A fast method for determining the voltage stability limit

8/7/2019 A fast method for determining the voltage stability limit

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E L S E V I E R Electr ic Power Systems Research 32 (1995) 35-43

E L E O T R I CP O I ,tJ E R $

A f a s t m e t h o d f o r d e t e r m i n i n g t h e v o l t a g e s t a b i l i t y l i m i t o fa p o w e r s y s t e mM . H . H a q u e

Department o[" Electrical and Comp uter Sys tem s E ngineering, Monash University', Clayton, Vic. 3168, AustraliaAccepted 19 August 1994

A b s t r a c t

The vo l tage s tabil i ty problem of a pow er system is associated with a rapid v ol tage dro p due to heavy sys tem load, an d i t occursbecause of inadequate react ive power support a t some cri t ical buses . One of the serious consequences of the vol tage s tabi l i typroblem is a sys tem blackout , and this problem has received much at tent ion in recent years . This paper proposes a fas t methodfor f inding the m aximu m load, especial ly the react ive pow er dema nd, at a part icular load bus before reaching the vol tage s tabi li tyl imit . The m ethod uses the base-case system info rma tion to f ind special two-bus equivalents of the sys tem for analyzing the vol tages tabil ity problem . The m etho d was tes ted on the IE EE 14-, 30-, and 118-bus sys tems and the resul ts o btained were compar ed withthose found by some o ther methods .Keywords: Voltage stability; Network reduction; Power system security

1 . I n t r o d u c t i o n

T r a n s m i s s i o n l i n e s i n a p o w e r s y s t e m a r e l o a d e dm o r e h e a v i l y t h a n e v e r b e f o r e t o a v o i d t h e c a p i t a l c o s to f b u i l d i n g n e w l i n e s. F o r a s h o r t l in e , t h e l o a d i n gc a p a b i l i t y m a y b e r e s t r i c t e d b y t h e t h e r m a l l i m i t .H o w e v e r , f o r a l o n g l i n e , t h e l o a d i n g c a p a b i l i t y m a yb e d i c t a t e d b y t h e v o l t a g e s t a b i l i t y r a t h e r t h a n t h et h e r m a l o r t r a n s i e n t s t a b i l i t y l i m i t . W h e n a p o w e r s y s -t e m a p p r o a c h e s t h e v o l t a g e s t a b i l i t y l i m i t , t h e v o l t a g eo f s o m e b u s e s r e d u c e s r a p i d l y f o r s m a l l i n c r e m e n t s i nl o a d a n d t h e c o n t r o l s o r o p e r a t o r s m a y n o t b e a b l e t op r e v e n t t h e v o l t a g e d e c a y . I n s o m e c a s e s , t h e r e s p o n s eo f c o n t r o ls o r o p e r a t o r s m a y a g g r a v a te t h e s i tu a t io na n d t h e u l t i m a t e r e s u l t i s v o l t a g e c o l l a p s e . V o l t a g ec o l l a p s e h a s b e c o m e a n i n c r e a s i n g t h r e a t t o p o w e rs y s t e m s e c u r i t y a n d r e l i a b i l i t y . M a n y i n c i d e n t s o f s y s -t e m b l a c k o u t s b e c a u s e o f v o l t a g e s t a b i l i t y p r o b l e m sh a v e b e e n r e p o r t e d w o r l d w i d e ( s e e R e f s . [ 1 , 2 ] f o r as a m p l e o f t h e l i t e r a t u r e ) . N o w a d a y s , a p r o p e r a n a l y s i so f t h e v o l t a g e s t a b i l i t y p r o b l e m h a s b e c o m e o n e o f t h em a j o r c o n c e r n s i n p o w e r s y s t e m o p e r a t i o n a n d p l a n -n i n g s t u d i e s .

T h e m a i n r e a s o n f o r v o l t a g e i n s t a b i l i t y i n a p o w e rs y s t e m i s i n a d e q u a t e r e a c t i v e p o w e r s u p p o r t a t s o m ec r i t i c a l b u s e s . V o l t a g e i n s t a b i l i t y i s a r e a c t i v e p o w e rp r o b l e m . U n l i k e a c t i v e p o w e r , i t i s v e r y d i f f i c u l t t oe s t i m a t e t h e r e a c t i v e p o w e r m a r g i n r e q u i r e d t o a c h i e v e0378-7796/95/$09.50 ~ 1995 Elsevier Science S.A. All rights reservedS S D I 0378-7796(94 ) 00893 -9

a c e r t a i n d e g r e e o f v o l t a g e s e c u r i t y . W h e n t h e v o l t a g eo f a s y s t e m s t a rt s t o d e c r e a s e , t h e c u r r e n t , a n d h e n c et h e r e a c t i v e p o w e r l o s s i n t r a n s m i s s i o n l i n e s a n d t r a n s -f o r m e r s , i s i n c r e a s e d . O n t h e o t h e r h a n d , a d e c r e a s e i nv o l t a g e r e d u c e s t h e r e a c t i v e p o w e r s u p p l y b y t h e l i n ec h a r g i n g a n d s h u n t c a p a c i t o r s . T h u s t h e v o l t a g e r e d u c -t i o n h a s a c u m u l a t i v e e f f e c t u n l e s s a m p l e r e a c t i v e p o w e rs o u r c e s o r s o m e a p p r o p r i a t e c o n t r o l s a r e a v a i l a b l e t or e g u l a t e t h e v o l t a g e a n d m a i n t a i n t h e r e a c t i v e p o w e rb a l a n c e .

E v e n t h o u g h t h e s y m p t o m o f i m m i n e n t v o l t a g e c o l -l a p s e i s r a p i d v o l t a g e r e d u c t i o n , t h e v o l t a g e m a g n i t u d ei t se l f i s n o t a g o o d i n d i c a t o r o f t h e p r o x i m i t y o f v o l ta g ec o l l a p s e [ 3 ] . S e v e r a l a p p r o a c h e s f o r a n a l y z i n g t h ev o l t a g e i n s t a b i l i t y p r o b l e m h a v e b e e n r e p o r t e d i n t h el i t e r a t u r e . V e n i k o v e t a l. [ 4] e s t i m a t e d t h e v o l t a g e s t a b il -i ty li m it f r o m t h e c o n v e rg e n c e o f t h e N e w t o n - R a p h s o n( N R ) l o a d f l o w c a l c u l a t i o n . H o w e v e r , i n t h e v i c i n i t y o ft h e v o l t a g e i n s t a b i l i t y p o i n t , t h e d i v e r g e n c e o f t h e N Rm e t h o d m a y b e c a u s e d e i th e r b y n u m e r ic a l p r o b l e m s o rb y t h e f a c t t h a t t h e i n s t a b il i ty c o n d i t i o n h a s a l r e a d y b e e nr e a c h e d . T h e m e t h o d i s v e r y ti m e c o n s u m i n g b u t i t g iv e sr e a s o n a b l y g o o d r e s u l t s . T h e m i n i m u m s i n g u l a r v a l u e o ft h e J a c o b i a n m a t r i x o f t h e N R l o a d f lo w e q u a t io n s h a sa l s o b e e n u s e d b y s o m e r e s e a r c h e r s t o d e t e r m i n e t h ev o l t a g e s t a b i l i t y i n d e x [ 5 , 6 ] .

