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Power system stability studies using Liapunov methods Metwally, Magda Mohsen 1971

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POWER SYSTEM STABILITY STUDIES USING LIAPUNOV METHODS by MAGDA MOHAMMED ABDEL-LATIF MOHSEN (METWALLY) B.Sc. Ain-Shams U n i v e r s i t y , Cairo, 1967 A THESIS SUBMITTED IN PARTIAL FULFILLMENT 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 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 Jul y , 19 7.1 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 t he 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 ee t h a t t he 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 ag ree 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 rpo se s may be g r a n t e d by t he 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 . Department o f The U n i v e r s i t y o f B r i t i s h Co l umb i a Vancouve r 8, Canada ABSTRACT The transient s t a b i l i t y of power systems is investigated using Liapunov's direct method. Willems' method i s applied to three-and four-machine power systems with the effect of damping included. The distribution of damping among the machines of a multi-machine system i s studied, and optimum ratios are derived. An extension of Willems' method is used to include governor action in the system representation. Finally, the effect of flux decay on s t a b i l i t y regions is studied using Chen's method. i i TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i LIST OF ILLUSTRATIONS v ACKNOWLEDGEMENT v i NOMENCLATURE v i i INTRODUCTION .1 CH. 1: GENERALIZED POPOV'S CRITERION AND WILLEMS' METHOD .4 1.1 Generalized Popov's C r i t e r i o n 4 1.2 Willems' Method 7 1.3 S t a b i l i t y Regions 12 1.4 Numerical Example 12 CH. 2: OPTIMUM DISTRIBUTION OF DAMPING FOR MAXIMUM TRANSIENT STABILITY REGION 18 2.1 System Equations 18 2.2 Construction of Liapunov Function 20 2.3 Numerical Example 20 2.4 Optimum Damping D i s t r i b u t i o n 22 CH. 3: EXTENSION OF WILLEMS' METHOD TO INCLUDE GOVERNOR ACTION.. 26 3.1 System Equations .. 26 3.2 Construction of Liapunov Function 27 3. 3 Numerical Example . . . 29 3.4 Concluding Remarks 30 CH. 4: A LIAPUNOV FUNCTION FOR A POWER SYSTEM INCLUDING FLUX DECAY (CHEN'S METHOD) 32 4.1 Chen's Method 32 4.2 Estimation of S t a b i l i t y Regions 34 4.3 System Equations 36 i i i Page 4.4 Construction of Liapunov Function 36 4.5 Numerical Example 37 4.5.1 The Fi r s t Approximation 39 4.5.2 The Second Approximation 40 4.5.3 The Third Approximation 40 CONCLUSIONS 45 REFERENCES - ' 46 iv LIST OF ILLUSTRATIONS Figure Page 1.1 Automatic Feedback Control System Containing Single Memoryless N o n l i n e a r i t y 5 1.2 Nonline a r i t y Confined to a Sector of the F i r s t and Thi r d Quadrants 5 1.3 Automatic Feedback Control System With M u l t i l i n e a r i t y .... 6 1.4 Automatic Feedback Control System With M u l t i l i n e a r i t y .... 6 1.5 A Three-Machine Power System 13 1.6 S t a b i l i t y Region V = V m f o r X± = X,, = Xg = 0 17 2.1 A Four-Machine Power System 21 2.2 S t a b i l i t y region V = V m for X± = X 2 = = X 4 = 0 23 4.1 Single Machine-Infinite Bus 38 4.2.1 F i r s t Approximation of S t a b i l i t y Region 42 4.2.2 Second Approximation of S t a b i l i t y Region 43 4.2.3 Third Approximation of S t a b i l i t y Region 44 v ACKNOWLEDGEMENT I wish to express my deep gratitude to Dr. M.S. Davies, my supervisor, for his continued guidance, encouragement and understanding. Thanks are due to Dr. Y.N. Yu for reading the manuscript. The careful proof reading of the f i n a l draft by Mr. H.A. Moussa and Mr. A.A. Metwally i s duly appreciated. The financial support from the National Research Council is gratefully acknowledged. v i NOMENCLATURE x Vector of state variable x Time derivative of x V Liapunov function V" Time derivative of V V Value of V defining s t a b i l i t y region m t Time 6 Angle between quadrature axis of synchronous machine and i n f i n i t e bus or a reference frame rotating at synchronous speed in the case of multimachine systems Steady state value of 6 Value of 6 at the unstable equilibrum position Inertia constant i n KW -Sec/KVA H/(TTf ) System frequency = 60c/s Damping coefficient a/M, Relative damping constant of synchronous machine Mechanical power input to synchronous machine El e c t r i c a l power output of synchronous machine Instantaneous voltage proportional to f i e l d flux of synchronous machine Eg Voltage of i n f i n i t e bus E Steady state internal voltage of synchronous machine Total reactance between synchronous machine and i n f i n i t e bus Transient reactance of synchronous machine X Reactance of transmission line e v i i 6 o 6 U H M f a R P m P e E' q Synchronous reactance of synchronous machine Open circuit transient time constant of synchronous machine (X + X,)/T' (X + X') e d o e d (X - X')E /T'(X o + X') d d B o e d The null matrix The unit matrix Laplace operator Product of three matrices, X and Y are n x n matrices and 1 is an n x n matrix with a l l elements equal to 1. v i i i INTRODUCTION Since the early days of a.c. electric power generation and u t i l i z a t i o n , oscillations of power flow between synchronous machines have been known to be present. The possiblity of such oscillations and the tendency of a system to lose synchronism appears to be more prevalent i n large systems. The s t a b i l i t y characteristics of a power system during transient disturbances may be assessed from i t s mathematical model: a set of nonlinear differential equations, known as the swing equations. These equations describe the power system dynamics, their order depending on the detail of representation used for the synchronous machines and associated control apparatus. Several