POWER SYSTEM TRANSIENT STABILITY ANALYSIS USING CATASTROPHE THEORY
by
SIDHESHWAR PRASAD
THESIS SUBMITTED
IN FULFILMENT OF THE REQUIREMENTS FOR THE AWARD OF THE DEGREE OF
DOCTOR OF PHILOSOPHY
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Centre for Energy Studies
INDIAN INSTITUTE OF TECHNOLOGY, DELHI
NEW DELHI - 11 0 016 INDIA May, 1994
CERTIFICATE
This is to certify that the thesis entitled " Power System Transient Stability Analysis Using Catastrophe Theory " being submitted by Mr. Sidheshwar Prasad to the Indian Institute of Technology, Delhi, for the award of the degree of Doctor of Philosophy is a record of the bonafide research work carried out by him. He has worked under our guidance and supervision and has fulfilled the requirements for the submission of this thesis which, to our knowledge, has reached the requisite standard. The thesis, or any part thereof, has not been submitted to any other University or Institute for the award of any degree or diploma.
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Prof. S. C. Tripathy
Sr. Scientific Officer, Gr.I Professor
Centre for Energy Studies, Centre for Energy Studies, Indian Institute of Technology, Indian Institute of Technology, New Delhi - 110 016 India New Delhi - 110 016 India
ACKNOWLEDGEMENTS
I express my deep sense of gratitude and respect to Prof. S.C.
Tripathy and Dr. T.S. Bhatti, 'my supervisors for their invaluable guidance, encouragement, constructive criticism, and help in planning, execution and completion of the research work and this thesis.
I am highly grateful to Prof. S.P. Sabberwal, Prof. D.P.
Rothari, Prof.C.S. Indulkar, Prof. Madan Gopal, Dr. Y.P.
Singh, Dr. R. Balasubramanian, Dr. (Mrs) Ratna Choudhary, Dr.
N.D. Kaushik, Dr.Y.P. Satya, Dr. Bhim Singh, and other faculty members of the Centre for Energy Studies and Electrical Engineering Department of this Institute for various types of help and support during the period of my study and research at this Institute.
I am equally grateful to the Head C E S, Prof. H. P. Garg and the Director, Prof. N. C. Nigam for providing all the necessary infrastructural facilities for research at this Institute.
I express my sincere thanks and appreciation to the Head, System Engineers, and the technical staffs of Computer Services Centre, I.I.T., Delhi for providing needful help and facilities while carrying out research at the main frame computer systems of this Institute.
I convey my sincere thanks to all Research Scholars, and personally to Mr. V.K. Jain, Sr. Technical Assistant, CES for providing timely and invaluable help at the CES Computer Centre.
I am highly indebted to Dr. D.S. Balain, Director, and Dr.14L. Yadav, Head, Div. of L.P.T., I.V. Research Institute, Izatnagar (U.P.) for granting study leave and their administrative support and help during the entire period of my study and research at the CES, I.I.T., Delhi.
It gives me immense pleasure to record my sincere thanks to Dr. Murari Prasad, Dr. Sushil Kumar, Dr. B.N. Kowle, Dr. V.K.
Rao, and other friends of mine at the IVRI, Izatnagar for their full cooperation and timely help all the time, without which it was not possible to concentrate in the research work with my utmost peace of mind and satisfaction.
I would like to express my deep sense of veneration and gratitude to my father, father-in-law and mother-in-law for their love, affection and encouragement for this study. I owe special indebtedness and respect to my brother-in-law, Dr.Abhay Pratap4 and Bhabhi, Dr. (Mrs.-) C- Pratap for their.' ever-cherishing love, inspiration and help during the period of this study and research.
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I wish to record a word of-acknowledgement to my wife, Vimla for her full cooperation, dedicated services, and bearing many invisible difficulties while accomplishing this arduous task.
My daughter, Tanuja, and sons, Siddhartha and Saurabh encouraged me with their love and _affection, and extended their full cooperation for early completion of this work.
I realize to,dedicate this work to the memory of my beloved mother who left for heavenly abode on April 23, 1993 during the period of this study and encouraged me even during her last days on this earth. Oh ! Almighty, I humbly pray YOU daily for making me the son of such a great mother.
(
4 1 11/11,1-0
(SIDHESHWAR PRASAD) New Delhi, May 1994
ABSTRACT
The "Catastrophe Theory" is a new mathematical tool and its application to power system transient stability analysis is a recent development. Three models based on this theory have been used for transient stability analysis of the single machine- infinite bus power system and multimachine power systems. A few typical power system examples have been selected to demonstrate the simulation results with the help of a computer and these results have been compared with the results obtained by the standard step-by-step numerical method.
The material inside the thesis is arranged in seven chapters with the following abstract of their contents:
Chapter I introduces transient stability problem in a power system with a critical review of past research work done in this area. Some important methods of power system transient stability analysis have been briefly stated and grouped as (a) classical methods, (b) energy function methods, and {c) other new methods. The need, justification and motivation for the present work have been clearly stated.