T h e n o n l i n e a r l o a d f l o w e q u a t i o n s h a v e m a n y s o l u -t i o n s . O n e o f t h e s o l u t i o n s i s c a l l e d th e s t a b l e o r


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36 M.H . Haque /E lec t r ic Pow er Sys tems Research 32 (1995) 35-43h i g h - v o l t a g e s o l u t i o n . T h e r e s t o f t h e s o l u t i o n s a r ec a l l e d t h e u n s t a b l e o r l o w - v o l t a g e s o l u t i o n s . S o m e r e -s e a r c h e r s h a v e u s e d m u l t i p l e l o a d f l o w s o l u t i o n s t od e t e r m i n e t h e v o l t a g e s t a b i l i t y l i m i t [ 7 , 8 ] . A f o r m a lm a t h e m a t i c a l w a y o f a n a l y z i n g t h e v o l t a g e s t a b i l i t yp r o b l e m u s i n g m u l t i p l e l o a d f l o w s o l u t i o n s i s b a s e d o nb i f u r c a t i o n p h e n o m e n a [ 9 ] . R e f s . [ 1 0 , 1 1 ] u s e d e n e r g ym e t h o d s t o a s s e s s t h e v o l t a g e s t a b i l i t y . E n e r g y m e t h o d sr e q u i r e b o t h t h e h i g h - a n d l o w - v o l t a g e s o l u t i o n s o f t h el o a d f l o w e q u a t i o n s . T h e v o l t a g e s t a b i l i t y p r o b l e m c a na ls o b e a n a l y z e d b y th e Q - V a n d P - V c u r v es [3 ]. T h er e l a t i o n s h i p b e t w e e n t h e e n e r g y b a s e d v o l t a g e s e c u r i t ya n d t h e s e c u r v e s i s d e m o n s t r a t e d i n R e f . [ 1 2 ] .

C h e b b o e t a l . [ 1 3 ] u s e d t h e c o n c e p t o f t h e t w o - b u st h e o r y t o e s t i m a t e th e m a x i m u m l o a d in g c a p a b i l it y o f ap a r t i c u l a r l o a d b u s i n a p o w e r s y s t e m . I n t h i s m e t h o dt h e p o w e r s y s t e m i s f i r s t r e p l a c e d b y t h e T h 6 v e n i nt h e o r y t o g e t t h e t w o - b u s e q u i v a l e n t m o d e l . T o i n c l u d et h e e f fe c ts o f n o n l i n e a r i t y o f l o a d s a n d g e n e r a t o r s i n t h ee q u i v a l e n t c i r c u i t , s o m e r e p e t i t i v e c o m p u t a t i o n i n t h eo r i g i n a l s y s t e m i s r e q u i r e d t o g e t t h e u l t i m a t e r e s u l t s . I ne a c h s t ep , t h e m e t h o d r e q u i r e s th e s o l u t i o n o f t h e l o a df lo w e q u a ti o n s a n d c o m p u t a t i o n o f th e Z m a t r i x f o r t hee n t i r e s y s te m . T o f i n d t h e l o a d i n g c a p a b i l i t y o f a d i f f e r-e n t l o a d b u s , t h e e n t i r e p r o c e d u r e h a s t o b e r e p e a t e d .T h i s s e e m s t o b e v e r y t i m e c o n s u m i n g a n d m a y n o t b ev e r y u s e f u l i n p r a c t i c e . W h e n a p o w e r s y s t e m i s r e p r e -s e n t e d b y s u c h a n e q u i v a l e n t m o d e l , t h e m e t h o d o fR e f s . [ 14 , 15 ] c a n a l s o b e u s e d t o e s t i m a t e t h e m a x i m u ml o a d i n g c a p a b i l i t y .

T h i s p a p e r d e t e r m i n e s t h e m a x i m u m l o a d i n g c a p a -b i l i t y o f a p a r t i c u l a r l o a d b u s i n a p o w e r s y s t e mt h r o u g h t h e T h 6 v e n i n e q u i v a l e n t c i r c u i t . T h e e q u i v a l e n tc i r c u i t s o f a l l l o a d b u s e s a r e o b t a i n e d i n a s i n g l e s h o t .S p e c i a l c a r e h a s b e e n t a k e n i n m o d e l i n g t h e g e n e r a t o r st o r e f l e c t a c t u a l o p e r a t i o n , e v e n f o r a c h a n g e i n o p e r a t -i n g c o n d i t i o n s . U n l i k e t h e o t h e r m e t h o d s [ 1 2 , 1 3 ] , t h ep r o p o s e d a p p r o a c h c a n p r o v i d e v e r y g o o d r e s u l t s w i t hl e s s c o m p u t a t i o n u s i n g t h e b a s e - c a s e s y s t e m i n f o r m a -t i o n . T h e m e t h o d h a s b e e n t e s t e d o n t h r e e d i f f e r e n tI E E E s t a n d a r d t e s t s y s te m s f o r a n u m b e r o f c as e s.

2 . Problem formulat ion

T h e o b j e c t i v e o f t h i s s e c t i o n i s t o d e m o n s t r a t e t h ec o n c e p t o f th e v o l t a g e s t a b i li t y p r o b l e m i n a v e r y si m p l et w o - b u s s y s t e m . T h e s a m e c o n c e p t i s t h e n a p p l i e d t o ag e n e r a l p o w e r s y s t e m t o d e t e r m i n e t h e m a x i m u m l o a d -i n g c a p a b i l i t y o f a p a r t i c u l a r l o a d b u s w i t h i n t h ev o l t a g e s t a b i l i t y l i m i t . T h i s r e q u i r e s a s p e c i a l t w o - b u se q u i v a l e n t m o d e l o f t h e g e n e r a l p o w e r s y s t e m . A p r o c e -d u r e f o r f i n d i n g th e e q u i v a l e n t m o d e l o f a p o w e r s y s t e mi s d e s c r i b e d i n t h e f o l l o w i n g s e c t i o n s .

C o n s i d e r a s i m p l e t w o - b u s s y s t e m a s s h o w n i n F i g .1 . T h e g e n e r a t o r a t b u s 1 t r a n s f e r s p o w e r t h r o u g h a

R jx

F i g . I . A s i m p l e t w o - b u s s y s t e m .