methods are available for the solution of the transient s t a b i l i t y problem. For simple configurations under the usual assumptions of constant input, no damping and constant voltage behind transient reactance,the equal area criterion or the phase plane method may be used. When the study involves a large number of machines or when i t is necessary to take into account such refinements as transient saliency, f i e l d decrement, exciter action and damping, s t a b i l i t y studies are usually investigated through step-by-step numerical integration of the system differential equations unt i l the c r i t i c a l switching time is found. Such a method is cumbersome and very costly since an almost prohibitive amount of computation is required in i t s execution. Thus the need increases for the development of more direct methods for studying s t a b i l i t y . During the past few years the application of the second method of Liapunov to the problem of power system transient s t a b i l i t y using models of varying degree of complexity for the power systems has been found useful and straight forward. The approach involves choice of a suitable Liapunov function to estimate the region of asymptotic s t a b i l i t y around the equilibrum state of 2 the post f a u l t system 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 d i f f i c u l t y i n the a p p l i c a t i o n of Liapunov's d i r e c t method i s that i n general there i s no obvious way to choose a s u i t a b l e Liapunov function. In many cases i n v o l v i n g a p h y s i c a l (mechanical or e l e c t r i c a l ) system the energy stored i n the system appears to be a natural candidate. Gless [13] studied 1~, 2-, and 3- machine systems representing the machines i n the simple form of a constant voltage behind synchronous reactance, neglecting a l l l o s s e s , damping, f l u x decaying and considering a constant input. El-Abiad •- and Nagappan [14] considered a multi-machine system i n c l u d i n g In t h e i r model losses and constant damping. Siddique [16] considers a s i n g l e machine system taking i n t o account f i e l d decrement and s i m p l i f i e d governor and regulator action. Other ap p l i c a t i o n s were made using formalized construction procedures, Yu and Vongsuriya [15] employed Zubov's method to develop a Liapunov function f o r one machine i n f i n i t e bus system using a second order model f o r the machine and i n c l u d i n g a damping c o e f f i c i e n t which i s a function of the angular displacement of the machine. Rao [17] used - Cartwright's [20] procedure to construct a V-function f o r a s i n g l e machine taking i n t o account the transient s a l i e n c y e f f e c t , a constant damping f a c t o r and a governor action represented by a si n g l e time constant. Rao also applied t h i s method to a s i m p l i f i e d 3-machine system. The v a r i a b l e gradient method [21] was applied by Rao and Desarkar [19] to a one-machine system i n c l u d i n g the e f f e c t of the f i e l d - f l u x linkage changes. Pa i , Mohan and Rao [18] applied Popov's theorem on the absolute s t a b i l i t y of nonlinear systems using Kalman's procedure [4] to construct a Lure-type Liapunov function f o r a one machine system with and without governor action. The generalized Popov c r i t e r i o n [8] for m u l t i v a r i a b l e 3 feedback systems was used by J.L. Willems.[9, 10] to develop a Liapunov function f o r n-machine power system. In t h i s thesis the s t a b i l i t y of single-machine as w e l l as m u l t i -machine power systems i s i n v e s t i g a t e d using two d i f f e r e n t procedures to construct s u i t a b l e Liapunov functions. In Chapter I Willems' method i s applied to a three machine power system taking into account the damping e f f e c t . A four machine system i s considered i n Chapter II and the best d i s t r i b u t i o n of damping r a t i o s i s obtained by maximizing the hypervolume enclosed by the Liapunov function. Willems' method i s extended i n Chapter III to study a three.machine system i n c l u d i n g governor act i o n . In Chapter IV Chen's method i s applied to a s i n g l e machine i n f i n i t e - b u s system taking into account the decay i n f i e l d f l u x linkage. 4 CHAPTER I GENERALIZED POPOV'S CRITERION AND WILLEMS1 METHOD The s t a b i l i t y study of automatic feedback control systems containing single memoryless nonlinearities, figure 1.1, was initia t e d by Lure. Normally the nonlinearity is confined to a sector of the f i r s t and third quadrants as shown in figure 1.2. Popov [1] made a most important contribution to the problem by giving sufficient conditions for absolute s t a b i l i t y which are completely dependent on the frequency response of the linear part of the system. A procedure for constructing Liapunov functions, for such systems was introduced by Kalman [4]. Recently Anderson [6], [8] developed a theorem generalizing Popov's criterion and Kalman's procedure to investigate the st a b i l i t y of feedback control systems containing more than one nonlinearity. The theorem relates the concept of a positive real matrix to the concept of minimal realization of a matrix of transfer functions [7]. Liapunov functions based on Anderson's theorem were constructed by Willems [10] for multimachine power system s t a b i l i t y studies. Willems' method i s applied i n this chapter to a three machine power system. 1.1 Generalized Popov's Criterion [8] Automatic feedback control systems with multi-nonlinearities, figure 1.3 and figure 1.4, can be descirbed mathematically in state / variable form by x = Ax - Bf(E) (1.1) e = Cx where x n vector e m vector 5 r=o N-L KS) G(s) F i g . 