMIB Test System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.6 Modelling of Multimachine System . . . . . . . . . . . . . . . . . . . . . . . 23

2.6.1 Rotor Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.6.2 Stator Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.6.3 Inclusion of Generator Stator in the Network . . . . . . . . . . . . . . 26

2.6.4 Load Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.6.5 Network Equations for Multimachine . . . . . . . . . . . . . . . . . . 28

2.7 Multimachine Test System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.8 Linearized 1.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 Genetic Algorithm: An Overview 32

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2 What is Genetic Algorithm? . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3 Working Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3.1 Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3.2 Fitness Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3.3 GA Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3.4 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.4 Implementation of genetic algorithm . . . . . . . . . . . . . . . . . . . . . . 36

3.5 Mathematical Model of SGAs . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4 Proposed Stabilization Technique: Single Machine System 42

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.3 Proposed Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.4 Application to SMIB System . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.4.1 Control Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.4.2 GA Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Contents iii

4.4.3 Operating Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.4.4 GA Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.4.5 Performance Analysis of Proposed GA Based PSS . . . . . . . . . . . 48

4.4.6 Robustness Test and Eigen Value Plots . . . . . . . . . . . . . . . . . 48

4.4.7 Simulation Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.4.8 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5 Proposed Stabilization Technique: Multimachine System 64

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.2 Performance Evaluation of the Stabilizer in Multimachine System . . . . . . 64

5.2.1 Control Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.2.2 GA Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2.3 Loading Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2.4 GA Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.2.5 Robustness Test and Eigen Value Plots . . . . . . . . . . . . . . . . . 67

5.2.6 Operating Points For Simulation Studies . . . . . . . . . . . . . . . . 69

5.2.7 Simulation Results and Discussion . . . . . . . . . . . . . . . . . . . . 70

5.2.8 Computational Requirements . . . . . . . . . . . . . . . . . . . . . . 73

5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6 Conclusions 85

A Calculation of Initial Conditions 87

B Heffron-Philips Model of the SMIB System 88

C Data for SMIB and Multimachine System 90

D Tuning Guidelines for the CPSS 92

E Mapping From a Binary String to a Real Number 98

F Derivation of Equation 4.1 99

References 101

List of Tables

4.1 Control Parameters Bounds . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2 GA Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.3 Initial and Final Values of PSS Parameters . . . . . . . . . . . . . . . 48

5.1 Control Parameters Bounds . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2 GA Parameters For Multimachine Case . . . . . . . . . . . . . . . . . 65

5.3 Loading Range of 3 Machine, 9 Bus System . . . . . . . . . . . . . . 66

5.4 Optimal stabilizer parameters of PGAPSS . . . . . . . . . . . . . . . . 67

5.5 Operating points of generators on a 100 MVA base . . . . . . . . . . 69

C.1 Generator Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

C.2 AVR Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

C.3 Generator Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

C.4 AVR Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

iv

List of Figures

1.1 D-contour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1 External two port network . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Excitation system block diagram. . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Block diagram of PSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 Single machine infinite bus system . . . . . . . . . . . . . . . . . . . . . . . . 22

2.5 Schematic of a multimachine system . . . . . . . . . . . . . . . . . . . . . . 23

2.6 Generator equivalent circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.7 3 machine, 9 bus power system model, single line diagram. . . . . . . . . . . 29

3.1 Single point crossover operation . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 A single mutation operation . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3 The general structure of genetic algorithms . . . . . . . . . . . . . . . . . . . 37

4.1 Flow Chart representation of the proposed method of tuning stabilizer . . . 45

4.2 Open loop poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.3 Closed loop poles with CPSS . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.4 Closed loop poles with PGAPSS . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.5 A step change of Tm = 0.1 pu, St = 0.5 + j0.1, Xe = 0.3 . . . . . . . . . . . 55





4.10 A step change of Tm = 0.1 pu, St = 0.5 + j0.1, Xe = 0.6 . . . . . . . . . . 56


v

List of Figures vi









4.20 A step change of Tm = 0.1 pu, St = 0.5 j0.2, Xe = 0.3 . . . . . . . . . . . 604.21 A step change of Tm = 0.1 pu, St = 0.8 j0.2, Xe = 0.3 . . . . . . . . . . . 604.22 A step change of Tm = 0.1 pu, St = 1.0 j0.2, Xe = 0.3 . . . . . . . . . . . 604.23 A step change of Tm = 0.1 pu, St = 0.5 j0.2, Xe = 0.6 . . . . . . . . . . . 614.24 A 3 to ground fault for 100 ms at generator terminal, St = 1.0+j0.2, Xe = 0.3 614.25 A 3 to ground fault for 100 ms at generator terminal, St = 0.8j0.2, Xe = 0.3 614.26 A 3 to ground fault for 100 ms at generator terminal, St = 1.0+j0.5, Xe = 0.6 624.27 A step change of Tm = 0.1 pu, St = 1.0 + j0.2, Xe = 0.3, H

= H/4 . . . . 62

4.28 A step change of Tm = 0.1 pu, St = 0.8 j0.2, Xe = 0.3, H = H/4 . . . . 624.29 A step change of Tm = 0.1 pu, St = 1.0 + j0.5, Xe = 0.6, H

= H/4 . . . . 63

4.30 A step change of Tm = 0.1 pu, St = 0.6 j0.15, Xe = 0.65 . . . . . . . . . 63

5.1 Open loop poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.2 Closed loop poles with CPSS . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.3 Closed loop poles with PGAPSS . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.4 A step change of Tm1 = 0.1 pu at unit 1, under SOP. . . . . . . . . . . . . 75

5.5 A step change of Tm1 = 0.1 pu at generator 1, under SOP. . . . . . . . . . 75





5.10 A 3- to ground fault for 100 ms at P, under SOP. . . . . . . . . . . . . . . 77

5.11 A 3- to ground fault for 100 ms at P, under SOP. . . . . . . . . . . . . . . 77

5.12 A step change of Tm1 = 0.1 pu at generator 1, under HOP. . . . . . . . . . 77


List of Figures vii





5.18 A 3- to ground fault for 100 ms at P, under HOP. . . . . . . . . . . . . . 79

5.19 A 3- to ground fault for 100 ms at P, under HOP. . . . . . . . . . . . . . 80

5.20 A step change of Tm1 = 0.1 pu at generator 1, under LOP. . . . . . . . . . 80






5.26 A 3- to ground fault for 100 ms at P, under LOP. . . . . . . . . . . . . . . 82

5.27 A 3- to ground fault for 100 ms at P, under LOP. . . . . . . . . . . . . . . 82

5.28 A step change of Tm2 = 0.1 pu at generator 2, under OOP. . . . . . . . . . 83






B.1 Heffron-Philips model of the SMIB system . . . . . . . . . . . . . . . . . . . 88

D.1 Phase angle plots of GEP(s), PSS(s) and GEP(s).PSS(s) . . . . . . . . . . . 93

D.2 Phasor diagram representation of synchronizing and damping torques . . . . 93

D.3 A typical root locus plot for SMIB system with the lead Compensator . . . . 94

D.4 Relationship governing VT (s), PSS(s), and EXC(s) . . . . . . . . . . . . . 96

D.5 Phase shift of Tpss for a speed input PSS . . . . . . . . . . . . . . . . . . 97

F.1 D-contour in x y plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Chapter 1

Introduction

1.1 Low Frequency Oscillations in Power System

Small oscillations in power systems were observed as far back as the early twenties of

this century. The oscillations were described as hunting of synchronous machines. In a

generator, the electro-mechanical coupling between the rotor and the rest of the system

causes it to behave in a manner similar to a spring-mass-damper system which exhibits

oscillatory behaviour following any disturbance from the equilibrium state.

