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 , C a i r o , 1967 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n t h e Department o f Electrical We a c c e p t t h i s t h e s i s as conforming to t h e required Research Engineering standard Supervisor Members o f Committee Head o f Department Members o f the Department o f Electrical THE UNIVERSITY Engineering OF BRITISH COLUMBIA J u l y , 19 7.1 In presenting this thesis an a d v a n c e d d e g r e e a t the L i b r a r y I for scholarly by h i s of this written the U n i v e r s i t y s h a l l make f u r t h e r agree that permission of p u r p o s e s may be g r a n t e d It requirements B r i t i s h Columbia, is understood of Columbia I agree r e f e r e n c e and this that copying or not for that study. thesis by t h e Head o f my D e p a r t m e n t for financial gain shall The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada the for extensive copying of permission. Department fulfilment of it freely available for representatives. thesis in p a r t i a l or publication be a l l o w e d w i t h o u t my ABSTRACT The transient s t a b i l i t y of power systems i s investigated using Liapunov's direct method. Willems' method i s applied to three-and four- machine power systems with the e f f e c t of damping included. The d i s t r i b u t i o n of damping among the machines of a multi-machine system i s studied, and optimum r a t i o s are derived. An extension of Willems' method i s used to include governor action i n the system representation. F i n a l l y , the e f f e c t of flux decay on s t a b i l i t y regions i s studied using Chen's method. ii TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i LIST OF ILLUSTRATIONS v ACKNOWLEDGEMENT vi NOMENCLATURE v i i INTRODUCTION .1 CH. 1: .4 CH. 2: CH. 3: CH. 4: GENERALIZED POPOV'S CRITERION AND WILLEMS' METHOD 1.1 G e n e r a l i z e d Popov's C r i t e r i o n 4 1.2 W i l l e m s ' Method 7 1.3 Stability Regions 12 1.4 N u m e r i c a l Example 12 OPTIMUM DISTRIBUTION OF DAMPING FOR MAXIMUM TRANSIENT STABILITY REGION 18 2.1 System E q u a t i o n s 18 2.2 C o n s t r u c t i o n of Liapunov Function 20 2.3 N u m e r i c a l Example 20 2.4 Optimum 22 Damping D i s t r i b u t i o n EXTENSION OF WILLEMS' METHOD TO INCLUDE GOVERNOR ACTION.. 26 3.1 System E q u a t i o n s 26 3.2 C o n s t r u c t i o n o f Liapunov Function 3. 3 N u m e r i c a l Example 3.4 C o n c l u d i n g Remarks .. 27 ... 29 30 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 4.3 System E q u a t i o n s Regions 34 36 iii Page 4.4 Construction of Liapunov Function 36 4.5 Numerical Example 37 4.5.1 The F i r s t Approximation 39 4.5.2 The Second Approximation 40 4.5.3 The Third Approximation 40 CONCLUSIONS REFERENCES 45 - ' iv 46 LIST OF ILLUSTRATIONS Figure 1.1 1.2 Page A u t o m a t i c Feedback C o n t r o l System C o n t a i n i n g Memoryless N o n l i n e a r i t y Single 5 N o n l i n e a r i t y C o n f i n e d t o a S e c t o r o f the F i r s t and T h i r d Quadrants 5 1.3 Automatic Feedback C o n t r o l System With M u l t i l i n e a r i t y .... 6 1.4 A u t o m a t i c Feedback C o n t r o l System With M u l t i l i n e a r i t y .... 6 1.5 A Three-Machine Power System 1.6 S t a b i l i t y Region V = V 2.1 A Four-Machine Power System 2.2 S t a b i l i t y region V = V 4.1 S i n g l e M a c h i n e - I n f i n i t e Bus 38 4.2.1 F i r s t A p p r o x i m a t i o n o f S t a b i l i t y Region 42 4.2.2 Second A p p r o x i m a t i o n o f S t a b i l i t y Region 43 4.2.3 T h i r d A p p r o x i m a t i o n o f S t a b i l i t y Region 44 m m 13 for X ± 17 21 for X ± v = X,, = Xg = 0 = X 2 = = X 4 = 0 23 ACKNOWLEDGEMENT I wish to express my deep gratitude to Dr. M.S. Davies, my supervisor, f o r h i s 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 f i n a n c i a l support from the National Research Council i s g r a t e f u l l y acknowledged. vi NOMENCLATURE x Vector of state v a r i a b l e 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 i n the case of multimachine systems Steady state value of 6 6 o 6 Value of 6 at the unstable equilibrum p o s i t i o n U I n e r t i a constant i n KW -Sec/KVA H H/(TTf) M System frequency = 60c/s f Damping c o e f f i c i e n t a a/M, Relative damping constant of synchronous machine R Mechanical power input to synchronous machine P m P e E l e c t r i c a l power output of synchronous machine Instantaneous voltage proportional to f i e l d flux of E' q synchronous machine Eg Voltage of i n f i n i t e bus E Steady state i n t e r n a l 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 e Reactance of transmission l i n e vii Synchronous reactance of synchronous machine Open c i r c u i t transient time constant of synchronous machine (X e (X d + X,)/T' (X + X') d o e d - X')E d B /T'(X o e o + X') d The n u l l matrix The unit matrix Laplace operator Product of three matrices, X and Y are n x n matrices and 1 i s an n x n matrix with a l l elements equal to 1. viii INTRODUCTION Since the early days of a.c. e l e c t r i c power generation and u t i l i z a t i o n , o s c i l l a t i o n s of power flow between synchronous machines have been known to be present. The p o s s i b l i t y of such o s c i l l a t i o n s 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 c h a r a c t e r i s t i c s of a power system during transient disturbances may be assessed from i t s mathematical model: set a of nonlinear d i f f e r e n t i a l equations, known as the swing equations. These equations describe the power system dynamics, their order depending on the d e t a i l of representation used for the synchronous machines and associated control apparatus. Several methods are available f o r 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 c r i t e r i o n or the phase plane method may be used. When the study involves a large number