some clarifications in the transient

8/4/2019 Some Clarifications in the Transient

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E L S E V I E R

E l e c t r ic a l P ow e r & E ne r gy Sy s t e m s V o l . 1 8 , N o . 1 , p p . 6 5 -7 2 , 1 9 9 6Co p y r i g h t © 1 9 9 6 E l sev ie r S c i ence L t d

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Some c lar i f icat ions in the t ransienten ergy func tion m ethod

M A P a i, M L a u f e n b e r g a n d P W S a u e r

Dep ar tm en t o f E l ect ri ca l and Com pu te r E ng i nee r i ng ,

Un ivers i ty o f I l li no is , 1 406 W Green St , Urbana,I L 6 1 8 0 1 , U S A

The purpose o f t h i s pape r i s to c lar i f y t he eva lua tion o f p a t hdependen t i n t egra l s in t he energy func t ion me thod fo rs tab i l i t y ana ly s i s i n pow er sy s t em s . In t he l i t e ra ture t heseare hand led in an app rox im ate m anner through s t ra igh t l ineapprox imat ion l ead ing to c losed orm ana ly t i c express ionso f t h e e n e r g y f u n c t i o n s . T h i s m a y n o t a l w a y s b e a c c u r a te .Here i t is compa red "wi th t he t rapezo ida l me th od o fin t egra t ion a long the . f au l t ed t ra j ec tory a s or ig ina l l y

p r o p o s e d b y A t h a y e t a l. T h e p a p e r a l so e m p h a s i z e s s o m et u t o r ia l a sp e c ts f o r e x p l a in i n g t h e P E B S a n d t h e B C Um e t h o d .

K e y w o r d s : t r a n s ie n t e n e r g y f u n c t i o n , p o w e r s y s t e mstab i l i t y

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

H i s t o r i ca l l y , t h e en e r g y f u n c t i o n m e t h o d f o r m u l t i -m ach i n e p o w er s y s t em s t ab i li t y an a l y si s u s ed t h e c l a ss i ca lm ach i n e r ep r e s en t a t i o n w i t h t h e i n t e rn a l n o d e m o d e l an dn eg l ect i n g t ran s f e r co n d u c t an ces . U n d e r t h e s e co n d i ti o n swe get a mathemat ica l :model which i s conservat ive ande i t h e r t h e L y ap u n o v b as ed m e t h o d o r t h e f i r s t i n t eg r a lm e t h o d g i v es an eq u i v a l en t en e r g y f u n c t io n . A s t r an s fe rco n d u c t an ces r ep r e s en t t h e e f fec t o f co n s t an t i m p ed an celoads , ignor ing them g ives er roneous r esu l t s w i th r espectto cr i t i ca l c lear ing t imes . Var ious ef fo r t s a t approx i -m a t i n g t h e t r an s f e r co n d u c t an ce t e r m s an a l y t ica l l y h av eb e e n m a d e , t h e m o s t p o p u l a r a m o n g t h e m b e i n g t h es t r a ig h t l in e ap p r o x i m a t i o n o f t h e f au l t ed t r a j ec t o r y 2 . As o m ew h a t o b s cu r e , b u t n o t s o o b v i o u s ap p r o x i m a t i o n , i st h e a s s u m p t i o n t h a t t h e p o s t f au l t s .e . p, i s t h e s am e a s t h epre- f au l t s .e .p , fo r comput ing th i s in tegra l . Th is po in t i sa l so c lar i f ied in th i s paper . I t i s a l so shown tha t the

m e t h o d d u e t o A t h a y e t aL l o f u s in g t h e t r ap ezo i d a lm e t h o d i s t h e co r r ec t o n e , a s t h e p a t h o f i n t eg r a t io n i sk n o w n f r o m t h e f au l ted t r a j ec t o r y . Wh i l e it m ay b e t r u etha t us ing the s t r a igh l ; l ine approx imat ion does no tin t roduce s ign i f ican t e r ro r s whi le us ing c lass ica l modelsf o r l a rg e s y s t em s , i t m ay , i f th e en e r g y f u n c t i o n m e t h o d i s

Rece ived 14 June 1994; rev ised 11 May 1995; accepted 1 June1 9 9 5

u s ed w h en d e t a i led m o d e l s a r e co n s i d e r ed o r i f t h ey a r eap p l i ed t o r ed u ced o r d e r an d h en ce s m a l l e r s y s t em s . I ns t ruc tu re p reserv ing energy funct ions (SPEF) (o r thes p a r s e t r an s i en t en e r g y f u n c t i o n ( T E F) m e t h o d ) p a t hd ep en d en t i n t eg r a l s ex i s t d u e t o v o l t ag e d ep en d en t r ea ll o ad s o r co n t r i b u t i o n s f r o m t h e e l ec t ri c a l v a ri ab l e s o f t h em ach i n e 3 . A v a r i a t i o n o f t h e p o t en t i a l en e r g y b o u n d a r ys u r f ace ( PE B S) m e t h o d w h i ch o b v i a t e s th e n eed t o co m -

pu te the pos t - f au l t s tab le equ i l ib r ium p o in t ( s .e .p .) is a l sop r o p o s ed . T h i s m i g h t r ed u ce t h e co m p u t a t i o n a l b u r d enin a qu ick screen ing o f con t ingencies .