Small oscillations were a matter of concern, but for several decades power system engineers

remained preoccupied with transient stability. That is the stability of the system following

large disturbances. Causes for such disturbances were easily identified and remedial measures

were devised. In early sixties, most of the generators were getting interconnected and the

automatic voltage regulators(AVRs) were more efficient. With bulk power transfer on long

and weak transmission lines and application of high gain, fast acting AVRs, small oscillations

of even lower frequencies were observed. These were described as Inter-Tie oscillations. Some

times oscillations of the generators within the plant were also observed. These oscillations

at slightly higher frequencies were termed as Intra-Plant oscillations.

The combined oscillatory behaviour of the system encompassing the three modes of oscil-

lations are popularly called the dynamic stability of the system. In more precise terms it is

known as the small signal oscillatory stability of the system.

A power system is said to be small signal stable for a particular steady-state operating

condition if, following any small disturbance, it reaches a steady state operating condition

which is identical or close to the pre-disturbance operating condition.

1

Chapter 1. Introduction 2

The oscillations, which are typically in the frequency range of 0.2 to 3.0 Hz., might be

excited by disturbances in the system or, in some cases, might even build up spontaneously.

These oscillations limit the power transmission capability of a network and, sometimes, may

even cause loss of synchronism and an eventual breakdown of the entire system. In practice,

in addition to stability, the system is required to be well damped i.e. the oscillations, when

excited, should die down within a reasonable amount of time.

Reduction in power transfer levels and AVR gains does curb the oscillations and is often

resorted to during system emergencies. These are however not feasible solutions to the

problem.

The stability of the system, in principle, can be enhanced substantially by application of

some form of close-loop feedback control. Over the years a considerable amount of effort

has been extended in laboratory research and on-site studies for designing such controllers.

There are basically three following ways by which the stability of the system can be improved,

(1) Using supplementary control signals in the generator excitation system.

(2) Making use of fast valving technique in steam turbine.

(3) Impedance Control-resistance breaking and application of the FACTS devices, etc.

The problem, when first encountered, was solved by fitting the generators with a feedback

controller which sensed the rotor slip or change in terminal power of the generator and fed

it back at the AVR reference input with proper phase lead and magnitude so as to generate

an additional damping torque on the rotor [1]. This device came to be known as a Power

System Stabilizer (PSS).

Damping power oscillations using supplementary controls through turbine, governor loop

had limited success. With the advent fast valving technique, there is some renewed interest

in this type of control [2].

There can also be other kinds of controls applied to the system for counteracting the oscil-

latory behaviour - for instance FACTS devices can be fitted with supplementary controllers

which improve the system stability.


Power system stabilizers are now routinely used in the industry. However, the complex,

constantly changing nature of power systems has severely restricted the efficacy of these

devices.

1.2 Fixed Parameter Controllers

Over the years, a number of techniques have been developed for designing PSSs and other

damping controllers [3]. Some of these stabilizing methods have been briefly described in

this section. The main motivation for including this rather brief exposition of the existing

techniques is to introduce the need for the application of robust control techniques in power

systems. Some of references cited here include a more comprehensive coverage of the topic.

1.2.1 Conventional Stabilizers

The earlier stabilizer designs were based on concepts derived from classical control theory

[4-8]. Many such designs have been physically realized and widely used in actual systems.

These controllers feedback suitably phase compensated signals derived from the power, speed

and frequency of the operating generator either alone or in various combination as input

signals so as to generate an additional rotor torque to damp out the low frequency oscillations.

The gain and the required phase lead/lag of the stabilizers are tuned by using appropriate

mathematical models, supplemented by a good understanding of the system operation.

The principles of operation of this controller are based on the concepts of damping and

synchronizing torques within the generator. A comprehensive analysis of these torques has

been dealt with by deMello and Concordia in their landmark paper in 1969 [1]. These

controllers have been known to work quite well in the field and are extremely simple to

implement. However, the tuning of these compensators continues to be a formidable task

especially in large multimachine systems with multiple oscillatory modes. Larsen and Swann,

in their three part paper [6], describe in detail the general tuning procedure for this type of

stabilizers.

PSS design using this method involves some amount of trial and error and experience on

part of the designer. Further these controllers are tuned for a particular operating condi-

tions and with change in operating conditions they require re-tuning. Robustness issues are

also not adequately addressed in this classical setting. The problem associated with these

controllers is more fully described later in this chapter.


1.2.2 Other Fixed Parameter Controllers

There have also been numerous attempts at applying various other control strategies -

in particular -modal control [9-11] and LQ optimal control [12, 13] techniques for designing

damping controllers. These attempts exemplify the growing preference for algorithmic con-

troller design methods as opposed to the classical intuitive ones. They call for a lesser amount

of engineering judgement and experience on part of the designer. The ill-suitedness of the

quadratic performance index used in LQR/LQG to the problem has motivated researchers to

define alternative performance indices which aptly capture the magnitude of system damping

[14, 15]. Such indices can be optimized using standard numerical optimization techniques

[16].

These techniques have the advantage of being straight forward and algorithmic with lit-

tle ambiguity in the recommended procedure. A few extensions of these methods tried to

incorporate some robustness by optimizing some additional index such as eigen value sensi-

tivities. Sensitivity minimization in this form, though, quite helpful as a means of providing

robustness in the absence of better methods is essentially a qualitative approach and hence

does not guarantee performance preservation in the face of modal inaccuracies [17].

1.2.3 The Drawbacks of Conventional Fixed Parameter Controllers

The main drawback of the above controllers is their inherent lack of robustness. Power

systems continually undergo changes in the load and generation patterns and in the trans-

mission network. This results in an accompanying change in small signal dynamics of the

system. The fixed parameter controllers, tuned for a particular operating condition, usually

give good performance at that operating condition. Their performance, at other operating

conditions, may at best be satisfactory, and may even become inadequate when extreme

situations arise. However such stabilizers have been very useful in system that could be

represented by single machine infinite bus models. In interconnected multimachine systems

the dynamic instability can manifest itself in the form of poorly damped oscillation of one

particular unit with the rest of the system or a group, or a group of machines oscillating

against another group of machines. Thus, a generating unit in a multimachine environment

often participates in both local and inter-area modes of oscillations simultaneously. The

spectral and temporal distributions of these modes are largely determined by the rest of the

system. As the operating conditions and system configuration are constantly changing in

actual power system the performance of the fixed parameter stabilizers can not be always


guaranteed.

1.3 Adaptive Controllers

The problem of changing system dynamics due to changes in the operating conditions can

be handled by the application of adaptive control [18, 19]. The power system can be con-

tinuously monitored and the controller parameters can be updated in real time to maintain

specified performance inspite of changes in the system dynamics. All three standard methods

of adaptive control listed below have been tried for designing power system stabilizers.

(a) Model reference adaptive control (MRAC) [20, 21]

(b) Self tuning control (STC) [22-24]

(c) Gain scheduling adaptive control (GSAC) [25]

In MRAC, the desired behavior of the closed loop system is incorporated in a reference

model. With the plant and the reference model excited by the same input, the error between

the plant output and the reference model output is used to modify the controller parameters,

such that the plant is driven to match the behavior of the reference model.

In STC, at every sampling instant, the parameters of an assumed model for the plant are

identified using some suitable algorithms, such as Recursive least squares (RLS) or Maximum

likelihood estimator etc. The identified parameters are then used in control laws which could

be based on various popular techniques such as pole-shifting, pole placement etc.

In GSAC, the gains of the controller are adjusted according to a variety of innovative

control strategies depending upon the plant operating conditions and important system

parameters. The gains could be computed either off-line or on-line.