of machines or when i t i s 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 d i f f e r e n t i a l equations u n t i l the c r i t i c a l switching time i s found. Such a method i s cumbersome and very costly since an almost p r o h i b i t i v e amount of computation i s required i n i t s execution. Thus the need increases for the development of more d i r e c t 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 f o r the power systems has been found useful and s t r a i g h t 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 s w i t c h i n g time can be o b t a i n e d by c a r r y i n g o u t o n l y one forward i n t e g r a t i o n o f t h e swing The is difficulty that i n general function. there equations. i n the a p p l i c a t i o n o f Liapunov's d i r e c t method i s no o b v i o u s way to choose a s u i t a b l e L i a p u n o v I n many cases i n v o l v i n g a p h y s i c a l (mechanical o r e l e c t r i c a l ) system t h e energy s t o r e d i n the system appears to be a n a t u r a l Gless in all [13] s t u d i e d 1~, 2-, and 3- machine systems r e p r e s e n t i n g the s i m p l e form o f a constant voltage [14] c o n s i d e r e d l o s s e s and constant Siddique field t h e machines b e h i n d synchronous r e a c t a n c e , l o s s e s , damping, f l u x d e c a y i n g and c o n s i d e r i n g •- and Nagappan candidate. a constant input. neglecting El-Abiad a multi-machine system i n c l u d i n g I n t h e i r model damping. [16] c o n s i d e r s a s i n g l e machine system t a k i n g i n t o account decrement and s i m p l i f i e d governor and r e g u l a t o r Other a p p l i c a t i o n s were made u s i n g p r o c e d u r e s , Yu and V o n g s u r i y a action. formalized construction [15] employed Zubov's method t o develop a L i a p u n o v f u n c t i o n f o r one machine i n f i n i t e bus system u s i n g 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 f u n c t i o n o f the a n g u l a r d i s p l a c e m e n t o f t h e machine. [20] p r o c e d u r e t o c o n s t r u c t a V-function f o r a s i n g l e machine t a k i n g account the t r a n s i e n t s a l i e n c y e f f e c t , a c o n s t a n t governor a c t i o n r e p r e s e n t e d Rao [17] used - C a r t w r i g h t ' s into damping f a c t o r and a by a s i n g l e time c o n s t a n t . Rao a l s o 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 g r a d i e n t method [21] was a p p l i e d by Rao and D e s a r k a r [19] to a one-machine system i n c l u d i n g the e f f e c t o f the f i e l d - f l u x l i n k a g e changes. Pai, Mohan and Rao [18] a p p l i e d Popov's theorem on the a b s o l u t e s t a b i l i t y of nonlinear systems u s i n g Kalman's p r o c e d u r e [4] to c o n s t r u c t a L u r e - t y p e Liapunov f u n c t i o n f o r a one machine system w i t h and w i t h o u t governor a c t i o n . The g e n e r a l i z e d Popov c r i t e r i o n [8] f o r m u l t i v a r i a b l e 3 feedback systems was function used by J.L. W i l l e m s . [ 9 , 10] to develop a L i a p u n o v f o r n-machine power system. In t h i s t h e s i s the s t a b i l i t y machine power systems construct o f s i n g l e - m a c h i n e as w e l l as m u l t i - i s i n v e s t i g a t e d u s i n g two s u i t a b l e Liapunov f u n c t i o n s . d i f f e r e n t p r o c e d u r e s to In Chapter I W i l l e m s ' method i s applied to a t h r e e machine power system t a k i n g i n t o account the damping effect. A f o u r machine system i s c o n s i d e r e d i n Chapter I I and the b e s t d i s t r i b u t i o n o f damping r a t i o s i s o b t a i n e d by maximizing the e n c l o s e d by the Liapunov f u n c t i o n . hypervolume W i l l e m s ' method i s extended i n Chapter I I I to study a three.machine system i n c l u d i n g governor a c t i o n . In Chapter IV Chen's method i s a p p l i e d to a s i n g l e machine i n f i n i t e - b u s system into account the decay i n f i e l d flux linkage. taking 4 CHAPTER I GENERALIZED POPOV'S CRITERION AND WILLEMS METHOD 1 The s t a b i l i t y study of automatic feedback control systems containing single memoryless n o n l i n e a r i t i e s , figure 1.1, was i n i t i a t e d by Lure. and t h i r d Normally the n o n l i n e a r i t y i s confined to a sector of the f i r s t quadrants as shown i n figure 1.2. Popov [1] made a most important contribution to the problem by giving s u f f i c i e n t conditions f o r absolute s t a b i l i t y which are completely dependent on the frequency response of the l i n e a r part of the system. A procedure for constructing Liapunov functions, f o r such systems was introduced by Kalman [4]. Recently Anderson [6], [8] developed a theorem generalizing Popov's c r i t e r i o n and Kalman's procedure to investigate the s t a b i l i t y of feedback control systems containing more than one n o n l i n e a r i t y . The theorem relates the concept of a p o s i t i v e r e a l matrix to the concept of minimal r e a l i z a t i o n of a matrix of transfer functions [7]. Liapunov functions based on Anderson's theorem were constructed by Willems [10] f o r 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 C r i t e r i o n [8] Automatic feedback control systems with m u l t i - n o n l i n e a r i t i e s , figure 1.3 and figure 1.4, can be descirbed mathematically i n state / variable form by x = Ax - Bf(E) (1.1) e = Cx where x n vector e m vector 5 r=o F i g . 1.1 N-L KS) A u t o m a t i c Feedback C o n t r o l G(s) System C o n t a i n i n g S i n g l e Memoryless Nonlinearity f(S) Fig. 1.2 Nonlinearity C o n f i n e d to a S e c t o r Quadrant o f the F i r s t and T h i r d 6 Fig. 1.3 Automatic Feedback C o n t r o l System With Multi-nonlinearity iws) N.L A F i g . 