"Applied Science, Faculty of"@en .
"Electrical and Computer Engineering, Department of"@en .
"DSpace"@en .
"UBCV"@en .
"Sharkawi, Mohamed Ali Ahmed Ali El"@en .
"2010-03-23T19:47:03Z"@en .
"1980"@en .
"Doctor of Philosophy - PhD"@en .
"University of British Columbia"@en .
"The growing size and complexity of modern electric power systems necessitate the need to develop dynamic equivalents of the external system for local system dynamic studies. An estimation technique is developed in this thesis to identify the dynamic equivalents of external systems. An introduction is given in Chapter 1. A Basic multi-machine model for dynamic studies is developed in Chapter 2. The dynamic equivalent is estimated from information measured locally due to an intentional disturbance. To reduce computational requirements, a weighted least-squares algorithm with adaptive\r\nstep size scheme is used, and a proper model for the external equivalent\r\nis chosen in Chapter 3. Applying the developed techniques, equivalent parameters are estimated for three test systems, and the results are included\r\nin Chapter 4. In Chapter 5, the dynamic equivalents are first verified\r\nby comparing the dynamic interacting effect of the external system on the study system for both original and equivalent systems. The equivalents are further verified by three-phase short-circuits on machine buses of the study system. The dynamic responses of the original systems and the equivalent\r\nsystems are compared. Conclusions are drawn in Chapter 6 that the dynamic\r\nequivalent derived by the technique developed in this thesis is unique that the estimated equivalent is a good representation of the original external system, and that it can be developed for on-line identification."@en .
"https://circle.library.ubc.ca/rest/handle/2429/22367?expand=metadata"@en .
"ESTIMATION OF DYNAMIC EQUIVALENTS OF EXTERNAL ELECTRIC POWER SYSTEMS by MOHAMED ALI AHMED ,EL-|SHARKAWI B.Sc, Cairo High I n s t i t u t e of Technology, 1971 M.A.Sc, University of B r i t i s h Columbia, 1977 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (Department of E l e c t r i c a l Engineering) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1980 T) Mohamed A l i Ahmed El-Sharkawi, 1980 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f-an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e tha the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d that c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h Co lumbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 \u00C2\u00BB... 3l/^r/n9Q ABSTRACT The growing size and complexity of modern e l e c t r i c power systems necessitate, the need to develop dynamic equivalents of the external system for l o c a l system dynamic studies. An estimation technique i s developed i n t h i s t hesis to i d e n t i f y the dynamic equivalents of external systems. An introduction i s given i n Chapter 1. A Basic multi-machine model f o r dynamic studies i s developed i n Chapter 2. The dynamic equivalent i s estimated from information measured l o c a l l y due to an i n t e n t i o n a l disturbance. To reduce computational requirements, a weighted least-squares algorithm with adap-t i v e step s i z e scheme i s used, and a proper model for the external equiva-lent i s chosen i n Chapter 3. Applying the developed techniques, equivalent parameters are estimated f o r three test systems, and the;.results are i n -cluded i n Chapter 4. In Chapter 5, the dynamic equivalents are f i r s t v e r i -f i e d by comparing the dynamic i n t e r a c t i n g e f f e c t of the external system on the study system for both o r i g i n a l and equivalent systems. The equivalents are further v e r i f i e d by three-phase s h o r t - c i r c u i t s on machine buses of the study system. The dynamic responses of the o r i g i n a l systems and the equiva-lent systems are compared. Conclusions are drawn i n Chapter 6 that the dy-namic equivalent derived by the technique developed i n t h i s thesis i s unique that the estimated equivalent i s a good representation of the o r i g i n a l exter n a l system,,and that i t can be developed for on-line i d e n t i f i c a t i o n . i i i TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i LIST OF TABLES v i i LIST OF ILLUSTRATIONS v i i i ACKNOWLEDGMENT x NOMENCLATURE x i 1 . INTRODUCTION 1 1-1 Dynamic Equivalencing Techniques 2 1-1-1 The I n f i n i t e Bus Approach 2 1-1-2 Modal Approach . 2 1-1-3 Coherency Approach 3 1- 1-4 I d e n t i f i c a t i o n Approach 4 1-2 A Survey of the I d e n t i f i c a t i o n Studies 4 1-3 Outline of the Thesis 7 1- 4 D e f i n i t i o n s of Some Important Terminologies 8 2. A BASIC MULTI-MACHINE POWER SYSTEM 9 2- 1 Voltage Equations of Synchronous Machines 9 2-2 Electromechanic Torque Equation of the Synchronous Machine . 13 2-3 A Basic Model for Multi-Machine Power Systems 14 2- 3-1 Torque Equation 1 8 2-3-2 Internal Voltage Equation 19 2-3-3 Terminal Voltage Relation 20 2-3-4 Voltage Regulator and E x c i t e r System 22 2-4 The Complete System Equations 24 2-5 Output or Measurement Equations' 24 i v Page 2-6 Machine Parameters 25 2- 7 Models of Reduced Order for the External Equivalent Machines , 26 3. IDENTIFICATION TECHNIQUE AND ALGORITHM 29 3- 1 Estimation Technique 29 3-1-1 Error Function 29 3-1-2 A Review of Estimation Techniques 30 3-1-3 Convergence Schemes 30 3-2 Algorithm 34 3-3 Computer Simulation of the Dynamic Response of the O r i g i n a l System 36 3-3-1 Load Bus Elimination 36 3-3-2 Inte n t i o n a l Disturbance . 37 3-4 Boundary of the Study System and External Equivalent Model . 38 3-4-1 Number of Dynamic Equivalent Machines 38 3- 4-2 The Equivalent Machine Model and Number of Parameters 40 3- 5 Estimation of Equivalent Machine Voltages . 43 4. CASE STUDIED AND RESULTS 46 4- 1 Power Systems Data 46 4- 1-1 Data of the Five-Machine System . 50 4-1-2 Data of the Thirteen-Machine System 51 4-2 Number of Dynamic Equivalent Machines . . 52 4-3 Choice of the Equivalent Machine Models 53 4-3-1 Selection of the Equivalent Model for Test System 1 . 53 4-3-2 Selection of the Equivalent Model for Test System 2 . 54 4-3-3 Selection of the Equivalent Model for Test System 3 . 56 4-4 Estimated Parameters 56 4-4-1 Cost Function 58 V Page 4-4-2 Machine Responses Due to Pulsed E x c i t a t i o n 63 4-5 Parameter Estimation Using Other Machines' Responses . . . . 63 4-6 Parameter Estimation with Di f f e r e n t Intentional Disturbances 68 4-7 Measurement Noise E f f e c t 71 4-8 Off-Line and On-Line I d e n t i f i c a t i o n 74 4- 9 Conclusions of Chapter 4 79 5. VERIFICATION OF THE VALIDITY OF THE EQUIVALENTS 81 5- 1 Model Constants of the Study System 82 5-2 The E l e c t r i c Torques 87 5-3 Three-Phase Fault Test 88 5-4 Conclusions of Chapter 5 102 6. CONCLUSIONS 103 APPENDIX 105 Al Self E l e c t r i c Torque Equations 105 A2 Dynamic Interacting E l e c t r i c Torque Equations 107 REFERENCES I l l v i LIST OF TABLES Table Page 2.1 Equivalent models with d i f f e r e n t order and d i f f e r e n t number of parameters 28 4.1 General information of the test systems 46 4.2 Machine data of the f i v e machine power system 50 4.3 Machine data of the th i r t e e n machine power system . . . . . 51 4.4 -4.6 I d e n t i f i e d equivalent parameters of test systems 1, 2 and 3, respectively . . . . . . 57-58 4.7 Case studies for parameter i d e n t i f i c a t i o n from other machines' responses 66 4.8 -4.9 Estimated parameters from machine #2 responses of te s t systems 1 and 2, respectively . 66-67 4.10 Estimated parameters from machine #2 and machine #3 responses of test system 3 67 4.11 Parameter i d e n t i f i c a t i o n from the response of test system 1 due to ramp torque disturbance . 71 4.12-4.14 E f f e c t of measurement noise on the estimated parameters of test systems 1, 2 and 3, respectively 73-74 4.15 Parameter updating of test system 1 . 79 5.1 -5.3 Constants of test systems 1, 2 and 3, respectively . . . . 86-87 5.4 -5.6 E l e c t r i c torques of test systems 1, 2 and 3, respec-t i v e l y 89-91 5.7 Three-phase f a u l t tests 92 v i i LIST OF ILLUSTRATIONS Figure Page 2.1 Winding representation of the synchronous machine by d and q axes . . . . . . . . . . 10 2.2 A synchronous machine phasor diagram . . . 13 2.3 An interconnected system 15 2.4 Voltage regulator and e x c i t e r system . . . . . 22 2.5 The i - t h synchronous machine model of a multi-machine power system 22 2.6 A t h i r d order model for the dynamic equivalent 26 2.7 A second order model for the dynamic equivalent 27 3.1 Estimation algorithm . 35 3.2 The configuration of the dynamic equivalent 39 3.3 Algorithm for the choice of a suitable model for the equivalent machines 41 3.4 The cost function vs. two parameters of a fourth-order twelve-parameter model 42 3.5 The cost function vs. the damping c o e f f i c i e n t of a dynamic equivalent machine 42 4.1 Test system 1 47 4.2 Test system 2 48 4.3 Test system 3 49 4.4 Number of the dynamic equivalents 52 4.5 The cost function vs. some parameters of test system 2; a t h i r d order five-parameter model . \u00E2\u0080\u00A2 55 4.6 -4.8 Cost function vs. equivalent parameters of test systems 1, 2 and 3, re s p e c t i v e l y 59-62 4.9 -4.11 Responses of test systems 1, 2 and 3, re s p e c t i v e l y , due to pulsed e x c i t a t i o n voltage 64-65 4.12-4.14 Responses of machine //2 of test systems 1, 2 and 3, resp e c t i v e l y . . . . . . 69-70 v i i i Figure Page 4.15 Responses of machine #3 of te s t system 3 70 4.16 Ramp mechanical torque disturbance . . . . . 71 4.17 Response of test system 1 due to mechanical torque disturbance 72 4.18-4.20 Responses of test systems 1, 2 and 3, respectively, with noise added to the o r i g i n a l system responses 75-76 4.21 Responses of test system 1 af t e r updating the dynamic equivalent . . . . . . . . . 78 5.1 Speed-torque r e l a t i o n . . . . . 84 5.2 Internal voltage representation 85 5.3 -5.19 Responses of d i f f e r e n t cases of three-phase f a u l t test . . 93-101 A . l The mechanical loop of the i - t h machine 105 A.2 The e l e c t r i c a l loop of the i - t h machine 106 A.3 Interaction of the mechanical loop of machine j on machine i . . . . 107 A.4 Interaction of the e l e c t r i c a l loop of machine j on machine i . . . 109 i x ACKNOWLEDGMENT I would l i k e to express my g r a t e f u l thanks and deepest g r a t i -tude to Dr. Yao-nan Yu and M. D. Wvong, supervisors of t h i s project, f o r t h e i r continued i n t e r e s t , encouragement and guidance during the research work of t h i s t h e s i s . The guidance of Dr. Yu during the l a s t year of t h i s study, while Dr. Wvong i s on s a b a t i c a l leave, i s g r a t e f u l l y acknowledged. I am g r a t e f u l to Dr. M. S. Davies and Mr. A. G. Fowler for t h e i r valuable comments and advice during the course of t h i s study. The reading of the d r a f t by Mr. A. Yan i s duly appreciated. I wish to extend my sincere thanks to Dr. T. H. Lee and Mr. M. Mobarak of the New Brunswick E l e c t r i c Power Commission for supplying the data used i n t h i s study. Thanks are due to Ms. K. Brindamour for typing t h i s t h e s i s . The f i n a n c i a l support from the Univ e r s i t y of B r i t i s h Columbia and the National Research Council of Canada are g r a t e f u l l y acknowledged. I am g r a t e f u l to my parents and my wife Fatma for t h e i r encourage-ment throughout my graduate program. X NOMENCLATURE Power System B = Local load susceptance D = Damping factor E ^ = Equivalent f i e l d voltage E' = Internal voltage q G = Local load conductance I = Machine current i n system coordinates i = Machine current i n i n d i v i d u a l coordinates KA = Voltage regulator gain M = I n e r t i a constant r = Resistance TA = Voltage regulator time constant T' = Synchronous machine f i e l d time constant do V = Voltage x = Reactance CJ = Per unit speed to = 2irf o w = Natural frequency of mechanical mode o s c i l l a t i o n s \u00C2\u00A3 = Damping c o e f f i c i e n t P = d/dt Subscripts b = Base values d,q = Individual machine coordinates D,Q = System common coordinates 1 1. INTRODUCTION Modern e l e c t r i c power systems are characterized by the growing s i z e and complexity that enormous number of generating u n i t s , loads, and trans-mission l i n e s are included i n the system. For example, there are about 300 major e l e c t r i c machines i n the North America Northeastern E l e c t r i c Power System, and also about 300 major e l e c t r i c machines i n the WSCC System [1]. The s t a b i l i t y study of such large systems i s very d i f f i c u l t , i f not impos-s i b l e , because of two major reasons: 1) The time -required for the computation i s almost p r o h i b i t i v e , very uneconomical i f every contingency at a l l s t r a t e g i c points of such systems are investigated; 2) The exact configuration and parameters of the whole system at any given moment may not be a v a i l a b l e to l o c a l operators because the data a c q u i s i t i o n of such systems would be extremely d i f f i c u l t . In recent years, extensive studies have been directed toward sim-p l i f y i n g the representation of the power system, as some parts of the system are replaced by f i c t i t i o u s low order equivalents, or \"system equivalencing\". In doing so, a power system must be divided i n t o two subsystems: 1) The \"Study System\" which i s of d i r e c t i n t e r e s t , and may include a l l or part of the l o c a l power system which may be represented i n d e t a i l . 