A Thesis. Submitted to the School of Graduate Studies. in Partial Fulfilment of the Requirements. for the Degree of _. Doctor of PhilosQphy

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1 MOIJEL RFllUCTION KETllOIJS APPLIED TO POWER S fsieks by IBRAHIM A. EL-NAHAS A Thesis Submitted to the School of Graduate Studies in Partial Fulfilment of the Requirements for the Degree of _ Doctor of PhilosQphy o December McMaster University

2 ... J KODEL REDUCTION ~ODS APPLIED TO POWER SYSTEKS

3 DOCTOR OF PHILOSOPHY (1983) (Electrical & Computer Engineering) M~~aster University Hamilton. Ontario TITLE Model Reduction Methods Applied to Power Systems AUTHOR: Ibrahim A. El-Nahas. B.Sc. (Elect. Eng., Cairo Univ.) M.Sc. (Elect. Eng., Cairo Univ.) SUPERVISORS:' R.T.H. Alden, Ph.D., P.Eng. N.K. Sinha, Ph.D., P.Eng. NUMBER OF PAGES: xvi, 237

4 ACKNOWLEDGEMENTS I I would like to express my sincere appre~iatlon to my supervisors Drs. R.T.H. Alden.a.nd N.K. Sinha. fo, having made this,",ork a pleasurable exchang~ of information. Their complementary experlise in I the power systems and.control areas. respectively."was invaluable in my development. Spe~ial thanks also go Dr. P. Taylor for hrs inte~est and ~ helpful advice. I am also grateful t~ my associates. Dr. M. Abu-EI-Magd, Dr. O. Ibrahim, Mr. F. Qureshy and Mr. M. EI-Sobki for their useful discussions", both on mathematical techniques and physical Interpretations_ Appreciation is expressed to the Natural Sciences and Engineering Research Council of Canada and M~Master Unviersity for their financial support. I,",ould like to sincerely thank Mrs. Dianne C. Crabtree of Dianne's Word Processing Service; Burlington. Ontario, for her typing and cheerful cooperation in preparing this manuscript... iii.. ~..'

5 ) ABSTIl.ACT This thesis presents a continuation in the process of rationalizing, unifying and improving existing model reduction techniques. Thus a method of reduction is developed which combines the method of aggrega~ion and partial Pade approximation in such a way as,to maintain their separate advantages while simultaneously removing their disadvantages. The i~portant aspects associated with the reduced-order models, obtained are: guaranteeing the stability of the reduced-order models,,' saving computation time, retainin~ the invariance property under state variable feedback conditions and matching some of the original system time,moments. Also, a criterion is proposed for,selecting the state variables of the original system to be retained in the reduced-order model. This criterion leads to developing a reduction; technique which can be '.regarded as a combination of the methods of akgregation and singular perturbation. Therefore, the reduced-order model obtained retains the physical significance of the state variables a!,d the dominant eigenvalues of the original system. Furthermore, a procedure is developed for obtaining dynamic equivalents of multimachine systems. This procedure utilizes the iv

6 ,,.-i'" concept of component cost analysis for identifying the coherent groups of generators..- Verification of the methods developed in the thesis is established using a variety of realistic power system models including a single synchronous machine connected to an infinite bus, a threemachine system and a lo-machine system. These applications include simulation, analysis and simp~e.controller design. I ) v

7 '. LIST OF PRINCIPAL SYMBOLS - stator voltages 1n direct- and quadrature-axis ciro cutts. respectively. - stator voltage. - stator currents in direct- and quadrature-axis circutts, respectively_ - stator flux linkages in direct- and.quadrature-ax1s circuits, respectiv,ely. - synchronous reactances in direct- and quadrature axis circuits, respectively - self reactances of field and direct-axis damper windings. - self reactances of quadrature-axis damper windings... - stator field mutual reactance. - stator-rotor mutual reactances with damper windings. R a - stator resistance. Rf,Rkdl,Rkql,Rkq2 field and damper winding resistances. \ if,ikdl,ikql,ikq2 - currents in field and damper windings. vi