1.4 Automatic Feedback C o n t r o l System With Multi-nonlinearity 7 A n x n asymptotically stable matrix B n x m matrix C m x n matrix f(e) m vector s a t i s f y i n g the sector condition 0 < f . (E.) < k . — 1 f (0) i 1 1 2 E 1 =0 i = 1,2, ,m Theorem [6] If there exist r e a l 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 is + (N + Qs) W(s) a p o s i t i v e real matrix where W(s) = C ( s l - A) ^B (n x n) matrix of stable r a t i o n a l transfer function and W (°°) = 0 then the system i s stable. (1.1) can be determined by the Lure type Liapunov rn Cx <V V(x,e) = x Px + 2Q / f ( e ) de 0 where P i s a p o s i t i v e d e f i n i t e symmetric matrix The s t a b i l i t y of system function (1.2) determined T T PA + A P = -LL PB = C N - L W Q + A C Q T W W T Q 0 T (1.3) T = 2NK + QCB + B C Q T T where L, WQ are a u x i l i a r y matrices of order ( n x n ) , 1.2 (n x m). Willems' Method [10] Willems applied the above technique to estimate the transient s t a b i l i t y regions f o r multimachine power systems. 8 Assuming that 1. The flux linkages are constant during the transient period 2. The damping power i s proportional to the s l i p v e l o c i t y 3. The mechanical power inputs to the machines are 4. Armature and transmission l i n e The d i f f e r e n t i a l equations describing the motion of the machines can be resistances are constant neglected, put i n the form d 6. M. — — i ,2 dt 2 d 6. + a, -r~I dt + P . - P . = 0 ei mi with P . = G.E + E ei i i , i = 1,2 E.E.Y.. s i n (6. - &*) i ] l] I J 2 where for = 1 n , (1.4) i = 1,2, n E. = i n t e r n a l voltage of the i t h machine i G. = l o c a l load conductance l Y., At = transfer admittance between the i t h and the i t h machine equilibrum d 6. dt Let - x = 1 d 6. 2 = w. = 0 , i " —=i ' dt 2 = d). = 0, I ' P . = P ml el 2n vector where to, a are column vectors with components to = [to^, u>2 > • • • • a = [a^, o > 6 n n •••• a ] 2 o, — 6T 1 1 1 ti) ] , o, = 6„ - 6„ , .... 2 2 2 a =6 n - 6° n n Although the state v a r i a b l e vector x has 2n components the actual order of the system i s (2n - 1) since only the differences between the rotor angles appear i n the system equations. 9 Let M = diag (MJ R = diag D =. an (-cO (m x n) m a t r i x such that (n x n) matrix (nxn) matrix the v e c t o r e = Da (1.5) has i t s components - - o> 2 n 2 ~ °1 ~ °3' z 2 3 n+1 2 a n-1 4' ' 1 m n-1 n n - " ( - > m n where 1 m D e f i n e the f u n c t i o n f ( e ) as f i. ( ie . ) = E p E q Y pq ( s i n (e. - s i n e?) i + e?) i I 1 ,• _ i,z,...m i o p, q are the i n d i c e s o f the component o f a on which c e? l be A = the v a l u e o f M R 0 1 I B = M -1 nn 0 n D e f o r 6. = 26? and i i (2n x 2n) Where wnere m (1.6) i s dependent. Let d e f i n e the m a t r i c e s A, B and C as matrix nn T (2n x m) matrix (m x 2n) matrix (1.7) nm L The D mn differential e q u a t i o r s (1.4) become e q u i v a l e n t to x = Ax - Bf'(e) e = The form (1.1) Cx s t a b i l i t y o f system (1.1) i s determined by a L i a p u n o v f u n c t i o n o f the (1.2). The time d e r i v a t i v e V i s g i v e n by V = -(x L - f(Cx) W ) T T T Q ( L x - W f(Cx)) T Q 2x C Nf(Cx) T T (1.8) 10 The next step i s to f i n d the.matrix P of equation (1.2). Since by d e f i n i t i o n T T CB = B C = 0mm and choosing N = 0 mm m then s u b s t i t u t i n g i n equation (1.3) results i n W i) = 0 0 (1.9a) mm PA + A P = -LL T PB = ii) T T AC Z(s) = sC(sI - A) are iii) (1.9b) (1.9c) -1 B i s p o s i t i v e r e a l i f a l l the damping constants nonnegative equation (1.8) reduces to T T V - -x LL x Let which i s negative semidefinite P = (1.10) where P^, P P^ are (n x n) square matrices. 2> Thus equation (1.9c) i s equivalent to P M P.M 2 - I T T D = D -1 T D = 0 (1.11a) nm (1.11b) and from the negative semidefinitness of PM PA + A P = X 1 R + RM P 1 P M R + _1 2 P 1 + P + P 2 3 -1 T KM T P 2 + P 3 nn we get P Since matrix 3 = -P M 2 -1 R = -RM contains -1 P m = ^~ T (1.12) 2 —^ 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 = u l where y i s a scalar constant and 1 i s an (n x n) matrix with a l l elements equal to 1. Applying the above reasoning to (1.11a) results i n -1 -1 -1 T (M P.M - M )D = 0 1 nm P 1 = M + yMlM (1.13) which i s p o s i t i v e d e f i n i t e i f y >\i where y i s the solution of the 1 o o det,/M + u M1M/ = 0 o -1 1=1 (1.14) 1 from equations (1.11b) and (1.12) -1 R PR 3 0 -1 T D = 0 nm and hence P 3 = yRlR P 2 = -yRIM (1.15) where y i s a scalar constant and i s taken equal to zero P = hence 0 nn 0 nn (1.16) 0 nn The matrix PA + A P i s negative semidefinite i f , and only i f , the matrix Z(y) = 2R + y(MlR + RIM) i s negative semidefinite Z(y) i s negative semidefinite f o r certain values of y (1.17) where u.. i s the solution of the det | z ( u ) | = 0 which i s equivalent to n n 7 (M. J i=l j=i+l 4 I - M. l a. n - u(I M ) - 1 i=l i (1.18) Equation (1.18) has a p o s i t i v e and a negative solution f o r y, the negative 12 one being u Substituting the value of P i n equation T T V(x) = to Mco +y .to MIMco + 2/ C x f(e) 0 T (1.2) we obtain de (1.19) with i t s d e r i v a t i v e V(x) = 2coRio + 2uco M1R T 1.3 (1.20) T S t a b i l i t y Regions Since the derivative of the Liapunov function i s 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, a:v(x) _ n (1.21) n 3.6. x The f i r s t equation gives co 1 = to z =....= to =0. n The second equation gives the closest equilibrum state (necessarily unstable) x . U to the origin The region bounded by the closed surface V(x) = V(x ) U and containing the o r i g i n i s a stable region. 1.4 Numerical Example Consider the three machine system The d i f f e r e n t i a l equations shown i n figure 1.5. describing the motion of the system are 13 F i g . 