I I . M a t h e m a t i c a l m o d e l 4 -7

Wi t h t h e u s u a l n o t a t i o n , t h e m a t h em a t i ca l m o d e l f o ran m m ach i n e s y s t em w i t h co n s t an t v o l t ag e b eh i n dr eac t an ce r ep r e s en t a t i o n an d co n s t an t i m p ed an ce l o adap p r o x i m a t i o n i s g i v en i n t h e C en t r e o f In e r t ia ( C O I )n o t a t i o n a s :

Oi ~--'03i (1 )

M iM g ~ i : P m i - P e g - ~ P C O l

6= .(O ) i = 1 , 2 , . . . , m (2 )

The r igh t -hand s ide in equat ion (2 ) has d i f f er en tparameter va lues ( i .e . G ij a n d B ij v a l u es ) i n co m p u t i n gPeg an d P c o l for th e f aulte d p er io d (0 _< t _< tot) and thep o s t - f au l t p e r i o d ( t > td) . T h e en e r g y f u n c t i o n f o r t h epos t - f au l t sys tem is cons t ruc ted as :

1 m m f O

: " - - ~ " ~ g i ~ - ~ _ ~ I f i ( O ) d O i2 g _ _ ~ /= 1 J O :

+ v ? e ( o ) A: V T O T (3 )

w h e r e 0 i and c?1 a r e t h e v a r i ab le s f r o m t h e f au l t edt r a j ec to r y . I n t h e ab s en ce o f tr an s f e r co n d u c t an c e t e r m sG i j ( i C j ) , the express ion fo r Vpe(O ) can b e ex p r e s s edanaly t ica l ly in a c losed fo rm 4 '5 . O therw ise the G ij t e r m s

6 5



6 6 C l a r i f i c a t io n s i n t r a n s i e n t e n e r g y f u n c t i o n m e t ho d ." M . A . P a i e t a l .

c o n t r i b u t e a p a t h d e p e n d e n t t e r m a s f o l lo w s '

m m - 1 ~ .~ IV p E ( O ) = - - Z e i ( O i - - O s ) - - Z C i j ( c o s Oij - - CO S 0 / ~)

i = l i =1 j = i + l

_ [ o , + o , ]J°~+°7 D i j c o s 0 q d ( O i + O j )

= V e o s + V M A 6 + V D (4 )

w h e r e C i j ~-- E i E j B i j , D i j = E i E j D i j a n d P i = P m i -I E i l 2 G i i . I n c o m p u t i n g e q u a t i o n ( 4), 0 i s o b t a in e d f r o m

t h e f a u l t e d t r a j e c t o r y a n d 0 ~ i s t h e p o s t - f a u l t s . e . p . W e

f o c u s o n t h e e v a l u a t i o n o f th e t e r m s o n t h e r i g h t - h a n d

s i d e o f ( 4) . T h e t h i r d t e r m i s o b v i o u s l y p a t h d e p e n d e n t .

I II. D i f f e r e n t m e t h o d s o f e v a l u a t in g t h ep a t h d e p e n d e n t t e r m

D e f i n e

[ 0i+0s

I i j = J07+07 D i j c o s O i j d ( O i + O j ( 5 )

II1.1 A n a l y t i c a l a p p r o x i m a t i o n

I n t h e b u l k o f t h e l i te r a t u r e , b y a s s u m i n g a s t r a i g h t l in e

p a t h o f i n t e g r a ti o n I i j i s a p p r o x i m a t e d a n a l y t i c a l l y a s :

= ( O , - O ) + ( O ; - O ~I i ; ( 0 i 0 s ) - - ( O j ~ . D i j ( s i n Oi j - - sin O S j ) ( 6 )

T h e o r e t i c a l l y t h i s is i n c o r r e c t a n d t h e a c t u a l p a t h n e e d s t o

b e t a k e n i n t o a c c o u n t a s p o i n t e d o u t i n R e f e re n c e 1 . T h e

s t r a i g h t l i n e p a t h i s u s e d o n l y i n c o m p u t i n g V~r w h i l eu s i n g t h e c o n t r o l l i n g u . e . p , m e t h o d . I n c o m p u t i n g t h e

p o t e n t i a l e n e r g y t e r m ( 4) , i t i s s h o w n t h a t i f t h e P E B S

m e t h o d i s u s e d t o c o m p u t e V ~r, h e n i t is p o s s ib l e t o a v o i d

c o m p u t i n g t h e p o s t - f a u l t s . e . p , a l t o g e t h e r b y p r o p e r

i n i t i a l i z a t i o n o f V e e ( O ) . T h e s e t w o i s s u e s a r e n o w

di scussed .

