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The Application of Lyapunov function to power systme stability analysis and control Vongsuriya, Khien 1968

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THE APPLICATION OP LYAPUNOV FUNCTION 10 POWER SYSTEM STABILITY ANALYSIS AND CONTROL by KHIEN VONGSURIYA B . E . , Chulalongkorn University, Thailand, I960-M . A . S c , University of Br i t i sh Columbia, 1964 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of E l e c t r i c a l Engineering We accept this thesis as conforming to the required standard Research Supervisor Members of the Committee Head of the Department =,., Members of the Department of E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA February, .19 68 In p r e s e n t i n g t h i s t h e s i s i n 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 o r an advanced deg ree 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 ag ree t h a t 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 tudy . I f u r t h e r ag r ee 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 pu rposes 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 . I t i s u n d e r s t o o d t h a t 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 . The U n i v e r s i t y o f B r i t i s h Co lumb ia Vancouve r 8, Canada Depa r tment Date F - 4 r - ^ S \ * ABSTRACT 'Lyapunov functions are applied to the power system studies. Three types of power system problems are investigated, namely, the determination of asymptotic s tab i l i t y regions of a nonlinear power system for fault and switching transient s t a b i l i t y studies, the systematic optimum parameter setting of power system controllers, and the determination of l inear and nonlinear optimum s tabi l i s ing signals as functions of state variables for both nonlinear and l inearized power systems. To investigate and construct the transient s tab i l i t y region of a synchronous machine connected to in f in i te bus through a transmission system after fault and switching, high degree Lyapunov function series generated by Zubov's method is applied. For the optimum parameter setting of a power system, a computation technique based on the method of gradient has been developed to adjust the system parameters simultaneously so as to minimise a system performance function which is eva-luated from a Lyapunov function of the second degree. For the computation of the second degree Lyapunov function a method based on the concept of s imi lar i ty transformation has been developed and applied so that the simultaneous solution of a large number of algebraic equations can be avoided. To determine the optimum s tabi l i z ing signals for a'power system, the concept of the "Lyapunov function of the optimum system" is applied. To compute the Lyapunov function of the optimum nonlinear power system, a general i terat ive scheme and algorithm have been developed. i i TABLE OP CONTENTS Page LIST CP ILLUSTRATIONS v i i ACKNOWLEDGEMENT . . . ix 1. INTRODUCTION 1 2. THE LYAPUNOV FUNCTION AND SOME APPLICATIONS 5 2.1 Lyapunov Functions and Stabi l i ty Cr i t er ia of the Direct Method 6 2.2 Asymptotic Stabi l i ty Region Determination by Lyapunov Function and Zubov's Method 8 2.3 Determination of a System Performance Function by a Quadratic Lyapunov Function 11 2.4 Lyapunov Function for an Optimum Nonlinear System 13 2.5 Lyapunov Function for an Optimum Linearized System 17 3. THE POWER SYSTEM EQUATION IN STATE VARIABLES 19 3.1 Synchronous Machine Equations for Power System Stab i l i ty Studies 19 3.2 Multi-machine System Equations in State Variables 23 3.3 One Machine-Infinite Bus System Equations in State Variables 28 3.3.1 A F i f th Order Model 28 3.3.2 A Third Order Model 33 3.3.3 A Second Order Model 39 3.4 Voltage-Regulator Equations 40 3.5 Governor Equations 4 l 4. - TRANSIENT STABILITY REGION STUDY OF A POWER SYSTEM 43 4.1 System Equations in State Variables' 44 • « • 1 1 1 Page 4.2 Lyapunov Function Series . . . . . . . . . . . . . . . . . . . . . 47 . 4.5 Numerical Example 48 .4.5.1 System Equations 48 4.5.2 S tab i l i ty Region Computation 50 4.5.5 Test of S tab i l i ty of the Power System 51 5. POWER SYSTEM PARAMETER SETTINGS BY MEANS OF A LYAPUNOV FUNCTION 56 5»1 A System Performance Function . . 56 5.2 Minimization of the Performance Function . . . . 59 5.5 Methods for Computing Matrix of Lyapunov Function 6 l 5»5»1 Direct Settings Algebraic Equations . . 6 l 5.5.2 S imi lar i ty Transformation 62 5.4 Linearized Power System 65 5.5 Selection of the System I n i t i a l Conditions and the Performance Function 69 5.6 Numerical Linearization 71 5.7 A Numerical Example 72 6. AN ITERATIVE METHOD FOR COMPUTING LYAPUNOV ' FUNCTIONS 79 6.1 Development of the Method 80 6.2 Convergence of the Iterative Process 85 6.5 Example of Application 86 6.4 A Method of Improving the Speed of Convergence 88 7. OBTAINING THE OPTIMUM STABILIZING SIGNAL FOR POWER SYSTEMS BY MEANS OF LYAPUNOV FUNCTIONS 92 7.1 Optimum Governor Stabi l iz ing Signals for a Hydro-electric System 92 7.1.1 Governor Model of a Hydro-electric System 92 iv Page 7,1...2 Proposed Configuration of Governor with Optimum Stabilizing Signals ..... 9 5 7 . 1 . 3 Linear Optimal Control Problem Based on the Proposed Configurations ....... 96 7 . 1 . 4 Numerical Study of Governor Optimum Stabilizing Signals and u 2 99 7 . 2 Optimum Governor and Voltage-Regulator Stabilizing Signals for a Hydro-electric System 110 7 . 2 . 1 System Equation Including Voltage-Regulator, Governor and Optimal Controls 110 7 . 2 . 2 Numerical Study of the Effect of Governor and Voltage-Regulator Optimal Stabilizing Signals 116 7 . 3 Nonlinear Governor Stabilizing Signal 126 7 . 3 . 1 Optimal Control Problem of a Third Order Nonlinear Power System. 126 7 . 3 . 2 Development of the Computation Algor-ithm 129. < 7 . 3 . 3 Numerical Study of the Nonlinear Stabilizing Signal u for a Third Order Power System 135 8. CONCLUSION 146 APPENDIX 3A Nomenclature for Synchronous Machine and System 148 APPENDIX 3B Park's Equations in the First Order Equations Form . 151 APPENDIX 3C Transformation of Coordinates of Synchronous.Machine into a Common Reference ..... 154 APPENDIX 4A Determination of I n i t i a l Operating Con-ditions E' and 5 for the Power q ' o Output Expression of the Synchronous Machine l 6 l APPENDIX 4B ^Expansion of D'(6) and R(s) in Power Series of State Variables 165 v Page APPENDIX 5A I n i t i a l Value3 of the Steady State Con-d i t i o n s of the Synchronous Machine In-f i n i t e Bus System 167 APPENDIX 5B D e r i v a t i o n of A l g o r i t h m f o r S e t t i n g e M a t r i x ..... 169 APPENDIX 5C A l g o r i t h m f o r S o l v i n g 1/ M a t r i x of (5-25) 173' APPENDIX 5D The Eigenvalue Confinement of L i n e a r Dynamic System by Systematic Adjustment of i t s Parameters . . . . 1 7 6 APPENDIX 6A V e r i f i c a t i o n s of the R e l a t i o n s h i p Y l = Y 2 3 C l = ^2 W h e n ^1 = ^2 1 8 5 APPENDIX 7A Determination of c r- 187 APPENDIX 7B Determination of a ^ f 1 ^ - 192 REFERENCES •,» 195 v i LIST OP -ILLUSTRATIONS Figure Page 3-1 A Schematic diagram of the synchronous machine infinite-bus system wir.h voltage regulator and speed governor 20 3- 2 Synchronous machine infinite-bus system 29 4- 1 A typica l power system 45 4-2 Numerical example 1 / 45 4-3 S tab i l i t y study by Zubov's method using various truncated Lyapunov function series 53 4-4 Determination of s tab i l i ty region by Zubov's method 54 4- 5 Comparison of s tab i l i ty boundaries with various choices of cp - functions 55 5- 1 Optimum parameter settings with various values of the specified minimum degree of s tab i l i ty y 78 6- 1 Vector relationship between the eigenvalue X of matrix A and the eigenvalue 6' of matrix 8 l 7- 1 Block diagram of a conventional hydraulic governor 94 7-2 An optimum stabi l iz ing signal u-^  of the governor 94 7-3 An optimum stabi l i z ing signal u_, of the governor 94 7-4 Power system responses with and without optimum s tab i l i z ing signal u-^  105 7-5 Comparison between power system responses with • s tab i l i z ing signals u-^  and U g 106 7-6.. System responses for governor s tabi l iz ing signals derived from different performance functions . 107 7-7 • Effect of change in T_ in the optimum feedback loop on system responses 108 7-8 System responses with s tabi l iz ing signals with different governor s e t t i n g s . . . . . . . . . . . . . . 109 • v i i . Figure Page 7-9 An optimum s tabi l i z ing signal u-, in the voltage-regulator I l l 7-10 Comparison of responses "between system with u 2 and u-, (—) and system with some terms of U g equals zero and u-, = o ( ) 121 7-11 System responses with the optimum stabi l i z ing signal U g and u-^  122 7 - 1 2 System responses with some terms in U g equals zero and = o 123 7-13 Effect of further ignoring g in U g on system responses 124 . 7-14 Effect of further ignoring g and h i n U g on system responses 125 7-15 Comparison of responses of a nonlinear system . with and without l inear optimum.stabilizing signals ' l 4 2 7 - l 6 a Nonlinear system responses with s tabi l iz ing signal u(N) , for N = 1 , 2 and 3 . . . . 143 7 - l 6 b Response of W(x,u(N)) for N = 1, 2, and 3 143 7 - 1 7 a Nonlinear system responses with s tabi l iz ing signal u(N) , for N = 1, 5 and 9 144 7-17b Responses of. W(x,u(N)) for N = 1 ,5 and 9 . . 144 7 - l 8 a Nonlinear svstem responses with s tabi l iz ing signal u(N) and with a small 5^  . 145 7 - l 8 b Responses of W(X,U(N)) with a small 5^  . . . . 145 3C Transformation of synchronous machines d-q coordinates.into a common reference 160 4A . Phasor diagram of a salient-pole synchronous , • • . machine 164 5A-1 The specified sector of eigenvalues 177 5A-2 Relationship between X-plane and p . ' -p lane . . . . - 1 7 7 v i i i • ACKNQ VJLEDGEl'IEH T The author wishes to express his most grateful thanks to his supervising professor, Dr. Y.N. Yu, who has inspired him throughout the course of the research. He also wishes to express his appreciation and thanks to Dr. P. Noakes for his encouragement, and to Dr. E .V. Bohn and Dr. M.S.'Davies for their valuable suggestions on this thesis. The author would l ike to extend"his sincere grat i -tude to his parents for their constant sp i r i tua l support and encouragement, and to Miss C. Siricharoen for her assistance In preparing the i l lustrat ions in this thesis. Grateful acknowledgements are given to the National Research Council for the f inancial support of this project under Grants 67-3626, to the University of Br i t i sh Columbia' for the Graduate Fellowships,September 1964 - May 1966, and to the Colombo Plan Administration in Canada for the scholar-ship given to the author since September 1966. ix 1. 1. INTRODUCTION Power system s tab i l i ty may be devided into steady (1 2) state s t a b i l i t y and transient stabil ity^ 3 The former refers to the s tab i l i t y of the system when subjected to small disturbances, while the lat ter applies when i t is subjected to -large disturbances. When a power system, in normal operation, is sub-jected to a small disturbance, such as a small change in load, i f the synchronous generator of the system does not lose synchronism because of the disturbance, the system is. said-to have the steady state s tab i l i ty with respect to the particular operating condition. Sometimes a power system may be subjected to large disturbances such as a short-circuit in the system, usually of re la t ive ly short duration. Under such a condition the electro-mechanical energy balance of the synchronous ma-chines is severely disturbed causing large changes in the relat ive power angles among the machines. If, after clearing the fault either with the fault section permanently isolated or with the fault cleared and the section restored to the f i n a l system, the generator does not lose the synchronism, then the disturbed i n i t i a l system is said to have transient s t a b i l i t y . In practice, the s tab i l i ty of a power system is usu-a l l y investigated by step-by-step numerical integration of system d i f f erent ia l equations using a d i g i t a l computer. Expl ic i t solutions of the system variables as a function of time are obtained and the s tab i l i t y is predicted from the trend of the time variations of the synchronous machine speeds and rotor angles, or the swing curves.' S t a b i l i t y study by the method of expl ic i t solu-t ion is not entirely"satisfactory because a large amount of numerical data must be accumulated before s tab i l i ty can be predicted. This is especially true when the system has a large, number of variables. It is e^en more tedious when dealing with the problem of power system design and synthesis involving the choice of the best voltage-regulator and speed governor settings for a given power system. In such a case, a large number of swing curves corresponding to many values of each parameter must be computed and compared before the most appropriate ones can be found. In search of a more convenient method than the exist-ing one for power system s tab i l i ty and design studies, the direct method of Lyapunov has been regarded by many investiga-tors as having great potent ial i ty . Gless^^ used the energy integral as a Lyapunov function of the direct method and studied one, two and three machine s tab i l i ty problems. El-Abiad and Nagappanv ' also using energy concepts, studied the transient s t a b i l i t y of a multi-machine system with losses. U n d r i l l ^ ) applied a quadratic form as a Lyapunov function to determine a power system performance function which was defined as the time integral of a quadratic form in a power system parameter settings study. In this study the performance function values with each system parameter varied were observed. "'. By means of a properly chosen Lyapunov function, the direct method of Lyapunov permits direct investigation of the s t a b i l i t y of a dynamic system. The Lyapunov f u n c t i o n can a l s o be a p p l i e d to system syn t h e s i s and design to f i n d the optimvm parameter s e t t i n g s , or to determine the optimum c o n t r o l s i g n a l f u n c t i o n . The major problem i n the a p p l i c a t i o n of the method, however, i s the determination of a s u i t a b l e Lyapunov f u n c t i o n . In t h i s t h e s i s , Lyapunov f u n c t i o n s are a p p l i e d t o the power system s t u d i e s . Three types of power system problem are i n v e s t i g a t e d , namely, the determination of asymptotic s t a b i l i t y regions of a n o n - l i n e a r power system f o r f a u l t and s w i t c h i n g t r a n s i e n t s t a b i l i t y s t u d i e s , the systematic f i n d i n g of the optimum parameter s e t t i n g s f o r power systems, and the determination of l i n e a r and n o n l i n e a r optimum s t a b i l i z i n g s i g n a l s as f u n c t i o n s of s t a t e v a r i a b l e s f o r both l i n e a r i z e d and n o n l i n e a r power systems. • Chapter 2 o u t l i n e s the Lyapunov D i r e c t Method and discusses the c o n s t r u c t i o n of the Lyapunov f u n c t i o n r e l e v a n t t o the s t u d i e s i n t h i s t h e s i s . Chapter 3 develops the equations d e s c r i b i n g a power system and c o n t r o l l e r s u s i n g s t a t e v a r i a b l e s . Chapter 4 i n v e s t i g a t e s the asymptotic s t a b i l i t y r e g i o n of a power system u s i n g Zubov's method to f i n d a s u i t a b l e Lyapunov f u n c t i o n . In Chapter 5 a Lyapunov f u n c t i o n of. the second degree i s a p p l i e d to determine the optimum, parameter s e t t i n g s f o r a power system. A computation technique based on the method o f g r a d i e n t s i s developed f o r a d j u s t i n g the system parameters simultaneously as t o minimize a system performance f u n c t i o n . A ' g e n e r a l i t e r a t i v e scheme and a l g o r i t h m f o r the computation of the Lyapunov f u n c t i o n i s developed i n Chapter 6 w i t h an example of i t s a p p l i c a t i o n . In Chapter 7, an optimum Lyapunov f u n c t i o n i s a p p l i e d to determine the optimum s t a b i l i z i n g s i g n a l f u n c t i o n s f o r a power system. The i t e r a -t i v e method developed i n Chapter 6 i s a p p l i e d i n the compu-t a t i o n of the f u n c t i o n s of the n o n l i n e a r system. 2 . THE LYAPUNOV FUNCTION AND SOME.APPLICATIONS 5. The Lyapunov d i r e c t method f o r dynamic system s t a b i l i t y study was f i r s t introduced i n l 8 9 2 \ " '. Since then the theory of the method has been extended and a p p l i e d t o many types of problem, f o r example, systems w i t h time d e l a y ^ ^ and d i s c r e t e s y s t e m ^ 1 0 ) . In t h i s chapter methods f o r the s t a b i l i t y study of an autonomous system w i l l be out-l i n e d . For the s t a b i l i t y study of the dynamic system, equa-t i o n s are w r i t t e n i n s t a t e v a r i a b l e form, • y±= F±(yl9y2> •••) > i = i , 2 , . . . n ( 2 - 1 ) or i n v e c t o r form y = P(y) ( 2 - 2 ) ,t inhere y = ( y x y 2 y ? . . . y n ) F(y) =.(}?! F 2 F n ) t ( 2 - 3 ) I t i s assumed t h a t the s o l u t i o n s of (2-2) e x i s t and are unique 'with respect to every i n i t i a l c o n d i t i o n , y(o) , i n a c e r t a i n r e g i o n Q of the s t a t e space. By s h i f t i n g the o r i g i n t o a p o i n t of e q u i l i b r i u m , which i s found from a steady s t a t e s o l u -t i o n 'by s e t t i n g (2-2) equal t o zero, a new set of equations Can be obtained which has the same form as (2-2), x = f ( x ) (2-4) ' . 6. but with f(o) = o (2-5) (2-5) is called the equation of the undisturbed motion, while (2-4) is called the equation of disturbed motion; the i n i t i a l condition x(o) is the disturbance. The problem of the s t a b i l i t y study of (2-1) w . r . t . i t s equilibrium now be-comes the s t a b i l i t y study of (2-4) w . r . t . i t s or ig in . The s tab i l i ty concept of Lyapunov may be described as follows. If for any given region A of the state space, there can be found another region X within A , including the or ig in , where every motion started within the region X stays within A a l l the time, the origin of (2-4) is said to be stable. Next , i f every solution of (2-4) started within the s t a b i l i t y region X , comes back ultimately to the or ig in , x = o , then the equilibrium is asymptotically stable. The precise mathematical definitions for s t a b i l i t y can be found in Lyapunov's work^""1"0^. 2.1 Lyapunov Functions and the Stab i l i ty C r i t e r i a of the  Direct Method The main problem of Lyapunov direct method is the determination of a Lyapunov function with certain properties in a region T which includes the origin of the state space and is contained in the region fi where the unique solution of (2-4) exists w .r . t . every i n i t i a l conditon x(o) . The Lyapunov function is usually denoted by V(x) and has the following properties, a. V(x) is a sign-definite function. b. V(x) i s continuous and has continuous f i r s t p a r t i a l , d e r i v a t i v e s w i t h respect to x . c. V(x) = • ^ • f ( x ) i s a s i g n s e m i - d e f i n i t e or s i g n - d e f i n i t e function.. A s i g n s e m i - d e f i n i t e f u n c t i o n V(x) i s c a l l e d p o s i t i v e (negative) s e m i - d e f i n i t e ^ 1 0 ^ i f V(o) = o and V(x) J> 0/•(<. °) i n T , i n c l u d i n g the case t h a t V(x) v a n i -shes i d e n t i c a l l y . I f V(o) = o , and V(x) > o (< o) f o r x ^ o i n r the f u n c t i o n i s p o s i t i v e (negative) d e f i n i t e . The o r i g i n a l s t a b i l i t y c r i t e r i a of the d i r e c t method given by Lyapunov i n the form Of theorems^\ can be summarized as f o l l o w s : \ I . The e q u i l i b r i u m corresponding t o the o r i g i n x = o of (2-4) i s s t a b l e , i f there e x i s t s a p o s i t i v e (negative) d e f i n i t e Lyapunov f u n c t i o n V(x) whose d e r i v a t i v e V(x) i s negative ( p o s i t i v e ) s e m i - d e f i n i t e . I I . . The e q u i l i b r i u m i s a s y m p t o t i c a l l y s t a b l e , i f there e x i s t s a p o s i t i v e (negative) d e f i n i t e Lyapunov f u n c t i o n V(x) whose d e r i v a t i v e V(x) i s negative ( p o s i t i v e ) d e f i n i t e . I I I . The e q u i l i b r i u m i s unstable, i f there e x i s t s a p o s i t i v e (negative) d e f i n i t e Lyapunov f u n c t i o n , V(x) whose• d e r i v a t i v e V(x) i s a l s o p o s i t i v e (negative) d e f i n i t e . The s t a b i l i t y r e g i o n of I and I I i s a bounded (Q) r e g i o n which may be described by^ ' . V(x) < c , c = constant , (2-9) W i t h i n which the c o n d i t i o n s f o r V and V must be s a t i s f i e d . Note th a t the c o n d i t i o n f o r V i n I I may be r e -laxed by r e q u i r i n g V(x) t c be negative ( p o s i t i v e ) semi-d e f i n i t e , provided that f o r any s o l u t i o n x ( t ) s t a r t e d i n the s t a b i l i t y r e g i o n (2-9) f o r x(o) ^  o , V ( x ( t ) ) does not vani s h a l l the time a f t e r a l l t > t Q f o r any t^o. This c o n d i t i o n w i l l always be s a t i s f i e d ^ ) i f ! _ { v - ( x ) } = a f S c - 3 . f ( x ) + o ( 2 - 1 0 ) holds on V ( X ) = M | l . f ( x ) = o , x ^ o , ( 2 - 1 1 ) i n the r e g i o n ( 2-9). ^ 2 . 