2) The \"External System\" which i s interconnected to the study system, i t s configuration and parameters may not be known to the study system operators. By system equivalencing, the external system may be modelled by a hypothesized structure much simpler than the actual system, but i t i s also necessary that the equivalent be f a i t h f u l l y 2 simulating the same i n t e r a c t i n g e f f e c t of the o r i g i n a l external system on the study system. The equivalencing techniques are being used i n both steady state and dynamics studies. While the steady state equivalencing techniques have been very w e l l developed and i n use [2-5], the dynamic equivalencing tech-niques remain to be improved. 1-1 Dynamic Equivalencing Techniques Many dynamic equivalencing techniques have been developed. The p r e v a i l i n g approaches may be summarized as follows: 1-1-1 The I n f i n i t e Bus Approach This i s the oldest and simplest approach. C l a s s i c a l l y , a large external system i s represented by an i n f i n i t e bus of constant voltage and constant frequency. The representation i s not accurate, since the dynamic i n t e r a c t i n g e f f e c t of the external system on the study system i s completely neglected. 1-1-2 Modal Approach [6-9] From a complete set of nonlinear d i f f e r e n t i a l equations describing the e n t i r e system, the system equations are l i n e a r i z e d , eigenvalues and eigen-vectors are analyzed, the system matrix i s diagonalized, and those modes which have n e g l i g i b l e e f f e c t s on the study system are eliminated. The major d i f f i c u l t i e s i n applying the technique are: 1) It requires f u l l knowledge of the whole system which i s not neces-s a r i l y a v a i l a b l e . 2) It requires long time f o r computation of the eigenvalues and eigen-3 vectors for the diagonalization of a large system matrix. 3) The new variables obtained usually do not correspond to s p e c i f i c v a r i a b l e s of the system. 4) Not su i t a b l e for on-line a p p l i c a t i o n . 1-1-3 Coherency Approach [1, 10-14] The objective i n t h i s approach i s also to reduce the order of ex-t e r n a l system. Major steps are as follows: a) Identify the coherent groups through extensive transient s t a b i l i t y analysis; a l l units which swing together at the same frequency and at close angles are i d e n t i f i e d as one group. b) A l l generating units of each coherent group are connected to an equivalent bus through i d e a l complex r a t i o transformers. c) Each group i s then replaced by one equivalent generating unit by a dynamic aggregation method, including synchronous machines, e x c i t a -t i o n systems, governor-turbine systems, and power s t a b i l i z e r s . The merit of the coherency approach i s that, i n the equivalent model, the same power plant structure with meaningful parameters i s retained. However, the technique also has some disadvantages: 1) I t requires f u l l knowledge of the e n t i r e e l e c t r i c power system. 2) The number of coherent groups i s usually very large. 3) The coherent groups may change with the contingency. 4) Heavy computation i s required f or both transient s t a b i l i t y analysis and dynamic aggregation. 5) Due to the long computation time, the on-line a p p l i c a t i o n i s s t i l l d i f f i c u l t . 4 1-1-4 I d e n t i f i c a t i o n Approach [15-20] The i d e n t i f i c a t i o n schemes have been proposed i n recent years f o r the determination of dynamic equivalents of the external systems. Data obtained only from within the study system are used. The approach comprises generally three steps: a) The external system i s represented by a hypothesized structure much simpler than the o r i g i n a l . b) An error function i s formulated to compare the equivalent system response with the o r i g i n a l system response. c) An algorithm i s developed that the parameters of the hypothesized model are being adjusted while the error function i s minimized. The advantage of the i d e n t i f i c a t i o n approach may be summarized as follows: 1) I t requires no information about the external system. 2) The responses used f o r i d e n t i f i c a t i o n are measured l o c a l l y . 3) The external system order can be greatly reduced. 4) The computation time can be economized and the on-line a p p l i c a t i o n i s f e a s i b l e . 1-2 A Survey of the I d e n t i f i c a t i o n Studies The a p p l i c a t i o n of i d e n t i f i c a t i o n technique to power systems i s not new. Many studies have been made and useful algorithms developed [21-25] . As f o r the dynamic equivalencing of power systems, important con-t r i b u t i o n s are as follows. I t was Masiello and Schweppe [15] who f i r s t provided an algorithm to i d e n t i f y the equivalent external system. Their technique was based on a 5 hypothesized structure for the external system, and the unknown parameters were i d e n t i f i e d by using the normal power system f l u c t u a t i o n s which were observed within the l o c a l system. The paper has addressed the problem i n depth, except the choice of the equivalent external system model which was not c l e a r . They recommended that the model chosen by representative f o r the bandwidth of the dynamic range concerned. According to t h i s suggestion, the equivalent may not be unique; several models with d i f f e r e n t orders may be representative. In t h e i r study, the v a l i d i t y test of the estimated model was not given. P r i c e et a l . [16-17] also used normal fl u c t u a t i o n s for the i d e n t i -f i c a t i o n . The maximum l i k e l i h o o d estimate i s applied to i d e n t i f y the exter-n a l equivalent parameters. The c r i t e r i o n for convergence i s based upon the s i m i l a r i t y i n magnitudes between the o r i g i n a l and equivalent system r e -sponses. In t h e i r study, not a l l the unknown parameters can be i d e n t i f i e d . The l i k e l i h o o d function of one parameter i s f l a t without a well-defined maximum. The adequacy of the equivalent model was not examined and the ap p l i c a t i o n of the technique to a 14-generator external system proved un-s a t i s f a c t o r y . No v a l i d i t y test of the equivalent system was given. Ibrahim et a l . [18] used an i n t e n t i o n a l disturbance for the dyna-mic equivalent i d e n t i f i c a t i o n . A deterministic power system was assumed. The dynamics of the external system were represented by stochastic l i n e a r difference equations, and a recursive least-square algorithm was used to es-timate the parameters. The model used for the equivalent external system, however, was purely mathematical, not a power system structure with meaning-f u l parameters, and the external equivalent model was found to be dependent on the nature and l o c a t i o n of the disturbance. 6 G i r i et a l . [19] applied the coherency approach for o f f - l i n e equiva-lencing and the i d e n t i f i c a t i o n approach f o r o n - l i n e equivalencing. Due to the coherency approach, a f u l l knowledge of the external system was required and the number of dynamic equivalent machines could be very large. In an e a r l i e r paper by Yu, El-Sharkawi and Wvong [20], the e n t i r e external system was represented by one large machine with meaningful para-meters . An i n t e n t i o n a l disturbance was used and a recursive least-squares algorithm was developed for i d e n t i f i c a t i o n of the unknown external equivalent parameters. Although the algorithm was e f f i c i e n t , the example was too simple, and no v a l i d i t y t e s t was given. Moreover, many other problems were not addressed, e.g., the uniqueness of the equivalent model, the a p p l i c a t i o n of the estimation technique to a much larger system, etc. It i s rather important to mention that i n some of the above studies [15-17], normal fluctuations are used for i d e n t i f i c a t i o n instead of inten-t i o n a l disturbances. The use of these f l u c t u a t i o n s , however, draw c r i t i c i s m among power engineers for two major reasons: 1) The normal f l u c t u a t i o n s which occur on a power system can be cate-gorized by two broad classes; changes i n loads, and hunting among the various generators and loads. The l a t t e r i s l a r g e l y a three phase phenomenon and i s probably well represented by the given model. But, the load v a r i a t i o n s which lead to the other type of f l u c t u a t i o n s on the power system have a s i g n i f i c a n t s i n g l e phase component and cannot be represented by a simple model. 2) Stanton [24] had worked on the problem of i d e n t i f y i n g power system dynamic models from normal operating data. From his experience, he advised against drawing any firm conclusions regarding p r a c t i c a l ap-p l i c a t i o n from the simulated r e s u l t s using normal operating data for 7 the i d e n t i f i c a t i o n . The c o l l e c t i o n and processing of the data to i d e n t i f y dynamic equivalents would be extremely d i f f i c u l t i n large interconnected systems. Also, the estimated parameters may not pro-vide accurate representation during contingencies with large d i s t u r b -ances . 1-3 Outline of the Thesis Research on the estimation of dynamic equivalent of e l e c t r i c power systems i s continued i n this t h e s i s . Taking into consideration the many un-solved problems i n previous works, the i n v e s t i g a t i o n . i s o u t l i n e d as follows: 1) To develop a basic multi-machine power system model for the dynamic studies. 2) To develop an estimation technique and fast convergent algorithm applicable to both noisy and deterministic systems. 3) To develop a technique choosing the equivalent model for the external system. 4) To apply the equivalency technique to d i f f e r e n t power systems. 5) To test the uniqueness of the i d e n t i f i e d equivalent external system by changing the l o c a t i o n and type of disturbance within the study system, and by using d i f f e r e n t machine responses of the study system for i d e n t i f i c a t i o n . 6) To v e r i f y the dynamic equivalent by comparing the model constants and the damping and synchronizing torques of the study system, before and a f t e r dynamic equivalencing. 7) To test the dynamic equivalent for severe contingencies by assuming three phase f a u l t s within the study system. 8 1-4 D e f i n i t i o n s of Some Important Terminologies For convenience, important terminologies used throughout t h i s thesis are defined as follows: Equivalent External System: ref e r s to the s i m p l i f i e d network representing the external system. Equivalent System: ref e r s to the system composed of the study system and the equivalent external system. Equivalent Machine: ref e r s to a machine of the equivalent external system. Equivalent Parameter: ref e r s to a parameter of the equivalent machine model. Boundary Line: r e f e r s to a f i c t i t i o u s l i n e drawn between the study system and the external system. Boundary Bus: r e f e r s to a bus on the boundary l i n e connected to both the study system and the external system. 9 2. A BASIC MULTI-MACHINE POWER SYSTEM MODEL An adequate synchronous machine model i s required to simulate the dynamic performance of a power system. Moreover, the behaviour of an i s o -l a t e d synchronous machine of a power system i s quite d i f f e r e n t from i t s behaviour i n a multi-machine power system because of the i n t e r a c t i o n s between machines. In t h i s Chapter, a basic multi-synchronous-machine power system model i s developed, for the dynamic studies i n general and for the estimation study i n p a r t i c u l a r . The dynamic i n t e r a c t i o n s between the machines are c l e a r l y represented, and the contribution of these i n t e r a c t i o n s to the s t a -b i l i t y of any machine i n the system i s considered. 2-1 Voltage Equations of Synchronous Machines In 1928, R.H. Park [26] developed the two reaction theory of the synchronous-machine, which has been the basis of the modern e l e c t r i c power system and synchronous machine studies. Modifications are, however, neces-sary to adapt the general theory to s p e c i f i c a p p l i c a t i o n s . The theory i s based on the transformation of armature winding of the synchronous machine on the three phase axes into a two winding equiva-lent on the d i r e c t (d) and quadrature (q) axes, F i g . 