1.5 A Three-Machine Power System 14 d6 dS 2 M + 1 dt K 2 1 dr d 6, + el P = P m l d6, 2 2 ~1~ A dt + P dT 2 + (1.22) e2 " m2 d6 M + 3 dt ' 2 a 3 l T + e 3 P = P m 3 Let the s t a t e v a r i a b l e vector be dS. x = ( dt d6. d6. ' dt o ' dt S T Following the steps d e s c r i b e d i n s e c t i o n 1.2, the system equations(1.22) become x' = Ax - Bf(e) e = Cx where r A = — a l 0 \ 0 a 0 0 B = ? ~ M 0 0 0 0 0 0 0 0 0 0 0 2 J "M 3 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1_ 1_ 1 M M 2 0 0 "M C = 2 1 3 1 "M 3 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 1 0 -1 0 0 0 0 1 -1 15 w = f (e ) = 2 2 E 1 2 12 ( s l n E 1 3 13 ( s l n E Y E Y f (c ) - E E Y 3 2 3 with 3 u l u +E) X e 2 = 4 " 6 X X e o l -< o 2 0 e 3 = 5 " 6 X X = 2> =3> 3 e X - sin 2 + e ) - sin 3 = 4 " 5 sin " } 2 (sin ( e 2 3 E 1 + e -«2 "I 3 - 5 -«3 2 The L i a p u n o v f u n c t i o n i s then g i v e n by V(x) = M^-rM X 2 + 2[E E Y 1 2 1 2 + M X 2 3 (cos e 3 + [M X _ + M X u 1 ] 2 ° - cos ( X ~ X 4 + E E Y 1 3 ( c o s ° - cos (X^ " + E E Y 2 3 ( c o s e° - cos (X - X 1 2 3 3 x e 6 5 1 = 1.174 [22.64° p p.u.. E 2 = 0.996 12.61 E 3 = 1.06 [-11.36° 0 p . . + M ^ ] 2 + e°) - % - s i n e°) + e°) - (X^ - X ) s i n ° ) g g The system s t u d i e d has the f o l l o w i n g E 2 + °) - ( x e $ e - X ) s i n e°) ] g (1.23) data Pir^ = 0.8 . . u p . . u H-. = 3 KW. sec/KVA l — H 2 — u Pm 2 = 0.3 . . Pm 3 = -l.lp. . p u u a 1 = 10 Y, _ = 1.13375 p.u. 12 = 7 Y = 0.52532 p.u. 13 a 2 H = 7 KW. sec/KVA M 2 a ~r = 3 = 8 KW. sec/KVA M 3 circuit The u n s t a b l e e q u i l i b r u m s t a t e n e a r e s t X. - X, = 2.95275 r a d 4 6 X 1 = X 2 = X 3 = 0 '° = 3.11850 p.u. t o ground o c c u r s on the t r a n s - c o n n e c t i n g machines 2 and 3 o f F i g u r e found to be a t X, - X, = 2.61168 r a d 4 5 0 0 23 3 A sudden 3-phase s y m m e t r i c a l s h o r t mission l i n e Y t o the o r i g i n (1.5) c l o s e to bus 3. i s c a l c u l a t e d and was 16 with V m = 3.36. The c r i t i c a l c l e a r i n g time o b t a i n e d from the above V - f u n c t i o n was to be between 14-15 Figure found cycles. (1.6) shows the f u n c t i o n V= V p l o t t e d i n the two dimensions m (X. - X ) , (X. - X,) f o r X = X„ = X„ = 0. 4 5 4 b 1 2 3 The a c t u a l critical c c l e a r i n g time o b t a i n e d from forward i n t e g r a t i o n o f the swing e q u a t i o n s u s i n g Runge K u t t a method i s 20 c y c l e s . 18 CHAPTER II OPTIMUM DISTRIBUTION OF DAMPING FOR MAXIMUM TRANSIENT STABILITY REGION In this Chapter Willems' method described i n Chapter I i s applied to a four machine power system and a study i s made to f i n d the optimum d i s t r i b u t i o n of damping that maximizesthe region of s t a b i l i t y . 2.1 System Equations Under the same assumptions made i n Chapter I the system equations are ~ d 6 M: —-ri ,2 dt 6 d + a .— I dt + P .. = P . ei mi i = 1,2,3,4 ' (2.1) where A P e2 = A e3 = <i "V 6 sin el " l P + sin l A A V + A 4 sin <4 e4 = 3 - y + A 5 sin 2 5 A S <1 - 6 - 5 ) + 6 sin 4 - A sin 2 <3 P P + < 6 2 ( 5 3 sin (5 4 3 A 5 r ) + y sin A - v + A 6 sin 1= 1 2 12 A E E Y A 4 " 2 3 23 E E Y A 2 5 = E sin v + 6 sin <4 - - 5 A = E Y A 2 4 24 E V (2.2) 1 3 13 E "V. -V sin and A <-\"V Y A 3 6 = 1 4 14 = E E E E 3 Y 4 34 Y Following the same steps described i n section 1.2 to represent (2.1) i n the form x = Ax - Bf(e) (2.3) e = Cx the results are x = e < U 03 l = (6 - 6 E = (x 1 5 co 2 - x CJ 3 2 6 6 x -x X 5 -6 ? 4 3 6 1 -6° <5 -6° 6 3 -6° 5 1 -6 6 6 2 -6 x - x 5 g 4 2 2 -63 x - x 6 ? x -x 6 -6°) 4 63 - S ^ 8 x -x ) ? 8 T ( ( T 2 2 4 5 ) ) (2.6) 19 A.(sin ( e . + e ? ) - s i n e ° ) A = 0 M (2.7) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 l 0 M„ 0 a. 0 0 C= 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0. 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 1 0 -1 0 0 0 0 0 1 0 0 -1 0 0 0 0 0 1 -1 0 0 0 0 0 0 1 0 -1 0 0 0 0 0 0 1 -1 1_ 1_ M, rr- 0 0 0 - zr- 0 1_ 0 0 77- • 0 1_ 1_ - rr0 0 1_ M, -1_ - M ^ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (2.8) 20 2.2 Construction of Liapunov Function According to the expression for P given i n section 1.2 nn nn nn where P. = M + U M 1 M r M 2 2 +pM yM M 3 yM M 4 1 1 2 M 2 +yM 2 y MM uKjM 2 yM M 2 2 M 3 yM^ 3 2 +yM 2 yM M 3 .P M M 2 3 4 2 2 M. +yM. 4 4 4 The Liapunov function i s then given by • vl vl+ M £< + +2 [A^(cos - cos ( X h M 4 4 X 2 5 " 6 X + ± - - <5 X + e) (x 5 +A (cos e' ~ COS ( x 5 " 8 + E ) - (x 5 3 +A^(cos e,.- COS 6 6 X X (X 6 " 7 ( X 6 " 8 X +A .(cos E - COS (X X 2 3 + + e 4 } " e) 5 ^ 6> 0 ? " 8 X 2.3 Numerical Example The system chosen as an example i s shown i n Figure 2.1. = 1.004 |0.0013 rad p.u. = 1.0410 10.103 rad p.u. = 1.1900 J0.197 rad p.u. E. = 1.070 0.0772 rad 4 ' p.u. E e) + " 7 2 +A, (cos e. - COS 4 4 2 + 1 5 3 E • u [M^X +A (cos e - COS ( x 2 E 4 + E " < X 6 ( X 6 (X ? M 2 2 X M X 3 - Vs i n - Vs i n - Vs i n -v s i n - Vs i n - Vs i n + M X ) 3 4 £ 1 £ 2 2 4 } ) e) 3 Z °\ l? e )] 6 (2.9) 22 P mi = °- 3 3 2 p - - \ u = 75350 P.u. l D = 1 , 0 P ' " u P^ = 0.1 P.u. M 2 = 1130 P.u. D 2 = 12.0 P.u. P =0.3 M 3 = 2260 P.u. D 3 = 2.5 P.u. m 3 P.u. P„, = 0.2 P.u. M, = 1508 P.u. A sudden .3-phase symmetrical short c i r c u i t D, = 6.0 P.u. . to ground o c c u r s c l o s e to bus 3 on.the t r a n s m i s s i o n l i n e The f o l l o w i n g t a b l e g i v e s the s t a b l e e q u i l i b r u m s t a t e o f the p o s t - f a u l t s y s t and the u n s t a b l e e q u i l i b r u m s t a t e c l o s e s t 6, Internal bus The exact was found c o n n e c t i n g machines 3 and 4 o f F i g u r e 2.1. t o the s t a b l e one. radians(stable) 6, radians 1 0.05630 0.06610 2 0.15013 0.20136 3 0.21430 3.0820 4 0.02497 -0.02425 critical (unstable) c l e a r i n g time was c a l c u l a t e d u s i n g Runge K u t t a and to be a t 30 c y c l e s . When c a l c u l a t i n g the v a l u e o f the L i a p u n o v f u n c t i o n a t the u n s t a b l e 1 e q u i l i b r u m s t a t e and X, = X_ = X„ = X, = 0 . 0 1 2 3 V^ = 3.155 which g i v e s a c l e a r i n g Figure, we o b t a i n a v a l u e f o r 4 time o f 25 c y c l e s . 