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

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POWER SYSTEM STABILITY STUDY BY SZEGO'S METHOD AND- A MAXIMIZED LIAPUNOV FUNCTION by ALY B.Sc,  ABDEL HAMEED METWALLY  AIN-SHAMS UNIVERSITY, CAIRO, EGYPT, 1965  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS  FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE  in  the Department o f  Electrical  We a c c e p t  this  Engineering  t h e s i s as conforming to t h e  required  standard  Research S u p e r v i s o r Members o f the Committee  Head of Department  Members o f the Department of Electrical  Engineering  THE UNIVERSITY OF BRITISH December, 1970  COLUMBIA  In  presenting  this  an a d v a n c e d  degree  the L i b r a r y  shall  I  f u r t h e r agree  for  scholarly  by h i s of  this  written  thesis at  the U n i v e r s i t y  make  that permission  of  ' t ^ c .  2?  * j  Columbia  JV7  by  shall  the  requirements  B r i t i s h Columbia, for  I agree  r e f e r e n c e and copying of  this  that  not  copying  or  for  that  study. thesis  t h e Head o f my D e p a r t m e n t  is understood  financial gain  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, C a n a d a  Date  It  permission.  Department  of  for extensive  p u r p o s e s may be g r a n t e d  for  fulfilment of  it freely available  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  In t h i s stability order  t h e s i s L i a p u n o v ' s d i r e c t method i s a p p l i e d t o t r a n s i e n t  s t u d y o f power systems.  power system i n c h a p t e r two  Szego's method i s a p p l i e d to a second and a q u a d r a t i c  to the same system i n c h a p t e r t h r e e . quadratic  V-function  i s maximized.  Liapunov f u n c t i o n  The hypervolume e n c l o s e d  F i n a l l y a maximized m o d i f i e d  from a t e n t a t i v e q u a d r a t i c  the  Changes i n the time d e r i v a t i v e o f the quad-  r a t i c V f u n c t i o n a r e then made to meet the c o n d i t i o n s functions.  by  applied  o f L i a p u n o v V and V  Liapunov f u n c t i o n i s  f u n c t i o n f o r a three-machine  ii  constructed  system.  TABLE OF CONTENTS Page ABSTRACT  1  TABLE OF CONTENTS  .  1  i i i  LIST OF ILLUSTRATIONS.. .  i  v  ACKNOWLEDGEMENT  v  NOMENCLATURE.  •  1.  INTRODUCTION  2.  A POWER SYSTEM STABILITY STUDY BY SZEGO'S METHOD. _ •2.1 ' 2.2 2.3 2.4 2.5  3.  •  V  1  1 3  Power System Equations...... ........ Szego's Method Algorithm Maximum Value of the Liapunov Function Numerical Example  3 5 7 9 10  MAXIMIZATION OF A LIAPUNOV FUNCTION  15  3.1  Constraints On a Quadratic V For a 2nd Order Power System . 3.2 Hypervolume Bounded By v=x'Ax_ . 3«3 Optimization Technique 3«4 Numerical Example 3.5 A Modified Liapunov Function 3.6 Concluding Remarks 4.  A MAXIMIZED LIAPUNOV FUNCTION FOR A 3-MACHINE POWER SYSTEM 4.1 . 4-2 4-3 4-4 4.5  5.  Equations Of a 3-Machine System Conditions To Ensure Negative Definiteness Of ^ .... Construction Of Liapunov Function And Maximization.. Numerical Example Concluding Remarks  x  15 ^ 19' 21 21 28 29 29 32 40 42 43  CONCLUSION  46  APPENDIX I.'.  47  APPENDIX I I  .  APPENDIX I I I REFERENCES  49 51  ;  V  iii  54  (  LIST OP ILLUSTRATIONS  Figure  Page  2.1  A Typical Power System  6  2.2  Phasor Diagram of Salient Pole Synchronous Machine...  6  2.3  Equivalent Power System  11  2.4  Numerical Example  11  2.5  S t a b i l i t y Region By Szego's Method  13  2.6  Plow Chart For Szego's Method  14  3.1  S t a b i l i t y Region By-A' Maximized Quadratic -V-Punction.  22  3.2  Plow Chart For Maximizing A Quadratic V-Punction  3.3  Choosing V  -23 26  m 3.4  S t a b i l i t y Region By Modified V-Function  27  4.1  Three-Machine Power System  30  4.2  V  44  For Three-Machine System. With x =x =x..=o ......... m 4 5 6  iv  ACKNOWLEDGEMENT  I w i s h t o express project,  my g r a t i t u d e t o Dr. Y. N. Yu, s u p e r v i s o r o f t h i s  f o r t h e guidance g i v e n throughout t h e p r e p a r a t i o n o f t h i s  thesis.  H i s h e l p and encouragement have been i n v a l u a b l e . Thanks a r e due t o Dr. M. S. Davies f o r many h e l p f u l d i s c u s s i o n s . Moiissa and Mr. B. P r i o r i s d u l y The of B r i t i s h  support  The p r o o f r e a d i n g o f t h e f i n a l d r a f t by Mr. H. appreciated.  from t h e N a t i o n a l R e s e a r c h C o u n c i l and t h e U n i v e r s i t y  Columbia i s g r a t e f u l l y I am d e e p l y  f o r r e a d i n g t h e m a n u s c r i p t and  acknowledged.  g r a t e f u l to my w i f e Magda f o r h e r continuous  ment and u n d e r s t a n d i n g .  v  encourage-  NOMENCLATURE Vector of State Variables Time d e r i v a t i v e of x State v a r i a b l e equation vector Liapunov function Value of V d e f i n i n g s t a b i l i t y region Time derivative of V Series resistance of transmission system Series reactance of transmission system Transformer reactance Transmission  system shunt susceptance  Shunt conductance representing l o c a l load Active power E l e c t r i c a l power output of synchronous machine Mechanical power input to synchronous machine Reactive  power  Terminal voltage of synchronous machine I n f i n i t e bus voltage . Equivalent impedence of l o c a l load and transmission system Equivalent resistance of l o c a l load and transmission system Equivalent reactance of l o c a l load and .'• • 'transmission system • • :  • . >  Equivalent i n f i n i t e bus voltage  vi  "  D  Damping c o e f f i c i e n t  6  Angle between quadrature axis of synchronous machine- and i n f i n i t e bus voltage or a reference frame r o t a t i n g at synchronous spread i n the case of multi-machine  H  I n e r t i a constant i n KW.  M  n/(%f)  f  System frequency = 60 c/s  systems.  Sec. /KVA  Time Internal voltage of synchronous machine  r  B  B  2' 3' V B  3 a n d y  Constants i n the expression f o r P  g  Steady state values f o r <s ;  Value of 6 at the unstable equilibrium  us  position C o e f i c i e n t s of expanded system equations Q(x),  Scaler functions of x  g(S(x))  The prime on a vector or matrix i n d i c a t e s the transpose Square Symmatric matrix with' v a r i a b l e  A(x)  elements a  i j (  Elements of A(x)  x)  a(x ), K (x )  Polynomials i n x^  a., b.  C o e f f i c i e n t s of  1  1  A(  1  a(x^) and  ^(x^) r e s p e c t i v e l y  1 X ; L  "d ?  ), B ( ) , X l  C(  X;L  )  Polynomials i n x^ D i r e c t axis synchronous reactance ' D i r e c t ' a x i s transient reactance  vii  II  Direct axis subtransient  reactance  Quadrature axis synchronous reactance Quadrature axis subtransient  reactance  Direct axis transient open-circuit time constant II  T  .  D i r e c t axis subtransient  open-circuit  time constant II  T  qo  Quadrature axis subtransient  open-circuit  time constant 'A a  H»  • a  Square symmetric constant matrix  i 2 ' 22  Elements of A  a  2  Second degree terms i n V  ^1' §2'""*' ^6  Constraint  equations  *i  Eigenvalues of A  *  Hypervolume enclosed  7 —  Augmented state v a r i a b l e vector  4> (z)  Object function  <J> (z) a  Augmented object  jg_(z)  Vector of constraint equations  v  Vector of Lagrange m u l t i p l i e r s  by V=x'Ax  function  %  g  -  6  6  g . . =7;— J Z, _g and <f> r e s p e c t i v e l y  A matrix of elements  6<fi a  Increments i n  a  x -z p  A vector of components . • . Projection matrix  y  Unit matrix  8  6fc  . ..'Step size  Y maximum  3Z  ...  Maximum value of V describing a closed surface  v  m  •  •  Value of V tangent to V=o Internal voltages of respective machines Component functions of v  1. Oscillations  INTRODUCTION  i n the power flow between synchronous machines  have l o n g been known to be p r e s e n t . t r u l y i n the steady system has  to be  s t a t e and  S i n c e no r e a l power system i s  t h e r e are always d i s t u r b a n c e s ,  c o n t i n u a l l y a d j u s t i n g to meet new  In o t h e r words the power system has  the  operating conditions.  to have adequate t r a n s i e n t s t a b i l i t y  margins. The  stability  sient disturbances e n t i a l equations equations  c h a r a c t e r i s t i c s o f a power system d u r i n g t r a n -  are u s u a l l y a n a l y z e d  from a s e t of n o n l i n e a r  known as the swing e q u a t i o n s .  The  o r d e r of  associated controllers.  i s u s u a l l y o b t a i n e d by s t e p - b y - s t e p d i s t u r b a n c e u n t i l the c r i t i c a l The  present  o r d e r to r a i s e u t i l i t y  The  s o l u t i o n of these  f a c t o r s and  to improve the l o a d f a c t o r s and  A need a r i s e s  [1],  f o r a more economic  [2], i s v e r y u s e f u l .  