POWER SYSTEM 
STABILIZATION CONTROL 
USING FUZZY LOGIC 
 
 
 
By 
D.K.P.U. Gunathilake 
 
 
 
A thesis 
Submitted to the Department of Electrical 
Engineering in partial fulfillment of the 
requirements for the Degree of 
Master of Engineering 
at the 
University of Moratuwa 
Sri Lanka 
 
 
 
2004 
 
80499 
 
 
 
 
  
Abstract 
 
A reliable and continuous supply of electric energy is essential for the functioning of 
today's complex societies. Due to a combination of increasing energy consumption 
and impediments of various kinds concerning the extension of electric transmission 
networks, utilities are forced to operate the systems closer and closer to system 
stability limits. This in turn requires use of special control aids to improve damping 
of low frequency electromechanical oscillations. 
 
The small signal stability problem is associated with modes of oscillations affecting a 
single machine or a small group of relatively closely connected machines. This 
problem has got a very high attention during the last three decades and many power 
system stabilizers based on classical and modern control theories have been 
developed to improve system damping. 
 
In the recent years, fuzzy logic has emerged as a powerful tool and is starting to be 
used in various power system applications. The application of fuzzy logic control 
technique appears to be the most suitable one whenever a well- , defined control 
objective cannot be specified, the system to be controlled is a complex one, or its 
exact mathematical model is not available. The control strategy depends upon a set 
of rules which describes the behaviour of the controller. 
 
