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Power system stability study by Szego's method and a maximized Liapunov function Metwally, Aly Abdel Hameed 1970

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POWER SYSTEM STABILITY STUDY BY SZEGO'S METHOD AND- A MAXIMIZED LIAPUNOV FUNCTION by ALY ABDEL HAMEED METWALLY B. S c , AIN-SHAMS UNIVERSITY, CAIRO, EGYPT, 1965 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of E l e c t r i c a l Engineering We accept t h i s thesis as conforming to the required standard Research Supervisor Members of the Committee Head of Department Members of the Department of E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA December, 1970 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 a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g 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 p u r p o s e s may be g r a n t e d by t he Head o f my Depar tment 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 . Depar tment o f 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 V a n c o u v e r 8, Canada Date ' t ^ c . 2 ? * j JV7 ABSTRACT In t h i s t h e s i s Liapunov's d i r e c t method i s applied to transient s t a b i l i t y study of power systems. Szego's method i s applied to a second order power system i n chapter two and a quadratic Liapunov function applied to the same system i n chapter three. The hypervolume enclosed by the quadratic V-function i s maximized. Changes i n the time d e r i v a t i v e of the quad-r a t i c V f u n c t i o n are then made to meet the conditions of Liapunov V and V functions. F i n a l l y a maximized modified Liapunov function i s constructed from a t e n t a t i v e quadratic function f or a three-machine system. i i TABLE OF CONTENTS Page ABSTRACT 1 1 TABLE OF CONTENTS . i i i LIST OF ILLUSTRATIONS.. . i v ACKNOWLEDGEMENT v NOMENCLATURE. • • V 1 1. INTRODUCTION 1 2. A POWER SYSTEM STABILITY STUDY BY SZEGO'S METHOD. _ 3 •2.1 Power System Equations...... ........ 3 ' 2.2 Szego's Method 5 2.3 Algorithm 7 2.4 Maximum Value of the Liapunov Function 9 2.5 Numerical Example 10 3. MAXIMIZATION OF A LIAPUNOV FUNCTION 15 3.1 Constraints On a Quadratic V For a 2nd Order Power System . 15 3.2 Hypervolume Bounded By v=x'Ax_ x ^ . 3«3 Optimization Technique 19' 3«4 Numerical Example 21 3.5 A Modified Liapunov Function 21 3.6 Concluding Remarks 28 4. A MAXIMIZED LIAPUNOV FUNCTION FOR A 3-MACHINE POWER SYSTEM 29 4.1 Equations Of a 3-Machine System 29 . 4-2 Conditions To Ensure Negative Definiteness Of ^  .... 32 4-3 Construction Of Liapunov Function And Maximization.. 40 4-4 Numerical Example 42 4.5 Concluding Remarks 43 5. CONCLUSION 46 APPENDIX I.'. 47 APPENDIX II . 49 APPENDIX III 51 REFERENCES ; V 54 i i i ( LIST OP ILLUSTRATIONS Figure Page 2.1 A Typical Power System 6 2.2 Phasor Diagram of Salient Pole Synchronous Machine... 6 2.3 Equivalent Power System 11 2.4 Numerical Example 11 2.5 Stability Region By Szego's Method 13 2.6 Plow Chart For Szego's Method 14 3.1 Sta b i l i t y Region By-A' Maximized Quadratic -V-Punction. 22 3.2 Plow Chart For Maximizing A Quadratic V-Punction -23 3.3 Choosing V 26 m 3.4 Stability Region By Modified V-Function 27 4.1 Three-Machine Power System 30 4.2 V For Three-Machine System. With x =x =x..=o ......... 44 m 4 5 6 iv ACKNOWLEDGEMENT I wish to express my gratitude to Dr. Y. N. Yu, supervisor of th i s p r o j e c t , for the guidance given throughout the preparation of t h i s t h e s i s . His help and encouragement have been invaluable. Thanks are due to Dr. M. S. Davies for reading the manuscript and f o r many h e l p f u l discussions. The proof reading of the f i n a l d r a f t by Mr. H. Moiissa and Mr. B. P r i o r i s duly appreciated. The support from the National Research Council and the U n i v e r s i t y of B r i t i s h Columbia i s g r a t e f u l l y acknowledged. I am deeply g r a t e f u l to my wife Magda for her continuous encourage-ment and understanding. v NOMENCLATURE Vector of State Variables Time derivative of x State variable equation vector Liapunov function Value of V defining s t a b i l i t y region Time derivative of V Series resistance of transmission system Series reactance of transmission system Transformer reactance Transmission system shunt susceptance Shunt conductance representing local load Active power El e c t r i c a l power output of synchronous machine Mechanical power input to synchronous machine Reactive power Terminal voltage of synchronous machine Infinite bus voltage . Equivalent impedence of local load and transmission system Equivalent resistance of local load and transmission system Equivalent reactance of local load and .'• • 'transmission system: • • • . > " Equivalent i n f i n i t e bus voltage v i D 6 H M f B r B 2 ' B 3 ' V 3 a n d y ;us Q(x), g(S(x)) A(x) a i j ( x ) a ( x 1 ) , K (x 1) a., b. 1 1 A( X ; L), B ( X l ) , C( X ; L) "d ? Damping coefficient Angle between quadrature axis of synchronous machine- and i n f i n i t e bus voltage or a reference frame rotating at synchronous spread i n the case of multi-machine systems. Inertia constant i n KW. Sec. /KVA n/(%f) System frequency = 60 c/s Time Internal voltage of synchronous machine Constants i n the expression for P g Steady state values for <s Value of 6 at the unstable equilibrium position Coeficients of expanded system equations Scaler functions of x The prime on a vector or matrix indicates the transpose Square Symmatric matrix with' variable elements Elements of A(x) Polynomials in x^ Coefficients of a(x^) and ^(x^) respectively Polynomials i n x^ Direct axis synchronous reactance ' Direct'axis transient reactance v i i II Direct axis subtransient reactance Quadrature axis synchronous reactance Quadrature axis subtransient reactance Direct axis transient open-circuit time constant T II . Direct axis subtransient open-circuit time constant T II qo Quadrature axis subtransient open-circuit time constant ' A • Square symmetric constant matrix a H » a i 2 ' a22 Elements of A 2 Second degree terms in V ^1' §2'""*' ^6 Constraint equations * i Eigenvalues of A * Hypervolume enclosed by V=x'Ax 7 — Augmented state variable vector 4> (z) Object function <J> (z) Augmented object function a jg_(z) Vector of constraint equations v Vector of Lagrange multipliers % g - A matrix of elements g . . = 7 ; — J 6 6 6<fi Increments i n Z, _g and <f> respectively a a x A vector of components -z . • . 