A few non standard adaptive control schemes have also been reported [26, 27] which do

not fit into any of the above categories. These schemes have been shown to work quite well

through simulations and laboratory experiments.


Adaptive controllers totally avoid the problem of tuning since that is taken care of by the

adaptation algorithm. The trade off is the larger on-line computational requirement. The

stabilizers are difficult to design and are also susceptible to problems like non-convergence

of parameters and numerical instability. Due to these reasons practical implementation of

adaptive stabilizers in actual plants has not been popular.

There have been numerous non conventional approaches including feedback linearization,

variable structure or sliding mode control and, in more recent times, schemes involving neural

networks, fuzzy systems and rule based systems [3] for designing stabilizers. Many of these

non-conventional approaches have been shown to work quite well in simulated power system

models.

Some of the above approaches have also been applied for designing supplementary stabi-

lizing controls for FACTS devices. Most of the modern control theoretical techniques use a

black box model for the plant. Hence, identical procedures can be adopted for the design of

power system stabilizers and other damping controllers.

1.4 Fuzzy Logic Controllers

In recent years, Rule based [28, 29], Artificial Neural Network (ANN) based [30, 31] and

Fuzzy Logic based [32-37] controllers have been suggested for PSS design. These are model-

free controllers i.e. precise mathematical model of the controlled system is not required. Here

control strategy depends upon a set of rules which describe the behavior of the controller.

Here lies, both the strength and weakness of this design philosophy. FLC controllers are

well-suited for PSS design as system and its interrelations are not precisely known as they

keep constantly changing with changes in both system and operating conditions. However,

as the design is rule and experience based, there can not be a unique design procedure.

1.5 Robust Control

The last 15 years have seen major developments in the field of robust control. This topic

deals with the analysis and design of feedback systems subject to incomplete knowledge of

the plant dynamics and accompanying uncertainties in the model of the plant. Such an

uncertainty in the plant model could arise due to various reasons, for instance - deliberate


approximations in the modelling procedure, measurement inaccuracies, parameter drifts and

time varying nature of certain systems. In the case of power systems, the nonlinear system

equations when linearized about an operating point result in a linear model with parameters

which vary with the operating condition.

Some of the major approaches to robust control are:

1. Loop transfer recovery for LQG designs [38]

2. H optimal control [39-48]

3. analysis and synthesis [49-50]

4. 11 optimal control [51]

5. Quantitive feedback theory (QFT) [52]

6. Parameter space methods [53]

In all the above approaches, except the first, the uncertainty in the plant is explicitly

modeled and is incorporated into the design process so as to guarantee good performance

in the presence of model uncertainties. Methods 2 and 4 above deal with norm bounded

descriptions of the uncertainty whereas 5 and 6 deal with bounds on parameter variations

in the system. analysis encompasses both kinds of descriptions.

Francis B.A. provides a theoretical basis to deal with uncertainties in a system control

design [39]. The parameter uncertainty was first addressed by Kartinov [54]. Doyle in his

frame work of H brought out a procedural approach to handle perturbations which are

norm bounded and time invariant [40].

H is a optimization technique which can be used to optimize the PSSs parameters. H

control theory provides a powerful tool to deal with robust stabilization and disturbance

attenuation problem. However the standard H control theory does not guarantee robust

performance under the presence of all uncertainties in the plant. This is specially true


in power systems where the plant parameters may change considerably with variations in

operating conditions.

There has been some effort in uncertainty modelling and treatment of plant with struc-

tured uncertainties. The problem is to find a controller such that infinity norm for the closed

loop system is satisfied for all the uncertainties in a given bounded set [55]. In the context

of power system the system uncertainties have to be identified, modelled and bounded, be-

fore guaranteed robust stabilizer can be designed. H optimal control design minimizes

the worst case energy gain(H norm) of certain suitably weighted closed loop transfer ma-

trix. With properly selected weighting functions, the controllers have good performance in

case of uncertainties in plant modelling and/or disturbances; moreover, trade-offs between

performance and robustness can be studied in this framework. Chen and Malik [43] have

developed a PSS based on H optimization method with an uncertainty description which

represents the possible perturbation of a synchronous generator around its normal operating

point.

Ashgharian [44] applies H theory to guarantee non degradation of torsional phenomenon

by considering the high frequency unmodelled dynamics in the system. Ohtsuka et.al. [41]

apply H optimization theory to improve the disturbance attenuation performance of LQ

optimal controllers. The changes in the operating conditions are not considered and there is

no explicit uncertainty modelling.

Chen and Malik [50] have applied synthesis for PSS design. The uncertainty in the

system is modelled in terms of variations in the values of the parameters K1 to K6 in the

Heffron-Philips model of a single machine infinite bus system. Bounds on these variations

are found and a controller is synthesized using the D-K iteration technique [56].

Almost all the above references are concerned only with robust stability of the closed loop

system which criterion is not sufficient for power system applications [45, 47]. Some of them

include disturbance attenuation specifications. Such specifications are not very relevant in

this application and are introduced to fit the problem to existing theory which has been

developed primarily for applications other than power system control.

Gibbard [57] suggests a PSS tuning method which is shown to be robust through an

example. The argument in this paper depends strongly on the invariance of the P-Vr charac-


teristics of the generators in spite of the variations in the operating conditions. Fatehi et.al.

[58] have applied loop transfer recovery to obtain a robust controller for power systems.

Khammash et.al. [59, 60] have used a non-negative matrices test for checking robust

stability in the presence of variations in the elements of the system matrices. Pai et.al. [61]

apply a Hurwitzness test for interval matrices to check the robust stability of power systems

in the presence of parameter variations. Werner et.al. [62] use LMI techniques for robust

PSS design. These papers deal primarily with robustness analysis of power systems.

Rao and Sen [63-65], have proposed a method based on quantitative feedback theory

(QFT) for designing a robust controller for a power system in a single input single output

(SISO) framework. These authors extended their work to multi-variable case also [66].

The increasing presence of FACTS devices in power systems now provides an alternative

control loop for further improving the stability of the system. It is known that well designed

controller of any FACTS device can enhance the system damping [67, 68]. The simulta-

neous application of PSSs and FACTS devices can be used to further enhance the small

signal dynamic performance of a power system. However, the distributed nature of power

systems requires the application of a decentralized control strategy wherein only locally mea-

surable signals are used for feedback at the various control inputs to the system. A robust

decentralized damping controller has been proposed by Rao and Sen [66].

Robust controllers are less sensitive to changes in operating conditions than conventional

controllers. They provide adequate damping over a wide operating range of power system.

There have also been a few other miscellaneous publications dealing with robustness issues

in power systems which are relevant to the present work.

1.6 Application of Genetic Algorithms to PSS Design

Genetic algorithm has recently attracted the attention of Power System Stabilizer design-

ers [69-75]. The advantage of GA technique is that it is independent of the complexity of

the performance index considered. It suffices to specify the objective function and to place

finite bounds on the optimized parameters. GA provides greater flexibility regarding con-

troller structure and objective function considered. Further more, GA based optimization


problem can readily accomplish control performance constraints, such as required closed

loop minimum performance. Introduction of GA helps to obtain an optimal tuning for all

PSS parameters simultaneously, which thereby takes care of interaction between different

PSSs, hence eigen value drift problem associated with sequential tuning methods can be

eliminated.

Several techniques of tuning of PSS using genetic algorithms have been reported in re-

cent literature. Magid and Abido [70] have applied GA to tune the hybridizing rule based

PSS. Advantage of rule based PSS is its robustness, less computational burden and ease of

realization.