2.1, plus a zero-axis winding which i s not shown i n the f i g u r e . 10 F i g . 2.1 Winding representation of the synchronous machine by d and q axes. the form, The synchronous generator equations of F i g . 2.1 can be written i n V f = r f i f + P \u00C2\u00A5 f V, = - r i J + P\u00C2\u00A5, - oif d a d d q V = - r i + P ? q a q q d where 11 and *d 1 CO o . V q 1 . V o X d X d f XdD 2 X d f X f X f D 2 XdD X f D *D i f V . X x . q qQ - i q 3 . 2 XqQ XQ : . V Some parameters and va r i a b l e s i n Park's equation have secondary e f f e c t s and may be neglected f or dynamic studies [27-32]. The following assumptions are usually made: 1) The e f f e c t of the damper windings i s much smaller than the damping provided by the power system s t a b i l i z e r s , loads, ... e t c . 2) The voltages P\u00C2\u00A5, and PY due to the rate of change of the fl u x l i n k -d q ages i n the respective d and q windings are n e g l i g i b l e as compared to the r o t a t i o n a l voltages , and a)\u00C2\u00A5 . d q 3) The per unit armature resistance of the large machines i s very small as compared to the per unit reactances; the resistance voltage i s n e g l i g i b l e . Applying these assumptions, Park's equations for a synchronous generator are reduced to V f = r f i f + p\u00C2\u00A5 f (2.1) V, = -cof d q V = q d (2.2) = ( _ x d x d + x d f V 7 \" 4f = -x i /u> (2.3) q q q * f = ( x f * f \" l x d f V 7 \" Substituting the values of V\ and \u00C2\u00A5 of equations (2.3) into (2.2), y i e l d s d q V J = x i d q q (2.4) V = E - x, i , q d d where E ^ x A f i . (2.5) dr r or the steady state equivalent f i e l d voltage which can be written E = E' + (x - x') i . (2.6) q d d d where E' = o)V. x../x. (2.7) q f df f and ' = - - ^ x ^ JXc (2.8) X d X d 2 \" d f ' ~ f From equations (2.4) and (2.6), the quadrature-axis voltage can t written V = E' - x' i , (2.9) q q d d Substituting the value of of equation (2.7) into equation (2.1) y i e l d s T! PE' = E \u00C2\u00A3 , - E (2.10) do q f d where T] = x J u r , do f f and (2.11) E,_, = x,_ V / r fd df f f Equation (2.10) can be written as a function of E^, as follows E q + Tdo P E q \" E f d \" ( X d \" XcP h (2.12) Equations (2.4), (2.9) and (2.12) form the basic voltage equations of synchronous machines. A phasor diagram also can be drawn as F i g . 2.2. F i g . 2.2 VfJ A synchronous machine phasor diagram. 2-2 Electromechanic Torque Equation of the Synchronous Machine In steady state, the mechanical power input to the generator from the turbine equals the e l e c t r i c a l power output plus machine l o s s e s . Any change i n input or output i s a disturbance and would cause energy i n e q u i l i b -rium of the system. During the period of disturbance, the speed of the generating u n i t changes i n r e l a t i o n to the magnitude of the difference between the mechanic power and e l e c t r i c power, the i n e r t i a of the u n i t , and the damping power of the system. The general electromechanic torque equation of the synchronous machine may be written M Pw + DOJ = T - T (2-13) m e where M = Jw?/P, (2-13a) b b For small,purturbations, t h i s equation can be l i n e a r i z e d as follows M PAco + DAoo = AT - AT (2-14) m e 2-3 A Basic Model f o r Multi-Machine Power Systems A synchronous machine model for small purturbations was developed by Heffron and P h i l i p s [27], and deMello and Concordia [28]. The dynamic i n t e r a c t i o n s between the machines were ignored i n the model. These i n t e r -actions, however, could have large e f f e c t s on large and small machines i n a multi-machine system, e s p e c i a l l y with strong t i e s between them. Therefore, a multi-machine power system model including these in t e r a c t i o n s i s required. Yu et a l . [30-31] developed a multi-machine model for non-salient pole synchronous machines for t h e i r dynamic i n t e r a c t i o n s t udies. The model i s s u i t a b l e f o r steam-electric units, but may not be accurate enough for h y d r o - e l e c t r i c units i n which the saliency e f f e c t may not be n e g l i g i b l e . The basic multi-machine model presented i n t h i s thesis has s i m i l a r outlook l i k e that of reference [30], with the dynamic in t e r a c t i o n s between machines c l e a r l y represented. However, the model i n t h i s thesis i s s u i t a b l e f o r both s a l i e n t and non-salient pole synchronous machines. 15 F i g . 2.3 shows the i - t h machine i n a multi-machine power system. I t i s connected to the other machine buses by transmission l i n e with imped-ances Z.., Z.. , etc., and has a l o c a l load with an admittance y... Machine currents can be expressed i n terms of bus voltages of the system i n matrix equation as follows, where [y^] i s an n x n complex symmetric matrix for an n-machine system, and n [i] = [ y t l [v] (2.15) y t i i (2.15a) y t i j The bar refe r s to a phasor value N F i g . 2.3 An interconnected system. Equation (2.15) represents system with a l l load buses eliminated. 16 According to F i g . 2.2, the voltage vector [V] can be expressed i n terms of [E'] and machine currents as follows, q [V] = [E' e j 6 ] - [ j x l ] [I] - [(x - x!) I e j ( 6 + 9 0 ) ] (2.16) q d q d q su b s t i t u t i n g the value of [V] from equation (2.16) into equation (2.15), y i e l d s [I] = [Y] ([E' e j S ] - [ ( x q - x!|) l_ e j ( 6 + 9 0 ) ] ) (2.17) where Y = ( [ y t ] _ 1 + [ j x j ] ) \" 1 (2.17a) Referring the i n d i v i d u a l machine current to i t s own machine coordinates y i e l d s n j ( B . .-6 ..+90)' j ( 3 . .-6 . .+180) i . = Z Y.. ( E \ e 1 3 1 J - (x .-x' ) I . e 1 J 1 3 ) (2.18) i j = i i J qj qj dj qj where j (90 -6 ) i . = I. e 1 x j 3 , , (2.18a) e i j i j Y. . = Y. . 1 J 6. . = 6. - 6. i J 1 3 The d i r e c t and quadrature axis components of the i ^ t h machine are I,. = Re ( i . ) = Z Y. . (-E\ S. . + (x .-x' ) I . C. .) dx x / x xj qj xj qj dj qj xj I . = Im ( i . ) = Z Y.. (E'. C.. + (x .-x' ) I . S..) q i i j = 1 I J qj I J qj dj qj xj (2.19) where C. . = cos (6.. - 6 . . ) S. . = s i n ( 3 . . - 6 . . ) (2.19a) where The l i n e a r i z e d equations f o r small o s c i l l a t i o n s may be written (2.20) Therefore where and where [AI d] = [Q d][AEM + [P d][A6] + [M d][AI ] [L ][AI ] = [Q ][AE'] + [ P l [ A 6 ] q q q q q Q ,. . = - Y . . S . . xdxj xj xj P \u00E2\u0080\u009E . = - Y.. (E'.C.. + (x .-x!.) I . S..) d i j ^ x 1 J qj i j qj dj qj xj P = - \u00C2\u00A3 P d i i ... d i j M... = Y. . (x .-x' ) C.. dxj xj qj dj xj L . . = - Y.. (x .-x' ) S.. qxj . \u00C2\u00A5 \u00C2\u00B1 xj qj dj xj ( 2 > 2 ( ) a ) L .. = 1 - Y (x .-x' ) S. . qxx j_\u00C2\u00B1 qx dx xx Q . . = Y .. C. . q i j 1 J 1 J P . . = - Y. . ( E \ S. . - (x . - x' ) I . C. .) q i j 13 q3 13 q3 dj qj xj P . . = - \u00C2\u00A3 P . . q n ^ \u00C2\u00B1 qiJ [Al ] = [Y ][AE'] + [F ] [A6 ] (2.21) q q q q [Y q] = [ L q ] - l [ Q q ] [F q] = [ L q ] \" 1 [ P q ] (2.21a) [Al.] = [Y ] [ AE' ] + [F , ] [ A 6 ] (2.22) d d q d 18 [ Y d ] = [ Q d ] + [M d][Y q] [F d] = [ P d ] + [M d][F q] (2.22a) 2-3-1 Torque Equation Since the e l e c t r i c power P^ approximately equals the e l e c t r i c torque i n per uni t , T . = R (V. i.) e i e l l = E \ I . + (X .-x' ) I,. I . (2.23) qx qx qx dx dx qx The l i n e a r i z e d torque equation becomes [AT ] = [S][AE'] + [T][AI,] + [0][AI ] (2.24) e q d q where [S], [T] and [0] are diagonal matrices, and S. . = I . xx qx T.. = (x .-x' ) I . (2.24a) xx qx dx qx 0.. = E'. + (x .-x' ) I,. xx qx qx dx dx Substituting the value of [Al ] and [Al,] from equations (2.21) and (2.22), q d respectively, into equation (2.24) gives [AT ] = [K1][A6] + [K2][AE'] (2.25) e q where [Kl] = [ T ] [ F J + [0][F ] d q [K2] = [T][Y ] + [0][Y 1 + [S] d q [Kl] and [K2] are not symmetrical matrices. (2.25a) 19 2-3-2 Internal Voltage Equation L i n e a r i z a t i o n of the i n t e r n a l voltage equation (2.10) gives [W][AE' ] = [AE ] - [XD][AI,] (2.26) q rd d where [W] and [XD] are diagonal matrices; e.g. W. . = 1 + S T' , x i dox and XD.. = x J . - x'. xx dx dx S i s for the Laplace transformation. Substituting the value of [1^] from (2.22) into equation (2.26) gives [J][AE'] = [AE.,] - [K4][A6] (2.27) q rd where [J] = [W] + [XD][Y d] (2.27a) [J] can be p a r t i t i o n e d into two matrices [J] = [ f ( ^ j ) ] + [g(^0] (2.27b) where f. . (^r) = (1 + S T' .K3. .)/K3. . xx K3 dox xx xx K3. . = (1 + (x - x ' ) Y,. .) 1 xx dx dx dxx K3.. = ((x..-x* ) Y,..) 1 xj dx dx dxj (2.27c) (2.27d) [f(-~r)] i s a diagonal matrix representing .the e f f e c t of AE' of i n d i v i d u a l K3 q machines, and [g(-^r)]\"\"is a matrix with zero diagonal elements representing K3 the dynamic i n t e r a c t i n g e f f e c t of AE^ of other machines. Equation (2.27) can be written 20 [AEM = [ f ( ^ ) ] 1 ( [ A E f d ] - [ g ( ^ - ) ] [ A E M - [ K 4 ] [ A 6 ] ) (2.28) [K3] and [K4] are not symmetrical matrices. 2-3-3 Terminal Voltage Relation The terminal voltage may be expressed by i t s d and q components V 2. = V 2. + V 2. (2.29) t i d i q i where V . = v.. + j V . (2.29a) tx dx qx L i n e a r i z a t i o n of (2.4) gives [AV d] = [ X ] [ A I d ] [AV ] = [AE'] - [X!][AI ] q q d d (2.30) where [X ] and [X'] are diagonal matrices. L i n e a r i z a t i o n of equation (2.29) q d and s u b s t i t u t i o n of [ A V j , [ AV ] and [Al,] into (2.29) y i e l d s d q a [AV] = [K5][A6] + [K6][AEM (2.31) where [K5] = [V ][X ][f ] - [ V ][X'][F,] u q q Q d d [K6] = [ V I [ X ] [ Y ] + [ V ] ( I - [ X ! ] [ Y ] ) D q q Q d d [V^ ] and [Vq] are diagonal matrices where Dxx dx x V_. . = V ./V. Qxx qx x [K5] and [K6] are not symmetrical matrices. (2.31a) (2.31b) The expressions of these constants i n terms of the machine para-meters and steady state values are summarized as follows: K l . . = (E'. + (x . - xi.) 1j\u00E2\u0080\u00A2) F .. + ((x . - x!.) I .) F,.. i j , q i . q i d i / dx y qxj v v q i dx y q i ' d i j K 2 i i = I q i + ( E q l + (Xqi \" x d i ) I d \u00C2\u00B1 ) Y q \u00C2\u00B1 i + ( ( x q \u00C2\u00B1 - x d \u00C2\u00B1 ) I q i ) Y d i \u00C2\u00B1 K2.. = (E' . + (x . - x' ) I,.) Y . . + ((x . - x l ,) I .) Y, . . xj qx qx. dx y dx y qxj v v qx d i y q i y d i j K3.. = (1 -T ( x d . -x'.) Y d . . ) - 1 K 3 i j ^ \u00C2\u00B1 \u00C2\u00AB X d i - X d i > Y d i j > _ 1 K4. . = (x,. - x' .') F, . . xj dx dx 7 dxj 3\u00E2\u0080\u0094 1 5 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 * \u00E2\u0080\u00A2 1 j \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 n K5.. = (V,. x . F .. - V . x1. F,..)/V. xj v dx qx qxj qx dx dxj' x 1 l j \u00E2\u0080\u00A2 a. a i 1 ) \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 TI K6.. = (V,. x . Y .. + V . (1 - x' Y ))/V. xx dx qx qxx qx dx dxx x K6.. = (V,. x . Y .. - V . x1. Y,..)/V. xj ^ i v dx qx qxj qx dx d x j 7 x (2.32) where F q, F d, Y and Y d are given by equations (2.21a) and (2.22a) 22 2-3-4 Voltage Regulator and E x c i t e r System A voltage regulator and e x c i t e r system i s shown i n block diagram as F i g . 2.4 VOLTAGE REGULATOR/EXCITER 't -o 'ref KA A E f d ) \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 1*STA F i g . 2.4 Voltage regulator and e x c i t e r system. A s o l i d state e x c i t e r with n e g l i g i b l e time constant i s assumed so that A E f d can be expressed by a f i r s t order d i f f e r e n t i a l equation. AV. . = -(1 + S TA.) AE,../KA. t l 1 I Q l 1 (2.33) Equations (2.14), (2.25), (2.28), (2.31) and (2.33) form the i - t h machine model of a multi-machine power system. The corresponding block diagram i s shown i n F i g . 2.5. a E K3ji 1 * S T < t o K 3 i i _KAj k 1*S TAj F i g . 2.5 The i - t h synchronous machine model of a multi-machine power system. The dynamic i n t e r a c t i o n e f f e c t s of other machines on the i - t h machine due to t h e i r angle and voltage changes are e x p l i c i t l y expressed. Every machine i s affected by other machines i n the system. However, the magnitude of the e f f e c t depends upon A6 and AE^- of other machines and the coupling gains K l , , which, i n turn, depend upon the steady state operation and the configuration of the system. For example, the stronger the t i e between machines, the l a r g e r i s the dynamic i n t e r a c t i o n . 2-4 The Complete System Equations In preceding analysis, each machine i s represented by a fourth order model with the state variables A6., AOJ., AE' . and AE_... Therefore, 1 1 q i f d i the complete system equations of the power system including a l l machines and interti.es can be expressed i n the matrix form [X] = [A][X] + [D] d ' (2.34) where [X] = [A6 l 5 , A6 n, A W l, , Ao)n, A E ^ , , AE^, A E f d l , , A E f d n ] T (2.34a) d i s a disturbance s c a l a r , and [D] i s the disturbance input vector. The system matrix A i s given i n equation (2.35) . 