determine the s t a b i l i t y  c a r r y i n g out o n l y one  the c r i t i c a l forward  The b a s i c d i f f i c u l t y  direct  The method enables  s o l u t i o n of the system's d i f f e r e n t i a l e q u a t i o n s . c o n s t r u c t e d L i a p u n o v f u n c t i o n the s t a b i l i t y  studies  and  F o r t h i s , the  o f the e q u i l i b r i u m s t a t e w i t h o u t  so  complexity  method f o r s t a b i l i t y  s t r a i g h t f o r w a r d method f o r s t u d y i n g s t a b i l i t y .  can be e s t a b l i s h e d and  the  t r e n d towards i n t e r c o n n e c t i o n of power systems i n  costly.  method of Liapunov,  after  s w i t c h i n g time i s found.  power systems making the s t e p - b y - s t e p  more t e d i o u s and  equations  i n t e g r a t i o n d u r i n g and  a c h i e v e more economical o p e r a t i o n , i n c r e a s e s the s i z e and  to  these  depends on the d e t a i l of r e p r e s e n t a t i o n of the synchronous  machines and  of  differ-  one  actual  With a s u i t a b l y  r e g i o n o f a power system  s w i t c h i n g time can be o b t a i n e d  i n t e g r a t i o n of the swing  by  equations.  i n the a p p l i c a t i o n i s the absence of a  2 unique method f o r c o n s t r u c t i n g Liapunov f u n c t i o n s a l t h o u g h some f o r m a l i z e d methods, [3] to [12], have been developed f o r c e r t a i n classes of functions. to  power systems.  Some o f these methods have been a p p l i e d  Yu and V o n g s u r i y a , [ 1 5 ] , employed  also  Zubov's method  and a t r u n c a t e d power, s e r i e s o f V - f u n c t i o n s t o study a one-machinei n f i n i t e bus system.  Rao,  [ 1 7 ] , used C a r t w r i g h t ' s p r o c e d u r e , [ 5 ] , t o  s t u d y one-machine and three-machine  systems.  A p p l y i n g Popov's  theorem  and Kalman's p r o c e d u r e , [ 4 ] , P a i , e t . a l . , [ 1 8 ] , s t u d i e d a one-machine system i n c l u d i n g , governor a c t i o n . . .The v a r i a b l e g r a d i e n t ^method d e v e l o p e d by G i b s o n and S c h u l t z , [20],  a p p l i e d by Rao  t o a t h i r d o r d e r model o f a one-machine system.  Popov c r i t e r i o n [22],  [ 6 ] , was  f o r m u l t i v a r i a b l e feedback systems was  ..  _-  and D e s a r k a r ,  The  generalized  used by W i l l e m s ,  t o d e v e l o p a L i a p u n o v f u n c t i o n f o r multimachine systems.  Others,  c o n s t r u c t e d L i a p u n o v f u n c t i o n s f o r one-machine, [14] [ 1 6 ] , as w e l l as m u l t i m a c h i n e systems,  [ 1 3 ] , [ 1 9 ] , [ 2 1 ] , based on energy  integrals.  T h i s t h e s i s i s an e x t e n s i o n of the t r a n s i e n t s t a b i l i t y for  power systems  studies  through the a p p l i c a t i o n o f L i a p u n o v ' s d i r e c t method.  Szego's method, [ 7 ] , i s a p p l i e d i n c h a p t e r 2 t o e s t i m a t e the t r a n s i e n t stability  r e g i o n of. a power system.  i n the form o f a power s e r i e s .  The L i a p u n o v f u n c t i o n o b t a i n e d i s  A q u a d r a t i c form L i a p u n o v f u n c t i o n i s  c o n s i d e r e d i n c h a p t e r 3, and the hypervolume i s maximized  e n c l o s e d by t h i s  s u b j e c t t o c e r t a i n c o n s t r a i n t s on the L i a p u n o v  V, and i t s time d e r i v a t i v e , V.  function  function,  The r e s u l t s are f u r t h e r improved  by  e l i m i n a t i n g the i n d e f i n i t e terms i n V and by m o d i f y i n g the q u a d r a t i c V-function.  In c h a p t e r 4 a L i a p u n o v f u n c t i o n f o r a  three-machine  system i s c o n s t r u c t e d . •••Starting w i t h a q u a d r a t i c V - f u n c t i o n , ' t h e time', d e r i v a t i v e V i s o b t a i n e d and a d j u s t e d to be n e g a t i v e d e f i n i t e . a c t u a l V - f u n c t i o n i s then formed and f i n a l l y the  q u a d r a t i c p o r t i o n o f t h i s new  The  the volume e n c l o s e d by  V - f u n c t i o n i s maximized.  3  2.  A POWER SYSTEM STABILITY STUDY BY SZEGO's METHOD  Based on Zubov's work [23], [ 2 4 ] , Szego [7] suggested a c o n s t r u c t i o n procedure  t o o b t a i n Liapunov  f u n c t i o n s f o r systems w i t h  n o n l i n e a r i t i e s r e p r e s e n t a b l e i n p o l y n o m i a l form. i n t h i s c h a p t e r t o determine  The method i s a p p l i e d  t h e s t a b i l i t y r e g i o n o f a second  order  n o n l i n e a r power system. The  e q u a t i o n s o f a d i s t u r b e d power system a f t e r f i n a l s w i t c h i n g  are w r i t t e n i n s t a t e v a r i a b l e form, w i t h t h e f i n a l . e q u i l i b r i u m a t t h e o r i g i n , as f o l l o w s  x = f(x) The  stability  ,  f(0) = 0  (2-1)  r e g i o n i s expressed  i n t h e s t a t e space by i t s boundary  s u r f a c e as V - V m  (2-2)  where V i s a Liapunov  f u n c t i o n and  d e s c r i b e s a c l o s e d s u r f a c e tangent  2.1.  POWER SYSTEM EQUATIONS  i s the maximum v a l u e o f V t h a t t o V = 0.  .  A t y p i c a l power system i s shown i n F i g . 2-1. I t c o n s i s t s o f a s a l i e n t p o l e synchronous g e n e r a t o r  connected  through a h i g h v o l t a g e t r a n s m i s s i o n l i n e .  t o an i n f i n i t e bus  The t r a n s m i s s i o n system  i s r e p r e s e n t e d by a s e r i e s r e s i s t a n c e r and r e a c t a n c e x. i s r e p r e s e n t e d by a r e a c t a n c e x^. local  The t r a n s f o r m e r  The c h a r g i n g e f f e c t o f t h e l i n e and  r e a c t i v e power are r e p r e s e n t e d by a susceptance  B and t h e l o c a l  l o a d i s r e p r e s e n t e d by a conductance G a t the machine t e r m i n a l . The total The  power output  o f the machine i s P + j Q a t . a t e r m i n a l v o l t a g e V • .  i n f i n i t e bus has a c o n s t a n t v o l t a g e V .  The following assumptions are made for the power system under study: a -  The i n t e r n a l induced voltage of the synchronous machine i s constant,  b -  The flux linkages i n the rotor c i r c u i t s of the synchronous machine are  constant.  c -  The mechanical input to the synchronous machine i s constant,  d -  The armature resistance i s neglected.  The synchronous machine dynamics are represented by a second order d i f f e r e n t i a l equation with, the voltage-.relations as shown i n F i g . 2^2. . ' Applying Thevenin's theorem, the system shown i n F i g . 2-1 can be reduced to the simpler form of F i g . 2-3 where Z  = 1 / [G + jB +  eq  J  — ] r+j (x+x ) 1  V [r+j(x+x )] V  = V  o  -  o  r+j(x+x )+l/(G+jB)  Thus /A  r  e  = {G[r +(x+x ) ]+r}  x  £  = {(x+x )-B[r +(x+x ) ]} /A  V  2  2  t  2  2  t  = V  o  A  o  (2-3)  t  //T  - 1 +  2[B(x+x )(l-Gr)+Gr]+(B +G )[r +(x+x ) ] 2  2  2  t  2  t  Although the damper winding c i r c u i t s are not included i n the machine equations, the damping e f f e c t i s approximated [27], [15] by D(6) = D D  1  1  = V  to o  2 Cos 6 +D  II  2  =  V  o  .  I ( x  d"V  o  II  (x-' - x-) x q q q  IO D  2 sin 6  2  II  o  /(x +x )• e q  i n  II T  d  / ( x  o  e  + X  d>  ••  *  :  "  . . . •  ••• ( 2  _  4 )  5  Including the energy conversion power output, P (6)> which i s derived i n appendix I, the swing equation of the machine has the form  £  2  ^-4  M  dt  D(6)  +  2  +  (6)  P  -  (2-5)  P.  d t  where P  e  (6) = B . E '  Let the  F B „ Cos(6+g)+B. s i n ( 6 + Y ) ] E ' + B . sin(6+y) Cos(6+8)  +  2  i q  '  5  L  dS -rdt  6 = 6,' o'  q  4  (2-6) 2 do — „ = o i n the steady state, and l e t ,2 dt  o and  state variables be chos"en as x, = 6- 6 .  1  x  o  (2-7)  d6 dt  2  The system equations i n state variable form can be written as *1 X  =  X  2  2 - \  [  P  i" e P  ( x  l  + 6  o  )  "  D ( X  1  + 6  o  )  X  2  ]  ( 2  "  8 )  which can be expanded into a power series to give  For  x  l  =  X  2  =  X  2  E . Pi i=l  X  l  +  X  2  E  i l i=l q  X  ( 2 _ 9 )  the s t a b i l i t y study the series may be truncated [15] a f t e r N terms.  The d e t a i l s of expansion are given i n appendix I I .  2.2. SZEGO'S METHOD A b r i e f summary of Szego's method i s given as follows.  A  system represented by (2-1), i f stable, w i l l be either g l o b a l l y or locally  stable.  According to Szego, the s u f f i c i e n t condition for  l o c a l s t a b i l i t y i s that the time derivative of the Liapunov function by  6  • INFINITE BUS  SALIENT POLE SYNCHRONOUS MACHINE  B  F i g . 