Taranto et.al. [72] have presented a method for simultaneous tuning of damping controllers

using modified GA operators. Tuning of fixed structure conventional PSS is reported in this

paper.

Zhang and Coonick [73] have proposed a new method based on the method of inequali-

ties applied to GA for the coordinated synthesis of PSS parameters in multimachine power

systems.

Andreoiu and Kankar Bhattacharya [74] have proposed Lyapunovs method based genetic

algorithm for robust PSS design.

Robust stability of closed loop system can be achieved using genetic algorithms. Abdel-

Magid et.al. [75] apply genetic algorithms to tune the parameters of a PSS such that robust

stability is achieved over a range of operating conditions. Taranto et.al. [76] have applied

parameter optimization using genetic algorithms for synthesizing a robust controller for

power systems. These papers focus upon the robust closed loop rotor mode location as is

the case in this thesis.

1.7 Robust PSS design using Genetic Algorithms: the

present approach

In this thesis a new method has been proposed for tuning of PSS using genetic algorithm.

Proposed method guarantees a robust performance over a set of operating conditions. A

more elegant approach to robust stabilizer design is used, in which fixed gain robust PSSs


have been designed to guarantee a minimum performance inspite of variations in the plant

operations, due to changes in load, line switching, transformer tap-changing and other oc-

casional disturbances. Based on system experience minimum performance requirements of

PSS have been decided and an attempt has been made to achieve it over a wide range of

operating conditions. The performance requirements of the PSSs are more fully described

in the next section.

In the present approach the power system operating at various loading is treated as a

finite set of plants. The problem of selecting the parameters of PSS which simultaneously

stabilize this set of plants is converted to a simple optimization problem which is solved by

genetic algorithm and an eigen value based objective function.

1.8 Performance Requirements of Power System Damp-

ing Controllers

There exists considerable ambiguity in current literature about the performance require-

ments of stabilizers and other damping controllers. This section attempts to establish, in

clear and precise term, the closed loop specifications required of any power system damping

controller.

Practical considerations merely require that the troublesome low frequency oscillations,

when excited, die down within a reasonable amount of time. No advantage is gained by

having excessive damping for these system modes. In fact, it has been noted [6] that aggres-

sive damping of oscillations can have detrimental effects on the system. Hence, rather than

maximizing the damping at some particular operating condition, it seems more appropriate

to decide upon the minimum amount of damping or minimum performance required of the

closed loop and attempt to achieve this over the entire range of operating conditions which

the system experiences. This set of operating conditions, which any given power system

might experience, is always known a priori in terms of maximum and minimum values of

power generations, transmissions and loads and all possible values of the network impedances.

It is therefore possible to model this bounded variation in the system as an uncertainty and

attempt to synthesize a PSS delivering the required performance over this entire range of

variation.


1.8.1 How Much Damping Do We Need?

A damping factor of around 10% to 20% for the troublesome low frequency electrome-

chanical mode is considered adequate. For a second order system =10% results in system

oscillations decaying to within 15% of the initial amplitude in 3 cycles of the oscillations. (for

=20%, the decay is to within 2.1% of initial amplitude in 3 cycles.) A damping factor of

10% would be acceptable to most utilities and can be adopted as the minimum requirement.

Further, having the real part of rotor mode eigen value restricted to be less than a value,

say , guarantees a minimum decay rate . A value = - 0.5 is considered adequate for an

acceptable settling time. The closed loop rotor mode location should simultaneously satisfy

these two constraints for an acceptable small disturbance response of the controlled system.

The frequency of oscillation is related to synchronizing torque and hence the imaginary

part of the rotor mode eigen value should not change appreciably due to feed back.

If any new modes arise as a result of closing the controller loop (e.g. exciter mode), these

should also be well damped i.e. they should satisfy the same constraints on the real part

and damping factor as the rotor mode. Real poles close to the origin can result in a sluggish

response and persistent deviations of the system variables from their steady state values and

hence should be avoided.

5 4 3 2 1 0 120

15

10

5

0

5

10

15

20

real

imag

= 0.5

=10%

Figure 1.1: D-contour


If all the closed loop poles are located to the left of the contour shown in Figure 1.1,

then the constraints on the damping factor and the real part of rotor mode eigen values

are satisfied and a well damped small disturbance response is guaranteed. This contour is

referred as the D-contour [63]. The system is said to be D-stable if it is stable with respect

to this D-contour, i.e. all its pole lie on the left of this contour. This property is referred to

as generalized stability in control literature. This generates a neat specification- the closed

loop should be robustly D-stable i.e. D-stable for the entire range of operating and system

conditions. Hence, in this thesis a system is said to be robust, if, inspite of changes in

system and operating conditions, the closed loop poles remain on the left of the D-contour

for specified range of system and operating conditions.

1.8.2 Performance Evaluation of a PSS

Many of the design methods suggested in literature have been accompanied by comparisons

between different types of stabilizers. Such comparisons usually consider the amount of

damping enhancement provided by each PSS. It is clear from the discussion in the previous

section that a more aggressive damping is not particularly beneficial. In fact, in view of

the other considerations, it would be more fruitful to have the rotor mode damping closer

to the minimum requirements. Thus a comparison of two different stabilizers on grounds

of the amount of damping they contribute at some particular operating condition is not

very appropriate. A better PSS would be one which guarantees the minimum acceptable

performance over a wider range without adversely affecting the large disturbance response

of the system.

1.9 Scope of Present Work

The objective of the present work is to show that even a properly tuned fixed parameter

controller can guarantee a robust minimum performance over a wide range of operating

conditions, if it is properly tuned. Since fixed parameter PSS is simple in structure and

widely used by most utilities, an attempt is made to tune the fixed parameter PSS to ensure

its robustness.

A new method has been proposed for robust PSS design, which includes several operat-

ing conditions and system configurations simultaneously in the design process and works

well with equal effectiveness in single and multimachine environments. PSS parameters are

obtained using genetic algorithm.


A simple objective function based on eigen values is formulated for robust PSS design in

which robust D-stability of the closed loop is taken as primary specification.

The efficacy of the proposed PSS in damping out low frequency oscillations have been

established by extensive simulation studies on single and multimachine systems. The details

of the proposed method are given in Chapter 4.

1.10 Organization of Chapters

The thesis chapters are organized as under,

Chapter 1

This chapter introduces the problem of low frequency oscillations and defines the closed loop

performance requirements for power system damping controller.

Chapter 2

In this chapter mathematical models of power system have been developed. Non-linear differ-

ential equations required for more accurate simulation of single machine infinite bus system

and multimachine system are given.

Chapter 3

Chapter 3 reviews the basic ideas of genetic algorithms , genetic operators and mathematical

model of simple genetic algorithm which are needed to support controller design tuning of

power system stabilizer.

Chapter 4

This chapter deals with the formulation of objective function based on D-contour and mini-

mum performance requirement criterion. A new method is proposed and robustness of the

PSS is tested on the single machine infinite bus system.

Chapter 5

This chapter illustrates the application of proposed method to multimachine power system.

The performance of the stabilizer has been promising over a range of system and test condi-

tions. Due to its simple structure and ease of design, the proposed stabilizer appears to be

well suited for application in real plants.


Chapter 6

This concluding chapter gives a brief summary of the work done and also includes a section

on the scope of future work relating to design of power system stabilizers.

Chapter 2

Mathematical Modelling of Power

System

2.1 Introduction

For stability assessment of power system adequate mathematical models describing the

system are needed. The models must be computationally efficient and be able to represent

the essential dynamics of the power system.