2.2 shows the f u n c t i o n V(x) = V ^ p l o t t e d i n the t h r e e d i m e n s i o n a l space (X^ - X ^ ) , (X^ - X ^ ) , (X,.- Xg) w i t h X = X = X = X. = 0.0. 1 2 3 4 2.4 by by t h e components 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 o f damping r a t i o s ( — ) i s obtained a. i f i n d i n g the r e l a t i v e v a l u e s o f — to maximize the hypervolume e n c l o s e d i the Liapunov f u n c t i o n t h a t d e f i n e s the s t a b i l i t y r e g i o n o f t h e system. C o n s i d e r i n g the V - f u n c t i o n (2.9) f o r a f o u r machine system 23 Fig. 2.2 S t a b i l i t y Region V = V for X = X = X = X = 0 m 1 2 3 4 24 T V = to (M + yMlM)to + 2 f T f (e) de C x where y i s given by f u " Ak -4 = 0 2 R lU y which gives u = 2 ( ) R where MM F R " l1 V2 V ( l " R „ MM + OT 13 MM RJ*7 2 4 + k.. = M, + M 1 1 2 + M ( ( R V 2 " R MM V 1 " + < R T 14 MM 2 + ( R OT 3-V 34 ( R 1 " V MM + R ¥ 2 " 23 ( R V ? 2 + M 4 3 I t i s n o t i c e d that the i n t e g r a l part of V does not depend on the damping c o e 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] H = TT 2 V p 2 /2/\. q 1 where T = to Aid V q A = M + y M1M |A| = determinant of matrix A = k ( l + y k^) 2 k = MMMM 2 1 2 3 4 V = N yN 2 2 2 2 N.. = M, to, + M oi + M_to + M.to, 1 11 22 33 44 ( M ^ + M u + M a) + M O J N, q 1 + 2 0 2 Thus H p 2 = JL_ ( 4 2 + yN ) V A N l 2 2 4 ) 2 +y k ^ For maximum volume " ' 3H 3 ^ - 0 1 i b U t 8H p 3R, i = 3H p 3y " 1.2,3, 4 • 3f 3y R ' 3f_ * 3R. R l ( 2 . 1 0 ) 25 Thus t o s a t i s f y equation (2.10) H £ 9u = 0 or J - = 0 3f 3f 77^ = 3R or B 1 K 3H i f 2 +u» ) k (N • t k <*i»* 2 which when e q u a t i n g N * = " T x 1 o ^ r " + 4 N 2 " k l V t o zero g i v e s - 4 N k N 2 1 = 1,2,3,4 0 3k 1 2 l N o Both answers a r e r e j e c t e d s i n c e t h e v a l u e o f u state variable Zu F R - 2 The t h i r d 3f ^ 3f R 3R„ 3 3 F M M 3 2 R„ 3 af l+ R F 2 + l k ( ^ i s that M ? ^ M 1R! - J 2 l -= A possibility M = ^ k , M „ \ " 2> „ M. ( l" ^ R ^ R M <" RT 1 depends on the v a l u e s o f M ( R 1 " 4 l ^2 R 1 1 2, M y 2 R ) } = ° 'JM ( R 3 2 " 3> ,2 R2 2 v R 17 + ( 3 - 4»= 4 R ° R 2, . 4 ^ 2 .2 3 R. 3 4 4 M y M M M R ,{- —1 ,J1 ^R.) 2 2 1 ,JL ^R,) 2 3 ,„2 -„2v (R, TT (Ro ~ " 7 2 R . l 4' R ' 2 4 R^ (R, 3 . 4P }= 0 R. 1 2 3 4 which g i v e s R 3R. 4 M M 4 IT + " 4 N R. = R 1 2 0 = R X 3 y v = R. 4 Thus f o r a maximum r e g i o n o f s t a b i l i t y the damping s h o u l d be e q u a l . ; ratios o f a l l machines 26 CHAPTER I I I EXTENSION OF WILLEMS In this 1 METHOD TO INCLUDE GOVERNOR ACTION c h a p t e r W i l l e m s ' c o n s t r u c t i o n procedure i s extended to develop a L i a p u n o v f u n c t i o n f o r m u l t i m a c h i n e power systems i n c l u d i n g governor a c t i o n . 3.1 System E q u a t i o n s Assuming t h a t f l u x l i n k a g e s damping power i s proportional be r e p r e s e n t e d are constant, to s l i p v e l o c i t y resistances are neglected, and governor response may by a s i n g l e time l a g t r a n s f e r f u n c t i o n , AP _m -K + T, 1 (3.1) The e q u a t i o n s o f the i t h machine a r e d6. dt M x ± 7dT dAP = " V i .' "HF where P , mi - " e i 1 P moi • m i - Y7 Pmoi i s the v a l u e [to , to,, and the m a t r i c e s B = U i 1 o f the m e c h a n i c a l i n p u t a t steady a) „ » AP,^' w,9» ,m2 AP 1 state. M Y Z 0 nn I 0 nn 0 nn _1 -1 T D nm nm A P „„> mn 5 i " 5 i > " 6 o> 5 „ ~ „] 6 o,T n (3.3) M R M (3.2) mi the v e c t o r X= A = A P I A P 1 Defining + K. " TT = + 1 0 nn C = 0 mn 0 mn D (3.4) 27 where Y = diag [- ] i Z = diag M, [- Tjr~ ] i R and D a r e as d e f i n e d i n c h a p t e r I . Equation (3.2) takes t h e form = Ax - B f ( e ) k (3.5) e = Cx 3.2 C o n s t r u c t i o n o f Liapunov F u n c t i o n A p p l y i n g Popov's g e n e r a l i z e d c r i t e r i o n , system if. (N + Qs)C ( I s - A) (3.5) i s s t a b l e i s p o s i t i v e r e a l matrix. Taking N = 0 and Q = I mm m then s C ( I s - A) "*"B i s p o s i t i v e r e a l i f t h e damping c o n s t a n t s a r e n o n n e g a t i v e . A s u i t a b l e L i a p u n o v f u n c t i o n f o r such a system i s Cx V = x Px + 2 / f ( e ) d£ o T where P i s determined PA + A P from the requirement that be n e g a t i v e s e m i d e f i n i t e and t h a t (3.6a) T T PB = A C Let P P P P T 12 T P 13 P 2 12 13 P Substituting 23 (3.7) 23 (3.7) i n t o - I T D P M P M 12 = -1 i n T D -1 T P M D 13 1 0 Substituting V^M (3.6b) (3.6b) g i v e s T D (3.8) = 0 nm 0 nm = f o r m a t r i x A i n e q u a t i o n (3.6a) + P 3+M R P + Y P + P 1 3 1 X P 12 " P 13 " M M l R + P l R + P 2 Y + P 23 23 Y + P 3 + M 1 " l p i + Z P 12 12 one has P M +P Z+M 1 L 1 r p 12 P 12 " 1 + P P 13 " 1 + P M M 2 Z + M 23 Z 1 " l p 2 i2 + Y P 2 T + Z P + P 2 23 M _ l R P M - I nn 13 + Y P T P 1 3 + 23 ZP T 2 3 (3.9) + P 3 28 Setting a l l o f f diagonal elements of (3.9) equal to zero give P . M - + 1 1. 13 M _ 1 r P P + *Z Y P P 1 + 1 Z P + 2 2 3 = 0 nn (3.10) = 0 nn (3.11) = o (3.12) + y M1M M = Y X P 1 3 = Y 2 P 3 2 ^ M1M M - Y 1 ^ Y (YZ = Y = a r 2 _ 1 _ 1 M 2 r*- P _1 - Y (M1MZ 1 + M RM1M) - y M l _1 I) 2 0 nn = J®~- 2 /-J-) ] -y E Mj_ - 1 = 0 " i i=l n - ^ M 0 nn 0 nn 0 nn + y M1M = - Y CiMl + I ) 1 2 (3.14) to be equal to zero, matrix P reduces to P _ l M R)M1M + "j 0 nn l 0 nn 0 nn P _ 1 e constant scalarsand y i s given by Choosing y^ and Y p M 1 - = I 2 [S S (M, i = l j=i+l * P ^ (Y Z- 1M 1 n (3.13) M 2 and Y where y = to give 2 P where 23 nn 1 P y P 3 P P ' n + 3 P ? 2 (3.8), (3.10), (3.11) and (3.12) are solved f o r P . , P „ , P „ , P , 12' and P 3 YP_ T 3 Equation P + 12 23 - I T M M ^ R P * + 12 Thus f o r a three machine system, the Liapunov function i s (3.15) 29 V(x) = M X + M X 2 1 T T 9 1 X X + 3 K ; X + M X 2 6 + ^ X + X 5 + 2E E Y 1 3 (cos e 2 - cos ( X - X + 2E E Y 2 3 (cos e 3 - cos ( X - X 1 2 2 3 V + 4 (cos e - cos ( X - X ± 3 3 M 1 2 2 .