3Z p Projection matrix 8 y Unit matrix 6fc . ..'Step size . . . • Y Maximum value of V describing a closed maximum surface • v m Value of V tangent to V=o Internal voltages of respective machines Component functions of v 1. INTRODUCTION O s c i l l a t i o n s i n the power flow between synchronous machines have long been known to be present. Since no r e a l power system i s t r u l y i n the steady state and there are always disturbances, the system has to be c o n t i n u a l l y adjusting to meet new operating conditions. In other words the power system has to have adequate transient s t a b i l i t y margins. The s t a b i l i t y c h a r a c t e r i s t i c s of a power system during tran-s i e n t disturbances are usually analyzed from a set of nonlinear d i f f e r -e n t i a l equations known as the swing equations. The order of these equations depends on the d e t a i l of representation of the synchronous machines and associated c o n t r o l l e r s . The s o l u t i o n of these equations i s u s ually obtained by step-by-step i n t e g r a t i o n during and a f t e r the disturbance u n t i l the c r i t i c a l switching time i s found. The present trend towards interconnection of power systems i n order to r a i s e u t i l i t y f a c t o r s and to improve the load factors and so achieve more economical operation, increases the s i z e and complexity of power systems making the step-by-step method f o r s t a b i l i t y studies more tedious and c o s t l y . A need arises f or a more economic and straightforward method for studying s t a b i l i t y . For t h i s , the d i r e c t method of Liapunov, [1], [2], i s very u s e f u l . The method enables one to determine the s t a b i l i t y of the e q u i l i b r i u m state without a c t u a l s o l u t i o n of the system's d i f f e r e n t i a l equations. With a s u i t a b l y constructed Liapunov function the s t a b i l i t y region of a power system can be established and the c r i t i c a l switching time can be obtained by carrying out only one forward i n t e g r a t i o n of the swing equations. The basic d i f f i c u l t y i n the a p p l i c a t i o n i s the absence of a 2 unique method f o r constructing Liapunov functions although some formalized methods, [3] to [12], have been developed for c e r t a i n classes of functions. Some of these methods have been applied also to power systems. Yu and Vongsuriya, [15], employed Zubov's method and a truncated power, s e r i e s of V-functions to study a one-machine-i n f i n i t e bus system. Rao, [17], used Cartwright's procedure, [5], to study one-machine and three-machine systems. Applying Popov's theorem and Kalman's procedure, [4], P a i , et. a l . , [18], studied a one-machine system including, governor action.. .The v a r i a b l e gradient ^method .. _ -developed by Gibson and Schultz, [6], was applied by Rao and Desarkar, [20], to a t h i r d order model of a one-machine system. The generalized Popov c r i t e r i o n f o r m u l t i v a r i a b l e feedback systems was used by Willems, [22], to develop a Liapunov function f or multimachine systems. Others, constructed Liapunov functions f or one-machine, [14] [16], as w e l l as multimachine systems, [13], [19], [21], based on energy i n t e g r a l s . This thesis i s an extension of the transient s t a b i l i t y studies f o r power systems through the a p p l i c a t i o n of Liapunov's d i r e c t method. Szego's method, [7], i s applied i n chapter 2 to estimate the transient s t a b i l i t y region of. a power system. The Liapunov function obtained i s i n the form of a power s e r i e s . A quadratic form Liapunov function i s considered i n chapter 3, and the hypervolume enclosed by t h i s function i s maximized subject to c e r t a i n constraints on the Liapunov function, V, and i t s time d e r i v a t i v e , V. The r e s u l t s are fur t h e r improved by e l i m i n a t i n g the i n d e f i n i t e terms i n V and by modifying the quadratic V-function. In chapter 4 a Liapunov function for a three-machine system i s constructed. •••Starting with a quadratic V-function,'the time', d e r i v a t i v e V i s obtained and adjusted to be negative d e f i n i t e . The a c t u a l V-function i s then formed and f i n a l l y the volume enclosed by the quadratic portion of t h i s new V-function i s maximized. 3 2. A POWER SYSTEM STABILITY STUDY BY SZEGO's METHOD Based on Zubov's work [23], [24], Szego [7] suggested a construction procedure to obtain Liapunov functions f o r systems with n o n l i n e a r i t i e s representable i n polynomial form. The method i s applied i n t h i s chapter to determine the s t a b i l i t y region of a second order nonlinear power system. The equations of a disturbed power system a f t e r f i n a l switching are w r i t t e n i n state v a r i a b l e form, with the f i n a l . e q u i l i b r i u m at the o r i g i n , as follows x = f(x) , f(0) = 0 (2-1) The s t a b i l i t y region i s expressed i n the state space by i t s boundary surface as V - V (2-2) m where V i s a Liapunov function and i s the maximum value of V that describes a closed surface tangent to V = 0. 2.1. POWER SYSTEM EQUATIONS . A t y p i c a l power system i s shown i n F i g . 2-1. I t consists of a s a l i e n t pole synchronous generator connected to an i n f i n i t e bus through a high voltage transmission l i n e . The transmission system i s represented by a s e r i e s resistance r and reactance x. The transformer i s represented by a reactance x^. The charging e f f e c t of the l i n e and l o c a l r e a c t i v e power are represented by a susceptance B and the l o c a l load i s represented by a conductance G at the machine terminal. The t o t a l power output of the machine i s P + jQ at. a terminal voltage V • . The i n f i n i t e bus has a constant voltage V . The following assumptions are made for the power system under study: a - The internal induced voltage of the synchronous machine is constant, b - The flux linkages in the rotor circuits of the synchronous machine are constant. c - The mechanical input to the synchronous machine is constant, d - The armature resistance is neglected. The synchronous machine dynamics are represented by a second order dif f e r e n t i a l equation with, the voltage-.relations as shown in Fig. 2^2. . ' Applying Thevenin's theorem, the system shown in Fig. 2-1 can be reduced to the simpler form of Fig. 2-3 where Z = 1 / [G + jB + 1 — ] eq J r+j (x+x ) V [r+j(x+x )] V = V -o o r+j(x+x )+l/(G+jB) Thus r e = {G[r 2+(x+x t) 2]+r} /A (2-3) x £ = {(x+x t)-B[r 2+(x+x t) 2]} /A V = V //T o o A - 1 + 2[B(x+x t)(l-Gr)+Gr]+(B 2+G 2)[r 2+(x+x t) 2] Although the damper winding circuits are not included in the machine equations, the damping effect i s approximated [27], [15] by 2 2 D(6) = D 1 Cos 6 +D2 sin 6 t o I I . I I o . . . D = V (x-' - x-) x /(x +x )• •• * : " ••• 1 o q q q o e q • ( 2 _ 4 ) I O I I I I I i n D 2 = V o ( x d " V Td / ( xe + Xd> o 5 Including the energy conversion power output, P (6)> which is derived in appendix I, the swing equation of the machine has the form 2 M ^-4 + D(6) £ + P (6) - P . (2-5) d t 2 d t where P (6) = B . E ' 2 + F B „ Cos(6+g)+B. sin (6+Y ) ] E ' + B . sin(6+y) Cos(6+8) e i q ' L 5 q 4 (2-6) 2 dS d o Let 6 = 6,' -r- o and — „ = o in the steady state, and let o' dt ,2 dt the state variables be chos"en as (2-7) x, = 6- 6 . 1 o d6 x2 dt The system equations in state variable form can be written as *1 = X2 X2 - \ [ P i " P e ( x l + 6 o ) " D ( X 1 + 6 o ) X 2 ] ( 2 " 8 ) which can be expanded into a power series to give x l = X2 X2 = E . Pi X l + X2 E q i X l ( 2 _ 9 ) i=l i=l For the s t a b i l i t y study the series may be truncated [15] after N terms. The details of expansion are given in appendix II. 2.2. SZEGO'S METHOD A brief summary of Szego's method is given as follows. A system represented by (2-1), i f stable, w i l l be either globally or locally stable. According to Szego, the sufficient condition for local s t a b i l i t y i s that the time derivative of the Liapunov function by 6 SALIENT POLE SYNCHRONOUS MACHINE B • INFINITE BUS F i g . 2-1 A T y p i c a l Power System J'xq 'q i r \ jxd id id F i g . 