2-1 A T y p i c a l Power System  J' q 'q x  r  \  jxd  i  id  id Fig.  2-2 '  Phasor Diagram o f S a l i e n t P o l e Synchronous Machine  <  7  (2-1),  v i r t u e of e q u a t i o n = 0(x)  V(x)  where 0(x) any  and  identically  (2-1),  as an  =  = 0 and  g(u)/u >0  do not  [7],  fact  the  or f a m i l y f o r u^O.  of the  x.)}  , a . . ( x . , x.) 13 i J  3  that  Differing indefinite  limit  limit  n  .  The  latter  (2-11)  Following  By v i r t u e of  nonlinear  (i=l,2,...,  n-1).  by  (2-1),  changing  to  the  (2-10).  d e s i r e d form g i v e n by  o / \ [2a(x )x  1  the .  r  1  Szego, the L i a p u n o v f u n c t i o n c o n s i d e r e d  + ^(x )x x  2  (2-9) 1  + x  2  A  {  \  da (x ;  2  . _/ 1_ + ( x ) x n  ±  1  +  (2-12)  2  N  C  2x ][ 2  E i=l  1  2  . +  X  l  x  d - ^-  2  . P  i  x J  5 (x1  ) •• ] x  2  1.  . . . .  N [ C(x )x  is  time d e r i v a t i v e of V i s  o  + x  1  dx^  1  elements  r e a l i n t e r s e c t i o n s with  f o r V which 'is then a d j u s t e d the  the  [25]  ALGORITHM  ) x  the  assumption i s - j u s t i f i e d  i s then d i f f e r e n t i a t e d , u s i n g  c o e f f i c i e n t s a..(x.,x.) to get ij i 3  X ; L  and  c y c l e s of the most g e n e r a l  each of the h y p e r p l a n e s x^ = c o n s t a n t  V = a(  cycle.  function  = a..(x., x.) Ji i 3  system i n the phase space have at most two  g i v e an e x p r e s s i o n  of  (2-11)  contain x  Equation  on  indefinite  t h a t V be  approximate i d e n t i f i c a t i o n  i s chosen f o r the L i a p u n o v  (a..(x., ij i  a . . ( x . , x.) ij i j  +  is  function with variable c o e f f i c i e n t s called  V-function  where A(x)  • V =  g(x)  = X'A(X) x  V(x)  2.3.  and  e q u a l to zero  = 0 i s a. c l o s e d s u r f a c e  i . e . £(x)  A quadratic generating  f u n c t i o n not  Zubov's methods, Szego r e q u i r e d  on a c l o s e d s u r f a c e  by  (2-10)  i s such t h a t g ( 0 )  g(u)  from L a S a l l e ' s  form  s o l u t i o n f f t h e system  on a c l o s e d s u r f a c e , and  the  • g(5(x))  i s a semidefinite  nontrivial  surfaces  has  N + x  2  I q i=l  x^" ] 1  ±  '  (2-13)  8  There a r e i n g e n e r a l two s t e p s i n Szego's method. a suitable  form f o r V  established  the a c t u a l V i s c o n s t r u c t e d .  and, s e c o n d l y , from t h i s form  However, a d i r e c t  L e t a(x^)  p o s s i b l e i n our c a s e .  First,  and £ ( x ^ )  c a l c u l a t i o n of V i s  have the g e n e r a l form  Art  a(x ) =  I i i=l a  1  x  i-l  i  (2-14) C(x  ) =  Substituting  xj  Z b i=l  1  (2-14) i n t o  V(x) = A ^ x  (2 13) gives r  + B(  2  X l  )x  2  + C(x )  (2-15)  1  where CO  A(x ) = 1  oo  .  Z b. x] i=l 1  +  1  1  1  C(  X ; L  ) =  Z  +  1  N  0 0  1  + 2 Z i=l  .  1  1  1  1  1  3  J  1  ...  Z b p . x^  .  3  k  (2-16)  3  (2-15) i s o f the second degree i n x  V = 0 will  q.4"  I ( i - l ) a . x ^ + 2 Z p.x^ + Z Z b.q.x^" i=2 i=l i=l j=l 1  1=1 j = l Equation  Z (i-l)b.x^" i=2 1  B(x ) = 2 I a.x] i=l  N  r  2  and hence the e q u a t i o n  d e s c r i b e two c u r v e s i n the s t a t e space.  Now  i f A(x^),  B(x^)  and C(x^) a r e chosen such t h a t B ( 2  X ; L  ) = 4A(x )C(x ) 1  then the two c u r v e s w i l l any l i n e p a r a l l e l  (2-17)  1  c o i n c i d e and V w i l l not change s i g n  to the x^ a x i s .  Condition  by s e t t i n g b o t h A(x^) and B(x^) i d e n t i c a l l y constant  term,  coeficient coefficient  b ^ + 2q^  o f x^, 2 o f x^,  =0  2b^ + 2 q  2  3b^ + 2q^  = 0 =0  (2-17) can be equal to zero.  along  satisfied A(X^)H  Q gives  coefficient  o f x!? \ nb +2q I n n  thus b  -2 = — q ,  ±  i = 1,2,..., N  ±  0  ,  i>N  (2-18)  B(x^) =0 g i v e s coefficient  o f x^, 2a^ + 2p^ + b - ^ i = 0 2  coefficient  o f x^, 3a^ + 2 p  + b^q + b q^ = 0  coefficient  3 o f x^, 4a^ + 2p^ + ^-^^  2  2  2  +  b  2 2 q  +  3 l  b  q  =  ®  thus a. = l  -1  O+iy  (  2  i  p  .l  +  i«i-j+l>  b  ±  > i-1.2;.-,., 2N-1 , i *2N  0  Equations  (2-19)  (2-18) and (2-19) p r o v i d e the a l g o r i t h m f o r c a l c u l a t i n g the  coefficients  o f the Liapunov  f u n c t i o n (2-12).  The time d e r i v a t i v e V  now becomes V = C(  X l  )  2N-1 Z  -  i-l  (2-20) N E  ... b.p. x f  ,  3  J-l  1  3  ( 2  . "  2 1 )  1  2.4. ' MAXIMUM VALUE OF THE LIAPUNOV FUNCTION' u s i n g the Liapunov  f u n c t i o n d e r i v e d i n the p r e v i o u s  the maximum v a l u e o f V d e s c r i b i n g a c l o s e d curve tangent determined  as f o l l o w s .  Consider equation  e q u i l i b r i u m p o s i t i o n 6= 6  , x^ = 6 -  U S  U S  6  section,  to V = 0 i s  (2-21), a t t h e u n s t a b l e q  and  N lr  M r e [ p  P  ( 6  ) ] =  v  .\  (2-22)  0  1  1=1  Thus V = 0 i s a s t r a i g h t through  P- i = X  1  l i n e p a r a l l e l t o the x US  the p o i n t x^ = 6  - 6 .  2  a x i s and p a s s i n g  Solving equation  (2-12) f o r x  one gets  10  -ax )x +4 (x )x 2  =  x 2  1  1  ^1^1-  - 4 a ( x ) x + 4V  2  2  1  ^  v  1  1  1  "  (2-23)  v  2 For  the curve V =  to be tangent to V • 0 , the value of the square  root must be equal to zero at x- = 6 1 V  = [ct(6 m .  US  - 6 ) - 7 5 (6 o 4 2  U S  us  - 6 , thus o  - 6 )](6 -6 ) o o U S  (2-24)  2  2.5. NUMERICAL EXAMPLE Szego's method i s now applied to study the s t a b i l i t y of a p a r t i c u l a r power system.  The synchronous machine under study has the.  following p a r t i c u l a r s : i  xd  x, = 1.0 p.u. d  x , = 0.04 d o  X  q  =  q  and  °-  6  P  - U  = 9 sec.  T "  qo,  sec.  = 0.07 sec.  }  x, = 0.22 d  p.u.  x  p.u.  q  o  »  = 0.29  i s delivering  H =4  KW sec/KVA  a power of 0.753 + j 0.03 p.u. to the system at an  i n i t i a l terminal voltage of 1.05 p.u. are  i  x '='0.27 p.u. d  The transmission system p a r t i c u l a r s  shown on F i g . 2-4. A sudden three-phase symmetrical short c i r c u i t to ground occurs  at  (x) on one of the transmission lines near the generator end causing bus  A to ground.  The faulty l i n e i s disconnected from the system at both  ends a f t e r a f a u l t duration of 8 cycles.  The f a u l t i s then cleared  and the l i n e restored. From the given i n i t i a l terminal voltage  and power output  F j Q j the i n i t i a l operating conditions determined, [26], are +  11 V.  Fig.  2-3  E q u i v a l e n t Power System  X jn(.  SALIENT SYNCHRONOUS MACHINE  2r  POLE  t  B  p.u.  = 0.7488 p.u. X =. r = 0- 75 p.u. B == 0. 057 p.u. G = 0.18 p.u.  y  2X  INFINITE  Fig.  x = 0.013  1 1  2-4  Numerical  Example  V  t  - 1.05 p.u.  Xd = 0.27 p.u. P= 0.753 p.u. Q =0.03 p.u.  ^ / BUS  12  V  E  r  = 0.989 p.u.  o i  = 1.053  6  p.u.  ,  o  = 0.942 radians  = 53.9  degrees  q 6  US  = 3.04  radians  = 174.28 degrees For the system considered the maximum value of V i s found to be  = 73 and i t gives a c r i t i c a l r e c l o s i n g time of 23 cycles.  swing curve equations  The  (2-8) are integrated forward using Runge-Kutta method  From the r e s u l t s i t is'found out that the c r i t i c a l r e c l o s i n g time i s 24 cycles.  F i g . 2-5  shows the actual region of s t a b i l i t y for the system  considered along with the s t a b i l i t y region defined by the Liapunov function and a system t r a j e c t o r y for a f a u l t duration of 8 cycles and l i n e reclosure a f t e r 23 cycles. used i s shown i n F i g .  2-6.  A flow chart for the computer program  x =00 2  —~  SYSTEM TRAJECTORY LIAPUNOV STABILITY BOUNDARY • ' "ACTUAL STABILITY . . BOUNDARY . > '  10  V=o'  1 V<o  6.  V) o  0.25 .175 -\5 -1.25 -1.0 -0.75-0.5 -0.25  \  0.5 0.75 1.0-  0 -2.  \ \ \  -4.  \ -6.  '-10F i g . 2-5  STABILITY REGION BY SZEGO'S .METHOD. SYSTEM TRAJECTORY LIAPUNOV STABILITY BOUNDARY ... ACTUAL STABILITY BOUNDARY  1.25 1.5  1.75^2:0 2.25  14  F i g . 2-6 FLOW CHART FOR SZEGO'S METHOD  15  3.  MAXIMIZATION OF A QUADRATIC LIAPUNOV FUNCTION  In this chapter a quadratic Liapunov function of the form V =  (3-1)  X'AX  i s considered, where A i s a p o s i t i v e d e f i n i t e symmetric matrix.  The  hypervolume enclosed by (3-1) i s sought and maximized subject to constraints a r i s i n g from conditions on the Liapunov function and i t s time d e r i v a t i v e .  The s t a b i l i t y region thus obtained for a power system  i s very r e s t r i c t i v e .  Since this does not serve the object of this study,  a new Liapunov .function i s then sought by eliminating the i n d e f i n i t e • terms i n V a n d modifying V accordingly.  The  new V-furtction thus obtained  gives a very good estimate of the s t a b i l i t y region for a power system. j  3.1.  