V ~ L S ~

2

S=P +jQ

t r a n s m i s s i o n l i n e h a v i n g a n i m p e d a n c e o f Z = R + j Xt o a l o a d c e n t e r a t b u s 2 . B u s 1 is c o n s i d e r e d a s a s w i n gb u s w h e r e b o t h t h e v o l t a g e m a g n i t u d e V , a n d a n g l e 6 ,a r e k e p t c o n s t a n t . F o r a g i v e n v a lu e o f V , , t h e r e l at i o n -s h i p b e t w e e n t h e l o a d v o l t a g e m a g n i t u d e V2 a n d t h el o a d p o w e r S = P + j Q c a n r e a d i l y b e w r i t t e n a s

p z + Q 2= II22 + 2 ( R P + X Q ) + (R 2 + X 2) - - (1 )V2a s s u m i n g x = V 22 , t h e a b o v e e q u a t i o n c a n b e w r i t -i n a q u a d r a t i c f o r m a s f o l l o w s :

V 12B yte na l x 2 + b , x + C l = 0w h e r ea l = lb l = 2 ( R P + X Q ) - V , 2c, = (R 2 + X 2 ) ( P 2 + Q 2 )

(2 )

T h e p o s i t i v e v o l t a g e m a g n i t u d e s o f b u s 2 c a n b e o b -t a i n e d f r o m t h e s o l u t i o n o f E q . ( 2 ) a n d a r e g i v e n b yV ~ = ( - b , ~ - d l / 2 ) 1/2

2a l (3a )V ~ ( - b l - d l / 2 ~ 1/2= \ 2a~ / (3 b)w h e r e t h e d i s c r i m i n a n t d i s g i v e n b yd = b , 2 - 4 a ' c l

= V . 4 + 4 [ 2 P Q R X - V , Z ( R P + X Q ) - R Z Q 2 - X 2 P 21(4 )

H e r e , V 2 " i s c a l l e d t h e h i g h - v o l t a g e o r s t a b l e s o l u t i o nw h i l e V ~ i s c a l l e d t h e l o w - v o l t a g e o r u n s t a b l e s o l u t i o n .F o r z e r o l o a d ( P = Q = 0 ) , V ~ a n d V ~ b e c o m e V l a n d0 , r e s p e c t i v e l y . A s t h e l o a d ( a t n o r m a l p o w e r f a c t o r ) i si n c r e a s e d f r o m z e r o , V 2n d e c r e a s e s w h i l e V ~ i n c r e a s e s .T h i s p r o c e s s c o n t i n u e s u n t i l a p o i n t i s r e a c h e d w h e r eb o t h V ~ a n d V ~ b e c o m e t h e s a m e . T h i s o c c u r s w h e nt h e v a l u e o f d i n E q . ( 4 ) b e c o m e s z e r o . T h e l o a d p o w e rf o r w h i c h V ~ = V ~ i s c a l l e d th e c r i t i ca l p o w e r a n d t h ec o r r e s p o n d i n g v o l t a g e i s c a l l e d t h e c r i t i c a l v o l t a g e . I t i ss a i d t h a t t h e s y s t e m h a s r e a c h e d t h e v o l t a g e s t a b i l i t yl im i t a n d i t is n o t c a p a b l e o f t r a n s f e r r in g a n y a d d i t i o n a lp o w e r . F o r h i g h e r l o a d p o w e r , t h e r e a l s o l u t i o n o f E q .( 2 ) ( a n d h e n c e t h e m a g n i t u d e o f I /'2 ) w i ll c e a s e t o o c c u rb e c a u s e o f t h e n e g a t i v e v a l u e o f d .


3/9

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+ + + - 2 ! ,I I i I / j ~/]0.4 i t/ J// l / / // /

i / , , / / /, ~ . I ],'+

O* O I I I I i i i i i [ ~ i i i i i i i i i i i i i i i i 4 i i i i i i i i i i i I0.0 2.0 4 .0 6.0 8.0L o a d a p p a r e n t p o w e r , p u

Fig. 2. Variation of load vo ltage against the load apparen t powe r forvarious power factors.

T y p i c a l v a r i a t i o n s o f l o a d v o l t a g e a g a i n s t t h e l o a da p p a r e n t p o w e r f o r v a r i o u s p o w e r f a c t o r s ( P F s ) a r esho wn in F ig . 2 . Th e f igure i s p lo t t ed for V1 = 1 .0 p .u . ,R = 0 . 0 1 p . u . a n d X = 0 . 1 p . u . T h e h i g h - v o l t a g e o r s t a -b l e s o l u t i o n i s r e p r e s e n t e d b y f u l l c u r v e s w h i l e t h el o w - v o l t a g e o r u n s t a b l e s o l u t i o n i s r e p r e s e n t e d b y b r o -k e n c u r v e s . T h e s e t w o c u r v e s o r v o l t a g e s m e e t a t t h ec r i t i c a l p o i n t ( m a r k e d b y " < i n t h e f i g u r e ) . I t c a n b eo b s e r v e d i n t h e f i g u r e t h a t b o t h t h e m a x i m u m l o a da p p a r e n t p o w e r a n d c r it ic a l v o l t a g e i n c r e as e a s t h e lo a dP F c h a n g e s f r o m l a g g i n g t o l e a d i n g . T h e o b j e c t i v e i s t of i n d t h e m a x i m u m l o a d a p p a r e n t p o w e r a n d t h e c o r r e -s p o n d i n g v o l t a g e . O b v i o u s l y , t h e c o n d i t i o n o f t h e m a x -i m u m l o a d a p p a r e n t p o w e r ( S i n ) c a n b e o b t a i n e d b ys e t t i n g t h e v a l u e o f d i n E q . ( 4 ) t o z e r o . T h i s g i v e s t h ef o l l o w i n g q u a d r a t i c e q u a t i o n :a2 Sm 2 _L b2 Sm + C2 = 0 (5 )w h e r ea 2 = 4 [ R X s in(2 0) - R 2 s in20 - X 2 cos20]b 2 = - 4 V ~ ( R cos 0 - - X s in 0)C2= V 14I n d e r i v i n g E q . ( 5 ) , i t is c o n s i d e r e d t h a t P = S c o s 0a n d Q = S s i n 0 , w h e r e 0 i s t h e P F a n g l e . T h e v a l u e o fS m c a n b e o b t a i n e d f r o m t h e s o l u t i o n o f E q . ( 5 ) :

Vt 2 Z - (R co s 0 + X s in 0)S i n - 2 ( R s in 0 + X c o s 0 ) 2 ( 6 )H e r e Z = ( R 2 + X 2) ~ /2 . N o t e t h a t t h e o t h e r s o l u t i o n o fE q . ( 5 ) i s n o t f e a s i b l e b e c a u s e o f i t s n e g a t i v e v a l u e .O n c e t h e v a l u e o f S m i s k n o w n , t h e c o r r e s p o n d i n gc r i t ic a l v o l t a g e V c r c a n b e o b t a i n e d f r o m e i t h e r E q . ( 3 a )o r ( 3 b ) b y s e t t i n g d = 0 a n d e v a l u a t i n g t h e c o e f f i c i e n t b la t t h e m a x i m u m l o a d a p p a r e n t p o w e r S i n . T h i s g i v e s