2-5 Output or Measurement Equations The output of the system can be expressed by [y] = [H][X] (2.36) where [y] i s an output vector which contains measureable variables such as powers, voltages, frequencies, currents, e t c . In the study of t h i s AS 1 A S N A fc^ Au>H A E ' q i A E ' q N A E f d 1 A E f d 1 v ) 0 0 0 3 7 7 0 0 0 0 0 0 0 0 I i i i 0 0 0 0 \ X N 0 0 0 0 0 0 0 A * N 0 0 0 0 0 3 7 7 0 0 0 0 0 0 A ^ -K111 M 1 - K l i N M 1 - D 1 M 1 0 0 - K 2 n M 1 - K 2 I N M 1 0 - 0 0 1 1 I 1 1 1 1 I I I 1 1 1 0 \ \ \ 0 1 1 1 1 1 1 I I I 0 0 0 - K 1 N 1 M N _ : - K 1 N N M N 0 0 c - D N M N - K 2 N 1 M N : - K 2 N N M N 0 0 0 A E q i - K 4 1 N - 1 - 1 1 0 0 Tdo1 Tdo1 0 0 0 Tdo1 K 3 11 Td 0 1 K 3 1 N Tdo1 1 1 1 I I I I 1 I I I 0 0 0 I I I 1 1 1 1 1 1 0 \ N \ \ 0 A E ' q N - K A N 1 - K 4 N N 0 0 0 - 1 - 1 0 1 TdoN TdoN TdoN K 3 N1 TdoN K 3NN 0 Y doN A E f d 1 TA-] 1 1 - K A , TA, ^ -1 TAi 0 0 0 TA, \" 6 l 1 0 0 l i 1 1 1 1 1 1 1 I I I 0 0 0 1 1 1 1 ! 1 1 1 1 0 \ \ \ \ 0 A E f d N - K A N T A N N1 -KAN T A N ^ N 0 0 0 \u00E2\u0080\u0094 T A N ^ N N 0 0 - 1 T A N 25 th e s i s , the speed change Aio (which i s r e l a t e d to the frequency), the active power change A p e \u00C2\u00BB a n Q the terminal voltage change AV are included i n the output vector. Other measurements, such as rea c t i v e powers, currents, etc., also can be included i f required. From the block diagram of F i g . 2.5, the measurements of machine #1 can be expressed as follows, Aw \u00E2\u0080\u00A2 AP ei AV t i 0 i l 0 O i O 0 IO 0 K l 11 K l In K5 11 0 0 K 2 n K 2 i n 0 0 K5 l n,0 0 i K 6 n K 6 l n i 0 0 J (2-37) A6X t Aw AE' I q l I i AE' qn AE f d l AE fdn 2-6 Machine Parameters Every machine model requires the following machine parameters, 1) Synchronous machine inductances x,, x and x' J d q d 2) I n e r t i a constant M 3) Damping c o e f f i c i e n t \u00C2\u00A3 4 ) F i e l d open c i r c u i t time constant T l do 5) Voltage regulator gain KA and time constant TA and the following network parameters 26 6) Local load conductance G and suseptance B 7) I n t e r t i e resistance R and reactance X 2-7 Models of Reduced Order for the External Equivalent Machines For the dynamic equivalencing, the external system i s u s u a l l y rep-resented by a much smaller number of machines than the o r i g i n a l , and by a r e -duced-order model with l e s s number of parameters for the external equivalent machine i f i t i s proved s a t i s f a c t o r y . Two examples are shown below: a t h i r d order model i s obtained by assuming a constant f i e l d voltage F i g . 2.6, and a second order model by assuming a constant quadrature axis i n t e r n a l voltage E', F i g . 2.7. q F i g . 2.6 A t h i r d order model for the dynamic equivalent. 27 Fig. 2.7 A second order model for the dynamic equivalent. The number of parameters can also be reduced i f i t i s proved s a t i s f a c t o r y . General considerations of the equivalent model reduction are as follows: 1) R and X of the t i e l i n e of the equivalent machine can be eliminated i f a d i r e c t connection of the equivalent machine to the boundary bus i s assumed. 2) G and B of the l o c a l load of the equivalent machine bus can be eliminated and the e f f e c t usually i s included i n the machine equiva-lent model i t s e l f . 3) A c y l i n d r i c a l rotor machine may be assumed for the equivalent machine, = x^; the saliency e f f e c t i s usually not noticeable. 4) KA and TA may be omitted r e s u l t i n g i n the third-order model, i f the voltage regulator e f f e c t i s not noticeable. 5) KA, TA, T' and x, may a l l be neglected i f the e x c i t a t i o n system do d ef f e c t and armature reaction are not noticeable, r e s u l t i n g i n a second-order model with a constant quadrature axis i n t e r n a l voltage. Table 2.1 Equivalent models with d i f f e r e n t order and. di f f e r e n t number of parameters. MODEL ORDER STATE VARIABLES NUMBER OF PARAMETERS R X G B *d X d X q M ? T ' do KA TA F u l l Fourth-order AS, Ao), AE', AE ' ' q fa 12 ft * * * * * ft ft ft * ft Third-order A6, Aio, AE' q 7 * * * * * * ft Third-order A6, Aw, AE^ 5 * * * ft ft Second-order A6, AOJ 3 ft * ft ho oo 29 3. IDENTIFICATION TECHNIQUE AND ALGORITHM 3-1 Estimation Technique The main objective of t h i s study i s to f i n d the dynamic equiva-lents f o r an external system through i d e n t i f i c a t i o n . The equivalent i s represented by a model of unknown parameters. The f i r s t step i s to f i n d a most s u i t a b l e model f o r the equivalencing, and the second step i s to minimize the errors of the equivalent system responses during i d e n t i f i c a -t i o n as compared with the pre-recorded o r i g i n a l system responses f o r the same disturbance. Many estimation techniques are a v a i l a b l e [37-41] . The technique, however, i s preferred to be simple, unbiased, r e l i a b l e and econ-omical i n computation. 3-1-1 Error Function The equations describing the dynamic equivalent system are ex-pressed s i m i l a r to those of the o r i g i n a l system, equations (2.34) and (2.36), but with a s i m p l i f i e d structure and less parameters. x(a) = A(a) x(a) + Dd (3.1) y(a) = H(a) x(a) (3.2) Where y.(ct) represents the responses of the dynamic equivalent system for the % Oi rb 1J same disturbance d of the o r i g i n a l system, x, y, A and H are a l l functions of a, the equivalent system parameter vector to be i d e n t i f i e d . Therefore, an e r r o r function may be defined by e(a) = y - y(a) (3.3) where y represents the responses of the o r i g i n a l system, which are d i r e c t l y measureable. 30 3-1-2 A Review of Estimation Techniques Motivated by todays' high-speed d i g i t a l computers, modern sequen-t i a l - p r o c e s s i n g approach i s applied to the estimation. Two techniques w i l l be b r i e f l y reviewed, the generalized least-squares estimation and the Markov (unbiased minimum-variance) estimation. The generalized weighted least-squares estimation can be used for the deterministic system. From the error function given by (3.3), which i s a non-linear function of the parameter vector a, a cost function of the quadratic form may be chosen t J.(a) = f / e'(a) W e(a) dt (3.4) t o where W i s a diagonal p o s i t i v e d e f i n i t e weighting matrix, and i t s elements are chosen that the responses i n smaller magnitudes be given larger weight-ing f a c t o r s . The unbiased minimum-variance estimation i s s u i t a b l e to systems with measurement noise and with the noise covariance known a - p r i o r i . This i s a s p e c i a l case of the generalized least-squares estimate, where W equals R - 1, and R i s a p o s i t i v e d e f i n i t e matrix representing the white measurement noise covariance. 3-1-3 Convergence Schemes To guarantee the convergency with minimum computational require-ment and reasonable time, the most popular techniques used i n power engineer-ing are based on the gradient methods, e s p e c i a l l y Newton's methods or steep-est-descent . 31 Taylor expansion of cost function or error function i s used i n Newton-Raphson and Gauss-Newton methods. The parameter updating scheme i s i n the form of a = a 0 - J \" 1 J (3.5) u act a where J i s the gradient of J, and J i s the second der i v a t i v e of J . Since a aa J(a) = - / t f [ ( y - y ( a ) ) ' W (y-y(a))] dt t o then and t J a = \" ; W ( y - ^ a > ^ d t t t ^ J = / f [y* W y\"] dt - / f [y' W (y-y(a))] dt (3.6) aa a a aa J o o The difference between Newton-Raphson and Gauss-Newton methods i s that the l a t t e r neglect the second term of equation (3.6). The computation of the second d e r i v a t i v e J and i t s inverse J - 1 r aa aa are time consuming e s p e c i a l l y f o r large systems with a large number of para-meters to be estimated. Moreover, the convergence i s not guaranteed and an adjustable step s i z e k may be necessary, a = an - k J - 1 J (3.7) u aa a and the value of k cannot be e a s i l y chosen. The steepest descent method adjusts the parameter vector a accord-ing to the scheme a = an - K J (3.8) u a where K i s a diagonal matrix. Depending on the chosen value of K, the con-vergency may be slow but smooth, or fast but o s c i l l a t o r y . Moreover, the estimate may diverge for the large values of K. Since the convergency be-comes slower and slower when the optimum i s approached due to the small value of J , K must increase as J decreases, but must be constrained to a a prevent any divergency of the estimated parameters. The steepest descent with step-size control i s also based on the algorithm a = ag + 6a where 6a = - K J (3.9) a But to guarantee a fast convergency without divergence, the step-size i s constrained by g = (6a) ' G (6a) - C 2 (3.10) where g = 0 (3.10a) G i s a p o s i t i v e d e f i n i t e diagonal weighting matrix, and C i s a constraint value which w i l l not allow the step-size becoming too large that the e s t i -mate diverges or too small to slow the convergence. Introducing a Lagrange m u l t i p l i e r v, the cost function and the constraint equations can be combined i n one equation J*(a) = J(a) + v g (3.11) and r e s u l t i n g J = J + v g = 0 a a a J a = - 2 v G (6a) (3.12) 33 Substituting 6a of equation (3.9) into (3.12) gives K = ( i ) G\"1 (3.13) 2v and 6a = (- \u00E2\u0080\u0094 ) G _ 1 J 2v a Solving equation (3.10) y i e l d s = C [J' G _ 1 J ] \" % (3.14) 2v a a Substituting the value of (~) of equation (3.14) into (3.13) gives the solu-t i o n of K K = C [ J ' G\"1 J ' ] \" * 5 G - 1 (3.15) a a Among these techniques, the steepest descent with step-size con-t r o l i s found superior to the others f o r the following reasons: 1) It does not require the c a l c u l a t i o n of the Hessian matrix J and ^ aa i t s inverse J - 1 . aa 2) The step-size i s automatically adjusted at each step a f t e r the con-s t r a i n t value C has been chosen. 3) A d i f f e r e n t step-size can be chosen for each parameter by adjusting the weighting matrix G. This i s e s p e c i a l l y h e l p f u l i f the cost function i s more s e n s i t i v e to some parameters than others. For the above reasons, the steepest descent with automatic step-s i z e control i s used i n t h i s thesis study. The value of the step-size con-s t r a i n t C also may be chosen according to the cost function behaviour during i d e n t i f i c a t i o n process. C i s reduced whenever the cost function tends to diverge and v i s e versa. 3-2 Algorithm An algorithm to estimate the dynamic equivalent of the external system i s shown i n F i g . 3.1, which consists of the following major steps: 1) Computer simulation of the dynamic response of the i - t h machine of the study system i n the o r i g i n a l system when an i n t e n t i o n a l d i s t u r b -ance i s applied to the same machine. The simulation i s an a l t e r n a -t i v e to actual measurements of a r e a l power system, which are not a v a i l a b l e i n t h i s study. 2) The s e l e c t i o n of the boundary of the study system and the external equivalent model, including the number of equivalent machines, type of models and number of parameters . 3) Computer simulation of the dynamic response of the i - t h machine of the equivalent system for the same i n t e n t i o n a l disturbance. 4) C a l c u l a t i o n and minimization of the cost function and sequentially updating the parameter vector a which has been discussed i n the l a s t s e c t i o n . ORIGINAL SYSTEM CALCULATION OF y = Hx 4 \u00E2\u0080\u00A2 EQUIVALENT MODEL CALCULATION OF y( a ) = fl(a)x(a) CALCULATION OF f ~ T [y-y(a)] w[y-y(a)] dt. UPDATE = \u00C2\u00AB o - k J a NO F i g . 3.1 Estimation algorithm 36 3-3 Computer Simulation of the Dynamic Response of the O r i g i n a l System The o r i g i n a l system equations are generally written x = Ax + D d (3.16) y = Hx + v (3.17) where x i s the state v a r i a b l e vector, d the disturbance, y the response vector, v a white measurement noise of covariance R, and A, D, and H matrices can be calculated based on the information of Chapter 2. These matrices are functions of the system i n i t i a l steady state values calculated from a load flow study [42] . A f t e r the disturbance d i s chosen, the d i f f e r e n t i a l equation (3.16) can be integrated numerically. The response y of the i - t h machine i s com-puted and stored for consecutive i n t e r v a l s of time. The measurement noise v can be added afterwards i f desired. For the simulation of the dynamic response of the o r i g i n a l system due to an i n t e n t i o n a l disturbance, considerations must be given to 1) load bus elimination 2) i n t e n t i o n a l disturbance. 3-3-1 Load Bus Elimination Equations (3.16) and (3.17) are written for an n-machine system with the load buses eliminated. The elimination of load buses w i l l not change the i n i t i a l steady state values of the machines, since the sum of i n -put and output currents at each load bus equal zero. Constant impedance load buses are assumed. Assume a power system of an n machine bus and an m load bus. The current equation may be written 37 y 1 2 _ 0 - ? 