111.2 I n i t i a l i z a t i o n o f V p E ( O a n d i t s u s e in t h e P E B S

m e t h o d

W h i l e i n t e g r a t i n g t h e f a u l t e d t r a j e c t o r y i n e q u a t i o n s ( 1 )

a n d ( 2) , t h e i n i ti a l c o n d i t i o n s o n t h e s t a t e s a r e O i ( O ) = 0 °

an d ~b i(0) = 0 . In t he ene rgy func t ion , t he r e fe rence ang le

an d v e loc i ty va r i ab l e s a re 0/~ an d ~b i(0) = 0 . Thus a t t = 0 ,

w e e v a l u a t e V e E ( O ) i n e q u a t i o n ( 3) a s :

V p E ( O ° ) = - ~ -'~ [ f f f i (O ) d O ii= 1 a im r n 1 ~ _ _ ~

= - Z e , ( ° ° - ° s ) -

i = 1 i = 1 j = i + l

(7 )

F[ c , j ( c o s - c o s o b )

I. .

-- d0/~+~[°+0j° O i j c o s O i j d ( O i + O j )]

= K (a con s t an t ) (8 )

I f th e p o s t - f a u l t n e t w o r k i s th e s a m e a s t h e p r e - f a u l t

n e t w o r k , t h e n K = 0 . O t h e r w i s e t h e v a l u e o f K i n ( 8 )

s h o u l d b e i n c l u d e d i n t h e e n e r g y f u n c t i o n . T h e p a t h

i n t e g r a l t e r m i n ( 8 ) i s e v a l u a t e d u s i n g t h e t r a p e z o i d a l

ru l e a s :

I i j ( o ) = D , j[ c o s( O ? - 0 ; ) + c o s ( 0 s - 0 7 ) 1

x [(0 ° + 0; ) - (0 s + 07) ] (9)

T h i s i s a g o o d a p p r o x i m a t i o n i f 0 s i s c l o s e t o 0 ° . F o u a d

a n d S t a n t o n 8 r e c o g n i z e d t h i s f a c t i n t h e i r w o r k a n d c a l le d

i t t h e V c o r r e c t i o n t e r m .I f o n e u s e s t h e p o t e n t ia l e n e r g y b o u n d a r y s u r fa c e

( P E B S ) m e t h o d 9 , t h e n e v e n i f t h e p o s t - f a u l t n e t w o r k i s

n o t e q u a l t o t h e p r e - f a u l t n e t w o r k , t h i s t e r m c a n b e

s u b t r a c t e d o u t o f th e e n e r g y f u n c t i o n , i .e .

v ( o , m ) = V K E ( ) + V E ( O ) - - V E ( O ° ) ( 1 0 )

H e n c e t h e p o t e n t i a l e n e r g y c a n b e d e f i n e d w i t h 0 ° a s t h e

d a t u m a s:

f E ( o ) - v p e ( o ° ) =

d o i ll O l d O , - , I > l

= - f i ( O ) dO ii = l 0 °

m m - 1 m F

= - ~ - ' ~ P i ( O ~ - O ° ) - z ~ [ C i j ( c o s O i j - c o s O ~ )i = l i = 1 j = i + l [

f 0 , % ]- I D q c o s O i j d ( O i + O j ) (11)

j0?+~o

A t t h e P E B S c r o s s in g 0 " , l J 'e E ( O * ) g i v es a g o o d a p p r o x i -

m a t i o n t o V c r . T o d e t e c t th e P E B S c r o s si n g , w e c a n

m o n i t o r t h e q u a n t i t y f T ( O ) . ( 0 - - 0 s ) wh e re f ( O ) r e fe r s

t o t h e p o s t - f a u l t p a r a m e t e r s a n d 0 s i s t h e p o s t - f a u l t s . e. p .