V c r = I V l 2 - 2 S m ( R C o s O + X s in O )lJ /22 (7)E q s . ( 6 ) a n d ( 7 ) r e p r e s e n t t h e m a x i m u m l o a d a p p a r e n tp o w e r a n d t h e c o r r e s p o n d i n g v o l t a g e , r e sp e c t iv e l y , f o r ag i v e n l o a d P F a n g l e 0 . T h e m a x i m u m a c t i v e p o w e rl o a d i n g P m ( w i t h Q = 0 ) a n d t h e c o r r e s p o n d i n g v o l t a g ec a n b e o b t a i n e d f r o m E q s . ( 6 ) a n d ( 7 ) , r e s p e c t i v e l y , b yse t t ing 0 = 0 :

V I z ( Z - - R )P m - 2 X 2 ( 8a )= ( 8 b )

S i m i l a r l y , t h e m a x i m u m r e a c t i v e p o w e r l o a d i n g Q m a tl a g g i n g P F ( w i t h P = 0 ) a n d t h e c o r r e s p o n d i n g v o l t a g ec a n b e w r i t t e n a s ( b y s e t t i n g 0 = 9 0 )

V I 2 ( Z - X )Q m - 2 R 2 ( 9 a )V cr = ( V '2 - -~ 2QmX ) ' / z ( 9 b )I t c a n a l s o b e n o t i c e d i n F i g . 2 t h a t t h e c r i t i c a l v o l t a g ef o r u n i t y a n d l a g g i n g P F l o a d s i s s i g n i f i c a n t l y l e s s t h a nt h e n o m i n a l v a l u e o f 1 .0 p . u . a n d i t m a y n o t b e a c c e p t -a b l e i n p r a c t i c e . I n p o w e r s y s t e m o p e r a t i o n , i t m a y b en e c e ss a ry t o k n o w t h e m a x i m u m l o a d a t w h i c h t h e l o adv o l t a g e s h o u l d n o t d r o p b e l o w a s p e c i f i e d v a l u e V 2 sp( > V o r) . T h e c o r r e s p o n d i n g l o a d S sp c a n b e o b t a i n e df r o m E q . ( 1 ) b y s e t t i n g V2 = V z sp . T h i s i n v o l v e s t h es o l u t i o n o f t h e f o l l o w i n g q u a d r a t i c e q u a t i o n :a3S sp 2 -}- b 3 S s p -F C3 : 0 10 )w h e r ea 3 = R 2 _]_ X 2b 3 = 2 V 2w2(R cos 0 + X s in O)C3 V2 sp4 2 2= - - V 1 V 2 s pT y p i c a l v a r i a t i o n s o f l o a d a p p a r e n t p o w e r S + p a g a i n s tt h e l o a d P F a n g l e f o r v a r i o u s s p e c if i ed l o a d v o l t a g e s a r es h o w n i n F i g . 3 . I t c a n b e s e e n i n t h e f i g u r e t h a t t h el o a d a p p a r e n t p o w e r d e c r e a s e s a s t h e l a g g i n g P F a n g l ei s i n c r e a s e d . F o r a g i v e n P F , t h e l o a d a p p a r e n t p o w e ri n c r e a s e s a s t h e s p e c i f i e d l o a d v o l t a g e i s d e c r e a s e d .

3 . V o l t a g e s t a b i l i t y l i m i t o f a g e n e r a l p o w e r s y s t e mI n g e n e r a l , t h e d e t e r m i n a t i o n o f t h e v o l t a g e s t a b i l i t y

l i m i t o f a g e n e r a l p o w e r s y s t e m i s a v e r y d i f f i c u l t t a s k .A q u i c k e r w a y t o f i n d t h e v o l t a g e s t a b i l i t y l i m i t o f ap o w e r s y s t e m i s t h r o u g h t h e t w o - b u s o r T h 6 v e n i ne q u i v a l e n t c i r cu i t o f th e s y s t em . N o t e t h a t t h e o p e r a t i n gp o i n t o f t h e g e n e r a t o r s a t t h e v e r g e o f v o l t a g e s t a b i l it y


4/9

38 M.H. Haque /Electric' Power Systems Research 32 (1995) 35 435 . 0

~ 4 . 0Q)0~ . 3 . 0

~ 2 . 00

1 .0

0 .0 ' ' ' ' ' ' ' ' ll ' ' ' l ' l ' llllll 'l 'l ' lll '~ll, , , , ll, , , lllll ]0 1 0 2 0 3 0 4 0 5 0L a g g i n g p f a n g l e , d e g r e e

F i g . 3 . V a r i a t i o n o f l o a d a p p a r e n t p o w e r a g a i n s t t h e l o a d P F a n g l ef o r var ious s pec i f i ed va lues o f l oad vo l t age .

m a y d i f f e r s i g n i f i c a n t l y f r o m t h e b a s e - c a s e o p e r a t i n gp o i n t . T h u s , t h e T h 6 v e n i n e q u i v a l e n t c ir c u i t o b t a i n e d a tt h e b a s e c a s e w i t h a c o n v e n t i o n a l g e n e r a t o r m o d e l m a yn o t r e p r e s e n t a g o o d e q u i v a l e n t c i r c u it t o d e t e r m i n e t h ev o l t a g e s t a b i l i t y l i m i t u n l e s s s o m e s p e c i a l c a r e i s t a k e ni n m o d e l i n g t h e g e n e r a t o r s . Ch e b b o e t a l . [ 1 3 ] o v e r c o m et h i s p r o b l e m f i r s t b y f i n d i n g a n e q u i v a l e n t c i r c u i t a t t h eb a s e c a s e a n d p r e d i c t i n g t h e m a x i m u m l o a d i n g c a p a b i l -i t y o f a p a r t i c u l a r l o a d b u s i n t h e e q u i v a l e n t c i r c u i t . T h el o a d o f t h a t b u s i s t h e n i n c r e a s e d t o t h e p r e d i c t e d v a l u ea n d a s e c o n d e q u i v a l e n t c i rc u i t is o b t a i n e d f o r t h e n e wo p e r a t i n g p o i n t . T h e p r o c e s s i s r e p e a t e d u n t i l t h e l o a df l o w s o l u t i o n o f t h e o r i g i n a l n e t w o r k f o r t h e p r e d i c t e dl o a d f a i l s t o c o n v e r g e . T h e m e t h o d u l t i m a t e l y p r o d u c e sa n e q u i v a l e n t c i r c u i t t h a t h a s b e e n o b t a i n e d f o r a l o a dc o n d i t i o n w h i c h i s v e r y c l o s e t o t h e v o l t a g e s t a b i l i t yl i m i t . T h e r e p e t i t i v e c o m p u t a t i o n s i n t h i s m e t h o d s e e mt o b e v e r y t i m e c o n s u m i n g . I n t h i s p a p e r s p e c i a l a t t e n -t i o n h a s b e e n g i v e n i n m o d e l i n g t h e g e n e r a t o r s t o a v o i dt h e s e r e p e t i t i v e c o m p u t a t i o n s a n d t h e T h 6 v e n i n e q u i v a -l e n t c i r c u i t s o b t a i n e d a t t h e b a s e - c a s e o p e r a t i n g p o i n tm a y b e f a i t h f u l l y u s e d t o d e t e r m i n e t h e v o l t a g e s t a b i li t yl imi t .3 .1 . G e n e r a t o r m o d e l