2 1 y 2 2 - - V 2 -(3.18) where and I l f r e s p e c t i v e l y , are the voltage vector and current vector of the machines, both of order n. V 2 i s an m vector of load bus voltages. The p a r t i t i o n e d matrices y n , yi2> Y21 a n a Y22 represent the admittances of the system, therefore [111. = t y n l [v 2] (3.19) where y = y - y y - ^ y (3.20) y 11 y 1 1 y 12 \u00E2\u0080\u00A2x22 yZl The elimination of the load buses w i l l not change the values of the currents or the steady state values of the machines, but does change the transmission system. 3-3-2 Intentional Disturbance To i n t e n s i f y the dynamic i n t e r a c t i n g e f f e c t of the external system on the study system f o r parameter i d e n t i f i c a t i o n , an i n t e n t i o n a l disturbance d i s necessary. The disturbance could be mechanical or e l e c t r i c a l . The mechanical disturbance, e.g. a change i n power output of the turbine for a ce r t a i n period of time, i s rather inconvenient to apply. On the other hand, the e l e c t r i c a l disturbance i s easy to control and the time constant i s small. Therefore, a pulsed e x c i t a t i o n voltage of c e r t a i n magnitude and duration within the power system voltage regulation l i m i t i s chosen for t h i s thesis study. 38 3-4 Boundary of the Study System and External Equivalent Model The boundary l i n e which separates the study system and the exter-nal system can be drawn i n a number of ways for the dynamic equivalent study. The number of equivalent machines depends upon the choice of the boundary l i n e . The order and the number of parameters of the external equivalent machine model are chosen so that the i d e n t i f i e d parameters w i l l be unique and the cost function minimum. 3-4-1 Number of Dynamic Equivalent Machines The boundary l i n e determines the number of dynamic equivalent machines of the external system. The retained buses at the boundary l i n e , or the \"boundary buses\", are connected to both sides, the study system and the external system, as shown i n F i g . 3.2(a). Each boundary bus of the equivalent system i s connected to one external equivalent machine as shown i n F i g . 3.2(b). The boundary l i n e may be drawn from geographic or other considera-t i o n s , but a l l parameters of the study system must be known. Boundary^ Line STUDY SYSTEM External System #1 External System #2 \/\/ / / \ \ External System \" #3 Transmission Line (a) , STUDY SYSTEM Equivalent Machine ' ( b ) F i g . 3.2 The configuration of the dynamic equivalent .(a) the o r i g i n a l system, (b) the dynamic equivalent system. Co 3-4-2 The Equivalent Machine Model and Number of Parameters 40 To obtain unique dynamic equivalents for the external system, a su i t a b l e model with an adequate number of parameters for the equivalent must be chosen. An over-represented high-order equivalent model with too many parameters w i l l not give unique estimated parameter values. In t h i s thesis study, equivalent models of d i f f e r e n t orders with d i f f e r e n t number of para-meters are l i s t e d i n Table 2.1, Section 2-7. They are l i s t e d i n the order that less important parameters may be removed f i r s t , and more important and meaningful parameters for the dynamic studies are to remain. F i g . 3.3 shows an algorithm of the reduction process. A f u l l -order equivalent model may be assumed i n the beginning. The v a r i a t i o n of the cost function J with respect to a l l equivalent parameter v a r i a t i o n s are then examined. If the cost function i s i n s e n s i t i v e to some parameters, enter the reduction loop and use the next lower-order model and/or fewer-parameter for the equivalent model u n t i l the cost function J becomes s e n s i -t i v e to a l l remaining parameters. The next step i s to i d e n t i f y these parameters for d i f f e r e n t i n i t i a l guesses. The r e s u l t s should be unique. If not, another model reduction i s necessary. Once unique r e s u l t s are ob-tained, an a d d i t i o n a l model reduction may not a f f e c t the uniqueness, but the cost function may not be minimum. F i g . 3.4 shows the v a r i a t i o n of the cost function with wide v a r i a -t i o n of two parameters i n a study. In t h i s p a r t i c u l a r study, a fourth-order twelve-parameter equivalent machine model was assumed to begin with. The cost function was s e n s i t i v e to the transmission l i n e resistance R, but i n s e n s i t i v e to the voltage regulator time constant TA. The model reduc-t i o n process was c a r r i e d out u n t i l the cost function was s e n s i t i v e to a l l 41 FULL MODEL CHECK J vs. CV NO REDUCTION PROCESS ESTIMATE OC FOR DIFFERENT INITIAL GUESSES NO REDUCTION PROCESS YES SUITABLE MODEL F i g . 3.3 Algorithm f o r the choice of a s u i t a b l e model f o r the equivalent machines. 42 a o 0.0 ( a ) l'.O - r ~i r 2.0 3.0 ZETRI/ZETRIX \u00E2\u0080\u0094l fl.O = 9 -cr 0\u00C2\u00A3O S 0.0 ( b ) ZErRJ/ZETfilX. - 1 4.3 F i g . 3 . 5 The cost function vs. the damping c o e f f i c i e n t of a.dynamic equivalent machine a) a third-order five-parameter model b) a second-order three-parameter model. 43 remaining parameters and the estimated parameters were unique, r e s u l t i n g a third-order five-parameter equivalent model. D e t a i l of th i s study w i l l be included i n Chapter 4. It i s important to mention that the i n s e n s i t i v i t y of the cost function to c e r t a i n parameter does not j u s t i f y the elimination of that p a r t i c u l a r parameter because of two reasons: 1) The cost function i n s e n s i t i v e to ou i n a higher order model may be-come s e n s i t i v e to the same parameter i n a lower order model. F i g . 3.5 shows that the cost function was found i n s e n s i t i v e to the damp-ing c o e f f i c i e n t of the third-order five-parameter model, F i g . 3.5(a), but s e n s i t i v e to the same parameter of a reduced second-order three-parameter model, F i g . 3.5(b). 2) The parameter may be e s s e n t i a l to represent the dynamic cha r a c t e r i s -t i c of the synchronous machine, such as transient reactance or i n e r -t i a constant. 3-5 Estimation of Equivalent Machine Voltages For the simulation of the dynamic response of the equivalent sys-tem, the terminal voltages of the equivalent machines must be estimated when a transmission l i n e are included i n the equivalent model. There w i l l be no changes i n topology, parameters, and steady state values of the study system. Let the currents, voltages and admittances of the study system and the equivalent system r e s p e c t i v e l y , i d e n t i f i e d by the subscripts s and e. The current equation of the o r i g i n a l system may be written ^ss v s e - v s -X e - - y e s v e e - v e -(3.21) 44 and that of the equivalent system y s s y s e I. es ^ee - r V s i - - V (3.22) Since the order of the external system and the number of buses have been greatly reduced, the admittances' matrices of equations (3.21) and (3.22) are d i f f e r e n t , but the off-diagonal elements of the submatrix y'gg are the same as those of y s s . Let the study system current of equation (3.22) be w r i t t e n -I*. = y\" V s se e (3.23) where XS = I s - yss V s If y g e i s a square and nonsingular matrix, then ^ = $-1 I* e J se s (3.24) (3.25) But, i f y s e i s not square matrix and equation (3.23) i s over-determinant, i . e . the number of buses of the external equivalent system i s l e s s than the number of buses of the study system, a l i n e a r least-square estimate of V can be obtained = (y y ) e J se Jse' -1 x * J se s (3.26) When the number of buses of the external equivalent system i s more than the number of buses of the study system (under-determinanat system), a l i n e a r t\j least-square estimate of V e may be obtained by assuming a cost function of the form J = y ( V e - V e ) ' W (V e - V e) (3.27) - nj where V e is chosen value for V e from engineering judgement according to the system topology, and W is a weighting matrix. A constraint g i s used ,-== I* - y v\" = 0 (3.28) & s Jse e Introducing a Lagrange multiplier X to equation (3.27) and minimizing J yields V e = V e - W\"1 y s e A (3.29) Solving equations (3.28) and (3.29) together gives ft = V +W\" 1^ 0 W\"1 ) ( I * - y V ) (3.30) e e ; s e w s e 'se s \u00E2\u0080\u00A2'se e 46 4. CASES STUDIED AND R E S U L T S Two p o w e r s y s t e m s , o n e f i v e - m a c h i n e s y s t e m a n d o n e t h i r t e e n -m a c h i n e s y s t e m a r e u s e d t o t e s t t h e d y n a m i c e q u i v a l e n c i n g t e c h n i q u e s d e -v e l o p e d i n t h e e a r l i e r p a r t o f t h i s t h e s i s . T h r e e d i f f e r e n t t e s t s y s t e m s a r e c o n s i d e r e d : 1) T e s t s y s t e m 1: T h e f i v e - m a c h i n e p o w e r s y s t e m w i t h t w o m a c h i n e s i n c l u d e d i n t h e s t u d y s y s t e m , a n d t h r e e i n t h e e x t e r n a l s y s t e m , F i g . 4.1. 2) T e s t s y s t e m 2: T h e t h i r t e e n - m a c h i n e p o w e r s y s t e m w i t h t w o m a c h i n e s i n t h e s t u d y s y s t e m , a n d e l e v e n i n t h e e x t e r n a l s y s t e m , F i g . 4.2. 3) T e s t s y s t e m 3: T h e same t h i r t e e n - m a c h i n e p o w e r s y s t e m b u t w i t h t h r e e m a c h i n e s i n c l u d e d i n t h e s t u d y s y s t e m , a n d t e n i n t h e e x t e r n a l s y s t e m , F i g . 4.3. I n a l l t h r e e t e s t s y s t e m s , d a s h e d l i n e s a r e u s e d i n t h e f i g u r e s t o s h o w t h e b o u n d a r i e s , w h i c h s e p a r a t e t h e s t u d y s y s t e m s f r o m t h e e x t e r n a l s y s t e m s . G e n e r a l i n f o r m a t i o n o f t h e s e t e s t s y s t e m s a r e s h o w n i n T a b l e 4.1. T a b l e 4.1 G e n e r a l i n f o r m a t i o n o f t h e t e s t s y s t e m s T e s t s y s t e m N u m b e r o f m a c h i n e s Numbe r o f t r a n s m i s s i o n l i n e s N u m b e r o f l o a d s Numbe r o f b o u n d a r y b u s e s Numbe r o f m a c h i n e s o f t h e s t u d y s y s t e m Numbe r o f l o a d s o f t h e s t u d y s y s t e m #1 5 6 5 1 2 2 #2 13 18 12 2 2 1 #3 13 18 12 3 3 2 4-1 P o w e r \" S y s t e m s D a t a T h e p o w e r s y s t e m s d a t a a r e g i v e n i n p e r - u n i t o n t h e 10,000 MVA a n d 500 kV b a s i s f o r t h e f i v e - m a c h i n e p o w e r s y s t e m , a n d o n 100 MVA a n d 345 kV b a s i s o n b u s ( a ) ' f o r t h e t h i r t e e n - m a c h i n e p o w e r s y s t e m , e x c e p t t h e t i m e c o n -s t a n t s i n s e c o n d s , a n d t r a n s m i s s i o n i m p e d a n c e s o f t e s t s y s t e m s 2 a n d 3 a r e i n p e r c e n t a g e . 47 TRANSMISSION LINES R R c R , R, R,. K ab af cf de df e-f 0.120 0.030 0.032 0.150 0.186 0.240 X x \u00E2\u0080\u00A2 X . X, X X ab af cf de df et 2.397 0.597 0.639 2.996 3.710 4.790 LOCAL LOADS G G \u00E2\u0080\u00A2 G G, G a b c ~d e 0.0098 0.0192 1.0884 0.6598 1.2690 B B u B B, B a b c d e -0.0049 -0.0092 -0.5271 -0.3195 -0.6149 F i g . 4.1 Test System 1 r i 0 H 1 0-42 I +J3.26 STUDY Y^STEM 0.41+i4.47 0-35*j3.S6 H 0.04 e 1.87*j12.13 1.95+J13-24 0-28 \u00E2\u0080\u00A2j349 1.23+J8.00 H R - \u00C2\u00A9 v P 0-32 0-07 V +jA.82 +J2.20 Iv. 022 n J *j6.59 .m 0. 60 \u00E2\u0080\u00A2J2-50 ,n 0-47+J5-30 0-71+J4-97 0.+J1.54 0.23 0 +J6-76 ,P .3 0.09+j0.95[ 0-04 jl.17 r 0-92.J10-14 He-0-22 \u00E2\u0080\u00A2j'6.60 0-06 J1.5 0.17 W 0.32 *j47 +J5.Q8 |2-67+j12A6 1.68+j7.76|\u00E2\u0080\u0094{-][ 0.04 1 iH\u00C2\u00AE 0-04 4.74+j23.23-Q.50+J4.90J Ujf\u00E2\u0080\u0094j 0.10 i 2.95! +jl-5 *j1.32 H H \u00C2\u00A9 0-09 w \u00E2\u0080\u00A2j'2.8 Fig. 4.3 Test system 3, 50 Some machines of the test systems are a c t u a l l y coherent equiva-lents of a large number of generating u n i t s . Nevertheless, the test systems serve as good examples f or developing and te s t i n g the dynamic equivalencing techniques. To generate dynamic responses of the o r i g i n a l system with an inten-t i o n a l disturbance, complete data of the en t i r e system are required. How-ever, the data of the external system are not required for the estimation of the dynamic equivalents. 4-1-1 Data of the Five-Machine System The data of transmission l i n e s and l o c a l loads are given i n F i g . 4.1, the machine data i n Table 4.2. In addition, each machine i s assumed to have 10% damping by i t s e l f i n terms o f the damping c o e f f i c i e n t t, o f the nor-malized torque equation, and a s o l i d state e x c i t e r with time constant TA of 0.05 seconds. Table 4.2 Machine data of the five-machine power system. Machine Number MW M X d X d X q Tdo KA K T ) 600 0.460 3.20 16.8 16.6 4.00 50.0 2(H) 900 1.10 1.60 8.18 4.62 7 .76 50.0 3(H) 13000 7.41 0.250 1.28 0.713 7.76 50.0 4(T) 7000 0.280 0.799 4.30 4.23 5.40 50.0 5(T) 14000 . 0.320 0.890 4.65 4.50 5.54 40.0 51 4-1-2 Data of the Thirteen-Machlne System The data of transformers and transmission l i n e s are given i n Figs. 4.2 and 4.3, i n percentage, the load admittances are Y = 5 . 0 0 + j 2 . 1 9 Y = 3.06 + j l . 1 5 a J e J Y- = 3.40 + jO.85 Y. = 3.20 + j l . 2 8 g J Y f = 2.20 + jO.60 Y k = 2.68 + j l . 0 0 Y = 7.70 + j l . 