In this thesis, a fuzzy-logic-based power system stabilizer to maintain stability and 
enhance closed-loop performance of a power system is developed. Simulation studies 
on a single machine infinite bus power system show that the proposed controller 
proves its effectiveness and improves the system damping compared to a 
conventional lead-lag power system stabilizer and an optimal power system 
stabilizer. 
DECLARATION 
I hareby declare that this submission is my own work and that, to the best of 
my knowledge and behalf, it contains no material previously published or 
written by another person nor material, which to substantial -extent, has been 
accepted for the award of any other academic qualificatio~f an university or 
institute of higher learning except where acknowledgement is made in the 
text. 
)J(W Dr.~cru 
Thesis Advisor 
January, 2004 
AKNOWLEDGMENTS 
It is a great pleasure to express my sincere gratitude to my supervisor, 
Dr. Nalin Wickramarachchi of the University of Moratuwa, Sri Lanka for his 
valuable advice, directions, encouragements and suppor,t throughout the 
program. My enthusiasm in this subject is inspired· by his profound 
knowledge in this area. I am truly honoured to work oft this thesis under his 
supervision. 
I also wish to thank to Professor J.R. Lucus, Head of Department of Electrical 
Engineering, all the lecturers and support staff in the Department of Electrical 
Engineering, the University of Moratuwa, Sri Lanka, for their help during my 
studies there. 
I extend my sincere gratitude to the Ceylon Electricity Board for offering me 
this opportunity to take part in this course. 
My special thanks to all my friends, felrow students and all other people 
around me for their valuable support and advice. 
" 
Finally, I extend my sincere thanks to my family, daughters Sathma and 
Sandalie, Son Priyanath, and wife Shyama for their commitment, sacrifice and 
encouragement for my studies. 
iv 
TABLE OF CONTENTS 
Abstract ...................................................................................................................... iii 
Acknowledgments ................................................................................................... iv 
Table of Contents ...................................................................................................... v 
List of Tables .......................................................................................................... viii 
List of Figures ........................................................................................................... ix 
1 Introduction ......................................................................................................... 1 
1.1 The Problen1 Statement .......................................................................................... l 
1.2 Power System Stabil izers ....................................................................................... 3 
1.2.1 Conventional Power Syst('m Stabilizers ................................................. 3 
1.2.2 Power System Stabilizers based on Modem Contro l Theory .............. 3 
1.3 Drawbacks of the PSSs based on Classical and Modern Control Theories .... 3 
1.4 New Techniques .................................................................. :. : ................................ 4 
1.5 Thesis Objective ............................................................... !. .................................... 4 
1.6 Thesis O rganiza tion ................................................................................................ 5 
2 Low Frequency Oscillations ............................................................................. 6 
2.1 Definition ................................................................................................................. 6 
2.2 Analysis ................................................................................................ ....... ............. 7 
2.3 Damping Controls .................................................................................................. 7 
3 Fundamental Concepts ...................................................................................... 9 
3.1 Power System Stability .......................................................................................... 9 
3.1.1 Classification of Instabilities .................................................................... 9 
3.1.2 Angular or Synchronous Instabili ty ........................................................ 9 
3.1.3 hequency Instability ............................................................................... l O 
3.1.4 Voltage Instabili ty .................................................................................... lO 
3.2 State Space Representation .................................................................................. ll 
3.2.1 State Space Model .................................................................................... ll 
3.2.2 Stabi lity of a Dynamic System ............................................................... 13 
3.2.3 Linearization ............................................................. ................................ 13 
3.3 Eigenvalues and Stabil ity Analysis ............................... .................... ................. 15 
3.3.1 Eigenvalues ........................................................ .. .... .. ............................... 15 
3.3.2 Eigenvectors ......................................................... .............. ...................... 17 
3.3.3 Modal Matrices ............................ ..... ....................................................... 17 
3.3.4 Free Motion of a Dynam ic System ........................................................ 18 
3.3.5 Participation Factors ................................................................................ 20 
3.3.6 Mode Shape ...................................................... .......... ................. ............. 20 
3.3.7 Controllabi lity and Observability .......................................................... 20 
4 Power System Modeling ................................................................................. 22 
4.1 Park's Transformation .......................................................................................... 22 
4.2 Two Axis Model .................................................................................................... 24 
4.3 Heffron-Phillips Model ........................................................................................ 25 
4.3.1 The Classical Model of Generator ......................................................... 26 
v 
4.3.2 Effects of Synchronous Machine Field Circuit Dynamics .................. 27 
4.3.3 Effects of Excitation System ...................................... ... .......................... 35 
4.3.4 Power System Stabi lizer ........................................................................ .40 
5 Methods of Damping Control ........................................................................ 45 
5.1 Conventional Methods of Damping Control... ................................................. 45 
5.1.1 Stabilizer based on Shaft Speed Signal ................................................. 46 
5.1.2 Delta-P-Omega Stabilizer ....................................................................... 46 
5.1.3 Frequency-based Stabilizer .................................................................... 48 
5.2 Damping Controllers based on Modem Control Theory ............................... 49 