2 Asymptotic S t a b i l i t y Region Determination by Lyapunov  F u n c t i o n and Zubov's Method. (ii-ik) According to Zubov, v ' t h e Lyapunov f u n c t i o n g i v i n g the exact boundary of asymptotic s t a b i l i t y of n o n l i n e a r system can be found by s o l v i n g the f o l l o w i n g p a r t i a l d i f f e r -e n t i a l equation s=l d X s s wher;e V i s the Lyapunov f u n c t i o n , x the s t a t e v a r i a b l e s x = f _ ( x ) and cp a p o s i t i v e d e f i n i t e or s e m i d e f i n i t e f u n c t i o n ^ ' of s t a t e v a r i a b l e s , which must be chosen before s o l v i n g ' t h e d i f f e r e n t i a l equation.. For a p o i n t t o be w i t h i n the asymptotic s t a b i l i t y r e g i o n i n the s t a t e space the f o l l o w i n g c o n d i t i o n , ( 2 - 1 2 ) o < V < 1 " ( 2 - 1 3 ) must be s a t i s f i e d and the s t a b i l i t y boundary may be determined by V = 1 (2-14) The choice o f cp , however, does not a f f e c t the s t a b i l i t y boundary. A cl o s e d form s o l u t i o n f o r V can be found from the p a r t i a l d i f f e r e n t i a l equation ( 2 - 1 2 ) o n l y f o r s p e c i a l cases. In g e n e r a l , by choosing cp as a p o s i t i v e d e f i n i t e or semi-d e f i n i t e q u a d r a t i c form, v v / V can be determined i n a form of I n f i n i t e s e r i e s , V = V 2 + + V 4 + ... + V M + ... ( 2 - 1 5 ) which converges at l e a s t f o r a s u f f i c i e n t s m all r e g i o n ' th around the o r i g i n . V M i n ( 2 - 1 5 ) i s of M degree homogeneous i n x n x _ . . , x , i . e . to 1 2 n 3 . • • M V yxl3 yx23'' ' 3 Y X n ) = Y V x l ' X23'''3 X n ) ( 2 - 1 6 ) f o r a constant y . When the V f u n c t i o n i s computed n u m e r i c a l l y , due to the l i m i t e d c a p a b i l i t y of a d i g i t a l computer,the i n f i n i t e s e r i e s ( 2 - 1 5 ) must be truncated, e.g., up t d the degree N f o r a- s u f f i c i e n t l a r g e N , 1 V(N). = V 2 + + ... + V N ( 2 - 1 7 ) The approximate asymptotic s t a b i l i t y r e g i o n i s 1 0 . determined by the minimum value of V(N) ' V(N) = c , c > o ( 2 - 1 8 ) tangent t o the surface V(N) = o , ? | y M f ( x ) = 0 ( 2 - 1 9 ) s=l s Thus the r e g i o n of asymptotic s t a b i l i t y determined by ( 2 - l 8 ) i s e n t i r e l y enclosed w i t h i n the exact asymptotic s t a b i l i t y r e g i o n of the system^ 1 1* . 'The n o n l i n e a r d i f f e r e n t i a l equation of the system may b e . w r i t t e n as dx n ^ m, m ? m, diT = * a s i x i + n L P s ( ^ V ^ V ' - - m n ) x l x 2 ••' xk 1 - 1 ( t m > l ) . . k=l K • = fs ( x l ^ x 2 ^ ' * X J > s = 1 , 2 , 3 , . . . n ( 2 - 2 0 ) The f i r s t summation term i n the r i g h t hand side of ( 2 - 2 0 ) i s the l i n e a r i z e d p a r t of the system; a g ^ are r e a l constants. The second summation term i n the r i g h t hand side of ( 2 - 2 0 ) contains terms higher than the f i r s t degree from the s e r i e s expansion of the n o n l i n e a r system i n s t a t e v a r i a b l e s x.x 0...x^ j the P are a l s o r e a l constants. x d n s The asymptotic s t a b i l i t y of the l i n e a r i z e d p a r t must be'assured f i r s t . In other words roots of the chara-c t e r i s t i c equation ' • | A - \ U | = 0 ( 2 - 2 1 ) must have non-zero negative r e a l parts.. A i s an nxn matrix w i t h element a . . s i S u b s t i t u t i n g ( 2 - 1 7 ) i n t o ( 2 - 1 2 ) and comparing terms' of l i k e degree i n x i x 2 " , x n > f o l l o w i n g recurrence formula were found by Zubov n • a V Q n E TFT < E a i k x k ) = " * ' ' ( 2 - 2 2 ) 1=1 B x i k=l 1 K K and n 3V n m £ ( £ a., x, ) = R (x, , x 0 i . . .x ) (2-23) i = l i k=l 1 K n m = 3 , 4 , 5 . . . . where i s the Lyapunov f u n c t i o n of the l i n e a r i z e d system, i and R m i s a f u n c t i o n of m degree homogeneous i n x i x 2 v , ' x n whose c o e f f i c i e n t s are determined, from the known V~,V_,,...V _ 2' 3 m - l obtained.In p revious computations. The determination of the c o e f f i c i e n t s o f V , m = 2 , 3 3 . . . N - amounts to the s o l u t i o n of sets of l i n e a r a l g e b r a i c equations obtained from the compari-son of c o e f f i c i e n t s of x ^ x 2 * • , x n °^ l i k e form i n the two sid e of ( 2 - 2 2 ) and ( 2 - 2 3 ) . Zubov's method f o r the determination o f s t a b i l i t y r e g i o n of a power system w i l l be i n v e s t i g a t e d i n Chapter 4 . 2.3 Determination of a System Performance Function by a  Quadratic Lyapunov Function^ The l i n e a r i z e d p a r t of system ( 2 - 2 0 ) i s w r i t t e n i n matrix form as . • .12. .• x •= Ax (2-24) By choosing cp(x) = x^Cx ( 2 - 2 5 ) where C is a positive semi-definite symmetric matrix, and assuming a quadratic form Lyapunov function V 0(x) = x*Lx " ( 2 - 2 6 ) where L is an nxn symmetric matrix to be determined, (2-22) yields A*L + LA = - C ( 2 - 2 7 ) For Vg to be a Lyapunov function of the system (2-24) which is asymptotically stable, L . vaxsji- be a positive definite matrix. The solution for L amounts to solving a system of n(n+l)/2 talgebraic equations for the unknown elements of L . Integrating dV (x) . • results in V 2(x(o)) = J%>(x)dt ( 2 - 2 9 ) since V2(x(oo)) o ( 2 - 3 0 ) The integral in the right hand side of (2-29) may.be regarded as a system performance function. For a given C and i n i t i a l condition x(o) , the performance function can be; evaluated from V 2 ( x ( o ) ) through ( 2 - 2 6 ) and ( 2 - 2 7 ) . I f A i s considered the f u n c t i o n of a set of parameter a = ( a ^ ...a^.)^ i . e . , A = A(a) , then V 2 .= V 2 ( x ( o ) , a ) = x t ( o ) L ( a ) x ( o ) , and the system performance can he minimized w i t h respect to a'.. Based on t h i s f o r m u l a t i o n the optimum parameter s e t t i n g s f o r power system w i l l be st u d i e d i n Chapter 5 . 2 . 4 Lyapunov Fu n c t i o n f o r an Optimum Nonlinear System. Consider a n o n l i n e a r system • x = f ( x , u ) . = Ax + g(x) + Bu ( 2 - 3 1 ) where x = ( x ^ X g . . , x n ) ^ i s the s t a t e v a r i a b l e v e c t o r , u = (u-^ u 2 . . . u k ) ^ a c o n t r o l v e c t o r , A an nxn matrix, B an nxk mat r i x , and g(x) = (g-^(x) g 2 ( x ) . . . g n ( x ) ) ^ i s a v e c t o r whose element g^(x) i s e i t h e r zero or a n o n l i n e a r f u n c t i o n expanded i n power s e r i e s beginning w i t h terms of the second degree i n x . ' I t i s r e q u i r e d t o f i n d such a c o n t r o l u(x) whereby the system ( 2 - 3 1 ) i s a s y m p t o t i c a l l y s t a b l e , and wherewith the f o l l o w i n g i n t e g r a l I G(x,u) dt ( 2 - 3 2 ) w i l l have a minimum along the t r a j e c t o r i e s of ( 2 - 3 1 ) . G(x,u) i s a given f u n c t i o n which may have the form 14. : G(x,u) = - i f ' E MxJ+u^Ru] ( 2 - 3 3 ) i=2 • where R i s a kxk p o s i t i v e d e f i n i t e m a t rix, each ijr^(x), i = 2 , 3 , . ..n, i s a homogeneous f u n c t i o n of t h x l X 2 ' ' * X n o f 1 degree, and tyg(x) i s a p o s i t i v e d e f i n i t e or p o s i t i v e s e m i - d e f i n i t e f u n c t i o n of qua d r a t i c form. For the s o l u t i o n of the problem, i t Is s u f f i c i e n t t o f i n d two f u n c t i o n s V(x) and u°(x) t h a t s a t i s f y the f o l l o w i n g c o n d i t i o n s ' " ^ ) 1. V(x) i s a Lyapunov f u n c t i o n of ( 2 - 3 1 ) , which s a t i s f i e s the c o n d i t i o n s of asymptotic s t a b i l i t y . - - G(x,u°(x)) ( 2 - 3 4 ) 0 dV d ' dt u=u°(x) 3 . The f u n c t i o n H(x,u(x)) t + G(x,u(x)) u=u(x) has a minimum at every p o i n t x i n the neighbourhood of the o r i g i n f o r u = u°(x) , the optimal c o n t r o l . From the c o n d i t i o n s ( 2 - 3 4 ) i t i s s u f f i c i e n t t o f i n d V and u t h a t s a t i s f y Min [|| + G(x,u)] = o ( 2 - 3 5 ) u(x) which, y i e l d s the f o l l o w i n g equations to be solved | | . f ( x , u ) + G(x,u) = o '• ( 2 - 3 6 ) » 3 V • a G _ ( X j U ) _ n - i 2 k ) 15. V(x) i n ( 2 - 3 6 ) i s sometimes r e f e r r e d t o as the optimum Lyapunov f u n c t i o n ^ \ or the Bellman-Lyapunov f u n c t i o n ^ \ . S u b s t i t u t i n g ( 2 - 3 l ) and ( 2 - 3 3 ) i n t o ( 2 - 3 6 ) ono has ||. {Ax+g(x)+Bu} + | { ^ -!ri(x)+utRu} = o ( 2 - 3 7 ) S o l v i n g f o r u from the second equation and s u b s t i t u t i n g i t i n t o the f i r s t of ( 2 - 3 7 ) gives -g.{Ax +g(x)} - ^ i V V l f ) - - ^J2*±(x) ( 2 - 3 8 ) u = - R ' V - g . ( 2 - 3 9 ) Thus i f V can be solved from (2-38), u(x) can be obtained from ( 2 - 3 9 ) . Closed form s o l u t i o n s f o r V and u of ( 2 - 3 8 ) and ( 2 - 3 9 ) , or e q u i v a l e n t l y ( 2 - 3 6 ) , are not p o s s i b l e i n (17) • genera l . A l ' b r e k h t v ' t r e a t e d t h i s problem a n a l y t i c a l l y by assuming s e r i e s s o l u t i o n s of the forms 2 3 u = u 1 + u 2 + ... ( 2 - 4 0 ) When ( 2 - 4 0 ) i s s u b s t i t u t e d i n t o ( 2 - 3 7 ) , the f o l l o w i n g general r e l a t i o n may be obtained f o r the determinations of V , m > 2 . 3V m ax m X J where ^ ( x ) i - s a homogeneous f u n c t i o n of x of m degree whose c o e f f i c i e n t s are determined from the known V~,v%,...V 23 3 m-1 p r e v i o u s l y determined and A = Ax + Bu 1 (2-42) where u^ i s determined from BV 0 , a v 0 * , . bV, -1 t S V 2 u, = - R B 2 ( 2 - 4 3 ) u i l " ax Note that ( 2 - 4 3 ) can be obtained from ( 2 - 3 8 ) and ( 2 - 3 9 ) by s e t t i n g g(x) and S 'lf.(x) t o zero. These are the equat-• : 1=3 1 ions t o be solved t o o b t a i n the optimal c o n t r o l f o r the l i n e a r i z e d system of ( 2 - 3 1 ) , x = Ax + BU-L ( 2 - 4 4 a ) w i t h the performance f u n c t i o n \ |°0{,!;2(x)+U^Ru1}dt ( 2 - 4 4 b ) I f the equation ( 2 - 4 3 ) of the l i n e a r i z e d system has so l u t i o n s - ^ v"2 and u^ t h a t s a t i s f y c o n d i t i o n s of asymptotic s t a b i l i t y of ( 2 - 4 4 a ) , then each term and u m _ ] _ ? o r .m > 2 can be determined u n i q u e l y and the convergence of the s e r i e s ( 2 - 4 o ) . . 1 7 . (17 19) i s a l s o assured^ 3 y 1 at l e a s t f o r a s u f f i c i e n t l y s m a l l r e g i o n around the o r i g i n x --- o . 2 . 5 Lyapunov Function f o r an -Optimum L i n e a r i z e d System. For the optimal c o n t r o l of the l i n e a r i z e d system (2 - 4 4 a ) w i t h the performance f u n c t i o n (2 - 4 4 b ) , i t i s known that the s o l u t i o n of the problem e x i s t s and can be found by s o l v i n g (2 - 4 ^ ) i f the system i s completely c o n t r o l l a b l e , i . e . i n the f o l l o w i n g matrix Z , Z = {B,AB,A2B, . . .A31"1!}} (2 - 4 5 ) there must be at l e a s t n l i n e a r l y independent column ( 20 21) v e c t o r s , v > ' where A i s an nxn , B an nxk , and Z an nxkn matrix. In order to determine Vg and u^ , s u b s t i t u t i n g V_(x) = x ^ x 4. and ilf 0(x) = x Cx (2 - 4 6 ) i n t o (2 - 4 3 ) where K i s an nxn symmetric matrix, and C i s a p o s i t i v e d e f i n i t e or p o s i t i v e semi--definite symmetric matrix, the f o l l o w i n g equations and - K B F f ^ K + A*K + KA •+ C = o (2 - 4 7 ) u x = - R~ 1B tKx (2 - 4 8 ) are obtained from which K and u^ can be determined. Note t h a t (2 - 4 7 ) c o n s i s t s of a set of n o n l i n e a r a l g e b r a i c equations which can always be solved n u m e r i c a l l y by s o l v i n g the f o l l o w i n g ' . 1 8 . ( 21) matrix d i f f e r e n t i a l equation.v /. |^ = KA + A*K - KBR" 1B tK + C ( 2 - 4 9 ) w i t h i n i t i a l c o n d i t i o n K(o) = o - ( 2 - 5 0 ) I t i s known tha t the constant matrix K i s the steady s t a t e s o l u t i o n of ( 2 - 4 9 ) , i . e . K = l i m K( T) ( 2 - 5 1 ) T-» e>o In Chapter 7 , the optimum s t a b i l i z i n g s i g n a l s f o r a power system w i l l be st u d i e d u s i n g the f o r m u l a t i o n i n t r o -duced i n s e c t i o n s . 2 - 4 and 2 -5 • 3 . THE POWER SYSTEM EQUATION IN STATE VARIABLES A power .system c o n s i s t s of a number of synchronous generators supplying power t o the system through t i e l i n e s . For the s t a b i l i t y study of a p a r t i c u l a r synchronous machine, the remaining p a r t s of a l a r g e system are normally considered an i n f i n i t e bus to which the machine i s connected through t i e l i n e s . . A schematic diagram of a t y p i c a l i n f i n i t e - b u s synchron-ous machine system with voltage r e g u l a t o r and speed governor i s shown i n F i g . 3 - 1 . In t h i s chapter the synchronous machine i n a power .system i s f i r s t described by Park's equations, and then r e -w r i t t e n as a set of the f i r s t order d i f f e r e n t i a l equations i n st a t e v a r i a b l e form. They are no n - l i n e a r d i f f e r e n t i a l equations with constant c o e f f i c i e n t s . The complete system equations i n c l u d i n g speed governor and voltage r e g u l a t o r are derived f o r a multi-machine power system as w e l l as f o r a one machine i n f i n i t e - b u s system. 3-1 Synchronous Machine Equations f o r Power System S t a b i l i t y  Studies. Park's equation f o r a synchronous machine i n d-q - (22) coordinates a r e v ' v d = - t q P e - Ri d.+ P * d V q = f d p 0 -- R I q + p t q .. fc = x a d ( 1 + TD-fP> vfd X d ( l + T d p ) ( l + T d p ) i d ' d •(l+T' d op)(l+Tj qp)" . O I I + ^ U + T ^ ) ( 3 - 3 ) ( 3 -D ( 3 - 2 ) SPEED FEEDBACK VOLTAGE REGULATOR F i g . 3-1 A schemat ic d i a g r a m o f .the synchronous machine i n f i n i t e - b u s sys tem w i t h v o l t a g e r e g u l a t o r and speed governor 2 1 . x ( 1 - I - T " P ) • -For d e t a i l s of symbols see Appendix JA . A d d i t i o n a l equations are as f o l l o w s . The torque equation may be w r i t t e n •= J p 2 6 + Dp5 •+ T e ( 3 - 5 ) where e = aj t + 6 , ( 3 - 6 ) the energy conversion torque i s T e - V d " V q ' • the t e r m i n a l voltage and the power and r e a c t i v e power output • • P = v.i . + v 1 (3- 9 a ) d d q q x ' Q = v 1 , v , i ( 3 - 9 h ) • q d d q where v, i s the t e r m i n a l v o l t a g e , T the electro-mechanical energy conversion torque, P the power output, and Q the r e a c t i v e power output of the machine. For a s t a b i l i t y study of a pov/er system the sub-t r a n s i e n t s may be neglected. By s e t t i n g the su b t r a n s i e n t time constants T", » T'1, T" and T" , t o zero (3-l) to do a q qo v ( 3 - 7 ) may be rearranged into' the f o l l o w i n g form: • • p * d = v d + R i d + t q % ( 1 + A w ) • P^q = v q •'' R i q " ^^o0^) - (3-10) P * E = V F - V F R 2 2 . pAw = (T.-DpS-T ) ( 3 - 1 1 ) where v-,-,, i - and i can be solved from FR d q *F + T D V F *d do - Tdo( x 'd- x 'd) 0 1 UL) UL). 0 FR l '• ( 3 - 1 2 ) and RT •p — = v f r j J ^ e f i e l d voltage ^ad F x F ad = f&> "the f i e l d f l u x l i n k a g e ( 3 - 1 3 ) FR — = the f i e l d current x a d f d The d e r i v a t i o n s are. given i n Appendix 3B. After- e l i m i n a t i n g Vp^, ar>d i . u s i n g ( 3 - 1 2 ) the ' system described by. ( 3 - 1 0 ) and ' ( 3 -11 )' can be w r i t t e n i n . terms • of s t a t e v a r i a b l e it ,', it * 6 and Aiu d q F V . 2 3 , 3 • 2 Multi-machine System Equations i n State V a r i a b l e s . For a power system c o n s i s t i n g of n synchronous th machines, the k machine can he represented by ( 3 - 1 0 ) , ( 3 - H ) and ( 3 - 1 2 ) w i t h respect to i t s own d-q coordinates as pilf d d ' "d (k) = y ( k ) + R ( k ) . ( k ) _ ' u u 0 ( l + A J k ) ) # ( k ^ 0 ( l + A ^ k ) ) (3-14) r , , ( k ) F FR 1, 2 , .> .n p 6 ^ k ^ = iw oA .u/ k ^ p A ( n « = 1 { T ( k ) _ D ( k ) p 5 ( k ) _ T ( k ) } k = 1, 2,.. .n ( 3 - 1 5 a ) T<k> « , ^ 1 ^ - ^ i ^ e 'd q r q d • ( 3 - 1 5 h ) A matrix equation s i m i l a r to ( 3 - 1 2 ) can be w r i t t e n from which the f o l l o w i n g r e s u l t s are obtained.' i ( k ) q • d do o - w. (4k )-d ( k )) i o - UJ ° x W q ^ ( k ) + T ( k ) y ( k ,00 q ( 3 - 1 6 ) k = l , 2 , . 2h. In and ( 3 - 1 5 a ) . ^ \ 6<» (]r) and A(JJV 1 3 k = 1 , 2 , .. .n , are the power system s t a t e (k) fk) f k l fkV v a r i a b l e s , whereas i \ , i \ . , v*,r> and T.V ; can be •» d 3 q. * FR i -expressed e x p l i c i t l y i n terms of the s t a t e v a r i a b l e s as ( 3 - 1 5 b ) and ( 3 - 1 6 ) . v' k^. ' and . T.'k^ are t r e a t e d as constants at .the moment but w i l l be considered s t a t e v a r i a b l e s when the v o l t a g e - r e g u l a t o r and governor equations are introduced. I t VkV (k) remains necessary t o express v^ ' and vK ' i n terms of the s t a t e v a r i a b l e s through the e x t e r n a l network r e l a t i o n s . For power system s t a b i l i t y s t u d i e s , the t e r m i n a l network through which the machines are connected i s normally considered a s t a t i c network at system f r e q u e n c i e s ^ ^ ' . The t e r m i n a l v o l t a g e s and cur r e n t s can be r e l a t e d i n the f o l l o w i n g form or where [ I ] = [ Y ] [ V ] [ v ] \ = [ Y ] - 1 . ! ] = [ Z ] [ I ] ( 3 - 1 7 a ) ( 3 - 1 7 b ) [V] = r d m (*)__"• (k) r u m ( 3 - 1 8 ) [ I ] = p+ J d ( l ) r m J(k) , ,/k) xr + J 1m ( 3 - 1 9 ) 2 5 -[ Z ] : = r l l + J ' X l l . r 1 2 + J X 1 2 * * r 2 1 + J X 2 1 r 2 2 + 0 " X 2 2 * * ( > 2 0 ) and [Y] = g l l + J ' b l l g 1 2 + J " b 1 2 g 2 1 + j b 2 1 S 2 2 + J b 2 2 9 m o ( 3 - 2 1 ) The components of [V] and [ I ] i n ( 3 - 1 8 ) and ( 3 - 1 9 ) are complex phasor q u a n t i t i e s i n common complex coordinates r o t a t -ing at the synchronous speed uoQ . I t i s shown i n Appendix 30 that the p.u. d-q components of terminal voltage v^ 'to and the current i W « n d i ( k ) and v to can be t r a n s -formed i n t o the components i n the common complex coordinates according t o the f o l l o w i n g t r a n s f o r m a t i o n and e o s S ^ s i n S ^ " to vv 'r ^ s i n 6 ^ k ^ c o s S ^ " to r c o s S ^ - s i n S ^ to vv ' m s i n S ^ c o s S ^ . , k = 1, 2, . . .n (3-22) , k = l , 2 , . . . n ( 3 - 2 3 ) S i m i l a r equations can b e ' w r i t t e n f o r the current t r a n s f o r m a t i o n . Since '(3-17b') together w i t h ( 3 - 1 8 ) / ( 3 - 1 9 ) and ( 3 - 2 0 ) can he w r i t t e n as 0 " ) + J - v ( k ) = £ / r +3v)(±(lhj±(l\)' k = l,2,..'n r 0 m 4=1 ^ R M or r n *=1 m : k £ " xk-t r k t m ^ lc — 1^ 2^ • • • n (3-24) s u b s t i t u t i n g ( 3 - 2 2 ) and- (3 -23) i n t o (3-24) y i e l d s v v 00 \ cos 6 ^ - s i n 6 ^ k ) - s i n e ^ V cos6' k) ' n ' ^ = 1 r k £ ~xk-e, L X k t rKi J hence c o s ' ^ - s i n s W x d s l n 6 ^ ^ c o s S ^ q ~(k)~ v d n E . £ = 1 v<k> q l_ cos ( 6 ^ - 6 ^ ) sin(6«-6<*>) •sm ( 6 ( k ) - 6 ( ^ ) C O s ( 6 ( k ) - 6 ( ^ ) x d 27 . ,-: k = 1, 2 , . . .n q (3-25) . By defining 03 rk^ + J" xk^ = z k A ^ = z k ^ c o s ^ + J s i n 2 w ( 3 - 2 6 ) i . e . rk£ _xk£_ , TtL-,1 = 1,2, . . . n (3-27) (3-25) can be further simplified into v (k) n E z -1=1 k-t \ cos ( 6 ( k ) - S ( ^ - ^ ) s i n ( 6 (k ) - 6 ( ^ - 0 k ) - s m ( kt ; ± ( k ) ,(k) , k = 1, 2, .. .n ( 3 - 2 8 ) Substituting ( 3 - l 6 ) into ( 3 - 2 8 ) results in V n c o s ( 6 ( k ^ - y " s i n ^ 1 ^ - ^ , ) 28 . d do. x 0 o f P F D , U ) , k = 1, 2 , . .n (3-29) where 6 ( k t ) ^ 6 ( k ) _ 6 ( ^ ) k, £ — 1, 2 ,. • .n, ( 3 - 3 0 ) When ( 3 - l 6 ) and (3^-29) are used t o e l i m i n a t e v(,^), v ( k ) and vW, (5-14), ( 3 - 1 5 a ) and ( 5 - 1 5 b ) become a set of f i r s t order d i f f e r e n t i a l equations i n s t a t e v a r i a b l e s * q k ^ * ^ k \ 6 ^ a n d A u / k ) , k = 1 , 2 , ...n . 3 . 3 One M a c h i n e - I n f i n i t e Bus System Equations i n State V a r i a b l e s . ' 3 . 3 . 1 A F i f t h Order Model. F i g . 3 - 2 shows a one-li n e diagram of a synchronons machine connected to an i n f i n i t e bus through e x t e r n a l im-pedance. A shunt admittance i s included at the machine t e r m i n a l F i g . 3-2 Synchronous machine I n f i n i t e - b u s system t o represent l o c a l loads and the l i n e charging e f f e c t s i n a long t r a n s m i s s i o n system. The . i n f i n i t e bus voltage mag-nitude v i s assumed constant, o Equation ( 3-14), ( 3 - 1 5 a ) , ( 3 - 1 5 b ) and ( > l 6 ) wit h -out s u p e r s c r i p t can be used t o describe the synchronous machine, whi l e the e x t e r n a l voltage and current r e l a t i o n s h i p takes the form ( 3 - 1 7 a ) as G+JB+ r+jx r+jx r+jx r+jx vr^' vm v +,tv or 0 om V^m ( 3 - 3 1 ) From the f i r s t of ( 3 - J l ) , comparison of r e a l and imaginary p a r t s gives -1-xB+rG -(rB+xG) xG+rB 1-xB+rG v. v_ m v or v. om r -x x r m ( 3 - 3 2 ) Using ( 3 - 2 2 ) and ( 3 - 2 3 ) , ( 3 - 3 2 ) becomes l^xB+rG: -(rB-i-xG) xG+rB r-xB+rG cos6 -sin6. sinS cos6 v. v. V . or v. om -f r -x x r cos6 -sin6 sin6 cos§ ( 3 - 3 3 ) According t o Appendix 3C, F i g . 3 C , 6 i s the angle between the machine d i r e c t a x i s and the r e a l a x i s of the complex reference c o r d i n a t e s . Let the i n f i n i t e bus volta g e l i e along the r e a l a x i s v + i v = v. + JO or ° om o • v (3-34) and l e t 6 be re d e f i n e d as the angle between the machine quadrature a x i s and the i n f i n i t e bus v o l t a g e , i . e . r e p l a c -i n g 6 by 5 - 9 0 ° . By s u b s t i t u t i n g ( 3 - 3 4 ) i n t o ( 3 - 3 3 ) , and r e p l a c i n g 6 by 6 - 9 0 ° , the s o l u t i o n f o r v, and v - Si are where v d v q k l k 2 " k 2 k l v sin6 o v cos6 o + k-^r+kgX k ^ r - k ^ x k-j^x—kgr k^r+kgX ( 3 - 3 5 ) k x =\ (1-xB+rG)/{(!-xB+rG) 2+(xG+rB)^} k„ = (xG+rB)/{(l-xB+rG) 2+(xG+rB) 2} ( 3 - 3 6 ) S u b s t i t u t i n g ( 3 - 1 6 ) and ( 3 - 3 5 ) i n t o ( 3 - 1 4 ) , ( 3 - 1 5 a ) and ( 3 - 1 5 h ) y i e l d s 32 Pt'd = P t q _ 1 -X, T , „ d d£- x d T do d do K2 T D-t • d do 0 0 k l k2 " k2 k l v sinS o v cos6 o + d do K 1 x 7 T 7 X d T do K , x V ' d o (*d-x'd> ° x 7 UJ , ( i _ u, ( i _ _2_) ° x 7 >° x 7 . X d X q K UJ, F * d J L + \ (3-37a) p 6 = UJQA UJ (3-37b) pAw J o {T^Dpa-Tg} 33 . ( 3 -37c ) (3-37d) Hence ( 3 - 3 7 a ) , (3-37b) and (3 -37c) c o n s t i t u t e a system of f i v e f i r s t order d i f f e r e n t i a l equations i n s t a t e v a r i a b l e s 6, AUJ, I ^ , \|fd and . i|f . The magnitude of t e r m i n a l v o l t a g e can be expressed i n terms of the s t a t e v a r i a b l e s through ( 3 - 8 ) , (3-34) and ( 3 - l 6 ) . 3 . 3 . 2 . A T h i r d Order Model. The f i f t h order synchronous machine i n f i n i t e - b u s model may be reduced to a t h i r d order model w i t h the f o l l o w -i n g assumptions: a. The induced v o l t a g e s . p ^ and p\!r , and \ the v o l t a g e s due to speed v a r i a t i o n s Awf^ and A U J ^ y i n (3-37a) are neglected because they are sm a l l compared t o the speed vo l t a g e s %• uu0 ^d^o d u e ^° c r o s s e x c i t a t i o n s . b. The time constant T-^^ and armature r e s i s t a n c e R , are neglected^being r e l a t i v e l y s m a l l . By s e t t i n g p^,' p* '.Acoijr^  Aaj.|;d , T d ^ and R equal to zero, ( 3 - 3 7 a ) , (3-37b) and (3-37c) can be w r i t t e n i n the form it) ur •' • ^-,+ T,-, ,, v T., where - = R + "k^r +.k 2x x d +p X V T', d do ( X d ~ X , d ) °' d x' L 3 4 . ( 3 - 3 8 a ) p6 = . u)QA,u) (3-38b) T x d' X . ) -Sh d do ( 3 - 3 8 c ) where / O V / 7 d ' d KA K, U) ( 1 — - X - i 0 x' O X n X d q *d k l k 2 •k 2 k 1 v Q s i n 6 v cos6 o + X 'do x' ( 3 - 3 8 d ) ( 3 - 3 8 d ) can be put i n the form \_ A-^sinP A ncos0 A^cosy A^siny sin6 + cos 6 \ 'do ( 3 - 3 9 ) i . e . F *d = A 1 c°s(S-3)+A 2-^ * q = A 5 s i n ( 6 - Y ) + A 4 - ^ 'do F 'do ( 3 - 4 0 ) where 5 5 . V UJ / : : : A l = ^ K l k l + ( K 2 - X q ) k 2 } + f K l k 2 - ( K 2 - X q ) k l ^ A and Wo fw.2. 