+ M X ) 2 2 2 + 2E E Y 1 3.3 T 2 K^ 3 + + u (ML^ 2 3 9 K^ 4 + + M X 2 2 ? g X 4 M + - k 2 T - X MT 2 2 5 3 3 l f + + e°) - ( X - X ) s i n e°) ? g + e°) - (X_, - X ) s i n e°) ? g ^ T 1 1 g + e°) - ( X - X ) s i n e°) g g g N u m e r i c a l Example The same n u m e r i c a l example o f Chapter I i s c o n s i d e r e d . With governor a c t i o n taken i n t o account the e q u a t i o n s d e s c r i b i n g the machine dynamics are • . • dfi. I - to. dt i dto. ~ - = -a.to. - P . + P . + AP . dt i x ei mox mx dAP K. mx x 1 di— " TT i " TT mi x x = u A P 1 = . _ 1 » » 2 3 System Data E. = 1.174 1 E 0 2 |22.64° ' I2.61 = 0.996 P.u. P.u. -f- = 10.0 M^ ~ = M 7.0 2 a E P P 0 3 = 1.006 1-11.36 mol P.u. —=3.0 M 3 = 0 . 8 p.u. r _ = 0.3 p.u. mo2 r T, = 0.2 s e c 1 Y,. = 1.13375 p.u. 12 T. = 0.22 s e c 2 Y, „ = 0.5232 p.u. 13 30 P. _ = -1.7p.u. mo3 T - K K ± 2 = K 3 = 0 3 = 0.25 sec Y„„ = 3.11856 P-u.. 23 0.0 The unstable equilibrum state close to the stable one i s calculated by s o l v i n g 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 1 = X 2 = X 3 = X 4 = X. = X. = 0.0 5 6 Equation (3.19) gives X., - X„ = 2.61168 rad / o X, - X with V m = = 2.95275 rad 3.36 The c r i t i c a l clearing time obtained from the above V-function was to be between 15~16 found 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 i s obvious from the material presented i n t h i s chapter that the generalized Popov's c r i t e r i o n can be successfully applied to power systems including governor action. On the other hand the same method f a i l e d when applied to a power system taking into account the flux decay i n the f i e l d c i r c u i t s of the synchronous machines. The reason behind this f a i l u r e i s that the n o n l i n e a r i t i e s introduced when considering flux decay are d i f f e r e n t i n form from those considered by Popov. Thus i t i s concluded that Popov's theorem i s applicable to higher order power systems as long as the n o n l i n e a r i t i e s r e t a i n the same form. 32 CHAPTER IV A LIAPUNOV FUNCTION FOR A POWER SYSTEM INCLUDING FLUX DECAY (CHEN'S METHOD) When including the e f f e c t of flux decay i n the f i e l d c i r c u i t of synchronous machines the power system can not be represented i n a suitable form f o r application of the generalized Popov's c r i t e r i o n . A new method, developed by Chen, based on the use of an a u x i l i a r y exact d i f f e r e n t i a l equation derived from the given nonlinear d i f f e r e n t i a l equation representing the system, i s applied i n this chapter. The method i s employed to construct a Liapunov function f o r a t h i r d order model of a synchronous machine connected to an i n f i n i t e bus with the e f f e c t s of flux decay i n 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 d i f f e r e n t i a l equations <*-D x = f(x) 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 p a r t i a l derivatives are defined and continuous i n some region Q of the state space E^ and the point x = 0 i s an equilibrum point also i n 11. Define 4-u then g. = f, + f . + l 1 2 n _ . + f. - f.^ l - l l+l f n (4.2) T which gives g x = 0 or g dx = 0 (4.3) Equation (4.3) i s said to be an exact d i f f e r e n t i a l equation i n Q i f there i s some single-valued d i f f e r e n t i a b l e function U(x) defined and continuous together with i t s f i r s t p a r t i a l derivatives i n some neighborhood point i n Q such that of every 33 dU(x) = g dU(x) • dx T „„, ,T . T = VU(x) • x = g • x —rr- (4.4) which results i n 3U(x) = 3 i x (4.5) S i i , j = 1,2, ..'..n 3X. 3X. i 3 and U(x) = / g c T which i s independent of U(x). • dx of any integration path C contained i n the domain Thus equation (4.5) i s a necessary and s u f f i c i e n t condition f o r the exactness of (4.3). exist. If equation (4.3) i s not exact the U(x) does not T A function h(x) can be added to g(x) such that (g + h) • dx i s an exact d i f f e r e n t i a l which pives dU(x) / . ,NT = (g + h) -x = (g dU(x) dt ~ h) + (4.6) • f T T h f For equation (4.6) to be an exact d i f f e r e n t i a l equation, then 3U(x) "3XT I = 8 3(g + h ) ± a n d j + ,, i h 3(g. + h.) ± rrr^- i = "gx 1 i» j = 1» 2, .. . .n (4.7) i The function U(x) can be evaluated by the l i n e i n t e g r a l U(x) = / (g + h ) C T dx (4.8) For an integration between l i m i t s o and x, (4.8) gives x U(x) - U(o) = / (g(y) + h ( y ) ) . dy o T (4.9) It remains then to select h(x) such that U(x) has the c h a r a c t e r i s t i c s of a Liapunov function. These are given by 34 a) (g + h) »x i s an exact d i f f e r e n t i a l 9(g. + h ) 3(g. + h ) ax. dU(x) dt T hf c) U(x) Let h l = i,j = ax. j b) dr i s negative 1,2, d e f i n i t e or semidefinite i s positive definite 6(g) + (4.10) ViK*) where 6(g) i s a known function of g(x). For n=3 3g-, 3go 38-, 3g / [ ( 6(g) = - /C axT 3^> 3 " ^ " ixf> " 6(g) i s given by J 3g, 3g. 9 d X 3g - /C 3X„ 2 3g 1 l c 3 ^ 3 " ix^ ) d X 3 ] dX 2 3 3X> d X 3 L (4.11) and i|)(x) i s a scalar function that has b and c are s a t i s f i e d . to be selected such that Substituting for h(x) from equation (4.10) gives X X U(x) = / (g(y) + h(y)) dy T U(x) = W(x) conditions X = / (g + 0(g) ) dy + / V<Ky) dy T T + ^(x) - lj, (0) If ;jj(x) i s chosen such that U(x) = W(x) (4.12)(0)= 0, then + i>(x) (4.13) conditions b and c can be restated as b) U(x) = W(x) + TJ;(X) i s negative c) U(x) = W(x) + (JJ(X) is positive definite where W(x) i s d i r e c t l y evaluated semidefinite from the system equation (4.1) with ^(x) serving as correction function to give U(x) i t s desired c h a r a c t e r i s t i c s . 4.2 Estimation of S t a b i l i t y Regions For locally stable systems a closed form solution for the 35 undetermined coefficients case the s t a b i l i t y in a series form. Thus W(x) In t h i s r e g i o n i s e s t i m a t e d by g e n e r a t i n g a L i a p u n o v function 62 where F j ( x ) expanding can be w r i t t e n = F .