A realistic power system is seldom at steady-state, as it is continuously acted upon by

disturbances which are stochastic in nature. The disturbances could be a large disturbance

such as tripping of generator unit, sudden major load change and fault switching of trans-

mission line etc. The system behavior following such a disturbance is critically dependent

upon the magnitude, nature and the location of fault and to a certain extent on the system

operating conditions. The stability analysis of the system under such conditions, normally

termed as Transient-stability analysis is generally attempted using mathematical models

involving a set of non-linear differential equations.

In contrast to this disturbance-specific transient instability, there exists another class of

instability called the Dynamic Instability or more precisely Small Oscillation Instability,

described in Chapter 1. As the small oscillation stability concerns itself with small excursions

of the system about a quiescent operating point, the system can be sometimes approximated

by a linearized model about the particular operating point. Once valid linearized model is

available, powerful and well established techniques of the linear control theory can be applied

for stability analysis and performance evaluation of various power system stabilizers.

16

Chapter 2. Mathematical Modelling of Power System 17

Nonlinear models on the other hand have more realistic representation of the power sys-

tems. Designing controllers for such nonlinear systems are understandably more difficult.

In this chapter, non-linear models of single and multimachine power systems have been

developed. Linear models have been obtained from these nonlinear models for designing

conventional power system stabilizers that are used for comparative performance analysis.

2.2 SMIB Model in Non-Linear Form

Consider the system shown in Figure 2.1. This shows the external network with two ports.

One port is connected to the generator terminals while the second port is connected to a

voltage source Eb 6 0. Assuming both the magnitude Eb and phase angle of the voltage source

to be constant, and neglecting the network transient, the system can be modelled using rotor

mechanical equations, rotor electrical equations and excitation system model.

tV^

External

Two Port

Network

^a

E b

+

I

.

0

Figure 2.1: External two port network

2.2.1 Rotor Equations

Rotor Mechanical Equations

The mechanical equations in per unit can be expressed as

Md2

dt2+ D

d

dt= Tm Te (2.1)

where, M = 2HB

, and H, D, , Tm and Te are inertia constant, rotor damping, rotor angle,

mechanical and electrical torques respectively. The above equation can be expressed as two


first order differential equations as:

d

dt= B(Sm Smo) = o (2.2)

2HdSmdt

= D(Sm Smo) + Tm Te (2.3)where, per unit damping D and generator slip Sm are given by:

D = BD (2.4)

Sm = B

B(2.5)

o and B are the synchronous and the base speed of the system.

Rotor Electrical Equations

Since the stator equations 2.12 and 2.13, stated later, are algebraic (neglecting stator tran-

sients) and rotor windings either remain closed (damper windings) or closed through finite

voltage source (field winding), the flux linkages of these windings cannot change suddenly.

Hence, it is not possible to choose stator currents id and iq as state variables (state variables

have to be continuous functions of time). The obvious choice for state variables are rotor

flux linkages or transformed variables which are linearly dependent on the rotor flux linkages

(Chapter 6 of [84]).

In a report published in 1986 by an IEEE Task Force [85], many machine models are

suggested based on varying degrees of complexity. Higher order models of machine in general

provide better results but it is adequate to use model (1.1) if the data is correctly determined.

In case studies cited in this report, only 1.1 model has been considered where two electrical

circuits are considered on the rotor i.e. a field winding on the d-axis and one damper winding

on q-axis. Differential equations for rotor and the electrical torque and are:

dE qdt

=1

T do

[E q + (xd xd)id + Efd

](2.6)

dE ddt

=1

T qo

[E d (xq xq)iq

](2.7)

Te = diq qid (2.8)= E did + E

qiq + (x

d xq)idiq (2.9)


where, vd, vq=d-q components of generator terminal voltage

id, iq=d-q components of armature current

Efd=voltage proportional to field voltage

E d=voltage proportional to damper winding flux

E q=voltage proportional to field flux

T do=d-axis transient time constant

T qo=q-axis transient time constant.

2.2.2 Stator Equations

The stator equations in Parks reference frame are expressed in per unit, these are

1B

ddt

Bq Raid = vd (2.10)

1B

qdt

Bd Raiq = vq (2.11)

It is assumed that the zero sequence in the stator are absent. If stator transients are to

be ignored, it is equivalent to ignoring the the pd and pq terms in above equations. In

addition it is also advantageous to ignore the variations in the rotor speed . If the armature

flux linkage components (pD and pQ), with respect to a synchronously rotating frame, are

(rotating at speed o) constants, then transformer e.m.f. terms (pd and pq) and terms

induced by the variations in the rotor speed cancel each other (chapter 6 of [84]). Then the

above equations 2.10 and 2.11 are reduced to

(1 + Smo)q Raid = vd (2.12)(1 + Smo)d Raiq = vq (2.13)

where, Smo is the initial operating slip, which, in most of the cases is assumed to be zero.

For the 1.1 model of the generator (field circuit with one equivalent damper winding on the

q-axis) the flux linkages are given by:

d = xdid + E

q (2.14)

q = xqiq E d (2.15)


Neglecting stator transients and letting Smo = 0, and substituting equation 2.15 in 2.12 and

equation 2.14 in 2.13, we get:

vd = Ed xqiq Raid (2.16)

vq = Eq + x

did Raiq (2.17)

2.2.3 Network Equations

It is assumed that the external network connecting the generator terminals to the infinite

bus is linear two port. The loads are assumed to be of constant impedance type. The voltage

there can be expressed as:

Vt = h11Ia + h12Eb = VQ + jVD (2.18)

h11 = zR + jzI , h12 = h1 + jh2 (2.19)

where, h11 is the short circuit self impedance of the network, measured from the generator

terminals, and h12 is a hybrid parameter (open circuit voltage gain). Equation 2.18 is

multiplied with ej which can be expressed as:

(vq + jvd) = (zR + jzI)(iq + jid) + Ebej(h) (2.20)

where, E b =

(h21 + h22)Eb, and tan h = h2/h1.

Equating real and imaginary parts of equation 2.20 separately, we can get: zR zIzI zR

id

iq

=

vd

vq

+ E b

sin( h) cos( h)

(2.21)

From Equation 2.21 we can get d-q component of stator currents. By using all the equations

in Section 2.2 model (1.1) can be simulated.

2.3 Excitation System Model

The excitation system is represented by a first order model. Let Ka and Ta be the AVR gain

and its time constants respectively. The block diagram of AVR is shown in figure 2.2 and

the equation describing it can be written as:

dEfddt

=1

Ta[Ka(Vref + Vs Vt) Efd] (2.22)

Efdmin Efd Efdmax (2.23)


K

1 + sTa

Efdmax

Efd

min

Efd

Vt

SV

refVa

Figure 2.2: Excitation system block diagram.

2.4 PSS Model

For the simplicity a conventional PSS is modelled by two stage (identical), lead/lag network

which is represented by a gain KS and two time constants T1 and T2. This network is

connected with a washout circuit of a time constant Tw, as shown Figure 2.3.

sT1 + w

wsT 1sT1 +

sT1 +

2

Ks2

VSmax

VSmin

VSmS

Figure 2.3: Block diagram of PSS

2.5 SMIB Test System

For the SMIB test system, the synchronous machine is assumed to be connected to an

infinite bus of voltage Eb through a transmission line of impedance Ze = jXe, as shown in

Figure 2.4. Since Re = 0 for this system hence ZR = 0.0, Zi = Xe, h1 = 1.0, h2 = 0.0,

h = 0.0.