1.1 Automatic Feedback Control System Containing Single Memoryless Non l i n e a r i t y f(S) F i g . 1.2 Nonlinearity Confined to a Sector of the F i r s t and Third Quadrant 6 F i g . 1.3 Automatic Feedback Control System With M u l t i - n o n l i n e a r i t y iws) N.L A F i g . 1.4 Automatic Feedback Control System With M u l t i - n o n l i n e a r i t y 7 A n x n asymptotically stable matrix B n x m matrix C m x n matrix f(e) m vector satisfying the sector condition 0 < f. (E.) < k. E 2 — 1 1 1 1 f i(0) =0 i = 1,2, , m Theorem [6] If there exist real diagonal matrices N = diag (n , n , . . . . , n ) m Q = diag .(q , q , , q ) i / m K = diag (k , k , , k ) l I , m with n > 0 , q > 0 , n +q >0 such that m — m — m m Z(s) = NK"1 + (N + Qs) W(s) is a positive real matrix where W(s) = C(sl - A) ^ B (n x n) matrix of stable rational transfer function and W (°°) = 0 then the system is stable. The st a b i l i t y of system (1.1) can be determined by the Lure type Liapunov function rn Cx <V V(x,e) = x Px + 2Q / f(e) de (1.2) 0 where P i s a positive definite symmetric matrix determined T T PA + A P = -LL PB = CTN - LWQ + A TC TQ (1.3) WQTW0 = 2NK + QCB + B TC TQ where L, WQ are auxiliary matrices of order ( n x n ) , (n x m). 1.2 Willems' Method [10] Willems applied the above technique to estimate the transient s t a b i l i t y regions for multimachine power systems. 8 Assuming that 1. The flux linkages are constant during the transient period 2. The damping power is proportional to the s l i p velocity 3. The mechanical power inputs to the machines are constant 4. Armature and transmission line resistances are neglected, The differential equations describing the motion of the machines can be put in the form d 26. d 6. M. — — + a, -r~- + P . - P . = 0 for i = 1,2 n i ,2 I dt ei mi , dt (1.4) with P . = G.E2 + E E.E.Y.. sin (6. - &*) i = 1,2, n ei i i , = 1 i ] l ] I J where E. = internal voltage of the ith machine i G. = local load conductance l Y., = transfer admittance between the ith and the ith machine At equilibrum d 6. d26. - 1 = w. = 0 , — = i = d). = 0, P . = P dt i ' 2 I ' ml el dt Let x = " 2n vector where to, a are column vectors with components to = [to^, u>2 > • • • • t i ) n ] a = [a^, o2> •••• a ] o, — 6T - 6n , o, = 6„ - 6„ , .... a =6 - 6° 1 1 1 2 2 2 n n n Although the state variable vector x has 2n components the actual order of the system is (2n - 1) since only the differences between the rotor angles appear in the system equations. 9 Let M = diag (MJ (n x n) matrix ( n x n ) matrix D =. an (m x n) matrix such that the vector e = Da has i t s (1.5) R = diag (-cO components - - o2> z2 ~ °1 ~ °3' n-1 1 n n 2 3 n+1 2 a4' ' m - " ( n - 1> m - -m n-1 n where Define the function f(e) as f.(e.) = E E Y ( s i n (e. + e?) - s i n e?) ,• _ i o m Where i i p q pq i i I 1 i,z,...m wnere (1.6) p, q are the indic e s of the component of a on which c i s dependent. Let e? be the value of e for 6. = 26? and define the matrices A, B and C as l i i A = B = M 1R 0 nn I 0 n nn -1 T M D nm (2n x 2n) matrix (2n x m) matrix L mn D (m x 2n) matrix The d i f f e r e n t i a l equatiors (1.4) become equivalent to x = Ax - Bf'(e) e = Cx (1.7) (1.1) The s t a b i l i t y of system (1.1) i s determined by a Liapunov function of the form (1.2). The time d e r i v a t i v e V i s given by V = - ( x T L - f(Cx) TW Q T) (L Tx - W Qf(Cx)) - 2x TC TNf(Cx) (1.8) 10 The next step is to find the.matrix P of equation (1.2). Since by definition T T CB = B C = 0 mm and choosing N = 0 mm m then substituting in equation (1.3) results in i) W = 0 0 mm PA + ATP = T T PB = A C -LL -1 (1.9a) (1.9b) (1.9c) i i ) Z(s) = sC(sI - A) B is positive real i f a l l the damping constants are nonnegative i i i ) equation (1.8) reduces to T T V - -x LL x which is negative semidefinite Let P = where P^, P 2 > P^ are (n x n) square matrices. Thus equation (1.9c) is equivalent to - I T T P M D = D -1 T P.M D = 0 2 nm and from the negative semidefinitness of PA + A P = PXM 1R + RM 1 P 1 + P 2 + P T P 2M _ 1R + P 3 -1 T K M P 2 + P 3 nn we get -1 -1 T P 3 = -P2M R = -RM P 2 (1.10) (1.11a) (1.11b) (1.12) Since matrix contains m = ^ ~ —^  columns with each column containing only two nonzero elements,+1 on the i t h row and -1 on the j t h row, the 11 solution of the equation YD =0 , where Y i s an unknown symmetric nm (n x n) matrix,is Y = ul where y is a scalar constant and 1 is an (n x n) matrix with a l l elements equal to 1. Applying the above reasoning to (1.11a) results in -1 -1 -1 T (M P.M - M )D = 0 1 nm P 1 = M + yMlM which is positive definite i f y >\i where y is the solution of the 1 o o det,/M + u M1M/ = 0 o 1=1 1 -1 (1.13) (1.14) from equations (1.11b) and (1.12) -1 -1 T R P0R D = 0 3 nm and hence P 3 = yRlR P 2 = -yRIM where y is a scalar constant and is taken equal to zero hence P = 0 nn 0 0 nn nn (1.15) (1.16) The matrix PA + A P is negative semidefinite i f , and only i f , the matrix Z(y) = 2R + y(MlR + RIM) is negative semidefinite Z(y) is negative semidefinite for certain values of y (1.17) where u.. is the solution of the det |z(u)| = 0 which is equivalent to n I n i=l j=i+l 7 (M. 4 J - M. l a . n - u(I M ) - 1 i=l i (1.18) Equation (1.18) has a positive and a negative solution for y, the negative 12 one being u Substituting the value of P in equation (1.2) we obtain T T C x T V(x) = to Mco +y .to MIMco + 2/ f(e) de (1.19) 0 with i t s derivative V(x) = 2coTRio + 2uco TM1R (1.20) 1.3 Stability Regions Since the derivative of the Liapunov function is negative serridefinite [9] the boundary of the transient s t a b i l i t y region can be obtained by solving the equations = 0 o CO . 1 for i = 1,2, n (1.21) a:v(x) _ n 3.6. x The f i r s t equation gives co = to =....