2-2 ' Phasor Diagram of S a l i e n t Pole Synchronous Machine < 7 v i r t u e of equation ( 2 - 1 ) , has the form V(x) = 0 ( x ) • g(5(x)) ( 2 - 1 0 ) where 0(x) i s a semidefinite function not i d e n t i c a l l y equal to zero on any n o n t r i v i a l s o l u t i o n f f t h e system ( 2 - 1 ) , and g(x) i s i n d e f i n i t e on a closed surface, i . e . £(x) = 0 i s a. closed surface or family of surfaces and g(u) i s such that g ( 0 ) = 0 and g(u)/u >0 for u^O. D i f f e r i n g from LaSalle's and Zubov's methods, Szego required that V be i n d e f i n i t e on a closed surface as an approximate i d e n t i f i c a t i o n of the l i m i t c y c l e . A quadratic function with v a r i a b l e c o e f f i c i e n t s c a l l e d the generating V-function i s chosen for the Liapunov function V(x) = X'A(X) x ( 2 - 1 1 ) where A(x) = (a..(x., x.)} , a..(x., x.) = a..(x., x.) and the elements i j i 3 13 i J J i i 3 a..(x., x.) do not contain x . The l a t t e r assumption i s - j u s t i f i e d i j i j n [7], by the f a c t that l i m i t cycles of the most general nonlinear system i n the phase space have at most two r e a l i n t e r s e c t i o n s with each of the hyperplanes x^ = constant ( i = l , 2 , . . . , n - 1 ) . Equation ( 2 - 1 1 ) i s then d i f f e r e n t i a t e d , using ( 2 - 1 ) , to give an expression for V which 'is then adjusted by changing the c o e f f i c i e n t s a..(x.,x.) to get the desired form given by ( 2 - 1 0 ) . i j i 3 2 . 3 . ALGORITHM [ 2 5 ] Following Szego, the Liapunov function considered i s V = a ( X ; L ) x 2 + ^ ( x 1 ) x 1 x 2 + x 2 ( 2 - 1 2 ) By v i r t u e of ( 2 - 9 ) the time d e r i v a t i v e of V i s o A { \ d 5 (x ) • r o / \ . 2 d a ( x n ; . _ / N . - - 1 •• V = [ 2 a ( x 1 ) x 1 + x± 1 _ + C ( x 1 ) x 2 + X l x 2 ^ - ] x 2 dx^ . . . . 1 . N . N + [ C ( x 1 ) x 1 + 2 x 2 ] [ E P i x J + x 2 I q ± x ^ " 1 ] ' ( 2 - 1 3 ) i = l i = l 8 There are i n general two steps i n Szego's method. F i r s t , a s u i t a b l e form for V established and, secondly, from t h i s form the ac t u a l V i s constructed. However, a d i r e c t c a l c u l a t i o n of V i s po s s i b l e i n our case. Let a(x^) and £ ( x ^ ) have the general form Art i - l a ( x 1 ) = I ai xi i = l C(x ) = Z b x j 1 i = l (2-14) Su b s t i t u t i n g (2-14) i n t o (2 r13) gives V(x) = A ^ x 2 + B ( X l ) x 2 + C(x 1) (2-15) where CO . oo r N A(x ) = Z b. x] + Z ( i - l ) b . x ^ " 1 + 2 Z q.4" 1  1 i = l 1 1 i=2 1 i = l B(x ) = 2 I a.x] + I ( i - l ) a . x ^ + 2 Z p.x^ + Z Z b.q.x^" J  1 i = l 1 1 i=2 1 1 i = l 1 1 i = l j = l 1 3 1 0 0 N . ... C( X ; L) = Z Z b kp. x^ 3 . (2-16) 1=1 j = l 3 Equation (2-15) i s of the second degree i n x 2 and hence the equation V = 0 w i l l describe two curves i n the state space. Now i f A(x^), B(x^) and C(x^) are chosen such that B 2 ( X ; L ) = 4A(x 1)C(x 1) (2-17) then the two curves w i l l coincide and V w i l l not change sign along any l i n e p a r a l l e l to the x^ a x i s . Condition (2-17) can be s a t i s f i e d by s e t t i n g both A(x^) and B(x^) i d e n t i c a l l y equal to zero. A ( X ^ ) H Q gives constant term, b^ + 2q^ = 0 c o e f i c i e n t of x^, 2b^ + 2q 2 = 0 2 c o e f f i c i e n t of x^, 3b^ + 2q^ = 0 c o e f f i c i e n t of x!? \ nb +2q I n n thus -2 b ± = — q ± , i = 1,2,..., N 0 , i>N (2-18) B(x^) =0 gives c o e f f i c i e n t of x^, 2a^ + 2p^ + b - ^ i = 0 2 c o e f f i c i e n t of x^, 3a^ + 2p 2 + b^q 2 + b 2 q ^ = 0 3 c o e f f i c i e n t of x^, 4a^ + 2p^ + ^-^^ + b 2 q 2 + b 3 q l = ® thus -1 a. = l O+iy ( 2 p i + .l± bi«i-j+l> > i-1.2;.-,., 2N-1 0 , i *2N (2-19) Equations (2-18) and (2-19) provide the algorithm for c a l c u l a t i n g the c o e f f i c i e n t s of the Liapunov function (2-12). The time d e r i v a t i v e V now becomes V = C ( X l ) (2-20) 2N-1 N ... , . - Z E b.p. x f 3 ( 2 " 2 1 ) i - l J - l 1 3 1 2.4. ' MAXIMUM VALUE OF THE LIAPUNOV FUNCTION' using the Liapunov function derived i n the previous s e c t i o n , the maximum value of V describing a closed curve tangent to V = 0 i s determined as follows. Consider equation (2-21), at the unstable e q u i l i b r i u m p o s i t i o n 6= 6 U S , x^ = 6 U S- 6 q and N l r v P - X i = 0 (2-22) M [ p r P e ( 6 ) ] = .\ 1 1 1=1 Thus V = 0 i s a s t r a i g h t l i n e p a r a l l e l to the x 2 axis and passing U S through the point x^ = 6 - 6 . Solving equation (2-12) f o r x one gets 10 -ax 1)x 1 +4 2(x 1)x 2 - 4a(x 1)x 2 + 4V x2 = ^ 1 ^ 1 - ^ v 1 1 " v (2-23) 2 For the curve V = to be tangent to V • 0 , the value of the square us root must be equal to zero at x- = 6 - 6 , thus 1 o V = [ct(6 U S - 6 ) - 7 5 2 ( 6 U S - 6 )](6 U S-6 ) 2 (2-24) m . o 4 o o 2.5. NUMERICAL EXAMPLE Szego's method is now applied to study the s t a b i l i t y of a particular power system. The synchronous machine under study has the. following particulars : i i x '='0.27 p.u. xd = 9 sec. d o » x, = 1.0 p.u. x , = 0.04 sec. d d o Xq = °- 6 P- U- T " = 0.07 sec. }o q q , x, = 0.22 p.u. H =4 KW sec/KVA d x = 0.29 p.u. q and is delivering a power of 0.753 + j 0.03 p.u. to the system at an i n i t i a l terminal voltage of 1.05 p.u. The transmission system particulars are shown on Fig. 2-4. A sudden three-phase symmetrical short c i r c u i t to ground occurs at (x) on one of the transmission lines near the generator end causing bus A to ground. The faulty line is disconnected from the system at both ends after a fault duration of 8 cycles. The fault i s then cleared and the lin e restored. From the given i n i t i a l terminal voltage and power output F +jQj the i n i t i a l operating conditions determined, [26], are 11 V. F i g . 2-3 Equivalent Power System X SALIENT POLE SYNCHRONOUS MACHINE xt = X =. r = B = G 0.013 p.u. = 0.7488 p.u. 0- 75 p.u. = 0. 057 p.u. = 0.18 p.u. j n ( . B F i g . 2-4 Numerical Example Vt - 1.05 p.u. Xd = 0.27 p.u. P= 0.753 p.u. Q =0.03 p.u. 2r 1 1 y ^ 2X / INFINITE BUS 12 r , V = 0.989 p.u. 6 = 0.942 radians o o i E = 1.053 p.u. = 53.9 degrees q 6 U S = 3.04 radians = 174.28 degrees For the system considered the maximum value of V is found to be = 73 and i t gives a c r i t i c a l reclosing time of 23 cycles. The swing curve equations (2-8) are integrated forward using Runge-Kutta method From the results i t is'found out that the c r i t i c a l reclosing time is 24 cycles. Fig. 2-5 shows the actual region of s t a b i l i t y for the system considered along with the s t a b i l i t y region defined by the Liapunov function and a system trajectory for a fault duration of 8 cycles and line reclosure after 23 cycles. A flow chart for the computer program used is shown in Fig. 2-6. —~ SYSTEM TRAJECTORY LIAPUNOV STABILITY BOUNDARY • ' "ACTUAL STABILITY . . BOUNDARY . > ' 10 1 6. .175 -\5 -1.25 -1.0 -0.75-0.5 -0.25 \ \ \ \ \ 0 -2. -4. -6. '-10-x2 =00 V=o' V<o V) o 0.25 0.5 0.75 1.0- 1.25 1.5 1.75^2:0 2.25 Fig. 2-5 STABILITY REGION BY SZEGO'S .METHOD. SYSTEM TRAJECTORY LIAPUNOV STABILITY BOUNDARY ... ACTUAL STABILITY BOUNDARY 14 Fig. 2-6 FLOW CHART FOR SZEGO'S METHOD 15 3. MAXIMIZATION OF A QUADRATIC LIAPUNOV FUNCTION In this chapter a quadratic Liapunov function of the form V = X ' A X (3-1) is considered, where A is a positive definite symmetric matrix. The hypervolume enclosed by (3-1) is sought and maximized subject to constraints arising from conditions on the Liapunov function and i t s time derivative. The s t a b i l i t y region thus obtained for a power system i s very res t r i c t i v e . Since this does not serve the object of this study, a new Liapunov .function is then sought by eliminating the indefinite • terms in Vand modifying V accordingly. The new V-furtction thus obtained gives a very good estimate of the s t a b i l i t y region for a power system. j 3.1. CONSTRAINTS ON A QUADRATIC V FOR A 2 — ORDER POWER SYSTEM To establish asymptotic s t a b i l i t y the Liapunov function must satisfy the following conditions a - V is positive definite. b - V is negative definite in an open region around the origin, c - V tangent to V s 0. m Consider the second order power system (2-8). Let A of (3-1) be A = a l l a12 a12 a22 (3-2) then one has V = a n x 2 + 2 a 1 2 X l x 2 + a ^ x 2 (3-3) The Sylvester conditions for V to be positive definite are ..