(x) + F W(x) I|J(X) does not e x i s t . form a f t e r o f the function- o 3 (x) + the system n o n l i n e a r i t i e s as the sum o f homogeneous p o l y n o m i a l s F ( o x (4.14) ) m i s a j t h degree homogenous p o l y n o m i a l . 0 into polynomial • Thus U(x) = F ( x ) + F ( X ) + .... F ^ x ) + <Kx) o 2 Q 3 and = - G U(x) where-G J o 2 (x) - i s also Q degree o f f ( x ) . G o 3 (x) ^ G m + S <K > + (4.15) . X a j t h degree homogeneous p o l y n o m i a l and s i s the h i g h e s t Then the f i n a l L i a p u n o v f u n c t i o n i s o b t a i n e d i n t h e f o l l o w i n g steps 1. Starting ^(x) . G 2. w i t h ip(x) = 0, check the p o s i t i v e I f both a r e p o s i t i v e definite I f F „ ( x ) and G „ ( x ) a r e n o t p o s i t i v e o2 o2 function IL(x) U x) definite, 1 2 1 3 (x) + F o G 1 ( m + s ) o f ij> (x) The unknown c o e f f i c i e n t s consider a quadratic coefficients, .(x) + .... F o3 .... of F n 2 = -G (x). - G l ( then U'(x) ij^(x) = F ^ ( x ) w i t h undetermined = [F (x) + F (x)] o2 12 1 (x) and o2 i s a Liapunov function. definitness thus (x) m (x) a r e determined from i) F 2(x) + F^ (x) i s positive definite ii) G^(x) i s positive definite 3. F o r a second a p p r o x i m a t i o n a homogeneous t h i r d o r d e r p o l y n o m i a l F ^ C x ) 0 is 2 added U (x) 0 2 U(x) -The 4. to U^(x) = [F (x) 2 u = -G to give ^ 1 2 lz (x) - G coefficients (x)] + [ F + F 2 3 oi (x) of F (x) + F „ . ( x ) ] + F 2 3 .... G (x) 2 ( n t f s ) Li (x) + .... F Q4 om (x) a r e determined by s e t t i n g F u r t h e r a p p r o x i m a t i o n can be made as r e q u i r e d . G 2 3 (x) = 0 .(x) 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 e f f e c t of flux decay i n the f i e l d X 1 X 2 4 = X3 X 2 m l n X = 6 - where X 'x-', 12 d ---X ( F ~ l 3 ~ = [16] are 1 9 n ( s-(X q c o s ( 2 6° X i + 6 °) " c o s 6 1+ 6°)) °) X = X = 6 z 1 (4.16) X 0 3 = E"' - E q In order to apply Chen's method,equations(4.16) are expanded as follows : Put = E /M(X h B K Thus + X'^ 12 = K E 2 1 X X = X 1 2 X 3 2 " 1 3 = K X S ± n ( X 1 " + = - T ^ X - n (cos(X 2 1 K ( s i n ( X 2 + 6°) 1 + " s i n 6 °> - cos 6 ° ) which when expanded gives X 1 = 2 X X 2 °° . °° = - E p xj- - X_ E 1=1 1=1 1 1 3 i - l q.X 1 1 (4.17) CO X, = 3 n,X_ 1 3 K 2 -7j 0 where P». . s m - E r . xj" . , i 1 1=1 ,fO , (6 + ITT. — ) l K q r 4.4 ± = n i l . ,.o ^ ( i - l ) . + - — ' - ir) T y s m (6 2 i — ! • /*° ^ (1 ~ 1) s + - = >2 — r — ' -" v , >) s m ^ (6 u T Construction of Liapunov Function Applying Chen's method to the system equations (4.17) gives (4.18) 37 g(x) E (p. r X + i=l + 3 J X q n i l X 1 ^ 3 + 1=1 (4.19) 1 i=l , n -E l r l X + Pi l 6(g) = 2 + ~ 3 X i=l X X n Z W(x) = -| X n+1 + E i=2 + 2 2X X 2 (2X r._ 2 - X ^i + The n )x 1 2 2 i 3 x + ^ i - i ) Example [16] data M = 147 x 1 0 ~ 4 P.u. X- ' = 0.3 P.u. d f = 1.03 P.u. .= 1.0 P.u. X e = 0.2 P.u. on base 25 MVA X, = 1.15 P.u. d To' I i l 3 " q X l r + i V X l _ 1 (4.20) + 1 r. Pi-1 , i=2 1 ^ ) r ( r -s , z xj x q i - i r (4.21) I system taken as an example i s shown i n F i g u r e 4.1 w i t h the 6° = 0.42 r a d q + 3 x 1 3 X " Y q E = 1.02 P.u. E' ) i=J Numerical following q 1 3 4.-1 + X„ ' "2 . i l i=l " ? , "«i i=l (l 4.5 l 3 X = 6.6 s e c 38 F i g . 4.1 S i n g l e Machine - I n f i n i t e Bus Power System 3° A three phase short c i r c u i t at the middle of one transmission c i r c u i t as shown i n Figure 4.1 occurs, the corresponHit-p-.V W( ) = | X 2X X 2 X + 2 - 3 ? q i X 2 + E C - . i=2 1 1 n+1 + E (2X„r. - 2X„p. ) X* . 2 l - l 3 l - l 1 i=2 1 2 0 - q ^ r ^ - 2)X - X 3 3 n+1 + 2X; E ( i - 1) i=2 n+1 + X 3 E ( l i-l q r i=2 9 + 2X E i=3 3 ( n i + 2 ( r i l ) *J 1 M E r i=2 2 X* 1 n+1 x{~ - X X 2 V - _ n+1 i-2 - X E q. . X. 3 . „ i-1 1 i=3 0 W(x) = - 2 n X X 1 and V-functions 'are 2 (2(1 - DP.., 3 q.^xj" 1 i=2 i-1 l " ^ i - l * - Dq.^ X^ X l - X X 2 3 E (1 - 2) q i=3 X^ 3 It i s seen that F-oZ „ and G'ol „ are not p o s i t i v e d e f i n i t e and successive r approximations are needed. 4.5.1 The F i r s t Approximation T A quadratic function F ^ W for { F ( x ) + and G F 12 2( ^ x 0 1 2 = X Ax i s added to W(x) . When solving ^ positive definite positive definite we get A = -1.45 -0.109 153 -0.109 -0.01 -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 i s obtained by c a l c u l a t i n g ct = min {U (x)/U (x) = 0, except the origin} 1 The c r i t i c a l 1 1 where IJ^Cx) = W(x) + F ^ x ) . clearing time i s calculated from the above V-function and was at 0.05 sec. 40 4.5.2 The Second Approximation A t h i r d order homogeneous polynomial ^.^Cx) "*" ^ * ^° U^( ) sa < F 23 ( X ) V l ' = + a 2 2 l X X + a 3 2 l X X + a 4 2 X 3+ a e c x 5 3 l X X 2 3 2 2 4- a^X^X^ + a^X^ + 3gX2X +.3^X2X3 + ^iQ^^^2 3 X 3 Solving f o r the unknown c o e f f i c i e n t s introduced by F2 (x) from the con3 dition G (x) = 0 23 gives a x = 0.1472 a 6 = 71.03 a 2 = -0.31947 a = 1.9328 a 3 -6 = 0.224456:. x 10 7 a 8 = 0.08046 = -0.042757 ,~--3 a. = -0.46473 x 10 9 4 a a = -15.752 5 a 10 = -0.04443 = The second approximate s t a b i l i t y region boundary i s obtained by c a l c u l a t i n g = min {U (x)/u" (x) = 0, except the origin} 2 2 where U^Cx) = U^(x) + F ( x ) 2 3 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 ) i s added to U (x) 2 F ( x ) = a^X + a X X 4 2 + a xJx X 6 + a + a 1 0 2 1 2 14 2 3 X + 2 + a X 3 X X X X + a ^ X j ^ a^X^ + 3 3 4 7 3 + a a ± / 3 l X + a^X -+ a X X 3 3 2 12 2 3 a^X 2 + .a^X^ a^x] + 15 3 X The unknown constants introduced by F ^(x) are calculated from the condition 3 G ( x ) = 0.0 34 41 which gives a 1 = -40.694 a, = 0.41651 a D a a 2 3 = 0.047175 = -118.09 a. = 0.64395 4 a 5 = -112.98 a ? = -0.40783 a g = -67.118 a 9 = 0.058045 a 1 Q = -0.56244 ll a a = 0 0 2 5 4 0 8 = 0.003362 1 2 13 -°- = ~°- 6 1 2 9 2 a.. = 0.72708 14 a 5 = -26.650 Also the t h i r d approximate s t a b i l i t y region boundary i s obtained by computing a where = min {U (x)/U (x) = 0 , except the origin} 3 3 U (x) = U (x) + ^ ( x ) 2 which gives a c r i t i c a l clearing time of 0.083 sec. The actual critical clearing time i s calculated by integrating the system equations using the Runge-Kutta method and i s found to be at 0.5 sec. Figures4.2.1, 4.2.2 and 4.2.3 show- the function V(x) = Y . f o r the f i r s t , second and t h i r d max approximations respectively plotted i n the three dimensional space X l 5 x 2 and X . N3 F i g . 