= to =0. The second equation 1 z n gives the closest equilibrum state (necessarily unstable) to the origin x U. The region bounded by the closed surface V(x) = V(x U) and containing the origin i s a stable region. 1.4 Numerical Example Consider the three machine system shown in figure 1.5. The differential equations describing the motion of the system are 13 F i g . 1.5 A Three-Machine Power System 14 d 26 M 1 d t 2 dS + K1 d r + P e l = P m l d 26, d6, 2 ~1~ + A2 dT + P (1.22) M dt 3 dt 2' e2 " m2 d6 + a 3 l T + P e 3 = P m 3 Let the state variable vector be dS. d6. d6. x = ( o S T dt ' dt ' dt Following the steps described i n section 1.2, the system equations(1.22) become x' = Ax - Bf(e) e = Cx where r a l A = — \ 0 a ? 0 0 0 0 0 ~ M 2 0 0 0 0 0 0 J "M 3 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 B = 1_ 1_ 0 1 M 2 M2 0 1 1 "M 3 "M 3 0 0 0 0 0 0 0 0 0 C = 0 0 0 1 -1 0 0 0 0 1 0 -1 0 0 0 0 1 -1 15 w = E 1 E 2 Y 1 2 ( s l n u1 + E l } " sin f 2 ( e 2 ) = E 1 E 3 Y 1 3 ( s l n u2 + E 2 ) - s in f 3 ( c 3 ) - E 2 E 3 Y 2 3 (sin ( e 3 + e 3) - s in with = X4 " X 5 o e l -< -«2 e2 = X4 " X 6 o e2 "I e3 = X 5 " X6 0 e 3 - 52 -«3 = 2> =3> The Liapunov function i s then given by V(x) = M ^ - r M2X2 + M 3X 3 +u [M1X]_ + M2X2 + M ^ ] 2 + 2 [ E 1 E 2 Y 1 2 ( c o s e ° - cos (X 4 ~ X + e°) - % - s i n e°) + E 1 E 3 Y 1 3 ( c o s e ° - cos (X^ " x 6 + e°) - (X^ - X g) s i n e ° ) + E 2 E 3 Y 2 3 ( c o s e° - cos (X 5 - X g + e°) - ( x $ - X g) s i n e°) ] (1.23) The system studied has the following data E1 = 1.174 [22.64° p . u . Pir^ = 0.8 p . u . E 2 = 0.996 12.61 0 p.u.. Pm2 = 0.3 p . u . E 3 = 1.06 [-11.36° p . u . Pm3 = - l . l p . u . a l H-. = 3 KW. sec/KVA — = 10 Y, _ = 1.13375 p.u. 1 12 a2 H = 7 KW. sec/KVA — = 7 Y = 0.52532 p.u. 2 M2 13 a H = 8 KW. sec/KVA ~r = 3 Y 0 0 = 3.11850 p.u. 3 M3 23 A sudden 3-phase symmetrical short c i r c u i t to ground occurs on the trans-mission l i n e connecting machines 2 and 3 of Figure (1.5) close to bus 3. The unstable equilibrum state nearest to the o r i g i n i s calculated and was found to be at X, - X, = 2.61168 rad 4 5 X. - X, = 2.95275 rad 4 6 X 1 = X2 = X 3 = 0 ' ° 16 with V = 3.36. m The c r i t i c a l c l e a r i n g time obtained from the above V-function was found to be between 14-15 cycles. Figure (1.6) shows the function V= V p l o t t e d i n the two m dimensions (X. - X c ) , (X. - X,) f o r X = X„ = X„ = 0. 4 5 4 b 1 2 3 The actual c r i t i c a l c l e a r i n g time obtained from forward i n t e g r a t i o n of the swing equations using Runge Kutta method i s 20 cycles. 18 CHAPTER II OPTIMUM DISTRIBUTION OF DAMPING FOR MAXIMUM TRANSIENT STABILITY REGION In this Chapter Willems' method described in Chapter I is applied to a four machine power system and a study is made to find the optimum distribution of damping that maximizesthe region of s t a b i l i t y . 2.1 System Equations Under the same assumptions made in Chapter I the system equations are ~ d 6 d 6 M: —-r- + a . — - + P .. = P . i = 1,2,3,4 (2.1) i ,2 I dt e i mi ' dt where P e l " A l sin <6i "V + A2 sin <61 - 6 ) r + A3 sin <-\ "V Pe2 = A l sin + A4 sin < 6 2 - 5 ) y + A5 sin "V. Pe3 = A2 sin <53 - V + A4 sin ( 53 - v + A6 sin - V Pe4 = A3 sin <S4 - y + A5 sin ( 54 - v + A6 sin <54 - V (2.2) and A 1 = E 1 E 2 Y 1 2 A 2 = E 1 E 3 Y 1 3 A 3 = E 1 E 4 Y 1 4 A4 " E2 E3 Y23 A5 = E2 E4 Y24 A6 = E 3 E4 Y34 Following the same steps described i n section 1.2 to represent (2.1) in the form (2.3) x = Ax - Bf(e) e = Cx the results are x = < U l 03 2 c o 3 C J 4 6 1 -6° <52 -6° 6 3 -6° -6°) T ( 2 4 ) e = (6 1 - 6 2 6 X -6 3 5 1 -6 4 6 2 -63 6 2 -6 4 63 -S^ ( 2 5 ) E = (x 5 - x 6 x 5 - x ? x 5 - x g x 6 - x ? x 6 - x 8 x ? - x 8 ) T (2.6) 19 A = A.(sin ( e . + e ? ) - sin e ° ) M l 0 0 1 0 0 0 0 M„ 0 a. 0 0 1 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 0 0 0 0. 0 0 1 0 0 0 0 (2.7) C = 0 0 0 0 0 0 1_ 0 0 0 0 0 0 0 0 0 0 0 0 1_ M, - zr- 0 1_ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 rr- 0 1_ 1 1 1 0 0 0 1_ - rr- 0 77- 0 -0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 -1 0 0 1 -1 1 0 0 1 • 0 1_ M, -1_ M^ 0 0 0 0 0 0 -1 0 -1 -1 (2.8) 20 2.2 Construction of Liapunov Function According to the expression for P given in section 1.2 where nn nn nn P. = M +U M1M r 2 2 M +pM 1 yM1M3 2 2 M2 +yM 2 yM2M3 yM2M4 uKjM y M2M3 M2 + y M2 . P M3M4 y M ^ yM3M4 2 2 M. +yM. 4 4 The Liapunov function i s then given by • vl+ vl + M £< h M4 X4 2 4 • u [M^ X 1 + M2 X2 + M3X3 + M 4X 4) 2 +2 [A^(cos - cos ( X5 " X 6 + e±) - <X5 - V sin £ 1 } +A2(cos e 2 - COS ( x 5 " X7 + e2) - ( x 5 - V sin £ 2 ) +A3(cos e'3~ COS (x 5 " X8 + E3) - ( x 5 - V sin e 3) +A, (cos e. -4 4 COS ( X6 " X 7 + e4 } " < X 6 -v sin °\ Zl? +A^(cos e,.- COS ( X6 " X8 + e 5) - ( X6 - V sin +A6.(cos E6- COS (X ? " X8 + 0 ^ E6> " (X ? - V sin e 6)] (2.9) 2.3 Numerical Example The system chosen as an example is shown in Figure 2.1. E = 1.004 |0.0013 rad E 2 = 1.0410 10.103 rad E = 1.1900 J0.197 rad E. = 1.070 0.0772 rad 4 ' p.u. p.u. p.u. p.u. 22 Pmi = ° - 3 3 2 p - u - \ = 75350 P.u. D l = 1 , 0 P ' u " P ^ = 0.1 P.u. M2 = 1130 P.u. D 2 = 12.0 P.u. P m 3 =0.3 P.u. M3 = 2260 P.u. D 3 = 2.5 P.u. P„, = 0.2 P.u. M, = 1508 P.u. D, = 6.0 P.u. . A sudden .3-phase symmetrical short c i r c u i t to ground occurs close to bus 3 on.the transmission l i n e connecting machines 3 and 4 of Figure 2.1. The following table gives the stable equilibrum state of the p o s t - f a u l t syst and the unstable equilibrum state clo s e s t to the stable one. Internal 6, radians(stable) 6, radians (unstable) bus 1 0.05630 0.06610 2 0.15013 0.20136 3 0.21430 3.0820 4 0.02497 -0.02425 The exact c r i t i c a l c l e a r i n g time was c a l c u l a t e d using Runge Kutta and was found to be at 30 cycles. When c a l c u l a t i n g the value of 1 the Liapunov function at the unstable equilibrum state and X, = X_ = X„ = X, =0.0 we obtain a value f o r 1 2 3 4 V^ = 3.155 which gives a c l e a r i n g time of 25 cycles . Figure, 2.2 shows the function V(x) = V^plotted i n the three dimensional space (X^ - X^), (X^ - X^), (X,.