8 ~f ~kdl '~kql'~kq2 - flux linkages with field and damper windings. E fd - field voltage..~ x e - total reactance between generator terminal and bus- bar. R ( w 0 - tota~ resistance -between generator terminal and busbar. ~ rotor angle. - angular frequen~y of infinite bus., H or M - inertia constant. T m w P,Q E E' d - input torque to generator shaft. - angular speed of rotor. - active and~reactlve power. - voltage behind synchronous impedance. - voltage proportional to quadrature-axis flux link- age.,. E' q ( x' T',., qo T'. do D - voltage proportional to direct-axis flux linkage. - stator transient reactance. - quadrature-axis transient open-circuit time constant. - direct-axis transient open-circuit time constant. - damping coefficient. Excitation System - voltage sensor output. - amplifier outp4t voltage... vii

9 .J, stabilizer output voltage. - voltage sensor time constant. - amplifier time constant. - stabilizing loop time constant., - exciter time constant. - amplifier gain. - stablizing loop gain - exciter gain. V ref - reference voltage. Kisce"llaneous - prescript denoting incremental change. \ T or t -1 s r - superscript denoting differentiation with respect to time. - subscript denoting ve~r quantity. - superscript denoting matrix or vector transpose. - superscript denoting matrix inverse. - Laplace operator. viii,.,."

10 . TABLE OF CONTENTS PAGE ACKNOWLEDGEMENTS ABSTRACT iii iv LIST OF LIST OF LIST OF PRINCIPAL SYMBOLS FIGURE TABLES vi xiii xvi CHAPTER 1 INTRODUCTION 1.1 The Bas ic Problem 1.2 Thesis Structure CHAPTER 2 FORMULATION OF THE APPROXIMATION PROBLEM 2.1 Introduction 2.2 Frequency Domain Formulation., , Ti~e Domain Formulation 2.4 Acc~acy Criteria : Introduction Relative Impulse Error Relative Step Error Modal Cost Analysis CHAPTER 3 A REVIEW OF EXISTING METHODS OF MODEL REDUCTION 3.2 Reduction Methods Based on Pade Approximation Definition of the Pade Approximation Method Continued Fractions Limitations of the Pade Approximation ix I.

11 TABLE OF CONtENtS (eon~idued) PAGE 3.3 Reduc~ion Me~hods Based on Error Minimization 3.4 Reduction Methods Based on Aggregation Principle Aggregation Principle Determination of the Aggregation Matrix General Case Aggregation Matrix with a.steady-state Constraint Optimal Aggregation Matri~ Determination of the Op.timal Output Matrix Control of a System via the Aggregated Model Stability Analysis \ Approximate Aggregation " Reduction Methods Based on Singular Perturbations Introduction Transformation of Physical Mod~ls into Singularly Perturbed Form A Two-Time-Scale Property Control Design Using Simplified Modele State Feedbaek by Eigenva~ue Assignment Linear Quadratic Control Deisgn 46 46, CHAPTER Conclusions A REVIEW OF METHODS OF PRODUCING SIMPLIFIED POWER SYSTEM "DYNAMIC MODELS Introduction The Form of the Linearized Model of Multi-Machine Power Systems State Space Reference State Space Frame Model Using Model Coherency-Based Dynamic Equiv~lentB l Machine Angle as in Center of Angle Reference 57 ' x I

12 TABLE OF CONtENtS (continued) PAGE Structural Conditions Under Which a Group of Machines Behaves as a Single Machine ~4.5 Modal Dynamic Equivalents Technique Summary CHAPT"lR 5 A UNIFIED ALGORITHM FOR MODEL INVARIANT DYNAMICAL SYSTEMS. REDUCTION OF LINEAR TIME Introduction Routh Approximation Method Frequency Domain Routh Approximation State-Space Routh Approximation A Proposed Procedure for Obtain~ng Reduced-Order Models Case of Single-Input Systems Case of Multi-Input System Applicat~n to Power ~ystems A Synchrbnous Machine Connected to an Infinite Bus A Synchronous Machine Connected to an Infinite Bus,Through a Tansmission Line.with an Exciter Conclusions 134 CHAPTER 6 ALGORITHMS FOR DETERMINING DYNAMIC EQUIVALENTS OF MULTIMACHINE SYSTEMS Introduction' A'New Algorithm for Model Reduction , , The Proposed Algorithm \ Simplified Models of Power Systems Single Machine Infinite Bus Multimachine System... System xii I I I I \