5.2.1 Optimal Power System Stabilizer .......................................................... 49 
5.2.2 Pole Placement Stabilizer ........................................................................ 51 
5.2.3 Adaptive Stabilizer .................................................................................. 52 
6 Fuzzy Logic Controller ............................................................. ; ....................... 54 
. 
6.1 An Overview ........................................................................ ~: .............................. 54 
;I 6.2 Fuzzy Set Fundamentals .................................................. : ................................... 55 
6.2.1 Fuzzy Sets ................................................................................................. 55 
6.2.1.1 Membership Functions- Fundamental Definition ................ 55 
6.2.1.2 Fuzzy Set Operations ................................................................. 56 
6.2.2 Fuzzy Numbers ........................................................................................ 58 
6.2.3 Linguistic Variables and Values ............................................................ 59 
6.2.4 Approximate Reasoning ......................................................................... 59 
6.2.5 Fuzzy Conditional Statements ............................................................... 61 
6.3 The Basic Structure of Fuzzy Controller ............................................................ 62 
7 Design of Fuzzy Logic PSS ............................................................................. 66 
7.1 Input and Output Variables ................................................................................. 66 
7.2 Membership Function Definition ....................................................................... 67 
7.3 Rule Creation and Inference ................................................................................ 68 
7.4 Defuzzification ...................................................................................................... 72 
7.5 Creation of Fuzzy Logic Inference System for PSS using the Matlab 
Fuzzy Logic Toolbox and Simulink ................................................................... 72 
8 Simulation Studies ........................................................................................... 75 
8.1 Construction ofSimulink Models ...................................................................... 75 
8.1.1 Building the Simulink Model of the Power System without PSS ..... 75 
8.1.2 Building the Simulink Models of the Power System including 
the Power System Stabilizers .................................... ............................. 76 
8.2 Development of the Optimal Feedback PSS ..................................................... 76 
8.3 Training the Fuzzy PSS wilh the Optimal PSS ................................................. 76 
8.4 Simulation Studies at Various Operating Conditions of the System ............ 77 
8.4.1 Nominal Operating Condition (P=0.9pu and Q=0.3pu) .................... 77 
8.4.2 Leading Power Factor Operation (P=1.0pu and Q=-0.2pu) ............... 83 
8.4.3 Operating at Light Load Condition (P=0.4pu and Q=O) .................... 85 
8.4.4 Operating at I Ieavy Load Condition (P=1.0pu and Q=O.Spu) .......... 88 
8.5 Transient Performance ......................................................................................... 91 
8.6 Eigenvalues at Different Operating Conditions ............................................... 92 
VI 
9 Conclusions ...........................................•............................................................ 94 
9.1 Summary of Results .............................................................................................. 94 
9.2 Future Work .......................................................................................................... 95 
9.2.1 Application of FLPSS in Multi-Machine Power Systems ................... 95 
9.2.2 Generalization of llcffron-Phillips Model for Multi-Machine 
Power Systems ......................................................................................... 95 
9.2.3 Other PSS Input Signals .......................................................................... 95 
9.2.4 FACTS Devices and Coordinated Systems .......................................... 96 
References .................................................................................................................. 97 
Appendices ................... ...................... ...................................................... ... 99 
Appendix A: Single Machine Infinite Bus Power Syste~Data ................ 100 
Appendix B ........................................................................ -;~-: .............................. 101 
B1 Modeling of Single Machine Infinite Bus Power System at 
aminal Operation (P=0.9pu and Q=0.3pu) .................................................. 102 
B2 Development of Optimal Feedback Controller. .............................................. 107 
B3 Simulink Models at Nominal Operating Condition and MATLAB 
Routines to find Responses ............................................................................... 109 
Appendix C .......................................................................................................... 112 
C1 Modeling of Single Machine Infinite Bus Power System at Leading 
Power Factor Operation (P=l.Opu and Q=-0.2pu) ......................................... 113 
C2 Simulink Models at Leading Power Factor Operation .................................. 118 
Appendix D ................................................. ._. ....................................................... 120 
D1 Modeling of Single Machine Infinite Bus Power System at Light 
Load Operating Condition (P=0.4pu and Q=0) .............................................. 121 
D2 Simulink Models at Light Load Operating Condition .................................. 126 
Appendix E ........................................................................................................... 128 
E1 Modeling of Single Machine Infinite Bus Power System at Heavy 
Load Operating Condition (P=1.0pu and Q=0.5pu) ....................................... 129 
E2 Simulink Models at I Ieavy Load Operating Condition ................................. 134 
Appendix F ........................................................................................................... 136 
F1 Determination of Eigenva lues at Nominal Operating Condition 
(P=0.9pu and Q=0.3pu) ...................................................................................... 137 
F2 Determination of Eigenvalues at Leading Power Factor Operation 
(P=l.Opu and Q=-0.2pu) ..................................................................................... 140 
F3 Determination of Eigenvalues at Light Load Operation 
(P=l.Opu and Q=O.Spu) ...................................................................................... 144 
F4 Determination of Eigenvalues at Heavy Load Operation 
(P=1.0pu and Q=O.Spu) ...................................................................................... 148 
VII 
Table 7.1 
Table 8.1 
Table 8.2 
Table 8.3 
Table 8.4 
Table A.1 
LIST OF TABLES 
j 
Rules table .......................................................................................... 71 
System eigenvalues at P=0.9pu and Q=0.3pu ............................... 92 