CONSTRAINTS ON A QUADRATIC V FOR A 2 — To e s t a b l i s h  ORDER POWER SYSTEM  asymptotic s t a b i l i t y the Liapunov function must  s a t i s f y the following conditions a - V i s positive definite. b - V i s negative d e f i n i t e i n an open region around the o r i g i n , c - V tangent to V s 0. m Consider the second order power system (2-8). A =  a  ll  a  12  a  12  a  22  Let A of (3-1) be  (3-2)  then one has V = a x  + 2a  2  n  1 2 X l  x  2  + a^x  (3-3)  2  The Sylvester conditions for V to be p o s i t i v e d e f i n i t e are  ..•  a  i  a  l  °  >  •,  2 l l 2 2 " 12 a  a  > 0  , , ... .  .... ,.. . - ,  .... ,  (a-D (a-2)  16  Next, u s i n g e q u a t i o n  V = 2{(a  (2-8), the time d e r i v a t i v e V i s D(x ) 1  - a  1 2  2  2  ) x  — r r -  2  + [ (a^  2  i " e  p  +  a  12 l X  P  (  x  l  )  p  —jr-)^  - a ^  M  (  D(  X ] L  +  * <  i "  p  22  (  x  e  M  l  )  )3  )  )  (3-4)  }  To s a t i s f y c o n d i t i o n b, i t i s r e q u i r e d t h a t V be n e g a t i v e a l o n g t h e two axes x^ and x,^, and a l s o t h a t V^, negative d e f i n i t e .  i " e M  P  ••  The  V  =  2  l 2 l  a  X  -(2T7  P  (  term x, ( 1  Along  —  l  x  • '  >  -6  US  Xl  o  ..'  '  '  '  "  i s negative f o r  + 5 )< <5  U S  degree terms o f V, a r e  the x ^ - a x i s , V i s g i v e n by  )  )  M  - 6  (  the second  I  o  thus one must have a Along  > 0  1 2  the x  *  =  2  (  a  2  '  .•  (b-1)  a x i s , V i s g i v e n by  i2  1r  -  D ( x  i  ) ) x  2  which i s n e g a t i v e i f  where  £  n m  i s the minimum v a l u e o f t h e damping c o e f f i c i e n t , g i v e n by  n  D +D D  - i n mm  m  ... .  (  X  1 > =  I  ^  x^  To f i n d . V  the use o f e q u a t i o n • V = 2{ ( n a  D-,- o D  -T Z  { A  A  +  Z  equation  2 >  Cos(2x 1 +26 o)} = D Z  (,3-4.) i s . expanded i n t o a power s e r i e s by  (2-9) t o g i v e N  x  + a 1  i2 2^ 2 ^ 12 l x  x  +  a  X  + a  22 2^^ i=l X  E  P  'v i i l 2 X  + X  • N i l i=l  2  q  X  . . i—1 ^  '  t  17  The second degree terms may be w r i t t e n  V  2  =  2(  X l  x ) 2  a  P  2  ll  ( a  and t h e c o n d i t i o n s  12 l  a  p  <  . .. 1 2 l a  P  2  12 l  + a  12 l q  for  + a  22 l P  )  (a  ( a  1 1  +a  12  + a  1 2  q +a 1  22 l q  t o be n e g a t i v e d e f i n i t e a r e  12  a  +  a  22 l -i q  )  finally  (  a  ll  +  a  l2 l q  +  a  22 l P  )  2  >  ...  0  (c-1) 3V _ SV 3V 3x 3x 8  31  3x * 1  2  2  =  (c-2)  Q  X l  S i n c e p^ i s n e g a t i v e , condition  P  (a a  l  (b-4)  condition C implies  V= 0  -  1  (b-3)  • ••' •  Also  p )  )  0  (  2 2  n  2  2  conditions  (b-1) and (b-3) a r e t h e same,  (b-4) can be r e w r i t t e n as  -a  )  2 1  2  - ^ a ^ a ^ - a ^ ) >0 2  which a u t o m a t i c a l l y  satisfies condition  To summarize, the f i n a l  (a-2).  set of constraints are:  = V = 0 9V  a  8  4  =  a  u  12 a  9V 3x,  3V 3x,  3XT'  3V  =  3x,  >0  > 0  22 2 D  ^ -r-M-.~. :=  a  12  > A  i(a +a q -a p ) >0 2  H  =  " l P  ( a  ll 22~ 12 a  a  }  1 1  1 2  1  2 2  1  (3-5)  18  3.2.  HYPERVOLUME BOUNDED BY V = x Ax i  The hypervolume bounded by the surface V = x_Ax, which i s to be maximized, i s found as follows.  Since the matrix A of equation  (3-1) i s p o s i t i v e d e f i n i t e , i t s eigenvalues X^, i=l,2,...,n are a l l p o s i t i v e and the surface described by this equation i s a closed one. The problem i s then reduced n  V =  to finding the volume bounded by  y 1 1^2y«a»£ XI  X. x l l 2  1=1  (3-6)  Let (3-7)  C. = vty/A. then n  - 2 x. (3-8)  i = l c. I  and the required volume i s given by / - 2 ,/c, 2 ./1-x,  - ,A-i n-1 . i=l n  r 1 = 2  -2, 2 x./c. i i  r^~  n  J  2 -c„vl-x,/c. 1' 1 '2* /,  -2.  resulting i n  / r  2  : .v1—E n-1 . T i=l  . , i-l  2  2,2 Jc. li l  dx  ,dx „. n-1 n-2 (3-9)  -  2 2 / x./c. l l  n-1 n odd  2  n  (—)  n  /  2  1  where |A| denotes the determinant i n appendix I I I .  n even  2.4.6. . .n of matrix A.  (3-10)  The d e t a i l s are given  19  3.3.  OPTIMIZATION TECHNIQUE Of t h e s i x c o n s t r a i n t s i n e q u a t i o n ( 3 - 5 ) , the l a s t f o u r a r e  i n e q u a l i t y c o n s t r a i n t s . ' These can be t r a n s f o r m e d i n t o e q u a l i t y c o n s traints  [ 2 8 ] , [ 2 9 ] , by i n t r o d u c i n g some new f r e e v a r i a b l e s as f o l l o w s .  Let  3  8  H  g  =  5  g  a  =  a  -  6  ir r  a  =  y  e = 0  12- 2" y  ^2 22~M "  e = 0  2 a  12- 3 y  - P ^ a ^ - a ^  _ C = 0  2  '  )  - f C a n + a ^ - a ^ )  2  ^  - e=0  (3-11)  where y^, y^, y^ and y^ a r e f r e e v a r i a b l e s and e i s a s m a l l p o s i t i v e constant.  -  ...—  An augmented  _—  ..  -._  _  -  _.  s p a c e , Z, c o n s i s t i n g of the x - s p a c e , t h e a-space  and t h e y-space i s c o n s i d e r e d .  The components of t h i s new space a r e  g i v e n by  •z=  (z  z ... z >' 9  2  ±  = (x x a 1  2  1 1  a  1 2  a  2 2  y ...y ) 1  The problem i s now d e f i n e d <j>(Z) = - I ( Z ) , s u b j e c t g r a d i e n t method  (3-12)  4  as f o l l o w s : '  minimize the cost  t o t h e c o n s t r a i n t s g(Z) = 0.  [ 3 0 ] , [ 3 1 ] , [ 3 2 ] , as b e s t e x p l a i n e d  to s o l v e t h i s problem.  The  function  projected  i n [ 3 2 ] , i s employed  The method i s summarized as f o l l o w s .  C o n s i d e r an augmented  cost  function  (j, (Z) = <|>(Z) + & ' ( Z ) v a — where v i s a v e c t o r of Lagrange m u l t i p l i e r s . Then  (3-13)  20  i  t  6<j> = (* +8 v) 6Z a  Z  (3-14)  Z  where <jj , v_ and 6_Z are vectors and g see nomenclature).  i s a matrix  (For notation  The steepest descent move i s given by  fi"= - k ^ v )  (3-15)  where k i s the step s i z e , k>0.  The question now i s how to choose v_.  The increment 6Z must be chosen so that the new point i s i n the cons*t r a i n t surface defined by £=0.  &Z must be chosen so that  nominal value of  % or  I f a f u l l c o r r e c t i o n i s used for a  = s «z = z  -s.  •  (~ > 3  16  6£ + £ = 0  Substituting (3-15) into (3-16) and solving f o r v_ y i e l d s  v = (g g ) (^/k - g * )  (3-17)  _ 1  z  z  z  z  substituting (3-17) into (3-15) gives  = -kP<J> - g ( g g ) & - 1  g  z  z  z  z  (3-18)  where P„ g  4  T  y  U - g'(g g') V Z Z Z Z 6  V 6  f t 7  y  where U i s the unit matrix. 2 64 yielding  (3-19)  6  Let the desired step size be  , thus  ' = 6Z  6Z  (3-20)  2 ' ' -1 k= /Sft - ^ Z Z 1 - g  g  •zV'z  g  ;  (3-21)  21  3.4.  NUMERICAL The  EXAMPLE  same n u m e r i c a l  example i n c h a p t e r  maximized q u a d r a t i c Liapunov f u n c t i o n s t a b i l i t y  2 i s used here f o r t h study.  The r e s u l t s  obtained a r e :  430  1.002  1.002 V  = 86  m  12.02 _  , tangent  X-L = 0.394  to V = 0  and  x =  at  1.238  2  This V-function gives a c r i t i c a l  r e c l o s i n g time o f 4 c y c l e s a f t e r  f a u l t o c c u r a n c e which i s a v e r y  restrictive result  A comparison between the s t a b i l i t y  as shown i n F i g . 3  r e g i o n d e f i n e d by t h i s  Liapunov  f u n c t i o n and the a c t u a l r e g i o n o b t a i n e d by i n t e g r a t i n g the system equations 3-2,  u s i n g Runge-Kutta method i s g i v e n i n the same f i g u r e . F i g .  shows a flow  3.5.  c h a r t of the program,  used  A MODIFIED LIAPUNOV FUNCTION From f i g . 3-1, i t i s n o t i c e d t h a t the r e a s o n  f o r t h e poor  estimation of s t a b i l i t y  r e g i o n i s due to the f a c t  i s near t o the o r i g i n .  A b e t t e r e s t i m a t e w i l l be o b t a i n e d  curve  i s shifted  D(  V=2t[a  1 2  -a  2 2  X ; L  +  a  the e x p r e s s i o n  )  ¥  i " e P  5  12 l< X  + t(a -a  2  (  u  x  l  1  2  p.-p x  X l  P  i "  p  (  x  e  S  l  )  ) ]*  2  )  "  out i n ( b - 2 ) ,  <"  3 4)  (  a 1  2  _ a  1 2 22—M—^ 2 (  X  )  x  i  s  a  l  w  a  y  s  negative.  (x.)  - io n(~—i—~) 12 1 M a  )  —rr-)^ +^  D  As p o i n t e d  i f this  f o r V,  D(  2  - -]x  P  V=0  away.  Consider i .  t h a t the curve  Also '  i  s  negative  for  -(2TT-6  U S  + 6 )<x <6 -6 . o 1 o US  The  X = 6-6Q ]  V>o  \  V <o TRUE STABILITY  BOUNDARY  ro Fig- 3-1 STABILITY REGION BY A MAXIMIZED QUADRATIC V-FUNGTIOM  N>  23  C  START  ~)  « READ SYSTEM DATA, I  FTTKT  \  <™ST?^>£  , E , V  q  °  CALC. B , Y , D , D B2. B , B 3  B  1  No  FIRST?  I  2 >  4  CALC.  P  1  .  Q  I  1  FORM V , V , G , G , P z  Z -  Z + DZ  BA-se^-g'Cg' g') g -1  D Z 1 , D Z 2 , DZ  l  STORE Z I N Y  _  K-0  BA ^0? No  o* a  Z - Y & C A L C . V,  SET E Q N . TO FAULT  COND.  I SET I N T G .  INTG.  .  CHANGE PARAMETERS TO FINAL  CHANGE PARAMCTERS TO POSTFAULT  T I K E TO CLEAR T I M E  SYSTEM E Q N . C A L C . V , $  THIRD  SET I N T G . T I M E TO FTNAI. T I M E  SET I N T G . T I M E TO RECLOSING TIME  SECOND  No  (  Fig. FLOW  CHART  FOR  STOP  )  3-2 MAXIMIZING  A  QUADRATIC  V-FUNCTION  24  r e m a i n i n g two terms, however, change s i g n a c c o r d i n g to v a l u e s o f x^ and  x^.  