6 0 Y = 358.0 + jO.90 Y = 2.00 + jO.53 e J n J o J Y = 3.97 + j l . 2 5 Y = 3.33 + j l . 1 4 r J s Y = 8.45 + J2.16 u J and the data of the synchronous machines are given i n Table 4.3. A damping c o e f f i c i e n t of 10% i s assumed for each machine. Table 4.3 Machine data of the thirteen-machine power system. Machine Number MW M X d X d X q T l do KA TA 1 960 65 .46 .0208 .1675 .1675 6.7 100 .02 2 600 55.20 .0560 .3030 .2820 5.5 100 .02 3 660 64.56 .0440 .1715 .1023 6.1 100 .02 4 100 16.64 .1269 1.192 1.192 5 .6 18.5 .20 5 135 6.52 .2467 .8667 .5207 3.5 40 .06 6 390 38.36 .0386 .3158 .2624 4.3 160 .03 7 184 27.94 .0789 .4993 .4819 3.3 18.5 .20 8 35047 3500 .0010 .0010 .0010 (constant E') q 9 600 78.00 .0179 .1285 .1230 4.0 50 .02 10 800 68.40 .0579 .2106 .2050 4.8 400 .02 11 140 16.10 .1060 1.540 1.490 7.9 45 .06 12 691 70.42 .0285 .1801 .1376 5.5 (no V.R.) 13 563 56.72 .0392 .3366 .3270 5.5 160 .02 52 4-2 Number of Dynamic Equivalent Machines The number of the dynamic equivalents of the external system i s determined af t e r a boundary l i n e has been drawn; each boundary bus i s to be connected to one external equivalent machine. Since the boundary of te s t system 1 passes through one boundary bus ( f ) , that of t e s t system 2 passes through the buses a and d, and that of the t h i r d t e s t system passes through the boundary buses a, c and d, there are one, two and three dynamic equivalents r e s p e c t i v e l y , as shown i n F i g . 4.4 (a), (b) and (c) . V ~ \u00E2\u0080\u00A2 T ! STUDY (a) Test system 1 , SYSTEM (E1J i (b) Test system 2 (c) Test system 3 F i g . 4.4 Number of the dynamic equivalents. 53 4-3 Choice of the Equivalent Machine Models The algorithm presented i n Section 3-4-2 i s used to select the order and number of parameters of the equivalent machines' models. For the model chosen, the estimated parameters must be unique, and the dynamic per-formances of the o r i g i n a l system and the equivalent system should be i n good agreement. In the algorithm, the equivalent model i s chosen by observing the v a r i a t i o n of the cost function when the parameters v a r i e d . The cost func-t i o n i n t h i s study i s calculated from the deviation of the equivalent system responses as compared with those of the o r i g i n a l system for the same pulsed e x c i t a t i o n disturbance. The same pre-recorded o r i g i n a l system responses due to the pulsed e x c i t a t i o n are used also for parameter i d e n t i f i c a t i o n . 4-3-1 Selection of the Equivalent Model for Test System 1 The equivalent model of test system 1 i s f i r s t assumed to be a f u l l model of fourth-order and twelve-parameter. The model i s given i n Table 2.1 and shown i n F i g . 2.5. However, the cost function i n s e n s i t i v i t y to some of these parameters indicates that the model i s over-represented. Examples were shown i n F i g . 3-4. In the f i g u r e , the cost function i s i n s e n s i t i v e to the e x c i t a t i o n time constant TA, but s e n s i t i v e to the transmission l i n e resistance R. The next reduced model l i s t e d i n Table 2.1 i s then tested. I t i s a third-order and seven-parameter model i n which the transmission l i n e and voltage regulator are eliminated, and a non-salient pole machine i s assumed. Although the model i s adequate according to the cost f u n c t i o n - v a r i a t i o n t e s t , i t does not give unique estimated parameters. A further reduction step i s 54 then e s s e n t i a l to s a t i s f y the uniqueness condition. The reduction achieved by removing the l o c a l load of the equivalent machine r e s u l t i n g i n the t h i r d -order and five-parameter model given i n Table 2.1 and shown i n F i g . 2.6. The model proves adequate to simulate the dynamic int e r a c t i o n s of the exter-nal system and the estimated parameters are unique. 4-3-2 Selection of the Equivalent Model for Test System 2 Test system 2 has two external dynamic equivalents as shown i n F i g . 4.4(b). The same model i s assumed for the two equivalent machines. The rough choice of the equivalent model i s somehow rela t e d to the s i z e of the external system; the larger the external system, the simpler the equivalent model could be. Examination of the data of test system 2 shows that the external system i s quite large. In order to save computation time, i t i s decided that the third-order five-parameter model be used as a s t a r t -ing point since i t i s adequate for test system 1. The cost f u n c t i o n - v a r i a t i o n test shows that the cost function according to t h i s model i s i n s e n s i t i v e to some parameters. Typical examples are shown i n F i g . 4.5. Therefore, one more reduction step i s c a r r i e d out, re-s u l t i n g i n the second-order three-parameter model which i s given i n Table 2.1 and shown i n F i g . 2.7. It i s a constant voltage behind transient reactance which proves to be the most suitable model. 55 F i g . 4.5 The cost function vs. some parameters of test system 2; a t h i r d order five-parameter model. 56 4-3-3 Selection of the Equivalent Model for Test System 3 Test systems 2 and 3 are constituted from the same power system but with d i f f e r e n t boundaries. There are three machines included i n the study system, and three dynamic equivalents are assumed for test system 3, as shown i n F i g . 4.4(c). The same model i s chosen for each equivalent machine. Moreover, the equivalent model chosen for te s t system 2 i s found also adequate for test system 3. 4-4 Estimated Parameters The parameters of the equivalent models are i d e n t i f i e d by the algorithm developed i n e a r l i e r chapters of t h i s t h e s i s . The computer pro-gram i s supplied with the f u l l data of the study system, the topology and the hypothesized model of the equivalent external system, the pre-recorded o r i g i n a l system responses due to disturbance, and the i n i t i a l guesses of parameters. The responses of machine #1 due to a pulsed e x c i t a t i o n voltage of one second duration applied to the same machine are recorded for two seconds. The responses are used for the choice of equivalent model and the i d e n t i f i -cation of the unknown parameters. For the parameters to converge to t h e i r optimum values, the number of i t e r a t i o n s v a r i e s enormously and depends very much upon both the step s i z e and the i n i t i a l guess. Tables 4.4, 4.5 and 4.6 show the estimated parameters of test systems 1, 2 and 3, respectively, f o r three d i f f e r e n t i n i t i a l guesses for each system. Table 4.5 shows the parameters of the two equivalent machines while Table 4.6 includes the parameters of the three equivalent machines. 57 It i s c l e a r from the tables that a l l the parameters i n each case study con-verge c l o s e l y to the same values regardless of the i n i t i a l guesses. I t i s also shown that d i f f e r e n t boundaries y i e l d s d i f f e r e n t equivalents f o r the same o r i g i n a l system. Table 4.4 I d e n t i f i e d equivalent parameters of test system 1 ( I n i t i a l guesses are i n parenthesis) . Pulsed, excitation.on machine #1 Case I Case II Case I I I 9 .950 (10.0) 9.939 (5.00) 9.989 (20.0) CE1 0 .1207 (0.25) 0.1212 (0.20) 0.1195 (0.30) X d E l 0 .5725 (0.25) 0.5713 (0.15) 0.57 61 (0.60) X d E l 2 .198 (1.20) 2.183 (0.80) 2.210 (2.00) T' do E l 5 .360 (5.50) 5.157 (3.50) 5.400 (7.80) Table 4.5 I d e n t i f i e d equivalent parameters of test system 2. Pulsed e x c i t a t i o n on machine #1 Case I Case II Case III hi X d E l 39.48 (50.0) 1.055 (0.50) 0.01352 (0.20) 39.25 (60.0) 1.051 (1.0) 0.01311 (0.05) 39.52 -(80.0) 1.058 (1.5) 0.01341 (0.1) ^ 2 CE2 y' dE2 154.8 (50.0) 2.989 (0.5) 0.1868 (0.6) 154.15 (130.0) 2.995 (2.5) 0.1871 (0.2) 154.41 (100.0) 2.973 (2.0) 0.1861 (0.4) 58 Table 4.6 I d e n t i f i e d equivalent parameters of test system 3. Pulsed e x c i t a t i o n on machine #1 Case I Case II Case III hi x d E i 45.89 (30.) 1.886 (.70) .0570 (.30) 45.82 (75.) 1.882 (1.4) .0572 (.04) 45.88 (50.) 1.901 (2.5) .0571 (.08) ?E2 XdE2 137.0 (50.) 2,519 (.80) .3411 (.40) 137.2 (80.) 2.523 (1.2) ;3409 (.10) 137.1 (100) 2.517 (2.0) .3397 (.25) ^ 3 CE3 XdE3 117.3 (70.) 1.989 (.30) .4363 (.50) 117.5 (85.) 1.995 (.60) .4358 (.30) 117.4 (100) 1.991 (2.0) .4364 (.40) 4-4-1 Cost Function F i g s . 4.6, 4.7 and 4.8, res p e c t i v e l y , show the cost function r a t i o versus the equivalent parameters of test systems 1, 2 and 3. In the figures JMIN represents the optimum value of the cost function and the asterisked parameter i s the i d e n t i f i e d value. A l l minima of the cost functions cor-respond to the best f i t of the dynamic responses of the equivalent system with the i d e n t i f i e d parameters' values to those of the o r i g i n a l system. For some parameters, such as M and T ^ of test system 1, the con-vergence i s f a s t e r when the i n i t i a l guesses are on the small side. (3 . cr O \u00E2\u0080\u00A2 o (_) F i g . 4.6 Cost function vs. equivalent parameters of tes t system 1 . i i i i i i i i i i i i i i i i i i ) i i i i i \u00C2\u00B0-\u00C2\u00B0 K\u00C2\u00B0 '^WtV* 4\"\u00C2\u00B0 5\"\u00C2\u00B0 6-\u00C2\u00B0 (a) Equivalent machine #1 (b) Equivalent machine #2 i g . 4.7 Cost function vs. equivalent parameters of test system 2. (a) Equivalent machine #1 (b) Equivalent machine #2 4.8 Cost function vs. equivalent parameters of t e s t system 3. a M3/M3X CM (c) Equivalent machine # 3 4-4-2 Machine Responses Due to Pulsed E x c i t a t i o n 63 F i g s . 4.9, 4.10 and 4.11 show the responses of machine #1 due to the pulsed e x c i t a t i o n voltage on the same machine for test systems 1, 2 and 3, r e s p e c t i v e l y . The equivalent system responses are shown i n dashed l i n e s which are based on the estimated r e s u l t s of case I of each test system, as compared with those of the o r i g i n a l system which are shown i n s o l i d l i n e s , for the same i n t e n t i o n a l disturbance. In a l l these studies, the pulsed ex-c i t a t i o n i s applied for one second, the dynamic responses of the o r i g i n a l system are recorded for two seconds, and the system s e t t l e s down i n about four seconds. A l l r e s u l t s indicate that the simulated responses of the dynamic equivalent systems and those of the o r i g i n a l systems are very close i n both magnitude and frequency. 4-5 Parameter Estimation Using Other Machines' Responses The responses of machine / / l due to an i n t e n t i o n a l disturbance on the same machine are used for the i d e n t i f i c a t i o n i n Section 4-4. However, the dynamic equivalent should be i d e n t i f i e d by machine responses anywhere i n the study system, and the uniqueness of the external equivalent system should not be affected i f enough measurements and suitable equivalent model are used Table 4.7 shows four more case studies f or the three t e s t systems. The i n t e n t i o n a l disturbance i s applied to a machine other than / / l i n the study systems, and the dynamic responses are recorded from the same dlsr. turbed machine. o F i g . 4.9 Responses of test system 1 due to 10% pulsed ex-c i t a t i o n voltage. F i g . 4.10 Responses of test system 2 due to 5% pulsed ex-c i t a t i o n voltage. Fig.' 4 .11 Responses of to 5% pulsed test system 3 due e x c i t a t i o n voltage. 66 Table 4.7 Case studies f o r parameter i d e n t i f i c a t i o n from other machines' responses. Machine responses used for estimation Machine #2 Machine #3 Test system \" Case Intentional disturbance Case Intentional disturbance 1 IV 7% U E - -2 V 5% U E -3 VI 5 % U E VII 5% U E Tables 4.8 and 4.9, re s p e c t i v e l y , show the r e s u l t s of cases IV and V. The f i r s t columns of Tables 4.4 and 4.5, respectively, are also included i n each table for comparison. Table 4.1 shows the r e s u l t s of cases VI and VII. The l a s t column from Table 4.6 i s also included for comparison. Table 4.8 Estimated parameters from machine #2 responses of te s t system 1. Pulsed e x c i t a t i o n on machine //2 Pulsed e x c i t a t i o n on machine #1 Case IV Case I ME1 9.154 (10.0) 9.95 5E1 0.1449 (0.25) 0.1207 X d E l 0.5833 (0.25) 0.5725 X d E l 2.186 (1.20) 2.198 T' doEl 5.394 (5.50) 5.360 Table 4.9 Estimated parameters from machine #2 responses of test system 2. Pulsed e x c i t a t i o n on machine #2 Pulsed e x c i t a t i o n on machine ill Case V Case I \"EI 39.94 (50.0) 39.48 hi 1.069 (0.50) 1.055 . XdEl* 0.0141 (0.20) 0.01352 ^ 2 155.12 (50.0) 154.8 h2 2.992 (0.5) 2.989 XdE2 0.201 (0.6) 0.1868 Table 4.10 Estimated parameters from machine #2 and machine #3 responses of test system 3. Pulsed e x c i t a t i o n on machine #2 Pulsed e x c i t a t i o n on machine #3 Pulsed e x c i t a t i o n on machine #1 Case VI Case VII Case I \l 45.99 (40.) 46.12 (40.) 