T h e P E B S c r o s s i n g 0 * i s t h e p o i n t w h e r e t h i s q u a n t i t y1

c h a n g e s f r o m n e g a t i v e t o p o s i t i v e . T h i s r e q u i r e s a

k n o w l e d g e o f 0 s. T h e m e t h o d p r o p o s e d b y K a k i m o t o e t

a l . 9 d e t e c t s t h e P E B S c r o s s i n g a s t h e p o i n t w h e r e t h e

p o t e n t i a l e n e r g y V e e r e a c h e s a m a x i m u m v a l u e . H e n c e

o n e c a n d i r e c t l y m o n i t o r _ Vee a n d t h u s a v o i d h a v i n g t o

m o n i t o r t h e d o t p r o d u c t f r ( 0 ) • ( 0 - 0 s ). T h i s l e ad s to a n

i m p o r t a n t a d v a n t a g e o f n o t h a v i n g t o c o m p u t e 0 s a t a ll .

I n f a s t s c r e e n i n g o f c o n t in g e n c i e s , t h i s c o u l d r e s u l t i n a

s ! g n if ic a n t s a v i n g i n c o m p u t a t i o n . T h e t e c h n i q u e o f u s i n g

V e e ( o ) i n th e P E B S m e t h o d h a s n o t b e e n w i d e l y u s e d s o

f a r i n th e l i t e r a t u r e a n d m e r i t s f u r t h e r i n v e s t i g a t i o n .

11 1.2 .1 T r a p e z o i d a l a p p r o x i m a t i o n

F o r t > 0 , i n s te a d o f c o m p u t i n g I i j , b y e q u a t i o n ( 6 ) , w e

c o m p u t e i t b y t h e t r a p e z o i d a l r u l e a s ( l e t t i n g n - -k A t , k >_ 1) :

I i j ( n ) = I i j ( n - 1) + 1 D i j [ c o s ( O i ( n ) - O j ( n ) )

"}- CO S(0 /Q"/ - - 1) - O j ( n - 1))]

× [ O i ( n ) + O j ( n ) - O i ( n - 1) - O j ( n - 1)],

n _> 1 (12 )

w i t h I i j ( O ) = 0 . T h i s i n i t i a l i z a t io n a s s u m e s t h a t t h e c o n -

s t a n t K i n (8 ) h a s b e e n e v a l u a t e d u s i n g ( 9) a n d a d d e d t o

t h e p o t e n t i a l e n e r g y f u n c t i o n i n ( 11 ). E q u a t i o n ( 1 2 ) is

u s e d i n R e f e r e n c e 1 f o r t h e p a t h d e p e n d e n t i n t eg r a lw i t h o u t a d d i n g e q u a t i o n ( 9) .

T h u s a s O i ( k A t ) , ~ i ( k A t ) , ( k > _ 1 ) a r e e v a l u a t e d f r o m

t h e f a u l t - o n t r a j e c t o r y w e o b t a i n V ( O , C o) as ( l e t t i ng



Clarifications in transient energy function method." M. A. Pai e t a l . 67

k A t = n):

m W/

V ( O ( n ) , & ( n ) ) = ~ Z , M i w 2 ( n ) - Z P i (O i (n ) - 0 ° )i = l i = 1m - 1 £ [

- ~ : C q ( c o s O i j ( n ) - cos0~j)i = 1 j = i + l

m - - I m "]

- - I i j ( n + K

j 1

( 1 3 )

w h e r e I i j ( n ) i s g iven by eq ua t ion (12) . 0 ~ i s u sed i n

c o m p u t i n g K a n d i s al s o n e e d e d t o c o m p u t e V ~ d i s c u s se d '

i n t h e n e x t s e c t i o n . E q u a t i o n ( 1 3) i s e q u i v a l e n t t o V ( O , &)i n e q u a t i o n ( 4 ) a n d i s u s e d i n t h e f o l l o w i n g s e c ti o n s .

I V . C o m p u t a t i o n o f V c r

IV.1 C o n t r o l l i n g u . e . p , m e t h o dI f t h e c o n t r o l l i n g u . e . p , m e t h o d I i s u s e d , t h e r e i s a n e e d t o

c o m p u t e 0 s. A n e f fi ci en t w a y o f c o m p u t i n g t h e c o n t r o l -

l i n g u . e . p , i s t h r o u g h t h e b o u n d a r y c o n t r o l l i n g u . e . p .( B C U ) m e t h o d 1 °'1 4. I n t h i s m e t h o d , o n e c o n t i n u e s t o

i n t e g r a t e a f t e r t h e P E B S c r o s s i n g u s i n g t h e f o l l o w i n g

r e d u c e d s e t o f e q u a t i o n s o f t h e p o s t - f a u l t s y s te m .