C o n s i d e r a g e n e r a l p o w e r s y s t e m a s s h o w n i n F i g .4 ( a ) . Bu s e s 1 t o m a r e t h e g e n e r a t o r b u s e s w h e r e t h ev o l t a g e m a g n i t u d e s a r e k e p t c o n s t a n t . Bu s e s m -4- 1 t o na r e t h e l o a d b u s e s . T h e a i m i s t o f i n d t h e m a x i m u ml o a d i n g c a p a b i l i t y o f a p a r t i c u l a r l o a d b u s k w i t h i n t h ev o l t a g e s t a b i l i t y l i m i t . I n g e n e r a l , t h e g e n e r a t o r s i n ap o w e r s y s t e m a r e m o d e l e d b y i n t e r n a l v o l t a g e s o u r c e sE g w i t h s e r i e s r e a c t a n c e s X g , a s s h o w n i n t h e f i g u r e .W h e n t h e s y s t e m l o a d o r o p e r a t i n g p o i n t c h a n g e s , t h eg e n e r a t o r i n t e r n a l v o l t a g e s E g a r e a d j u s t e d a c c o r d i n g l y

V = 0 . 7 5 p uV = 0 . 8 0 p uV = 0 , 8 5 p u

V = 0 . 9 0 p u

V = 0 . 9 5 p u

jx ,~m

J X g l

~ Es~(a)

m + l

jXs. - 0 g gmm

j X I = 0 V c ,

I( b )

F i g . 4 . G e n e r a t o r m o d e l i n a p o w e r s y s t e m .

m +l

n - - - ~k

t o m a i n t a i n c o n s t a n t t e r m i n a l v o l t a g e s Vg a s l o n g a s t h er e a c t i v e p o w e r g e n e r a t i o n o f t h e g e n e r a t o r s i s w i t h i nt h e l i m i t s . Be c a u s e o f t h e v a r i a b l e n a t u r e o f E g , t h eT h 6 v e n i n e q u i v a l e n t c i r c u it o b t a i n e d f o r a g i v e n o p e r a t -i n g p o i n t , w i t h t h e a b o v e g e n e r a t o r m o d e l , m a y n o t b eu s e d f o r a d i f f e r e n t o p e r a t i n g p o i n t u n l e s s t h e y a r ev e r y c l o s e . H o w e v e r , t h e v o l t a g e i n s t a b i l i t y p o i n t i s , i ng e n e r a l , f a r a w a y f r o m t h e b a s e - c a s e o p e r a t i n g p o i n t .S o m e r e s e a r c h e r s h a v e m o d e l e d t h e g e n e r a t o r s b yc o n s t a n t i m p e d a n c e s w i t h a n a p p r o p r i a t e s i g n . T h o s ei m p e d a n c e s a r e a g a i n o b t a i n e d f r o m t h e b a s e - c a s e l o a df l o w s o l u t i o n . A b e t t e r g e n e r a t o r m o d e l f o r t h e a n a l y s i so f t h e v o l t a g e s t a b i l i t y p r o b l e m i s p r o p o s e d i n t h ef o l l o w i n g .

A s m e n t i o n e d e a r li e r, t h e g e n e r a t o r i n t e r n a l v o l t a g e sE g a r e a d j u s t e d w i t h t h e s y s t e m l o a d t o m a i n t a i n c o n -s t a n t t e r m i n a l v o l t a g e s V g . T h e d i f f e r e n c e b e t w e e n E ga n d V g i s t h e v o l t a g e d r o p a c r o s s X g . W i t h t h i s i n m i n d ,t h e g e n e r a t o r s c a n b e c o n s i d e r e d a s E g = Vg = c o n s t a n tw i t h X g ~ - 0 , a s s h o w n i n F i g . 4 ( b ) . W h e n r e a c t i v ep o w e r g e n e r a t i o n i s c h a n g e d d u e t o t h e c h a n g e i nr e a c t i v e p o w e r d e m a n d i n t h e s y s t e m , t h e a n g l e o f V g


5/9

M . H . H a q u e / E l e c t r i c P o w e r S y s t e m s R e s e a rc h 3 2 (1 9 95 ) 3 5 4 3 3 9

( o r E g ) w i l l n o t ch a n g e s i g n i f i can t l y b ecau s e t h e an g l em a i n l y d e p e n d s o n a c t i v e p o w e r . T h u s t h e g e n e r a t o r s i nF i g . 4 ( b ) can b e co n s i d e r ed a s i d ea l v o l t ag e s o u r cesw h e r e b o t h t h e v o l t a g e m a g n i t u d e s a n d a n g l e s a r e m o r eo r l e ss c o n s t a n t a n d i n t e r n a l i m p e d a n c e s a r e n e g l i g ib l e .T h i s m o d e l p r e s e r v es t h e t e r m i n a l ch a r ac t e r i s t i c s o f t h eg e n e r a t o r s e v e n f o r a c h a n g e i n o p e r a t i n g c o n d i t i o n s .S u c h a m o d e l c a n b e f a i t h f u l l y u s e d t o d e t e r m i n e t h ev o l t ag e s t ab i l i t y l i m i t d u e t o t h e ch an g e i n r eac t i v ep o w e r d e m a n d a s l o n g a s t h e r e a c t i v e p o w e r g e n e r a t i o no f t h e g en e r a t o r s i s w i t h i n t h e l i m i t s . H o w ev e r , i f i tex ceed s t h e l i m i t s f o r s o m e g en e r a t o r b u s es , t h o s e b u s esc a n b e c o n s i d e r e d a s l o a d b u s e s b y s e t t i n g t h e v a l u e o fQ t o t h e l i m i t i n g v a l u es .

W h e n t h e a c t i v e p o w e r l i m i t o r t h e a p p a r e n t p o w e rl i m i t ( f o r n o n z e r o P F s ) is d e t e r m i n e d w i t h t h is g e n e r a -t o r m o d e l , s o m e e r r o n e o u s r e s u l t s m a y o c c u r . I n t h i sc a s e t h e v a r i a t i o n o f a c t i v e p o w e r g e n e r a t i o n w i l lc h a n g e t h e a n g l e o f V g . T h u s , a g e n e r a t o r c a n b ec o n s i d e r e d a s a s o u r c e w i t h c o n s t a n t t e r m i n a l - v o l t a g em a g n i t u d e b u t v a r i a b l e a n g l e . S i m u l a t i o n r e s u l t s i n d i -ca t ed t h a t t h e e r r o r i n v o l v ed i n t h i s ca s e i s n o t s i g n i f i -can t b ecau s e t h e ch an g e i n an g l e i s n o t a s h i g h a s i n t h ec a s e o f a n g l e i n s t a b i l i t y . I t m a y a g a i n b e m e n t i o n e dh e r e t h a t t h e v o l t ag e s t ab i l i t y p r o b l em i s a r eac t i v ep o w e r p r o b l e m a n d t h e m a i n o b j e c t i v e o f t h i s p a p e r i st o d e t e r m i n e t h e r eac t i v e p o w er l i m i t .