• a i l > ° • , , , ... . .... ,.. . - , ..  , (a-D 2 a l l a 2 2 " a12 > 0 (a-2) 16 Next, using equation (2-8), the time d e r i v a t i v e V i s D(x 1) 2 D( X ] L) p i " p e ( x l ) V = 2 { ( a 1 2 - a 2 2 — r r - ) x 2 + [ (a^ - a ^ — j r - ) ^ + *22 < M )3 p i " P e ( x l ) + a 1 2 X l ( M ) } (3-4) To s a t i s f y condition b, i t i s required that V be negative along the two axes x^ and x,^ , and also that V^, the second degree terms of V, are negative d e f i n i t e . Along the x^-axis, V i s given by P i " P e ( x l ) • •• V = 2 a l 2 X l ( M > ' ..' ' ' ' " The term x, ( — ) i s negative f o r 1 M -(2T7 - 6 U S + 5 )< X l<5 U S - 6 o I o thus one must have a 1 2 > 0 ' . • (b-1) Along the x 2 a x i s , V i s given by * = 2 ( a i 2 - 1r D ( xi ) ) x2 which i s negative i f where n m £ n i s the minimum value of the damping c o e f f i c i e n t , given by D +D D-,-Do Dm - i n ( X 1 > = ^ {- AT A + Cos(2x +26 )} = D mm I x^ Z Z 1 o Z ... . To fin d . V 2 > equation (,3-4.) i s . expanded i n t o a power se r i e s by the use of equation (2-9) to give N • N . . ' • 'v i i—1 V = 2{ ( a n x 1 + a i 2 x 2 ^ x 2 + ^ a 1 2 X l + a 2 2 X 2 ^ ^  E P i X l + X 2 2 q i X l ^ i = l i = l t 17 The second degree terms may be written V 2 = 2 ( X l x 2 ) a 1 2 P l 2 ( a l l + a 1 2 q l + a 2 2 P l ) 2 ( a 1 1 + a 1 2 q 1 + a 2 2 p 1 ) ( a 1 2 + a 2 2 q l ) and the conditions f o r to be negative d e f i n i t e are (b-3) (b-4) (c-1) (c-2) a 1 2 p l < 0 . . . a 1 2 P l ( a 1 2 + a 2 2 q l ) - i ( a l l + a l 2 q l + a 2 2 P l ) 2 > 0 ... • ••' • f i n a l l y condition C implies V= 0 3 1 3V _ SV 3V = Q 3x 1* 3x 2 3 x 2 8 X l Since p^ i s negative, conditions (b-1) and (b-3) are the same, Also condition (b-4) can be rewritten as - P l ( a n a 2 2 - a 1 2 2 ) - ^ a ^ a ^ - a ^ ) 2>0 which automatically s a t i s f i e s condition (a-2). To summarize, the f i n a l set of constraints are: = V = 0 9V 3V 3 X T ' 3x, 9V 3x, 3 V = 3x, a u > 0 84 = a 1 2 > 0 a22 D2 ^:=-r-M-.~. a 1 2 > A H = " P l ( a l l a 2 2 ~ a 1 2 } i ( a 1 1 + a 1 2 q 1 - a 2 2 p 1 )2 > 0 ( 3 - 5 ) 18 3.2. HYPERVOLUME BOUNDED BY V = x Ax i The hypervolume bounded by the surface V = x_Ax, which is to be maximized, is found as follows. Since the matrix A of equation (3-1) is positive definite, i t s eigenvalues X^, i=l,2,...,n are a l l positive and the surface described by this equation i s a closed one. The problem is then reduced to finding the volume bounded by V = n 1=1 X. x 2 l l y 1 1^2y«a»£ XI (3-6) Let C. = vty/A. (3-7) then - 2 n x. i=l c. I (3-8) and the required volume is given by r 1 = 2 J resulting in / - 2 , 2 ./1-x, /c, / , - 2 . 2 -c„vl-x,/c. ' 2 * 1' 1 A-i -2, 2 - , - x./c. n-1 . n i i i=l r^~2 n . , i i - l / r 2 - 2 / 2 : .v1—E x./c. n-1 . T l l i=l 2,2 dx ,dx „. J c . n-1 n-2 l l (3-9) n-1 n odd 2 n ( — ) n / 2 1 2.4.6. . .n n even (3-10) where |A| denotes the determinant of matrix A. The details are given in appendix III. 19 3.3. OPTIMIZATION TECHNIQUE Of the s i x c o n s t r a i n t s i n equation (3-5), the l a s t four are i n e q u a l i t y c o n s t r a i n t s . ' These can be transformed i n t o e q u a l i t y cons-t r a i n t s [28], [29], by i n t r o d u c i n g some new f r e e v a r i a b l e s as f o l l o w s . Let 8 3 = a i r y r e = 0 H = a 1 2 - y 2 " e = 0 ^2 2 g 5 = a22~M " a 1 2 - y 3 _ C = 0 ' g 6 - - P ^ a ^ - a ^ 2 ) - f C a n + a ^ - a ^ ) 2 ^ - e=0 (3-11) where y^, y^, y^ and y^ are f r e e v a r i a b l e s and e i s a s m a l l p o s i t i v e constant. - ...— _ — . . -._ _ - _. An augmented space, Z, c o n s i s t i n g of the x-space, the a-space and the y-space i s considered. The components of t h i s new space are given by • z = (z± z 2 ... z9>' = ( x 1 x 2 a 1 1 a 1 2 a 2 2 y 1 . . . y 4 ) (3-12) The problem i s now defined as follows:' minimize the cost f u n c t i o n <j>(Z) = - I ( Z ) , subject to the c o n s t r a i n t s g(Z) = 0. The p r o j e c t e d gradient method [30], [31], [32], as best explained i n [32], i s employed to s o l v e t h i s problem. The method i s summarized as f o l l o w s . Consider an augmented cost f u n c t i o n (j, (Z) = <|>(Z) +&'(Z)v (3-13) a — where v i s a v e c t o r of Lagrange m u l t i p l i e r s . Then 20 i t 6<j>a = (*Z+8Z v) 6Z (3-14) where <jj , v_ and 6_Z are vectors and g is a matrix (For notation see nomenclature). The steepest descent move is given by fi"= - k ^ v ) (3-15) where k is the step size, k>0. The question now is how to choose v_. The increment 6Z must be chosen so that the new point is in the cons*-traint surface defined by £=0. If a f u l l correction i s used for a nominal value of &Z must be chosen so that % = sz«z = -s. • (3~16> or 6£ + £ = 0 Substituting (3-15) into (3-16) and solving for v_ yields v = ( g z g z ) _ 1 ( ^ / k - g z* z) (3-17) substituting (3-17) into (3-15) gives = -kPg<J>z - g z ( g z g z ) - 1 & (3-18) P„ 4 U - g'(g7g') V (3-19) where T g y 6Z V 6Z f tZ y 6Z where U is the unit matrix. Let the desired step size be , thus 2 ' 64 = 6Z 6Z (3-20) yielding 2 ' ' -1 k= /Sft - g- ^ gZ gZ ; 1 (3-21) •zV'z 21 3.4. NUMERICAL EXAMPLE The same numerical example i n chapter 2 i s used here f o r th maximized quadratic Liapunov function s t a b i l i t y study. The r e s u l t s obtained are: 430 1.002 1.002 12.02 _ V = 86 , tangent to V = 0 at m X-L = 0.394 and x 2 = 1.238 This V-function gives a c r i t i c a l r e c l o s i n g time of 4 cycles a f t e r f a u l t occurance which i s a very r e s t r i c t i v e r e s u l t as shown i n F i g . 3 A comparison between the s t a b i l i t y region defined by th i s Liapunov function and the act u a l region obtained by in t e g r a t i n g the system equations using Runge-Kutta method i s given i n the same f i g u r e . F i g . 3-2, shows a flow chart of the program, used 3.5. A MODIFIED LIAPUNOV FUNCTION From f i g . 3-1, i t i s noticed that the reason for the poor estimation of s t a b i l i t y region i s due to the f a c t that the curve V=0 i s near to the o r i g i n . A bet t e r estimate w i l l be obtained i f t h i s curve i s s h i f t e d away. Consider the expression for V, i . D ( X ; L ) 2 D ( X l ) P i " p e ( x l ) V = 2 t [ a 1 2 - a 2 2 - ¥ - ] x 2 + t ( a u - a 1 2 — r r - ) ^ + ^  S ) ]*2 P i " P e ( x l ) + a 1 2 X l < 5 " <3"4) D ( X 1 ) 2 As pointed out i n (b-2), ( a 1 2 _ a 2 2 — M — ^ x 2 i s a l w a y s negative. Also p.-p (x.) ' - a i o x n ( ~ — i — ~ ) i s negative f o r - ( 2 T T - 6 U S + 6 )<x <6 U S-6 . The 12 1 M o 1 o V>o \ V <o X] = 6-6Q TRUE STABILITY BOUNDARY Fig- 3-1 STABILITY REGION BY A MAXIMIZED QUADRATIC V-FUNGTIOM ro N> 23 C START ~) « READ SYSTEM DATA, I FTTKT \ < ™ S T ? ^ > £ , E , V q ° C A L C . B , Y , D , D 2 > B 1 B 2 . B 3 , B 4 F I R S T ? No I C A L C . P 1 . Q 1 I Z - Z + DZ FORM V , V , G , G z , P D Z 1 , D Z 2 , DZ BA-se^-g'Cg' g') - 1g STORE Z I N Y l _ K-0 o * a BA ^0? No Z - Y & C A L C . V, SET E Q N . TO FAULT COND. I SET I N T G . T I K E TO CLEAR T I M E I N T G . SYSTEM E Q N . C A L C . V , $ . CHANGE PARAMETERS TO F I N A L SET I N T G . T I M E TO FTNAI. T I M E T H I R D CHANGE PARAMCTERS TO POSTFAULT SET I N T G . T I M E TO R E C L O S I N G T I M E SECOND No ( STOP ) F i g . 