- Xg) with the components X = X = X = X. = 0.0. 1 2 3 4 2.4 Optimum Damping D i s t r i b u t i o n a. The optimum d i s t r i b u t i o n of damping r a t i o s ( — ) i s obtained a. i by f i n d i n g the r e l a t i v e values of — to maximize the hypervolume enclosed i by the Liapunov function that defines the s t a b i l i t y region of the system. Considering the V-function (2.9) for a four machine system 23 Fig. 2.2 Stability Region V = V for X = X = X = X = 0 m 1 2 3 4 24 T C x T V = to (M + yMlM)to + 2 f f (e) de where y i s given by f R u 2 " AklU -4 = 0 which gives y u = 2 ( ) R where MM „ MM MM MM ? F R " lV ( R l " V + O T ( R 1 " V + < R T ( R 1 " V + R ¥ ( R2 " V 1 2 1 3 1 4 2 3 MM MM + RJ*7 ( R 2 " V 2 + O T ( R 3 - V 2 2 4 3 4 k.. = M, + M + M + M 1 1 2 3 4 I t i s noticed that the in t e g r a l part of V does not depend on the damping coe f f i c i e n t s and therefore the hypervolume enclosed by the quadratic part alone i s to be considered.. This volume i s given by [22] where H = TT2 V 2 / 2 / \ . p q 1 T V = to Aid q A = M + y M1M |A| = determinant of matrix A = k 2 ( l + y k^) k 2 = M1M2M3M4 V q = N 1 + y N 2 2 2 2 2 N.. = M, to, + M oi + M_to0 + M.to, 1 1 1 2 2 3 3 4 4 N, ( M ^ + M 2u 2 + M a) + M 4 O J 4 ) 2 2 Thus H p = JL_ ( N l + y N 2) V A2 + y k ^ For maximum volume " ' 3H 3 ^ - 0 1 " 1.2,3, 4 ( 2 . 1 0 ) i b U t 8H 3H • 3f p = p 3y R 3R, 3y ' 3f_ * 3R. i R l 25 Thus to s a t i s f y equation (2.10) H 3f £ = 0 or J - = 0 or 77^ = 0 1 = 1,2,3,4 9u 3 f B 3R K 1 3H 2 k (N +u» ) i f • t k 2 < * i » * + 4 N2 " k l V which when equating to zero gives N x k 1N 1 - 4 N 2 * = " T2 o r ^ " 3 k l N o Both answers are rejected since the value of u depends on the values of state v a r i a b l e -= Zu FR - 2 k l A l + FR + 2 k l ^ The t h i r d p o s s i b i l i t y i s that 3f M M ? M „ „ M. ^ = ^  ^ ( \ " R2> ( R l " ^ ^ " R 4 ) } = ° 3f M M M 'JM 1R! - J <" RT ( R 1 " 4 + I T ( R 2 " R3> + 17 ( R3 - R 4 » = ° 2 1 3 4 3 FR M 3 , M l ^ 2 2, M 2 ,2 2, . M 4 ^ 2 .2 3 3R„ 2 R 1 y R2 v 2 3y R. 3 4 3 R„ 1 4 af M M M M R M4 , 1 ,J1 ^2 N 2 ,JL ^2 X 3 ,„2 „2v {- — (R, - R.) - TT1 (Ro ~ R,) " 7^ (R, - R P }= 0 3R. 2 R . l 4' R ' 2 4y R v 3 . 4;4 R. 1 2 3 4 which gives R. = R 0 = R = R. 1 2 3 4 Thus f or a maximum region of s t a b i l i t y the damping r a t i o s of a l l machines  should be equal. 26 CHAPTER III EXTENSION OF WILLEMS1 METHOD TO INCLUDE GOVERNOR ACTION In t h i s chapter Willems' construction procedure i s extended to develop a Liapunov function for multimachine power systems i n c l u d i n g governor action. 3.1 System Equations Assuming that f l u x linkages are constant, resistances are neglected, damping power i s proportional to s l i p v e l o c i t y and governor response may be represented by a s i n g l e time l a g transfer function, AP _m -K 1 + T, (3.1) The equations of the i t h machine are d6. where dt x M ± 7dT = " V i " P e i + Pmoi + A P m i dAP .' , K. mi 1 • I " H F - = " TT A P m i - Y7 U i 1 1 (3.2) Pmoi i s the value of the mechanical input at steady s t a t e . Defining the vector o,T X = [to1 , to,, a) „» AP,^' A Pw,9» A P „ „ > 5 i " 5 i > " 6o> 5 „ ~ 6 „ ] ,m2 mn n B = M _ 1R M 1 0 and the matrices A = Y Z I nn 0 nn 0 0 nn nn -1 T M D nm nm C = 0 0 D mn mn (3.3) (3.4) 27 where Y = diag [- ] i Z = diag [- Tjr~ ] i M , R and D are as defined i n chapter I. Equation (3.2) takes the form k = Ax - Bf(e) e = Cx 3.2 Construction of Liapunov Function (3.5) Applying Popov's generalized c r i t e r i o n , system (3.5) i s stable i f . (N + Qs)C (Is - A) i s p o s i t i v e r e a l matrix. Taking N = 0 and Q = I then sC(Is - A) "*"B i s p o s i t i v e r e a l i f the mm m damping constants are nonnegative. A s u i t a b l e Liapunov function f o r such a system i s T Cx V = x Px + 2 / f(e) d£ o where P i s determined from the requirement that PA + A P be negative semidefinite and that T T PB = A C Let P T T P P 12 13 P P 12 2 P13 P23 23 S u b s t i t u t i n g (3.7) into (3.6b) gives - I T T P M D = D -1 T P i nM D = 0 12 nm -1 T P 1 0M D = 0 13 - nm Subst i t u t i n g f o r matrix A i n equation (3.6a) one has (3.6a) (3.6b) (3.7) (3.8) V^M + PX3+M 1RP 1+YP 1 2+P13 P LM 1+P 1 2Z+M 1 r p 1 2 + Y P 2 + P 2 3 P 1 2 M " l R + P 2 Y + P 2 3 + M " l p i + Z P 1 2 P 1 3 M " l R + P 2 3 Y + P 3 P 1 2 M " 1 + P 2 Z + M " l p i 2 T + Z P 2 P 1 3 M " 1 + P 2 3 Z M _ l R P 1 3 + Y P 2 3 + P 3 - I T T M P 1 3 + Z P 2 3 nn (3.9) 28 Setting a l l off diagonal elements of (3.9) equal to zero give P . M - 1 + P *Z + M ^ R P * + Y P _ + P = 0 1. 12 12 2 23 nn M _ 1 r P 1 3 + Y P23 + P 3 = 0 nn (3.10) (3.11) - I T T M P 1 3 + Z P 2 3 = o nn (3.12) Equation (3.8), (3.10), (3.11) and (3.12) are solved for P . , P „ , P „ , P , 12' P 1 3 and P 2 3 to give P = M + y M1M P 1 2 = Y X M1M P 1 3 = Y 2 M 1 M ' P ^ - Y ^ ^ I M P3 = Y 2 ( Y Z _ 1 M _ 1 + M_1R)M1M P 2 = Y _ 1 ( Y 2 Z - 1 1 M - Y 1 (M 1M Z + M_1RM1M) -yMl - I ) where y and Y 2 a r e constant scalarsand y is given by ? n n 1 r*- J®~- 2 n y [S S - (M, - ^  / - J - ) ] - y E Mj_ - 1 = 0 i=l j=i+l * " j " i i=l (3.13) (3.14) Choosing y^ and Y 2 to be equal to zero, matrix P reduces to P = where p l 0 nn 0 nn 0 nn P2 0 nn 0 nn _ 0 nn 0 nn (3.15) P l = M + y M1M P 2 = - Y 1CiMl + I ) Thus for a three machine system, the Liapunov function i s 29 V(x) = M 1X 2 + M 2X 2 + M 3X 2 + u (ML^ + M 2X 2 .