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A thesis by PhD student
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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
4.6 A step change of Tm = 0.1 pu, St = 0.8 + j0.2, Xe = 0.3 . . . . . . . . . . . 55
4.7 A step change of Tm = 0.1 pu, St = 0.8 + j0.4, Xe = 0.3 . . . . . . . . . . . 55
4.8 A step change of Tm = 0.1 pu, St = 1.0 + j0.2, Xe = 0.3 . . . . . . . . . . . 56
4.9 A step change of Tm = 0.1 pu, St = 1.0 + j0.5, Xe = 0.3 . . . . . . . . . . . 56
4.10 A step change of Tm = 0.1 pu, St = 0.5 + j0.1, Xe = 0.6 . . . . . . . . . . 56
4.11 A step change of Tm = 0.1 pu, St = 0.8 + j0.2, Xe = 0.6 . . . . . . . . . . . 57
v
List of Figures vi
4.12 A step change of Tm = 0.1 pu, St = 0.8 + j0.4, Xe = 0.6 . . . . . . . . . . . 57
4.13 A step change of Tm = 0.1 pu, St = 1.0 + j0.2, Xe = 0.6 . . . . . . . . . . . 57
4.14 A step change of Tm = 0.1 pu, St = 1.0 + j0.5, Xe = 0.6 . . . . . . . . . . . 58
4.15 A step change of Tm = 0.1 pu, St = 0.5 + j0.0, Xe = 0.3 . . . . . . . . . . . 58
4.16 A step change of Tm = 0.1 pu, St = 0.8 + j0.0, Xe = 0.3 . . . . . . . . . . . 58
4.17 A step change of Tm = 0.1 pu, St = 1.0 + j0.0, Xe = 0.3 . . . . . . . . . . . 59
4.18 A step change of Tm = 0.1 pu, St = 0.5 + j0.0, Xe = 0.6 . . . . . . . . . . . 59
4.19 A step change of Tm = 0.1 pu, St = 0.8 + j0.0, Xe = 0.6 . . . . . . . . . . . 59
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
5.6 A step change of Tm2 = 0.1 pu at generator 2, under SOP. . . . . . . . . . 75
5.7 A step change of Tm2 = 0.1 pu at generator 2, under SOP. . . . . . . . . . 76
5.8 A step change of Tm3 = 0.1 pu at generator 3, under SOP. . . . . . . . . . 76
5.9 A step change of Tm3 = 0.1 pu at generator 3, under SOP. . . . . . . . . . 76
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
5.13 A step change of Tm1 = 0.1 pu at generator 1, under HOP. . . . . . . . . . 78
List of Figures vii
5.14 A step change of Tm2 = 0.1 pu at generator 2, under HOP. . . . . . . . . . 78
5.15 A step change of Tm2 = 0.1 pu at generator 2, under HOP. . . . . . . . . . 78
5.16 A step change of Tm3 = 0.1 pu at generator 3, under HOP. . . . . . . . . . 79
5.17 A step change of Tm3 = 0.1 pu at generator 3, under HOP. . . . . . . . . . 79
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
5.21 A step change of Tm1 = 0.1 pu at generator 1, under LOP. . . . . . . . . . 80
5.22 A step change of Tm2 = 0.1 pu at generator 2, under LOP. . . . . . . . . . 81
5.23 A step change of Tm2 = 0.1 pu at generator 2, under LOP. . . . . . . . . . 81
5.24 A step change of Tm3 = 0.1 pu at generator 3, under LOP. . . . . . . . . . 81
5.25 A step change of Tm3 = 0.1 pu at generator 3, under LOP. . . . . . . . . . 82
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
5.29 A step change of Tm2 = 0.1 pu at generator 2, under OOP. . . . . . . . . . 83
5.30 A step change of Tm3 = 0.1 pu at generator 3, under OOP. . . . . . . . . . 83
5.31 A step change of Tm2 = 0.1 pu at generator 2, under OOP. . . . . . . . . . 84
5.32 A step change of Tm2 = 0.1 pu at generator 2, under OOP. . . . . . . . . . 84
5.33 A step change of Tm3 = 0.1 pu at generator 3, under OOP. . . . . . . . . . 84
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.
Chapter 1. Introduction 3
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.
Chapter 1. Introduction 4
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
Chapter 1. Introduction 5
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.
Chapter 1. Introduction 6
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
Chapter 1. Introduction 7
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
Chapter 1. Introduction 8
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-
Chapter 1. Introduction 9
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
Chapter 1. Introduction 10
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
Chapter 1. Introduction 11
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.
Chapter 1. Introduction 12
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
Chapter 1. Introduction 13
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.
Chapter 1. Introduction 14
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 1. Introduction 15
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
Chapter 2. Mathematical Modelling of Power System 18
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)
Chapter 2. Mathematical Modelling of Power System 19
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)
Chapter 2. Mathematical Modelling of Power System 20
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)
Chapter 2. Mathematical Modelling of Power System 21
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.
Chapter 2. Mathematical Modelling of Power System 22
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)
Chapter 2. Mathematical Modelling of Power System 23
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
Chapter 2. Mathematical Modelling of Power System 24
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)
Chapter 2. Mathematical Modelling of Power System 25
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)
Chapter 2. Mathematical Modelling of Power System 26
[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.
Chapter 2. Mathematical Modelling of Power System 27
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)
Chapter 2. Mathematical Modelling of Power System 28
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
Chapter 2. Mathematical Modelling of Power System 29
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.
Chapter 2. Mathematical Modelling of Power System 30
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
Chapter 2. Mathematical Modelling of Power System 31
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
Chapter 3. Genetic Algorithm: An Overview 34
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.
Chapter 3. Genetic Algorithm: An Overview 35
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
Chapter 3. Genetic Algorithm: An Overview 36
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.
Chapter 3. Genetic Algorithm: An Overview 37
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
Chapter 3. Genetic Algorithm: An Overview 38
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
Chapter 3. Genetic Algorithm: An Overview 39
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)
Chapter 3. Genetic Algorithm: An Overview 40
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)
Chapter 3. Genetic Algorithm: An Overview 41
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