I f these i n d e f i n i t e  terms a r e e l i m i n a t e d from V by s u b t r a c t i n g  t h e i r i n t e g r a l , w i t h r e s p e c t t o time, from the o r i g i n a l V - f u n c t i o n , then V w i l l be n e g a t i v e d e f i n i t e .  2x [(a -a 2  n  V  "(x^ - - ) x  t  /  1 2  Since  ¥  + a  1  P  (  2 2  e  (  l )]dt  x  )  M  o  -2/ [(V 12T~ l 22~li \  a  ( ! ) )x + a  D  x  p  (i " e ^) ]l ^d x p  x  Q  V 2 D  = ( a  ir i2~2Ma  .... -  2  i  ) x  -  f  (  x  i  l :  }  ( 3  ~  2 2 )  where a  f(x,)1  i2  v  ( D  rV  -  . „ — — [Cos(2x +26 ) + 2 x i n ( 2 x +26 )-Cos 26 ] 4M 1 O 1 1 O O 1  1  n  l S  2a_„ , + - T T = - {E [ B ( s i n ( x + 6 +3)-x. Cos(6 +3)-sin(6 +3)) M q 2 i o l o ' o 0  n  B,(Cos(x +6 +y) + x. s i n ( 6 +y) - Cos(6 + y ) ) ] - j B ,  [Cos(2x +26^+3+ ) + 2x, Sin(26 +3+y) - Cos(26 +6+y)]} ± o ' 1 o o 1  (3-23)  Y  let l  D  V=x Ax - (  1 1  l 2 12 ~2M~ D  = a  a  +  ~  a 1 2  2 2M  +  2 l  D  X  +  2  D  (  ) x  l  '  W  f  2 +  2 a  12 l 2 X  x  + a  22 2 X  +  f  (  x  l  }  (  3  _  2  4  )  Then a V = 2  < 12 a  " ~  D(x ) 2  1  i r -  P,~P (x ) e  )  4  +  which i s n e g a t i v e d e f i n i t e  2 a  12 l X  (  for -(2IT-6  1  M U S  ( 3  +V6 )<X^ < ( I § q  US  ~ 6 ) q  "  2 5 )  25 From (3-24) one has  - l2 a  x  =  Ju l±  X  / C a  •D,+D,  1  1 2 " 12 22 ~ W a  a  ]  X  2  1  " 22 a  f  (  x  l  )  +  &  22  V  (3-26)  :  2  a  22  For any curve V = c o n s t a n t e x i s t a v a l u e f o r x^ such  t o be a c l o s e d one,  t h a t t h e square  in  t h e r e must  r o o t term equals  zero  resulting  2 a v - ' f r i ' . +  (D +D ) _ a  -ig  ' x l M  1  (3  -  27)  Thus the maximum v a l u e o f V d e s c r i b i n g a c l o s e d curve i s o b t a i n e d from (3-27) by d i f f e r e n t i a t i n g x^  and e q u a t i n g  t h e r i g h t hand s i d e w i t h r e s p e c t to  to zero,  ^ V ^ l ^ 1  a  '  (3-28)  12^ 22 ^ l^~ 12 ^ a  D  X  a  The v a l u e o f x^ o b t a i n e d  M  from (3-28) i s then s u b s t i t u t e d back i n (3-27)  to g i v e V ." . maximum The v a l u e o f V tangent two  to V-0,  V^, i s found by s o l v i n g t h e  equations V=0  3V  and  3V  8V  3V  ^  (  the v a l u e of V to be used f o r s t a b i l i t y  3  _  2  9  )  i s V or V . whichT maximum m  ever i s s m a l l e r , as shown i n F i g . 3-3. Applying give ° and Fig.  V  t h e above procedure  . >V maximum T rr  ,V  m  T  to t h e same example,  = 878.4 which i s tangent °  (2-8),  t o V=0 a t x =2.1 1  x =0. 2  3-4, shows the r e s u l t i n g s t a b i l i t y r e g i o n to be v e r y c l o s e t o  the a c t u a l r e g i o n o b t a i n e d  earlier.  Fig.  3-3  CHOOSING V ON  io x  2  =  w  61 4-  V  <o  2.  -15: !  1  -1.0 h  0.5  0.5  0\  7-0  — i —  -2-  -4..  V  m  = 878. 4  .-6. -8. •• . .  TRUE STABILITY  -10\ F i g . 3-4 STABILITY REGION. BY MODIFIED V-FUNCTION  BOUNDARY  28  3.6.  CONCLUDING REMARKS A l t h o u g h a m a x i m i z a t i o n t e c h n i q u e f o r q u a d r a t i c Liapunov  f u n c t i o n s has been r e p o r t e d r e c e n t l y by D a v i s o n and Kurak work o f t h i s c h a p t e r i s independant.  [33], the  There a r e f u r t h e r m o r e two major  d i f f e r e n c e s i n our works 1.  To ensure t h a t V i s n e g a t i v e everywhere i n s i d e and on the s u r f a c e  V, i n D a v i s o n and Kurak's in  s e a r c h i n g procedure  they c o n s t r u c t a g r i d  the n - d i m e n s i o n a l space, c a l c u l a t e V" at e v e r y p o i n t where the g r i d  intersects  the n o r m a l i z e d s u r f a c e x. Ax=V, 0<V<1 and then c o n s t r a i n the  maximum v a l u e o f V to be n e g a t i v e .  I n our case the f o l l o w i n g a)  are  imposed f o r the same purpose  are  t o be n e g a t i v e d e f i n i t e and  2.  D a v i s o n and Kurak s t a t e i n t h e i r paper t h a t the q u a d r a t i c Liapunov  b)  The second degree  constraints  V=0 i s tangent t o V.  f u n c t i o n y i e l d s good e s t i m a t e s of s t a b i l i t y tmes.  terms of V, V^t  We found t h e r e g i o n s u n s a t i s f a c t o r y  regions f o r nonlinear s y s i n the case o f a power  system u n l e s s o t h e r terms were added t o make s u r e that V i s n e g a t i v e definite. region.  The r e s u l t s then g i v e a b e t t e r e s t i m a t e o f the s t a b i l i t y  29  4.  A MAXIMIZED LIAPUNOV FUNCTION FOR A 3-MACHINE POWER SYSTEM  In the previous  chapter i t was found that a quadratic Liapunov  function does not y i e l d a good estimate of the s t a b i l i t y region f o r a power system.  I t i s also noticed that the expression of V has a great  e f f e c t on the r e s u l t i n g s t a b i l i t y  region.  In the following a Liapunov function f o r a multimachine power system i s constructed function. adjusted  s t a r t i n g with a tentative quadratic Liapunov  After the time d e r i v a t i v e of this function i s obtained, to be negative  d e f i n i t e i n a region around the o r i g i n .  it is  The  actual Liapunov function i s then formed, checked for p o s i t i v e definiteness and,  as a f i n a l step, the quadratic portion i s maximized.  4.1. - - EQUATIONS OF A 3-MA CHINE SYSTEM  :  ._.  Consider a three-machine system as that of F i g . 4-1. In addition to the assumptions of chapter 2, the following assumptions are made. a)  The damping power i s proportional to the s l i p frequency.  b)  Resistance  of transmission l i n e s i s neglected.  With these assumptions, the d i f f e r e n t i a l equations describing the motion of the system are d 6  .  2  1  M. ' 2 1 dt d 6  + D, 1  2  M 2 n  ?  ,2 dt  dF"  + P  el  = P.. i l  + P = P e2 12  U  d&  2  dt  ±  d6„ + D — 2 dt  d 6  *S ~TT~  d&  +  D  3 dT"  +  P  e3  =  P  i3  ( 4  "  THREE MACHINE.POWER SYSTEM  31  where el  P  P  =  s i n  (  ( S  <S  = ^  g  k  l  k  2  k  3  =  6  d<5 — — dt  1  E  1 2 E  E  Y  =  l  E  E  <S  Y  =  6  1  '  6  2  +  k  dt  2  2  I 3  =  k  - S3) <  1  s i n  ( 2~ 3^ < S  < S  sin(6  -6 )  3  2  ,  "  2 )  p o s i t i o n we have  6  2  '  6  3  =  3  6  df  = o  '  u  =0  dt  (4-2) and (4-3) i n t o  i2  6  u  2 d^5 — j " - = 0 and dt  di.  =  (  ( 4  =  — -  '  = 0  s i n  1 3  equilibrium  u  P  3  23  d6  Substituting  k  12  Y  3  E  = n  — ^ dt  +  sinCS^-iS-^) +  3  " 2 3  the s t a b l e  P  " 2^ 5  1  = k.^ s i n ( 2~ 1^  g 2  P 2  At  l  k  2 d 6 ~ ± ' = 0 dt  (4-3)  (4-1) y i e l d s  sin(6° - 6°) + k^ sin(<S° - 6°)  l  = k  s i n  3  ( 2 ~ 1^ 6  5  +  k  2  s  i  n  sin(6° - 6°) + k  (  2  5 2  ~ ~  5  3^  sin(6° - 6°)  .  (4-4)  32  Let the state variables be  x  l  X  2  =  o 1 " «1  6  6  o 2  o  3  X  4  dt  X  5  dt d6  and The  63  X  X  3  dt  6  system equations become *1 x  x  =  4  - x  2  x  3  =  X  4  =  X  5  6  M { il k  s i n  <Si ~ 2 6  sin(x  X  5  =  M ^ ]_t k  s i n  6  =  M  t  - x  s  i  n  (  x  i  " 2 x  +  6  s i n  ( 3 6  2  - x  3  +  k  3^  S ± n ( |  5l" S3 |  )  3  - x  D  X 2  5^  + 6° - 6°)] + k ^ s i n C ^ -5°)  1  +6° - 6 )] 2  x  2  -  2  )]  + 6° - 6°)] + k [sin(6° - 6°)  ±  + 6° - 63)] -  s  - x  6  + 6° " 63)] ~ i 4 }  ~ in(x  _  3  1 " 2 D  3  2  sin(x  4.2.  "  ( 6 2 " <S°) " s i n ( x ~ x  sin(x  X  1  )  D  3 g^ x  (4-5)  CONDITIONS TO ENSURE NEGATIVE DEFINITENESS OF V Consider a tentative quadratic Liapunov function of the form  33 (4-6)  V = xAx For the three  where A i s a p o s i t i v e d e f i n i t e symmetric matrix, machine case one has a  l  a  2 3 4 5 6 a  a  a  a  a  2 7 8 9 10 l l  a  3 8 12 l 3 !4 l 5  a  4 9 1 3 l 6 17 18  a  a  a  a  a  a  a  a  a  a  a  3  a  a  a  a  a  5 1 0 1 4 17 19 20 a  6  a  a  a  a  (4-7)  l l 1518 20 21  a  a  a  a  a  D i f f e r e n t i a t i n g (4-6) with respect to time by v i r t u e of (4-5) y i e l d s  (4-8)  V = 2[ j, (x) + <f,(x) + <j,(x) + <},(x)] <  1  2  3  4  where (  j) ( ) = (x^ x x ) 1  X  2  3  ir e^  P  M-,  10  z  n  a  p  -p * i 2 e2 p  M„  13  l  M  L l M  (x  <j) (x) 2  x  4  5  x ) T/x 6  \ i3- e3 P  3_  P  (4-9)  4\  (4-10) where T=  h  U  {  2 5 U  (a  4 "  h 2 5 U  I, I 6 (a  a  16  +  a  D  + M  9  a  (a  a  2  M D +M D 13" — — M  ;  1 3 M  i l o  18 )  ;  3  a  , 1 , 2 ll 14 U  M  2 3 D  + M  3 2  + a  v ;  M„D„+MJ), 2"3'"3"2 . MM 20 x  1  A  a  2  3  }  ( a  .  D  &  ll  ^ 2 V ~11 ~"14 ( a i1  U  +  j  2  3  ;  3  D  3  X  .  a  1 2 2 1 , , ^2 9 " MM 17 10" M 19 M  +  }  M D +M D M D +M D . 1 , _ 13 31 1 2 2 1 17 2 6 13 MM 18 M,M; 12  20  ;  D. "3 l 5 M 21 a  }  3  (4-10a)  34  tj> (x) = ( x x x ) 3  1  2  "(a  3  1  "  D  (a  )  l  M  2  4 •  a  M  2  (a  V  _ 3  "  a  l 3  U  3  M  V  3  }  2  M  8  ( a  M  8  ?  