3.2. Thkvenin equivalent circuit beh ind a load busT h e n o - l o a d o r T h 6 v e n i n e q u i v a l e n t v o l t a g e V t h o fl o a d b u s k c a n b e o b t a i n e d f r o m t h e l o a d f l o w s o l u t io n .

T h e l o a d f l o w s o l u t i o n is to b e d e t e r m i n e d b y c o n s i d e r -i n g a l l l o ad s i n t h e s y s t em ex cep t a t b u s k . T h e T h 6 v -e n i n e q u i v a l e n t i m p e d a n c e Z t h o f b u s k i s t h e k t hd i a g o n a l e l e m e n t o f t h e Z m a t r i x . I n c a l c u l a t i n g t h e Zm a t r i x , l o ad s o f a l l b u s es ( ex cep t a t b u s k ) a r e t o b er e p l a c e d b y c o n s t a n t i m p e d a n c e s a n d t h e g e n e r a t o r s a r et o b e r ep l aced b y n eg l i g i b l e r eac t an ces . T h e r e l a t i o n s h i pb e t w e e n t h e v o l t a g e , i m p e d a n c e a n d l o a d o f b u s i is

I v ' 1 2 ( l l )P, - JQi

N o t e t h a t , a t t h e v e r g e o f v o l t a g e s t a b i l i t y , t h e c o n -s t a n t - i m p e d a n c e l o a d m o d e l m a y r e p r e s e n t l e s s l o a dt h a n i t s h o u l d b e c a u s e o f v o l t a g e r e d u c t i o n a t s o m eb u s es . T h u s , t h e T h 6 v en i n eq u i v a l en t c i r cu i t w i t h t h ea b o v e l o a d m o d e l m a y p r o v i d e p e s s i m is t ic r e su l t s a t t h ev e r g e o f v o l t ag e s t ab i l i t y . H o w ev e r , t h e e r r o r i s n o ts i g n i f i can t b ecau s e t h e l o ad i m p ed an ces a r e , i n g en e r a l ,m u c h h i g h e r t h a n t h e l i n e i m p e d a n c e s o r g e n e r a t o ri m p e d a n c e s ( X g ~ 0 ). T h u s , t h e l o a d i m p e d a n c e s h a v ev e r y l it t le e f fec t o n t h e d i ag o n a l e l em en t o f t h e Z m a t r i xo r t h e T h 6 v e n i n i m p e d a n c e . I n o r d e r t o f i n d t h e T h 6 v -en i n eq u i v a l en t c i r cu i t b eh i n d a d i f f e r en t b u s , t h e en t i r e

jX g~ =--0

jXsj = 0

m N

m + l

F i g . 5 . T h ~ v e n i n i m p e d a n c e o f b u s l o a d k .

:b

Z~ ,

p r o ce d u r e h as t o b e r ep ea t ed . T h i s ~s a v e r y t i m ec o n s u m i n g p r o c e s s .

A v e r y f a s t a p p r o a c h t o d e t e r m i n e t h e T h 6 v e n i neq u i v a l en t c i r cu i t s o f a l l l o ad b u s es i n a s i n g l e s h o t i sp r o p o s e d i n t h e f o l l o w i n g s e c ti o n s. T h e p r o p o s e d a p -p r o ach u s e s t h e r e s u l t s o f a s i n g l e l o ad f l o w s o l u t i o na n d t h e s y s t e m Z m a t r i x . B o t h t h e l o a d f l o w s o l u t i o na n d t h e Z m a t r i x a r e o b t a i n e d b y c o n s i d e r i n g a l l t h el o a d s i n t h e s y s t e m . T h e v o l t a g e a n d i m p e d a n c e o f t h eT h 6 v e n i n e q u i v a l e n t c i r c u i t a r e t h e n o b t a i n e d b ys l ig h t ly m o d i f y i n g t h e l o a d f l o w s o l u t i o n a n d t h e d i a g o -n a l e l em en t s o f t h e Z m a t r i x i n o r d e r t o n u l l i f y t h ee f f e c t s o f l o a d i m p e d a n c e a t t h e c a n d i d a t e b u s .3.2.1. Determ inat ion o f the Thkvenin impedanceL e t Z kk b e t h e k t h d i a g o n a l e l e m e n t o f t h e Z m a t r i xw h en a l l l o ad s a r e co n s i d e r ed . O u r a i m i s t o f i n d t h eT h 6 v e n i n i m p e d a n c e Z t h o f b u s k w h e n i t s l o a d i si g n o r ed . T h es e t w o i m p ed an ces ( Z k k an d Z t h ) a r es h o w n i n F i g . 5 . I t c an b e o b s e r v ed i n t h e f i g u r e t h a t

LZk Z ,~ (12)= z # . l l Z , , , - z # . + z , , ,H e r e Z ~ is t h e l o a d i m p e d a n c e o f b u s k T h u s , t h eT h 6 v e n i n i m p e d a n c e Z , h c a n r e a d i l y b e w r i t t e n a sZ t h = 2 ~ (13)k3.2.2. Determ inat ion o f the Thkvenin vol tage

L e t Vx. b e t h e v o l t ag e a t b u s k o b t a i n ed f r o m t h el o ad f l o w s o l u t i o n w h en a l l l o ad s i n t h e s y s t em a r eco n s i d e r ed . T h e o b j ec t i v e i s t o f i n d t h e T h + v en i nv o l t ag e V~h o f b u s k w h en i t s l o ad i s i g n o r ed . F i g . 6 ( a )a n d ( b ) s h o w s t h e T h a v e n i n e q u i v a l e n t c i r cu i t s o f F i g . 5a t p o i n t s aa ' an d b b ' , r e s p ec t i v e l y . B y co m p ar i n g F i g .6 ( a ) an d ( b ) , t h e v a l u e o f V th can r ead i l y b e w r i t t en a s

Z~h \V , h= l + ~ y | V k (1 4)Z, /


6/9

4 0 M.H. Haque /E lec tr ie Power Sys tems Research 32 (1995) 35-43

a'(a) (b)

F i g . 6 . T h ~ v e n i n e q u i v a l e n t c i rc u i t s o f l o a d b u s k .

Z~ Z~a aI I /------]

Fig. 6(b) represents the Th6venin equivalent circuit ofbus k and the maximum loading capability of this buscan be determined by varying the load impedance Z~.When the Th6venin equivalent circuit of a generalpower network behind a particular load bus is ob-tained, all the equations derived in Section 2 can beused to analyze the voltage stability problem associatedwith that bus.

4 . S i m u l a t i o n r e s u lt s

The proposed method of determining the maximumreactive power loading of a load bus in a general powersystem has been vigorously tested on the followingthree power systems: the IEEE 14-bus system, the IEEE30-bus system, and the IEEE 118-bus system.