2 = 7-r~ ^ I + ( K 2 - X q ) K 2 3 Ax ,x i d q V UJ j : — • : : — h = " ~ r - 7 f K i k 2 - ( K 2 - x / d ) k i 3 2 + ^ i k i + ( K 2 - x / d ) k 2 ^ A l x d UJ A ^ = ^ t - ¥ i + ( K 2 - x V K i } A X d K..k +k (K„-x ) p , a r c t a n ^ i ^ - J - ^ } K.k +k ( K p - x ' ) Y = a r c t a n { 1 1 2 2 — K 1 V k l ( K 2 " x / d ) ft* , ^ ( K 2 - x q ) ( V x / d ) ' (0* A = — ° - . ' X ' d X q X > q Using ( 3 - ^ 0 ) t o e l i m i n a t e * d and * , * ( 3 - 3 8 a ) , ( 3 - 3 8 b ) and ( 3 - 3 8 c ) become P* F = V F " ^1 ~ + TI 2COS(6-0) (3-42a) • Tdo p 6- = UJQA UJ ( 3 - 4 2 b ) 3 6 . K *w- 2T e = -^—7 CB 1 s in(6-Y)+B 2 CQs(6-P)}+B 5 s in(6-Y)cos(6-P) + B 4 ( - i _ ) Tdo T do' ( 3 - 4 2 c ) where „ xd - ( xV xd> . (* d-*' d) 4 x d B. 1 o 2 3 V / x ' 1- _ 1^ . ^ ( 3 - 4 3 ) x d q x B 2 = i „ A 1 A 4 { - i - - i - J X d q 3 o 1 3 L / x J  x d q x d q x d p l [ ^ X / d - X ^ d ) ^ d O ^ , . v g ^ V ^ ^ T V o B ^ ( * V X ) 2 ( X ' q + X ) 2 ( 3 - 4 3 a ) where the f i r s t term a m in the damping coefficient is due to the mechanical damping and the second term the e l ec tr i ca l damping. Thus (3-42), (3-42b) and (3 - 4 2 c ) constitute the equations f o r the t h i r d order model of the system w i t h 6,Au and ifrj, as s t a t e v a r i a b l e s . The st a t e v a r i a b l e ^ can be expressed i n terms of the p h y s i c a l q u a n t i t i e s i f d and 6 by s u b s t i t u t i n g (3-40) i n t o the f i r s t of (3-16) and s e t t i n g = o y which r e s u l t s i n - 7 - = r T 2 + TT C O S ( 6 ^ ) T do 1 1 ^ The i d e n t i t y , = x a d i f d 5 i S t n e n s u b s t i t u t e d i n t o (3-44) t o give ~T- - V + ^ ! C O B ( 6 - P ) ( 3 - 4 5 ) do Another form f o r t h i r d order model w i t h Vp R, i n s t e a d of typ i n con junction w i t h 6 and Aw as s t a t e v a r i a b l e s may a l s o be deriv e d by s u b s t i t u t i n g (3-44) i n t o ( 3-42a), (3-42b) and ( 3-42c), which y i e l d s • n p V F R = " 7 ~ ( V ' W + M 2sin(6-p)p6 (3-46a) do p6 = uu Am (3 -46b) -^P {T.-DpS-T } UJQJ I ^ e J T e = VFR ( c 1 sin(6-Y)+C 2cos(6-p)}+C^sin(6-Y)cos(6-p) + . 38 . + C 4 ^ 1 R + c 5 c o s 2 ( 6 - p ) ( 3 - 4 6 c ) where C l = C 2 = B 2 / r , l l + 2 B 4 + 2 ^1 ^2 C„ = B-,*— + B 3 " " I i ^ ' ^3 . (3~46d) ^ 2 / V S u b s t i t u t i n g ( 3 - 4 4 ) i n t o ( 3 - 4 o ) y i e l d s * d = A c c o s t 6-p) + A ' 2 v p R *q = A' 5'cos(6 -p ) .+ A ' ^ v F R •+ A 5 s i n ( 5 - v ) where ^2 ' A 1 = A l + A2^f A ' 2 = A g / ^ A' - A,^ 2  A 3 X A ' 4 = A J / T ^ (3 -47 ) (3 -48) • 39. Based on the assumption a. and b. in the beginning of this section, (3-l4a) and (3-14b)- give the relations v d = ~ V o (5-49),. Thus can be expressed in terms of the state variables through ( 3 - 8 ) , (3-49) and (3~4o) or ( 3 - 4 7 ) . 3 . 3 - 3 A Second Order Model. Since the f i e ld resistance is very small in compari-son with the inductance, the flux linkage tends to remain nearly constant during transient period when the terminal conditions of the machine are subjected to sudden (34) changes. Some investigations^^ ' indicated that the assumption of constant f i e l d flux linkage in the s tab i l i ty study of syn-chronous machines tended to .y ie ld conservative results as com-pared to those obtained using a fu l l y represented model. With o r *p treated as constants, equation (3-42b) and (3-42c) constitute a second order model of the synchronous machine-infinite bus system with 6 and Au) as state variables, The second order model based on the constant flux linkage assumption can also be derived from a phaser diagram as in Chapter 4 . By this model the voltage proportional-to • A f i e ld flux lingage typ/i"'^Q = E' • and the i n i t i a l 6 angle can be determined. (p"5) • 3.4 Voltage-Regulator Equations.. v ^' The equations f o r the voltage r e g u l a t o r can "be w r i t t e n as p v f d = T~ ^ e x ^ r ' v r ~ v t ) " K l e x v s 3 " " T ^ P " 5 2 ) e e p v s 8 8 ^ s t p v f d ~ T; p - 5 5 > where v i s the s t a b i l i z e r s i g n a l , v r the re f e r e n c e , and s the v o l t a g e - r e g u l a t o r parameters, u , \i&3 u r , u g t , T G and T are described i n Appendix 3A. By v i r t u e of ( 3 - 1 3 ) , equations (3 -52) and (3-53) can be w r i t t e n as P V F = T ^ t ( T r - v t ) - v ' a ] - £ (3-5») e e v' pv' s= * P v p - ^  (3-55) u s e A v where v ' s = v s / ( u a u r ) .. *'e = ^  W r C> 5 6 ) and ^'s = ^ex^st + 1 The t r a n s f e r f u n c t i o n corresponding t o (3 -54) a n d (3 -35) i s V P ^ / e ( 1 ' | T s P ) ~rz—sz y = : p - (3 -57) ^ r t ' 1+(T +T U' )p+T T p c  v e s s'^ e s^ S u b s t i t u t i n g ( 3 - 5 4 ) i n t o ( 3 - 5 5 ) the voltage r e g u l a t o r equa-t i o n s are w r i t t e n i n s t a t e v a r i a b l e form as p v p = — { ( v r - v t ) - v s 3 - — e e / b r / \ / "> X S ( 3 - 5 8 ) I-1 s e e 3 . 5 Governor Equations '. " ' The equation of a speed governor f o r a hydro-f o i l ' e l e c t r i c system v ' can be extended and w r i t t e n as PS = " -bkr C^pAuH^tSSl} ( 5 _ 5 9 ) ph = - 2pg - ( 3 - 6 0 ) 1 CO where (2m -1-1) g + 1 . 5 r — 3 h = T r T i o ( 3 - 6 1 ) The t r a n s f e r f u n c t i o n corresponding to ( 3 - 5 9 ) , ( 3 - 6 o ) and ( 3 - 6 1 ) i s • " - T . i - T i o j A T i _ 1 ( l + w ) ( l - i a 2 T u p ) . . AID A«J a (a+6,) w / £ " Q T r P + D ( i - w . 5 v ) Equation ( 3 - 6 2 ) i s w r i t t e n i n the general form. I t a p p l i e s to a hydro e l e c t r i c t u r b i n e governor i f - = 1 3 and (2S) a p p l i e s to a steam t u r b i n e governor v - " i f m-^  = m^  = o . S u b s t i t u t i n g (>4'2b) i n t o (3-59) and ' J ( 3 - 6 o ) , the governor equations are w r i t t e n i n s t a t e v a r i a b l e s and i n per u n i t as 1 m T 2m o+l Am4-n-cr p g a ' -5TT- ^ ^ i 0 + g + 1 . 5 ( — | - - ) h - T e - m a J } - ^ S . ] (3-63) o ' mn 2nu+l • , (3 -64) where D = cuQD (3-65) \ 4. TRANSIENT STABILITY REGION STUDY OF A POWER SYSTEM In t h i s chapter, Zubov's method i s a p p l i e d to cons t r u c t Lyapunov .functions f o r the determination of the reg i o n • of asymptotic s t a b i l i t y of a nonlin e a r power system,after f a u l t . The equations of the power system a f t e r f i n a l s w i t c h i n g are w r i t t e n as the f i r s t order d i f f e r e n t i a l equations i n st a t e v a r i a b l e s , x = f ( x ) , f ( o ) = O' (2-4) with the e q u i l i b r i u m of the f i n a l system as o r i g i n , the s t a b i l i t y r e g i o n i s then expressed by i t s boundary surface i n the n-dimensional state-space defined by, V = c (4-1) where V i s the Lyapunov f u n c t i o n and c i s a constant. To t e s t the s t a b i l i t y of a power system a f t e r f i n a l s w i t c h i n g , i . e . , with the' f a u l t s e c t i o n permanently i s o l a t e d or with the f a u l t c l e a r e d and the s e c t i o n r e s t o r e d to the system, the system equations i n s t a t e v a r i a b l e s are i n t e g r a t e d by the step-by-step method from the i n s t a n t of f a u l t u n t i l the f i n a l s w i t c h i n g . I f the s t a t e at the i n s t a n t of f i n a l s w i t c h -i n g f a l l s i n s i d e the s t a b i l i t y r e g i o n , then the s t a b i l i t y of the system i s assured. For a power system with c o n t r o l l e r s , such as voltage r e g u l a t o r s and speed governors, or with p o s i t i v e damping e f f e c t s , the system,.if s t a b l e , i s a s y m p t o t i c a l l y s t a b l e . The system w i l l f i n a l l y s e t t l e down at the s t a b l e e q u i l i b r i u m of the f i n a l system. Zubov's method of c o n s t r u c t i n g the Lyapunov f u n c t i o n f o r the s t a b i l i t y r egion study as described i n chapter 2, i n v o l v e s the s o l u t i o n of p a r t i a l d i f f e r e n t i a l equation §V f ( x ) = -cp(l-V) . (4-2) where V i s the Lyapunov f u n c t i o n , and cp a p o s i t i v e -d e f i n i t e or p o s i t i v e - s e m i - d e f i n i t o f u n c t i o n of s t a t e v a r i a b l e s . For the case under study i n t h i s chapter, s o l u t i o n of (4-2) In closed form i s not p o s s i b l e , and s e r i e s expansion of V i s necessary. In theory, the exact boundary of asymptotic s t a b i l i t y r e g i o n can always be determined by a p p l y i n g t h i s method u s i n g an i n f i n i t e number of terms of the V - s e r i e s . In p r a c t i c e , however, due t o the l i m i t e d c a pacity of a computer a truncated V - s e r i e s must be used. 4.1 System Equations i n State V a r i a b l e s : A t y p i c a l power system i s shown i n F i g . 4-1. A s a l i e n t pole synchronous generator i s connected to the i n f i n i t e bus through a high voltage long t r a n s m i s s i o n system. The r e s i s t a n c e of the l i n e conductor and the leakage conductance of the t r a n s m i s s i o n system are neglected, but not the charging e f f e c t of the t r a n s m i s s i o n l i n e which i s represented by a susceptance B at the machine t e r m i n a l . The machine supplies a power P + jQ at a t e r m i n a l voltage v^ t o the system, and the i n f i n i t e bus has a f i x e d voltage v . o The assumption i s made, according to p r a c t i c e , that the . synchronous machine f o r the s t a b i l i t y study can be represented by a second order mathematical model, with a constant voltage E^, a d i r e c t a x i s t r a n s i e n t reactance x^,_ and a quadrature a x i s reactance x n . The equation of motion of the synchronous E SALIENT POLE SYNCHRONOUS MACHINE Q X I'NFINITE BUS F i g . h-1 A t y p i c a l power system B SALIENT POLE SYNCHRONOUS MACHINE X X Y|2 1/ X Y J 2 X A Y J 2 Y|2 / Xd = 0 .27 P . U . Xt = 0.015 P.U. X = 0 . 74 8 8 P . U . Y | 2 = jO .0335 P .U . P = 0 . 735 P .U . Q = 0 . 05 P.U. INFINITE BUS F i g . k~2 Numerical example g e n e r a t o r - i n f i n i t e bus system i s w r i t t e n as 2 M ^-|+ D(6)§| + P m s i n 6 + P s s i n 26 = P. ' (4-3) dt where . V 2 ( X ' - X " ) T " s i n 2 6 V 2 ( X ' - X " ) T ' ' cos 26 o v d d' do o v g g' qp D(5) - ~— 1- ^ ~n • — (4-4) (x'+xf (x',+ x) 2 Pm = (xJx^xx'B) (^5) 2/ / \ v (x^ ,-x„) P - ,° d .q . / i i ^ s 2 ( x d - x - x x d B ) ( x +x-x5TB7 In' ( 4 - 3 ) , M = H/Vf, an i n e r t i a constant, D(&) the damping c o e f f i c i e n t , ( 1 ' 2 7 , 2 ^ P and P the power c o e f f i c i e n t s , ' m s 1 6 the angle of the quadrature a x i s of the synchronous machine with respect t o the i n f i n i t e bus, and P^ the mechanical power input which i s assumed constant. The mechanical damping i s neglected. The d e r i v a t i o n of P and. P and the determina-& m s • \ t i o n of the i n i t i a l values of E' and . v i n (4-5) and (4-6) f o r a given operating c o n d i t i o n are given i n Appendix 4A. The system equations can be normalized by i n t r o d u c i n g T = t/m" (4 -7) y i e l d i n g 2 d_6 + D / / 6 ) d_6 + g i n 6 + p / s i n 2 6 = p, ^ _ 8 ) dT a T s 1 47-. where D '(5) « D ( 6 ) ^ , K=TZ> P i = ( 4 - 9 ) m m At the e q u i l i b r i u m p o i n t of the f i n a l system, 6 = 6 , 6• = 0, 6 = 0 ( 4 - 1 0 j Hence 6 Q can be found from- s u b s t i t u t i n g ( 4 - 1 0 ) i n t o .(4-8) s i n 6 • + P' s i n 26 - P'" o S O 1 .. ( 4 - 1 1 ) The system equation ( 4 - 8 ) t r a n s f o r m e d to the new o r i g i n by l e t t i n g r e s u l t i n g 2 C , d^6 6' , 6 = 6 ' d6' d T | + D'-(6''). ™ • + R( 6') = 0 ( 4 - 1 2 ) ( 4 - 1 3 ) D e t a i l s of D'(6 7) and R(6') are given i n Appendix 4B. Equation ( 4 - 1 3 ) i s then reduced to two f i r s t order d i f f e r -e n t i a l equations by w r i t i n g d&' A - • = UJ (4-14) d T duj d T \ = - D ' ( 6 ' ) U J - R(s') 4 . 2 Lyapunov Function Ser i e s To construct the Lyapunov f u n c t i o n s e r i e s by Zubov's method according t o ( 4 - 2 ) f o r system ( 4 - l 4 ) , a p o s i t i v e q u a d r a t i c form i s chosen f o r cp, cp = a 6 ' 2 + pw 2, a > 0, p > 0 and a + p y 0 (4-15] The Lyapunov f u n c t i o n s e r i e s can be expressed as N m+1 D m-2 k,=l ' m-hl-k, k , - 1 <• I JL JL 6 UJ =' (D 2 1 6 ' 2 + D 2^ 2 6'UJ + D 2 5 U J 2 ) + (D 5 a l6 / 5"+.-D 5 2 6 ' 2 UJ .+ ^6'UJ 2 ' • + D , ^ 3 ) + ( . ) + ( ) + ... ( 4 - 1 6 ) The c o e f f i c i e n t s D , are determined from the re -currence formular ( 2 - 2 2 ) and ( 2 - 2 3 ) i n Chapter 2 , which can be deriv e d by s u b s t i t u t i n g ( 4 - 1 5 ) and ( 4 - l 6 ) i n t o ( 4 - 2 ) . For the d i g i t a l computation of the c o e f f i c i e n t s D m ^ of the Lyapunov f u n c t i o n s e r i e s V(N) f o r the system (4-14) a computer program i s w r i t t e n based on the al g o r i t h m (13} f i r s t given by M a r g o l i s . v " . 4.3. Numerical Example . 4 . 3 . 1 System Equations The synchronous machine under study, has the f o l l o w i n g parameters, *d = 0 . 2 7 p.u. T d o 9 , sec. x d = I. 0 p.. u. T d o - 0 . 0 4 sec. = 0 . 6 p.u. = 0 . 0 7 sec. *d = 0 . 2 2 p.u. H = 4 kw-sec/kva, x" q = \ 0 . 2 9 p.u. The t r a n s m i s s i o n system to which the synchronous machine i s connected i s shown i n F i g . 4 - 2 which can be reduced to F i g . 4 - 1 . The synchronous machine d e l i v e r s 0 . 7 3 5 + J 0 . 0 3 p-u. power to the system and the i n i t i a l t e r m i n a l voltage of the machine i s 1 . 0 5 p.u. A sudden three-phase symmetrical short c i r c u i t t o ground occurred at t r a n s m i s s i o n l i n e (x) near the end A which causes bus A to ground. The f a u l t e d l i n e s e c t i o n between A and B was disconnected from the system at both ends.of the l i n e at 5 c y c l e a f t e r the f a u l t and the l i n e was reconnected to the system at 24 c y c l e with the f a u l t c l e a r e d . For the given t e r m i n a l voltage v^ and power output P + jQ the i n i t i a l angle 6 and the i n t e r n a l voltage E' O C[ of the synchronous machine and the voltage V q of the i n f i n i t e bus are c a l c u l a t e d from (4A - 5 ) , (4A -9 ) and ( 4 A - 4 ) i n Appendix 4A. 6 o = 5 0 . 8 3 ° or 0 . 8 8 7 2 radians v = I . O 5 8 p.u. • '• • . • o E' = 1 . 0 5 5 p.u. To o b t a i n the system equations f o r the s t a b i l i t y study, the ' r e a l time t i s transformed t o T according t o ( 4 - 7 ) , T = 7 . 3 0 8 t The system equation during the f a u l t becomes ; j 2 / = P'. = 0 . 6 4 8 9 dT 1 . \ The equation a f t e r the f a u l t e d l i n e s e c t i o n disconnected i s 2 <L|L + ( 0 . 0 3 0 7 2 c o s 2 6' + 0 . 0 1 3 3 2 s i n 2 6 ' ) + dT . + O . 7 2 5 8 s i n 6' - 0 . 0 7 2 2 7 s i n 26' = 0 . 6 4 8 9 and the equation f o r the f i n a l system, i . e . , the system a f t e r the l i n e reconnected with the f a u l t c l e a r e d i s 2 ( 0 . 0 4 9 7 2 c o s 2 5' + 0 . 0 2 4 6 8 s i n 2 6') + s i n 6' -dT \ - O . 1 2 9 1 s i n 26' = 0 . 6 4 8 9 • 5 0 . To determine the region of asymptotic s t a b i l i t y - , the d i f f e r e n t i a l equation of the f i n a l system i s transformed i n t o two f i r s t order equations as ( 4 - l 4 ) . ^ = - D ' ( 6 ' ) u) - R ( 6 ' ) where D ' ( 6 ' ) i s c a l c u l a t e d from (4B-3) and (4B-4) i n Appendix 4B, ;P ' (&' ) = D Q + E (0.01252)- cos ( 2 6 Q + § ^ ) } 6 ' N where D = 0 . 0 3 7 2 + 0 . 0 1 2 5 2 cos 2 6 ~ o o^ 6 O = 0 . 8 8 7 2 , and R ( 6 ' ) from (4B-.5) and ( 4 B - 6 ) R(6') = ~ b i ^ ' o ^ ^ - 2 n ( 0 . 1 2 9 l ) cos ( 2 6 O 4 J ^ > ) } 5 ' In the computation i t i s found s u f f i c i e n t that D ' ( 6 ' ) and R ( & ' ) be truncated' at n = 3 0 . 4 . 3 . 2 S t a b i l i t y Region Computation The approximate boundaries of asymptotic s t a b i l i t y r e g i o n of the f i n a l system computed from the truncated Lyapunov f u n c t i o n s e r i e s V(N) by Zubov's method f o r N = 5 , 8 , 1 0 , 1 6 and 2 6 are shown In F i g . 4 - 3 - The cp f u n c t i o n chosen f o r the computation i s 2 2 cp = . a5 ' . + Pa) where a = p = D. = 0 . 0 3 7 2 + 0 . 0 1 2 5 cos 2 6 5 1 . The r e g i o n of s t a b i l i t y increases with N f o r N = 5,8 ,10 and l 6 but decreases f o r N 26 i n d i c a t e d that the approximate r e g i o n of s t a b i l i t y does not approach the true boundary raono-t o n i c a l l y with the increase of N. The s t a b i l i t y r e g i o n of N = 35 has a l s o been computed i n d i c a t e d no improvement over t h a t of N•= 1 6 . I t seems that un'jess a very l a r g e N i s used N = 16 i s the best t r u n c a t i o n f o r t h i s p a r t i c u l a r example. F i g . 4-4 shows an example of hov; the boundary of a s t a b i l i t y r e g i o n of V(8) = c i s determined. From the V(N) f u n c t i o n of ( 4 - l 6 ) with the D , c o e f f i c i e n t s determined, v ' m,K the p o i n t s on, V(N) = 0 are searched u n t i l the minimum value of V(N), or c, i s found. The V(N) = c gives the boundary * of s t a b i l i t y and i s tangent to V(N) = 0 . F i g . 4 -5 shows the e f f e c t of three d i f f e r e n t choices of cp f u n c t i o n s , cp1 D o ( 6 / 2 + iu 2) cp2 = D o ( 6 / 2 + R ^ 2 ) and \ 2 cp^ , = D UJ 3 o on the s t a b i l i t y boundaries f o r N = 5 and 16 . I t seems that the f i r s t choice gives the best r e s u l t . 4 . 3 - 3 " Test of S t a b i l i t y of the Power System. To t e s t the s t a b i l i t y of the system by Lyapunov's •V f u n c t i o n the system equations are i n t e g r a t e d from the i n s t a n t of f a u l t t i l l t h a t of f i n a l s w i t c h i n g . The i n t e g r a t i o n i s c a r r i e d out by Runge-Kutta's method. For a f a u l t c l e a r i n g a t 5 c y c l e (.) and a f i n a l s w i t c h i n g at 24 c y c l e (X) i n 5 2 . P i g . 4 - 4 . i t i s found that the f i n a l s t a t e of the system f a l l s i n s i d e the s t a b i l i t y r e g i o n of V ( i O ) , V ( l 6 ) and V (26) but outside those of V (5 ) and V ( 8 ) . The f i n a l power system i s s t a b l e according t o the s t a b i l i t y regions of V ( 1 0 ) , V ( l 6 ) , and V( 2 6 ) . \ 53. P i g . 4-3 S t a b i l i t y ' study by Zubov's method u s i n g various, truncated Lyapunov f u n c t i o n s e r i e s P i g . k-k Determination of s t a b i l i t y r e g i o n by Zubov's method 55. N = I6 THE TRUE STABILITY BOUNDARY $i = D 0 ( 8 + ul ) 2 2 <j>2= D 0 ( 8' +R|U ) • 2 ^ 3 = Do CJ P i g . 4 - 5 Comparison of s t a b i l i t y boundaries w i t h v a r i o u s choices of cp - fu n c t i o n s 56. 5. POWER SYSTEM PARAMETER SETTINGS BY MEANS O F . A LYAPUNOV FUNCTION I n t h i s c h a p t e r , a Lyapunov f u n c t i o n o f q u a d r a t i c form i s a p p l i e d t o determine the optimum parameter s e t t i n g s f o r the synchronous machine i n f i n i t e - b u s system d e s c r i b e d i n Chapter J . I t has a v o l t a g e - r e g u l a t o r and a speed governor and i s o p e r a t i n g a t a g i v e n c o n d i t i o n . A t ime i r i t e g r a l o f a q u a d r a t i c f u n c t i o n w i t h an e x p o n e n t i a l w e i g h i n g f a c t o r i s chosen as the system performance f u n c t i o n . Thus- the optimum parameters s e t t i n g s r e s u l t i n g from m i n i m i z a t i o n o f the p e r f o r m -ance f u n c t i o n w i l l y i e l d a system w i t h a p r e s c r i b e d minimum degree o f s t a b i l i t y . The method f o r m i n i m i z a t i o n o f the system performance f u n c t i o n w i t h the and o f the Lyapunov f u n c t i o n i s deve loped i n t h i s c h a p t e r , and a l g o r i t h m s are w r i t t e n f o r s y s t e m a t i c computa t ion o f Lyapunov m a t r i x . 5.1 A System Performance F u n c t i o n C o n s i d e r a l i n e a r i z e d system i n the form o f x = A ( a ) x \ a = ( a , a, ( 5 -D where t (5-2) • • • i s a se t o f m a d j u s t a b l e parameters o f A(a) , an nxn m a t r i x . L e t the f o l l o w i n g change o f v a r i a b l e s be made: x ( t ) = e" Y y ( t ) , t e [0,00) , Y > o (5-3) where y i s a r e a l p o s i t i v e c o n s t a n t . S u b s t i t u t i n g ( 5 - 3 ) i n t o ( 5 - 1 ) y i e l d s y = (A(a)+ YU)y ( 5 - 4 ) According to s e c t i o n 2.5 i n Chapter 2, i f the system of (5-4) i s a s y m p t o t i c a l l y s t a b l e , then, f o r a given p o s i t i v e s e m i - d e f i n i t e f u n c t i o n cp = ytcy (5-5) where C i s a. chosen nyn p o s i t i v e s e m i - d e f i n i t e matrix, the f o l l o w i n g r e l a t i o n must e x i s t , J y tCydt = y ^ L ( a ) y o o = - V ( y Q , a ) (5-6) where V(y ,a) i s the Lyapunov f u n c t i o n of system (5-4) , L(a) i s a p o s i t i v e d e f i n i t e matrix s a t i s f y i n g , (A t + v U)L + L(A+vIJ) = - C , (5 -7) and y Q i s a set of i n i t i a l c o n d i t i o n s of (5-4) at t = o . By v i r t u e of (5-3) , s u b s t i t u t i n g and y G = x o » t = ° \ y = 'eYtx , t > o i n t o (5-6) y i e l d s V(y n,a) = V(xn,a) = f e ^ V c x d t (5-9) J o From (5-9) , i t i s c l e a r that the system parameter a t o be chosen to minimize the performance f u n c t i o n (5-6) of system (5-4) , a l s o minimizes the i n t e g r a l (5-9) of (5-1) . The meaning of the constant y i n the i n t e g r a l ( 5 - 9 ) may be i n t e r p r e t e d as f o l l o w s . " L e t t h e e i g e n v a l u e s o f A(a) i n ( 5 - 1 ) be denoted by \ ± , i = 1, 2,.. .n where X i s a t i s f y | A - X U | = o ( 5 - 1 0 ) L e t X = p - Y ( 5 - H ) S u b s t i t u t i n g ( 5 - 1 1 ) i n t o ( 5 - 1 0 ) y i e l d s |(A+ YU) - pU|' = o ( 5 - 1 2 ) L e t p i •, i = 1, 2, ...n* be t h e e i g e n v a l u e s o f the m a t r i x A '+ Y U . Prom ( 5 - 1 1 ) , Rp± < o , i «= 1,2, . . . n (5-13) i f BX±- < - Y , • 1 = 1 / 2 , . . .n ( 5 - 1 4 ) S i n c e t h e degree o f s t a b i l i t y o f ( 5 - 1 ) can be d e f i n e d f rom t h e r e a l p a r t o f a s ( ^ 5 ) A Degree o f s t a b i l i t y o f ( 5 - 1 ) =. - Max(RX±) ( 5 - 1 5 ) U s i n g ( 5 - 1 4 ) , one has Degree o f s t a b i l i t y o f ( 5 - 1 ) > Y ( 5 - l 6 ) On t h e o t h e r hand, s i n c e p^ , i = 1 ,2 , . . .n a r e the e i g e n -v a l u e s o f t h e m a t r i x o f system ( 5 - 4 ) w h i c h i s d e r i v e d from ( 5 - 1 ) , the c o n d i t i o n ( 5 - 1 3 ) ensures the exi s t e n c e of. the performance i n t e g r a l which equals the Lyapunov f u n c t i o n as equation ( 5 - 9 ) . Thus w i t h the r e l a t i o n of ( 5 - l 6 ) the constant Y of ( 5 - 9 ) can be considered as the p r e s c r i b e d minimum decree of s t a b i l i t y o f ( 5 - 1 ) . Note that the performance f u n c t i o n of the form ( 5 - 9 ) has a l s o been used i n the s t a t e v a r i a b l e p a r t of the performance f u n c t i o n ( 2 - 4 4 b ) f o r s o l v i n g a l i n e a r o p t i m a l c o n t r o l problem^ - ^ K 5 . 