+ M 3X 3) 2 T T T M T M T M T 1 9 9 2 3 2 1 1 2 2 3 3 + K ^ X 4 + K ^ X 3 + K ; X 6 + ^ X 4 + X 5 + V ^ X 4 + - k 2 - X 5 + l f + 2 E 1 E 2 Y 1 2 (cos e± - cos (X ? - X g + e°) - (X ? - X g) s i n e°) + 2 E 1 E 2 Y 1 3 (cos e 2 - cos (X ? - X + e°) - (X_, - X g) s i n e°) + 2 E 2 E 3 Y 2 3 (cos e 3 - cos (X g - Xg + e°) - (X g - Xg) s i n e°) 3.3 Numerical Example The same numerical example of Chapter I i s considered. With governor action taken into account the equations describing the machine dynamics are • . • d f i . I - to. dt i dto. ~ - = -a.to. - P . + P . + AP . dt i x e i mox mx dAP K. mx x 1 . _ di— = " TT u i " TT A P m i 1 = 1 » 2 » 3 x x System Data E. = 1.174 |22.64° P.u. -f- = 10.0 1 ' M^ E 0 = 0.996 I2.61 P.u. ~ = 7.0 2 M 2 a E 0 = 1.006 1-11.36 P.u. — = 3 . 0 3 M 3 P =0.8 p.u. T, = 0.2 sec Y,. = 1.13375 p.u. mol r 1 12 P _ = 0.3 p.u. T. = 0.22 sec Y, „ = 0.5232 p.u. mo2 r 2 13 30 P. _ = -1.7p.u. T 0 = 0.25 sec Y„„ = 3.11856 P-u.. mo3 3 23 K± - K 2 = K 3 = 0.0 The unstable equilibrum state close to the stable one is calculated by solving the equations ^ X - ) = 0.0 (3.17) = 0.0 (3.18) =0.0 i = 1,2,3 (3.19) Equations (3.17) and (3.18) gives X = X = X = X = X. = X. = 0.0 1 2 3 4 5 6 Equation (3.19) gives X., - X„ = 2.61168 rad / o X, - X = 2.95275 rad with V = 3.36 m The c r i t i c a l clearing time obtained from the above V-function was found to be between 15~16 cycles while the exact clearing time obtained from the forward integration of the swing equations using Runge Kutta was at 20 cycles. 3.4 Concluding Remarks It is obvious from the material presented i n this chapter that the generalized Popov's criterion can be successfully applied to power systems including governor action. On the other hand the same method failed when applied to a power system taking into account the flux decay in the f i e l d circuits of the synchronous machines. The reason behind this failure i s that the nonlinearities introduced when considering flux decay are different in form from those considered by Popov. Thus i t is concluded that Popov's theorem is applicable to higher order power systems as long as the nonlinearities retain the same form. 32 CHAPTER IV A LIAPUNOV FUNCTION FOR A POWER SYSTEM INCLUDING FLUX DECAY (CHEN'S METHOD) When including the effect of flux decay in the f i e l d c i r c u i t of synchronous machines the power system can not be represented in a suitable form for application of the generalized Popov's criterion. A new method, developed by Chen, based on the use of an auxiliary exact differential equation derived from the given nonlinear differential equation representing the system, is applied in this chapter. The method is employed to construct a Liapunov function for a third order model of a synchronous machine con-nected to an i n f i n i t e bus with the effects of flux decay in the f i e l d included. 4.1 Chen's Method [11], [12] Consider a set of n f i r s t order autonomous differential equations x = f(x) <*-D where both x and f(x) are n dimension vectors, a l l f^(x), i = 1,2, ....n together with their f i r s t partial derivatives are defined and continuous in some region Q of the state space E^ and the point x = 0 is an equilibrum point also in 11. Define g. = f, + f. + + f. - f . ^ f (4.2) l 1 2 l - l l+l n 4-u n then _ . which gives T g x = 0 or g dx = 0 (4.3) Equation (4.3) i s said to be an exact differential equation in Q i f there is some single-valued differentiable function U(x) defined and continuous together with i t s f i r s t partial derivatives in some neighborhood of every point in Q such that 33 dU(x) = g T • dx dU(x) „„, ,T . T —rr- = VU(x) • x = g • x (4.4) which results in 3U(x) = 3 x i S i (4.5) i , j = 1,2, ..'..n 3X. 3X. 3 i T and U(x) = / g • dx c which is independent of any integration path C contained i n the domain of U(x). Thus equation (4.5) i s a necessary and sufficient condition for the exactness of (4.3). If equation (4.3) is not exact the U(x) does not T exist. A function h(x) can be added to g(x) such that (g + h) • dx i s an exact differential which pives dU(x) / . , N T = (g + h) - x (4.6) = (g + h ) T • f dU(x) T dt ~ h f For equation (4.6) to be an exact differential equation, then 3U(x) , , "3XT = 8 i + h i I 3(g ± + h ±) 3(g. + h.) a n d r r r ^ - = "gx 1 i» j = 1» 2, .. . .n (4.7) j i The function U(x) can be evaluated by the line integral U(x) = / (g + h ) T dx (4.8) C For an integration between limits o and x, (4.8) gives x U(x) - U(o) = / (g(y) + h ( y ) ) T . dy (4.9) o It remains then to select h(x) such that U(x) has the characteristics of a Liapunov function. These are given by 34 a) (g + h) » x is an exact differential dr i , j = 1,2, 9(g. + h ) 3(g. + h ) T b) hlf ax. j dU(x) dt ax. is negative definite or semidefinite c) U(x) is positive definite Let h = 6(g) + V i K * ) where 6(g) is a known function of g(x). For n=3 6(g) is given by (4.10) 6(g) = - /C 3g-, axT 3g. 3go 38-, 3g 9 J 3g, 3^> d X 3 " / [ ( ^ " ixf> " 1 l c 3 ^ 3 " i x ^ ) d X 3 ] d X2 - /C 3g 2 3g 3 3X„ 3X> d X 3 L (4.11) and i|)(x) is a scalar function that has to be selected such that conditions b and c are satisfied. Substituting for h(x) from equation (4.10) gives X X X U(x) = / (g(y) + h(y)) Tdy = / (g + 0(g) ) Tdy + / V<Ky)Tdy U(x) = W(x) + ^(x) - lj, (0) (4.12)-If ;jj(x) i s chosen such that (0)= 0, then U(x) = W(x) + i>(x) (4.13) conditions b and c can be restated as b) U(x) = W(x) + TJ;(X) is negative semidefinite c) U(x) = W(x) + (JJ(X) is positive definite where W(x) is directly evaluated from the system equation (4.1) with ^(x) serving as correction function to give U(x) i t s desired characteristics. 4.2 Estimation of Stability Regions For locally stable systems a closed form solution for the 35 undetermined c o e f f i c i e n t s of the function- I|J(X) does not e x i s t . In t h i s case the s t a b i l i t y region i s estimated by generating a Liapunov function i n a s e r i e s form a f t e r expanding the system n o n l i n e a r i t i e s i n t o polynomial form. Thus W(x) can be w r i t t e n as the sum of homogeneous polynomials W(x) = F 6 2 .