45.89 hi 1.811 (1.5) 1.790 (1.5) 1.886 X d E i .0568 (.10) .0560 (.10) .057 ^ 2 137.7 (130) 138.1 (130) 137.0 ^E2 2 .531 (2.0) 2.498 (2.0) 2.519 XdE2 .3382 (.30) .3321 (.30) .3411 ^ 3 116.9 (HO) 118.1 (110) 117.3 CE3 2.011 (2.5) 2.105 (2.5) 1.989 XdE3 .4410 (.40) .4215 (.40) .4363 68 A l l i d e n t i f i e d values of the parameters are very close and the dynamic equivalents are unique regardless of which machine responses of the study system are used for i d e n t i f i c a t i o n . The three measurements, speed, power and voltage, are adequate and no a d d i t i o n a l measurements are required. F i g s . 4.12, 4 .13, 4^14 and 4.15 compare the responses of the o r i g i -n a l system ( s o l i d l i n e s ) and dynamic equivalent systems (dashed l i n e s ) for case studies IV, V, VI and VII, r e s p e c t i v e l y . 4-6 Parameter Estimation with D i f f e r e n t Intentional Disturbances The pulsed e x c i t a t i o n s i g n a l used for i d e n t i f i c a t i o n i s considered to be very r e a l i s t i c and always recommended whenever i n t e n t i o n a l disturbance i s required, e s p e c i a l l y for on-line t e s t s . However, other types of d i s -turbances also s h a l l be investigated. In any case, the nature of the d i s -turbance should not a f f e c t the uniqueness of dynamic equivalents. In t h i s study, a ramp mechanical input torque s i g n a l , as shown i n F i g . 4.16, was applied on machine #1 of test system 1. The estimated para-meters due to t h i s disturbance are shown i n Table 4.11, together with those estimated by the pulsed e x c i t a t i o n s i g n a l of Table 4,4. F i g . 4.17 shows the responses of the machine due to t h i s mechanical torque disturbance, the s o l i d l i n e i s for the o r i g i n a l system and the dotted l i n e for the dynamic equiva-l e n t . Similar r e s u l t s were obtained from other test systems, but are not recorded here for b r e v i t y . F i g . 4.12 Responses of machine #2 of tes t system 1. F i g . 4.13 Responses of machine ill of t e s t system 2. F i g . 4.14 Responses of machine #2 of test system 3. F i g . 4.15 Responses of machine #3 of t e s t system 3. 71 A T M \" (RU.) 1-0 Time (sec.) F i g . 4.16 Ramp mechanical torque disturbance. Table 4.11 Parameter estimation from the response of test system 1 due to ramp torque disturbance. Ramp torque on machine #1 Pulsed e x c i t a t i o n on machine #1 Case I 9.81 (10.0) 9.95 5 E l 0.1235 (0.25) 0.1207 XQEI 0.5731 (0.25) 0.5725 X d E l 2.129 (1.20) 2.198 T' doEl 5.159 (5.50) 5.360 The discrepancy of the responses begin to show a f t e r two seconds. A l l the responses, however, converge u l t i m a t e l y . 4-7 Measurement Noise E f f e c t Random measurement noise i s added to the response of machine #1 due to the pulsed e x c i t a t i o n i n t e n t i o n a l disturbance. The noise i s assumed to be white [5], i t s magnitude i s within 10% of the response maximum value. The unbiased minimum variance estimate presented i n Section 3-1-2 i s used to i d e n t i f y the parameters. Tables 4.12, 4.13 and 4.14 ahow the estimated F i g . 4.17 Responses of test system 1 due to mechanical torque disturbance. 73 parameters of such cases of test systems 1, 2 and 3, r e s p e c t i v e l y . In the l a s t column of each table, the i d e n t i f i e d parameter values given i n Tables 4.4, 4,5 and 4.6, r e s p e c t i v e l y , obtained from the determinestic systems are also included. The r e s u l t s show that the added white measurement noise does not bias the estimate. Table 4.12 E f f e c t of measurement noise on the estimated parameters of test system 1. Pulsed e x c i t a t i o n on machine #1' with measurement noise Without noise Case I 9.962 (10.0) 9.95 hi 0.1199 (0.25) 0.1207 x d E i 0.5770 (0.25) 0.5725 X d E l 2.191 (1.20) 2.198 T' do E l 5.292 (5.50) 5.360 Table 4.13 E f f e c t of measurement noise on the estimated parameters of te s t system 2. Pulsed e x c i t a t i o n on machine #1 with Without noise measurement noise Case I 39.35 (50.0) 39.48 -CE1 1.06 (0.50) 1.055 X d E l 0.01335 (0.20) 0.01352 ^ 2 154.6 (50.0) 154.8 hi 2.981 (0.50) 2.989 XdE2 0.1863 (0.60) 0.1868 74 Table 4.14 E f f e c t of measurement noise on the estimated parameters of test system 3. Pulsed e x c i t a t i o n on machine #1 with Without noise measurement noise Case I \"EI 45.92 (80.) 45.89 ?E1 1.891 (3.0) 1.886 x d E i .0569 (.50) .0570 136.9 (30.) 137.0 CE2 2.511 (3.0) 2.519 dE2 .3421 (.60) .3411 117.7 (30.) 117.3 ?E3 1.995 (3.0) 1.989 XdE3 .4568 (.80) .4363 The responses of machine #1 of t e s t systems 1, 2 and 3 due to the pulsed e x c i t a t i o n are shown i n Figs. 4.18, 4.19 and 4.20, r e s p e c t i v e l y . The o r i g i n a l measurements with noises are shown by the s o l i d l i n e s and the responses of the estimated dynamic equivalents by dashed l i n e s . 4-8 Off-Line and On-Line I d e n t i f i c a t i o n For on-line applications, the dynamic equivalent must be i d e n t i -f i e d quickly with minimum i t e r a t i o n s , and as frequently as required. The computational time varies widely and depends strongly on the following f a c -tors : 1) The number of dynamic equivalent machines. 2) The order and number of unknown parameters of the equivalent machine model. F i g . 4.18 Responses of test system 1 with noise added to the o r i g i n a l system responses. F i g . 4.19 Responses of t e s t system 2 with noise added to the o r i g i n a l system responses. F i g . 4.20 Responses of test system 3 with noise added to the o r i g i n a l system responses. 3) The step s i z e . 4) The i n i t i a l guess of the unknown parameters. The number of dynamic equivalent machines i s decided by how many machines of the study system are of great concern. The model of equivalent machines are chosen from o f f - l i n e i d e n t i f i c a t i o n so that the estimated parameters are unique. The step siz e cannot be f r e e l y chosen and must be constrained as suggested i n the e a r l i e r part of t h i s t h e s i s . Therefore, the most important factor for reducing the computational time for on-line i d e n t i f i c a t i o n i s the i n i t i a l guess. The number of i t e r a t i o n s can be greatly reduced i f the i n i t i a l guess i s close to the f i n a l value. Based on these considerations, i t i s suggested that the i d e n t i f i -cation may proceed i n two steps. Step 1, Off-Line I d e n t i f i c a t i o n : When no p r i o r information of the external equivalent system i s a v a i l a b l e . I t i s a r e l a t i v e l y long computa-t i o n a l process i n which the topology of the external system i s hypothesized and the equivalent model i s chosen. The i n i t i a l guess of the equivalent model parameters could be f a r from the f i n a l value. Step 2, On-Line I d e n t i f i c a t i o n : The computational time process c could be very short a f t e r step 1; the o f f - l i n e i d e n t i f i c a t i o n . The dynamic equivalent can be frequently updated, where the i d e n t i f i e d parameters from the o f f - l i n e process can be used as the i n i t i a l guess, even i f there i s r e l a t i v e l y major change i n the external system. To demonstrate the f e a s i b i l i t y of on-line i d e n t i f i c a t i o n , major change i s assumed i n the external system of t e s t system 1, the transmission l i n e connecting machine #4 to the boundary bus i s assumed to be double l i n e , F i g . 4.21 Responses of test system 1 a f t e r updating the dynamic equivalent. 79 one of them i s tripped o f f due to a three phase f a u l t . The dynamic equiva-lent of the external system i s no longer the same as p r e f a u l t . To update , the dynamic equivalent parameters, the pref a u l t estimated values are used as the i n i t i a l guesses. Table 4.15 shows the updated parameters, i t takes only s i x i t e r a t i o n s f o r the i d e n t i f i c a t i o n . The responses of machine / / l due to pulsed e x c i t a t i o n are shown i n F i g . 4.21, The s o l i d l i n e i s for the o r i g i n a l system response while the dashed l i n e i s for the dynamic equivalent system response. Table 4.15 Parameter updating of test system 1 Parameter values before t r i p p i n g o f f After t r i p p i n g o f f 9.95 9.456 CE1 0.1207 0.1323 X a E l 0.5725 0.5539 X d E l 2.198 3.942 T' doE 5.36 5.605 4-9 Conclusions of Chapter 4 From the case studies i n t h i s chapter, the following conclusions may be drawn: 1) The dynamic equivalent can be uniquely estimated with proper choice for the equivalent model. 2) Regardless of the l o c a t i o n of the disturbed machine i n the study system, the external dynamic equivalent i s unique. 3) Different types of disturbances can be used to estimate the dynamic equivalent, and the r e s u l t s are unique. The white measurement noise does not create problems for the i d e n t i f i c a t i o n . The load fluctuations i n power systems can be ignored when the e f f e c t of an i n t e n t i o n a l disturbance i s much larger than these f l u c t u a t i o n s . On-line i d e n t i f i c a t i o n can be expedited when the parameter values from o f f - l i n e i d e n t i f i c a t i o n are used as the i n i t i a l guess. The dynamic equivalent can be frequently updated. 81 5. VERIFICATION OF THE VALIDITY OF THE EQUIVALENTS In Chapter 4, the structure of the equivalent system has been hy-pothesized, and i t s parameters have been i d e n t i f i e d . Although the machine responses of the o r i g i n a l system and the dynamic equivalent system due to the i n t e n t i o n a l disturbance are i n good agreement, more v e r i f i c a t i o n s to show the v a l i d i t y of the dynamic equivalent are necessary. Three more proofs of the v a l i d i t y of the estimated dynamic equiva-lent are provided i n t h i s t h e s i s : 1) The K,.., , K .. constants of the study system machines are almost I n 611 equal whether computed from the o r i g i n a l or the equivalent system parameters. 2) The e l e c t r i c torques of the study system machines simulated from the o r i g i n a l and dynamic equivalent systems are i n good agreement. 3) The study system machines' responses of the o r i g i n a l and dynamic equivalent systems due to a severe f a u l t within the study system are i n good agreement. A three-phase f a u l t i s tested on each machine bus of the study system for a l l cases. A l l these are aiming to prove the equivalents estimated by the techniques developed i n t h i s thesis are v a l i d . However, i n power industry the K constants of the o r i g i n a l system cannot be calculated unless f u l l i n -formation of the external system i s known, and the three-phase s h o r t - c i r c u i t t e s t i s not recommended for engineering p r a c t i c e . 82 5.1 Model Constants of the Study System The constants of a machine model i n multi-machine power system depend upon the configuration and parameters of the e n t i r e system. When the external system i s replaced by an equivalent, the state v a r i a b l e s and s e l f constants of the model w i l l be a l t e r e d , unless the external equivalent sys-tem has the same e f f e c t as the o r i g i n a l external system. The measured variables of the i - t h machine used for i d e n t i f i c a t i o n are speed Acu., power AP . and voltage AV.. The state variables of the i - t h l e i l machine are either d i r e c t l y measured or function of the measurements. Since the o r i g i n a l and the dynamic equivalent systems responses are in good agree-ment, so w i l l be the state variables Aw. , AS., AE,.,. and AE'. of the i - t h l l f d i q i machine of the study system, because Or Aw. = Au. ; measured va r i a b l e I l A6. = A6. ; AS. = Au./S X 1 1 1 AE^. - AE^. ; AE^ V. = -AV. KA./(1+S TA.) fdx fdx fdx x x x AE'. - AE'\u00E2\u0080\u00A2 qx qx Some elaboration of the l a s t equation i s necessary. Since P . = [ E 1 . V'. s i n S./x' ] + [(x'.-x .) V 2 s i n 2 6./(2 x'. x .)] ex qx x x dx dx qx x l dx qx AP . = Ci AE'. + Co AV + Cq AS ex 1 qx ^ x 3 x and Therefore, % % <\i a. AP . = C i AE'. + C 2 AV. + Co AS. ex 1 qx x \u00C2\u00B0 x AE'. - AE'. qx qx Ci, C2 and C3 are functions of the i - t h machine parameters and steady state values. The machine's s e l f constants K l . . , , K6.. have been found i n x i xx Chapter 2, equation (2.32). For s i m p l i c i t y , these constants can be rewrit-ten assuming a c y l i n d r i c a l rotor machine where x, equals x and E' equals d q q the voltage behind transient reactance [30], as follows: K l . . = E'. E E'.Y..S.. 1 1 q i ^ \u00C2\u00B1 qj IJ IJ K2.. = I . + E ' . Y . . C . . xx qx qx xx xx K3.. = [1 - (x -x' ) Y.. S..]\" 1 xx dx dx xx xx K4.. = (x -x' ) E E'. Y.. C.. n dx dx j\u00C2\u00B1 qj xj xj (5.1) K5.. = -x' [V,. E E'. Y.. S , + V . E E'. Y.. C..]/V. i i dx dx ^ \u00C2\u00B1 qj xj xj qx ^ \u00C2\u00B1 qj xj xj x K6.. = [V,. x' Y.. C . + V . (1 + x' Y.. S..)]/V. xx dx dx xx xx qx dx xx xx x where Y_\u00E2\u0080\u009E can be found from (2.17a). Since current equations of the i - t h machine can be expressed by the form I \u00E2\u0080\u009E = E' . Y. . S . . + E E'. Y. . S. . dx qx xx xx j\u00C2\u00B1 qj xj xj I . = E'.Y..C..