D  (a  V  2  D  .  x  (a  ' ' °3 12 M 15j  ,  14'  2  IV  3  (4-11)  3  and a  • <x)=ex x x ) 4  4  5  l6 M  a  a  I n o r d e r to ensure  a  a  !8  M  2  19  a  20  M  l  P  3  M  i l  " el\ P  P i2 r  3  - rP  e2  2 a  20  a  21  M  l  (4-12)  \ i3 -e3/ P  M  3  P  t h a t V i s n e g a t i v e d e f i n i t e i n a r e g i o n around t h e  i t s component f u n c t i o n s , <t>'s, must be e i t h e r n e g a t i v e d e f i n i t e or  semidefinite. 1.  l  18 M  17  M  17 M  origin  3  6  Examining e q u a t i o n s  <)>^(x) i s a f u n c t i o n o f x^, x  2  (4-9) to (4-12) we n o t i c e the f o l l o w i n g : and x  3  o n l y and can be made n e g a t i v e i n a  r e g i o n around the o r i g i n . 2.  <f> (x) i s a f u n c t i o n i n x^, x^ and x^ which i s n e g a t i v e d e f i n i t e i f the 2  m a t r i x T o f e q u a t i o n (4-10a) i s n e g a t i v e 3.  <f> (x) and <j> (x) a r e both 3  4  e i t h e r by s e t t i n g  indefinite.  each i d e n t i c a l l y  definite.  They can be e l i m i n a t e d from V  equal to zero or by I n t e g r a t i n g  them w i t h r e s p e c t t o time and then s u b t r a c t i n g the r e s u l t  from t h e  V-function. In the f o l l o w i n g each component of V i s examined t o develop c o n d i t i o n s f o r i t s e l i m i n a t i o n o r to ensure (J)^(x) can be made e i t h e r i d e n t i c a l l y Knowing t h a t the sum of the m e c h a n i c a l  the n e g a t i v e  equal  the n e c e s s a r y  definiteness.  to zero o r n e g a t i v e  definite.  i n p u t s to the system i s e q u a l t o  t h e sum o f t h e e l e c t r i c a l o u t p u t s , i f we f u r t h e r s e t  35  a  4  a  a  a  9  V  5  " M  l"  M  - M  2  io  a  a  13 M  a  l  il M  l~  M  '  3  14 M  3  and  3  15  2  M  (4-13)  3  i n (j)-^(x) , t h i s function w i l l be i d e n t i c a l l y equal to zero. Substituting f o r P^ and P_^ from (4-2) and (4<-4) respectively into (4-9)  one gets:  ^(x)  = kjjsinds^- 6 ) -sin(x -x +6 ~ 2  .  /  0  , 0  2  1  2  O 6  3  )][X  M  *  2  2  1 M7 " M3 u  2  3  ,  3  -S  a  3  a  )+X  1  (  a i 5  +  (  a  . 0  3  ll l4 M " • M > *3< M 2 a  ) + X  (  i3 M 1  a  ) + X  5  /  l0 \ " M 2 a  1 M7  T r  s  4  (  6 " 9 " ll 6° )][x (^ " M^ 2 MY " M -sm(x^-x +6^3 .  k [sin(6°-  +  3  2  2  )][X  1  -sxn(x ~x +6 -  k tsin(6°-  +  a  6  2  ) + X  3  a  3  (  13 M " 1  *">] M  3  setting A  - %  =  5 M ' 2 a  M  a  _ !i _ r  ,!l4 _ f i 5 v  n  6 M 3  a  M  1  13 M  =  a  1  and 3  i4 M  (4-14)  2  tj)^(x) becomes  4 ^ ( x ) = (r~ 3  a  5 0 0 0 ^~) ( k ( x - x ) [sin( 6°-6°)-sin(x -x +6° 1  ;L  2  +k (x ~x ) [sin(6 2  ...  .  ,  +  2  k (x x 3  r  3  3 )  2  1  2  s  0  -6 )] 2  -<5 °)-sin(x -x -t <$° - 6 ° ) ] 2  [ i n ( c j -6^)- in(x -x S  1  0 3 + 6  _ ° 6  )  J  }  )l  2  a  (  J  ( 4  _  1 5 )  36  which i s negative d e f i n i t e for values of x^, x,, and x^ s a t i s f y i n g - r-2(6°--6°)<(x -x )<ir -2(fiJ-5°) 1  2  1  -7r-2(6°-«53)<(x -x )<Tr -2(6°-6°) 3  2  - T T - 2 ( 6 ° - 6 ° ) < ( X - X ) < T T -2(6°-6°) 1  (4-16)  3  The expression <j>(x) can be made i d e n t i c a l l y equal to zero by 2  setting D a  4  l 16  =  2 M P  a  io  =  2  a  15  =  " 3 M  5  +  a  19  a  21  3  D  a  a  9  - D. 1 ,2 =  D  a  6  +  ll  a  a  13  +  a  (  =  M7  l4 =  (  l  D  3  M^IS  +  a  n  d  M^ M7 20 +  )a  ( 4  "  1 ? )  On the other hand, for <j>(x) to be negative d e f i n i t e , the conditions on 2  matrix T are  :  V U  a <0 16  l  D  D (  V  w «16  2 s h 10 H^ !^- 4 V D  ) ( a  l a  l6  ) [ ( a  + (  8  °2 l 0 " H~ l 9 a  2  ) ( a  1 2 9 " M  a  15  D  4 M  2 1 M D  M L  ^3 M 21 a  3  }  a  2  .2 17  1 4 ll ( a  ' ) >  '  0  M + a  l4  2 3 3°2 20 D  + M  &  37 JL, " 2 5  M  +  U  1 2 D  1  < a  6  + a  6  + a  13  2  u  a  3  M M  J  X  1  3  M +  V  M  2  18  )(  1  ll  a  +  a  l8 4  1 a  )[  D  3  2  a 3  M D +M D 1  ( a  5  2  2  D  + M  2 1 ^1^2  ,  D  2  20  }1  M^+M^  1  M^MT~ 1 7  9  + a  1 2  9  3  M M  14  a  1  M^MT  13  w 2 1  a 3  M D +M D a  3  U  1  3  M D +M D ( a  h. ~ M  r  M  2  +M D  3  "  ~  X  +  D  X  M D  l • l7 2 15  2 1  M M  x  4  + M  9  + a  a  )( a  ll  + a  14  "  J  MT^ 20 3  }  M D +M D  -|< 10-^ 19> a  a  ( a  6  + a  13-  \  1  ^  a  l8>J  < 0  "  ( 4  X 8 )  t)> (x) can be e l i m i n a t e d by s e t t i n g 3  a  a  —  -  ~  l  "• M  2 ~  -  M  • a  a  4  a  5 ~  2  '  M  ^3 M . 6  3  a  9  ! M  a  a 1  '  l l3  ~  ~  -  :  '  ^2 A  10  '  M7 l l  M ~  ?  M  D a  a  8  =  3 2  3  a  °2 14  =  a  a  n  d  12=M7 l5 a  ( 4  On.the o t h e r hand one may i n t e g r a t e 2cf> (x) w i t h r e s p e c t t o time. 3  noticed that  ----l - ^  -  D  2  (  a  l  D  2 ( a  7  -l 4  D  V  f  x  l 4 X  d  t  =  (  a  2  l " M7  a  2 )  x  2 7 " W l0  l  2  D  " Mj" 1 0 a  ) ;  X  2 5 X  d t  =  ( a  a  n  ) x  2  "  It is  1 9 )  38 l 1 2 " "ST; 1 5 D  2 ( a  a  D  )f X  2 2 ~ M^ 5  3 6 X  d t  =  l2" W  ( a  D  2 ( a  a  i  x  d  t  +  2  5  '  2  2 "  a  5  ) x  3  -  l 2 X  8 "  + 2 (  X  d t  °2 M "5 "  =  °1 V  a  2  7  X  2 4 X  d t  D a  D 2 ( a  2  ) x  X  D 2 ( 3  3 15 3  V '2 4  2  D 2 ( a  <  a  3  l  D  x  )f  a  6  )  7 X  l 6 X  d t + 2 ( a  3  "  13  a  3  } 7X  3 4 X  d t  =  °2 a  H / 2 6 )  x  X  2(a  d t + 2 ( a  8 "  a  !4  3 a )x x .4 " 3 n  8  2  ) / X  3 5 X  d t  •  =  D  a . ) fx .x dt  3  M  i f one sets  D  2  M D  2  " M  5.  3 3  3 M  l M  3  9  D  a M  l  D a  6  =  3  13  °2 " M l4  D  a  a  ll  (4-20)  a  2  one.has 2 /(fr (x)dt=(x x x ) 2  D,  D, (  a  i *1  ~  T T  M  / ) ( o ~4' "2  -  a  v  a  -  M  D, ( a  2 " M  L 3 ' ( a  a  2  5  )  ( a  M  3  D,  D, a  2  a  5  6  }  }  ( a  ( a  7  8 "  M  a  2  a  V  3  10  H  )  }  ( a  8  ( a  M  l2"  ll  a  3  a  )  15j  '3/ (4-21)  which w i l l be subtracted  from the tentative Liapunov function.  F i n a l l y <f>^(x) can be eliminated as follows.  Knowing that the  39 sum o f m e c h a n i c a l i n p u t s t o t h e system i s e q u a l t o the sum o f e l e c t r i c a l o u t p u t s , i n o r d e r to reduce ^ ( x )  x  to zero one may s e t  16  M.,  (4-22) On the o t h e r hand, s i m i l a r  -  a  to the c a s e o f <j>3(x), s e t t i n g  M  _1  18 " M  a 2  M a 2  17 1 6 +  (M  *19  -  2  M^a^  M,  3 M~ 1 7 M  a  20  =  a  a  n  M3 a  M  21  d  M (M - Mp 3  3  MM  l6  a  x  3  (4-23)  17  i n cb,(x) r e s u l t i n 16 2/<j> (x)dt = 2( M 3  a  +  k  2 o  2  X  + k_ / 3 o  l 7- )  M,  4  3  t  s i n  ( 2 6  x.. - x 1 2 _.._ o 0  {  V o  _ 6  [sin(6 r  3)-  s i n  /r  ( 2~ 3 x  x  1  + 6  o,  ,0 „o.  -6 )-sinXx -x +6 -6 )]d(x r  2  2  1  "^3)] ( d  x  2  2  - x  1  2  1  3)  1~ 3 X  [sin(<5° -6°)-sin(x.,-x;H-6° 1  3  -  1  3  1  -6°) ]d ( x . - x j } 3  which a g a i n w i l l be s u b t r a c t e d from the t e n t a t i v e Liapunov  1 3  function.  (4-24)  •  40  4.3.  CONSTRUCTION OF LIAPUNOV FUNCTION AND MAXIMIZATION There are sixteen d i f f e r e n t combinations of <)>^(x), <j>(x) , ^ ( x ) 2  and (^(x) that r e s u l t i n a negative d e f i n i t e or semidefinite V.  These  combinations are a l l investigated and a summary of the results obtained i s given i n table 4-1.  Out of these sixteen combinations only one results  in a p o s i t i v e d e f i n i t e V-function that has a negative d e f i n i t e time derivative. This i s the case where (j>^(x) and cj> (x) are made negative d e f i n i t e and cb^Cx) 2  and <f>(x) are integrated. 4  Combining  conditions (4-14), (4-18), (4-20) and  (4-23) we end up with V(x) = x Ax - 2/<j> (x)dt  (4-25)  x Ax = x Ax - 2/<j> (x)dt  (4-26)  4  where 3  and -2/tf> (x) dt i s p o s i t i v e d e f i n i t e and i s given by equation (4-24) . The 4  new matrix A i s given by  rvi M  a  1  4  V ? MM X  M ^ 17  a 2  V l7 M 4  a  X  ;  1 2  1  7  3  1 3 MM D  M  X  2  M  a  3  2  M2 l7 M V  a 2  ;  a  3 3~ 1 M^" 4 ^ L j " 1 7 D  M  a  3  1  7  %JV^l  M^V  M l7^ a  2  M„  3  16  a  D  a  M }  3 3 M^" 17 M^ 17 M  a  rr- a,  17 M  +  2  V ^ 1 2 3 M MM 17 D  +  x  !3  (a  a  1  a  2  3  M  VS  MM l7  , ^3 ^3 l7 M l7 M i7  0  M  2 3 l7 M ^ l7 D  a  M  a  ^3 M 17 2  M. 1 7 2  \  M  V  H 17  _2  +  M^V"  4  B  %V^1  M  h V  a  2  ' 1 2 M  i7  3  V 3  M  a 2  V ^ 1 \ V 3 M 1 7 MM 17  +  U  V3 M  MM  a  3  • M~ 17 n  M,-M,  M  M,  3 3 M^^' M  _ M  1 ' • • M 17J 3  2  (4-27)  ,  4o  No.  2  *L  3  \  V  V  1  0  0  .0  0  0  p.s.d.  2  0  0  0  /  0  p.s.d.  3  0  0  /  0  0  p.s.d.  4  0  n  0  0  n.s,d.  p.s.d.  5  n  0  0  0  n.s.d.  p.s.d.  6  0  0  /  f  7  0  n  0  f  n.s.d.  p.s.d.  8  0  n  /  0  n.s.d.  p.s.d.  9  n  0  0  f  n.s.d .  