UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

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

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1971_A7 M48_8.pdf [ 2.61MB ]
Metadata
JSON: 831-1.0101940.json
JSON-LD: 831-1.0101940-ld.json
RDF/XML (Pretty): 831-1.0101940-rdf.xml
RDF/JSON: 831-1.0101940-rdf.json
Turtle: 831-1.0101940-turtle.txt
N-Triples: 831-1.0101940-rdf-ntriples.txt
Original Record: 831-1.0101940-source.json
Full Text
831-1.0101940-fulltext.txt
Citation
831-1.0101940.ris

Full Text

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.  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0101940/manifest

Comment

Related Items