+ E E'.Y..C.. qx qx xx xx ^ \u00C2\u00B1 qj xj xj (5.2) K l . . , , K6.. can also be rewritten xx xx K l . = E'. ( I , . - E \ Y. . S. .) (5.3a) xx q i d i qx xx xx K2., = I . + E'. Y. . C. . (5.3b) xx qx qx xx i x K3.. = [1 - (x -x* ) Y.. S . . ] \" 1 (5.3c) xx dx dx xx xx K4.. = (x -x'-)(I . - E ' . Y.. C.) (5.3d) xx dx dx qx qx xx xx K5.. = x!; [V,. E'. Y.. S.. + V . E'.Y.. C.. - P .]/V. ix dx dx qx xx xx qx qx xx xx ex x (5.3e) K 6 . . = [x' V,. Y.. C . + V . (1-x' Y.. S..)]/V. (5.3f) xx dx dx xx xx qx dx xx xx x where P . = V I + V . I . e i dx dx qx qx As shown from equations (5.3), K I ^ J , K6_\u00E2\u0080\u009E of the study system machine of the o r i g i n a l and dynamic equivalent systems w i l l agree i f the following condition i s s a t i s f i e d . * j 3 i i Y. . e xx Y. . e ix ( 5 . 4 ) or and Y. . S . - Y S xx xx xx xx ( 5 . 5 ) Y.. C.. - Y C XX XX XX XX (5.6) Y of equation (2.17a) i s composed of impedances and transient reactances of the e n t i r e o r i g i n a l system, and Y composed of those of the ent i r e equivalent system. The cond i t i o n of equation (5.5) can be obtained by comparing the speed-torque r e l a t i o n of the i - t h machine i n the two systems, F i g . 5.1. Since AUK and AT_^ match AUK and AT_^, respectively, D^ and must be i n good agreement, which i s apparent from a comparison of F i g . 6.1 (a) and (b). A T ; 1 A T j 1 M j S + Dj Mj S* Dj A ^ i (a) O r i g i n a l system (b) Dynamic equivalent system F i g . 5.1 Speed-torque r e l a t i o n Since N.2 = (2 ?.) 2 u> K l . . M. X O XX X (5.7) 85 and o 'Xj Df = (2 ? . ) 2 OJ K l . . M. 1 1 O 11 1 (5.8) hence K l . . = K l . . i i i i 'Xj 'XJ Therefore, from (5.3a), Y.. S.. equals Y.. S.., consequently (5.5) i s v a l i d . i i i i ^ i i i i H The condition of equation (5.6) can also be obtained by comparing the i n t e r n a l voltage A E ^ of the two systems, F i g . 5.2. Since the voltage regulator e f f e c t i s usually much larger than the other dynamic i n t e r a c t i n g A E a i HJ (a) O r i g i n a l system A E f d . - K A j f AV t. i + US TAi K 5 j j (b) Dynamic equivalent system F i g . 5.2 Internal voltage representation e f f e c t through K3 . and K4 ., the i n t e r n a l voltage equations of the two sys-13 13 tems can be approximated by AE'. = K3.. ( A E r J . - K4.. A6.)/(l+S T\ . K3..) q i x i fdx xx x dox . xx AE *\j f\j ^ K3.. ( A E \u00C2\u00A3 J . - K4.. A6.)/(l+S T\ . K3..) xx fdx xx x dox xx (5.9) (5.10) Since K3__ approximately equals K 3 ^ according to equations (5.3c) and (5.5), K4_\u00E2\u0080\u009E approximately equals K 4 ^ according to equations (5.9) and (5.10), hence Y.. C.. approximately equals Y.. C.. according to equation xx xx . xx xx (5.3d). Since the r e l a t i o n given by (5.5),and (5.6) are proven to be true, a l l the model constants K l . . , , K6.. (5.3) of the o r i g i n a l and dynamic xx xx equivalent systems must be i n good agreement. Tables 5.1, 5.2 and 5.3 show the ca l c u l a t e d r e s u l t s of the study system constants, of the o r i g i n a l system and the dynamic equivalent system, f o r test systems 1, 2 and 3, re s p e c t i v e l y . Table 5.1 Constants of test system 1 Machine #1 Machine #2 Or i g i n a l System Dynamic Equivalent System O r i g i n a l System Dynamic Equivalent System K l 0.0809 0.0810 0.1440 0.1470 K2 0.1201 0.1210 0.13 02 0.1305 K3 0.2311 0.2313 0.4205 0.4200 K4 1.6013 1.6033 0.7100 0.7150 K5 0.0590 0.0580 -0.0045 -0.0051 K6 0.2111 0.1980 0.6419 0.6421 87 Table 5.2 Constants of test system 2 Machine #1 Machine ill O r i g i n a l System Dynamic Equivalent System O r i g i n a l System Dynamic Equivalent System Kl 9.7551 9.9010 5.1271 5.1501 K2 11.0510 11.0071 2.6520 2.5613 K3 0\2682 0.2711 0.2431 0.2405* K4 1.4721 1.5051 0.5101 0.5031 K5 0.1165 0.1201 0.0069 0.0091 K6 0.5711 0.5645 0.3112 0.2945 Table 5.3 Constants of te s t system 3 Machine ill Machine ill Machine #3 O r i g i n a l System Dynamic Equivalent System O r i g i n a l System Dynamic Equivalent System O r i g i n a l System Dynamic Equivalent System K l 9.7551 9.6212 5.1271 5.1351 6.931 6.2212 K2 11.0510 11.0023 2.6520 2 .5612 8.5310 8.1121 \"K3 0.2682 0.2711 0.2431 0.2461 0.4205 0.4611 K4 1.4721 1.4451 0.5101 0.4962 0.8121 0.7953 K5 0.1165 0.1185 0.0069 0.0089 -0.018 -0.028 K6 0.5711 0.5645 0.3112 0.3182 0.4781 0.5421 The K.. constants of the study system machines of the o r i g i n a l sys-tem are i n good agreement with those of the dynamic equivalent systems. 5-2 The E l e c t r i c Torques The e l e c t r i c torque of the i - t h machine i n a multi-machine power system consists of two parts, e l e c t r i c torque generated by the machine i t s e l f , and dynamic i n t e r a c t i n g torque due to other machines i n the system. Each torque has two components, the synchronizing component i n phase with the 88 torque angle, and damping component i n phase with the speed. The p e r f o r -mance and s t a b i l i t y of the i - t h machine depend very much upon these torque components; the e l e c t r i c torques of the study system machines of the o r i g i -n a l and dynamic equivalent systems must be i n good agreement;, i . e . the external equivalent system has the same dynamic i n t e r a c t i n g e f f e c t on the study system as the o r i g i n a l external system. The equations for the e l e c t r i c torque components are included i n the Appendix. Computer simulated r e s u l t s of the study system e l e c t r i c torque components of the o r i g i n a l and dynamic equivalent systems are given i n Tables 5.4, 5.5 and 5.6 for t e s t systems 1, 2 and 3, r e s p e c t i v e l y . The values are simulated f o r the systems o s c i l l a t i o n frequency of about 1.0 Hz. 5-3 Three-Phase Fault Test Although the dynamic equivalent system proves v a l i d for the inten-t i o n a l disturbance used for i d e n t i f i c a t i o n , i t i s necessary to be given a more severe t e s t . The contingency chosen for the t e s t i s a three-phase short c i r c u i t f o r s i x cycles i n the study system at a d i f f e r e n t l o c a t i o n each time. The dynamic responses of the equivalent system from computer simulation are compared with those of the o r i g i n a l system. Also included for comparison are the responses with the external system approximated by i n f i n i t e buses. When the f a u l t i s assumed at s p e c i f i c machine bus i n the study system, not only the responses of t h i s machine are observed but also the responses of a l l other machines i n the study system. The tests are i d e n t i f i e d i n Table 5.7. The t e s t r e s u l t s are shown i n F i g s . 5.3 to 5.19; the o r i g i n a l sys-tem response i s the s o l i d l i n e , the dynamic equivalent response i s the dotted l i n e and the i n f i n i t e bus equivalent response i s the dashed l i n e . Table 5.4 E l e c t r i c torques of test system 1 Machine #1 Machine #2 Or i g i n a l System Dynamic Equivalent System O r i g i n a l System Dynamic Equivalent System Self-damping torque 0.02795 A6X 0.02774 LS1 0.02774 AS 2 0.02699 A6 2 Interacting damping torque 0.00359 A6X 0.00313 ASX 0.00824 A<52 0.00783 AS2 Total damping torque 0.03154 LSi 0.03087 A5 X 0.03598 AS2 0.03482 AS2 Self-synchronizing torque 0.07097 AS! 0.07233 A5X 0.1414 A. T' . K3. . ) z j l i i i i l doi i i The synchronizing torque of the e l e c t r i c a l loop i s usually nega-t i v e , while the damping torque i s p o s i t i v e . However, the e f f e c t i s reversed i f K5.. i s negative and (K5.. + K4../KA.) i s \"less than zero, which may hap-l i i i i i l pen to systems with high impedances and heavy loadings. The t o t a l s e l f synchronizing and damping torques of i - t h machine res p e c t i v e l y , are ATS.. = ATS,,.. + ATS,, (A.3) ix Mxx Exx ATD. . = ATDW. . + ATD\u00E2\u0080\u009E . . xx Mxx Exx (A.4) A2 Dynamic Interacting E l e c t r i c Torque Equations F i g . A.3 shows the loop of torque contribution to machine i by other machines through t h e i r mechanical loops . A $ : K1\u00E2\u0080\u009E. r 1 CJL)O s K 1 i j J MECHANICAL LOOP OF MACHINE j ATi m y * F i g . A.3 Interaction of the mechanical loop of machine j on machine i . Since the i - t h machine i s disturbed, the dynamic i n t e r a c t i o n be-tween any other two machines i s small and can be ignored. The e l e c t r i c torque contribution to machine i by machine j through i t s mechanical loop i s given by 108 where K l . . K l . . to2. ' AT = - -7 20__ A 6 mij ... K l . . S 2 + 25. w . S + co2. i J3 J nj nj . co K l . . D. w \u00E2\u0080\u00A2 = \u00E2\u0080\u0094 ^ , and C. = -rrr^ ni M. i 2M. .co . 3 J n3 Substituting S by ju> and separating the synchronizing and damping torque components gives K l . . K l . . (co2. - co?) A T S m i j . j . K l . . ^ [(co 2. - co 2) 2 + (2C. co . ) 2 ] A 6 i K l . . K l . . 2\u00C2\u00A3. co . A T D m i j -. K l , , ^ [(co 2. - c o ^ + n - ( 2 c . co . ) 2 ] A * i The synchronizing torque contribution ATS .. i s usually negative, mij while the,damping torque contribution ATD .. i s p o s i t i v e . However, i f mij machine i has high i n e r t i a that co . < co., the e f f e c t i s reversed. nj I The dynamic i n t e r a c t i n g torque con t r i b u t i o n to machine i by a l l other machines through t h e i r mechanical loops are ATS\u00E2\u0080\u009E.. = E ATS .. Mij mij ATDW.. = E ATD . . Mil . , . mil There i s also torque contribution to the i - t h machine by other machines through t h e i r e l e c t r i c a l loops; F i g . A.4. KA. K2.. K3.. [K5.. + K4.. (1+S TA.)/KA.] T = _ _ J U J J l i 11 J J _ A 6 e i j ... (1+S T' . K3..)(l+S TA.) + KA. K3.. K6.. i iri doj 33 3 3 33 33 Assuming that T ^ 0 j > ; > IA. , and replacing S by jco_^, the two torque components are ELECTRICAL LOOP OF MACHINE j A.4 I n t e r a c t i o n of the e l e c t r i c a l loop of machine j on machine 110 KA. K2.. K3.. (K5.. + K4../KA.)(1 + KA. K3.. K6..) e i j (1 + KA. K3 . K6..) z + (to. T' . K3..) z \u00C2\u00B0i 3 33 33 i do3 21 KA. K2.. K3 2. (K5.. +K4../KA.) co. T' . A T T ) = 3 12 3J l i _ l i 3 i doj_ A ( S e i j (1 + KA. K3 . K6..) 2 + (oo. T* . K3..) z i 3 33 33 i doj 33 For (K5.. + K4../KA.) > 0, the synchronizing torque ATS .. has negative 3 1 3 1 3 6 ^ E I : ] e f f e c t and the damping torque T ^ e i j l a a s p o s i t i v e e f f e c t , and vice versa f o r (K5.. + K4../KA.) < 0. 31 3 i 3 The dynamic i n t e r a c t i n g component torque contributions to the i - t h machine by a l l other machines through t h e i r e l e c t r i c a l loops are ATS^.. = E ATS .. E i 3 i H e i j ATD\u00E2\u0080\u009E. . = E ATD .. E i 3 j H eaj The t o t a l i n t e r a c t i n g synchronizing and damping torque contribu-tions to the i - t h machine by a l l other machines through t h e i r mechanical and e l e c t r i c a l loops are \u00E2\u0080\u00A2ATS..' = ATS\u00E2\u0080\u009E.. + ATS .. (A.5) 13 Mi3 Exj ATD.. = ATD . + ATD\u00E2\u0080\u009E.. (A. 6) 13 Mi3 Exj F i n a l l y , the t o t a l e l e c t r i c torque components are ATS = ATS. . + ATS . . tx xx xj ATD = ATD.. + ATD.. tx xx xj I l l REFERENCES [ 1] EPRI, \"Development of Dynamic Equivalents for Transient S t a b i l i t y Studies\", prepared by Systems Control, Inc., EPRI EL-456, (Research Project 7 63) . [ 2] J.W. Ward, \"Equivalent C i r c u i t s for Power Flow Studies\", AIEE Trans., V o l . 68, 1949, pp. 373-383. [ 3 ] G. Contaxis and A.S. Debs, \" I d e n t i f i c a t i o n of External System Equiva-lents for Steady-State Security Assessment\", IEEE Trans, on Power Apparatus and Systems, March/April 1978, pp. 409-414. [ 4 ] H. Duran and N. A r v a n i t i d i s , \" S i m p l i f i c a t i o n f o r Area Security Analy-s i s : A New Look at Equivalence\", IEEE Trans, on PAS, March/April 1972, pp. 670-679. [ 5 ] T.G. D e V i l l e and F.C. 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P h i l l i p s , \" E f f e c t of a Modern Amplidyne Voltage Regulator on Underexcited Operation of Large Turbine Generators\", AIEE Trans, on PAS, August 1952, pp. 692-697. [28] F.P. deMello and C. Concordia, \"Concept of Synchronous Machine S t a b i l -i t y as Affected by E x c i t a t i o n Control\", IEEE Trans, on PAS, A p r i l 1969, pp. 316-328. [29] M.K. El-Sherbiny and D.M. Mehta, \"Dynamic System S t a b i l i t y , Part I\", IEEE Trans, on PAS, Sept./Oct. 1973, pp. 212-220. [30] H.A. Moussa and Yao-nan Yu, \"Dynamic Interaction of Multi-Machine Power System and E x c i t a t i o n Control\", IEEE Trans, on PAS, July/Aug. 1974, pp. 1150-1158. [31] Yao-nan Yu, \"Power System Dynamics\", Lecture Notes at UBC, 1976. 114 [32] V. Subbarao, R.E. Burridge and R.D. Findlay, \"Mathematical Models of Synchronous Machines for Dynamic Studies\", IEEE PES Winter Meeting, 1979. [33] P.C. Krause, F. Nozari, T.L. Skvarenina and D.W. 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Melsa, \"Estimation Theory with Applications to Communications and Control\", McGraw-Hill, 1971. [42] Glenn W. Stagg and Ahmed H. El-Abiad, \"Computer Methods i n Power System Analysis\", McGraw-Hill, 1968. "@en .
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