Chapter II discusses a general introduction of the Catastrophe Theory and its application to power system transient stability
analysis. Attempts have been made to explain the catastrophe theory in a simple way so as to have physical and mathematical understanding of the subject. The geometry of the fold, the cusp, and the swallowtail catastrophes as well as their bifurcation set have been discussed in detail with a focus on their application to power system transient stability problem. A critical review of the research work on power system transient stability problem using catastrophe theory and the allied models brings out the need for development of a new model.
chapter III presents the detailed mathematical modelling of single machine-infinite bus power system and multimachine power system using catastrophe theory. The energy function equation for single machine-infinite bus power system and multimachine power system have been derived in terms of the system parameters separately. The same energy function equation is used to derive the three distinct catastrophes viz. the fold, the cusp, and the swallowtail catastrophe models for the single machine-infinite bus power system. For the multimachine power system, the cusp and the. swallowtail catastrophe models have been formulated from the same energy function equation of the multimachine power system.In the end, the models for the detailed representation of the power system considering speed governor control, flux decay effect and excitation control systems have been briefly stated.
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Chapter IV presents the computer simulation results for single machine infinite bus power system. Three typical examples of single machine-infinite bus power system have been chosen for simulation of the fold, the cusp, and the swallowtail catastrophe models of the power system transient stability analysis. In two examples the mechanical input power is varied from 0.8 p.u. to 1.2 p.u. and the results have been compared with the standard step-by-step numerical method.
It has been demonstrated that the swallowtail catastrophe model is the appropriate and most suited for transient stability analysis of single machine infinite bus power system and multimachine power system. The critical fault clearing time and angle computed by this model for all the power system examples are very close to the results obtained by the standard step-by-step numerical method and the variations are within the acceptable limits. Secondly, the developed model is valid for any loading condition irrespective of the types of fault and fault locations in the power system. The method is direct and fast as the computations mainly involve a few control parameters which are mostly algebraic expressions.
Chapter V presents the simulation results of the cusp and the swallowtail catastrophe models of the multimachine power systems. Here, four typical examples of multimachine power
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systems consisting of the two examples of 2-machine power system, one example of 3-machine power system, and one example of 4-machine power system have been selected for transient stability analysis using the catastrophe theory.
For each example the critical fault clearing time and angle have been computed using the cusp and the swallowtail catastrophe models. The simulation results have been compared with the swing curve data obtained by the standard step- by-step numerical method. It has been demonstrated that the swallowtail catastrophe model is an appropriate catastrophe model for transient stability analysis of multimachine power system as it gives results very close to the results obtained by the standard step- by- step numerical method.
Chapter VI presents the modelling of the detailed representation of power system after incorporating the effect of speed governor action, flux-decay, and voltage regulator action. Two power system examples of single machine-infinite bus power system- have been selected to simulate the individual or combined effect of speed governor action, flux-decay and voltage regulator action, and also to demonstrate new transient stability limit of the power system under the specifiedconditions of control.
It has been demonstrated with the help of swallowtail catastrophe model that the flux-decay effect significantly
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decreases the critical fault clearing time and hence lowers the transient stability limit of a power system, whereas the combined effect of the flux-decay and AVR action considerably increases the stability limit of a power system. The combined effect of the flux-decay and AVR action and the speed governor control action increases the critical fault clearing time and hence enhances the transient stability limit of a power system significantly.
Chapter VII summarises the significant conclusions of the previous chapters and states the suggestions for future research work.
TABLE OF CONTENTS
Page
ABSTRACT vi
LIST OF SYMBOLS xviii
LIST OF FIGURES xxii
LIST OF TABLES xxvii
CHAPTER I: INTRODUCTION AND REVIEW OF LITERATURE (1-50)