2 M i n i m i z a t i o n of the Performance Function Let the p a r t i a l d e r i v a t i v e of ( 5 - 7 ) he taken w i t h respect to 9 i = 1, 2, .. .m as {(AVYU)L+L(A +YU)} = - |§- = o , i = l , 2 , . . . m from which one obtains (At+YU)|i_ + f_(A + YU) = L + L H-) (5-17) 1 1 1 2L where the unknown -~- , i = 1, 2, ...m , i s an nxn symmetric ma t r i x which can \e solved i f L i s a l r e a d y known. Note that ( 5 - 1 7 ) d i f f e r s s l i g h t l y from ( 5 - 7 ) i n f o r m . Once L i s determined the a l g o r i t h m f o r s o l v i n g L can be a l s o used f o r s o l v i n g w i t h r e l a t i v e l y l i t t l e e f f o r t . Based on (5-17) a simple gradient method f o r m i n i -m i z i n g ( 5 - 9 ) f o r a given C and y w i t h respect t o the para-meter a may now be o u t l i n e d as f o l l o w s 6 0 . 1. Choose a s t a r t i n g p o i n t a^ 0^ i n the parameter space such t h a t ( 5 - 1 * 0 i s s a t i s f i e d . 2 . Solve ( 5 - 7 ) f o r L(a(°)) . 3 > . Determine — T ~ T , i = 1 , 2 , ...m, by s o l v i n g ( 5 - 1 7 ) Note that y Q = x Q , and determine tBL_(a(°)) x i M x 0 , * { 0 ) ) 1 = 1 2 m f5_l8) o a a . x o da. 5 1 -L, m ^ ±oj x 1 1 A Nt 5. Determine the gradient v e c t o r a = ( a ^ a^ ...a m) 6. Determine a p o s i t i v e number e _> o such t h a t V(x , a ( e a ) a t t a i n s i t s minimum, i . e . |V(xoa(°)-ea) = ? M i E o l ^ l f L ) . (- a) = o ( 5 - 1 9 h ) 7 . Set a(-0 = a ( ° ) - e l ( 5 - 1 9 c ) and repeat step 2 to 7 over again by reg a r d i n g a("0 as a new 0 ^ ° ) 3 u n t i l the minimum of V(x ,a) i s a t t a i n e d , or u n t i l e « o . In step 6 , e may be found by g r a d u a l l y i n c r e a s i n g *e from zero w i t h a constant s t e p ' s i z e AS u n t i l (5-19b) changes s i g n from - . to + and then g r a d u a l l y decrease e w i t h a step s i z e reduced i n h a l f u n t i l ( 5 - 1 9 b ) changes s i g n from + 'to . .• . The step s i z e i s ' then f u r t h e r reduced i n h a l f and. the whole process i s repeated u n t i l e i t h e r ( 5 - 1 9 b ) i s s a t i s f i e d or A S smaller than a p r e s c r i b e d number. The a l g o r i t h m given above i s used f o r the study of the parameter s e t t i n g s of a power system. The a l g o r i t h m may a l s o be in c o r p o r a t e d i n use w i t h other more s o p h i s t i c a t e d algorithms of the gradient method v ' f o r power system s t u d i e s . In a p p l y i n g the above procedure, the i n i t i a J . s t a r t -i n g p o i n t must be chosen such t h a t ( 5 - 1 4 ) i s s a t i s f i e d . A systemmatic method f o r determining the i n i t i a l s t a r t i n g p o i n t a(°) i n the parameter space based on Zubov's a p p r o a c h ^ ^ ^ i s developed and i n c l u d e d i n Appendix 5D. The problem of.comput-i n g ( 5 - 7 ) i n step 2 w i l l be considered i n the next s e c t i o n . 5 . 3 Methods f o r Computing M a t r i x of Lyapunov Func t i o n . 5.3.1 D i r e c t S e t t i n g A l g e b r a i c Equations. In order to determine the -QC^ -O. unknowns of the symmetric matrix L of ( 5 - 7 ) , a system of r X — l i n e a r a l g e b r a i c equations are r e q u i r e d which may be put i n the form Rl = I ( 5 - 2 0 ) where 1 i s .a v e c t o r w i t h n ^ — ^ - unknown elements of L, f (n+l) i s a vector w i t h n - ^ — — u n k n o w n elements of matrix C which is--, chosen, and <R i s an n^ 1* 1) x nX^~ L matrix. A system-a t i c a l g o r i t h m f o r s e t t i n g ( 5 - 2 0 ) d i r e c t l y from ( 5 - 7 ) f o r any given matrix A + yU and "C i s developed in Appendix.5B. 6 2 . However, the computer memory requirement for stori n g the matrix iB increases r a p i d l y with n and thus the method i s r e s t r i c t e d to low order systems. 5 . 3 . 2 . S i m i l a r i t y Transformation. By s i m i l a r i t y transformation, the system equations (5- J0 can be reduced into a simpler form. Let y = Rz ( 5 - 2 1 ) Then ( 5 - 4 ) becomes z = R _ 1 { A + Y U } R Z ( 5 - 2 2 ) The transformation matrix R can be chosen such t h a t ^ 2 ^ 3 ^ ^ 3 8 ^ R ~ 1 { A + Y U } R =.B = 0 0 1 0 0 1 a l a 2 0 0 i a.. n ( 5 - 2 3 ) which has most zero elements. Let V' = z V z \ ( 5 - 2 4 ) be the Lyapunov function of (5-22), where L' s a t i s f i e s B*L' + L'B = - C' and A -f-C' = R CR ( 5 - 2 5 ) ( 5 - 2 6 ) Substituting ( 5 - 2 3 ) and ( 5 - 2 6 ) into ( 5 - 2 5 ) y i e l d s t t (A t+YU){R" 1.L'R" 1} + {R"1 L /R~ 1}(A+ YU) = - C (5-27) Comparing ( 5 - 2 7 ) with ( 5 - 7 ) one has L•= ( j f V r / t i r 1 ) ( 5 - 2 8 ) Thus L i s determined from (5-25) which has a sim p l e r form, and ( 5 - 2 8 ) gives the s o l u t i o n L of ( 5 - 7 ) . Only n equations f o r s o l v i n g the f i r s t n values of L i s needed and the r e s t can be computed by recurrence r e l a t i o n . This i s an improvement of the method described i n s e c t i o n 5 - 3 . 1 which r e q u i r e s s e t t i n g up n ( n + l ) x n(n-H_)i m a _ t r l x f o r the s o l u t i o n . The a l g o r i t h m i s r e l a t i v e l y complex since i t i n v o l v e s many steps of ope r a t i o n s , i . e . the s i m i l a r i t y t r a n s f o r m a t i o n d e f i n e d hy ( 5 - 2 3 ) , the s o l u t i o n of ( 5 - 2 5 ) f o r 1 / . ,. and the determ-i n a t i o n of L f o r ( 5 - 2 8 ) . However the numerical computations show th a t the method r e q u i r e s much l e s s computing time. The a l g o r i t h m f o r the s o l u t i o n of ( 5 - 2 5 ) f o r any C' matrix i s developed and given i n Appendix 5C. Note that the formula given by F e l ' b a u m ^ f o r computing ( 5 - 2 5 ) f o r C' was concerned v/ith a diagonal matrix which can not be a p p l i e d to t h i s case.• 5 . 4 L i n e a r i z e d Power System. The synchronous machine i n f i n i t e bus system equations ( 3 - 4 2 a ) , ( 3 - 4 2 b ) and ( 3 - 4 2 c ) , w i l l be considered. These and v.oltage-regulator and governor equations ( 3 - 5 8 ) , ( 3 - 6 3 ) and ( 3 - 6 4 ) and the t e r m i n a l voltage e q u a t i o n ' ( 3 - 8 ) , ( 3 - 4 9 ) and ( 3 - 4 0 ) are l i n e a r i z e d around a p a r t i c u l a r o p e r a t i n g p o i n t w i t h a power output P+jQ and a t e r m i n a l voltage . The l i n e a r i z e d equations are w r i t t e n In the form of ( 5 - 1 ) , where 64. x = h AiX) g A5 and t h e A matrix i s (5-29) 6 5 . CO H I 5) y <; H cO s + CM b CD ^ 0 | 0 CU 3 •p t-3 <; <x>' \ H s + O > H b < P •H I • -co 1 3. I H I »• co! cu CD CD t-O 3 < H s CM + £ CM CM -P CO ! T t> -P -L i D O 3 H CM •O 3 I -P cO + •+ H I K g CM +3 cO I t) o o o H I a) 13 > CD 0 o •ca i 5 Ul CM P" I O I LP* o r H | t -' 3 3 : 6 • «—> CM -P + b i H i t -'s 3 < +3 •O O 3 O CM t-• H + • " cO OJ b + CM. • • H N 6 s 3C H + CM -P 6 cO CM + ^"H s • s 1 t-—I o o o ca H t-H I ! O - CO t-3-. I. O O o o '-d^ o i 66. where A I" r ( B l s l n ( V Y ) + B 2 C O S ( 5 o - P ) ) , o p * P O l — — : — - + 2 B 4 . — J Tdo Tdo Awd = - ^ { - ^ [ B 1 c o s ( 6 0 - Y ) - B 2 s i n ( 6 0 - P ) } + B 3 c o s ( 2 5 0 - Y - P ) } Tdo A v e = - ^ W 2 / \ J o + * q o V T d o l A %2 V d V t o { - * d o A l s i n ( 6o J p') +*qoA3C O s ( 6 O " Y ) 3 ( 5 " 3 1 ) D tu 0 (X'-X")T'' ( X ' - X ' ' ) T " A _ ° _ or ,„*r q. q do ^  ci d' qo 1 The constants A's, B's and a m i n the r i g h t hand si d e of (5-31) are defined i n (3-41), (3-43) and (3-43a) r e s p e c t i v e l y . The i n i t i a l values of the steady s t a t e c o n d i t i o n , , JFo 5 *!do3 '•'qo-' v o So c a n 1 ° e determined from the given P+jQ and v^Q , i n clo s e d f orm, ' as given i n Appendix 5A. With the f o l l o w i n g changes of v a r i a b l e . T = p ' t , p ' = a 'do 67. n -0JO ALU where a' is' a chosen constant greater than one and x and B are constants shown i n F i g . 3 - 2 . Equations ( 5 - 1 ) , ( 5 - 2 9 ) and ( 5 - 3 0 ) can be put In the same form as ( 5 - 1 ) , | f - A(a)x ( 5 - 3 3 ) where x = h n ( 5 - 3 4 ) A 6 AV-Ai!r-F and the A matrix i s < CD +3 <0 H CD 3 JL. W 1 t- < b OJ •» CO H n s o ca -— OJ d • 3 ^ -P CD 3 t o - 'a < O CD t-> CD s. ca *• CD t- ca 3. ca I—I I ra CD t-•s. ca. i o o o CD t-ca rH ca > .— < -P ^—. CD <0 t-l 3 + 1 < b *- w rH 3. 6 ca OJ ca o ' . . — , -p b <o + w b OJ CSJ ca "2, CM ca CM ca w OJ T3 t) t-CM -P «o + ca + b co. o •a > < CD t-ca o ca i o CO •H w OJ ca o HI'*-' i 3 < 3 CM I +3 + b 3 =! I ca i 3 3 O OJ H + -P OJ •o 6 + OJ b s ca X y— .-—. -P H <o + - L i OJ b S3 OJ ca «H w S o o iH I CO 0 H | ca i o o o o I o II 6 9 . Equation ( 5 - 3 3 ) w i l l be used f o r the study of the parameter s e t t i n g s of the power system. 5 . 5 • -Selection of the System I n i t i a l Conditions and the Performance Function. • Since the performance f u n c t i o n ( 5 - 9 ) depends on the i n i t i a l value x , chosen f o r x a t time zero, i t i s decided f o r t h i s study t h a t x Q i s determined from a sm a l l load p e r t u r b a t i o n , AT a p p l i e d to the torque e q u i l i b r i u m of the power- system equations defined by ( 5 - 1 ) , ( 5 - 2 9 ) and ( 5 - 3 0 ) . A f t e r the new steady s t a t e i s reached, the l o a d , AT , i s suddenly removed t o permit the system v a r i a b l e x • t o go back to the o r i g i n . The i n i t i a l value x Q i s then taken to be the new steady s t a t e at which the a p p l i e d load i s suddenly r e l e a s e d . . When AT i s s u b s t i t u t e d i n t o ( 5 - 1 ) , ( 5 - 2 9 ) and ( 5 - 3 0 ) , one has dx = Ax + b ( 5 - 3 6 ) where x and A are defined i n ( 5 - 3 * 0 and (5-35) and b = 0 0 A T 0 0 0 0 ( 5 - 3 7 ) With the l e f t hand side of (5-36) set. to zero, the steady s t a t e a f t e r p e r t u r b a t i o n can be determined by s o l v i n g the 7 0 . system equations. The r e s u l t s are x - 0 o 0 0 (5 -33) 0 ( A * ) 0  ( A V o where f o r the optimum parameter s e t t i n g study ( v F ) 0 i s set t o zer and ( A S ) q = - ATrip/Det ( A^p)o = " A T r i 2 s i n ( 5 0-S)/Det Det = - T U A , + A Tusin(6 -p) '1 wd uue '2 v o ^' (5-39) Prom (5-39) one has the r e l a t i o n , ( A 6 ) q / ( A * f ) 0 = ^ / { u g S i ^ S ^ P ) } (5-40) Since (5-40) i s a l i n e a r r e l a t i o n and the V f u n c t i o n of (5 -9) i s quadratic i n xQ , the optimum parameter s e t t i n g s obtained from the m i n i m i z a t i o n of (5 -9 ) i s independent o f the magnitude and the s i g n of ( A 6 ) . The performance f u n c t i o n (5 -9 ) i s chosen which has the form o e 2 Y T £ q i ( Av t) *+q2( fs ) ^ ( h ) " + % ( n ) *+^( g) *+%{ A5 ) . ^ 2, 2,. + q 7 ( A v F ) c : + q 8 ( A ^ F ) 2 } d T (5-41) where y i s the s p e c i f i e d minimum degree of s t a b i l i t y q's are constants chosen, and AV^. Is the voltage e r r o r . Using ( 3 - 8 ) , (3-4-9), and (3-4-0), Av^ . can be expressed i n terms of A^J, and A 6. as A v t = ^ A S + C 2 A i ! r p ( 5 - 4 2 a ) where C l = " P ^ s i n ^ - p ) + C 4 A 5 c o s ( 6 0 - Y ) - C 3 G 4 C 2 = -4- A 2 + - 7 - A 4 Tdo Tdo ( 5 - 4 2 b ) and C 5 " v t o C 4 ^ to ( 5 - 4 2 c ) Thus the matrix C corresponding to (5-9) can be determined. 5 . 6 . Numerical L i n e a r i z a t i o n . ?>L For the purpose of computing matrix L and from A ( cx ) and " l ^ 0 ^ i n the m i n i m i z a t i o n of the performance f u n c t i o n , s e c t i o n 5 - 2 , i t i s found convenient to approximate A ( a ) and -ITT^- 0^ n u m e r i c a l l y from ( 5 - 3 3 ) which i s w r i t t e n i n the form, x ± = f ± ( x , a ) , 1 = 1 , 2 , .. .n ( 5 - 4 3 ) - • 7 2 ' Thus the element of A(a) i s computed by a., = >i(x+A.e k,a) - f^x-A-e^a) i k 2 A , i , k = 1,2,.. . n ( 5 - 4 4 ) where A i s a sma l l p o s i t i v e number, x i s the e q u i l i b r i u m of ( 5 - 4 J ) , f o r x = o and e^ i s the k column of an nxn u n i t ma t r i x . The p a r t i a l d e r i v a t i v e of A(a) i s obtained through ( 5 - 4 4 ) by si\f„\ A(a+A»e.) - A(a-A«e.) - 2, i = l , 2 , . . . k ( 5 - 4 5 ) Since f^(x,a) , i = 1 , 2 , ...n i n ( 5 - 4 3 ) are l i n e a r f u n c t i o n of x , the numerical o p e r a t i o n f o r o b t a i n i n g A(a) from (5-43) may be done a c c u r a t e l y by a computer. I t i s noted t h a t ( 5 - 4 4 ) and' ( 5 - 4 5 ) may a l s o be used to o b t a i n the approximate A(a) and -|^f a^ from the n o n l i n e a r equations which are w r i t t e n i n the same form as (5-43) except x 4 ° 3 i . e . the e q u i l i b r i u m p o i n t . o f the n o n l i n e a r system around which the system i s l i n e a r i z e d . . • ' 5 - 7 A Numeric al_Example. For,the p o w e r - s y s t e m . ( 5 - 3 3 ) a computer program i s w r i t t e n f o r the m i n i m i z a t i o n of the. performance f u n c t i o n ( 5 - 5 1 ) w.r.t. parameters of the v o l t a g e - r e g u l a t o r and the governor. The program determines the performance f u n c t i o n by s o l v i n g for L ' of ( 5 - 7 ) - w i t h the given C matrix of the performance f u n c t i o n and a s p e c i f i e d minimum degree, of s t a b i l i t y y • The s o l u t i o n of ( 5 - 7 ) i s obtained by the method of s i m i l a r i t y ' ' 7 3 -t r a n s f o r m a t i o n i n s e c t i o n 5 » 3 . 2 . Danilevsky's m e t h o d ^ i s used to o b t a i n R - 1, B and R of (5-23).' The l a s t column of l / , 1 , w i t h n elements, of ( 5 - 2 5 ) i s determined by a matrix equation of the form ' 1' = AT 1 I ' . where and "f are set up a u t o m a t i c a l l y by means of the a l g o r i t h m developed i n Appendix 5 C The other elements of L' are computed by the recurrence forumlas ( 5 C - 5 ) . Next the performance f u n c t i o n i s determined from ( 5 - 9 ) f o r the given . i n i t i a l value x Q , ( 5 - 3 8 ) , and a set of parameters a . A f t e r the matrix L(a) and the performance f u n c t i o n V (a,x o) are determined the matrix of | ^ - a ^ x o ^ , . i i i = 1 , 2 , . . . m of the gradient method as developed i n s e c t i o n 5 . 2 can be computed from ( 5 - 1 7 ) . Since the e q u a t i o n s ( 5 - 1 7 ) and ( 5 - 7 ) are i d e n t i c a l i n form i n the l e f t hand s i d e , the a l g o r i t h m used f o r s o l v i n g L(a) i s a l s o used f o r s o l v i n g ^ . Note th a t the matrices. R'^B,R and p'1 of the method of s i m i l a r i t y t r a n s f o r m a t i o n have already been computed along w i t h L(a) . Thus the determinations of -|IiL a) i = 1 , 2 , . . . m can be done w i t h r e l a t i v e l y l i t t l e e f f o r t . The numerical l i n e a r i z a t i o n method described i n s e c t i o n 5 . 6 i s used i n the computation. • •A s i m i l a r program to the above i s a l s o set up f o r the m i n i m i z a t i o n of the performance f u n c t i o n ( 5 - h i ) o f . t h e power system ( 5 - 3 3 ) - The only d i f f e r e n c e i s t h a t i n t h i s program the matrix L(a) i s solved d i r e c t l y from ( 5 - 7 ) u s i n g the a l g o r i t h m given, i n Appendix 5B. Numerical couputa-t i o n i n d i c a t e s t h a t the program f o r s o l v i n g L w i t h the method of s i m i l a r i t y t r a n s f o r m a t i o n r e q u i r e s o n l y h a l f , the computing time as r e q u i r e d by the method of d i r e c t s e t t i n g equations _ T . _ , . , . , • . . ~ • n ( n + l \ n ( n + l ) f o r s o l v i n g L which i n v o l v e s tne i n v e r s i o n of an J — — x -»— matrix. In the subsequent numerical examples the method of s i m i l a r i t y t r a n s f o r m a t i o n f o r s o l v i n g L(ct) w i l l be used. The parameter, s e t t i n g of a v o l t a g e - r e g u l a t o r w i l l , •be s t u d i e d as an example. The synchronous machine has the f o l l o w i n g p a r a m e t e r s ' ^ . x^ = x^ = 1 . 3 9 p.u. T d Q = 5 . 9 S 7 second x^ = 0 . 2 2 p.u. H = 6 kw-sec/kva The s u b t r a n s i e n t parameters of the synchronous machine are not known and t h e ' o v e r a l l damping c o e f f i c i e n t D Q i n ( 5 - 3 1 ) i s assumed t o be D Q = 3 P.u. v The parameters of the e x t e r n a l network according to P i g . 3 - 2 are r+jx = 0 . 0 + JO. 4 p.u., and G+JB = 0 . 0 The v o l t a g e - r e g u l a t o r and the s t a b i l i z e r have the f o l l o w i n g time constants 7 5 -T = 1 . 5 9 second and T = 1 . 5 9 second and t i e v o l t a g e ' r e g u l a t o r gain and the s t a b i l i z e r g a i n u' are t o be determined. The governor has the f o l l o w i n g t r a n s f e r f u n c t i o n A Tm , 2 0 Att) ( 1 + T i P ) ( 1 + T 2 P T > T l = T2 = ° * 6 5 s e c o n d < Comparing t h i s w i t h ( 3 - 6 2 ) AT m 1 ( l + T r m l P ) ( l - m 2 T w P ) Z 5 T = " a (a+5 t) ( > 6 2 ) {1+ H_ T p } { l + 0 . 5 T p } a r or J one has m l = m 2 " ° 3 Tw = 1 , 3 s e c o n d j T r = 0 . 0 3 0 9 5 second, •a = 0 . 0 5 and a + 6 t ' = 1 . 0 5 The synchronous machine d e l i v e r s 1 . 0 + JO.O p.u. power to the system and the t e r m i n a l voltage of the machine i s 1 . 0 p.u. The i n f i n i t e bus voltage v Q and the i n i t i a l values of the steady s t a t e c o n d i t i o n s determined a c c o r d i n g t o Appendix 5 A are n Q = 0 . 0 , 5 Q = 1 . 7 1 2 r a d i a n v p o = 1.323 p.u., v Q = 1.077 p.u. *Fo = i [*56'6 p.u., c j \ b d o .= 0.812 p.u. and co^Jr q o = 0 . 5 8 4 p.u. The i n i t i a l values of the steady s t a t e c o n d i t i o n s of the v a r i a b l e g and h i n the governor and v' i n the voltage-r e g u l a t o r are equal to zero. The number a"' i n ( 5 - 3 2 ) i s chosen equal t o 1.0, thus P' =6.442 A l l the elements of ( 5 - 3 5 ) can be found n u m e r i c a l l y from the above parameter values and i n i t i a l c o n d i t i o n s , except the terms t h a t i n v o l v e u\' and p' which are to be determined. • s s The f o l l o w i n g performance f u n c t i o n i s chosen accord-i n g t o (5-41) and ( 5 -42a) by s e t t i n g the q's t o zero, except q^ and q^, , the two p o s i t i v e constants to be chosen, ^ e 2 Y T { q 1 ( A V t ) 2 + q l f n 2 } d T OO = J e 2 Y T{q 1(C 1A6+C 2A,!r p) 2+q i |n 2}dT where. "C-^  and "Cg are determined from (5-42b) and (5-42C). The i n i t i a l value x Q i s determined from ( 5 - 3 8 ) and (5-40) w i t h ( A 5 ) set t o 1.0. S e v e r a l sets of the u' and u' s e t t i n g s are s obtained u s i n g v a r i o u s q^ and various s p e c i f i e d minimum degrees of s t a b i l i t y y • I t i s found that the optimum para-meter s e t t i n g s not only minimize .the given performance f u n c t i o n s but a l s o y i e l d high power c e i l i n g s . A high power c e i l i n g i s • d e s i r a b l e i n p r a c t i c e . P i g . 5-1 shows the s t a b i l i t y boundaries^ in the p' - u' plane f o r P + jQ =' 2 . 0 + jO.O p.u. and e s P = 1 . 8 + jO.O p.u. A l s o shown i n the same f i g u r e are the optimum parameter s e t t i n g s corresponding to P + jQ = 1.0-fjO.O p.u. The parameter s e t t i n g s f o r y - 0 . 0 , 0 . 0 2 , and 0 . 0 3 are shown by © i n the f i g u r e . The constants q-^  and q^ are both set to 1 . 0 i n t h i s computation. I t i s evident t h a t the increase i n y tends to give a parameter s e t t i n g t h a t decreases the power c e i l i n g . I t seems that f o r the case under study y < 0.03 gives a good compromise. I t i s a l s o noted t h a t when q^ Is set t o 1 . 0 and q 2 v a r i e d from 0 . 1 to 2 . 0 , the r e s u l t s of the optimum parameter s e t t i n g s does not d i f f e r s i g n i f i c a n t l y from those which appear i n P i g . 5 - 1 ' F ° r example, f o r y = 0 . 0 3 the numerical r e s u l t s are, = 3 . 8 0 , u^ = 1 2 5 . 3 8 f o r q 2 = 0 . 1 Us = 3 . 8 5 , \x'& = 1 1 5 . 3 0 ^ f o r q 2 = 1 . 0 Ug = 3 . 7 9 , = 1 0 9 . 1 3 f o r q 2 = 2 . 0 However the power c e i l i n g i s very s m a l l when q^ i s set i d e n t i c a l l y to zero. 0 1 CD-T = 0 . 0 3 2 0 4 0 6 0 8 0 I 2 0 I40 F i g . 5-1 Optimum parameter s e t t i n g s w i t h v a r i o u s v a l u e s o f the s p e c i f i e d minimum degree o f s t a b i l i t y y -3 CO 6 AN ITERATIVE METHOD FOR COMPUTING LYAPUNOV FUNCTIONS In t h i s chapter, an i t e r a t i v e method i s developed f o r the s o l u t i o n of the p a r t i a l d i f f e r e n t i a l equation, 3V • Ax = -<b ( 6 - 1 ) 9x rm v 1 where V and <}> are.both homogeneous f u n c t i o n s of degree m In n v a r i a b l e s x-^, x g , ..,x n, tj>m i s known, Vm i s to be determined, and A i s a given nxn matrix w i t h eigen-values i = l , 2 , ...n, a l l w i t h negative r e a l p a r t s . As stated i n Chapter 2 the s o l u t i o n of V of ( 6 - 1 ) can be used to construct Lyapunov f u n c t i o n - s e r i e s f o r s t a b i l i t y r egion s t u d i e s and f o r the determination of non-l i n e a r optimal c o n t r o l . I n ( 6 - 1 ) , the N unknown c o e f f i c i e n t s of V can be determined by s o l v i n g a system of N l i n e a r a l g e b r a i c equations, r e s u l t i n g from comparisons of c o e f f i c i e n t s of l i k e terms, i n x of the same degree from both sides of (6-1). For a homogeneous function' of m degree i n n v a r i a b l e s , N i s determined by ( 6 ) . N = (m+n-1)'./[(n-ltml] ( 6 - 2 ) In g e n e r a l , the work' f o r s e t t i n g up the N . l i n e a r a l g e b r a i c equations r e q u i r e s a l a r g e number of symbolic m u l t i -p l i c a t i o n s . For a l a r g e n, even f o r m = 2 , a l a r g e compute memory c a p a c i t y i s needed f o r s t o r i n g the NxN matrix of the l i n e a r equations. , . ' • ' I n the f o l l o w i n g , an i t e r a t i v e method f o r s o l v i n g ( 6 - 1 ) i s developed to avoid the s e t t i n g up NxN matrix e x p l i c i t l y , which r e s u l t s i n a uniform process and simple a l g o r i t h m f o r d i g i t a l computation w i t h l e a s t memory r e q u i r e -ment . 6.1 Development of the Method L e t A 1 = | + kU ( 6 - 3 ) where k = 1 -I- ^ and a i s a constant greater than or equal t o zero, and r i s the r a d i u s of a, c i r c l e w i t h the center at (-(r+ct), o) \ l y i n g i n the l e f t - h a l f of the complex X-plane. The c i r c l e encloses a l l the eigenvalues X-^  X2> •• o f m a t r i x A, as shown i n P i g . 