p.s.d.  10  n  0  /  0  n.s.d.  p.s.d.  11  n  n  0  0  n.d.  p.s.d.  12  0  n  .... /  • - • n.s.d.  p.s.d.  13  n  0  /  /  n.s.d.  14  n  n  0  I  n.d.  p .d.  15  n  n  f  0  n.d.  p.s.d.  16  n  n  f  /.  n.d..  p. d.  —  Table 4-1 ->• negative integrated n.s.d. -»- negative semidefinite n.d.  negative d e f i n i t e  p.s.d. -* p o s i t i v e semidefinite p.d.  ->-' p o s i t i v e d e f i n i t e  /  conditions contradict  0  —  .  -  ' p.s.d.  same as 14  - Q  41 which i s p o s i t i v e d e f i n i t e i f a >0 17  a  16 M  a  1  17 M  >0 2  (4-28)  >0 l  D  M  2  Also V = 2[ (, (x) + ^ ( x ) ] (  (4-29)  1  where a^ i7^i ^ ( x ) = (—- - — - — )  o o o o {k (x -x )[sin(fi ~6 )-sin(x -x +6 -6 )]  a  1  1  1  2  1  2  1  2  +k (x ~x )[sin(6° -6°)-sin(x -x +6 2  2  3  2  3  1  2  2  -6°)]  +k (x -x )[sin(6° -6°)-sin(x -x +6° -63)]} 3  1  3  1  3  (4-30)  and  •*z  ( x )  =  (x  4  X  5' 6 X  4\  l  ) T  (4-31)  V6 x  l " MT 1 6 4 1 D  T =  ( a  a  }  M a -D a 2  0  4  2  1 6  M^-M^ MM  M,  1  0  2  ^ 0  17' 3 4- 3 l6 ^ M, M  r  a  D  a  ,1 3 M  D  - 3°l >LM„ T*2 M  a  . l7  ;  (4-32) which i s negative d e f i n i t e i f M  M M 3 2 .1 D~ *D~ *D~ 3 2 1  , a n d  (4-33)  42  Thus we  have a p o s i t i v e d e f i n i t e Liapunov f u n c t i o n V(x)  (4-25) and i t s time d e r i v a t i v e V g i v e n by (4-29).  Equations  given  by  (4-28) and  (4-33) d e f i n e the r e l a t i o n between the t h r e e parameters a^, a^g and ^y]' Finally  the hypervolume e n c l o s e d by  the q u a d r a t i c p a r t of  s h a l l be maximized s u b j e c t to (4-28), (4-33) along w i t h  the tangent  V(x) conditions  V = .0  3 l _ 3V  3x  3V  4.4.  1  1  3V  3V 3x  1  2  3  = 0  = 0  34  4  3V 3x  1  3V x  3V 3x  5  3V 3x  x  6  3V 3  X  3  3V 8x  3V 3x  3V 3x  3V 1  3V 3^  3X  3  9 X  The  2  3V  9V  3V 8x  -  3V 3x  5  6  3V 3x  same a l g o r i t h m of c h a p t e r  = o = o  = 0  (4-34)  X;L  3 i s used.  NUMERICAL EXAMPLE The  three-machine system c o n s i d e r e d has  E^l.174  ^22.64° p.u.  E =0.996  z,2.61° p.u.  2  the f o l l o w i n g d a t a :  P =0.8  '  2  P  =  0  ,  p.u.  3  p  E =1.006 z.-11.36° p.u. K.W.sec./K.V.A.  ^1_ M  H =7 K.W.sec./K.V.A.  ^2  A sudden 3-phase symmetrical  2  ^3 M  =  1 0  l  M H„=8 I.W.sec./K.V.A.  ' p.u.  3  1^=3  u  ' =  ?  -  =  '  3  short c i r c u i t  . to ground occurs  on  43 the t r a n s m i s s i o n l i n e c o n n e c t i n g machines 2 and 3 of F i g . 4-1 c l o s e to bus  3.  The c r i t i c a l  18 c y c l e s .  clearing  time o b t a i n e d from  The a c t u a l c r i t i c a l  clearing  the above V - f u n c t i o n i s  time o b t a i n e d from  t h e system's  swing curves i s 20 c y c l e s . The r e s u l t i n g Liapunov following  f u n c t i o n , of t h e form  (4-25), has the  particulars V  m  =33.65 which i s tangent  to V=0 a t t h e p o i n t  x = 2.195, -0.137, -0.005, 0.112 x 10~ , -0.278 x 10~ , -0.595 x 10 3  A=T3.26  Fig. x , n  Fig.  5.32  2.28  0.326  0.76  0.868  5.32  8.69  3.72  0.532  1.24  1.42  2.28  3.72  1.6  0*228  0.532  0.608  0.326 0.532  0.228 0.129  0.076  0.087  0.76  1.24  0.532 0.076  0.402  0.203  _0.868 1.42  0.608 0.087  0.203  0.489_  4-2 shows the f u n c t i o n V ( x ) = V x„ and x  with  m  3  p l o t t e d i n the three dimensional  the o t h e r components x., x  r  and x  s e t to zero.  space This  shows the maximum d e v i a t i o n s i n t h e t h r e e r o t o r a n g l e s , w i t h r e s p e c t to  a r e f e r e n c e frame r o t a t i n g a i synchronous speed, w i t h o u t  losing  synchronism  w i t h each o t h e r .  4.5.. . CONCLUDING REMARKS " It  i s interesting  to n o t i c e t h a t when s e t t i n g  a. = 0 4 a  i7  = °  *16 = 1 M, in  t h e Liapunov  (4-35)  f u n c t i o n of e q u a t i o n (4-25), t h e r e s u l t i n g V - f u n c t i o n i s  45  e x a c t l y the same as t h a t of Willems [22] and i s g i v e n by V = (x. x x,) 4 5 6 c  2{k,/ l X  X  2  k  k  .  r  M O 0 M. 0  M  0  0  0 ~ 0  2  M  X  2  X  N  /x.\  X  3  +  5  l6 X  + 6 ° -S°)-sin(<5° -  2 3[sin(x -x +6 b  X 2 /  3  T  3  2  -6°)-sin(6° - 6 ° ) ] d ( x - x ) + 2  3  l 3 [ s i n ( x - x + 6 ° -6°)-sin(6°-6°)]d(x -x )} X  1  3  1  3  (4-36) j '  When a p p l y i n g  t h i s V - f u n c t i o n t o the same n u m e r i c a l  s e c t i o n 4.4., the r e s u l t i n g c r i t i c a l 18 c y c l e s c l e a r i n g time o b t a i n e d chapter.  example c o n s i d e r e d i n  c l e a r i n g time i s 14 c y c l e s compared t o  from t h e V - f u n c t i o n c o n s t r u c t e d  i n this  46  5.  CONCLUSIONS  The d i r e c t method of Liapunov has been applied to the study of transient s t a b i l i t y i n power systems. 1.  The following conclusions are drawn:  Although Szego's procedure has been applied successfully to construct  a Liapunov function f o r a second order single machine-infinite bus system, work remains to be done i n developing algorithms f o r applying this method to higher order systems. 2.  An expression  for the hypervolume enclosed by a quadratic form function  i s developed and employed 3.  i n maximizing the estimated s t a b i l i t y region.  A construction procedure f o r optimized Liapunov functions f o r power  systems has been developed.  I t s t a r t s with a quadratic form and i s modified  by the negative d e f i n i t e V constraints before maximization of the estimated s t a b i l i t y region.  The procedure has been applied successfully to a single  machine-infinite bus system as well as a three machine system. In general, i t remains to develop procedures f o r the construction of Liapunov functions f o r multimachine power systems i n which synchronous machines and c o n t r o l l e r s are represented i n great d e t a i l .  47 APPENDIX I Expression (2-6) f o r the e l e c t r i c a l power output i s obtained as follows.  From the phasor diagram of F i g . 2-2, one has  V,  -(I-l)  where sin6 +  = V o V i q  r  -x  e  cos6  r  X  e  L.  i  e  d' (1-2)  i , q  e J  also  +"0  = 0  q  t  E  V  x"  -x  q  i  0  d  Solving (1-2) and (1-3) f o r V^,  V, \ d = V  v i q o A  e .d e  q  and i ^ ,  i  d  d  i  and gets E' r x +-9- , e q A r . +x (x +x ) (1-4) e q e q  s i n cS  e q  i  (1-3)  q  e  q  cos 6  and = V  ' f-r e x +x, e d  sin 6  q  . A  -r  cos , < 5 _  where A = r  e  Let 1  A  2  (1-5)  r \  e (1-6)  + (x +x )(x +x.) e q e d  A =x Jr +(x +x') 1 q e e d  x +x e q  /A  = r x /A e q / 2 2 A- = x, /r +(x +x ) / . 3 d e e q A 2 0  1  A. = [r * + x (x +x )] / A 4 e e e q  48  A  5  = (x +x ) / A e q i  g = arc  x, +x / d e> an( )  r e  x +x.  V  Y = arc tan (—q  >)  e  3  (1-7)  Substituting (1-7) into (1-4) and (1-5) y i e l d s = V  sinS  A^ sing -A^ cosf Ag cosy  A^ siny  cosS  3 . — - r siny d  sin6  + E  and = V  3 —r cosy d A  _ A  X  +E  /  \  X  l " l • cos6 — sing — cosg L x x q q q The e l e c t r i c a l power output P (<5) i s given by A  A  (1-9)  P (6) = (V. V ) (1-10) Substituting  (1-8) and (1-9) into (1-10) gives  '  2  P (6) = B,EI e 1 qq 2  J  '  +  B-fcos (<5+g)+B„ sin(6+y)]E +B.sin(6+y)cos(6+g) (1-11) 2-j q 4  where B  l  =  A  2  V  (  A  4  /  X  q  )  B  2  = -V A (A A /x )  B  3  " o 2 3 x-  o  V  A  1  A  (  5+  4  q  " I'> q  B  4:= - v '  o  2  AA.(1 3 x  d q  x, d  (1-12)  49  APPENDIX II The  expression  (2-4) f o r the damping c o e f f i c i e n t  i s expanded as  follows ^D(6) • = i ( D  cos <S + D  D D 1 +  2  D l  -M(—2— Substituting  i  i  D  D  2  — 2 ~  +  C  x  i  + D  2  °  1  r  S  +  r  D  D  + D  l  -D  2  2  f o r 6 = -^ ^  i 2 —rr: 2M D  sin 6)  2  2  w  6  e  0  )  g  e t  2  D  2 rrr- [cos26 cos2x - s i n 2 5 sin2x. ] 2M o 1 o 1 D  2 1~ 2 (2x ) (2x ) ^rr- + ^ r r - [cos2<5 (1 r-i— + ,i , 2M 2M o 2. 4 •  +  D  D  D  2  4  x  " o  V  x  (2x )  3  x  -sin26 (2x -- - -+ o  -  I 1=1  q,x/  1  3  r  (2x )  -....)  5  x  —--••••)  ,  1  (II-l)  where D  q  At  i ~  i  +  D  2  D  15  r  D  2  2TT  • c o s 2 6  - - •  °  the e q u i l i b r i u m s t a t e :  P.=P l  e  (6 ) o  =B,E 1 q Thus:  +[B„ cos(6 +3)+B 2 o j  0  s i n ( 6 + y ) ] E +B.sin(6 +y)cos(6 +3) o q q o o  (II-3)  50  ^ ( P . - P ^ ) ) = i{B E^[cos(6 2  +S)-cos( +6o+e)] +  o  Xl  B„E [sin(6 +y)-sin(x.+6 +y)]+B.[sin(6 + Y ) C O S ( 6 +3) 3 q o 1 o 4 o o -sin(x..+6 + Y ) C O S ( X + 6 +6)]} 1 o 1 o 1  1 ' =rr{B E [cos(6 +3)-cos(x +6 +B)}+ M 2 q o 1 o 0  1  B E [sin(6 +Y)-sin(x +6 +y)]+ j q o 1 o 1  B, - -i-[-sin(26 +3+Y)-sin(2x +26 +3+Y) 2  o  1  ]}  o  = i { B E ' [ c o s ( 6 +3)(1-cosx..)+sin(6 + 3 ) s i n ] M 2 - i..' : O 1 O 1 q n  X l  t  +B„E [sin(6 + Y ) ( 1 - C O S X . , ) - c o s ( 6 +y)sinx ] 3 q o 1 o 1 v  1  1  +4 --— [sin(25 +3+Y) (l-cos2x-)-cos (26 + 3 + Y ) s i n 2 ] } 2 o 1 o 1 B  X l  E* 2 4 ..q x x = — [B„cos(6 +3)+B„sin(6 +y) ] [-± - -± + ...] M 2 o j o 2 . 