M i pO i = P m i - P e i - - - - ~ c o 1 ~ = f . ( O ) i = l , . . . , m ( 1 4 )

w i t h t h e i n it i a l c o n d i t i o n 0 ( 0 ) = 0 " , u n t i l I I f ( 0 ) l l i s m i n i -m u m . T h i s is th e m i n i m u m g r a d i e n t p o i n t ( M G P ) O u. T o

g e t t h e e x a c t u . e . p , w e s o l v e f ( O ) = 0 b y th e N e w t o n -R a p h s o n m e t h o d w i t h ~ u a s t h e i n it i a l g u e s s t o g e t 0 u . I n

m os t ca se s 0 u i s su f f i c i en t l y c lose t o 0 u .

H a v i n g o b t a i n e d 0 " , s y m b o l i c a l l y d e n o t e V~, =

V(0 u, &u) = V e e ( O ~ ) , as o3u = 0 . Howeve r , t he c r i t i c a l l y

c l e a r e d t r a j e c t o r y i n v o l v i n g b o t h t h e f a u l t e d a n d p o s t -

f a u l t s y s t e m t o e v a l u a t e V p e ( 0 u ) a r e n o t k n o w n , a s t ~ i s

n o t k n o w n ! H e n c e v a r i o u s m e t h o d s h a v e b e e n p r o p o s e d

t o c o m p u t e V ?E ( O ~ ) . A m o n g t h e m t h e s t r a i g h t l i n e

a p p r o x i m a t i o n f r o m 0 s t o 0 u i s t h e m o s t c o n v e n i e n ton e 1 '2. V eE ( O u) i s g iven b y

?n

V p E (O U ) = - - Z e i ( o u - O s )

i= 1

m - I m

- ~ ~ [ c . ( c o s O ~ - c os O b) ]i = 1 j= i+I

(0 u - Oi~) + (0~ - 0~) D i j (s in 0~ . - sin 0~)+ ( o u o : 1 - ( o ? o :1

(15)

T h e t h i r d t e r m o f e q u a t i o n ( 1 5 ) i s d e r i v e d a s f o ll o w s .A s s u m e a r a y f r o m 0 s t o 0 /u a n d t h e n a n y p o i n t o n t h e r a y

is 0, = 0 s + p ( O u - O S ) , ( 0 < _ p < 1 ) . T h u s d ( 0 g + 0 j ) =dp(0 u - 0s + 0/u - 0 ~ ). T h e p a t h d e p e n d e n t t e r m i n

e q u a t i o n ( 4) i s n o w e v a l u a t e d a t 0 u a s l :

v ~ ( o " ) = [ ( o ? - o h + ( o 7 - o : ) 1 D , . : c o s

× { (0¢ - 0 : ) + p [ ( 0 ? - 0 h - ( 0 y - 0 : ) ] } d p

_ (0 u - O ) +__ 0~ - -Oj_) Dq {s in(O ~ - 0~)( e ~ e l ) ( e ? - e : )

O S ' ~ l l l P = l+ p [ ( O u - e s ) - ( O ? - - j lJ Jl p= O

( e ? - o : ) + ( o 2 - o , ; )= -- 0.s) D ij (sin 0~ - s in 0/~)

( e u o : ) ( o ? :

(16)

V . N u m e r ic a l e x a m p l eT h e w e l l k n o w n t h r e e -m a c h i n e n i n e - b u s s y s t e m n w a s

c h o s e n t o i l l u s t r a t e t h e v a r i o u s o b s e r v a t i o n s i n S e c t i o n s

I I I a n d I V . F i g u r e 1 i s th e s i n g l e l i n e d i a g r a m .

S e v e r a l te s t s w e r e d o n e o n t h i s s y s t e m t o i l l u s t r a t e t h e

t w o m e t h o d s ( P E B S a n d B C U ) a s w el l a s th e a p p r o x i m a -

t i o n t e c h n i q u e s i n v o l v i n g D i j t e r m s .

T h e f i r s t e x a m p l e c a s e i s t h e * 7 - 5 c o n t i n g e n c y , t h i s

c o n s i s t s o f a f a u l t o n b u s 7 , w h i c h i s c le a r e d b y t h eo p e n i n g o f li n e 7- 5. U s i n g t h e s t e p - b y - s t e p i n t e g r a t i o n

m e t h o d , t h i s c o n t i n g e n c y w a s f o u n d t o h a v e a c r it ic a l

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t c , = 0 .21 s

w e r e o b t a i n e d u s i n g t h e M A T L A B p r o g r a m 12 w i t h t h e

P o w e r S y s t e m T o o l b o x 13.

F i g u r e 2 is a p l o t o f t h e g e n e r a t o r a n g l e s w i t h a c l e a r i n g

t i m e o f 0 .1 9 s , w h i c h s h o w s t h a t t h e s y s t e m d o e s i n d e e d

r e m a i n s t a b l e . F i g u r e 3 is t h e s a m e c o n t i n g e n c y c l e a r e d a t

0 .2 1 s , w h e r e t h e g e n e r a t o r a n g l e s a r e u n s t a b l e .