3-2 F L O W C H A R T F O R M A X I M I Z I N G A Q U A D R A T I C V - F U N C T I O N 24 remaining two terms, however, change sign according to values of x^ and x^. If these i n d e f i n i t e terms are eliminated from V by subtracting t h e i r i n t e g r a l , with respect to time, from the o r i g i n a l V-function, then V w i l l be negative d e f i n i t e . Since t " ( x ^ V P e ( x l ) / 2 x 2 [ ( a n - a 1 2 - ¥ - ) x 1 + a 2 2 ( M )]dt o \ D ( x ! ) p i " p e ^ x l ^ -2/Q [(Va12T~)xl + a22(~li ) ] d x l V D 2 2 .... - : = ( a i r a i 2 ~ 2 M - ) x i - f ( x i } ( 3 ~ 2 2 ) where a i 2 ( D r V -f ( x , ) - . „ — — [Cos(2x 1+26 )+2x l Sin(2x +26 )-Cos 26 ] v 1 4M 1 n 1 O 1 1 O O 2a_„ , + - T T = - {E [B 0(sin(x n+6 +3)-x. Cos(6 +3)-sin(6 +3)) M q 2 i o l o ' o B,(Cos(x +6 +y) + x. sin(6 +y) - Cos(6 +y))] - j B , [Cos(2x 1+26^+3+ Y) + 2x, Sin(26 +3+y) - Cos(26 +6+y)]} (3-23) ± o ' 1 o o l e t D l + D 2 2 ' V=x Ax - ( a 1 1 ~ a 1 2 2M X l + f W D l + D 2 2 2 = a 1 2 ( ~ 2 M ~ ) x l + 2 a 1 2 X l x 2 + a 2 2 X 2 + f ( x l } ( 3 _ 2 4 ) Then a D(x ) P,~P e(x 1) V = 2 < a 1 2 " ~ 2 1 i r - ) 4 + 2 a 1 2 X l ( M ( 3 " 2 5 ) which i s negative d e f i n i t e for - ( 2 I T - 6 U S +V6 q)<X^ <( I§ U S ~ 6 q) 25 From (3-24) one has •D,+D, Ju 1 2 - a l 2 X l ± / C a 1 2 " a12 a22 ~ W ] X 1 " a 2 2 f ( x l ) + & 2 2 V x = : (3-26) 2 a22 For any curve V = constant to be a closed one, there must e x i s t a value for x^ such that the square root term equals zero r e s u l t i n g i n 2 a (D +D ) _ a v - ' f r i ' . + ' x l M -ig1 (3-27) Thus the maximum value of V describing a closed curve i s obtained from (3-27) by d i f f e r e n t i a t i n g the r i g h t hand side with respect to x^ and equating to zero, ^ V ^ l ^ ' (3-28) 1 a 1 2 ^ a 2 2 D ^ X l ^ ~ a 1 2 M ^ The value of x^ obtained from (3-28) i s then substituted back i n (3-27) to give V . " . maximum The value of V tangent to V-0, V^, i s found by solving the two equations V=0 3V 3V 8V 3V ^ ( 3 _ 2 9 ) and the value of V to be used for s t a b i l i t y i s V m or V . which-T maximum ever i s smaller, as shown i n F i g . 3-3. Applying the above procedure to the same example, (2-8), give V . >Vrr , V m = 878.4 which i s tangent to V=0 at x =2.1 ° maximum T T ° 1 and x 2=0. F i g . 3-4, shows the r e s u l t i n g s t a b i l i t y region to be very close to the actual region obtained e a r l i e r . F i g . 3-3 CHOOSING V O N io x2 = w 61 4-2. V <o -15: ! 1 h -1.0 0.5 0.5 7-0 — i — 0\ -2--4.. .-6. -8. •• . . -10\ Vm = 878. 4 TRUE STABILITY BOUNDARY Fig. 3-4 STABILITY REGION. BY MODIFIED V-FUNCTION 28 3.6. CONCLUDING REMARKS Although a maximization technique f o r quadratic Liapunov functions has been reported recently by Davison and Kurak [33], the work of t h i s chapter i s independant. There are furthermore two major diff e r e n c e s i n our works 1. To ensure that V i s negative everywhere i n s i d e and on the surface V, i n Davison and Kurak's searching procedure they construct a g r i d i n the n-dimensional space, c a l c u l a t e V" at every point where the g r i d i n t e r s e c t s the normalized surface x. Ax=V, 0<V<1 and then constrain the maximum value of V to be negative. In our case the following constraints are imposed f o r the same purpose a) The second degree terms of V, V^t are to be negative d e f i n i t e and b) V=0 i s tangent to V. 2. Davison and Kurak state i n t h e i r paper that the quadratic Liapunov function y i e l d s good estimates of s t a b i l i t y regions for nonlinear sys-tmes. We found the regions unsatisfactory i n the case of a power system unless other terms were added to make sure that V i s negative d e f i n i t e . The r e s u l t s then give a bett e r estimate of the s t a b i l i t y region. 29 4. A MAXIMIZED LIAPUNOV FUNCTION FOR A 3-MACHINE POWER SYSTEM In the previous chapter i t was found that a quadratic Liapunov function does not yield a good estimate of the s t a b i l i t y region for a power system. It is also noticed that the expression of V has a great effect on the resulting s t a b i l i t y region. In the following a Liapunov function for a multimachine power system is constructed starting with a tentative quadratic Liapunov function. After the time derivative of this function is obtained, i t i s adjusted to be negative definite in a region around the origin. The actual Liapunov function i s then formed, checked for positive definiteness and, as a f i n a l step, the quadratic portion i s maximized. 4.1. - - EQUATIONS OF A 3-MA CHINE SYSTEM : ._. Consider a three-machine system as that of Fig. 4-1. In addition to the assumptions of chapter 2, the following assumptions are made. a) The damping power is proportional to the s l i p frequency. b) Resistance of transmission lines is neglected. With these assumptions, the di f f e r e n t i a l equations describing the motion of the system are d 26 1 . d&± M. ' 2 + D, dF" + P = P.. 1 dt 1 el i l d 26 ? d6„ M + D — - + P = P n2 ,2 U2 dt e2 12 dt d 26 d& *S ~TT~ + D3 d T " + Pe3 = P i 3 ( 4 " dt THREE MACHINE.POWER SYSTEM 31 where P e l = k l s i n ( ( S 1 " 5 2 ^ + k 3 s i n ( 6 1 - < S 3 ) P g 2 = k.^  s i n (<S2~<S1^ + k 2 s i n ( < S 2 ~ < S 3 ^ P g2 = ^ 3 sinCS^-iS-^) + s i n ( 6 3 -6 2) k l = E 1 E 2 Y12 k 2 " E 2 E 3 Y23 k 3 = E l E 3 Y 1 3 ( 4 " 2 ) At the sta b l e equilibrium p o s i t i o n we have 6 1 = 6 1 ' 62 = 62 ' 6 3 = 6 3 d<5 d6 df — — = n — - = o = 0 dt u ' dt u ' dt u 2 2 2 d i . d^5 d 6 — ^ = 0 , — j " - = 0 and ~ ± ' = 0 (4-3) dt dt dt Su b s t i t u t i n g (4-2) and (4-3) into (4-1) y i e l d s = sin(6° - 6°) + k^ sin(<S° - 6°) P i 2 = k l s i n ( 6 2 ~ 51^ + k 2 s i n ( 5 2 ~ ~ 5 3^  P I 3 = k 3 sin(6° - 6°) + k 2 sin(6° - 6°) . (4-4) 32 Let the state variables be x l = 6 1 " o «1 X2 o 6 2 o X3 63 X4 dt X5 dt d 6 3 X 6 dt and The system equations become *1 = x4 x 2 - x 5 x3 = X 6 X4 = M { k i l s i n < S i ~ 6 2 ) " s i n ( x i " x2 + 61 " 6 2 ) ] + k3^ S ± n ( |5l" |S3 ) sin(x 1 - x 3 + 6° " 63)] ~ D i x 4 } X 5 = M ^ k]_t s i n(62 " <S°) " sin(x 2~ x± + 6° - 6°)] + k 2[sin(6° - 6°) sin(x 2 - x 3 + 6° - 63)] - D 2 X 5 ^ X 6 = M t s i n ( 6 3 _ ~ s i n ( x 3 - x 1 + 6° - 6°)] + k ^ s i n C ^ -5°) sin(x 3 - x 2 +6° - 6 2)] - D3 xg^ (4-5) 4.2. CONDITIONS TO ENSURE NEGATIVE DEFINITENESS OF V Consider a tentative quadratic Liapunov function of the form 33 V = xAx where A is a positive definite symmetric matrix, machine case one has For the three (4-6) a l a 2 a 3 a 4 a 5 a6 a2 a7 a8 a9 a10 a l l a3 a8 a12 a l 3 a!4 a l 5 a4 a9 313 a l 6 a17 a18 a 5 a10 a14 a17 a19 a20 a6 a l l a15 a18 a20 a21 (4-7) Differentiating (4-6) with respect to time by virtue of (4-5) yields V = 2[<j,1(x) + <f,2(x) + <j,3(x) + <},4(x)] where (j) 1( X) = (x^ x 2 x 3) M-, l13 L M l M„ z10 a n M 3 _ P i r p e ^ p -p *i2 e2 \ Pi3- Pe3 (4-8) (4-9) <j)2(x) (x 4 x 5 x 6) T/x 4\ (4-10) where T = ( a4 " a16 } M D +M D h 1 2 2 U1 2 U 5 a 9 M,M; 1 2 M D +M D . 1 { , _ 1 3 3 1 v a17 ; 2 U 6 313 M M3 a18 ; h + M 1 D 2 + M 2 D 1 , , ^2 , 1 , + M 2 D 3 + M3 D2 . 