The results obtained by the proposed method werecompared with those found by the NR method. In theNR method, the maximum reactive power loading of aload bus is determined by gradually increasing thereactive power demand at the candidate bus in theoriginal unreduced system until the method fails toconverge in solving the load flow problem. The actualmaximum reactive power demand should not be lessthan the value obtained by the NR method because ofthe numerical problems of the me thod in the vicinity ofthe voltage instability point. As mentioned earlier, themaximum loading capability of a bus estimated by theproposed method is slightly higher than the correspond-ing actual value because of the constant-impedanceload model. Thus, the results obtained by the proposedand NR methods and the actual value can be ranked asfollows:Result obtained Actual Result estimated by<


7/9

M .H. Haque ~Electric Pow er System s Research 32 (1995) 35 43 41T a b l e 3M a x i m u m a p p a r e n t p o w e r l o a d in g ( a t P F = 0 . 8 l a gg i ng ) o f th e14-bus s ys temB u s n o . V ~, ( p . u . ) S m ( M V A ) o b t a i n e d E r r o r ( % )

P r o p o s e d N Rm e t h o d m e t h o d

9 0 . 5 7 9 7 2 9 9 2 7 5 2 8 0 8 . 7I I 0 . 5 4 9 4 2 0 6 2 0 0 2 0 5 3 . 01 4 0 . 5 4 8 0 1 31 1 2 5 1 3 0 4 . 8

T a b l e 4M a x i m u m r e a c t i v e p o w e r l o a d i n g o f t h e I E E E 3 0 - b u s s y s t e mB u s n o . V ~ ( p . u . ) Q m ( M V A r ) o b t a i n e d E r r o r ( % )

P r o p o s e d N Rm e t h o d m e t h o d

7 0 . 5 1 3 9 4 5 2 4 4 5 4 5 0 1 .614 0 .53 90 127 120 125 5 ; .821 0 .53 23 189 180 185 5 .02 8 0 . 5 0 8 9 4 4 6 4 3 5 4 4 0 2 . 53 0 0 .5 2 5 1 3 8 . 7 3 7 3 8 4 . 6

l o a d b u s e s a f t e r s e t t i n g t h e v a l u e o f Q f o r t h o s e b u s e st o t h e u p p e r l i m i t s . T h e p r o p o s e d a n d N R m e t h o d sw e r e t h e n a p p l i e d t o t h e m o d i f i e d s y s t e m t o d e t e r m i n et h e v a l u e o f Q m f o r s o m e r e m o t e b u s e s ( 9 - 1 4 ) . T h er e s u l t s o b t a i n e d a r e s u m m a r i z e d i n T a b l e 2 . I t c a na g a i n b e o b s e r v e d i n T a b l e 2 t h a t t h e r e s u l t s o b t a i n e db y t h e p r o p o s e d m e t h o d a r e s l i g h t l y h i g h e r , a s u s u a l ,t h a n t h o s e f o u n d b y t h e N R m e t h o d . T h e m a x i m u me r r o r f o u n d i n t h i s c a s e w a s l e s s t h a n 1 3 . 5 % ( o r1 3 M V A r ) , w h i c h is s li g h tl y h i g h e r t h a n t h e p r e v i o u sc a s e . B e c a u s e o f t h e a b s e n c e o f a v o l t a g e c o n t r o l b u s ,t h e v o l t a g e o f a l l b u s e s ( e x c e p t t h e s w i n g ) a t Q m w a sf o u n d t o b e m u c h l e s s t h a n t h e n o m i n a l v a l u e o f1 .0 p . u . H o w e v e r , v o l t a g e s a t t h e b a s e c a s e ( a r o u n d1 .0 p . u . ) w e r e u s e d t o r e p l a c e t h e l o a d s b y c o n s t a n ti m p e d a n c e s i n d e t e r m i n i n g t h e T h 6 v e n i n e q u i v a l e n t c i r -c u i t . B e c a u s e o f a s i g n i fi c a n t r e d u c t i o n o f v o l t a g e i n t h ee n t i r e s y s t e m , t h o s e i m p e d a n c e s r e p r e s e n t l o w e r l o a d sa t t h e c r i t i c a l p o i n t s . T h i s r e s u l t s i n a h i g h e r e s t i m a t i o no f r e a c t i v e p o w e r d e m a n d o b t a i n e d b y t h e p r o p o s e dm e t h o d , a s m e n t i o n e d e a r l i e r .

T h e m a x i m u m a p p a r e n t p o w e r l o a d i n g , S i n , a t 0 . 8l a g g i n g P F a n d t h e c o r r e s p o n d i n g v o l t a g e V cr w e r e a l sod e t e r m i n e d f r o m E q s . ( 6 ) a n d ( 7 ) , r e s p e c t i v e l y . T h er e s u l t s f o r s o m e b u s e s i n t h i s s y s t e m a r e g i v e n i n T a b l e

1 .2

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F i g . 7 . V o l t a g e a t b u s 4 i n t h e I E E E 1 4 - b u s s y s t e m fo r v a r i o u sr e a c t i v e p o w e r l o a d i n g s .

3 . T h e a c t i v e p o w e r g e n e r a t i o n o f b u s 2 w a s c o n s i d e r e dt o b e 8 0 , 70 , a n d 6 0 M W i n d e t e r m i n i n g t h e v a l u e o f Sma t b u s e s 9 , 1 1 , a n d 1 4 , r e s p e c t i v e l y . T h e a b o v e M Wv a l u e s w e r e c o n s i d e r e d i n o r d e r t o i n c r e a s e t h e g e n e r a -t i o n o f a l l g e n e r a t o r s b y a l m o s t t h e s a m e f a c t o r . I n t h i sc a s e , t h e a n g l e o f t h e g e n e r a t o r b u s v o l t a g e c h a n g e sb e c a u s e o f t h e c h a n g e i n a c t iv e p o w e r g e n e r a t i o n . T h u s ,t h e T h 6 v e n i n e q u i v a l e n t c i r c u i t o b t a i n e d i n S e c t i o n 3m a y g i v e s o m e e r r o n e o u s r e s u l t s , a s m e n t i o n e d e a r l i e r .H o w e v e r , t h e m a x i m u m e r r o r o b s e r v e d i n t h i s c a s e w a sn o t s i g n i f i c a n t a n d i n f a c t w a s l e s s t h a n 1 0 % .