N e x t , tc r w a s c a l c u l a te d u s i n g t h e P E B S m e t h o d . T h i s

m e t h o d l o o k s f o r t h e v a lu e o f t h e m a c h i n e a n g l e s a t t h e

p o i n t a t w h i c h t h e d o t p r o d u c t f r ( O ) . ( 0 - 0 s ) d i s c u s s e d

i n S e c t i o n I V . 1 c r o s s e s f r o m n e g a t i v e t o p o s i t i v e . T h i s d o t

p r o d u c t i s p l o t t e d i n F i g u r e 4, a n d s h o w s t h a t t h e P E B S i s

c rosse d a t t = 0 .356 s .O n c e t h e P E B S c r o s s in g i s f o u n d , t h e p o t e n t i a l e n e r g y

i s f o u n d a t t h e s a m e p o i n t , a n d i s d e f i n e d a s t h e c r i t i c a l

e n e r g y o f t h e s y s t e m , V ~r. T h e c a l c u l a t i o n s o f V t , o s a n d

V M nG i n e q u a t i o n ( 4) a t t h i s p o i n t a r e t r iv i a l. F o r t h e

c a l c u l a t i o n o f Vz~ o n e c o u l d u s e e i t h e r t h e t r a p e z o i d a lr u l e ( T R A P ) o f e q u a t i o n ( 12 ) o r t h e s t r a ig h t - l i n e

a p p r o x i m a t i o n ( S L ) o f e q u a t i o n ( 6 ) f o r th e e v a l u a t i o n

o f t h e p a t h - d e p e n d e n t i n t e g ra l . A s t h e p a t h o f i n te g r a -

t i o n i s a v a il a b l e , i t i s m o r e a c c u r a t e t o u s e t h e a c t u a l p a t hi n t h e e v a l u a t i o n u s i n g t h e t r a p e z o i d a l r u l e r a t h e r t h a n t o

a s s u m e a s t r a i g h t - l in e p a t h .

T o e x a m i n e h o w t h is s t r ig h t -l in e a p p r o x i m a t i o n a f fe c ts

t h e P E B S m e t h o d , VD w a s c a l c u l a te d b o t h w a y s . F o r t h e

* 7 - 5 c o n t i n g e n c y , t h e S L m e t h o d r e s u l t s i n a

V e t ( O * ) = V c r = 1 .2 85 3. T h e T R A P m e t h o d g a v e

V p E ( O * ) = V c r ~ - 1 . 3 2 6 9 . F i g u r e 5 s h o w s a n e x a m p l e o f

t h e f a u l t e d e n e r g i e s o n t h e s y s t e m . N o t e t h a t V c c f o r t h e

T R A P m e t h o d is ve r y c lo s e t o t h e m a x i m u m o f V e t in

F i g u r e 5 . T h i s i s v e r y t y p i c a l w h e n u s i n g t h e P E B S

m e t h o d . A q u i c k w a y t o e s t i m a t e tc r f r o m t h e g r a p h i s

t o d r a w a l in e f r o m t h e p e a k o f V e E pa ra l l e l t o t he x -ax i s

un t i l i t i n t e r sec t s V T " o r . The t ime a t t h i s i n t e r se c t i on i s tc r.

U s i n g t h e S L r e s u l t s f o r Vcr, tcr = 0 . 1 9 6 s , w h e r e a s

t c r - - 0 . 1 9 9 s u s i n g t h e T R A P m e t h o d . T h e r e s u l t s f o r

t h is c o n t i n g e n c y w e r e v e r y s im i l a r f o r b o t h m e t h o d s . T h e

l o a d i n g o f t h e s y s t e m w a s i n c r e a s e d t o 1 5 0 % o f th e

n o m i n a l c a s e , w i t h a l l t h e e x c e s s l o a d b e i n g t a k e n b yt h e s la c k b u s , a n d t h e P E B S m e t h o d w a s u s e d a g a in . T h e

l o a d i n g w a s t h e n p u s h e d t o 2 0 0 % o f th e n o m i n a l , a g a i n

w i t h t h e e x t r a l o a d t a k e n b y t h e s l ac k b u s a n d u s i n g th e