2 U 5 + a9 " MXM2 317 ; ( a10" M2 a19 j 2 U l l + a 1 4 &20 ; M„D„+MJ), D. I, . M D +M D i ll2"3'"3"2 . x "3 I ( a6 13" — — 3 l o ) ^ ( a i 1 1 A~ 18; 2V~11 "14 M1 M3 M2M3 a20 } ( a l 5 M3 a21 } (4-10a) 34 tj> 3(x) = ( x 1 x 2 x 3 ) and • 4<x)=ex 4x 5x 6) " ( a 1 " a 4 ) • ( a 2 ( a 2 D l . Mx V ( a ? _(a3 " a l 3 } ( a 8 a l 6 317 a!8 M l M2 M3 a17 a19 a20 M l M 2 M3 a18 a20 a21 M l M M3 M2 V U 3 M3 V D 8 M3 IV D2 , ' ' °3 M2 14' 12 M3 15j (4-11) P i l " P e l \ P - P r i 2 r e 2 \ P i 3 - Pe3/ (4-12) In order to ensure that V i s negative d e f i n i t e i n a region around the o r i g i n i t s component functions, <t>'s, must be e i t h e r negative d e f i n i t e or semid e f i n i t e . Examining equations (4-9) to (4-12) we notice the follow i n g : 1. <)>^ (x) i s a function of x^, x 2 and x 3 only and can be made negative i n a region around the o r i g i n . 2. <f>2(x) i s a function i n x^, x^ and x^ which i s negative d e f i n i t e i f the matrix T of equation (4-10a) i s negative d e f i n i t e . 3. <f>3(x) and <j>4(x) are both i n d e f i n i t e . They can be eliminated from V eit h e r by s e t t i n g each i d e n t i c a l l y equal to zero or by Integrating them with respect to time and then subtracting the r e s u l t from the V-function. In the following each component of V i s examined to develop the necessary conditions f o r i t s el i m i n a t i o n or to ensure the negative d e f i n i t e n e s s . (J)^(x) can be made ei t h e r i d e n t i c a l l y equal to zero or negative d e f i n i t e . Knowing that the sum of the mechanical inputs to the system i s equal to the sum of the e l e c t r i c a l outputs, i f we furth e r set 35 a4 a5 V M l " " M 2 - M3 ' a9 aio a i l M l ~ M3 a13 a14 315 M l M2 M3 and (4-13) in (j)-^ (x) , this function w i l l be identically equal to zero. Substituting for P^ and P_^  from (4-2) and (4<-4) respectively into (4-9) one gets: ^ ( x ) = kjjsinds^- 62) . / , 0 -sin(x 1-x 2 +6 1 ~ 0 a4 62)][X1(M7 M2 * 2 \ a l 0 a i 3 " M ) + X 3 ( M 2 1 - S J + k 2 t s i n ( 6 ° - -sxn(x 2~x 3 +6 2 - O s T r / 5 63)][X1(M7 " M ) + X 2 ( M " u 3 2 a l l a l 4 • M3 >+*3< M2 - a i 5 ) l + k 3 [ s i n ( 6 ° - . , . 0 -sm(x^-x3+6^- 6° 3)][x 1(^ a6 " a9 " " M^)+X2(MY " a l l a13 M ) + X 3 ( M " 3 1 *">] M3 setting A - % = _r!i _ = ,!l4 _ f i 5 v n a5 a6 M ' M 2 3 M M 1 3 a13 a i 4 and M1 M2 (4-14) tj)^(x) becomes 34 a5 0 0 0 0 ^ ( x ) = (r~ ^~) (k 1(x ; L-x 2) [sin( 6°-6°)-sin(x 1-x 2 +6° -6 2 )] +k 2(x 2~x 3) [sin ( 6 2 -<5 °)-sin(x2-x -t <$° - 6 ° ) ] ... . , + k 3 ( x r x 3 ) [ s i n ( c j - 6 ^ ) - S i n ( x 1 - x 3 + 6 0 _ 6 ° ) J } ( 4 _ 1 5 ) 36 which is negative definite for values of x^, x,, and x^ satisfying - 1r-2(6°--6°)<(x1-x 2)<ir -2(fiJ-5°) -7r-2(6°-«53)<(x2-x3)<Tr -2(6°-6°) - T T-2(6°-6°)<(X 1-X 3)<TT -2(6°-6°) (4-16) The expression <j>2(x) can be made identically equal to zero by setting a4 = D l 16 a i o = P 2 M 2 a19 a15 = D3 " M 3 a21 - D. a 5 + a 9 = 1 ,2 D l D3 a6 + a13 = (M7 + M ^ I S a n d a l l + a l 4 = (M^ +M7 ) a20 ( 4 " 1 ? ) On the other hand, for <j>2(x) to be negative definite, the conditions on matrix T are : V a 1 6<0 U D l w D2 s h + M 1 D 2 4 M 2 D 1 .2 ' ' «16 ) ( a10 H^ 8!^- 4 ( V a 9 " M L M 2 a17 ) > 0 D l °2 ^3 1 M 2 D 3 + M 3 ° 2 ( V a l 6 ) [ ( a l 0 " H~2 a l 9 ) ( a 1 5 M 3 a21 } 4 ( a l l + a l 4 &20 37 JL, + M 1 D 2 + M 2 D 1 u l r • h. w + M 1 D 2 + M 2 D 1 , " 2 U 5 + a 9 M X M 2 a l 7 M 2 U 1 5 ~ M 3 a 2 1 M V a 9 ^1^2 x M 1D 3 + M 3 D 1 M 2 D 3 + M 3 D 2 4 < a 6 + a 1 3 ~ " J M X M 3 a 1 8 ) ( a l l + a 1 4 M 2 M 3 a 2 0 } 1 X M 1 D 3 + M 3 D 1 1 M 1 D 2 + M 2 D 1 M ^ + M ^ + 2 ( a 6 + a 1 3 M ^ M T 1 a l 8 ) [ 4 ( a 5 + a 9 M ^ MT~ a 1 7 ) ( a l l + a 1 4 " J M T ^ 3 2 0 } D M D + M D - | < a 1 0 - ^ a 1 9 > ( a 6 + a 1 3 - 1 \ ^ a l 8 > J < 0 ( 4 " X 8 ) t)>3(x) can be eliminated by s e t t i n g a l "• M a4 ' a2 ~ M 2 a5 ~ M a9 ' — - ~ - • ^3 ! l ~ ~ - : a 3 M . a6 M 1 a l 3 ' ^2 A ? M 2 310 ' D 3 °2 a8 = M 7 a l l = M ~ a14 a n d a12=M7 al5 ( 4 " 1 9 ) On.the other hand one may integrate 2cf>3(x) with respect to time. I t i s noticed that ----- - --D l D l 2 2 ( a l - ^  V f x l X 4 d t = ( a l " M7 a 4 ) x l D 2 D 2 2 2 ( a 7 " Mj" a 1 0 ) ; X 2 X 5 d t = ( a7 " Wn a l 0 ) x 2 38 D l D3 2 2 ( a 1 2 " "ST; a15 ) f X 3 X 6 d t = ( a l 2 " W3 a15 ) x3 D2 D l 2 ( a 2 ~ M^a5)f x i x 5 d t + 2< a 2 ' V ' X 2 X 4 d t = D2 °2 °1 2 ( a 2 " a 5 ) x l X 2 + 2 ( M 2 " a5 " V 7 X 2 X 4 d t D D 2 ( 3 3 - a 6 ) 7 X l X 6 d t + 2 ( a 3 " a 1 3 } 7 X 3 X 4 d t = D3 °2 2 ( a 8 " a H ) / x 2 X 6 d t + 2 ( a 8 " a ! 4 ) / X 3 X 5 d t = • i f one sets 2(a 8 3 " M 3 a n)x 2x 3.4 D2 D l M 2 a5. " M 39 D3 D l M3 a6 = M a13 D3 M3 a l l °2 " M2 a l 4 D a . ) fx .x dt (4-20) one.has 2 /(fr (x)dt=(x x 2x ) ( a i ~ T T - a / ) ( a o -D, D, D, D, *1 M ~4' v"2 M 2 a 5 ) ( a3 M 3 V D, ( a2 " M 2 a 5 } ( a7 M 2 a10 } ( a8 M 3 a l l ) L ( a3 ' a 6 } ( a8 " a H ) ( a l 2 " a15j '3/ (4-21) which w i l l be subtracted from the tentative Liapunov function. Finally <f>^ (x) can be eliminated as follows. Knowing that the 39 sum of mechanical inputs to the system i s equal to the sum of e l e c t r i c a l outputs, i n order to reduce ^(x) to zero one may set x16 M., (4-22) On the other hand, s i m i l a r to the case of <j>3(x), s e t t i n g M - _1 a18 " M 2 a17 *19 M 2 a 1 6 + ( M 2 - M ^ a ^ M, M 3 a20 = M~ a17 a n d M3 M3(M3 - Mp a21 Mx a l 6 MM 317 (4-23) i n cb,(x) r e s u l t i n 2/<j>4(x)dt = 2( 316 a l 7 x.. -x 0 1 2r_.._/ro ro, ,0 „o. M M,-) { V o [ s i n(6 1 -6 2 ) -s inXx 1 -x 2+6 1-6 2 ) ]d(x 1 + k 2 o 2 3 t s i n ( 6 2 _ 6 3 ) - s i n ( x 2 ~ x 3 + 6 2 " ^ 3 ) ] d ( x 2 - x 3 ) X1~ X3 + k_ / [sin(<5° -6°)-sin(x.,-x;H-6° -6°) ]d ( x . - x j } (4-24) 3 o 1 3 - 1 3 1 3 1 3 • which again w i l l be subtracted from the ten t a t i v e Liapunov function. 40 4.3. CONSTRUCTION OF LIAPUNOV FUNCTION AND MAXIMIZATION There are sixteen different combinations of <)>^ (x), <j>2(x) , ^ ( x ) and (^(x) that result in a negative definite or semidefinite V. These combinations are a l l investigated and a summary of the results obtained i s given in table 4-1. Out of these sixteen combinations only one results in a positive definite V-function that has a negative definite time derivative. This is the case where (j>^ (x) and cj>2(x) are made negative definite and cb^ Cx) and <f>4(x) are integrated. Combining conditions (4-14), (4-18), (4-20) and (4-23) we end up with where V(x) = x Ax - 2/<j>4(x)dt (4-25) x Ax = x Ax - 2/<j>3(x)dt (4-26) and -2/tf>4(x) dt is positive definite and i s given by equation (4-24) . The new matrix A i s given by rvi V V S M1 a4 M ^ a17 M M2 a i 7 a4 H 317 M1M2 a l 7 V ? V + V ^ 1 \ V 3 B_2 M2 + V ^ 1 D2 M3 MXM2 a l 7 M X U4 M2 a17 ; MM 317 M2 a l 7 M V M a 1 7 MM 317 V 3 V 3 % + V ^ 1 , ^3 ^3 % J V ^ l , M1 M2 1 7 ' M1 M2 M^V" M2 a l 7 ; M2 a l 7 Mx a i 7 M^V M2 a l 7 ^ h ^3 M„ M. 317 M0 317 316 a17 rr- a,n • 2 2 M~ 17 V \ M !3 M M,-M, M, D1 M3 D2 M3 M3 D3~ D1 M3 M3 M3 M 3 _ M 1 '• MXM2 a l 7 M ^ a l 7 M^"(a4+ ^ L j " a17 } M^" a17 M^ 317 M ^ ^ ' M2 317J (4-27) 4o - Q No. *L 2 3 \ V V 1 0 0 .0 0 0 p.s.d. 2 0 0 0 / 0 p.s.d. 3 0 0 / 0 0 p.s.d. 4 0 n 0 0 n.s,d. p.s.d. 5 n 0 0 0 n.s.d. p.s.d. 6 0 0 / f 0 7 0 n 0 f n.s.d. p.s.d. 8 0 n / 0 n.s.d. p.s.d. 9 n 0 0 f n.s.d . p.s.d. 10 n 0 / 0 n.s.d. p.s.d. 11 n n 0 0 n.d. p.s.d. 12 0 n .... / — / — . - • - • n.s.d. p.s.d. 13 n 0 / / n.s.d. ' p.s.d. 14 n n 0 I n.d. p .d. 15 n n f 0 n.d. p.s.d. 16 n n f /. n.d.. p. d. conditions contradict same as 14 ->• negative integrated Table 4-1 n.s.d. -»- negative semidefinite n.d. negative definite p.s.d. -* positive semidefinite p.d. ->-' positive definite 41 which is positive