4 . ' 3 E x^ + - J [ B s i n ( 6 +3)-B cos(6 +y)][x • +. . . ] M 2 o 3 o 1 3 . B 2M  (2 -[sin(26 +3+Y)(—  4  X ; L  )  2  (2  )  4  +...)-COS(26  Q  2x )  X ; L  q  +3+Y)  3  OO  =  E p.x* i=l  (H-4)  1  where  i  p. = — {-J[B„cos(6 +r+ x l ., M 3 o r  2  u ) - B s i n ( 6 +3+V" 2 o 2  -i-l ._, , V B.cos(26 +3+Y+—• TT) + M 4 o 2  9  TT)]  (II-5)  51  APPENDIX I I I i  In equation (3-5) the hypervolume bounded by V=x_ Ax i s given by j( " 2 / 2 • 1 - X1/ C 1  / , n-1/ l r C_  2  C  l  C  1  n  1  - 2 , 2 £- X ^IC .l  _  2  i=l  1  1=2  -[c  - c / - c A-x /c 2  1  -c  2  1 1  2  n-1  Jl-  1  n-1 _2 2 1- y x,/c.]dx ,dx „...dx I i n-1 n-2 1=1 L  -2,2  . , x,/c. i=l I l  Consider the innermost integration -9 / nn-1-  x  -2  i  i=l  /  c  .2  i  A /L-  _ 2 y x.2/ c. dx T .^ i i n-1 i=l - 1  *1  c  " „  n-2 -c  n  L  n-1" i - 1 *-2± c. l  Let  sin 9 n-1  h ence x —— 2 c. l  n-2  dx  2  - = c T /L- y n-1 n-l . . 1=1 v  L  cos GdG  and _2 /n-2 x =/i_ E —rT 2 i=l c.  n-1 A-  x  x  -| c.  ±  i=l  n-2 C  (1  1  1  -2 2  c. l  TT  1" n n - l C  C  ,,  ( 1 _  h-2  co§  i  i>  •„ n-l -.S  2  -2 X.  ^  /  -v/2  .^ - i ) i=l „  cos 9 d9  J  (3-9)  52  For the next step of integration use the s u b s t i t u t i o n  :  ,  / - -2 n  3  n-2 L y x. =/l-. x c 1=1 — ^ n-2 2 c. l Zj  1  . s m 0n  0  / " -2 hence dx „ = c . / l - . * - . . x. n-2 n-2 i^l _i 2 c. l n  3  cos 9 d9  n-3 n-2 ( l - A xt)=:(l - J  -2 x ^r) cos  ±  / n-3 -2 c  0  n  /"  ~  / l -  2  9  - '  -  E i i = l T X  c. l  n-2 -2  2  c c (1- E i ) dx _ n n-1 . .. —rn-2 1  ^ -c  2  i=l  ..  c  -"  •  / n - 3 -2 V l -E  n-2  .  ..  1=1  V  —x-  2 1  n-3 x7  3/2  T c c ,c (1- E -± ) 2 n n-1 n-2 ., 2  TT/2  ' f  l  „  cos  J  e.de  —TT/2  n-3 x 2 IT /i H \ 3/2 = 2.— . — c c c „(1- I —TT ) 3 2 n n-j n-2 , , 2 1 = 1 c. 2  1  o  l This procedure i s repeadted. (n-1) times to give the f i n a l answer.  I t i s noticed  that as a result of the k — step, the term cos^^SdS appeares, thus TT/2 n -TT/2 the l a s t i n t e g r a l i s j. cos 9d9 which i s equal to -TT/2  TT/2  21  cos 9d9 n  o  1.3.5. . . (n-1) .TT. 2.4.6...n V  n even  2.4.6...(n-1) 1.3.5...n  n odd  The volume required i s thus given by r-l  I = 2 (h 2~n  2  1.3.5...n  n l)2 2 ^  n 2  . \ ,  r  v  (  i  but  n * i=l  i=l l 1  n odd  n  _1 2.4.6...n  v  n  .TT c.  c  i=l i  n even  /.n, '.\ Xi i=l n  c  i  =  /  v  and the product of the eigenvalues of a matrix i s equal to the matrix determinant.  Thus  c. =/v /\k\ l n-1 n  1=1  1  1  n n,Tr.2 ^ V  1 /v/|A| 2.4.1. ..n.  n even  54 REFERENCES Ro E.'Kalman,. J . E. B e r t r a m , " C o n t r o l System A n a l y s i s and D e s i g n V i a  1.  the  Second Method o f L i a p u n o v " ASME T r a n s . , J . o f B a s i c E n g i n e e r i n g ,  June I960, pp. 371-393. 2.  J . L a S a l l e , S. L e f s c h e t z , " S t a b i l i t y by L i a p u n o v ' s D i r e c t Method w i t h A p p l i c a t i o n s " , RIAS, B a l t i m o r e , M a r y l a n d , 1961, Academic P r e s s , N.Y.  3.  S. G. Margolis,.¥. G. Vogt, " C o n t r o l E n g i n e e r i n g A p p l i c a t i o n s o f V. I . Zubov's C o n s t r u c t i o n P r o c e d u r e f o r L i a p u n o v F u n c t i o n s " , IEEE T r a n s , on A u t o m a t i c C o n t r o l , A p r i l 1963, pp. 104-113  4.  R. E. Kalman, " L i a p u n o v F u n c t i o n s f o r t h e P r o b l e m o f L u r ' e i n A u t o m a t i c • C o n t r o l " P r o c . Nat. Acad. S c i . , U.S. 49, 2, 1963. pp. 201-205  5.  M. L. C a r t w r i g h t , "On t h e S t a b i l i t y o f S o l u t i o n s o f C e r t a i n E q u a t i o n s o f F o u r t h Order".-  6.  Differential  Quart. J . Mech. A p p l . Math., 1956, 9,  (2).  D. G. S c h u l t z , J . E. G i b s o n , "The- V a r i a b l e G r a d i e n t Method f o r G e n e r a t i n g L i a p u n o v F u n c t i o n s " , AIEE T r a n s , on A u t o m a t i c C o n t r o l , Sept. 1962.  7.  G. P. Szego, "A C o n t r i b u t i o n t o L i a p u n o v ' s Second Method:  Nonlinear  Autonomous Systems", J . o f B a s i c E n g i n e e r i n g , Dec. 1962. pp. 571-578 8.  W. J . Cunningham,  "An I n t r o d u c t i o n t o L i a p u n o v ' s Second Method^" AIEE  T r a n s , on A p p l . and I n d . , J a n u a r y 1962. pp. 325-332 9-  Y. H. Ku, N. N. P u r i , . "On L i a p u n o v F u n c t i o n s o f H i g h Order N o n l i n e a r Systems", J . o f F r a n k l i n I n s t . , V o l . 276, No. 5, Nov. 1963.  10.  N. N. P u r i , C. N. Weygandt, " L i a p u n o v and Routh's C a n o n i c a l Form", ibid.  11.  D. R. Ingwerson, "A M o d i f i e d L i a p u n o v Method f o r N o n l i n e a r S t a b i l i t y A n a l y s i s " , IRE T r a n s , on A u t o m a t i c C o n t r o l , May 1961. p p  12.  (  199-210  C. S. Chen, E. K i n n e n , " C o n s t r u c t i o n o f L i a p u n o v F u n c t i o n s " , J . o f F r a n k l i n I n s t . , V o l . 289, No. 2, Feb. 1970. pp. 133-146  55  13.  A. H. El-Abiad, K. Nagappan, "Transient S t a b i l i t y Regions of M u l t i machine Power Systems", IEEE Trans, on Power Apparatus and Systems,Vol.  14.  PAS-  85,  No.  2,  Feb.  1966.  pp. 169-179  G. E. Gless, " D i r e c t Method of Liapunov Applied to Transient Power System S t a b i l i t y " , i b i d . pp« 153-168  15.  Y. N. Yu,. K. Vongsuriya," Nonlinear Power System S t a b i l i t y Study by Liapunov Function and Zubov's Method". 12,  and Systems, V o l . PAS-86, No.  16.  M.'W.  PP-  1480-1485  I n t . J . of C o n t r o l , V o l . 8, No. 2, 1968, pp. 131-144.  N. D. Rao, "Routh-Hurwitz Conditions and Liapunov Methods f o r the IEE P r o c , A p r i l 1969.  Transient S t a b i l i t y Problem". 18.  1967.  Siddiquee, "Transient S t a b i l i t y of an A. C. Generator by Liapunov's  . D i r e c t Mdthod." 17-  Dec.  IEEE Trans, on Power Apparatus  PP- 537-547  M. A. P a i , M. A. Mohan, J . G. Rao, "Power System Transient S t a b i l i t y Regions u s i n g Popov's Method".  IEEE Summer Power Meeting, D a l l a s ,  Texas, June 1969. 19-  T. H. Lee, R. J . Fleming, "Power System S t a b i l i t y Studies by the D i r e c t Method of Liapunov". i b i d .  20.  N. D. Rao, A. K. DeSarkar, " A n a l y s i s of a Third Order Nonlinear Power System S t a b i l i t y Problem Through.the Second Method of Liapunov". IEEE Winter Power Meeting, N. Y., January 1970.  21.  G. Luders, "Transient S t a b i l i t y of Multimachine Power Systems V i a the D i r e c t Method of Liapunov".  22.  J . L. Willems, "Optimum Liapunov Functions and S t a b i l i t y Regions For Multimachine Power Systems".  23.  ibid.  IEE P r o c , V o l . 117'/ No. 3, March 1970.  V. I . Zubov, "Methods of A. M. Liapunov and Their Application",.U.S. Atomic Energy Commission, D i v i s i o n of Technical Information, 1957, AEC - t r - 4439 Physics.  56  24.  V. I . Zubov, "Mathematical Methods of I n v e s t i g a t i n g Automatic Regulation Systems".  25.  AEC-.tr-4494.  J . R. Hewit, C. Storey, "Numerical A p p l i c a t i o n of Szego's Method f o r Constructing Liapunov Functions". F e b !  26.  IEEE Trans, on Automatic C o n t r o l ,  pp. 106-108  Y. N. Yu, K. Vongsuriya, "Steady State S t a b i l i t y L i m i t s of a Regulated Synchronous.:. Machine Connected To an I n f i n i t e System".  IEEE Trans, on  Rower Apparatus and Systems, V o l . PAS-85, J u l y 1966, pp. 759-767. 27.  R. V. Shepherd, "Synchronizing and Damping Torque C o e f f i c i e n t s of Synchronous Machines".  AIEE Trans, on Power Apparatus and Systems,  Vol. 80, June 1961, pp. 180-18928.  A. P. Sage, "Optimum Systems C o n t r o l " .  P r e n t i c e H a l l , Inc., Englewood  C l i f f s , N. J . 29.  Lo S. Lasdon, A. D. Waren, "Mathematical Programming f o r O p t i m a l " Design".  30.  E l e c t r o Technology, Nov. 1967.' PP- 55-70  J . B. Rosen, "The Gradient P r o j e c t i o n Method For Nonlinear Programming. Part I:  Linear C o n s t r a i n t s " .  J . Soc. I n d u s t r i a l and A p p l i e d Math..,  No. 8, I960, pp. 181-217. 31.  J . B. Rosen, "The Gradient P r o j e c t i o n Method f o r Nonlinear Programming, Part I I :  32.  Nonlinear C o n s t r a i n t s " .  J . SIAM, Dec. 1961, pp. 514-532.  E. V. Bonn, "A S i m p l i f i e d Algebraic-Geometric Approach To Computational Techniques i n Systems Optimization".  Presented a t the 1970 N a t i o n a l  Conference on Automatic C o n t r o l , NRC A s s o c i a t e Committee on Automatic Control i n Cooperation with U. o f Waterloo. 33.  E. J . Davison, E. M. Kurak, "A Computational Method f o r Determining Quadratic Liapunov Functions For Nonlinear Systems". JACC, 1970.  Presented a t the  

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