T h e a c t u a l v o l t a g e o f b u s 4 f o r v a r i o u s r e a c t i v ep o w e r d e m a n d s a t t h a t b u s ( u p t o Q m ) w a s d e t e r m i n e db y t h e N R m e t h o d i n t h e o r i g i n a l s y s t e m . T h e c o r r e -s p o n d i n g v o l t a g e w a s a ls o e s ti m a t e d f r o m E q . ( 3 a ) a f t e rr e p l a c in g t h e r e s t o f th e s y s t e m b y t h e T h 6 v e n i n e q u i v a -l e n t c ir c u i t . F i g . 7 s h o w s t h e v a r i a t i o n o f t h e a c t u a l a n de s t i m a t e d v o l t a g e s o f b u s 4 a g a i n s t t h e r e a c t i v e p o w e rd e m a n d . I t c a n b e o b s e r v e d i n t h e f i g u r e t h a t t h e r e i sh a r d l y a n y d i f f e r e n c e b e t w e e n t h e s e t w o v o l t a g e s . T h i sv a l i d a t e s t h e e x c e l l e n t a c c u r a c y o f t h e g e n e r a t o r m o d e lu s e d i n S e c t i o n 3 . 1 i n d e t e r m i n i n g t h e T h ~ v e n i n e q u i v a -l en t c i rcu i t .4 . 2 . T h e I E E E 3 0 - b u s s y s t e m

T h e s i n g l e- li n e d i a g r a m a n d d a t a o f th e I E E E 3 0 - b u ss y s t e m a r e g i v e n i n R e f . [ 1 6 ] . T h e m a x i m u m r e a c t i v ep o w e r l o a d i n g , Q m , a n d t h e c o r r e s p o n d i n g v o l t a ge Vc,.,f o r s o m e l o a d b u s e s , o b t a i n e d b y t h e p r o p o s e d m e t h o da r e g i v e n i n T a b l e 4 . T h e m a x i m u m r e a c t i v e p o w e rf o u n d b y t h e N R m e t h o d i s a l s o g i v e n i n T a b l e 4 . I t i sc l e a r f r o m T a b l e 4 t h a t t h e v a l u e s o f Q m o b t a i n e d b yt h e p r o p o s e d m e t h o d a r e v e r y c l o s e t o t h e c o r r e s p o n d -i n g v a l u e s f o u n d b y t h e N R m e t h o d , b u t s l i g h tl y h i g h e r ,a s e x p e c t e d . T h e m a x i m u m e r r o r o b s e r v e d i n t h i s s y s -t e m w a s l e s s t h a n 6 % .4 . 3 . T h e I E E E 1 1 S - b u s s y s t e m

T h e s i n g le - li n e d i a g r a m a n d d a t a o f th e I E E E l l 8 -b u s s y s t e m a r e o b t a i n e d f r o m R e f . [ 1 7 ] . T h e p r o p o s e da n d N R m e t h o d s o f d e t e r m i n in g t h e m a x i m u m r e ac t iv e


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42 M.H. Haque /Electric Power Systems Research 32 (1995) 35-43Table 5Maximum reactive power loading of the IEEE ll8 -bus systemBus no. Vcr (p.u.) Qm (MVA) obtaine d Error (%)

Proposed NRmethod method

2 0.4948 495 485- 490 2.122 0.4911 184 175-180 5.130 0.4893 1128 1080 1085 4.444 0.5045 180 170-175 5.956 0.4913 708 650 655 8.962 0.5046 1249 1245 1250 0.388 0.5014 477 470 475 1.5

very close to the corresponding actual or true value,but slightly higher, as expected. The error was be-tween 2.8% and 6.7%. On the other hand, the errorobtained by Overbye's method [12] was significantlyhigh (between -30% and 72%). It is worth mention-ing here that the error in Overbye's method decreasesas the system load approaches the stability limit.However, this load limit is not known in advanceand the objective is to find it from the base-case infor-mation.

5 . C o n c l u s i o n s

power loading were applied to some load buses in thissystem. A summary of the results is given in Table 5. Theerror observed in this system was between 0.3% and8.9%.The maximum loading capability of bus 44 in thissystem is very interesting. In a recent article by Overbyeet al. [12], the active, reactive, and apparent power (atPF = 0.894 lagging) margins of bus 44 were found to be420 MW, 114 MVAr, and 165 MVA, respectively. Thesevalues were obtained by a different method for a baseload of about 4000 MW. Note that the method requiresboth the high- and low-voltage solutions of the load flowequations. The corresponding actual or true valuesreported in the paper were 237 MW, 167 MVAr, and183 MVA, respectively. In this paper, the values ofVth and Zth behind bus 44 were found to be 0.99734 p.u.and (0.04256+j0.13495) p.u., respectively. For thisTh6venin equivalent circuit, the maximum active, reac-tive, and apparent power loading of the bus were foundto be 270 MW, 180 MVAr, and 207 MVA, respectively.Note that the original load at bus 44 is (16 +j8) MVA.Thus, the margins obtained by the proposed methodwere 254 MW, 172 MVAr and 189 MVA. These resultsare summarized in Table 6.

It can be observed in Table 6 that the maximumloading capability obtained by the proposed method is

Table 6Maximum loading capability of bus 44 of the ll8-bus system ob-tained by the proposed and Overbye's methods

Active Reactive Apparentpower power powerlimit (MW) limit (MVAr) limit (MVA)

Actual value" 253 175 201Overbye's method 436 122 182.9(Perce ntage error) (72%) ( - 30%) ( - 9%)Proposed method 270 180 207(Percentage error) (6.7%) (2.8%) (3.0%)

a The actual values are obtained from Ref. [12] after consideringthe original load of (16 +j8) MVA.

A simple and fast method for analyzing the voltagestability problem of a general power system through atwo-bus equivalent has been described. The generatormodel used in this paper is very insensitive to thechange in operating conditions. Thus the two-busequivalents obtained at the base-case operating pointthrough the Th6venin theorem can be faithfully ap-plied to determine the voltage stability limit. Unlikethe other methods, the Th6venin equivalent circuits ofall load buses are efficiently obtained in a single shot.This requires the results o f the base-case load flowsolution and computation of the Z matrix when allloads in the system are considered. A minor modifica-tion to the bus voltages and the diagonal elements ofthe Z matrix is required to exclude the effects of theload at the candidate bus. Determination of the vari-ous quantities at the verge of voltage stability involvesthe solution of simple quadratic equations. The maxi-mum demand at a load bus to ensure a minimumspecified voltage can also be determined from the solu-tion of another quadratic equation. Because of theconstant-impedance model of the system load, theTh6venin equivalent circuit used in this paper mayprovide slightly overestimated results at the verge ofvoltage stability.

The proposed method of determining the voltagestability limit through the two-bus equivalent has beentested on the IEEE 14-, 30-, and ll8-bus systems for anumber of cases. Simulation results reveal that thevoltage stability limit obtained by the proposed methoddue to the change in reactive power demand is veryclose to the corresponding actual value. The methodmay provide slightly erroneous results when the activepower demand of the system is also changed. However,the error found in this case was not significant. It isworth mentioning here that the errors found in thesimulation results are higher than the actual valuesbecause of numerical problems with the NR method atthe verge of voltage stability. Unlike the other methods,the proposed method can provide much better andreliable results using the base-case system informationwith significantly less computation.


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M.H . Haque /E le c tr ic " Powe r Sy s te m s Re search 32 (1995) 35 -43 43

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A fast method for determining the voltage stability limit