definite i f a 1 7>0 a16 a17 M1 M2 >0 D l M2 >0 (4-28) Also V = 2[((,1(x) + ^ ( x ) ] (4-29) where a^ a i 7 ^ i o o o o ^ 1(x) = (—- - — -—) {k 1 ( x 1 - x 2 ) [ s i n ( f i 1 ~6 2)-sin(x 1-x 2+6 1 -6 2)] +k 2(x 2~x 3)[sin(6° -6°)-sin(x 2-x 3+6 2 -6°)] +k 3(x 1-x 3)[sin(6° -6°)-sin(x1-x3+6° -63)]} and (4-30) • * z ( x ) = ( x4 X 5 ' X 6 ) T T = D l ( a " MT a16 } 4 1 l 4\ Vx6 0 0 M 2a 4-D 2a 1 6 M ^ - M ^ ^ M, M1M2 17' (4-31) 0 rM 3 a 4 - D 3 a l 6 , M1 D3 - M3°l . ^ M, >LM„ a l 7 ; T*2 (4-32) which is negative definite i f M M M 3 2 .1 , D~ *D~ *D~ a n d 3 2 1 (4-33) 42 Thus we have a p o s i t i v e d e f i n i t e Liapunov function V(x) given by (4-25) and i t s time d e r i v a t i v e V given by (4-29). Equations (4-28) and (4-33) define the r e l a t i o n between the three parameters a^, a^g and ^y]' F i n a l l y the hypervolume enclosed by the quadratic part of V(x) s h a l l be maximized subject to (4-28), (4-33) along with the tangent conditions V = .0 3 l _ 3V 3x 2 - 3V 3^ 2 3V 3x 1 = 0 9V 3V 9 X 3 3V 3 X 3 3 V 3x 1 = 0 3V 8x 1 3V 3 X4 3V 3x 4 3V 3x 1 = o 3V 3V 3 x 5 3V 3x 5 3V 3x x = o 3V 8x 1 3V 3x 6 3V 3x 6 3V 3 X ; L = 0 (4-34) The same algorithm of chapter 3 i s used. 4.4. NUMERICAL EXAMPLE The three-machine system considered has the following data: E ^ l . 1 7 4 ^22.64° p.u. ' P =0.8 p.u. E2=0.996 z,2.61° p.u. P 2 = 0 , 3 p u ' E3=1.006 z.-11.36° p.u. p.u. 1^=3 K.W.sec./K.V.A. ^1_ = 1 0 M l ' H =7 K.W.sec./K.V.A. ^2 = ? M2 -H„=8 I.W.sec./K.V.A. ^3 = ' M 3 . A sudden 3-phase symmetrical short c i r c u i t to ground occurs on 43 the transmission l i n e connecting machines 2 and 3 of F i g . 4-1 close to bus 3. The c r i t i c a l c l e a r i n g time obtained from the above V-function i s 18 cycles . The actual c r i t i c a l c l e a r i n g time obtained from the system's swing curves i s 20 cycles. The r e s u l t i n g Liapunov function, of the form (4-25), has the following p a r t i c u l a r s V =33.65 which i s tangent to V=0 at the point m x = 2.195, -0.137, -0.005, 0.112 x 10~3, -0.278 x 10~3, -0.595 x 10 A=T3.26 5.32 2.28 0.326 0.76 0.868 5.32 8.69 3.72 0.532 1.24 1.42 2.28 3.72 1.6 0*228 0.532 0.608 0.326 0.532 0.228 0.129 0.076 0.087 0.76 1.24 0.532 0.076 0.402 0.203 _0.868 1.42 0.608 0.087 0.203 0.489_ Fi g . 4-2 shows the function V(x)=V m plotted i n the three dimensional space x n , x„ and x with the other components x., x r and x set to zero. This F i g . shows the maximum deviations i n the three rotor angles, with respect to a reference frame r o t a t i n g a i synchronous speed, without l o s i n g synchronism with each other. 4.5.. . CONCLUDING REMARKS " It i s i n t e r e s t i n g to notice that when s e t t i n g a. = 0 4 a i 7 = ° *16 M, = 1 (4-35) i n the Liapunov function of equation (4-25), the r e s u l t i n g V-function i s 45 exactly the same as that of Willems [22] and i s given by V = (x. x c x,) T M. 0 0 ~ /x.\ + 4 5 6 2{ k , / X l X 2 r . O N 0 M 2 0 X 5 0 0 M 3 l X6 +6° -S°)-sin(<5° -k 2 / X 2 X 3 [ s i n ( x 2 - x 3 + 6 2 -6°)-sin(6° -6°)]d(x 2-x 3) + b k 3 X l X3[sin(x 1-x 3+6° -6°)-sin(6°-6°)]d(x 1-x 3)} (4-36) j ' When applying t h i s V-function to the same numerical example considered i n s e c t i o n 4.4., the r e s u l t i n g c r i t i c a l c l e a r i n g time i s 14 cycles compared to 18 cycles c l e a r i n g time obtained from the V-function constructed i n th i s chapter. 46 5. CONCLUSIONS The direct method of Liapunov has been applied to the study of transient s t a b i l i t y in power systems. The following conclusions are drawn: 1. Although Szego's procedure has been applied successfully to construct a Liapunov function for a second order single machine-infinite bus system, work remains to be done in developing algorithms for applying this method to higher order systems. 2. An expression for the hypervolume enclosed by a quadratic form function i s developed and employed in maximizing the estimated s t a b i l i t y region. 3. A construction procedure for optimized Liapunov functions for power systems has been developed. It starts with a quadratic form and is modified by the negative definite V constraints before maximization of the estimated st a b i l i t y region. The procedure has been applied successfully to a single machine-infinite bus system as well as a three machine system. In general, i t remains to develop procedures for the construction of Liapunov functions for multimachine power systems in which synchronous machines and controllers are represented in great detail. 47 APPENDIX I Expression (2-6) for the el e c t r i c a l power output is obtained as follows. From the phasor diagram of Fig. 2-2, one has V, -(I-l) where = V o sin6 + r e -x e id' V i q cos6 X L. e r e J i , q also = 0 + " 0 x " q t -x d 0 V q E i q Solving (1-2) and (1-3) for V^, and i ^ , i and gets V, \ d = v i o V A q q e .d i e d e q i d e q E' sin cS +-9- , cos 6 A and = V ' f-r e q x +x, e d -r sin 6 . A x +x e q _ cos ,<5 r \ e where (1-2) (1-3) r x e q r . +x (x +x ) e q e q (1-4) (1-5) A = r + (x +x )(x +x.) e e q e d Let A =x Jr1 +(x +x') 2 / A 1 q e e d (1-6) A 0 = r x / A 2 e q 1 / 2 2 A- = x, /r +(x +x ) / . 3 d e e q A A. = [r * + x (x +x )] / A 4 e e e q 48 A = (x +x ) / A 5 e q i x, +x / d e> g = arc an( ) r e VY = arc tan (—3 ) x +x. q e> (1-7) Substituting (1-7) into (1-4) and (1-5) yields = V A^ sing -A^ cosf Ag cosy A^ siny sinS cosS + E and = V q A3 _ A 3 . —r cosy — - r siny X d X d A l " A l • — sing — cosg L x x q q sin6 cos6 +E / \ The e l e c t r i c a l power output P (<5) is given by P (6) = (V. V ) Substituting (1-8) and (1-9) into (1-10) gives I 2 Jq (1-9) (1-10) ' 2 ' P (6) = B,E + B-fcos (<5+g)+B„ sin(6+y)]E +B.sin(6+y)cos(6+g) (1-11) e 1  2-- j q 4 where B l = A 2 ( V A 4 / X q ) B 2 = -V oA 1(A 5 +A 4/x q) B 3 " Vo A2 A3 (x- " I'> q d B4:= - v ' 2 AA.(- -o 1 3 x x, q d (1-12) 49 APPENDIX II The expression (2-4) f o r the damping c o e f f i c i e n t i s expanded as follows ^D(6) • = i(D cos2<S + D 2 s i n 2 6 ) D 1 +D 2 D l-D 2 - M ( — 2 — + — 2 ~ C ° S 2 6 ) S u b s t i t u t i n g f o r 6 = x-^ +^ 0 w e g e t i i D i + D 2 D r D 2 D i + D 2 D r D 2 —rr: 1 rrr- [cos26 cos2x -sin25 sin2x. ] 2M 2M o 1 o 1 D l + D 2 D1~ D2 ( 2 x x ) 2 ( 2 x x ) 4 ^rr- + ^rr- [cos2<5 (1 r - i — + , i , -....) 2M 2M o 2. 4 • V " o ( 2 x x ) 3 ( 2 x x ) 5 - s i n 2 6 o ( 2 x 1 - - 3 - r - + — - - • • • • ) where - I q , x / 1 , ( I I - l ) 1=1 D i + D 2 D r D 2 • - - • q i ~ 1 5 2TT c o s 2 6 ° At the equilibrium s t a t e : P.=P (6 ) l e o =B,E +[B„ cos(6 +3)+B0 sin(6 +y)]E +B.sin(6 +y)cos(6 +3) (II-3) 1 q 2 o j o q q o o Thus: 50 ^ ( P . - P ^ ) ) = i { B 2 E ^ [ c o s ( 6 o +S)-cos( X l+6 o + e ) ] + where B„E [sin(6 +y)-sin(x.+6 +y)]+B.[sin(6 + Y ) C O S ( 6 +3) 3 q o 1 o 4 o o -sin(x..+6 + Y ) C O S ( X 1 + 6 +6)]} 1 o 1 o 1 ' =rr{B0E [cos(6 +3)-cos(x1+6 +B)}+ M 2 q o 1 o B E [sin(6 +Y)-sin(x 1+6 +y)]+ j q o 1 o B, -2-i-[-sin(26o+3+Y)-sin(2x1+26o+3+Y) ] } = i{B nE'[cos(6 +3)(1-cosx..)+sin(6 +3)sin X l] M 2 - i . . ' : O 1 O 1 q t +B„E [sin(6 + Y ) ( 1 - C O S X . , ) - c o s ( 6 +y)sinx 1 ] 3 q v o 1 o 1 1 + B4 --— [sin(25 +3+Y) (l-cos2x-)-cos (26 + 3 + Y ) s i n 2 X l ] } 2 o 1 o 1 E* 2 4 ..q x x = — [B„cos(6 +3)+B„sin(6 +y) ] [-± - -± + ...] M 2 o j o 2 . 