P E B S m e t h o d . T h e r e s u l t s a r e s h o w n i n T a b l e 1 . I n a l l

c a s e s t h e t r a p e z o i d a l r u l e i s m o r e a c c u r a t e t h a n t h e

s t r a i g h t - l i n e a p p r o x i m a t i o n . T h e d i f f e r e n c e b e t w e e n t h e

t w o m e t h o d s b e c o m e s m o r e p r o n o u n c e d a s t h e s y s t e m

b e c o m e s m o r e h e a v i ly l o a d e d . A t t h e h i g h e s t lo a d i n g t h e

d i f f e re n c e i n V~r f o r t h e t w o m e t h o d s i s a b o u t 1 0 % . T h e

c r i t i c a l c l e a r i n g t i m e i s a l s o m o r e a c c u r a t e u s i n g t h e

t r a p e z o i d a l m e t h o d , a s e x p e c t e d , a n d t h e d i f f er e n c e i n

tc r b e t w e e n t h e t w o m e t h o d s i s a l so g r e a t e r w i t h h i g h e r

l o a d i n g . T h e i n c r e a s e i n tc r w i t h i n c r e a s e d l o a d i n g i s so l e l yd u e t o t h e f a c t t h a t t h e e x t r a l o a d i s t a k e n u p b y t h e s l a c k

b u s . T h e p u r p o s e h e r e w a s o n l y to s h o w t h e d i ff e r e n c e

b e t w e e n th e T R A P a n d S L m e t h o d s . I f i n c r ea s e d l o a d is

s h a r e d b y t h e n o n - s l a c k b u s e s , t h e d i f f e r e n c e b e t w e e n t h e

T R A P a n d S L m e t h o d s s t i l l e x i s t s a n d a s e x p e c t e d

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Clarifications in transient energy function method." M. A. Pa i e t a l . 69

5 . . . . . . . . . . . . . : . . .. . .. .. . .. .. . .. .. . .. .. . .. .. . .. . .. .. . .. .. . .. .. . .. .. . .. .. . .. .. . .. ... . . . . . . . . . . . . . . . . . . . . .: : > . . .

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T i m e ( s c c )

F i g u r e 5 . ( a ) VTOT a n d ( b ) VpE u s i n g t h e t r a p e z o i d a l r u l e fo r " 7 - 5

T a b l e 1 . C o m p a r i s o n o f S L a n d T R A P u s i n g t h e P E B S m e t h o d , t o , i n s e c o n d s

S t r a i g h t - l i n e

S y s t e m l o a d i n g a p p r o x i m a t i o n T r a p e z o i d a l r u l e A c t u a l ( s t e p b y s t e p )

1 . 0 V c r = 1 . 2 8 5 3 V c r = 1 . 3 2 6 9 tc r = 0 . 2 0 1 s

t c r - - 0 . 1 9 6 s t c r = 0 . 1 9 9 S

1 . 5 V c r = 1 . 9 7 1 9 V c r = 2 . 1 1 6 5 t c r = 0 . 2 7 9 s

t c r = 0 . 2 7 2 S tc ~ = 0 . 2 8 0 S

2 . 0 V ~r - -- - 2 . 5 9 6 3 V cr = 2 . 9 5 9 2 t c r = 0 . 3 8 7 S

t c r = 0 . 3 6 0 S t c r = 0 . 3 7 7 S

O

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1 .1 . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . .. . .. . .. . .. . , . . . . . . i . . . . . . . . . . : i . . . . . . . . .

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0 . 80 . 5 0 . 6 0 . 7 0 . 8 0 . 9 I . I 1 . 2 1 . 3 1 . 4

. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . ! . . . . . . . . . . . .

F ig ur e 6 . P lo t o f th e m in imu m g r a d ie n t p o in t fu n c t io n

T i m e ( s e e )

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e~

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, . . . . . . . . . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -

0 . 8 ~

0 .6 . . . .. . .. . . . ! . . / . . . . . . .. . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . .. . . . . . . \ . . . . . . . . .

0.42 ~ '.....!... i i ......i'ii / ...........

0 0.2 0.4 0.6 0.8 1.2 1.4 1.6

T i m e ( s e c )Figure 7. Vror an d VpE fo r c r i t i ca l ly c leared sys tem u s ing (a) s t ra ight l ine approx imat ion (b) t rapezo ida l ru le

T h e B C U m e t h o d w a s a l s o t e s t e d f o r d i f f e r e n c e s

b e t w e e n t h e s t ra i g h t- l in e a n d t r a p e z o i d a l e v a l u a t i o n s o f

t h e e n e r g y f u n c t i o n . T h e f i r s t s t e p i n t h e B C U m e t h o d i s

i d e n t ic a l t o t h e f i rs t s t e p in t h e P E B S m e t h o d , i n t h a t t h e0 * is f o u n d . T h e n e x t s t e p i s t o i n t e g r a t e t h e p o s t - f a u l t

g r a d i e n t s y s t e m e q u a t i o n ( 1 4 ) u s i n g 0 * a s t h e i n i t i a l