6-1. As a r e s u l t the eigenvalues of A^ are a l l l e s s than one In absolute value . L e t the eigenvalues of A^ be 6'^, i = l , 2 , ...n. D i v i d i n g through (6-1) by r and s u b s t i t u t i n g ( 6 -3 ) i n t o (6-1) y i e l d s ^ . ( A r k U ) x = - ^ ( 6 - 4 ) A p p l y i n g E u l e r ' s theorem of homogeneous f u n c t i o n k - J . U x = kmVm ( 6 - 5 ) to ( 6 - 4 ) r e s u l t s i n V = i {* + 9 V m . A OX} ( 6 - 6 ) m m % Y m -ox where I t i s c l e a r t h a t the matrix A^ has eigenvalues smaller than or equal to those of A^. Equation (6-1) has now been t r a n s -formed i n t o ( 6 - 6 ) . Hence, the s o l u t i o n V that s a t i s f i e s ( 6 - 6 ) a l s o s a t i s f i e s ( 6-1). P i g . 6-1 Vector r e l a t i o n s h i p between the eigenvalue X of matrix A and the eigenvalue 6' of matrix A, From ( 6 - 6 ) , the i t e r a t i v e process f o r the s o l u t i o n V may be defined as m J ^ = I , 2 , ..., k+l, ... ( 6 - 8 ) where i s the approximate sol u t i o n ' of ( 6 - 6 ) a t i t e r a t i o n , and V^°^ the i n i t i a l approximation. 3 m • L e t the operator A 0x • —- be introduced and denoted by K, i . e . , -2. A „ 5 < K U K V = V ;. ax- . ( 6 - 9 ) m^ 2 ax For £ = k+l, the ( k + l ) t h i t e r a t i v e s o l u t i o n of V m i n ( 6 - 8 ) , can be expressed i n terms of A , A 0 and V^°^ as m 3 c vm-' 2 m f o l l o w s : m m 1 "m m J V ( 2 ) = I {,!; + KV^ 1) } = i {* + i K * } + m m 1 ,Jm m • m t¥m m •mJ m v ( k + l ) = 1 j 1 K 1 K 2 m m .m m Jm md • m ( 6 - i o ) l k , K k + 1 (°) m m I f the i t e r a t i v e process converges, ^j^00^ gives the s o l u t i o n o f . ( 6 - 6 ) . 6 . 2 Convergence of the I t e r a t i v e Process Assume t h a t A g has d i s t i n c t eigenvalues 6^, .1 = 1 , 2, ... n. I t can be shown t h a t the r e s u l t of the i t e r a t i v e process ( 6 - 1 0 ) based on ( 6 - 8 ) converges t o give the s o l u t i o n V of (6-6) f o r any m and n. m v ' L e t the eigenfunctions Q of the operator K be defined by U t = I = 1, 2 , ... N ( 6 - 1 1 ) where £ i s a homogeneous f u n c t i o n i n x^, Xg, ... x n , of degree m and l e t the eigenvalue (6) of K be 4 = 1 , 2 , ... N ( 6 - 1 2 ) where m. are p o s i t i v e i n t e g e r s which s a t i s f y the r e l a t i o n , n 2 m. = m, l = 1, 2 , ... N (6-13) 1=1 1 Then there e x i s t s N d i f f e r e n t sets of m. , 1 = 1, 2, ... N 1 V t h a t s a t i s f y (6-13) and thus the eigenvalue \x , determined by ( 6 - 1 2 ) , has a t most N d i f f e r e n t values ( 6 ) . The f u n c t i o n •it and V^°^ can be expressed as l i n e a r combinations of gm m ^ eigenfunc.tions Q ( 3 1 ) _ L et; and v ( ° ) = I YJ0) C, (6-14)' where y and Y»°^ a r e constants. By v i r t u e of (6-l4) and ( 6 - 1 1 ) , . 84. N N S i m i l a r l y one has N K S u b s t i t u t i n g (6-lfjA) and ( 6 - I 5 B ) i n t o ( 6 - 1 0 ) y i e l d s , y ( k + l ) _ 1 f I r , M . - ' H l , / % k 1 H , and hence. u ., n ( 1 " . + A v i 0 ) C ^ ) k + 1 ( 6 - 1 6 ) Now, i f ^ 1 < 1 , 4 = 1, 2 , ... N ( 6 - 1 7 ) Then, m 4-1 k -» 00 ( 6 - 1 8 ) To v e r i f y t h a t ( 6 - l 8 ) i s the s o l u t i o n of ( 6 - 6 ) l e t ( 6 - l 8 ) be s u b s t i t u t e d i n t o ( 6 - 6 ) , thus 8 5 . v ( * ) B i [ t + K [ U - ^ f - ] ] ( 6 - 1 9 ) m m -m mt=l u. J v ( 1 - . - £ ) • . m ' Using ( 6 - 1 1 ) and ( 6 - l 4 ) , ( 6 - 1 9 ) becomes ^ m ' = 1 E {- Y ^ } Ci- —) v m ' - by ( 6 - 1 8 ) . Hence, ( 6 - 1 8 ) i s the s o l u t i o n of ( 6 - 6 ) . The convergence of the i t e r a t i v e process ( 6 - 1 0 ) , or e q u i v a l e n t l y ( 6 - 8 ) , i s based on the assumption ( 6 - 1 7 ) . It remains to be shown that m From ( 6 - 1 2 ) | < 1 f o r I = 1, 2, ... N ( 6 - 1 7 ) ( 6 - 2 1 ) 1 m 1 1 m ' Ap p l y i n g t r i a n g l e I n e q u a l i t y 1^1 < ( m ^ - J + m 2 j 6 2 | 4- ... + m n j 6 n | ) / m ( 6 - 2 2 ) L e t U m a x | i max | 6 i | , i = 1 , 2, . . . n, (6-22a) ( 6 - 2 2 ) becomes 1^1 < I W < m i / m ) > •>* = 2> • • • N ( 6 " 2 5 ) < U m a x h by v i r t u e of ( 6 - 1 3 ) Hence |-^| < 1, by the c o n s t r u c t i o n of A 2 ( 6 - 2 4 ) Thus, the convergences of ( 6 - 1 0 ) and hence (6-8), are assured. • 8 6 6 . 3 Example of A p p l i c a t i o n The general i t e r a t i v e process ( 8 ) ' i s a p p l i e d t o develop an a l g o r i t h m f o r the computation of a quadratic f u n c t i o n Vg, i . e . m = 2 . Let V 2 = x tBx . t ( 6 - 2 5 ) i|r 2 = x Cx where . C and B are nxn symmetric matrices; C i s known and B i s t o he determined. S u b s t i t u t i n g ( 6 - 2 5 ) i n t o ( 6 - 8 ) y i e l d s = ^{xtCx + x t { A * B ^ " l ) + B ^ _ l ) A 2 } x . = x t { | [ C + A^B^" 1) + B(l-1h2]}x, 1 = 1 , 2 , . . . ( 6 - 2 6 ) Hence, B I = 1{0 + A^B^'-1) + B ^ V } , I = 1, 2, ... (6-27) i s the i t e r a t i v e s o l u t i o n of B. Without l o s s of g e n e r a l i t y B^0^ can be assumed equal to zero. By a p p l y i n g ( 6 - 2 7 ) repeatedly, the approximate s o l u t i o n a t ( k + l ) ^ i t e r a t i o n , B^ k +^^, can be expanded in terms of C and A 2 as t ( I - J ) j B(M) = E I (x) t i — ( 6 _ 2 8 ) 1=0 J=0 d 2 ^ + X ) where and £ (]•) = 2 1 ( 6 - 2 9 ) J=0 J Based on ( 6 - 2 8 ) , a norm, |j || of ( B ^ L B ( k + 1 ) ) i s used t o study the rat e of convergence of the i t e r a t i v e s o l u t i o n , ^ ( I - J ) j | | B(»)_ B(k+l) || || ~ 1 A 2 C A 2 } , p 1=0 J=0 0 2 t ( l - j ) j k I T A 2 C A 2 = 2 f(J) -3 || 3 1=0 J=0 0 JL+1 J 2 s 2 I + 1 2 } l l I=k+1 J=0 J 2 X + J -i ( ? ) ( I - J ) -< • £ £ C - ^ i ||A 2 | | 3 I=k+1 J=0 2 X + J - ^ ^ l l c iu s I {-4+r M A g l l i ) , <E=k+l J=0 2 X + J - ^ ^ since | | A * | | 5 = J | A 2 | | 5 ^ | C | U S 1|A, W 2 ' 3 5 8 8 . I . e . ; | i B C ~ ) - B ( k + : L > . | | < i||c||, s 11 A„| | | • ( 6 - 3 0 ) -> ~ d -> I=k+1 d -> Since (32) | 6 M A X | < | | A 2 | | 5 ( 6 - 3 1 ) where 6 maxl = m a x I 6 i ' 3 1 = 1 } 2> • , ; . n by assuming t h a t | |Ag| |, < 1 '(6-32) ( 6 - 3 0 ) becomes • I I A I I k + 1 •,,B(-).B(^ )||5<|||C||5ci^ |^ } ( 6 - 3 3 ) Based on ( 6 - 3 3 ) , the rat e of convergence of the i t e r a t i v e s o l u t i o n may be defined as t l | A 2 | | k + 1 n y = (i -n4n i • l | A 2 M 1 ( 6 " 3 l t ) 5 I n the case that f | A 0 l j / 1 but j6 m„ v| < 1, us i n g an 3 expression analogous to ( 6 - 3 * 0 , another r a t e of convergence ^max^ i r -7r-\ •n = T T T T T — T \ (6 - 3 5 ) may be defined as k+1 6 . 4 A Method of Improving the.Speed of Convergence A method f o r improving the r e s u l t of i t e r a t i v e s o l u t i o n of ( 6 - 8 ) s i m i l a r t o the concepts of L. A. L i u s t e r n i k ^ ; i s developed. Prom ( 6 - l 8 ) and ( 6 - l 6 ) , U k+1 V (-) (k+1) _ l l (m-) I C v ( o ) f ^ K + l ~ m 1=1 ( 1 _ % 1=1 1 1 K m m ' 8 9 • N ' a.C H. k + 1 = £ - i i i — ( 3 ) ( 6 - 5 6 ) where m '4 v^ m a. = Y ( ° } ( 1 - ^ ) , 4 = 1 , 2 , ... N ( 6 - 3 7 ) Without l o s s of g e n e r a l i t y l e t one assume t h a t — , - — > — are i n descending order i n magnitude. From (6-12) and m ( 6 - 2 3 ) , l e t - i = 6 • m max ( 6 - 3 8 ) r-*p m max where — and —=• are a complex conjugate p a i r and 6 as defined i n ( 6 - 2 2 a ) . For a s u f f i c i e n t l a r g e number of i t e r a -t i o n s , k +l, (6-36) may be approximated by the f i r s t two terms. By v i r t u e of ( 6 - 3 8 ) one has T ( ~ ) v ( k + 1 > - g i C ] - Ay k + 1 1 a& r V + 1 m m vm y u c vm ' ( 1 - - i ) ( 1 - — ) v m ; v m ' a \ h fik+l , a 2 C 2 •^max1 ( 6 " 5 9 ) I t i s shown i n Appendix 6A t h a t , as lo n g as ^ and u g i n (6-38) are complex conjugate p a i r s , so are Q1 and C 2 > 9 0 . and Y p 0 ^ i , e C 2 . = C * , Y 2 - Y i V and = Y ( 0 ) (6-4QA) Thus from ( 6 - 3 7 ) o 2 = ax (6-40B) Hence ( 6 - 3 9 ) can be w r i t t e n as max max >k+l . _. u.r * e * k + l talh.6max + a l h maxJ = (5max+ 6W+ 6*maxW -A6max maxJ H - (6 m + 6* ) + 6* 6 * max max' max max' ( 6 - 4 2 ) Now from ( 6 - l 6 ) , 4 where a;, 4 = 1 , 2 , ... N are given i n ( 6 - 3 7 ) . For.a l a r g e ( k + l ) , ( 6 - 4 3 ) may be approximated by the f i r s t two terms, i . e . V ^ - v W * ^ ( ^ ) k + a 2 C 2 • (6 - 4 4 ) S u b s t i t u t i n g (6-40A), (6-40B). and ( 6 - 3 8 ) i n ( 6 - 4 4 ) y i e l d s , 91. With k replaced by k+l, (6-45) becomes By s u b s t i t u t i n g (6-45) and (6-46) into (6-42), one has ,v(k+2) v ( k + l h v ( c o ) _ v ( k + i ) = ( V m - % >  m m { l - (6 +6* ) + 5 6* } v max max' max max 6 6* ( V ( k + 1 ) - v W ) max max v m m ' { l - ( 6 +6* )+6 6* } v max ma,xy max max (6-49A) ^1 In the case that —- = 6 „„ i s a re a l number, simi l a r argu-m max ments as above leads to r ( o o ) _ v(k+l) y(k+2). y(k+l) _vj ^-^vj (6-49B) m m (1 - 5 ) v max' Hence (6-49A) and (6-49B) give the closer approximation of °° ^  than that obtained by V^ k + 1^ alone. After a suf-f i c i e n t l y large number of i t e r a t i o n s being performed on (6-8), the i t e r a t i v e solution may be improved by using (6-49A) or (6-49B). . 9 2 7 . OBTAINING THE OPTIMUM STABILIZING SIGNAL .FOR POWER - • ' (42) SYSTEMS BY.MEANS OP LYAPUNOV FUNCTION ' In t h i s chapter, optimal c o n t r o l s as f u n c t i o n s of. power system s t a t e v a r i a b l e s are determined w i t h the a i d of optimum Lyapunov f u n c t i o n according t o the technique o u t l i n e d i n Chapter 2. The optimal c o n t r o l f u n c t i o n s f o r a power system are i n the form of s t a b i l i z i n g s i g n a l i n the speed governor, and i n the v o l t a g e - r e g u l a t o r loops. With system equations l i n e a r i z e d around a given e q u i l i b r i u m , the optimum s t a b i l i z i n g s i g n a l i s a l i n e a r f u n c t i o n of the s t a t e v a r i a b l e s For the governor two c o n f i g u r a t i o n s i n c l u d i n g the optimum s t a b i l i z i n g s i g n a l s are proposed. The e f f e c t s of the optimum s t a b i l i z i n g s i g n a l s on power system responses are i n v e s t i g a t e d A l s o i n v e s t i g a t e d i s a n o n - l i n e a r s t a b i l i z i n g s i g n a l f o r a n o n - l i n e a r power system, u s i n g the i t e r a t i v e method developed i n Chapter 6. Comparisons are made between the e f f e c t s of l i n e a r and n o n - l i n e a r s t a b i l i z i n g s i g n a l s on power'.system responses f o r d i f f e r e n t i n i t i a l d i s t u r b a n c e s . 7 . 1 Optimum Governor S t a b i l i z i n g S i g n a l s f o r a H y d r o - e l e c t r i c  System. 7 . 1 . 1 Governor Model of a H y d r o - e l e c t r i c System. For the study of governor c o n t r o l l e d power system, the second order synchronous machine i n f i n i t e - b u s equation i n Chapter 4 w i l l be used with.the.assumption th a t the i n t e r n a l v o l t a g e E' , which i s p r o p o r t i o n a l t o the f i e l d f l u x l i n k -Vi age, i s held constant, thus the voltage r e g u l a t o r a c t i o n w i l l be ignored. The synchronous machine i n f i n i t e - b u s equation (4-3) i s modified by i n t r o d u c i n g a term AP^ due to governor a c t i o n . The system equations are 9 3 . d V d t ~ % L w dA(u 1 ( 7 - l a ) where AP ± = g + 1 . 5 h ( 7 - l b ) A OO _ and . D(6') = S D 6 = D'( b'^^/MP n=o R(6') = E R n 6 ' n = R (6')PM n=l ( 7 - l c ) where M and P m have been defined a f t e r (4-3) whil e i)'(6') and R'(6') are i n c l u d e d i n 4-l4) and are-given : i n Appendix 4B i n d e t a i l s . The governor gate change g and the gate feed back g^ may be described by . • '. | | = _ I - {AtD+gf+Cg} .. \ ' ( 7 - 2 ) d g f . dg f f dt °t dt " T r and the head change h due t o water a c c e l e r a t i o n by dh _ 0 dg _ h , r / "eft ~~ •"" dt ~ T " - ^ { ' ? ) w The t r a n s f e r f u n c t i o n block diagrams of the governor corre-sponding to ( - 7 - l b ) , ( 7 - 2 ) and ( 7 - 3 ) i s shown i n F i g . 7 - 1 where the speed e r r o r -Au) i s a m p l i f i e d by a servo-motor '94, - A W - 4 0 - (T + Zg p l + 0.5? w p ( S t i r P l + Ig P F i g . 7 - 1 . Block diagram of a conventional h y d r a u l i c governor -aw +„ i 9 1 - I w P A P i t <r + zg p 1 + 0.5 I „ p 5 t T r p 1 + I r p 1 + Za p F i g . 7 - 2 An optimum s t a b i l i z i n g s i g n a l u-^  of the governor -AW +, 1 9 1 - l v f P ) < r + t g p 1 + 0.5 I*p l+ l a p U2 F i g . 7 - 3 An optimum s t a b i l i z i n g s i g n a l u 2 of the governor .'• " 9 5 . which causes gate change g. . The change of gate opening actuates a dashpot•mechanism to provide a feed back s i g n a l -g f . 7 . 1 . 2 Proposed C o n f i g u r a t i o n of Governor w i t h the Optimum  S t a b i l i z i n g S i g n a l s . P i g . 7 - 2 and F i g . 7 - 3 show two. proposed schemes w i t h s t a b i l i z i n g s i g n a l s based on F i g . 7 - 1 . In the f i r s t case, F i g . 7 - 2 , the optimum s t a b i l i z i n g s i g n a l u-j^  i s introduced through an a c t u a t i n g element w i t h a gain u a and a time constant T a . The dashpot i s con-t r o l l e d ' by the a c t u a t i n g element. According to F i g . 7 - 2 the governor equations now become | f = - y- {AuH-gf+org}' g d g f d g f x g f d g f -dt^^-T: > a u r s f n } a 1 In the second case, F i g . 7 - 3 , the optimum s t a b i -l i z i n g s i g n a l Ug i s introduced i n the same way as Fig... 7 - 2 , but w i t h the dashpot removed. Thus the governor equations become f§-= - T~ fou>+gf+ag) g dg f. 1 ( 7 - 2 b ) 96 . Equations ( 7 - l a ) and ( 7 - l b ) are l i n e a r i z e d and the optimum s t a b i l i z i n g s i g n a l i s determined as a l i n e a r combina-t i o n s of the s t a t e v a r i a b l e s i n the form • u l = u l l A 6 H 2 A ^ 1 3 ^ 1 4 ^ 1 5 % ^ ^ (7 -4 ) and U 2 = u 2 l A & + U 2 2 A , J ^ U 2 3 S + U 2 4 S f + U 2 5 h (7-5) The s t a t e v a r i a b l e s i g n a l s i n ( 7 -4 ) and ( 7 -5 ) must be a v a i l a b l e f o r the optimal c o n t r o l . 7 . 1 . 3 L i n e a r Optimal C o n t r o l Problem Based on the Proposed  C o n f i g u r a t i o n s . With the equations ( 7 - l a ) and ( 7 - l b ) l i n e a r i z e d and w r i t t e n as dA 6' . -- d t " = ^ = lb [~\ 5 V UJ"I"S-+1'5h 3 ( 7" l d ) together w i t h equations ( 7 -2a) and (7 -3 ) and w i t h the f o l l o w i n g changes of v a r i a b l e s , / P A / / P A wo n = A u ) . u ) 0 / = F " A u ) the system equations become dx ( 7-le) dT =. A x x -!• b ^ ( 7 -6 ) where 97.. A l =• 0 1 0 0 Vo V V . . o 2 H P 2 2 H 0 2 H P 2 0 1 a 1 ° g 0 0 0 I 0 0 0 0 0 2 2a' 2 Vg p g 0 0 F T a a 0 •l..'5«) c 2 H P 2 0 0 0 w (7 -7 ) x -- A5 n g g-P h and - A ^a b = u n 0 0 0 1.0 (7-8) (7 -9 ) a The optimal c o n t r o l i s t o be obtained from the m i n i m i z a t i o n of a performance f u n c t i o n J l = i L ^ i ( x ) + ' i 1 ^ i } d T 3 cp 1(x) = x t C 1 x (7-10) where C-^  i s a 6x6 p o s i t i v e d e f i n i t e or p o s i t i v e s e m i - d e f i n i t e m a trix and f a ^ i s a p o s i t i v e constant',which- may be chosen a r b i t r a r i l y . The system equations of the second case, u s i n g 9 8 . (7-Id), ( 7 - l e ) , (7-2b) and (7-3), can be w r i t t e n as cT7 = A 2 X + b 2 U 2 where A, 0 1 0 0 0 Vo °o 0 1 . 5 » 0 2 H p 2 2HP 2 H P 2 2 H P 2 0 1 a 1 0 - o g S ~ P ^ 0 0 0 1 " ?Ta 0 0 2 2d 2 2 o g P T P T „ g " T P w K (7 - 1 1 ) (7 - 1 2 ) x = I A 6 ' n g g f h - L ^a and u 2 = -^r - u 2 a b^ = 0 0 0 1 (7-13) (7-14) The optimal c o n t r o l u 2 i s t o be obtained to minimize the performance f u n c t i o n J 2 , 1 r 0 0 2 t J 2 = 2 J ( c P 2 ^ X ^ a 2 a 2 ^ d T > = X ° 2 X (7 - 1 5 ) • ' ' r w i t h respect to the system equations ( 7 - l l ) , where . is a 5x5 p o s i t i v e d e f i n i t e or p o s i t i v e s e m i - d e f i n i t e matrix and a 2 i s < p o s i t i v e constant which may be chosen a r b i t r a r i l y . A ccording t o Chapter 2 , the optimum s t a b i l i z i n g f u n c t i o n u^ i s given by u = - ~-.bjKx ( 7 - 1 6 ) a l where K i s a p o s i t i v e d e f i n i t e constant matrix of the optimum Lyapunov f u n c t i o n V of the system (7-6), where V - x ^ x and K may be obtained from the steady s t a t e s o l u t i o n of the matrix d i f f e r e n t i a l equation " . • # ( T ' ) = KA X + A^K - { K b ^ K } + C 1 . (7-17) d T ' . a l w i t h the i n i t i a l c o n d i t i o n , K(o) = o . ( 7 - 1 8 ) 7 . 1 . 4 Numerical Study of Governor Optimum S t a b i l i z i n g S i g n a l s , u^ and Ug . The pov/er system i n Chapter 4 , w i t h the same . system parameters and"operating c o n d i t i o n s w i l l be s t u d i e d . 'Using .D'(6') and R'( 6•'•'') computed i n Chapter 4 , from ( 7 - l c ) .one has • D Q = 2 . 0 2 6 4 , , and 1^= 0 . 7 7 5 I n order to determine the optimum s t a b i l i z i n g f u n c t i o n s the f o l l o w i n g governor s e t t i n g s w i l l ' be used throughout the computation unless otherwise s t a t e d , 6. = 0.8 , a = 0.045 , T = 1 1 , w . = 1 . 6 , T = 0 . 0 2 e. and T = 0 . 0 1 seconds. From ( 7 ~ l e ) a T = 7.308t , and n = y ^ g f = 53.62Aiu . From ( 7 - 7 ) - ( 7 - 9 ) , A l = 0 . 0 0 0 0 1 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 - 0 . 6 8 3 8 - 0 . 0 3 4 6 6 0 . 8 8 2 8 0 . 0 0 0 0 0 . 0 0 0 0 1 . 3 2 4 2 0 . 0 0 0 0 - 0 . 1 3 2 6 - 0 . 3 0 7 9 - 6 . 8 4 1 8 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 - 0 . 0 1 2 4 5 - 1 0 . 9 4 8 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 - 1 3 . 6 8 5 2 0 . 0 0 0 0 0 . 0 0 0 0 0 . 2 6 5 2 0 . 6 1 5 8 1 3 . 6 8 3 6 0 . 0 0 0 0 - 0 . 1 7 1 X = and A6' n g b. = u n = 1 0 0 7 . 3 0 8 ^ a u l 0 0 0 . 8 1 . 0 0 By choosing a.± = 1 . 0 and c 1 as a u n i t matrix or ^1 ~ L b + n + s + ? f + ^ + h • f the K matrix i s obtained from i n t e g r a t i o n of ( 7 - 1 7 ) w i t h zero i n i t i a l c o n d i t i o n . For example K ( T ' = 5 4 ) equals 3 . 0 1 2 0 . 6 9 2 - 6 . 8 2 1 - 2 3 . 8 0 3 1 8 . 8 1 1 - 3 . 6 3 4 0 . 6 9 2 4 . 9 0 2 0 . 4 6 6 - O . 9 8 3 2 . 6 5 0 2 . 7 6 4 K = - 6 . 8 2 1 0 . 4 6 6 64 , , 2 8 5 9 7 . 2 8 0 - 7 8 . 5 1 2 3 1 . 0 3 6 - 2 3 . 8 0 0 - O . 9 8 3 9 7 . 2 8 0 1 0 3 5 9 . 8 - 8 2 7 9 . 1 6 7 . 4 3 6 + 1 8 . 8 1 0 2 . 6 5 0 - 7 3 . 5 1 2 - 8 2 7 9 . 1 6 6 1 8 . 0 - 5 2 . 4 0 3 - - 3 . 6 3 4 2 . 7 6 4 3 1 . 0 3 6 6 7 . 4 3 6 - 5 2 . 4 0 3 . 1 7 . 4 9 5 from ( 7 - 1 6 ) u 1 = 0 .231A5' - 1 . 8 6 4 n + 0 . 6 8 8 g - 8 . 7 Q 2 g f + 5 . 2 8 5 g f . I l 1 . 5 4 5 n . . . In the second case, from ( 7 - 1 2 ) to ( 7 - l 4 ) one has A 2 = 0 . 0 0 0 0 1 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 - 0 . 6 8 3 8 - 0 . 0 3 4 6 0 . 8 8 2 8 0 . 0 0 0 0 1 .3242 0 . 0 0 0 0 - 0 . 1 3 2 6 - 0 , 3 0 7 9 - 6 . 8 4 1 8 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 - 1 3 . 6 8 3 6 0 . 0 0 0 0 0 . 0 0 0 0 0 . 2 6 5 2 0 . 6 1 5 8 1 3 . 6 8 3 6 - 0 . 1 7 0 0 X = A6' n g g f h b 2 = 0 0 1 0 and 1 0 0 u a u 2 u 2 ~ ~ T 7 3 0 S " with Cp chosen as a unit matrix , cp2 = A 6 / 2 + n 2 + g 2 + g| •+ h 2 and with a"g = 1 . 0 , the K matrix determined from (7-17) for T 1 = 2 7 . 5 i s 2 . 7 7 3 0 .724 - 6 . 2 9 7 - 0 . 0 9 8 - 3 . 1 6 9 0 . 7 2 4 4 . 2 6 8 O . 7 8 3 1 . 7 9 1 2 . 3 8 3 - 6 . 2 9 7 O . 7 8 3 6 2 . 7 6 5 - 0 . 9 2 5 3 0 . 1 4 7 - 0 . 0 9 8 1 . 7 9 1 - 0 . 9 2 5 1 . 7 0 0 1 . 3 0 6 - 3 . 1 6 9 2 . 3 8 3 30 . 1 4 7 1 . 3 0 6 1 6 . 3 8 5 From ( 7 - 1 6 ) one has u 2 = 0 . 0 9 8 A 6 ' - 1 . 7 9 1 n + 0 . 9 2 5 g - 1 . 7 0 0 g f - 1 . 3 0 6 h In the same manner, various u^ and Ug can he '. obtained f o r d i f f e r e n t governor parameter settings and cp^  and cpg . In order to study s t a b i l i z e d systems, time responses of the systems ( 7 - 6 ) and ( 7 - 1 1 ) are computed with i n i t i a l values of a l l the state variables set to zero except A 6 ' = 0 . 2 radians. Typical r e s u l t s are summarized i n F i g . 7-4 to F i g . 7 - 8 together with the' following observations:— In. F i g . 7 - 4 , dotted l i n e s show the time responses of the power system with conventional governor as shown i n Fi g . 7 - 1 . The parameter settings of the governors are 6^ = 0 . 8 p.u, a = 0.045 p.u. T G = 0 . 0 2 seconds, T R = 1 1 seconds and, the water time constant i s T W = 1 . 6 seconds. In the same figure solid l i n e s represent the time responses of the system with the s t a b i l i z i n g s i g n a l u n introduced as shown , 103 i n F i g . 7 - 3 . System i n s t a b i l i t y i s observed from the dotted l i n e s responses but not from the solid l i n e s responses. With, the s t a b i l i z i n g s i g n a l u-^  introduced the system i s not only ' made s t a b l e but a l s o settlesdown q u i c k l y . The computational d e t a i l for u^ has been given at the beginning of t h i s s e c t i o n i n an example. . F i g . 7 - 5 compares the e f f e c t s of the s t a b i l i z i n g s i g n a l s on the power system responses. The dotted l i n e s r e -present system responses w i t h u^ , (7-9), and the solid. l i n e s those w i t h u 2 , ( 7 - 1 1 ) . The two s t a b i l i z i n g s i g n a l s give e q u a l l y good system responses. When more emphasis i s placed on the r e d u c t i o n of the magnitude of gate change g , and that of head change h dur i n g t r a n s i e n t , optimal s t a b i l i z i n g s i g n a l can be determined by s p e c i f y i n g the f u n c t i o n cp-^  , ( 7 - 1 0 ) , i n the. f i r s t case and cp2 , (7-15), i n the second case w i t h l a r g e r c o e f f i c i e n t s f o r g and h . For example, when cp2 = A 6 / 2 + n 2 + 1 0 g 2 -1- 10g 2 + 1 0 h 2 , u 2 = -0.0242A 6 ' - 1 . 7 6 5 0 5 ^ 2 . 7 3 1 2 4 g - 3 . 4 8 0 4 g f - 2 . 1 9 2 h -the r e s u l t i n g system responses are p l o t t e d i n dotted l i n e s i n F i g . 7 - 6 . For another, example, when cp2 '= A 6 / 2 + n 2 + g 2 + g2.+ h 2 < u 2 = 0 . 0 9 8 A 6 , - 1 . 7 9 1 n + 0 . 9 2 5 g - 1 . 7 0 0 g f - 1 . 3 0 6 h the r e s u l t i n g . system responses are p l o t t e d i n solid l i n e s i n the same figure' f o r comparison. I t i s seen th a t the g and h have sm a l l e r amplitude by dotted l i n e than those by the solid l i n e s . However-'the reverse i s true f o r power angle v a r i a t i o n s A 6' and speed v a r i a t i o n n . The A 5' and n represented by the dotted l i n e s tend t o have l a r g e r amplitude of o s c i l l a t i o n s and take longer time t o s e t t l e down than those by the solid l i n e s . S i m i l a r phenomena are found i n the second case. P i g . 7 - 7 shows the e f f e c t on the power system r e -sponses when the feedback servo -time constant' T , i s v a r i e d . The dotted l i n e s represent responses of the system x^hen u^ i s determined from ( 7 - 1 1 ) as given i n the numerical example w i t h T & = 0 . 0 1 } and the s o l i d l i n e s represent those w i t h T = 0 . 