AVR

X

Efd

P , Q

e

Vt

E b

Control Input

Infinite bus

Figure 2.4: Single machine infinite bus system

Considering Ra = 0, the dynamic equations of the SMIB system considered can be sum-

marized as :

d

dt= BSm (2.24)

dSmdt

=1

2H[DSm + Tm Te] (2.25)

dE ddt

=1

T qo

[E d (xq xq)iq

](2.26)

dE qdt

=1

T do

[E q + (xd xd)id + Efd

](2.27)

dEfddt

=1

Ta[Ka(Vref + Vs Vt) Efd] (2.28)

vd

vq

=

E

d

E q

0 x

q

xd 0

id

iq

(2.29)

id

iq

=

0 XeXe 0

1

vd

vq

+ E b

sin cos

(2.30)


Te = Edid + E

qiq + (x

d xq)idiq (2.31)

2.6 Modelling of Multimachine System

Figure 2.5 shows the schematic of a multimachine system. This section describes the

dynamic equations represented by each block shown in the ith machine and external network.

It is assumed that power system consists of n number of generators and generators feed local

loads which are constant.

Loads

To Other Machines

InterfaceMachine

(Electrical)

AVR

Ij^

I^

I = Y V^ ^ ^

NVj^

Vi^

VDi

, VQi VqiVdi ,

IDi Qi

I,

Vref, i

E fdi

I Idi qi,

Machine

i

i

i

Figure 2.5: Schematic of a multimachine system

In multimachine system without infinite bus, it is necessary to take a reference angle to

compare all other rotor angles of generators. Conventionally the rotor angle of machine


having highest inertia is taken as a reference. Another reference which is also considered

very often is the center of inertia (COI) angle and speed deviation 0 and 0 and these are

defined as:

COI =1

MT

n

i=1

Mii (2.32)

0 =1

MT

n

i=1

Mii (2.33)

where, MT =

Mi is total inertia of n number of generators. In the case study the rotor

angle and slip of the machine having highest inertia are taken as reference.

2.6.1 Rotor Equations

Rotor Mechanical Equations

The mechanical equations for multimachine in per unit can be expressed as:

didt

= B(Smi Smio) = i io (2.34)

2HdSmidt

= Di(Smi Smio) + Tmi Tei (2.35)

where, H, Tmi and Tei are machine inertia, mechanical and electrical torque respectively of

ith machine. Per unit damping (Di), generator slip (Smi), and electrical torque (Tei) are

given by:

Di = BDi (2.36)

Smi =i B

B(2.37)

Tei = Ediidi + E

qiiqi + (x

di xqi)idiiqi (2.38)

In matrix form we can rewrite the equations 2.34 and 2.35 as:

d[]

dt= B[Sm] = [] [o] (2.39)

2[H]d[Sm]

dt= {[D][Sm] + [Tm] [Te]} (2.40)


where,

[H] = diag[

H1 H2 ... Hk ... Hn]

(2.41)

[D] = diag[

D1 D2 ... Dk ... Dn]

(2.42)

[Sm] =[

Sm1 Sm2 ... Smk ... Smn]t

(2.43)

[Tm] =[

Tm1 Tm2 ... Tmk ... Tmn]t

(2.44)

[Te] =[

Te1 Te2 ... Tek ... Ten]t

(2.45)

[] =[

1 2 ... k ... n]t

(2.46)

[] =[

1 2 ... k ... n]t

(2.47)

Rotor Electrical Equations

For 1.1 model, differential equations for the rotor flux linkages and voltages for rotor windings

for multimachine can be written as:

dE qidt

=1

T doi

[E qi + (xdi xdi)idi + Efdi

](2.48)

dE didt

=1

T qoi

[E di (xqi xqi)iqi

](2.49)

Above equations in matrix form are,

[T do]d[E q]

dt=

{[E q] + ([xd] [xd])[id] + [Efd]

}(2.50)

[T qo]d[E d]dt

={[E d] ([xq] [xq])[iq]

}(2.51)

where,

[T do] = diag[

Tdo1 Tdo2 ... Tdok ... Tdon]

(2.52)

[T qo] = diag[

Tqo1 Tqo2 ... Tqok ... Tqon]

(2.53)

[Efd] =[

Efd1 Efd2 ... Efdk ... Efdn]t

(2.54)

[E d] =[

E d1 Ed2 ... E

dk ... E

dn

]t(2.55)

[E q] =[

E q1 Eq2 ... E

qk ... E

qn

]t(2.56)

[id] =[

id1 id2 ... idk ... idn]t

(2.57)

[iq] =[

iq1 iq2 ... iqk ... iqn]t

(2.58)


[xd], [xq], [xd], [x

q] and [Ra] are diagonal matrices of same size, and one of them is shown as

below

[Ra] = diag[

Ra1 Ra2 ... Rak ... Ran]

(2.59)

2.6.2 Stator Equations

Stator equations are expressed in per unit with assumption of neglecting zero sequence and

stator transients, as in section 2.2.2, we have the equations:

(1 + Smio)qi Raiidi = vdi (2.60)(1 + Smio)di Raiiqi = vqi (2.61)

where, subscript i stands for ith machine; Smo is the initial operating slip, which, in most of

the cases is assumed to be zero and is defined as:

Smio =io B

B(2.62)

Neglecting stator transients and letting Smo = 0, equations 2.16 and 2.17 are rewritten for

multimachine as:

vdi = Edi xqiiqi Raiidi (2.63)

vqi = Eqi + x

diidi Raiiqi (2.64)

The above two equations can be represented in matrix form as: [vd]

[vq]

=

[E

d]

[E q]

[Ra] [x

q]

[xd] [Ra]

[id]

[iq]

(2.65)

where,

[vd] =[

vd1 vd2 ... vdk ... vdn]t

(2.66)

[vq] =[

vq1 vq2 ... vqk ... vqn]t

(2.67)

2.6.3 Inclusion of Generator Stator in the Network

The generator equivalent circuit can be drawn as in the Figure 2.6. It can be represented

in terms of a current source Ig and its internal admittance Yg such that armature current,

Ia = Ig YgVt. The equivalent circuit shown in the figure can easily be merged with the ACnetwork external to the generator.


I Yg g Vt

aI

Figure 2.6: Generator equivalent circuit.

Treatment of Transient Saliency

When transient saliency is neglected then the stator can be represented by a voltage source

(E q + jEd) behind an equivalent reactance (Ra + jx

). But if transient saliency is considered

then a stator cannot be represented by a single phase equivalent circuit. A generator can

be represented by a dependent current source, which is a function of the field and damper

winding flux and , to treat saliency.

The generator stator voltage can be re-expressed as a single equation in phasor quantities:

Vt = (vq + jvd)ej = [E q + j(E

d + E

dc)]e

j (Ra + jxd)Ia (2.68)where, E dc = (x

d xq)iq (2.69)

Equation 2.68 can be rearranged to represent the equivalent circuit of Figure 2.6 as:

Ig = YgVt + Ia (2.70)

where, Ig = Yg[Eq + j(E

d + E

dc)]e

j (2.71)

Yg =1

Ra + jxd(2.72)

This requires an iterative solution for the dependent current source and this problem of

iterative solution can be eliminated by considering a rotor dummy coil on q-axis which links

only with q-axis coil in the armature and considering E dc as a state variable. The differential

equation for E dc can be expressed as:

dE dcdt

=1

Tc

[(xd xq)iq E dc

](2.73)


where, Tc is the open circuit time constant of the dummy coil, which can be arbitrarily

selected. Tc should be small and it can be 0.01 sec for acceptable accuracy. This is of a

similar order as the time constant of high resistance damper winding.

2.6.4 Load Representation

Loads are represented as static voltage dependent models given by

PL = fP (VL) = a0 + a1VL + a2V2L (2.74)

QL = fQ(VL) = b0 + b1VL + b2V2L (2.75)

If load is represented by constant impedances then a0 = a1 = b0 = b1 = 0, and Yl is given by

Yl =PLo jQLo

V 2Lo(2.76)

where subscript o indicates operating values.