4 . ' 3 E x^ + -J[B sin(6 +3)-B cos(6 +y)][x • - +. . . ] M 2 o 3 o 1 3 . B 4 ( 2 X ; L ) 2 ( 2 X ; L ) 4 2 M-[sin(26 Q+3+ Y ) ( — + . . . ) - C O S ( 2 6 q +3+Y) 2x ) 3 OO = E p.x* (H-4) i = l 1 i p. = — {-J[B„cos(6 +r+ u)-B 9sin(6 +3+V" TT)] rx l ., M 3 o 2 2 o 2 - i - l ._, , V B.cos(26 +3+Y+—• TT) (II-5) + M 4 o 2 51 APPENDIX III i In equation (3-5) the hypervolume bounded by V=x_ Ax is given by C l C 2 1=2 j( " 2 / 2 / n _ 2 - 2 , 2 •1-X 1/C 1 C_ 1 / l - £ X ^ C . -[c 1 1 , n-1 - I l r i=l - c / - c A-x2/c2 -c Jl- 1 - 2 , 2 1 2 1 1 n-1 . , x,/c. i=l I l n-1 _2 2 1- y x,/c.]dx ,dx „...dx L I i J n-1 n-2 1=1 (3-9) Consider the innermost integration / n-*1 " 9 -2 . 2 n-1- x i / c i i=l -c n-2 „ n-1" i-1 * ± -2 c. l A - 1 _2 2 c /L- y x. / c. dx T n . L^ i i n-1 i=l Let n-1 sin 9 h ence and n-2 x 2 dx - = c T /L- y —— n-1 n - l v . L . 2 1=1 c. l cos GdG n-1 _2 /n -2 x 2 A- x x ± =/i_ E - | co§ i = l —rT c. 2 i = l i c. n-2 -2 ^ •„ Cn-l ( 1-.S i> / 1 1 2 -v/2 c. l cos 9 d9 h-2 -2 TT ,, X. 1" C n C n - l ( 1 _ . ^ - i ) i=l „ 52 For the next step of integration use the substitution : , / n - 3 - 2 n-2 L y x. . n =/l-. Z j 1 x sm 0-c 0 1=1 —^ n-2 2 c. l / n " 3 -2 hence dx „ = c . / l - . * - . . x. cos 9 d9 n-2 n-2 i ^ l _ i 2 c. l n-2 ( l - A xt)=:(l - J ± n-3 -2 x ^r) cos 9 / n-3 -2 - ' -c 0 / l - E X i n ~ 2 i = l T /" c. l n-2 -2 c c 1 (1- E i ) dx _ 2 n n-1 . .. —r- n-2 i = l 2 -" .. c • ^ /n-3 -2 -c V l - E V n-2 . .. —x-1=1 2 1 n-3 x7 3/2 TT/2 „ T c c ,c (1- E -± ) ' f cos J e.de 2 n n-1 n-2 . , 2 l —TT/2 n-3 x 2 o 2 IT / i H 1 \ 3/2 = 2.— . — c c c „(1- I —TT ) 3 2 n n-j n-2 , , 2 1 = 1 c. l This procedure i s repeadted. (n-1) times to give the f i n a l answer. It is noticed that as a result of the k — step, the term cos^^SdS appeares, thus TT/2 n -TT/2 the last integral i s j. cos 9d9 which is equal to -TT/2 TT/2 21 o cosn9d9 1.3.5. . . (n-1) .TT. 2.4.6...n V n even 2.4.6...(n-1) 1.3.5...n n odd The volume required i s thus given by r - l n I = 2n(hr2~- . \ , .TT1 c. n odd v2 1.3.5...n i=l l n ^ i n 2n( l ) 2 _ 1 c v2 2.4.6...n i=l i n even but n /.n, n * c i = / v '.\ Xi i=l i=l and the product of the eigenvalues of a matrix i s equal to the matrix determinant. Thus c. =/vn/\k\ 1=1 l 1 1 n-1 n n,Tr.2 1 /v/|A| n even ^ V 2.4.1. ..n. 54 REFERENCES 1. Ro E.'Kalman,. J . E. Bertram, " C o n t r o l System A n a l y s i s and Design V i a the Second Method of Liapunov" ASME Trans., J . of Bas i c Engineering, June I960, pp. 371-393. 2. J . L a S a l l e , S. L e f s c h e t z , " S t a b i l i t y by Liapunov's D i r e c t Method w i t h A p p l i c a t i o n s " , RIAS, Bal t i m o r e , Maryland, 1961, Academic Press, N.Y. 3. S. G. Margolis,.¥. G. Vogt, " C o n t r o l Engineering A p p l i c a t i o n s of V. I . Zubov's C o n s t r u c t i o n Procedure f o r Liapunov Functions", IEEE Trans, on Automatic C o n t r o l , A p r i l 1963, pp. 104-113 4. R. E. Kalman, "Liapunov Functions f o r the Problem of Lur'e i n Automatic • C o n t r o l " Proc. Nat. Acad. S c i . , U.S. 49, 2, 1963. pp. 201-205 5. M. L. Ca r t w r i g h t , "On the S t a b i l i t y of S o l u t i o n s of C e r t a i n D i f f e r e n t i a l Equations of Fourth Order".- Quart. J . Mech. Appl. Math., 1956, 9, (2). 6. D. G. S c h u l t z , J . E. Gibson, "The- V a r i a b l e Gradient Method f o r Generating Liapunov Functions", AIEE Trans, on Automatic C o n t r o l , Sept. 1962. 7. G. P. Szego, "A C o n t r i b u t i o n to Liapunov's Second Method: Nonlinear Autonomous Systems", J . of Bas i c Engineering, Dec. 1962. pp. 571-578 8. W. J . Cunningham, "An I n t r o d u c t i o n to Liapunov's Second Method^" AIEE Trans, on Appl. and Ind., January 1962. pp. 325-332 9- Y. H. Ku, N. N. Puri,. "On Liapunov Functions of High Order Nonlinear Systems", J . of F r a n k l i n I n s t . , V o l . 276, No. 5, Nov. 1963. 10. N. N. P u r i , C. N. Weygandt, "Liapunov and Routh's Canonical Form", i b i d . 11. D. R. Ingwerson, "A Mo d i f i e d Liapunov Method f o r Nonlinear S t a b i l i t y A n a l y s i s " , IRE Trans, on Automatic C o n t r o l , May 1961. pp ( 199-210 12. C. S. Chen, E. Kinnen, " C o n s t r u c t i o n of Liapunov Functions", J . of F r a n k l i n I n s t . , V o l . 289, No. 2, Feb. 1970. pp. 133-146 55 13. A. H. El-Abiad, K. Nagappan, "Transient S t a b i l i t y Regions of M u l t i -machine Power Systems", IEEE Trans, on Power Apparatus and Systems,-Vol. PAS- 85, No. 2, Feb. 1966. pp. 169-179 14. G. E. Gless, "Direct Method of Liapunov Applied to Transient Power System S t a b i l i t y " , i b i d . pp« 153-168 15. Y. N. Yu,. K. Vongsuriya," Nonlinear Power System S t a b i l i t y Study by Liapunov Function and Zubov's Method". IEEE Trans, on Power Apparatus and Systems, Vol. PAS-86, No. 12, Dec. 1967. PP- 1480-1485 16. M.'W. Siddiquee, "Transient S t a b i l i t y of an A. C. Generator by Liapunov's . Direct Mdthod." Int. J. of Control, Vol. 8, No. 2, 1968, pp. 131-144. 17- N. D. Rao, "Routh-Hurwitz Conditions and Liapunov Methods f o r the Transient S t a b i l i t y Problem". IEE P r o c , A p r i l 1969. PP- 537-547 18. M. A. Pai, M. A. Mohan, J. G. Rao, "Power System Transient S t a b i l i t y Regions using Popov's Method". IEEE Summer Power Meeting, Dallas, Texas, June 1969. 19- T. H. Lee, R. J. Fleming, "Power System S t a b i l i t y Studies by the Direct Method of Liapunov". i b i d . 20. N. D. Rao, A. K. DeSarkar, "Analysis of a Third Order Nonlinear Power System S t a b i l i t y Problem Through.the Second Method of Liapunov". IEEE Winter Power Meeting, N. Y., January 1970. 21. G. Luders, "Transient S t a b i l i t y of Multimachine Power Systems Via the Direct Method of Liapunov". i b i d . 22. J. L. Willems, "Optimum Liapunov Functions and S t a b i l i t y Regions For Multimachine Power Systems". IEE P r o c , Vol. 117'/ No. 3, March 1970. 23. V. I. Zubov, "Methods of A. M. Liapunov and Their Application",.U.S. Atomic Energy Commission, D i v i s i o n of Technical Information, 1957, AEC - t r - 4439 Physics. 56 24. V. I. Zubov, "Mathematical Methods of Investigating Automatic Regulation Systems". AEC-.tr-4494. 25. J. R. Hewit, C. Storey, "Numerical Application of Szego's Method for Constructing Liapunov Functions". IEEE Trans, on Automatic Control, F e b ! pp. 106-108 26. Y. N. Yu, K. Vongsuriya, "Steady State S t a b i l i t y Limits of a Regulated Synchronous.:. Machine Connected To an I n f i n i t e System". IEEE Trans, on Rower Apparatus and Systems, Vol. PAS-85, July 1966, pp. 759-767. 27. R. V. Shepherd, "Synchronizing and Damping Torque Coefficients of Synchronous Machines". AIEE Trans, on Power Apparatus and Systems, Vol. 80, June 1961, pp. 180-189-28. A. P. Sage, "Optimum Systems Control". Prentice H a l l , Inc., Englewood C l i f f s , N. J. 29. Lo S. Lasdon, A. D. Waren, "Mathematical Programming for Optimal" Design". Electro Technology, Nov. 1967.' PP- 55-70 30. J. B. Rosen, "The Gradient Projection Method For Nonlinear Programming. Part I: Linear Constraints". J. Soc. In d u s t r i a l and Applied Math.., No. 8, I960, pp. 181-217. 31. J. B. Rosen, "The Gradient Projection Method for Nonlinear Programming, Part I I : Nonlinear Constraints". J. SIAM, Dec. 1961, pp. 514-532. 32. E. V. Bonn, "A Si m p l i f i e d Algebraic-Geometric Approach To Computational Techniques i n Systems Optimization". Presented at the 1970 National Conference on Automatic Control, NRC Associate Committee on Automatic Control i n Cooperation with U. of Waterloo. 33. E. J. Davison, E. M. Kurak, "A Computational Method for Determining Quadratic Liapunov Functions For Nonlinear Systems". Presented at the JACC, 1970. 

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