1 ; u_ = - 0 . 3 0 8 3 3 A 5 / - 1 . 5 3 3 8 5 n + 1 . 7 3 8 4 3 g - 5 . 1 3 2 4 8 g „ - 0 . 5 7 0 0 4 h . a c. I Higher amplitudes of the g, h and n v a r i a b l e s of the f i r s t swing are observed w i t h T = 0 . 1 than those w i t h T = 0 . 0 1 . a a However the amplitude d i f f e r e n c e s become n e g l i g i b l e a f t e r 1 second. .The same cp^  f u n c t i o n cp2 = A 6 / 2 + n 2 •+ g 2 + g 2 + h 2 i s - u s e d i n both computations. Fig'. 7-8 shows the system responses w i t h s t a b i l i z i n g s i g n a l u^ determined from d i f f e r e n t governor s e t t i n g s . Dotted l i n e s ^ r e p r e s e n t the responses of system w i t h 6^  = 0 . 2 5 T R = 5 • In t h i s case F i g . 7-4 Power system r e s p o n s e s w i t h and w i t h o u t optimum s t a b i l i z i n g s i g n a l u . F i g . 7-5 Comparison between power system responses w i t h s t a b i l i z i n g s i g n a l s and Ug P . U . G A T E , H E A D , E T C *--d (ft I CO c+ CD 3 CD h i < CD P. Hj CD 4 tn O Ti 0 O p 4 m H* CD Hj CO Ms CD Hj H O CD 4 O CD O 3. o CD <! CD 3 o 4 m ct P H H-N H-O 3 .01 Co p" p (—1 CO o .150 i -.075 i _ .000 .075 i .150 i .225 i P . U . S P E E D X 5 3 . 6 2 .150 i - .075 i - .000 • .075 .150 i .225 i A N G L E RAD. .150 _.075 - .000 1 .075 i .150 .225 o o ''LOT H O O O 0 CO ro c+ CD 3 I H) as o c+ O •-»> o 13* 4 0) « ! o 0) ja CO CD c+ CD O c+ CD CD P< a1 P o -.150 ! - .075 i P . U . GATE , - .000 i HEAD , .075 i E T C . .150 i • .225 i _.I50 i - .075 i P . U . SPEED - . 000 i X 5 3 . .075 i 62 .150 i .225 i ,_.I50 - .075 A N G L E - . 0 0 0 R A D . .075 i .150 .225 *80I CM CM-o i n U l -UJ ~ «/•» <<=> LU < ° o 1 o CM CM "I CM CM O OA CM LTJ X Q LsJ o 111 O CL O to I 3 CL o i o 1 I . 0 0 0 500 1.000 1.500 T I M E S E C S -2 .000 2 .500 F i g . 7-8 System responses w i t h s t a b i l i z i n g s i g n a l s u-, w i t h d i f f e r e n t governor s e t t i n g s H O \x1 = 1 . 0 9 3 6 8 A 6 ' - 2 . 1 3 6 3 6 n - 0 ' . 9 l 6 7 7 g - 4 . 2 3 2 0 6 g f + 0 . 4 0 6 6 g f - 2 . 9 7 9 0 8 h Solid l i n e s represent the responses of system w i t h 6^  = 0 . 8 and T = 1 1 . The remaining governor s e t t i n g are the same as s t a t e d i i ; the beginning of t h i s s e c t i o n . The d i f f e r e n c e be-tween the dotted l i n e s and the solid l i n e s are r e l a t i v e l y s m a l l which i n d i c a t e s that the s t a b i l i z i n g s i g n a l gives good system responses f o r wide range of the 6^. and T p s e t t i n g . 7 . 2 Optimum Governor and Voltage-Regulator S t a b i l i z i n g  S i g n a l s ' f o r a H y d r o - e l e c t r i c System. 7 . 2 . 1 System Equation I n c l u d i n g Voltage-Regulator, Governor  and Optimal C o n t r o l s . . " In s e c t i o n 7 . 1 the s t u d i e s concerning the optimal s t a b i l i z a t i o n of a power system through a feed back i n the governor feedback loop are based on the second order model, of the synchronous machine i n f i n i t e - b u s equations and w i t h the v o l t a g e r e g u l a t o r a c t i o n ignored. In t h i s s e c t i o n an a d d i t i o n -a l o p t i mal s t a b i l i z i n g s i g n a l i n the v o l t a g e r e g u l a t o r feedback loop w i l l be introduced. The i n t e r a c t i o n of u^ 3 the s t a b i l i z i n g s i g n a l of the governor, and u^ , that of the v o l t a g e - r e g u l a t o r w i l l now be s t u d i e d . P i g . 7 - 9 shows the t r a n s f e r f u n c t i o n block diagram of a v o l t a g e - r e g u l a t o r w i t h an optimal s t a b i l i z i n g s i g n a l u-, i n the feedback loop through an a m p l i f i e r w i t h gain u s and time constant T . The equation f o r the r e g u l a t o r can be w r i t t e n as dAv xx. Av,? , a t ^ - t l-4v t-v s} (7-19) e e A V F ) 1 + Z e p u 3 I + Z s p P i g . 7-9 An optimum s t a b i l i z i n g s i g n a l u^ i n the v o l t a g e - r e g u l a t o r ^ The equations are l i n e a r i z e d w i t h the a i d of ( 3 - 8 ) , ( 3 - 4 - 9 ) and ( 3 - 4 0 ) which can be w r i t t e n i n the f o l l o w i n g form d A v F ^e A A . , , ^ e . A , A v F ^ e v d t — = T~ vd + — ve F " — " T- Vs e e e e dv„ v u ( 7 - 1 9 a ) dt T T 3 s s When the v o l t a g e - r e g u l a t o r i s i n c l u d e d , the t h i r d order synchronous machine i n f i n i t e - b u s model w i t h 6, Auu and ^ as s t a t e v a r i a b l e s , (3-4-2a) to ( 3 ~ 4 2 c ) , can be used. A f t e r l i n e a r i z a t i o n around an e q u i l i b r i u m p o i n t , 6 = 6 Q , A«J = o and i F = * p , ( 3 - 4 2 a ) to ( 3 - 4 2 c ) become ' ( 7 - 1 9 b ) Sr - ~ "2> + \e^F + h + -iirh d A * p — = - T i 2 s i n ( 6 q - P ) A 6 / - ThJUp + A v p The constants A y d , A y e , A ^ and A ^ i n ( 7 - 1 9 a ) and ( 7 - 1 9 b ) have be en defined i n ( 5 - 3 1 ) of Chapter 5 » They are f u n c t i o n s of i n i t i a l c o n d i t i o n s llr-- , i!r,„, ft and 6 ?F * '"do ' qo o o ^ which can be determined from the given P+jQ and V. accord-^o i n g to Appendix 5D. Note that-the governor a c t i o n g+1.5h i s a l s o i n c l u d e d i n ( 7 - 1 9 h ) . . With the f o l l o w i n g changes of v a r i a b l e s n = Awou /P' where a' i s . a constant, which may be chosen as 1.0 or J P m as i n ( 7 - l e ) , the system equations ( 7 - 1 9 a ) , ( 7 - 1 9 b ) , and ( 7 -2b) may be w r i t t e n as 4 ^ = A-,x + Bu Q. T j> where V o 0)0. o o we O g 0 S 2' 2HP' P 2 ' s l n ( 5 o - P ) 0 i 3L u e A v d . Q ^e Ave. ' 1 • V * ' ' T Q P ' ' . p ' T e o ' 6 . 0 . o o 0 0 0 0 0 . — 0 0 ° g 0 0 0 0 2HS 2 / 0 S 2 /2H 0 0 (7-22) 0 0 0 a P'T S I:: 0 o 2a 3'T 1 P'T a 0 P'T,, P'T, w A 6' ' > B =• ~0 0 n 0 0 0 0 A v p C 0 1 0 g • 0 0 g f 0 1 h 0 0 He U 3 = P'T. u, and = 3 2 a u. u = u 3 115. (7-23) (7-24) The optimal c o n t r o l u i s to be obtained from the mi n i m i z a t i o n of a performance f u n c t i o n OO J 3 = ! J {cp 3(x)+u tRu}dr , cp 5(x) - xtC^x (7-25) subject t o ( 7 - 2 1 ) , where i s a 8x8 p o s i t i v e d e f i n i t e or semi-d e f i n i t e m a t r ix, and R i s a 2x2 p o s i t i v e d e f i n i t e sym-metric matrix which may be chosen a r b i t r a r i l y . A ccording t o Chapter 2 the s o l u t i o n f o r u i s given by u = - R ' V K X (7-26) inhere K i s a p o s i t i v e d e f i n i t e constant matrix of the optimum Lyapunov f u n c t i o n which may be obtained from the steady s t a t e s o l u t i o n of — ( T ' ) = K A . , + A ^ K - K B R ' V K + C-, d<r' * 5 ^ (7-27) . /. . . . 1 1 6 . w i t h the i n i t i a l c o n d i t i o n K(o) = o ( 7 - 2 8 ) 7 . 2 . 2 Numerical Study of the E f f e c t of Governor and Voltage- Regulator Optimal S t a b i l i s i n g S i g n a l s . The pov/er system s t u d i e d i n s e c t i o n 7 . 1 . 4 has a steady s t a t e power output p'+jQ = 0 . 7 5 3 + J O . 0 3 and v t o ~ - " - ' ^ 5 p.u. The i n i t i a l c o n d i t i o n s computed by the method given i n Appendix 5A are *Fo = 9 . 4 8 , ^ d Q = 0 . 9 7 7 / t q p > - 0 . 3 8 % 6 Q = ; 0 . 8 8 7 , A««;='"o. 1 0 . , T S = 0 . 5 , T A = 0 . 0 1 , T g = 0 . 0 2 , 7 . 3 0 8 one has from ( 7 - 2 2 ) .With' T = 1 . 0 , u = e 3 "e a = 0 . 0 4 5 and p' = 0 . 0 0 0 • 1 . 0 0 0 - 0 . 6 8 J 8 - 0 . 0 3 4 6 6 - 0 . 0 8 3 3 3 0 . 0 0 0 0 0 . 1 2 9 7 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 - 0 . 1 3 2 6 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 2 6 5 2 0 . 0 0 0 0 . 0 0 0 - 0 . 0 8 1 9 0 . 0 0 0 - 0 . 0 2 5 3 7 0 . 1 3 6 8 - 0 . 1 0 6 4 6 - 0 . 1 3 6 8 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 8 8 2 8 0 . . 0 0 0 0 0 . 0 0 0 0 - 1 . 3 6 8 3 . 0 . 0 0 0 0 - 0 . 2 7 3 6 7 0 . 0 0 0 0 0 . 0 0 0 0 - 0 . 3 0 7 9 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 6 1 5 5 0.0000 0.0000 0.0000 1.3242 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -6.8418 0.0000 - 1 3 . 6 8 3 6 0.0000 1 ^ . 6 8 3 6 ,-0.1710 By choosing and R as u n i t matrices, the numerical values f o r K i s obtained from ' (7-27) u s i n g (7-28) i n the same way as In s e c t i o n 7 . 1 . 4 . In t h i s p a r t i c u l a r study K = 2.786 0.726 0.726 4.356 0.119 -O.969 0 . 142 - 0 . 0 6 6 -0 .134 - 0 . 0 0 3 - 6 . 6 0 9 1.827 - 0 . 0 8 2 1.790 - 3 . 3 0 8 2.902 0.119 0 . 142 -0 .134 - 6 . 6 0 9 -O.969 - 0 . 0 6 6 - 0 . 0 0 3 1.827 7 .464 1.111 - 0 . 7 2 8 - 9 . 1 0 7 1.111 I . 3 7 8-O . 9 6 3 -1 .590 - 0 . 7 2 3 - 0 . 9 6 3 1.652 1.104 - 9 . 1 0 7 -1 .590 1 .104 78 .946 -0.115 0.012 - 0 . 0 2 6 -1 .106 - 4 . 6 7 1 -0 .778 0.520 38.032 - 0 . 0 8 2 - 3 . 3 0 8 1.790 2.903 -0.115 - 4 . 6 7 1 0.012 - 0 . 7 7 8 - 0 . 0 2 6 0.521 -I . 1 0 6 38.032 I . 7 0 6 1.223 1.223 20.232 ana u . 0.134. 0 .003 0.728 0.963 -I . 6 5 2 -1 .104 0.026 - 0 . 5 2 1 0 .082 -1 .790 0.115 - 0 . 0 1 2 0.026 1.106 -1 .706 - 1 . 2 2 2 i.e. A 6 ' n Av P h u 5 = 0 . 134A 6 '+0.003n+0. 728A i|?p-0.963A v p- 1 . 6 5 2 v £ - 1.104g+ 0 . 0 26 g f - 0 . 5 2 1 u^ = 0.08 2A 6 ' - 1 . 790n+0 .115A <!r 0.012A v^+0.0 26v e +1.10 6g-1.70 6g_~-1. 222 d Si Si a X The f o l l o w i n g s t u d i e s are concerned w i t h the r e l a -t i v e e f f e c t s of u 2 and u^ on the system responses. The 1 1 9 . i n i t i a l c o n d i t i o n s are.set a l l equal to zero except A6' = 0 . 2 r a d i a n s . The r e s u l t s are summarized i n F i g . 7 - 1 0 to 7 - l 4 . It ha,s been found from t h i s study t h a t the i n t r o -d u c t i o n of an optimal s t a b i l i z i n g s i g n a l i n t h i s case has l i t t l e e f f e c t on the time responses of the system v a r i a b l e s g, h, n and A 6' . Furthermore, i c has been found that a l - * though the s t a b i l i z i n g s i g n a l s are f u n c t i o n s of a l l s t a t e v a r i a b l e s , some terms i n the f u n c t i o n s can be ignored without making much d i f f e r e n c e i n the time responses of g, h, n and A 6' . In F i g . 7 - 1 0 comparison i s made between system re-sponses of the case when some terms i n the s t a b i l i z i n g s i g n a l f u n c t i o n s are ignored, and those when s t a b i l i z i n g s i g n a l Ug and u^ are f u l l y employed. The dotted l i n e s represent the responses of system v a r i a b l e s g, h, n and A 6' f o r which u 2•= - 1 . 7 9 n + 1 . 1 0 6 g - 1 . 2 2 2 h T L = o . i . e . the s t a b i l i z i n g s i g n a l s t o the v o l t a g e - r e g u l a t o r and most of the terms i n the s t a b i l i z i n g s i g n a l s t o the governor are set to zero except f o r n, g and h . The solid l i n e s r e -present, the responses of system v a r i a b l e s g, h, n and A§' f o r u_ and TL, c o n t a i n i n g a l l s t a t e v a r i a b l e s . The d i f f e r -2 3. 0 ence between the solid and dotted l i n e s i s 'small. In the f i g u r e only f o u r s t a t e v a r i a b l e s are show.. The time responses of a l l eight s t a t e v a r i a b l e s corresponding t o the solid l i n e s of F i g . 7 - 1 0 are shown i n 1 2 0 . P i g . 7 - 1 1 and those to the dotted l i n e s i n F i g . 7 - 1 3 -From the comparison of F i g . 7 - 1 1 and F i g . 7 - 1 2 i t is evident that by s e t t i n g u. equal zero the mechanical v a r i a b l e g, h, n and A§' are not e f f e c t e d except f o r the e l e c t r i c a l v a r i a b l e s A typ, , A v p and v g . F i g . 7 - 1 3 f u r t h e r i n v e s t i g a t e s the e f f e c t of the v a r i a b l e h i n the u^ f u n c t i o n , by s e t t i n g h equal to zero i . e . u^ = -1.79n + 1.106g, u^ = o . The time response of g, h, n and A&' are shown i n dotted l i n e s i n F i g . 7-13-The responses of the system when u 2 = 1 . 7 9 n + 1 . 1 0 6 n - 1 . 2 7 2 h , u^ = o , i . e . w i t h h i n c l u d e d , are shown i n solid l i n e s . Comparison o f the two sets of curves i n d i c a t e s that the h v a r i a b l e s i n u^ i s important and must not be ignored. F i g . 7-14 shows the responses of a l l eight v a r i a b l e s when Ug = 1.79n , = o , I.e. both g and h are ignored i n u 2 . Comparison of the r e s u l t s w i t h those i n F i g . 7 - 1 2 , i.e. w i t h both g and h inc l u d e d i n u^ , shows th a t g and h are very important i n u^ . It Is i n t e r e s t i n g to observe t h a t , from a comparison of the mechanical system v a r i a b l e s g, n, h and A 6' i n solid' l i n e s of F i g s . 7 - 5 and 7 - 1 2 , the two sets of curves a l -most c o i n c i d e w i t h each other. Although F i g . 7 - 5 corresponds to the f i f t h order system i n c l u d i n g the second order machine model and F i g . 7 - 1 0 corresponds t o the eight order system i n c l u d i n g the t h i r d order machine model, the r e s u l t s i n d i c a t e 3 H-CO o S3 CD ct-(D h; co O ro CD p H CO N CD O P. •3 p-(ft I H O O O •§ P H> CO O O H> CD CO •d o 0 CO CD CO & CD ct .-< • H CD CD CO <<i CO c+ CD H-c+ 3-II o ro p . CO «i CO ct (D -.150 i P . U . GATE , HEAD , E T C . .075 _.000 .075 .150 .225 _.I50 i P. U . S P E E D X 53 .075 _.000 .075 6 2 .150 .225 150 -.075 A N G L E R A D . _.000 .075 . 150 .225 •1ST -.150 P . U . G A T E , H E A D , E T C . .075 - . 0 0 0 .075 .150 .225 i t—1 H co ct-P a* H-I—1 H-N (ft CO D2 CO c+ CD P H ro . P. 4 CO CO >d o CO CD CO s; ch 3* el-s' CD O •d cf-H« I 13 1 .150 i i - . 0 7 5 • P . U . SPEED - . 0 0 0 • » X 5 3 . 6 2 .075 i . 150 i i .225 i .150 - .075 i A N G L E - . 0 0 0 R A D . .075 . 150 .225 •c3c3I - 5 I H ro (O CO £ to to 3 N l-i CD CD 4 C O O 13 o P C O P- 6 to c+ P ' C O o CD c+ CD a C O I-" ro P . U . G A T E , H E A D , E T C . - . 1 5 0 _ . 0 7 5 _ . 0 0 0 .075 .150 I— 1 1 ! ! 225 P . U . S P E E D X 5 3 . 6 2 - . 1 5 0 - . 0 7 5 - . 0 0 0 .075 .150 « J i i i 225 , - . 1 5 0 - . 0 7 5 A N G L E R A D . - . 0 0 0 .075 50 • 1 2 6 . that the f i f t h order system i s as good as the eighth order system as f a r as t h i s study i s concerned. • For the eighth order system w i t h t h i r d order machine model,computations have a l s o been made to f i n d the system responses g, h, n and A5 ' u s i n g the c o n t r o l s i g n a l , u^ , de r i v e d from the f i f t h order system w i t h the second order machine model, u 2 =.098A6' - 1 . 7 9 1 n + 0 . 9 2 5 g - 1 . 7 0 0 g f - 1 . 3 0 6 h The r e s u l t s , which are not shown, are almost the same as those u s i n g u 2 = 0 . 0 8 2 A 6 ' - 1 . 7 9 0 n + 1 . 1 0 6 g - 1 . 7 0 6 g f - 1 . 3 0 6 h + 0 . 1 1 5 A * p -- 0 . 0 1 2 A v T ? + . 0 2 6 v F S as der i v e d from the ei g h t order system as shown i n the r e a l ' l i n e s of F i g . 7 - 1 0 . 7 . 3 Nonlinear Governor S t a b i l i z i n g S i g n a l 7 . 3 . 1 Optimal C o n t r o l Problem of a T h i r d Order Nonlinear  Power System. ' Consider a problem o f determining a c o n t r o l u(x) that minimizes a performance f u n c t i o n , w ' = \ r w(x,u)dT (7-29) o where W(x,u) - G(x) + a u 2 (7-30) 1 2 7 . and G(x) = £ Z H- {p . x ^ k l ^ ^ 2 l j 1=2 k-,=l k 0 = l ^ K l ^ K 2 1 1 • • ' k -1- k -1 X 2 X 3 1 ( 7 - 3 0 ) with respect to dx., _ I F = a l l x l + a12 x2 + a l 3 * 3 . + P l u dx ? _ dT = a 2 1 X l + a22 X2 + a23 X3 " 5 ( x l ) x 2 " Mx±) + ^ 2 U ( 7 - 3 1 ) dx^ , _ d P T = a31 Xx + a 3 2 x 2 + a33 x3 + P 3 U where • £(x x) ,= D Q + D 1x 1 + D x 2 + ... ( 7 - 3 2 ) R(x 1) = R 1x 1 + R 2 x i + *•* In (7-30) to ( 7 - 3 2 ) the p . , a , p's, D's and R's are J l 5 2 a constants. In the last' two equations, i f one i d e n t i f i e s x-^ = 6-^ ' , x 2 = n and x^ = g , and sets \ ~ a l l = a13 = a22 = a 3 1 = P l = P 2 = °> P3 = 1> ( 7 ~ 5 5 ) t> Ru) 2HP > - 2 Hp2 » Z 3 32 CU 0T g 33 /3 7 f i and with' - A u u = where D and R are defined i n ( 7 - . l c ) , then the problem 1 2 8 , of determing u of (7-29) becomes f i n d i n g a s t a b i l i z i n g s i g n a l u(x) f o r the power system described by ( 7 - l a ) , ( 7 - l " b ) aad ( 7 - l c ) w i t h A P ^ g and w i t h the f o l l o w i n g change* of v a r i a b l e s , i . e . , T = Bt n = i F oA ai as define d i n ( 7 - l e ) . The power system now has a simple governor c o n t r o l w i t h a forward t r a n s f e r f u n c t i o n AU) CT+T p ( 7 - 3 4 a ) and a feedback s t a b i l i z i n g s i g n a l u governor gate change g becomes _ ( A U H - U ) - (a+ T „ p ) With t h i s s i g n a l the ( 7 - 3 4 b ) form Equation ( 7 - 3 1 ) and ( 7 - 3 2 ) can be w r i t t e n i n the dx d T = f ( x , u ) ( 7 - 3 1 a ) where f = Ax + g(x) + bu ( 7 - 3 5 ) X = [ x x x 2 x^ a l l a 1 2 a n l p A = - ^ l + a 2 1 -D o+a 2 2 a 2 3 ( 7 - 3 6 ) a 3 1 a 3 2 a 3 3 g(x) and 0 - ! 2 ( ^ - i x i k " l x 2 + ^ x i ) 0 b = P-P, P: 1 2 9 ( 7 - 3 7 ) ( 7 - 3 8 ) 7 . 3 . 2 Development of the Computation A l g o r i t h m According to Chapter 2 , the optimal c o n t r o l u can be determined by s o l v i n g the p a r t i a l d i f f e r e n t i a l equation 3 ( 7 - 3 9 ) and E BV_ , 1 . aw i = l - ax. 2 au 1 au = o where V(x) i s the optimum Lyapunov f u n c t i o n . Equation ( 7 - 3 9 ) w i l l be solved by assuming the f o l l o w i n g truncated s e r i e s forms f o r V and u 3 v(N+i) =..v2(x) + v 5 ( x ) + ... + y N + 1 ( x ) u(N) = u 1 ( x ) + u 2 ( x ) + + u N ( x ) ( 7 - 4 0 ) The a l g e b r a i c equations f o r the determination of v 2 and u^ are . f i r s t obtained from comparisons of terms of l i k e degree i n x^, x 2 and x^ . The equations f o r the determination of v-z and u„ and so on are obtained i n s i m i l a r manner. 3 2 Equations f o r - t h e determinations of v and u , , m > 2 m m-L3 can be w r i t t e n i n a general form -§3T * A x. 8 3 m^ » m = 3 , 4 , . . . (7-41) and a th where cpm i s a known homogeneous form of m degree i n Xl3 X23 X 3 3 Ax = Ax + b u 1 (7-43a) where u-^  has to be determined f i r s t . The determination of Vg and u^ amounts to s o l v i n g the optimal c o n t r o l problem of the l i n e a r i z e d system of ( 7 - 3 1 b ) , i . e . | f = Ax + b ^ ( 7 - 4 3 b ) w i t h the quadratic p a r t s of ( 7 - 3 0 ) i n the performance f u n c t i o n ( 7 - 2 9 ) , . i . e . 1 -°° 3 3-(k- L-.l) 2 - ( k 1 - l ) - ( k 2 - l ) k x - l k - 1 _ _ 2 { 2 E P Q v v x i x o x ^ + a u i 5d o k-,^1 k 2 = l 2 ' K 1 ' K 2 1 2 ^ . 1 ( 7 - 4 3 c ) Let v m , and u m be expressed i n the forms 131 m + 1 m+l-(k 2-l) m - ( k n - l ) - ( k p - l ) (k.-1) ( L - l ) v = X 2 d T- x.. * x 0 ' -x, ^ m k =1 k 2 = 1 m > k i ^ 2 1 2 5 ra = 2,3,... ' (7-44) m+1 m+l-CV 1 5 , m - ( k r l ) - ( k 2 - l ) (k.,-1) (kg-l) U„ = 2 2 S . r-'. "IZ' X- X_. X-, , m k ^ l k 2=l 1 . 2 3 7 m = 2,3, .... and let m+1 m + 1 - ( V l } m-(k -l)-( k -1) ( k - l ) ( k p - l ) m k ^ l k 2=l n^!^' 1 2 3 . (7-45a) The coefficients of cpm can be expressed as m-1 C v m " ,k x,k 2 _ " k f 2 ^ k l " 1 ^ - l d m - k + l , k 1 , k 2 + kA dm-k+l,k ] L+l,k 2 5 - m-2 k+l k+l-(i-l) + 2 *k=2 f i = l j=l e k , i , j V k + l ^ - i + l ^ - j + l } - I P m ^ k , - (7-45b) where in (7-45b) the d's, e's are coefficients of the homogeneous form. Thus either d „ p or e „ « i s set equal to zero i f either a > Y+1 3 or p > Y+1 -(a-l) (7-45c) where a > o 9 p > o The d e r i v a t i o n of (7-45a) to (7-45c) i s given i n Appendix 7A : 132. Since a l l the eigenvalues of A (7-.43a), of the l i n e a r systera (7-43b) have negative r e a l - p a r t s , (7-41) may be solved by the i t e r a t i v e method developed i n Chapter 6 as f o l l o w s : • Let r be the radi u s of a c i r c l e l y i n g i n the l e f t h a l f of a complex plane whose circumference encloses a l l the eigenvalues of A , and l e t the centre of the c i r c l e l i e on the negative r e a l a x i s at a dis t a n c e r+a from the o r i g i n . Then (7-41) can be transformed i n t o T 3v where CO 'rn k r A l A 2 k A l = I + k U ex k = 1 + -r (7-46b) and U i s a u n i t matrix. As shov/n i n Chapter 6, the approx-imate s o l u t i o n of (7-46a) and hence (7-41) can be obtained by the i t e r a t i v e process where i s the J^* 1- i t e r a t i v e s o l u t i o n of v m which converges to the s o l u t i o n of (7-41) as t - °° ,. The a l g o r i t h m based on (7-47) i s as f o l l o w s : 133 . m k-^1 k2=l * i \ > * 2 1 . 2 5 ( 7 - 4 8 ) and • o v j ^ - 1 ) m + 1 m + l - l V ^ j M ) m-(Vl)-(k p-l) m * A^x = E £ d ^ T - T r X , - 1 - ^ 3 x 2 k 1 = l k 2 = l m , k l , k 2 ^ X (k - 1 ) (k - 1 ) • x 2 1 x^ ^ ( 7 - 4 9 j . S u b s t i t u t i n g ( 7 - 4 5 a ) i n t o the f i r s t expression of ( 7 - 4 6 b ) r e s u l t s i n m+1 m+l -C^-l) m - ( k n - l ) - ( k 0 - l ) ( E . - 1 ) ( k > l ) m k ^ l k 2 = l K r ^ k l ^ k 2 1 2 3 ( 7 - 5 0 ) . S u b s t i t u t i n g ( 7 - 4 8 ) , ( 7 - 4 9 ) and ( 7 - 5 0 ) i n t o ( 7 - 4 7 ) y i e l d s the f o l l o w i n g a l g o r i t h m , d i ^ 1 ; k 2 = I % ? c m , k 1 5 X 2 + ^ ^ 2 5 . f O T m = ' A S . . . ( 7 - 5 D Note t h a t In each i t e r a t i o n , the k^, and kg "must be v a r i e d over k^ - 1, 2 , *. .m+1 and ( 7 - 5 2 ) k 2 = 1, 2 , .. ,m+l-(k" 1-l) The d e t a i l s of c y- • . i n ( 7 - 5 1 ) are given i n m, K - J ^ , K 2 (7-^5), and d j ^ 1 ^ i s deriv e d In Appendix 7B which can be' summarized as, d^" 1 ' = a + ( E r 1 }0igs2t*45 ^  j - ^ . 1 + a ^ {Ea-1 }«< g E g where a l l the d ^ - " 1 ^ c o e f f i c i e n t s i n (7-53) have the form (Jt-l) dv"; 'r, and are set to zero i f e i t h e r m, p i a > m+1 9 or 3 > m+l-(a-l) where (7_c;2|.) • a > o and P > o and a l l the a'»s are the elements of A g . A f t e r v m i s determined from the i t e r a t i v e process, u m - l c a n t e o b t a i n e d from (7-42). Since u , =' m-1 1 S V r n S V m 3Vrr> x r m o , m a , , m l % x f V ^ ^ ^ (7-55) m M V 1 * V l 83 " = LS _ 2 [ P ^ m - ^ - l ) - ( - - l ) ) d ^ -a kx=l k2=l . 1 d m^ k-^ k 2 •2 -3 2 1 m^^+l/Kg H 5 A 2 m , k 1 , k 2 + l ] x 1 . ^ x? ^ X-(7-56) From the d e f i n i t i o n i n (7-44) _ A m ^ ( V 1 ) _ ( m _ i ) _ ( k r i ) _ ( k 2 - l ) ( k r l ) ( k 2 - l ) V l = k ^ l k 2 = l ^ l , ^ , ^ X 2 X 3 (7-57) . Hence V + P j V m ^ V 1 ' (7"58' where k-^  = 1 , 2,.. .m (7-59) k 2 = 1 , 2 , . . . ^ - ( k j - l ) Thus the general term of the n o n l i n e a r c o n t r o l s i g n a l f u n c t i o n u~m-i'.^  (7-4o) i s determined along w i t h the general term of the optimum Lyapunov f u n c t i o n v m • 7 . 3 . 3 Numerical Study of the Nonlinear S t a b i l i z i n g S i g n a l  u f o r a Th i r d Order Power System. The' system equations are x = f ( x , u ) where 1 3 6 . f ( x , u ) = Ax + g ( x ) + bu a x = 6 ' n F o r t h e s y n c h r o n o u s machine i n f i n i t e - b u s s y s t e m , t h e same s y s t e m p a r a m e t e r s and o p e r a t i n g c o n d i t i o n s as i n C h a p t e r 4 w i l l be u s e d . The p a r a m e t e r s e t t i n g s o f t h e g o v e r n o r n e w l y i n t r o d u c e d a r e T = 0 . 