2.6.5 Network Equations for Multimachine

The AC network consists of transmission lines, transformers, shunt reactors, capacitors in

series and shunt. It is assumed that the network is symmetric. Hence single phase repre-

sentation (positive sequence network) is adequate. The network equations can be expressed

using bus admittance matrix YN as

IN = [YN ]V (2.77)

where, V is a vector of complex bus voltages and IN is vector of current injections. The

generator and load equivalent circuits at all the buses can be integrated into the AC network

and the overall system algebraic equations can be obtained as follows:

I = [Y ]V (2.78)

where [Y] is the complex admittance matrix which is obtained from augmenting [YN ] by in-

clusion of the shunt admittance Yg (from generator equivalent circuit) and Yl at the generator

and load buses. Element Ygj or Ylj corresponding to jth bus is added to YNjj element of the

admittance matrix YN to obtain [Y]. I is the vector of complex current sources. Equations


2.78 can be rewritten as:

V = [Y ]1I = [Z]I (2.79)

[VQ + jVD] = [ZR + jZI ][IQ + jID]

VD

VQ

=

ZR ZIZI ZR

ID

IQ

(2.80)

2 3

4

5

7

8

6

1

P

j0.0586j0.06250.0085+j0.072

B/2=j0.0745

0.0119+j0.1008

B/2=0.1045230/13.818/230

j0.0

576

16.5

/230

Load C

0.03

2+j0

.161

0.01

0+j0

.085

Load A Load B

0.01

7+j0

.092

0.03

9+j0

.170

B/2

=j0

.179 9

B/2

=j0

.079

B/2

=j0

.153

B/2

=j0

.088

230 kV

G2 G3

G1

13.8 kV18 kV

16.5 kV

Figure 2.7: 3 machine, 9 bus power system model, single line diagram.

2.7 Multimachine Test System

The multimachine configuration considered for the purpose of study consists of 3 generators

[86, 87] interlinked as shown in Figure 2.7.


Substituting n = 3 in the equations developed in Section 2.6, the dynamic equations

representing this system can be summarized as :

COI =1

MT

3

i=1

Mii (2.81)

COI =1

MT

3

i=1

Mii (2.82)

d[]

dt= B[Sm] = [] [o] (2.83)

2[H]d[Sm]

dt= {[D][Sm] + [Tm] [Te]} (2.84)

[T do]d[E q]

dt=

{[E q] + ([xd] [xd])[id] + [Efd]

}(2.85)

[T qo]d[E d]dt

={[E d] ([xq] [xq])[iq]

}(2.86)

[T c]dE dcdt

={([xd] [xq])[iq] [E dc]

}(2.87)

[Ta]d[Efd]

dt= [[Ka]([Vref ] + [Vs] [Vt]) [Efd]] (2.88)

[id]

[iq]

=

[Ra] [x

q]

[xd] [Ra]

1

[Ed] [vd]

[E q] [vq]

(2.89)

VD

VQ

=

ZR ZIZI ZR

ID

IQ

(2.90)

Tei = Ediidi + E

qiiqi + (x

di xqi)idiiqi (2.91)

Ig = Yg[Eq + j(E

d + E

dc)]e

j (2.92)

Yg =1

Ra + jxd(2.93)

where, [ZR +jZI ] = [Z] = [Y ]1 and [Y] is the complex admittance matrix which is obtained


from augmenting bus admittance matrix YN by shunt admittance Yg of generator and load

admittances at the generator and load buses Yl.

2.8 Linearized 1.1 Model

Linearized 1.1 model for both, single machine and multimachine system was obtained us-

ing LINMOD facility available in MATLAB. Details of this model are given in ref. [84].

These models have been used in later chapter for simulating power systems equipped with

conventional and proposed PSSs to analyze the performance of the controllers at various

system and operating conditions.

Chapter 3

Genetic Algorithm: An Overview

3.1 Introduction

In the open access environment, the power utilities are often forced to work their system

far away from predesigned conditions. In this situation, the systems may be operating near

their stability limits. It is therefore necessary to re-approach the problem related to power

system stability with this perspective. Several recent major system blackouts in different

countries and voltage collapses have clearly indicated the need for better stabilization efforts

in the interconnected power systems. Conventional power system stabilizers are designed for

particular system and operating conditions and are therefore not effective throughout the

expected range of operation of such systems.

In contrast, application of Genetic Algorithms (GA) in power system stabilizer design

is an attractive proposition as it provides greater flexibility regarding controller structure

and objective function. In addition to the constraints on the parameter bounds, the GA

based optimization problem can readily accomplish control performance constraints, such

as required closed-loop minimum performance. Further more, GA helps to obtain an opti-

mal tuning for all PSS parameters simultaneously, which takes care of interactions between

different PSSs.

This chapter gives a brief and quick introduction to Genetic Algorithm. This is needed for

a better understanding of the GA based stabilizer design process dealt in the later chapters.

32

Chapter 3. Genetic Algorithm: An Overview 33

3.2 What is Genetic Algorithm?

Genetic Algorithms are adaptive methods which may be used to solve search and opti-

mization problems. Over many generations, natural populations evolve according to the

principles of natural selection and survival of the fittest. By mimicking the process, genetic

algorithms are able to evolve solutions to real world problems, if they have been suitably

encoded.

3.3 Working Principles

The basic principles of GAs were first laid down rigourously by Holland [77], in mid sixties.

Thereafter, many researchers have contributed to developing this field. To date, most of the

GA studies are available through a few texts [78-81]. There are many variations of the

genetic algorithm but the basic form is the simple genetic algorithm. The working principle

[82] of SGA can be described as:

3.3.1 Coding

Before a GA can run, a suitable coding for the problem must be devised. It is assumed

that a potential solution to a problem may be represented as a set of parameters. These

parameters (known as genes) are joined together to form a string of values (often referred

as chromosome or Individual). Binary coded strings having 1s and 0s are mostly used.

For example, if 10 bits are used to code each variable in a two-variable function optimization

problem, chromosome would contain two genes, and consists of 20 binary digits. Decoding

technique of binary coded strings in to function variables is given in Appendix E.

3.3.2 Fitness Function

As pointed out earlier, GAs mimic the survival of the fittest principle of nature to make

a search process. Therefore, GAs are naturally suitable for solving maximization problems.

Minimization problems are usually transformed in to maximization problems by suitable

transformation. In, general, a fitness function is first derived from the objective function

and used in successive genetic operations. Certain genetic operators require that the fitness

function be nonnegative, although certain operators do not have this requirement. For


maximization problems, the fitness function can be considered to be the same as the objective

function. For minimization problems, the fitness function is an equivalent maximization

problem chosen such that the optimum point remains unchanged.

3.3.3 GA Operators

The GA works with a set of individuals comprising the population. The initial popula-

tion consists of N randomly generated individuals where, N is the size of population. At

every iteration of the algorithm, the fitness of each individual in the current population is

computed. The population is then transformed in stages to yield a new current population

for the next iteration. The transformation is usually done in three stages by sequentially

applying the following genetic operators:

(1) Selection : In the first stage, the selection operator is applied as many times as there

are individuals in the population. In this stage every individual is replicated with a

probability proportional to its relative fitness in the population. The population of N

replicated individuals replaces the original population.

(2) Crossover: In the next stage, the crossover operator is applied with a probability pc,

independent of the individuals to which it is applied. Two individuals (parents) are

chosen and combined to produce two new individuals (offsprings). The combination is

done by choosing at random a cutting point at which each of the parents is divided into

two parts; these are exchanged to form the two offsprings which replace their parents

in the population. This is known as single point crossover. Figure 3.1 illustrates the

single point crossover operation.