(x) + F o 3 ( x ) + F o m ( x ) (4.14) • where F 0j ( x ) i s a j t h degree homogenous polynomial. Thus U(x) = F o 2 ( x ) + F Q 3(X) + .... F ^ x ) + <Kx) and U(x) = - G o 2 ( x ) - G o 3 ( x ) G m + S ^ + <KX> (4.15) . where-G Q J i s also a j t h degree homogeneous polynomial and s i s the highest degree of f ( x ) . Then the f i n a l Liapunov function i s obtained i n the following steps 1. S t a r t i n g with ip(x) = 0, check the p o s i t i v e d e f i n i t n e s s of F (x) and o2 G ^(x) . I f both are p o s i t i v e d e f i n i t e then U'(x) i s a Liapunov function. 2. I f F „ (x ) and G „ (x ) are not p o s i t i v e d e f i n i t e , consider a quadratic o2 o2 n function ij^(x) = F^ 2 (x) with undetermined c o e f f i c i e n t s , thus IL(x) = [F (x) + F (x)] + F .(x) + .... F (x) 1 o2 12 o3 o m U l ( x ) = -G 1 2 ( x ) . - G 1 3 ( x ) . . . . G 1 ( m + s ) ( x ) The unknown c o e f f i c i e n t s of ij> (x) are determined from i ) F 02 (x ) + F^ 2(x) i s p o s i t i v e d e f i n i t e i i ) G ^ ( x ) i s p o s i t i v e d e f i n i t e 3. For a second approximation a homogeneous t h i r d order polynomial F^ C x ) i s added to U^(x) to give U 0 (x) = [ F 2 ( x ) + F (x)] + [F (x) + F „ . ( x ) ] + F (x) + .... F .(x) 2 u ^ l z oi Li Q4 om U(x) = -G 1 2 ( x ) - G 2 3 ( x ) .... G 2 ( n t f s ) ( x ) -The c o e f f i c i e n t s of F 2 3 ( x ) are determined by s e t t i n g G 2 3 ( x ) = 0 4. Further approximation can be made as required. 36 4.3 System Equations For a single machine connected to an i n f i n i t e bus the system equatior© including the effect of flux decay in the f i e l d [16] are X1 = X2 X2 4 ( F m l - - - X 1 9 q ' x - ' , s - ( X 1 + 6 ° ) ) 12 d X 3 = ~ n l X 3 ~ n 2 ( c o s ( X i + 6°) " c o s 6°) (4.16) where X = 6 - 6 ° X 0 = X = 6 X = E"' - E z 1 3 q In order to apply Chen's method,equations(4.16) are expanded as follows : Put h = E B/M(X 1 2 + X'^ K 2 = K 1E Thus X 1 = X 2 X2 = " K1 X3 S ± n ( X1 + " K 2 ( s i n ( X1 + " s i n 6°> X 3 = - T ^ X - n 2(cos(X 1 + 6 ° ) - cos 6 ° ) which when expanded gives X1 = X2 °° . °° i - l X = - E p xj- - X_ E q.X 2 1=1 1 1 3 1=1 1 1 (4.17) CO X, = - n,X_ - E r. xj" 3 1 3 . , i 1 1=1 where P». l K 0 2 . ,fO , ITT. -7j s m ( 6 + — ) K l . ,.o ^  ( i - l ) . q ± = T y s m ( 6 + -— ' - ir) n2 • /*° ^  (1 ~ 1) s — s m ( 6 + -=>—r— ' -v , ) r i i ! ^ u T 2 " > (4.18) 4.4 Construction of Liapunov Function Applying Chen's method to the system equations (4.17) gives 37 g(x) E (p. + r X + X 3 J n q i X l 1 + ^ 3 i = l 1=1 1 , r l X l + X2 + n l X 3 i = l n -E i=l 4.-1 + X„ P i X l ~ X 3 Z . q i X l ' "2 i= l 6(g) = " ? , "«i + ^ i ) X 3 + I q i + l X 3 " l r i V X l _ 1 i=l ( l - n 1 ) x 3 (4.19) (4.20) W(x) = -| X2 + 2X 2X 3 1 Y2 "+ 1 Pi-1 2 q i x 3 + 1 , ( r i=2 r . ^ ) x j n+1 + E i=2 ( 2 X 2 r . _ 1 ^ i - i ) xr - s z, q i - i xr i = J (4.21) 4.5 Numerical Example [16] I The system taken as an example i s shown i n Figure 4.1 with the following data E = 1.02 P.u. 6° = 0.42 rad M = 147 x 10~ 4 P.u. Xf- ' = 0.3 P.u. d E' = 1.03 P.u. q .= 1.0 P.u. X = 0.2 P.u. on base 25 MVA e X, = 1.15 P.u. d To' = 6.6 sec 38 F i g . 4.1 Single Machine - I n f i n i t e Bus Power System 3° A three phase short circuit at the middle of one transmission circuit as shown in Figure 4.1 occurs, the corresponHit-p-.V and V-functions 'are W(X) = | X 2 + 2X 2X 3 - ? q i X 2 + E C 1 - 1 . 1 - 1 ) * J i=2 n+1 _ n+1 + E (2X„r. - 2X„p. ) X* - X 0 E q. . X. . 0 2 l - l 3 l - l 1 3 . „ M i - 1 1 i=2 i=3 W(x) = - 2n 1X 2X 3 - q ^ r ^ - 2)X 3 - X 2 E r X* 1 i=2 i-2 n+1 n+1 + 2X; E ( i - 1) V l x{~2 - X 2X 3 (2(1 - DP.., q . ^ x j " 1 i=2 i=2 n+1 i-1 + X3 E 9 ( q l r i - l + 2 ( r i l " ^ i - l * X l i=2 + 2X3 E ( n i - D q . ^ X ^ - X 2X 3 E (1 - 2) q X ^ 3 i=3 i=3 It is seen that F- „ and G' „ are not positive definite and successive oZ ol r approximations are needed. 4.5.1 The Firs t Approximation T A quadratic function F ^ W = X Ax is added to W(x) . When solving for {F 1 2(x) + F 0 2 ( x ^ ^ positive definite and G 1 2 we get A = positive definite -1.45 -0.109 -0.109 -0.01 153 -1.0 153 -1.0 50.5 "Die f i r s t approximate s t a b i l i t y region boundary is obtained by calculating ct1 = min {U 1(x)/U 1(x) = 0, except the origin} where IJ^Cx) = W(x) + F ^ x ) . The c r i t i c a l clearing time i s calculated from the above V-function and was at 0.05 sec. 40 4.5.2 The Second Approximation A third order homogeneous polynomial ^ .^Cx) "*"s a <^ e c* ^° U^( x) F 2 3 ( X ) = V l ' + a 2 X 2 X l + a 3 X 2 X l + a4 X2 3 + a 5 X 3 X l 2 3 2 2 4- a^X^X^ + a^X^ + 3gX2X3 +.3^X2X3 + ^iQ^^^2X3 Solving for the unknown coefficients introduced by F2 3(x) from the con-dition gives G 2 3(x) = 0 ax = 0.1472 a6 = 71.03 a 2 = -0.31947 a7 = 1.9328 -6 a 3 = 0.224456:. x 10 a8 = 0.08046 ,~--3 a. = -0.46473 x 4 10 a9 = -0.042757 a 5 = -15.752 a10 = = -0.04443 The second approximate s t a b i l i t y region boundary i s obtained by calculating = min {U2(x)/u"2(x) = 0, except the origin} where U^Cx) = U^(x) + F 2 3(x) which gives a c r i t i c a l clearing time of 0.1 sec. 4.5.3 The Third Approximation A complete fourth order homogeneous polynomial F^ C x ) is added to U 2(x) F 3 4(x) = a^X4 + a 2X 3X 2 + a ^ X j ^ a ^ X ^ + a^X2 + a 6 x J x 2 X 3 + a 7 X l X 3 + a ^ X 3 + . a ^ X ^ + a 1 0X 1X 2X 3 + a ± / 2 -+ a 1 2X 2 3X 3 + a ^ x ] + a14 X2 X3 + a15 X3 The unknown constants introduced by F 3^(x) are calculated from the condition G 3 4(x) = 0.0 41 which gives a 1 = -40.694 a 2 = 0.047175 a 3 = -118.09 a. = 0.64395 4 a 5 = -112.98 a, = 0.41651 D a ? = -0.40783 a g = -67.118 a 9 = 0.058045 a 1 Q = -0.56244 a l l = - ° - 0 0 2 5 4 0 8 a 1 2 = 0.003362 a13 = ~ ° - 6 1 2 9 2 a.. = 