0 2 s e c o n d s , a = 0 . 0 4 5 p . u . . g S i n c e H = 4 u}Q = 3 7 7 , a n d . 0 = 7.3 0 8 as i n s e c t i o n 7 . 1 . 4 . S u b s t i t u t i n g ( 7 - 3 3 ) i n t o ( 7 - 3 6 ) t h r o u g h ( 7 - 3 8 ) g i v e s A g(x) 0 . 0 1 . 0 L - O . 6 8 3 8 - 0 . 0 3 4 6 0 . 0 - 0 . 1 3 2 6 0 . 0 0 . 8 8 2 8 • 0 . 3 0 7 9 0 NX - 2 (A c ^ f i ' k - ^ + R f i ' * ) k = 2 . * . b =• 0 0 1 where % - l ( ^ 1 ) . ( 0 . 0 1 2 5 2 c o s ( 2 6 o + ( ^ ) 7 r ) ^ k = k l / { ^ ( S o ^ ^ ^ - ^ C O . ^ i J c o s C Q e ^ C ^ T T ) } 6 Q ' . = 0 . 8 8 7 2 and NX i s set equal to JO. By s p e c i f y i n g W(x,u) of the performance f u n c t i o n i n ( 7 - 3 0 ) as W(x,u) = x tCx + b u 2 where C i s a 3*3 u n i t matrix, the s o l u t i o n of u can be determined i n s e r i e s of the form of ( 7 - 4 0 ) . The f i r s t term i s obtained by s o l v i n g f o r where K i s determined from s o l v i n g the optimum c o n t r o l problem of the l i n e a r i z e d system ( 7 - 4 3 b ) f o r u^ which minimizes 4k , (x^Cx+u, ) d T d J o 1 corresponding t o ( 7 - 4 3 c ) one has F o l l o w i n g the same procedure as i n s e c t i o n 7 . 1 . 4 K = 2 .0670 0 . 7 1 2 7 -0 . 15913 0 .7127 3-3329 1.3546 •0 .15913 1.3546 1.5593 a n d u1 = 0 . 1 5 9 1 3 6 ' - 1.3546n - 1.5593g Then the A matrix i n (7 -43a) becomes 13* A = 0 . 0 1.0 - 0 . 6 8 3 8 - 0 . 0 3 4 6 6 0 . 1 5 9 1 3 - 1 . 4 8 7 2 0 . 0 0 . 8 8 2 8 - 1 . 8 6 7 2 whose eigenvalues are - 0 . 9 8 1 3 7 s and - 0 . 4 6 0 1 7 + JO.9728 Let the radius of the c i r c l e r 3 i n ( 7 -46b) "be chosen equal t o 2 . 7 . From ( 7 - 4 6 b ) , one has k = 1 A 2 ~ A l 1.0 0 . 3 7 0 3 7 0 -O .25326 0 . 9 8 7 1 6 0 . 3 2 6 9 7 0 . 0 5 8 9 4 - 0 . 5 5 0 8 3 0 . 3 0 8 4 3 The maximum of the absolute values of the eigenvalues of A, pmax 3 "*"s ^ o u n c * ^° b e approximately equal t o 0 . 9 1 . The s o l u t i o n f o r the successive terms of v and m u m _ ^ of the s e r i e s expansion of V and u i n ( 7 - 3 9 ) are then obtained by f i r s t computing the c o e f f i c i e n t s d r-3 I 3 2 of v m u s i n g the a l g o r i t h m given i n ( 7 - 5 1 ) , and then the c o e f f i c i e n t e -, . 7 - — m , . r r of u T u s i n g ( 7 - 5 8). In the r J - ' K p k ^ m-1 process v-^  and u^ are determined f i r s t and then v^ and u^ : and so on. In t h i s study computation has been c a r r i e d out f o r m up to 1 0 . The absolute value of the d i f f e r e n c e of two successive i t e r a t i o n s , Id^'^ ^ - d ^ v ^ r - I I s s e ^ 3 1 m^k-^kg m,k l ajs 2' 1 3 9 - h" smaller than 10 and the r a t e of convergence r\ s i m i l a r to ( 6 - 3 5 ) t o be te s t e d i s n - P max _ (0.91)^ , n n n l c -r) = - _ }~' ' v < 0.0015 I t i s found t h a t I = 1 0 0 gives th>? approximate s o l u t i o n . The numerical c o e f f i c i e n t s of v^ and u^ , f o r example, are given i n the f o l l o w i n g t a b l e . k l k 2 C ^ k l ^ 2 6 2 , k 1 , k 2 1 1 0 . 0 9 6 0 5 - 0 . 0 2 3 5 2 : - 0 . 1 1 6 9 7 1 2 O.I8252 0 . 1 6 6 4 7 - 0 . 0 6 7 7 5 1 3 0.00000 0 . 0 3 3 8 7 -O.O2506 1 4 • 0.00000 0 . 0 0 8 3 5 ; 2 1 0 . 4 6 6 6 7 0 . 1 6 7 6 9 -O . I7985 2 2 - 0 . 0 3 3 2 2 0 . 1 7 9 8 5 - 0 . 1 0 6 0 4 2 3 0.00000 0 . 0 5 3 0 2 3 1 0 . 0 8 1 7 3 0 . 1 7 9 1 0 -0.11511 3 2 0.00000 0 . 1 1 5 1 1 To study the e f f e c t of t r u n c a t i o n of the i n f i n i t e s e r i e s for the optimal s t a b i l i z i n g s i g n a l on the time responses of the power system, computations•are c a r r i e d out f o r u(N), w i t h . N up to 9 « To s t a r t the computation of the responses a l l I n i t i a l values of the state' v a r i a b l e s , are set to zero except 6' . T y p i c a l r e s u l t s are summarized i n P i g . 7 - 1 5 through F i g . 7 - 1 9 w i t h the f o l l o w i n g observations. In F i g . 7 - 1 5 comparison i s made between the time 140. responses of the n o n l i n e a r system w i t h a governor without a s t a b i l i z i n g s i g n a l ( s o l i d l i n e s ) , and those w i t h a l i n e a r s t a b i l i z i n g s i g n a l u(N=l) (dotted l i n e s ) . The s t a b i l i z i n g . s i g n a l has a d e f i n i t e e f f e c t of improving the t r a n s i e n t per-formance of the system. ' P i g . 7-l6a compares the time responses g,n, and 6' of the pov/er system w i t h d i f f e r e n t s t a b i l i z i n g s i g n a l s .u(N=l), u ( H = 2 ) and u(N=3) w i t h the same i n i t i a l c o n d i t i o n 6' = 1 . 5 r a d i a n . Prom the W(x,u(N)) f u n c t i o n , i A . . W(x,u(N)) = x tCx + u 2(N) N = 6 / 2+n 2+g 2,+{ E u . ( x ) } 2 .1=1 1 • p l o t t e d w i t h respect to time f o r N =1,2 and 3, i t i s ob-1 r°° -served t h a t the area under the curves, W(x,u(N))dt , d J o decreases w i t h the increase of N . Computations of time responses of the same system w i t h the same i n i t i a l c o n d i t i o n s are those o f P i g . 7 - l 6 are c a r r i e d out f u r t h e r f o r N = 5 and N = 9 . The r e s u l t s are p l o t t e d i n P i g . 7 - l S . I t i s observed t h a t the area 1 W(x,u(N))dt a l s o decreases w i t h the increase of N . 2 J o However when P i g . 7 ~ l 6 and P i g . 7 - 1 7 are p l o t t e d together f o r comparison,- which i s not shown, i t i s observed t h a t the.Increase of N from 2 to 9 does not make much d i f f e r e n c e on the time responses of W(x,u(W)) under the p a r t i c u l a r i n i t i a l .141. c o n d i t i o n s . F i g . 7 - l 8 a shows s i m i l a r r e s u l t s as those i n F i g . 7 - l 6 a w i t h the i n i t i a l c o n d i t i o n 6' = 1*25 i n s t e a d of 1.5« The d i f f e r e n c e between system responses w i t h the . l i n e a r c o n t r o l u ( l ) and those w i t h the n o n l i n e a r ones, u(2) and u(3) , are much l e s s than those when . 6 7 = 1 .5 as i n F i g . 7 - l 6 a . In F i g . 7 - l 8 b the time responses of ¥ ( x 5 u ( N ) ) f o r N = 1 52 and 3 corresponding to F i g . 7 - i 8 b are .shown. Again the d i f f e r e n c e s of the three cases are much l e s s than those i n F i g . 7 - l 6 b . S i m i l a r r e s u l t s are obtained f o r N = 5 and N = 9 although not shown. I t i s noted t h a t when 6 7 = 1.0 the l i n e a r and n o n l i n e a r s t a b i l i z i n g s i g n a l s u(N) '. f o r a l l N give almost I d e n t i c a l time responses of 6 3 n and g . . ' 5 . P i g . 7 -15 Comparison of responses of a n o n l i n e a r system w i t h and without l i n e a r optimum s t a b i l i z i n g s i g n a l s P.U. GATE , ETC I I H r o *1 (ft i H GY P 3 ca'gj c+ O P 3 C H H* H* H 3 H« CD N p H- 4 eg 0 P (ft to Pi " < J CO CO p H 4 CD CO S3 3 —'CO CD <» CO Hj p. O c+ 4 p* .1.000 1 -.500 ! - .000 1 .500 1.000 1 1. J .1.000 1 -.500 i P.U.SPEED -.000 i X 53 .62 .500 i 1.000 • 1. I 1.000 -.500 • ANGLE -.000 RAD . .500 1.000 1 1. H* (ft - 4 I H O CD S CO hd o I I CD H O \» Hj ro £ 1 £5 I— .000 W-CX,U[N)) FUNCTION 1.000 1.500 2.000 2.500 H: era. - 4 I H te; to ^ ci- O I! $» 3 a* H H H- H-H b H* CD H- 4 &  e r a w <<! co co H- ci-c r a fi> 0 3 fo H 4 CO ^ o - CO CD «• CO Hj p . O ri-4 0 1 H-era -3 I H O CD CO •d O CO II CD CO. H o VQv. -1.000 o o o m o o m o o o o O O P.U.GATE, ETC. .500 -.000 .500 I .000 I . 500 1 P. i U.SPEED X 53 . 62 j -1.000 1 -.500 i -.000 i .500 1.000 1 . j ANGLE RAD . 1 - .000 -.500 i • -.000 .500 1.000 i i 1 . z z z II II II •o in — O O era -<1 I H CO H- CD c+ Co d 2 CO CO CD B to H O i — 1 Ho C » S , ^1 te! ,_.ooo w ( x i . U ( N l ) F U N C T I O N 500 I . 000 I . 500 2 . 000 2 .500 146. 8. CONCLUSION The f o l l o w i n g conclusions are drawn from the s t u d i e s of t h i s t h e s i s : 1. Zubov's method f o r the determination of asympto-t i c s t a b i l i t y r e g i o n has been applied, to power system t r a n -s i e n t s t a b i l i t y r e g i o n s t u d i e s (Chapter 4.). However a t r u n -cated Lyapunov f u n c t i o n s e r i e s of high degree i s necessary. t o approximate the true s t a b i l i t y boundary. Although the method i s the most systematic one f o r c o n s t r u c t i n g Lyapunov f u n c t i o n , works remain to be done i n the development of a l g o r -ithm f o r digita.1 computations when t h i s method i s a p p l i e d t o a l a r g e complex power system. 2 . A quadratic Lyapunov f u n c t i o n can be used t o eva-l u a t e the performance f u n c t i o n of a power system f o r the optimum parameter s e t t i n g s (Chapter 5)- A systematic method f o r computing the L matrix of the Lyapunov f u n c t i o n has been developed based on s i m i l a r i t y t r a n s f o r m a t i o n concept. The computer memory requirements and computer time are s i g n i f i c a n t l y reduced by t h i s method. Once the Lyapunov f u n c t i o n i s computed , the gradient of the gradient method f o r m i n i m i z i n g the per-formance f u n c t i o n f o r the optimum parameter s e t t i n g s can be computed w i t h r e l a t i v e l y l i t t l e a d d i t i o n a l e f f o r t . 3. Optimum s t a b i l i z i n g s i g n a l s as f u n c t i o n s of s t a t e v a r i a b l e s can be determined from the performance f u n c t i o n s of n o n l i n e a r and l i n e a r i z e d power systems u s i n g the concept of optimum Lyapunov f u n c t i o n . A quadratic Lyapunov f u n c t i o n i s used to o b t a i n the optimum s t a b i l i z i n g s i g n a l s f o r the 147. linearized sj^steui. The results have been applied to Unear-th th th ized power system of 5 3 6 and 8 orders, in deriving the stabilizing signals for the governor and voltage-regulator feedback loops. It is found that the signals for the voltage-regulator feedback loop are not as significant as those for the governor feedback loops. It Is also found that, as far as the derivation of the optimum governor stabilizing signal is concerned, the second order synchronous machine model ob-tained by assuming constant f i e l d flux.linkage, i s as good as the third order synchronous machine model which allows changes of f i e l d flux linkage (Chapter 7, section 7.2.2). 4. A general iterative scheme for computing the Lyapunov function series terms for the nonlinear system has been developed and i t s convergence proved. A method of improving the speed of convergence has also been given (Chapter 6 ) . The results have been applied to a nonlinear power system of the third order in deriving the nonlinear stabilizing signals for the governor feedback control. It i s found that the linear terms of the signals have a dominant effect (Chapter 7, section 7. J>. J>). Works -remain to be done in the f i e l d . It is desirable, for example, to improve the computation of Lyapunov functions by Zubov's method, to develop algorithms for the computation of optimum stabilizing signals for nonlinear systems of high order, and to extend the studies of this thesis to inter-connected multi-machine power systems. This thesis is Intended to-provide a stepping-stone to the applications of some non-linear s t a b i l i t y and optimal control theories to power system studies. 148. Appendix 3 A Nomenclature f o r Synchronous Machine and System (k) s u p e r s c r i p t to denote a v a r i a b l e of k ^ machine i n the multi-machine system. ' o s u b s c r i p t to denote an i n i t i a l c o n d i t i o n A p r e f i x to denote a l i n e a r i z e d v a r i a b l e w.r.t. a steady s t a t e c o n d i t i o n , p = -g^ r time d e r i v a t i v e operator. i ^ , i armature currents i n d and q axes,. r e s p e c t i v e l y v d * v q .armature voltages i n d and q axes, r e s p e c t i v e l y v^ armature t e r m i n a l voltages v Q i n f i n i t e bus voltage 1 f d ; , v f d f i e l d c u r r e n t and f i e l d v o l t a g e , r e s p e c t i v e l y ^df*q armature f l u x l i n k a g e s i n d and q axes, r e s p e c t i v e l y f i e l d f l u x l i n k a g e s x^,x synchronous reactance i n d and q axes, ^ r e s p e c t i v e l y . x & d mutual reactance between the s t a t o r and r o t o r i n d-axis x / (^,x / t r a n s i e n t reactance i n d and q axes, q r e s p e c t i v e l y x " d,x" s u b t r a n s i e n t reactance i n d and q axes, ^ r e s p e c t i v e l y Rp f i e l d winding r e s i s t a n c e R armature winding r e s i s t a n c e i n d- or q-axis c i r c u i t x a d L Vp — v^^ = a voltage p r o p o r t i o n a l to f i e l d Voltage F 1^9. X a d A ftp T T ^ f d = a f l u x l i n k a g e p r o p o r t i o n a l to f i e l d P f l u x l i n k a g e A Vp^ x a d ^ f d = a v°ltage p r o p o r t i o n a l t o f i e l d current T / ^ O d i r e c t - a x i s t r a n s i e n t o p e n - c i r c u i t time constant. T ' , d i r e c t - a x i s t r a n s i e n t s h c r t - c i r c u i t time constant • d d i r e c t - a x i s damper leakage time constant T " D O 5 T " o o s u b t r a n s i e n t o p e n - c i r c u i t time constant i n d and q axes,, r e s p e c t i v e l y A E' ^F^^'do = synchronous machine i n t e r n a l v oltage p r o p o r t i o n a l t o f i e l d f l u x l i n k a g e . r+jx . t i e - l i n e impedance between the generator and the i n f i n i t e bus G+jB shunt admittance at synchronous machine t e r m i n a l f o r the synchronous machine i n f i n i t e bus- system v r j V m armature t e r m i n a l voltages i n the common reference, r and m-axes, r e s p e c t i v e l y . 6 power angle Tj mechanical torque input to the r o t o r T energy conversion torque P r e a l power output of the machine Q r e a c t i v e power output of the machine H .. i n e r t i a constant J moment of i n e r t i a a • mechanical damping c o e f f i c i e n t D over a l l damping c o e f f i c i e n t 6 instantaneous angular p o s i t i o n of r o t o r p8 angular v e l o c i t y • r a t e d angular v e l o c i t y •system frequency p9- cu o — : = p.u. change ~>f speed o e x c i t e r time constant s t a b i l i z e r time constant of the v o l t a g e r e g u l a t o r e x c i t e r gain a m p l i f i e r gain i n ' t h e voltage r e g u l a t o r s t a b i l i z e r gain i n the voltage r e g u l a t o r A ^ex^a^r = r e S u l a t o r S a i n A •(^"ad^P^e = o v e r a l l r e g u l a t o r gain A 1 + u u. , = o v e r a l l s t a b i l i z e r gain ' • A • v„/(u u ) = v o l t a g e p r o p o r t i o n a l to the s t a b i l i z v o l t a g e v s •' • permanent speed droop of governor temporary speed droop of h y d r a u l i c governor dash-pot time constant water time constant p.u. gate change p.u. head change 1 c i Appendix 3B Park's Equations i n the F i r s t Order Equation Form N e g l e c t i n g s u b t r a n s i e n t time constants"(3-3) becomes x a d ( 1 + W ) v f d X d ( ^ ' d ^ d V % ( l * T ' d o p ) . ^O ( l + T ' d o p ) t^W* f V f d , f V d f d i ^ d o i ^ d i - , (1^'doP) R ^ \ ; - ' "° » 0 ( l + T ' d o p ) * d X X • ? ' . ^ V » . + p { T n , - ^ V - . + X . ( T ' - T'' )1.'3 R„ f d ^ Di R_ f d d v do d y d J x, = ~ —— d ± » 0 ( l + T ' d o p ) or ~& UJQ w0 (3B-2;-where FR 7 7 - ^ : . -(3B-3) < 1 + T do p^ and L X a d V F = R f V f d • (3B-4) F I t can be shown that A T ' ' x / d = - 7 A - x d (5B-5) T do From (3B-3) 1 5 2 . VP = P [ T ' d o [ V F R - ( x d - x ' d ) i d ] - V F ] + v : FR or v F = P* F + v F R (3B - 6 ) where = T , d o [ v F R - ( X d - x / d ^ d ] - T m V D£ VF (3B -7) S u b s t i t u t i n g (3B-4) i n t o (3B - 6 ) gives (3B-8) Note th a t (3B-8) has the form of f i e l d v o l t a g e equation by i d e n t i f y i n g R F Fx ad = 4 f d , the f i e l d f l u x l i n k a g e . . (3B-9) and v-FR x ad l f d , the f i e l d current (3B - 1 0 ) C o l l e c t i n g ( 3 B - 7 ) , (3B - 2 ) and (3-4) gives * F + V V F do 1_ T ' ( x , - x ' ) do v d cr x o CO, o a V FR (3B-11) In ( 3 - 6 ) , l e t p9' = ID -I- p6 or p6 = *p0 - iu o 153 . or A p6 '= uj A'J) (3B-12) where Atu i s the per u n i t change i n speed p8-u) Au) = - j j p - ^ . (3B - 1 3 ) o Thus ( 3 - 5 ) can be w r i t t e n as pAto = ^ {T m-Dp6-T e} (3B-14) Appendix 3?C Transformation of Coordinates of Synchronous Machines i n t o a Coxomon Reference Park's t r a n s f o r m a t i o n matrix, modified by Yu has the form: ( 2 6 ) i a V s i c ccs9 - s i n 9 _1 - s i n ( 8-120) — A / 2 _1 -sin(9+120) j2 o (3C-1) cos( 8-120) cos(8+120) where i , i- . , i are armature currents i n phase coordinates, a 5 o3 c 3 and i , , i , i , the transformed v a r i a b l e s i n Park's d i r e c t , d3 q3 o3 3 quadrature and zero.coordinates r e s p e c t i v e l y . The same matrix i s a l s o used f o r transforming the phase v o l t a g e s and the f l u x l i n k a g e s . The inv e r s e t r a n s f o r m a t i o n i s of the form "o /2 43 cos 6 cos(8-120) -sinG -sin(8-120) _1 ^2 1_ 72 cos(8+120) •sin( 8+120) _1 /2 "a (3C -2) Since the tr a n s f o r m a t i o n i s . u n i t a r y , the power i s i n v a r i -ant, i . e . a a o b c c d d q q o o (3C-3) With the phase currents end vol t a g e s of the forms, 1 5 5 . i = icos(6+p) and i b = i c o s ( e+p-120). (3C-4) i c = i c o s ( 9+P+120-) v a = vcos(6+0-0) v b = vcos( e+P-C2f-120) (3C - 5 ) , v c = vcos(9+0-0+120) the t r a n s f o r m a t i o n (3C-2) y i e l d s , i d = A / 3 IcosP , - i = 7 3 I s i n p ( 3 C - 6 a ) v d = 73 Vcos(0-0) , V q = 73 Vsin(0-0) and i Q - o , v Q = o ( 3 C - 6 o ) where A A • V = ^- , I = •— (3C-7) S u b s t i t u t i n g (3C-6a) and ( 3 C - 6 b ) i n t o 3 C - 3 ) y i e l d s P = 3VIcos0 .. (3C-8) In the steady s t a t e , V and I are the r.m.s. values of voltage and current r e s p e c t i v e l y and P i s the t o t a l average, power. To o b t a i n the expression f o r the average power per phase, one may w r i t e 156. P = i ' . v ' + i ' v' a d d q q = VIcosc?f (3C-9) where 4 i , t i , i = I s i n p ^ (3C-10) i ' = — = IcosP , i - ' =  I s i n p d 73 q 73 A V / l A V v ' d = — = -Vcos(P-0), v' = -3- = Vsin(P-0O 73 73 and "*"'q* v ' d v ' q a r e ^ e components of r.m.s. cu r r e n t s and voltage of any one phase. When p e r - u n i t system i s adopted, (j5C-6a)' and (3C-10) can be w r i t t e n In the same form. For the three phase system one may choose I b a s e = X a s e = l q b a s e = ^ V a s e - n o m i n a l ^ vbase = V n Q = V Q = ^  Vphase-nominal ™ ^ - \ \ ) base ^base r ^base = total r a t e d MVA of the synchronous machine Equation (3C-6a) becomes p.u F H p . u p \ . u = V P . U C O S ( P - ^ * .Vqp u = V p . u s i n ( p - ^ For the single-phase system one may choose I = i ' = i ' = 1 "KA b a s e dbase qbase phase-nominal vbase = v d f e a s e = V ' q b a s e = Vphase-nominal K k (3C-12) (3C-13) -P'base = rated single phase MA To carry out the transformation, substituting (3C-4), .(3C-5) arid .(3C-7) in (3C.-1) results in J3 Icos(0+p) =' i,cos6 - i s in 8 • d q ' (3C-14) J3 Vcos( 9+p-s2f) = v,cos8. - v sin0 Dividing through (3C-14) by ,/3 and using the definition in (30-10), equation (30-14) becomes ' ~ Icos(9+0) = i'^cosS - . i ' sine (3C-15) Vcos( 8+p-i2f) = v dcos0 - v sinO In other words both three-phase and single phra.se transformations yields the same results in p.u. system pro-vided that the base quantities are chosen as (3C-11) and (3C-13) respectively. The results are ~L cos(8+p) = i , cos8 - i . sin8 p u p.u qp.u V „cos(G+P-^) = v, cos6 - v„ sin8 D . U \ r r~ / ^ Q y p.u. Hp.u The equation (3C-16) can be expressed as I e J ( e + ? ) = i , e^ 0 -hi "ettWV p ' u dp.u Q-p.u V e J ( 0 + P - ^ = v, e^ 9 + v e^S+90) P' u d P . u %.u where the real part of ( 3 0 - 1 7 ) corresponds to (3C-16) (3C-16) (3C-17) 1 5 8 . S u b s t i t u t i n g 6 = wQt + .6 ( 3 C - 1 8 ) i n t o ( 3 C - 1 7 ) one has P ' U 1 d p . U \ . U : ] v e J ( 6 + P - ^ ) / V = { e j 6 + v 3 ( 9 0 + 6 ) p , U . ' dp.u %.u ( 3 C - 1 9 ) The phasor diagram r e p r e s e n t i n g ( 3 C - 1 2 ) , ( 3 C - 1 6 ) , ( 3 0 - 1 7 ) , ( 3 C - 1 8 ) and ( 3 0 - 1 9 ) i s given i n P i g . 3 0 . • 3io t By dropping the term e i n . ( 3 C - 1 9 ) , the remaining p a r t s become the equations of phasors i n the new reference w i t h complex axes or-om-as i n F i g . 3 0 . The new reference i s r o t a t i n g at the angular v e l o c i t y uuQ . The phasor equations are j - e J(6+P) = i e J « + i e J ( 9 0 + 6 ) i ' xp.u d xa r J m F p.u Hp.u ( 3 0 - 2 0 ) - v d u e ^ + v q i Vr+jTa * p.u P ' u r where i r + j i m - , and v p + Jv are the complex t e r m i n a l current and voltage- of the synchronous machine r e s p e c t i v e l y w.r.t. the u n i f o r m l y r o t a t i n g reference frame. Comparing the r e a l and imaginary p a r t s of ( 3 C - 2 0 ) y i e l d s 1 r . cos6 -sin6 i m sin6 cos6 i q_ v r cos6 - s i n 6 . v n _ v m j s i n 6 cos 6 V q (3C- 21) Note that the p.u. s u b s c r i p t s are dropped, and that the i n i t i a l angle 5 Q of any machine w.r.t. a common r o t a t i n g r eference frame can be found from a steady s t a t e load f l o w study. In F i g . 3C the i n i t i a l v o ltage phase angle between the t e r m i n a l v o l t a g e phasor and the reference a x i s or can be found from the load flow s t u d i e s . Using the known t e r m i n a l voltage and power output, i . e . the i n i t i a l o p e r a t i n g condi-t i o n s , the angle 6 Q between the voltage phasor and the d-axis of the machine, 6 Q , can be determined (29) Thus the i n i t i a l angle 6 (k) .th of k 1 1 . k = 1.2. ...n machine i n a multi-machine system'w.r.t. a common r o t a t i n g reference frame can be determined. l 6 o . P i g . 3C Transformation of synchronous machines d-q coordinates i n t o a common reference 1 6 1 . Appendix 4A Determination of I n i t i a l Operating C o n d i t i o n s , E q , V q and 6 f o r the Power Output Expression of the Synchron-ous Machine. - At the l o c a l bus i n F i g . 4-1 . V t ( J B ) + •= I (4A-1) In dq components ( v d + j v ) ( j B ) + ( v d + j v q ) - ( v o s i n 6 + J V Q C O S 6) = i d + J i q Hence (4A-2) v Q S i n 6 = v d ( l - x B ) + x i q (4A-3) v cos 6 = v (1-xB) - x i , o q x a and V Q = A / { v d ( l - x B ) + x i q } ^ + { v q ( l - x B ) - x i d p " (4A->4) 6 = a r c t a n { V d ( l - x B ) + X 1 q ) (4A-5) Lv~ri^x^nnr:J q x d A phasor diagram f o r the synchronous machine i s shown i n F i g . 