(3) Mutation : In the final stage, the mutation operator changes the values in a randomly

chosen location on an individual with a probability pm. Figure 3.2 shows the mutation

operation.

3.3.4 Convergence

If the GA has been correctly implemented, the population will evolve over successive

generations so that the fitness of the best and the average individual in each generation

increases towards the global optimum. The algorithm converges after a fixed number of

iterations and the best individual generated during the run is taken as the solution.


Parent 1 1 0 1 0 0 1 0 1 1 0

Parent 2 1 0 1 1 1 0 1 0

Crossover Point

1 0 1 0 0 1 1 0 1 0

1 0

1 0 1 1 1 1 10 0 0

Offspring 1

Offspring 2

Crossover

Figure 3.1: Single point crossover operation

Offspring

OffspringMutated 1 0 1 0 0 1 0 0 1 0

1 0 1 0 0 1 1 0 1 0

Mutation

.

.

Figure 3.2: A single mutation operation


3.4 Implementation of genetic algorithm

The implementation of the simple genetic algorithm is as follows:

Input:l: length of each solution string

N : population size, number of strings in a population

pc: probability of crossover

pm: probability of mutation

MAXGEN: maximum number of generations

output:x: best string from the current population

Algorithm:

1. Generate N strings, each of length l, randomly to form the initial population.

2. Evalute each string in the current population and assign a fitness value to each

string.

3. Select a highly fit string using selection operator and repeat this process N times

to generate a new population of N strings for next generation.

4. Randomly choose pairs of these selected strings and perform crossover with a

probability pc to generate children strings. Crossover exchanges bit values between

the two strings at one or more locations.

5. Randomly choose some bit positions with a probability pm and mutate the bit

values. That is change 1 to a 0 and 0 to a 1.

6. Steps 2-5 constitute a generation. Repeat steps 2-5 till the number of generations

is MAXGEN and stop. Output the best string from the current population.

Figure 3.3 shows the general structure of the genetic algorithms.


10111010101100101010101110111000110110011100110001

11001010101011101110

0011011001

0011001001

1100101110

1011101010

0011001001

Solutions

encoding

New population

Roulettewheel

Selection

Chromosomes

Xover

Mutation

EvaluationOffspring

computation

Decoding

1100101110

Solutions

Fitness

Figure 3.3: The general structure of genetic algorithms


3.5 Mathematical Model of SGAs

This section describes the exact mathematical model [81] of simple genetic algorithms.

This model is based on above mentioned algorithm, with only difference that only one

offspring from each crossover survives.

If we define vectors p (t) and s (t), each of length 2l, where vector p (t) exactly specifiesthe composition of the population at generation t, ands (t) reflects the selection probabilitiesunder the fitness function, then these are connected via fitness.

Let F be a two-dimensional matrix such that Fi,j = 0 for i 6= j and Fi,i = f(i). Diagonalelements (i, i) of F which are nonzero give the fitness of the corresponding string i. Under

proportional selection,

s (t) = Fp (t)

2l1j=0 Fjjpj(t)

(3.1)

Where, pj(t) is the jth component of vector p (t). Thus, given p (t) and F , s (t) can be

easily found, and vice versa.

Now , define a single operator G such that applying G to s (t) will exactly mimic theexpected effects of running the GA on the population at generation t to t + 1:

s (t + 1) = Gs (t) (3.2)

Then iterating G on p (0) will give an exact description of the expected behaviour of theGA.

Let GA be operating with selection alone (no crossover or mutation). Let E(x) denote the

expectation of x. Then, since si(t) (ith component of vector s (t)), is the probability that i

will be selected at each selection step,

E(p (t + 1)) = s (t) (3.3)

Let xy denotes the scalar difference between x and y , i.e. x = ky , where, k is ascalar. Then, from Equation 3.1, we have


s (t + 1) Fp (t + 1)

which implies

E(s (t + 1)) Fs (t)

This is the type of relation of the form in Equation 3.2, with G = F for this case of selection

alone.

Crossover and mutation can be included in the model by defining G as the composition of

the fitness matrix F and a recombination operator M that mimics the effects of crossoverand mutation. One way to define M is to find ri,j(k), the probability that string k will beproduced by a recombination event between string i and string j, given that i and j are

selected to mate. If ri,j(k) were known, we could compute

E(pk(t + 1)) =

i,j

si(t)sj(t)ri,j(k) (3.4)

Once ri,j(0) is defined, it can be used to define the ri,j(k)

Term ri,j(0) can be expressed as a sum of two terms: the probability that crossover does

not occur between strings i and j and the selected offspring (i or j) is mutated to all zeros

(first term) and probability that crossover does occur and the selected offspring is mutated

to all zeros (second term).

The probability that string i will be mutated to all zeros can be given by:

p|i|m(1 pm)l|i| (3.5)

where, |i| is the number of ones in a string i of length l

Incorporating the above expression, the first term in the expression for ri,j(0) can be

written as

ri,j(0)1 =1

2(1 pc)[p|i|m(1 pm)l|i| + p|j|m (1 pm)l|j|] (3.6)


where, pc= The probability that crossover occurs between strings i and j

1 pc= The probability that crossover does not occur between strings i and jpm= The probability that mutation occurs at each bit in the selected offspring

1 pm=The probability that mutation does not at each bit in the selected offspring

The factor 12

indicates that each of the two offsprings has equal probability of being

selected.

Let h and k denote the two offspring produced from a crossover at point c (counted from

the right-hand side of the string). Since there are l 1 crossover points, so the probabilityof choosing point c is 1/(l c).Second term can be expressed as:

ri,j(0)2 =1

2

pcl 1

l1

c=1

[p|h|m (1 pm)l|h| + p|k|m (1 pm)l|k|] (3.7)

Let i1 be the substring of i consisting of l c bits to the left of point c, let i2 be thesubstring consisting of the c bits to the right of point c, and let j1 and j2 be defined likewise

for string j. Then |h| and |k| can be given by:

|h| = |i| |i2|+ |j2| (3.8)|k| = |j| |j2|+ |i2| (3.9)

Expression for i2 and j2 can be written as:

|i2| = |(2c 1) i| (3.10)|j2| = |(2c 1) j| (3.11)

where denotes bitwise and. Since 2c 1 represents the string with l c zeros followedby c ones, |(2c 1) i(orj) returns the number of ones in the rightmost c bits of i(orj).If we define:

i,j,c = |i2| |j2| = |(2c 1) i| |(2c 1) j| (3.12)


Then

|h| = |i| i,j,c (3.13)|k| = |j|+ i,j,c (3.14)

Now, a complete expression for ri,j(0) can be written as:

ri,j(0) =(1 pm)l

2[|i|(1 pc + pc

l 1l1

c=1

i,j,c) + |j|(1 pc + pcl 1

l1

c=1

i,j,c)] (3.15)

These results give expectation values only; in any finite population. In the limit of an

infinite population, the expectation results are exact. Let G(x ) = F M(x ) for vectorsx , where is the composition operator. Then, in the limit of an infinite population,

G(s (t)) s (t + 1)

Define Gp as

Gp(x ) = M(Fx /|Fx |) (3.16)

where |Fx | denotes the sum of the components of vector Fx . Then in the limit of aninfinite population,

Gp(p (t)) = p (t + 1) (3.17)

G and Gp act on different representations of the population, but one can be transformed

into other by simple transformation.

3.6 Conclusions

The simple genetic algorithm described in this chapter is applied for tuning the PSS

parameters for both single machine and multimachine power systems, disc

Power System Protection

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