13 TABLE OF CONTENTS (continued) PAGE 6.3 An Algorithm for Identifying Coherent Generators of Multivarious Systems Concepts of Cost Decomposition Disturbance Models Linear Differential Systems Driven by White Noise A Reduction Algorithm for Obtaining Dynamic Equivalents Identification of Coherent Generators Reduction of Generator Buses Dynamic Aggregation of Generating Unit Models Testing the Reduction Algorithm on the 39 Bus New England System Model / 6.4 Conclusions 195 CHAPTER 7 A COMPARISON OF MODEL REDUCTION METHODS FOR COMPENSATOR DESIGN 7.1 Introduction 7.2 System D~scription 7.3 Compensator Design Using ~implified Models The Design of a PID Controller The Design of a Lead or Lead-Lag Compensator Co.nclusions 214 CHAPTER 8 CONCLUSIONS 217 REFERENCES APPENDIX 1 APPENDIX Aggregation with Moment Matching 8.2 Aggregation Combined with Singular Perturbation 8.3 Contributions of the Thesis 8.4 Suggestions for Future Work xii

14 LIST OF FIGURES FIGURE 2.1 Comparison of Time Domain and Frequency Domain Formulations of the Approximation Problem PAGE Interconnected Power System Stru~ture Torque Angle Response Following a 107. Step Change in Mechanical Torque Terminal Vo ltage Res pense to a 107. Step Change in Field Voltage Terminal Voltage Response to a 107- Step Change in Reference Vo ltage ~ 6.1 Single Machine-Infinite Bus Configuration 145 ' Terminal Voltage Response Following a 107. Step Change in Field Voltage Torque Angle Response Following a 107. Step Change in Field Voltage Torque Angle Response FollowinR a 107. Step Change in Mechanical power - Nine-Bus System Impedance Diagram; All Impedances are in pu Based on a 100 ijya Ntne-Bus System Load-Flow DiaRram Showl ng Prefault Conditions; All Flows are in MW and MVAR Voltage (Proportional to Direct Axis Flux Linkage of the Second Generator) Response Following a 107. Step Change in the Field Voltage of the Third Generation 16h Voltage (Proportional to Direct Axis, Flux Linkage of the' Third Generator) Response Following a 107. Step Change in the Field Voltage of the Third Generator 162 xiii

15 LIST OF FIGURES (continued) FIGURE PAGE The Motion of the Centre of Inertia of Generators 2 and 3 with Respect to Generator 1 Response Following a 10% Step Change in the Field Voltage of the Third '-' Generator 163 Voltage (Proportional to Direct Axis Flux Linkage of the Second Generator) Response to a 10% Step Change in Mechanical Power of the Third Generator 164 Voltage (Proportional to Direct Axis Flux Linkage of the Third Generator) Response Following a 10% Step Change in Mechanical Power of the Third Generator 165 The Motion of Centre of Inertia of Generators 2 and 3 with Respect to Generator 1 Following a 10% Step Change in Mechanical Power of the Third Generator 166 The Standard New England Test System 185 Torque Angle Responses at Buses 1, 8 ~nd 9 Following a 10% Step Change in Mechanical Power at Bus 9 - Aggregation Levell" 190 Torque Angle Responses at Buses 1, 8 and 9 Following a 10% Step Change in Mechanical Power at Bus 9 - Aggregation Level Torque Angle Response~ at Buses 1, 8 and 9 Following a io% Step Change in Mechanical Power at Bus 9 - Aggregation Level Torque Angle Responses at Buses 1, 8 and 9 Following a 10% Step Change in Mechanical Power at Bus 9 - Aggregation Level Torque Angle Responses at Buses 1, 8 and 9 Following a 10% Step Change in Mechanical Power at Bus 9 - Aggregation LevelS 194 Original System and Reduced Model Responses Following a Step Disturbance Frequency Response of the Original System and the Reduced-O rder Models 203 ) xiv

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1 Unified Power Flow Controller (UPFC)

Power flow control with UPFC Rusejla Sadikovic Internal report 1 Unified Power Flow Controller (UPFC) The UPFC can provide simultaneous control of all basic power system parameters ( transmission voltage,