4A. The voltage and power r e l a t i o n s ^ a r e , d q q v •. = S' -x' i , q q d d . (4A -6 ) 2 - 2 2 v, v,+v t d q (4A - 7 ) P = v,1,+v i d d q q Q = V q V V q (4A-8) 1 6 2 . Given magnitude of P,.Q, and v t of the synchronous machine; ( 2 9 ) . as f o l l o w s 1 q* d' q d i , and E' are solved^ q p v, "q V < V + ( v t + x q Q ) ; X 1 q q 2 2 v -v t d (4A -9 ) QrfX i 2 q q ' V E' = v + x ' i . q q d d To f i n d the power of the synchronous machine (4A -6 ) i s s u b s t i t u t e d i n t o (4A-3) y i e l d i n g 1-xB+x/x, 0 1-xB+x/x' ' d 'd v sin6' o v cos6 + xE'/x' o q d Hence v sin6 o v d ~ (l-xB+x73c^T v q Tl-xB + x/xJT From (4A-8), (4A - 6 ) and (4A-11) v cos6+xE'/ x^ o q d v,i,+v 1 d d q q (4A-10) (4A-11) 3 ^ - WU P sin6 + P sin26 m s x (4A-12) 1 6 3 . where v E' P = o _g m (x+x^-xxJB) , v 2 (x d-x ) • P s = I T T ^ + x ^ x y B T ( ^ - x " - ^ x ^ (4A-13) F i g . 4A Phasor diagram of a s a l i e n t - p o l e synchronous machine 1 6 5 . Appendix 4B Expansion of D'(&) and R(6) i n Power Seri e s of State V a r i a b l e . From ( 4 - 4 ) and ( 4 - 9 ) D'.(6) may be w r i t t e n D'(6) = A cos 26 + B s i n 2 6 = + Azl cos 25 (4B-1 Hence from ( 4 - 1 2 ) D'(6') = ^  + c o s 2 ( 6 ' + 6 o) = ^  + ^ { c o s 2 6 o ( 1 .^ + 4^_ . . . ) - s i n 26 0(26'.- t ^ l + - ...)}. which can be w r i t t e n as D'(6') = , D + D 16 / + D 26' 2 + ... + D n5' n (4B-3) where n A+B , A-B n n o 5 , D o = ~ 2 ~ + — C o s 2 6 o S u b s t i t u t e 6 i n equation ( 4 - 1 2 ) and P^ i n ( 4 - 1 1 ) i n t o ( 4 - 8 ) t o compute R(6') r e s u l t s i n R(6'). = s i n ( 6 / + 5 Q ) - s i n 5 o + P g | s i n 2 ( 6 ' + 6 Q ) - s i n 2 6 o | r t ' 5 ,2 ' , 4 ,6 = c o s a o -Ir-^fr-- ...} + s * 6 o {- fy- + ^ " V + /"} + P;cos26 o{26' ' - + - • • •}• + P ^ i n 2 6 o { - % 6 ' + T T & T - + which can be w r i t t e n as •R(6') = R 16' + R 2 6 ' 2 + 166* (4B-5) where R n ^ { c o s { 6 o + ( ^ ) 7 r } + 2 n p ; c o s { 2 6 0 + ^ ) 7 r ) } / n i (4B-6) .67. Appendix 5A I n i t i a l Values of the Steady State Conditions of the Synchronous Machine I n f i n i t e Bus System The f o l l o w i n g equations f o r the steady s t a t e condi-t i o n of the synchronous machine i n f i n i t e bus system are w r i t t e n based on the assumptions given In the beginning of S e c t i o n 3 . 3 . 1 . From ( 3 - 8 ) , ( 3 - 9 a ) , ( 3 - 9 b ) , ( 3 - 1 0 ) , ( 3 - 1 1 ) , ( 3 - 1 2 ) and ( 3 - 3 5 ) one has 2 2 2 • V t o = v d o + vqo vdo - -+qo^ o V = ^do^o (Att.) Q = o P W d o + V V Q = vqo 1do ~ vdo 1qo V d o = V F o " V d o V q o " " V q o *PO = Tdo VFo " Tdo<Xd ( 5 A - 1 ) v s i n 6 ^ = (l-xB+rG)v,^ - (rB+xG)v - r i , + x i . o o x ' do v ' qo do v cos6^ = (x G + r B ) v ^ + (l-xB+rG)v n - x i , - r l o o v 7 do v 7 qo do qo do qo ~ - x i , ^ - r l qo do qo where P,Q,v t Q are given and v d Q , v q Q , v Q , v p o , * d o , # q Q, ,!,po, i d o , i q o , 6 Q and ( A W ) 0 are t o be determined. From the f i r s t nine equations the f o l l o w i n g s o l u t i o n s can be obtained^ ^  ^ (Aljj) 0 = o 168, t o Vdo = V q o vqo = Aso^do .2 q q° d o V V P O = V + x d W ( 5 A~ 2 ) *qo = - Vdo / u Jo + do = V q o / u J o i ' /n = T ' V „ - T ' ( x , - x ' ) i , Fo do Fo do v d d' do and from the l a s t two equations o f (5A-1) one obtains ro = J v - + v 2 — 2 -do + Vqo (5A -3 ) 6^  = a r c t a n f v , /v ) o v do qo' where vdo = ( l - x B + r G ) v d Q - (rB+xG)v q o - r ± d o + x i q o ( 5 A - 4 ) _ A. ' ' • v „ = (xG+rB)v, + ( l - x B + r G ) v _ - x i , - r i qo v ' do . v • ' qo do qo Thus (5A-2), (5A-s3) and (5A - 4 ) give the i n i t i a l values of the steady s t a t e c o n d i t i o n s of the synchronous machine i n f i n i t e bus system f o r a given power output P+jQ } and a given t e r m i n a l voltage v. . 1 6 9 . Appendix 5B D e r i v a t i o n of A l g o r i t h m f o r S e t t i n g the 8 M a t r i x In ( 5 - 7 ) l e t . * A + yU = A (53 - 1 ) then ( 5 - 7 ) can be w r i t t e n as A*L + LA = - C" (5B - 2 ) From (5B - 2 ) n ^ ^ i ^ j ^ i A j ^ = - c i j > ^ = l,2,....n (5B-3) where a Qp , and c a p , a , p = 1, 2 , ...n are elements of the matrix A 3 L and C r e s p e c t i v e l y . From (5B-3>) = - c ± j , i , j = 1 , 2 , ...n (5B-4) Being symmetric, only N I ~ i l elements df L need be deter-mined. Noting .j,ap = , (5B-4) y i e l d s " ~ c i j 5 1 = 1 , 2 , . . . ] ! . ' , j = l , . . . n (5B - 5 ) Equation (5B - 5 ) can be d i v i d e d i n t o two p a r t s , i . e . when j > i (5B - 5 ) becomes 1 7 0 . ^ i ^ O + & ^ + ^ 0 * + a J i ^ d \X± ^ i ^ - k i ~ l o-l n £ a, . + a. .f. . + £ a, + a + £ a, k = 1 ^ J k i i J H k = i + l * J l k 00 10 k=J+l^CJ l k = -c. . . i = 1 , 2 , . . . , n - 1 , j = i+ 1 ,...n , 10 hence ^ K i ^ J + a k J ^ i } + k ! f ^ f a k i ^ J + a k ^ i k } \ J + 1 { a k i ^ - k + a i , o ^ k 5 + {a..+a J j}^. J + a ^ ^ j j + a . ^ . . = -c±. , 1 = 1 , 2 , . . . n - 1 0" - . • .n (5B -6) and when j = 1 (5B-5) give s or. i - 1 n k f x 2 a k i < k i + k f 1 2 a k i * 1 k = - c i i > 1 = l , 2 , . . . n (5B - 7 ) Note t h a t the f i r s t summations of (5B - 6 ) and (5B - 7 ) do not e x i s t when i = 1 . Let a system of r e a l i n t e g e r s be d e f i n e d as M(a,p) = PiP-1) + a , a. = 1 , 2 , ....n 2 (5B-8) P = «, n 171 Hence the matrix elements ca^ and ta^ i n ( 5 B - 6 ) and (5B-7) can be transformed i n t o the corresponding v e c t o r elements "? M£ a and ~l^ra r e s p e c t i v e l y by means of (5B-8). Thus (5B-6) and (5B-7) can be w r i t t e n as i - 1 _ _ j-1 _ ^ { \ i V k } j ) + \ j % ( k , i ) } ^ J \ i V k } j O + ^ j V i 3 k ) 3 U l ^ ^ ^ ^ ^ j ^ ^ ) 5 + ( a i i + a 0 0 } ^ ( i , j ) + a J i ^ ( J, J) + a i d * M ( l , i ) = - \ ( i , j ) > 1 = 1,2,...,n-l ( 5 B _ g ^ j = i+1,...,n and i - 1 _ n ^ 2 a k i ' M ( k , i ) \ f . 2 a k i % ( i , k ) = ~ § M ( i , i ) > 1 = s l > 2 > " ' n .. (5B-9b) Thus (5B-9) can be w r i t t e n i n the matrix form BI = 1 where l and § are column ve c t o r w i t h n(n-l-l)/2 elements corresponding t o the ta^ and ea^ , p _> a s r e s p e c t i v e l y . The elements of e matrix D r g j r , s = 1,2,...n(n+l)/2 , and those of the vec t o r § can now be set up s y s t e m a t i c a l l y according to ( 5B-9) as f o l l o w s : Let i = 1, 2, ...n and J = i , i + l , ...n . Set • ? M ( i , j ) = - C i j ' P o r 1 < J S e t b,r/_. _-\ ,„/*s = a.. + a., J M ( i , j ) , M ( i , d ) ~ a i i JM(i,j),M( - a 0 i J M ( i , j ) , M ( i , i ) ~ a i J 172. For i < j and i - 1 > 1 set 3M(i , o),M(k , o ) ~ \l f o r k = 1,2,.. . i - 1 For i < j and i+1 _< J - l , set 5 M ( i , j),M(k, j ) ^ i 3 M ( i , j ) , M ( i , k ) = a^. f o r k = i+1,..., j - l For 1=3 and i - 1 > 1 , set b M ( i , j ) , M ( k , i ) = 2 a k i > f o r k = 1,2,...i-1 b M ( i , 0 ) , M ( i , k ) = 2 a k i > f o r k = 1,1+1,...n A f t e r the elements of 1 are obtained from (5E-10), the L m a t r i x can be determined from (5B-8) by i d e n t i f y i n g "aP ^M(a^p) > a - 1,2, . . . n P = a , n (5B-11) . 173. Appendix 5C . Algorithm for Solving L ' Matrix of (5-25) With (5-23), (5-25) can be expanded in the for m n ^ ^ i k ^ ' j k ^ i ^ c i j > i . = ± , * , . . . n ^ ^ . ^ where V , i = 1, 2, . . .n j = 1,2, . . . n f 1 , P = Y+l, Y < n Y = n PY 5 CYP = °PY (5C-2) and , b ^ and c ^  , Y,P = 1,2, . . . n are elements of matrix L ' , B and - C ' respectively. Equations (5C-1) and (5C-2) give n(n+l)/2 equations, ( i = 1, 2, . . .n (5C-3) J = 1 ,2 , . . . n which can be divided into two parts. ~ For c. . , i = J = a , i 0 (5C-3) gives 2 V l , a + 2 l a n & a = c a a > a = 1 * 2 , . . , n (5C-4) ^except -?-a_-j_ a = 0 w h e n ot-1 = o . For c^., i / j 3 replacing j by J+l , (5C-3) yields ^0+1,1-1 + lin&3+l + ^ '+ l ,n a i ~ Ci,0+1 or 1 7 4 . llj C i , j + l ~ ' -1-1,0+1 ~ a j + l * ' i , n " a i ^ + l , n 1 , 2 , . . . , n - l ( 5C-5) j = 1, i+ 1 ,... , n - l ,except ^ = o when i - 1 = o . By s u b s t i t u t i n g (5C - 5 ) i n t o (5C-4) repeatedly, f i r s t l e t i = a - 1 , j = a3 and then i = a-2, j = a+1, e t c . , u n t i l o n l y the elements t^n , i = 1, 2 , ...n are l e f t , (5C-4) becomes c = 2 £ ( - l ) 1 + 1 c . , . - 2 E ( - l ) 1 + 1 f a ,.t . +a 3 v y ™-i,a+i 7 1 a + i V i , n a-i''a+i,n J aa • -i ' a- i = l + 2 V a n + ^ - ^ ' a - J - ^ a + J > a = l, 2 , . . . n ( 5C-6) except £ a_j_i a + j = ° when a-J - 1 = o 8 1 1 ( 1 V j - l , a + J = V j - l , n w h e n a + J = n i . e . a - 1 i f n-a > a -1 n-a i f a - 1 > n-a J = Equation (5C-6) c o n s t i t u t e s a system of n a l g e b r a i c equat-ions i n n unknown i = 1 , 2 , ...n . By u s i n g (5C-6) a matr i x equation i n the form B'Z' = §' (5C - 7 ) i can be set up and solved f o r , i = 1,2,...n . The other elements of 1/ can then be obtained from (5C-5) by f i r s t • 175. l e t t i n g i = 1, j - 1,2, ...n-1, then 1 = 2 , j = 2,.. .n-1, e t c . and f i n a l l y i = n-1, j = n-1, u n t i l a l l the elements of L' are determined. 1 7 6 . Appendix 5D Eigenvalue Confinements of L i n e a r Dynamic System by .'Adjustment of i t s Parameters For a l i n e a r dynamic system x = A(a)x (5D -1 ) A . ^consider the problem of a d j u s t i n g the parameter a =-(a-^a .. . a m ) 1 of A(a) such t h a t the eigenvalues of A(ct), X^, i = 1,2, ...n, which s a t i s f y |A(a)-XU| = o (5D-2) l i e w i t h i n a s p e c i f i e d s e c t o r i n the l e f t h a l f of the complex \-plane. Let the s e c t o r be s p e c i f i e d by a constant y and angle P as shorn i n the shaded r e g i o n of F i g . 5D-L Let the r e a l and imaginary axes of the \-plane be transform i n t o those of the p'-plane, f i r s t by the transform-a t i o n of the o r i g i n , o , of the X-coordinates to the l e f t at d i s t a n c e y , and then r o t a t i n g the X-axes i n the a n t i -c l ockwise d i r e c t i o n by an angle p , as shown i n F i g . 5D-2. Then the p o i n t X of the X-plane can be expressed as a p o i n t p ' of p'-plane by the f o l l o w i n g r e l a t i o n , X = p'e 3* P - Y (5D-3) There w i l l be n values f o r p ' corresponding to the n-values of X . Since X^, i = 1,2,...n occur i n complex conjugate pairs,-.- i f a l l p f , i = 1,2, ...n l i e w i t h i n the l e f t h a l f of :p'-plane, then a l l X^ i = l , 2 , . . . n l i e i n the s p e c i f i e d s e c t o r of the X-plane. P i g . 5A-2 R e l a t i o n s h i p between \-plane and p'-plane 1 7 8 . The l e f t h a l f of the P'-plane i s f u r t h e r mapped i n t o a u n i t , c i r c l e on a complex p-plane by the tr a n s f o r m a t i o n , ( 1 1 / 1 S u b s t i t u t i n g (5D -3 ) i n t o (5D -4) y i e l d s -Hence i f a l l p^ , i = 1 , 2 , ...n l i e i n the u n i t c i r c l e i n p-plane, then a l l X^, i = ' 1 ,2 ,...n must l i e In the s p e c i f i e d s e c t o r . S u b s t i t u t i n g (5D -5 ) i n t o (5D -2) one obtains o = |A-{£=J e^- Y }u| = | { ( A + Y U ) e - ^ ( l t P ) - ( p - l ) U r r i | F y | = |{[(A+ YU)e- J" p+U] + p [ ( A + Y U ) e " J P - U ] } p | ? y | = I {f (A+yU) e" J P+U ] [ (A+ YU) e" J P-U ]" 1+pU} • {[ (A+ YU) e" U j}. = | {[(A+YU)e~ J" P-l-U][(A+ YU)e"^-U]" 1H-pU}| • o"P ^ [ . ( A + Y U j e - ^ U j ^ j f j j | Hence |B+pU| = 0 (5D-6) where 1 7 9 . B = {(A+vU)e" J-|3+U3 {(A+YU)e" J ^~U} 1 or B = U + 2{(A+YU)e" ^ - U } " 1 (5D -7 ) The eigenvalues of B are ~Pj_, i = 1 , 2 , ...n . Thus 1-P ± 1 I P i I , i = l , 2 , . . . n . . - ' '- (5D-8) Hence i f a l l the eigenvalues of B l i e i n the u n i t c i r c l e , the , i = 1,2,...n are i n the s p e c i f i e d s e c t o r . Now l e t a p o s i t i v e d e f i n i t e - H e r m i t i a n matrix H be constructed as H k = ( B k ) t X " ( B k ) (5D-9) where i s the k^*1 power of B . The eigenvalues of H^ ( "39} are r e a l and p o s i t i v e s a t i s f y i n g v ' J ' < l ( - p , - ) k | 2 < h , . 1 = 1 , 2 , ...n (5D -10) mm — ' X ^ I 1 — max 3 3 3 v where ^ m ± n ^max a r e ^ e s m a H - e s t and the l a r g e s t eigenvalues of H^ r e s p e c t i v e l y , and (-p^) k , i = 1 ,2 ,...n, are the eigenvalue of B^ . Prom (5D-10) i f then the eigenvalues of . B^ and those of B are i n the u n i t c i r c l e . (5D-11) Is s a t i s f i e d i f the sum of a l l the eigen-values of H k i s l e s s than u n i t y . Let > W, = Sum of the eigenvalues of H, K k (5D-12) = Sum of the diagonal terms of R k .. .180. From (5D-9) n n W. = £ £ b* b . . ( 5 D - 1 3 ) K 1 = 1 / = 1 l x 1 1 where are the e lements o f , and b ^ those o f ( B k ) * The p r o b l e m o f c o n f i n i n g the e i g e n v a l u e s \ ^ 3 I = 1 , 2 , . . . n i n the s p e c i f i e d s e c t o r now becomes t h a t o f m i n i m i z i n g W"k u n t i l W"k < 1 . • • (5D- .U) t h r o u g h a s y s t e m a t i c adjustment o f a se t o f parameter a = ( a ^ a g . . • a m ) t i n the m a t r i x A(a ) . G r a d i e n t method s i m i l a r t o t h a t g i v e n i n S e c t i o n 5 . 2 o f Chapter 5 may be used to m i n i m i z e w . r . t . a 3 and the components o f the g r a d i e n t o \ v e c t o r —— , i = 1 2 , . . . n can be de termined as f o l l o w s . From ( 5 D - 1 3 ) W n n ab 3 b * ' = A A C b J j + b^ } - 1 = l i 2 > - ' m ( 5 D _ 1 5 ) where -•• • ' M are the e lements o f - 2— , and - — t h o s e o f dou aa^ 5 acK a f B^) * a B^ a f B^) * • ^ 7 . The m a t r i x — — and • \ n ' can be de termined i n aa^ aa^ acu term's o f as f o l l o w s . From (5D-7) one has (A+yU)e J " P - U = 2 ( B - U ) ~ 1 ( 5 D - 1 6 ) and hence (A+ YU)e~ j P ( B - U ) . = (B+U) (5D-17) By t a k i n g p a r t i a l d e r i v a t i v e o f (5D-17) ^ _ f A + Y U ) e - 3 P ( B - U ) } = | | -o r E-JP H-(B-U) + {(A+YU)e-JP-U} | f - = o ( 5D-l8) By s u b s t i t u t i n g (5D-16) i n t o (5D-18) , one has = -(B-U) l ^ - ( B - U ) |" (5D-19) SB F o l l o w i n g t h i s -~— and ~ ± — c a n be d e t e r m i n e d . The m a t r i x B i n (5D-19) may be d e t e r m i n e d f r o m (5D-7) w h i c h can be w r i t t e n as B = U•+ (R+jM)" 1 (5D-20) where R '= (A+yU)cosp - U • M = -sinp(A+ v U ) From (5D-20) (5D- 21) U + {(R-jM)(R+jM)}"'1(R-jM) B x + J B 2 . (5D-22) where / 182. reduced t o Zubov's B, = U + 2 ( R 2 + M 2 ) - 1 R (5D-23) B p = - ( R 2 + M 2 ) - 1 M I t i s noted that when 8 = y = o s the method i s ,.(11) Appendix 6A " V e r i f i c a t i o n s of the R e l a t i o n s h i p Yi'= Y 2 * a n d ^1 = C 2* w h e n I-1}. = 1^ 2* o f A 2 Le t the eigenvector k corresponding to the eigenvalue 6 k of A* he v^., i . e . A 2 U k = 6 k U k (6A-1) A 2 v k = 6 k v k > k = 1, 2, ... n Since 6^ ., k = 1, 2, . . . n are assumed to be d i s t i n c t , and u^ can be normalized such that V u 0 = 6 i j 3 ± > J = 1 3 2 > "' n (6A-2) where 6 i ; j - 1 , i - J 6, , = 0 , i £ J 16 1) The e i g e n f u n c t i o n Q equal . C t = ( v ^ x ) * ( v g . x ) * ( v 5 - x ) ^ . . . ( v n . x ) (6A-3) where m. , i = 1, 2, ... n are Integers s a t i s f y i n g n • • E m . = m , m. _> 0 (6A-4) i - 1 xl xl Equation (6A-4) determines N d i f f e r e n t sets of m. where N i s given by (6-2) . Using (6A-3) and (6A-l), m l m 2 V 3C. v S C ( v 1 - x ) 4 ( v ? . x ) * . ; . ( v -x) A „ x • — - = A„x - i ^ S 2 3x 2 . dx m-^  -1 m 2 = m1 ( A 2 x - v 1 ) (vj^-x) 1 ( v 2 « x ) + L84, wnere m l , m 2 Z 1 m2 ( A 2x-v 2) (v 1 - x ) \v2>x) 1 ... + m l m 2 m 3 ( A x ' v 3 ^ v l ' x ) ^ V x ) * ( A 2 x ' v - 5 ) -1 m4 (vyx) 1 (vyx) * ... + ... = (m^ 6-^  + m 2 6 2 + . . . ) H T = M I 6 X + ra 6 2 + . . . + mn 6 n (6A-6) To show t h a t a homogeneous f u n c t i o n of m degree V m l m 2 m n V = L P W J m2' m n ) x i x 2 " ^ n (6A-7) n 2 m. =m 1=1-1 can be expanded i n the form N V - ^  ^ (6A-8) where a n d m2-> • • • m n ) a r e constants, l e t x ^ k , k = l , 2, ... n i n (6A-7) he w r i t t e n as x / = (e k.x) (6A-9) k t h where e i s the k column vector of an nxn u n i t matrix k Let e , k = -1, 2, ... n be expanded as a l i n e a r combination of v^, i = 1, 2, . . . n, ' i .e. 1 8 5 . i k i * ek = § c. kv. , k = 1 , 2 , . . . n (6A - 1 1 A ) 1=1 where c^. i s determined as f o l l o w s . By v i r t u e of (6A - 2 ) , ( e k . u i ) = ( ^ 0 . ^ . ) ^ . . = c i k (6A - 1 1 B ) hence from (6A-9), ' n .mk = 4 £ [(| i , - e K ) ( v . . - x ) ] } , k = 1 ,2 , . . ..n (6A-12) u i = l 1 1 J Note t h a t (v. *x) i n (6A-12) are f a c t o r s used to obtai n Q In ( 6 A - 3 ). When (6A -12) i s s u b s t i t u t e d i n t o (6A -7 ) and m u l t i p l i e d out, the c o e f f i c i e n t y of (6A -8 ) can be obtained. From ( 6 - 3 8 ) l e t Ii = m61, U 2 = m6 2 , .(6A-13) where 6 3 . " = = W (6 A" 1 4) From (6A -1 ) one has V 2 = V l u 2 = u J . (6A -15) From (6A-3) C-L = ( v 1 . x ) C 2 = ( v 2 - x ) m ( 6 A - 1 6 ) By v i r t u e of (6A-I5) and (6A-I6), C-L = C * (6A -17) • The c o e f f i c i e n t y^s y 2 i n (6A -8 ) are determined by s u b s t i t u t -i n g (6A -12) i n t o (6A - 7 ). Thus n k m Y1 = £ (u -e K) B k=i 1 * 186, n , m v = -Z (u -eK) B (6A-18) where 6 X = P ^ , O , o, . . . j j m1 = m A P 2 = P ( o , m2, o, . . .), m2 = m (6A-I9) I t i s c l e a r from ( 6 A - 1 8 ) and ( 6 A - I 5 ) t h a t Y L = Y 2 * (6A - 2 0 ) 1 8 7 . Appendix JA 'mJiVl'xv2 Determination of ' C ^ k ^ , k By s u b s t i t u t i n g ( 7 - 3 5 ) and ( 7 - 3 0 ) i n t o ( 7 - 3 9 ) one has • 0.(Ax+g+bu) + | ( x ) + § u 2' = 0 (7A-1) and l ^ . b + / a u = 0 (7A-2) ax where 0(x) - z _ ^ ^ ( V 1 ) P , - ; s ' - ( ^ r D - ( V - ) / . - V 1 £=2 k =1 k2=l ***VK2 1 x 2 x 3 . . . (7A-3) From (7A-2) one has u = - t - § . b ' (7A-M S u b s t i t u t i n g (7A-4) i n t o (7A -1 ) y i e l d s iZ.(Ax+5 : ) + £fx). _ 2^  ( ^ . b ) 2 = 0 (7A-5) Bx ^? A +°> + 2 2a ^ dx ' w ^ ; By s u b s t i t u t i n g the s e r i e s from ( 7 - 4 0 ) i n t o (7A-4) and ( 7 A - 5 ) the equation f o r the determination of the general terms v and u _• f o r m > 2 become m m-1 i av u . = - ±- — £ . b (7A-6) m-1 a ax v / and a.v m ax - - ^ W - | ; { 2 ( ^ - > > ) ( ^ . b ) } a v P a v m m-i av„ . , where 1 8 8 . g k = o 0 (A7-8) m - ( k 1 - l ) - ( k 2 - l ) T ^ - l k 2 - l m+1 m+l-(k 1-i) G J X ) L -J- _ •£ ' P m k ¥ x i • ~ " XP " x ^ - k 1 = l k 2 =l mjk-^kg l 2 3 (A7-9) m By using the recurrence relat ion (7A-6) in (7A-7), the results is rn ^ . (Ax+bu 1) = cp m (7A-10) where m-1 3 V _ m-2 «P™ = - ^ ^ . g\ + a E u,u , - 4'G ( X ) ' m - - « Y * k m _ k « m * ' k=2 - F i + F 2 + ir W (7A-11) (7A-12) In (7A-12) m-1 av, F, = - 2 1 k=2 a x m-k+1 ' g k (7A-13) P 2 = ( a / 2 ) k i 2 Uk Um-k (7A-14) By substituting (7-44)- and (7A-8) into (7A-13) one has , -- k _ i k m-k+2 m-k+2-(i=l) 1 = 1 j = l • . - r v , c + 1 , i , J ( ^ ) ^ ( 1 - 1 ) - ( j - 1 , - k + 1 4 " 2 *n After multiplying out, le t i - i+1 for the second term, then . 1 3 9 . m-1 m-k+2 m-k+2-(i-': ) • F, = - 2 { E E [ ( i - l)D. _d • , , , . .+IR, d , , v . ,-, .] 1 k = 2 ^ j=l- ' j=l - k - 1 m-k+l,i,j_ - k m-k+l, i+l ,j J x m _ ( i _ l ) _ ( j_l) X 0 - - 1 ) X U - 1 ) (7A -15 ) where the d coefficients of the homogeneous form of m-k degree has the form d „ which is set equal to zero i f Y J C X J P • either a > ' Y + 1 J or . p > Y + l - ( a - l ) • (7A -16 ) where a > 0 , 6 > 0 m+1 m+l-(k n -l) (T- A ( V n v k - 1 k^- l Let - E 2 1 s ¥ T x ^ l ^ M V .^x,,1 x 2 ( 7 A - 1 7 ) 1 k ^ l k 2 =l m ^ l ' k 2 1 d ? By comparing (7A--15) and (7A -17 ) for the l ike terms in x^,x 2 and x^ one has s m m ~ 1 f — - \" ,1c ,Tc2 = ' R f 2 | ( k l " 1 ^ - k - l d m r k + l , l c : L , ^ 2 + k l - k d m - k + 1 , ^ + 1 , ^ 2 ! ( 7 A - 1 8 ) where d coefficients of the form d a are set to zero i f (7A - 1 6 ) i s sat isf ied. Substituting (7-44) into (7A-14) gives m-"1 J l X 2 J 2 ~ _ ..i-2 rk+l k+l-( i - l ) m-k+1 m-k+l-( i p - l ) F„ =(a/2) E Z E 1 - E E ^ k=2 4 . , = ! j =1 i 0 = l j* =1 - m - ( i 1 + i 2 _ 2 ) - ( J 2 + o 1 - 2 ) k , : ^ , ^ m - k , i 2 , J 2 1 . x 2 1 + l 2 2 x^ l+V2}} (7A -19 ) 190. Let . ; m+1 m+l-(k -]) m - ( k 1 - l ) - ( k 2 - l ) "k^-1 k~2 (7A-20) ' compare (7A-20) with (7A-I9) and l e t k n = i + i g - l (7A - 2 1 ) one has ...-2 ,k+l k + l - ( i 1 - l ) rn-k+1 m-k+l-(i„-l) k £ = ( a / 2 ) 2 { 2 r X ' V K 2 - k=2 l i =1 j = t„ T: v =(a / 2 ) S -i 2 • 2 2 2 e =1 ^ 2 = ^ J * 2 = l k , i ^ 5 j - ^ where only terms th a t s a t i s f y (7A - 2 1 ) are used i n (7A - 2 2 ) . With (7A -21 ) the i n d i c e s i 2 , and J 2 can he e l i m i n a t e d f r o m ' e m - k , i 2 , j 2 ' t h u S _^ m-2 k+1 k + l - ( i 1 - l ) . . (7A - 2 3 ) where the l a s t element has the form e ft which i s s e t equal to zero i f e i t h e r a > y + l j or p > Y + l - ( a - l ) (7A-24) where a > o, p > o. By means of (7A - 1 2 ) , (7A-I7), (7A - l 8 > ,.(7A-20), ( 7 - 9 ) and ( 7 - 2 3 ) the c o e f f i c i e n t e v F d e f i n e d i n (7-45a). can m , K ^ , K 2 .now he w r i t t e n • 191. Cm.,Z ,K2 = " kf 2 " i^ kl" 1^-k-l dm-k+l J¥ 1,lc 2 + k l - k d m - k + l ,1^+1 ,k 2\ m-2 .k+l k + l - ( i 1 - l ) ^ subjected t o the c o n d i t i o n s (7A-16.) and (7A-24). 1 9 2 . AppenOix 7B Determination of d l V V Substituting (7-48) i i t o (7 - 4 9 ) yields ^m f**1) m+i ra+1"(kr1) • • A 2 x = S E [ { a - ^ + a ' 1 2 x 2 + a { 5 x 3 } K n — _L K„—-L 3X ,~2 : I * k 2 { m - ( k r l ) - ( V l ) 3 d ^ ¥ ' x ^ x ^ + ./ 1\ m - ( k n - l ) - ( k - 1 ) L - 2 k - 1 + { a 2 l X l ^ - x 2 x/ , . v m - ( k 1 - l ) - ( k p - l ) L - l Ts p - 2 + C ^ 1 x 1 + a ^ 2 ^ ' 5 5 x 5 3 { E 2 . l } d ^ ^ 2 X 1 x ^ x 5 f . ] m+1 ^"(V1) _ E ' [ { a { J C m - ( k 1 - l ) - ( k 2 - l ) } + a 2 2 { k 1 - l } + a ^ 3 n c 2 - l } } . , ,x m-(k.-l)-(k 0-l) k,-l k_-l 4 ^ ] k 2 x l 1 2 - 2 1 X / +Cai2{n.(TE1-lWV1)5 D U ~ D x ^ ^ ^ ^ x ^ x ^ S + r a ' fm fk i W k - i n d ^ - 1 ) -^ k ^ k / l X 2 X 3 } + [ a 1 3 { m " ( k l 1 } C k 2 ^ ^ m ^ k ^ k g m - ^ - l ) - ^ - ! ^ ^ _ 1 } d ( - t l ) _ ^ (V2)"^"1) x l ' X 2 x3 J + i a 2 1 i k l l j dm,k 1 3k 2^l ^ - 2 ^ - 1 m-(k -l)-(k -1) (k -2) k ? x g 1 x / J + C a ^ ^ - l } ^ ] ^ 1 • 2 x 2 ! x 3 2 } + 193. F l a 3 1 l k 2 ^ V^k^l . X2 X3 j + m - ( k . . - l ) - ( k 0 - l ) k n k -2 ^ ^ ^ ^ k ^ ^ l . . X 2 X 3 } ] . In the r i g h t hand side of (7B-1), l e t the dummy i n d i c e s , k^ -» k^+1 , kg = kg i n the second term, x : ~ k^ = k^ , kg -> k g - l i n the t h i r d term, k^ = k-^  , kg '-» kg+l i n the f o u r t h term, k^ -> k^+1 , kg -» k g - l i n the f i f t h term, k^ = k-^  , kg -> kg+l i n the s i x t h term, and k^ ~* k-^-l , kg -> kg+l i n the l a s t term, (7B-1) can be w r i t t e n .A^x = E a 1 2 + a 2 l ^ l ^ i , ^ i l , k 2 + ^ a ^ " 1 ^ ^ + a23^i } d i ,^ii,kg-i Sl^2^i^]kg+1 + a52^2}^l,ig+l + m-(krl)-(kg-l) (krl) (kg-l) (7B_2)  , X 1 X 2 X3 (Jt-1) where the d v ' c o e f f i c i e n t s o f the homogeneous form of m*h degree have the form d^~~l - which i s set t o zero i f m, a, B e i t h e r a > m+1 s or B > m+1 - ( a-l) where ( 7 3 - 3 ) a > o , and P > o By comparing ( 7 B - 2 ) with ( 7 - ^ 9 ) 3 the c o e f f i c i e n t d j ^ 1 ^ i n ( 7 - 5 3 ) i s determined. REFERENCES E.W. 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