System eigenvalues at P=1.0pu and Q=-0.2pu ............................. 92 
System eigenvalues at P=0.4pu and Q=O ...................................... 92 
System eigenvalues at P=l.Opu and Q=0.5pu ............................... 93 
Generator parameters used in the study ..................................... 100 
.._ 
"' 
viii 
Figure 4.1 
Figure 4.2 
Figure 4.3 
Figure 4.4 
Figure 4.5 
Figure 4.6 
Figure 4.7 
Figure 4.8 
Figure 4.9 
Figure 5.1 
Figure 6.1 
Figure 6.2 
Figure 6.3 
Figure 7.1 
Figure 7.2 
Figure 7.3 
Figure 7.4 
Figure 7.5 
Figure 8.1 
Figure 8.2 
Figure 8.3 
Figure 8.4 
Figure 8.5 
Figure 8.6 
LIST OF FIGURES 
Single machine connected to a large system through 
transmission lines .............................................................................. 25 
Equivalent circuit of lhe infinite bus system 
with classical generator model. ....................................................... 26 
Equivalent circuits relating to machine 
flux linkages and currents ............................................................... 28 
Phasor diagram of single machine connected 
to an infinite bus ........ ........................................................................ 30 
Block diagram representation with constant Efit ........................... 34 
Thyristor excitation system with A VR. ............................... :··········37 
Block diagram representation with exciter and AVR .. /. : ............. 39 
Block diagram representation with AVR and PSS ... /. .............. ..40 
Thyristor excitation system with AVR and PSS ......................... ..41 
Block diagram realization of del ta-P-Omega PSS ...................... ..48 
Membership functions p(T) used for describing 
the primary values LOW, MEDIUM, and HIGH, 
of the linguistic variable temperature ............................................ 59 
The basic structure of the FLC ........................................................ 64 
Three different defuzzification methods: 
COA, COS, and MOM ...................................................................... 65 
Block diagram representation of the exciter and 
the proposed FLC .............................................................................. 66 
Membership functions scaled from -1 f'o 1 .................................... 67 
Variation of error and derivative of error 
for a highly oscillatory system ........................................................ 68 
Rule generation by understanding the system dynamics ........... 69 
Phase plane for the controller fuzzy variables .............................. 70 
Dynamic response of /:i(J) to a 5% step rotor speed 
disturbance at P=0.9pu and Q=0.3pu ............................................. 78 
Dynamic response of 6.6 to a 5% step rotor speed 
disturbance at P=0.9pu and Q=0.3pu ............................................. 78 
Dynamic response of /:i(J) to a 5% step mechanical 
torque disturbance at P=0.9pu and Q=0.3pu .............. .. ................ 79 
Dynamic response of 6.o to a 5% step mechanical 
torque disturbance at P=0.9pu and Q=0.3pu ................................ 79 
Dynamic response of 6.fu with FLPSS to a 5% step 
rotor speed disturbance for different values 
of exciter gain at P=0.9pu and Q=0.3pu ........................................ 80 
Dynamic response of 6.15 with FLPSS to a 5% step 
rotor speed disturbance for different values 
of exciter gain at P=0.9pu and Q=0.3pu ........................................ 81 
ix 
Figure 8.7 Dynamic response of D.rJJ to a 5% step reference voltage 
disturbance at P=0.9pu and Q=0.3pu ........ ..................................... 82 
Figure 8.8 Dynamic response of D.c5 to a 5% step reference voltage 
disturbance at P=0.9pu and Q=0.3pu ............................................. 82 
Figure 8.9 Dynamic response of D.rJJ to a 5% step rotor speed 
disturbance at P=l.Opu and Q=-0.2pu ........................................... 83 
Figure 8.10 Dynamic response of D.b' to a 5% step rotor speed 
disturbance at P=l.Opu and Q=-0.2pu ........................................... 84 
Figure 8.11 Dynamic response of /).{J) to a 5% step mechanical 
torque disturbance a t P=l.Opu and Q=-0.2pu ............................... 84 
Figure 8.12 Dynamic response of D.€5 to a 5% step mechanical 
torque disturbance at P= l.Opu and Q=-0.2pu ............................... 85 
Figure 8.13 Dynamic response of D.rv to a 5<>'o step rotor speed 
disturbance at P=O..!pu and Q=O .................................................... 86 
Figure 8.14 Dynamic response of D.6 to a 5% step rotor speed· 
disturbance at P=0.4pu and Q=O ·······················1·: ........................ 86 
Figure 8.15 Dynamic response of D.rv to a 5% step mechrui.ical torque 
disturbance at P=0.4pu and Q=O .................................................... 87 
Figure 8.16 Dynamic response of D.6 to a 5% step mechanical torque 
disturbance at P=0.4pu and Q=O .................................................... 87 
Figure 8.17 Dynamic response of D.rv to a 5% step rotor speed 
disturbance at P=l.Opu and Q=0.5pu ................... ......... ................. 88 
Figure 8.18 Dynamic response of D.o to a 5% step rotor speed 
disturbance at P=1.0pu and Q=0.5pu ............................................. 89 
Figure 8.19 Dynamic response of {).(J) loa 5% step mechanical torque 
disturbance at P=l.Opu and Q=0.5pu ............................................. 89 
Figure 8.20 Dynamic response of D.6 to a 5%.._step mechanical torque 
disturbance at P=1.0pu and Q=0.5pu ............................................. 90 
Figure 8.21 Response of !).{J) to a..S0°1o step mechanical torque 
disturbance at P=0.9pu and Q=0.3pu ............................................. 91 
Figure 8.22 Response of D.o to a 50~o step mechanical torque 
disturbance at P=0.9pu and Q=0.3pu ............................................. 91 
Figure B3.1 Simulink model of the power system 
at P=0.9pu and Q=0.3pu ................................................................ 109 
Figure 83.2 Simulink model of the power system with lead-lag PSS 
at P=0.9pu and Q=0.3pu ................................................................ 109 
Figure B3.3 Simulink model of the power system with optimal 
feedback PSS at P=0.9pu and Q=0.3pu ........................................ l10 
Figure 83.4 Simulink model of the power system with the 
proposed FLPSS at P=0.9pu and Q=0.3pu ............. ..................... 110 
Figure B3.5 Simulink model inclusive of the all three controllers 
at P=0.9pu and Q=0.3pu ................................................................ 111 
Figure C2.1 Simulink model of the power system without PSSs 
at P=l.Opu and Q=-0.2pu ............................................................... 118 
Figure C2.2 Simulink model of the power system for P=1.0pu and 
Q=-0.2pu to study dynamic responses for the all three PSSs ... 119 
X 
Figure 02.1 Sirnulink model of the power system without PSSs 
at P=0.4pu and Q=O ........................................................ ................ 126 
Figure 02.2 Sirnulink model of the power system for P=0.4pu and 
Q=O to study dynamic responses for the all three PSSs ............ 127 
Figure E2.1 Sirnulink model of the power system without PSSs 
at P=1.0pu and Q=O.Spu ................................................................ 134 
Figure E2.2 Sirnulink model of the power system at P=1.0pu and 
Q=O.Spu to study dynamic responses of the system 
for the all three PSSs ....................................................................... 135 
.l 
;I 
xi