1.1 Introduction 1
1.2 Review of Literature 7
1.2.1 Some Methods of Transient Stability Analysis 8
1.2.1.1 Classical Methods 9
1.2.1.2 Energy Function Methods 14
1.2.1.3 Some Other New Methods 31 .
1.3 Problem Formulation 43
1.4 Organization of the Thesis 46
1.5 Conclusions 49
CHAPTER II: CATASTROPHE THEORY AND ITS APPLICATION TO
POWER SYSTEM (51-94)
2.1 Introduction 51
2.2 Physical Concept 53
2.3 The Zeeman Catastrophe Machine 54
2.4 Catastrophe Functions 59
2.4.1 Evolution of Catastrophe Theory 59
2.4.2 The Implicit Function Form 62
2.4.3 The Morse Forms 63
2.4.4 The Thom Forms 65
2.4.5 Catastrophe Function 67
2.5 The Seven Elementary Catastrophes 68
2.5.1 The Elementary Catastrophe Functions 69
2.5.2 The Fold Catastrophe 69
2.5.3 The cusp Catastrophe 72
2.5.4 The Swallowtail Catastrophe 76
2.6 Application of Catastrophe Theory to Power System 80
2.6.1 Principle 80
2.6.2 Construction of Bifurcation Set 83
2.6.2.1 Bifurcation Set of the Fold Catastrophe 83 2.6.2.2 Bifurcation Set of the Cusp Catastrophe 84 2.6.2.3 Bifurcation Set of the Swallowtail Catastrophe 85 2.6.3 Review of Power System Transient Stability Models
Using Catastrophe Theory 89
2.7 Conclusions - 93
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CHAPTER III : POWER .SYSTEM MODELS USING CATASTROPHE
THEORY (95-146)
3.1 Mathematical Modelling of Single Machine -
Infinite Bus Power System 95
3.1.1 Power Transfer Equation 95
3.1.2 Power Output of Salient-Pole Synchronous
Machine 98
3.1.3 Swing Equation and Inertia Constant 99
3.1.4 Energy Function Equation 106
3.1.5 The Fold Catastrophe Model 111
3.1.6 The Cusp Catastrophe Model 116
3.1.7 The Swallowtail Catastrophe Model 120 3.2 Mathematical Modelling•of Multimachine Power
Systems 124 •
3.2.1 Power Output and Energy Function Equation 124 3.2.2 Multimachine Power System Model Using
The Cusp Catastrophe 135
3.2.3 Multimachine Power System Model Using
The Swallowtail Catastrophe 139
3.3 Speed Governor Control, Flux-. Decay Effect, and
Excitation Control System Models 141
3.3.1 Speed Governor Control Model 142
3.3.2 Field Flux- Decay Model 143
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3.3.3 Flux- Decay and AVR Action Model 144 3.3.4 The Swallowtail Catastrophe Model 145
3.4 Conclusions 145
CHAPTER IV : COMPUTER SIMULATION OF SINGLE MACHINE-
INFINITE BUS POWER SYSTEM (147-176)
4.1 Introduction 147
4.2 Power System Example 4.1 147
4.2.1 The Fold Catastrophe Model 151
4.2.2 The Ctisp Catastrophe Model 151
4.2.3 The Swallowtail Catastrophe Model 153
4.3 Power System Example 4.2 159
4.3.1 The Cusp Catastrophe Model 161
4.3.2 The Swallowtail Catastrophe Model 164
4.4 Power System Example 4.3 16 8
4.4.1 The Cusp Catastrophe Model 168
4.4.2 The Swallowtail Catastrophe Model 171
4.5 Conclusions 171
CHAPTER V : COMPUTER SIMULATION OF MULTIMACHINE POWER
SYSTEMS (177-203)
5.1 Introduction 177
5.2 Power System Example 5.1 177
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5.2.1 The Cusp Catastrophe Model 178 5.2.2 The Swallowtail Catastrophe Model 182
5.3 Power System Example 5.2 184
5.3.1 The Cusp Catastrophe Model 184
5.3.2 The Swallowtail Catastrophe Model 188
5.4 Power System Example 5.3 191
5.4.1 The Swallowtail Catastrophe Model 191
5.5 Power System Example 5.4 194
5.5.1 The Swallowtail Catastrophe Model 197
5.6 Conclusions 201
CHAPTER VI : TRANSIENT STABILITY STUDY WITH SPEED GOVERNOR CONTROL, FLUX-DECAY EFFECT AND EXCITATION
CONTROL SYSTEMS (204-248)
6.1 Introduction 204
6.2 Speed Governor Control System 204
6.2.1 Problem Formulation 205
6.2.2 Speed Governor Control Model 206
6.2.3 Swallowtail Catastrophe Model 211
6.2:4 Results and Discussion 212
6.2.5 Conclusions 219
6.3 Flux-decay Effect 221
6.3.1 Field Flux-decay Model 222
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6.3.2 Swallowtail Catastrophe Model 223
6.3.3 Results and Discussion 226
6.3.4 Conclusions 228
6.4 Flux-decay Effect and AVR Action 231
6.4.1 Flux-decay and AVR Action Model 232
6.4.2 Swallowtail Catastrophe Model 235
6.4.3 Results and Discussion 236
6.4.4 Conclusions 242
6.5 Flux-decay Effect and AVR Action, and
Speed Governor Action 242
6.5.1 Swallowtail Catastrophe Model 243
6.5.2 Results and Discussion 245
6.5.3 Conclusions 248
CHAPTER VII: CONCLUSIONS (249-253)
7.1 Summary of Conclusions 249
7.2 Suggestions for Future work 253
REFERENCES 254
APPENDICES (261-277)
Appendix-1.1 261
Appendix-2.1 262
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Appendix-3.1 263
Appendix-4.1 267
Appendix-4.2 268
Appendix- 4.3 269
Appendix- 5.1 270
Appendix-5.2 271
Appendix-5.3 272
Appendix-5.4 274
Appendix-6.1 275
Appendix-6.2 276
Appendix-6.3 277
CURRICULUM VITAE 278
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