0.72708 14 a 5 = -26.650 Also the third approximate s t a b i l i t y region boundary i s obtained by computing a = min {U 3(x)/U 3(x) =0, except the origin} where U (x) = U 2(x) + ^ ( x ) which gives a c r i t i c a l clearing time of 0.083 sec. The actual c r i t i c a l clearing time i s calculated by integrating the system equations using the Runge-Kutta method and is found to be at 0.5 sec. Figures4.2.1, 4.2.2 and 4.2.3 show- the function V(x) = Y . for the f i r s t , second and third max approximations respectively plotted i n the three dimensional space X l 5 x 2 and X . Fig. 4.2.1 First Approximation of Stability Region N3 Fig. 4.2.2 Second Apprxoimation of Sta b i l i t y Region Co Fig. 4.2.3 Third Approximation of S t a b i l i t y Region 3 45 CONCLUSIONS Two methods for constructing Liapunov functions have been applied to study the transient s t a b i l i t y of power systems. In Chapter I Willems' method was applied to a three machine system in which each machine was represented by a second order nonlinear differential equation. The optimum distribution of damping ratios among different machines in a multimachine power system was investigated in Chapter II by maximizing the hypervolume enclosing by quadratic part of the Liapunov function. Governor action was included ..in- the representation of the power systems in Chapter III and Willems' method was extended to enable construction of Liapunov functions for such systems. The effect of f i e l d flux decay is considered i n Chapter IV and Chen's method was employed to construct a Liapunov function for a third order model of a single machine connected to an i n f i n i t e bus. From these studies i t is concluded that: 1. The efficiency of Willems' method, based on the generalized Popov criterion, i s not affected by the number of machines included i n the power system studied nor by the introduction of a governor 2. For a maximum region of s t a b i l i t y , the damping ratios of a l l the machines should be equal. 3. Willems' method cannot be applied when the effects of flux decay are included. 4. Chen's method is applicable when power systems are represented in detail but i t yields very restrictive results unless a large number of successive approximations is performed. It is also shown that the s t a b i l i t y region estimated using this method does not increase monotonically with the number of approximations . 46 REFERENCES 1. V.M. Popov "Absolute Stability of Nonlinear Systems of Automatic Control", Avt i telemekh 22, 961-979 (1961) and automatic and remote control vol. 22 No. 8, March 1962 pp. 857-875. 2. S. Lefschetz, "Stability of Nonlinear Control Systems" Academic Press, New York, 1965. 3. R.E. Kalman, J.E. Bertram,., "Control System Analysis and Control Via the Second Method of Liapunov", ASME trans,,J. of Basic Engineering, June 1960 pp. 371-393. 4. R.E. Kalman, "Liapunov Function for the Problem of Lure in Automatic Control", Proc. Nat. Acad. Sci. US, 49, 2, 1963 pp. 201-205. 5. J.A. Walker and N.H. McClamrock, "Finite Regions of Attraction for Problem of Lure 1, Int. J. Control, London, vol. 6, October 1967 No. 4 pp.331-336. 6. B.D.O. Anderson, "Stability of Control System With Multiple Non-li n e a r i t i e s " , J. Franklin. Inst, vol 282, No. 3, September 1966 pp. 155-160. 7. B.D.O. Anderson, "A System Theory Criterion for Positive Real Matrices", SIAM. J. 1967, 5, pp. 171-182. 8. Moore J.B. and B.D.O. Anderson, "A Generalization of the Popov Criterion", J. Franklin Inst. 1968, 285 pp. 488-492. 9. J.L. Willems and J.C. Willems, "The Application of Liapunov Methods to the Computation of Transient Stability Regions for Multimachine Power Systems", IEEE Trans, on Power Apparatus and Systems Vol. PAS-89 No. 5/6, May/June 19 70. 10. J.L. Willems, "Optimum Liapunov Functions and Stability Regions for Multimachine Power Systems", Proc. IEEE,vol 117, No. 3 March 1970. 47 11. C.S. Chen, E. Kinnen, '."Construction of Liapunov Function", J. Franklin Inst, v ol 289, No. 2, February 1970, pp. 133-146. 12. E. Kinnen and C.S. Chen, "Liapunov Functions Derived From Auxiliary Exact Differential Equations", Automatica, vol. 4, pp. 195-204, 1968. 13. G.E. Gless, "Direct Method of Liapunov Applied to Transient Power System Stability", IEEE Trans, on Power Apparatus and Systems, vol. PAS-85 No. 2 February 1966 pp. 158-168. 14. A.H. El-Abiad, K. Nagappan, "Transient Stability Regions for Multi-machine Power Systems", ibid. pp. 169-179. 15. Y.N. Yu, K. Vongsuriya, "Nonlinear Power System Stability Study by Liapunov Function and Zubov's Method", IEEE Trans. .on Power Apparatus and Systems vol. PAS 86, No. 12 December 1967 pp. 1480-1685. 16. M.W. Siddique, "Transient Stability of an A.C. Generator by Liapunov's Direct Method", Int. J. of Control svol. 8, No. 2, 1968, pp. 131-144. 17. N.D. Rao, "Routh-Hurwitz Conditions and Liapunov Methods for the Transient Stability Problem", Proc. IEEE„ April 1969 pp. 537-547. 18. M.A. Pai, M.A. Mohan, J.G. Rao, "Power System Transient Stability Regions Using Popov's Method", IEEE Summer Meeting, Dallas, Texas, June 1969. 19. N.D. Fao, A.K. Desarkar, "Analysis of a Third Order Nonlinear Power System Stability Problem Through the Second Method of Liapunov", IEEE Winter Power Meeting, New York, January 1970 20. M.L. Cartwright, "On the Stability of Solutions of Certain Differential Equations of Fourth Order", Quart. J. Mech. Appl. Math 1956, 9, (2). 21. D.G. Schultz and G.E. Gibson, "The Variable Gradient Method for Generating Liapunov Function", AIEE trans. On Automatic Control, September 1962. 48 22. A.A. Metwally, "Power System Stability by Szego's Method and a Maximized Liapunov Function", M.A.Sc. Thesis El e c t r i c a l Eng. U.B.C., 1970. 

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