4.2.1 F i r s t Approximation of S t a b i l i t y Region Fig. 4.2.2 Second Apprxoimation of S t a b i l i t y Region Co 3 F i g . 4.2.3 Third Approximation of S t a b i l i t y Region 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. method was In Chapter I Willems' applied to a three machine system i n which each machine was represented by a second order nonlinear d i f f e r e n t i a l equation. The optimum d i s t r i b u t i o n of damping r a t i o s among d i f f e r e n t machines i n a multimachine power system was investigated i n 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 i n Chapter III and Willems' method was for such systems. extended to enable construction of Liapunov functions The e f f e c t of f i e l d flux decay i s considered i n Chapter IV and Chen's method was employed to construct a Liapunov function for a t h i r d order model of a s i n g l e machine connected to an i n f i n i t e bus. From these studies i t i s concluded that: 1. The e f f i c i e n c y of Willems' method, based on the generalized Popov c r i t e r i o n , 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 r a t i o s of a l l the machines should be equal. 3. Willems' method cannot be applied when the e f f e c t s of f l u x decay are included. 4. Chen's method i s applicable when power systems are represented in detail but i t y i e l d s very r e s t r i c t i v e r e s u l t s unless a large number of successive approximations i s performed. It i s also shown that the s t a b i l i t y estimated using this method does not increase monotonically number of approximations . region with the 46 REFERENCES 1. V.M. Popov "Absolute S t a b i l i t y of Nonlinear Systems of Automatic Control", Avt i telemekh 22, 961-979 (1961) and automatic and remote control v o l . 22 No. 8, March 1962 pp. 857-875. 2. S. Lefschetz, " S t a b i l i t y of Nonlinear Control Systems" Academic Press, New York, 1965. 3. R.E. Kalman, J.E. Bertram,., "Control System Analysis and Control V i a the Second Method of Liapunov", ASME trans,,J. of Basic Engineering, June 1960 pp. 371-393. 4. R.E. Kalman, "Liapunov Function f o r the Problem of Lure i n Automatic Control", Proc. Nat. Acad. S c i . US, 49, 2, 1963 pp. 201-205. 5. J.A. Walker and N.H. McClamrock, " F i n i t e Regions of A t t r a c t i o n for Problem of L u r e , Int. J . Control, London, v o l . 6, October 1967 No. 4 1 pp.331-336. 6. B.D.O. Anderson, " S t a b i l i t y of Control System With Multiple Nonl i n e a r i t i e s " , J . Franklin. Inst, v o l 282, No. 3, September 1966 pp. 155-160. 7. B.D.O. Anderson, "A System Theory C r i t e r i o n for P o s i t i v e Real Matrices", SIAM. J . 1967, 5, pp. 171-182. 8. Moore J.B. and B.D.O. Anderson, "A Generalization of the Popov C r i t e r i o n " , 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 S t a b i l i t y Regions for Multimachine Power Systems", IEEE Trans, on Power Apparatus and Systems No. 10. V o l . PAS-89 5/6, May/June 19 70. J.L. Willems, "Optimum Liapunov Functions and S t a b i l i t y Regions f o r 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 o l 289, No. 2, February 1970, pp. 133-146. 12. E. Kinnen and C.S. Chen, "Liapunov Functions Derived From A u x i l i a r y Exact D i f f e r e n t i a l Equations", Automatica, v o l . 4, pp. 195-204, 1968. 13. G.E. Gless, "Direct Method of Liapunov Applied to Transient Power System S t a b i l i t y " , IEEE Trans, on Power Apparatus and Systems, v o l . PAS-85 No. 2 February 1966 pp. 158-168. 14. A.H. El-Abiad, K. Nagappan, "Transient S t a b i l i t y Regions for M u l t i machine Power Systems", i b i d . pp. 169-179. 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 v o l . PAS 86, No. 12 December 1967 pp. 1480-1685. 16. M.W. Siddique, "Transient S t a b i l i t y of an A.C. Generator by Liapunov's Direct Method", I n t . J . of C o n t r o l v o l . 8, No. 2, 1968, pp. 131-144. s 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", Proc. IEEE„ A p r i l 1969 pp. 537-547. 18. M.A. P a i , 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 Meeting, Dallas, Texas, 19. June 1969. N.D. Fao, 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, New York, January 1970 20. M.L. Cartwright, "On the S t a b i l i t y 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 S t a b i l i t y by Szego's Method and a Maximized Liapunov Function", M.A.Sc. Thesis E l e c t r i c a l Eng. U.B.C., 1970.
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Power system stability studies using Liapunov methods Metwally, Magda Mohsen 1971
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Title | Power system stability studies using Liapunov methods |
Creator |
Metwally, Magda Mohsen |
Publisher | University of British Columbia |
Date Issued | 1971 |
Description | The transient stability of power systems is investigated using Liapunov's direct method. Willems' method is 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 is 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 stability regions is studied using Chen's method. |
Subject |
Lyapunov functions |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-05-06 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0101940 |
URI | http://hdl.handle.net/2429/34307 |
Degree |
Master of Applied Science - MASc |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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