c o n d i t i o n , u n t i l a m i n i m u m o f l l f ( 0 ) l l i s f o u n d . T h i s i s

t h e m i n i m u m g r a d i en t p o i n t ( M G P ) . F i g u r e 6 s h o w s t he

r e s u lt s o f t h is i n t e g r a t i o n f o r t h e * 7 - 5 c o n t i n g e n c y a t

n o m i n a l l o a d i n g . T h e p o t e n t i a l e n e r g y is t h e n e v a l u a t e d

a t th i s M G P u s i n g th e s tr a ig h t - li n e a p p r o x i m a t i o n

( e q u a t i o n ( 1 5) ). T h e t r a p e z o i d a l r u l e i s n o t a n o p t i o n

h e r e , b e c a u s e t h e e x a c t p a t h f r o m 0 s to 0 u i s n o t k n o w n •

U s i n g F i g u r e 6 , t h e M G P w a s f o u n d a t t = 1 . 38 4 s , a n d

u s i n g e q u a t i o n ( 1 6 ) t h e c r i ti c a l e n e r g y V c r w a s c a l c u l a t e d

t o b e Vp e (Ou) = 1 . 3198 = V~r. A s t h i s i s c l o s e t o t he

v a lu e s o b ta i n e d b y t h e T R A P a n d S L m e t h o d s ( T a b le

1), ter s h o u l d b e b e t w e e n 0 .1 9 6 a n d 0 . 1 9 9 s . T h e B C U

m e t h o d i s t h e s a m e a s th e P E B S m e t h o d u n t il 0 * is

r e a c h e d .

B e f o r e t h e f a u l t i s c le a r e d t h e r e i s n o t m u c h d i f fe r e n c e

i n t h e e n e rg i es c o m p u t e d b y th e S L a n d T R A P m e t h o d s .

H o w e v e r , l a r g e v a r i a t i o n s s h o w u p i n t h e p o s t - f a u l tp e r i o d . I t is o n l y w h e n t h e S L a p p r o x i m a t i o n i s u s e d t o

c a l c u l a t e e n e r g i e s a f t e r tc r t h a t l a r g e d i f f e r e n c e s o c c u r , l c r

i s o f t h e o r d e r o f 0 . 2 s a n d t h e P E B S c r o s s i n g i s c l o s e r to

0 . 4 s . I n F i g u r e 7 , t h e d i f f e r e n c e i n e n e r g i e s o f t h e t w o

m e t h o d s i s q u i t e l a r g e a f t e r t h e f a u l t i s c r i t i c a l l y c l e a r e d •

I n f a c t, th e S L m e t h o d s h o w s t h a t t h e to t a l s y s t e m e n e r g y

i n c r e a s e s a f t e r t h e f a u l t i s c l e a r e d , w h e n i n r e a l i t y t h i s i s

1. 4

1.2

0.8

o .6

0.4

o.21

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0 0.05 O. 1 O. 15 0.2 0.25 0.3 0.35 0.4

F igure 8 . P otent ia l energy o f the l ine-c leared sys tem

Time( see)



Clari f icat ions in t ransien t energy fun ct ion method." M. A. Pa i et al . 71

2.5

1.5

0.5

0

i . " . . . / "

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Figure 9 . Poten t ia l ene rgy o f the se l f -c lea red system

impossible. VToTmust decrease afte r the faul t is clearedin a system that has damping. The trapezoidal method ofcomputing Iq correctly reflects this. This could be ofsignificance if the energy function is used for voltagedip calculations. The path integral terms initially are asmall percentage of the potential energy but increase toabout 30-40% near the PEBS crossing. However, largevariations show in in the post-fault period.

V .1 T h e e n e r g y f u n c t i o n a t t = 0

In Section IlI.2, the initialization of VpE at t = 0 wasdiscussed. In most energy funct ion plots the value o f VpEat the time when the fault is applied is shown to betypically zero. This is only true under specific circum-stances, namely, when the post- fault system is identical tothe pre-fault system, i.e., when the fault is self-clearing.Figures 8 and 9 show the difference between a fault tha tis switched out, resulting in a different system, and afaul t that is self-cleared, resulting in the same system.

Figure 8 is the *7-5 contingency, and Figure 9 is the *7contingency.

The only other way VpE could start at zero is if 0 ° issubstituted for 0 s in the potential energy equations. Thisprocedure is correct only when the PEBS method asdiscussed in Section III.2 is used.

V . 2 P o s t - f a u l t e n e r g y

Another point of clarification concerns the total systemenergy after the fault has been cleared. In the literature,the post-fault energy of the system is always port rayed asa constant. However, in a system with damping, theenergy must decrease over time. Note that in Figure 10,which portrays the three-machine system criticallycleared from a *7-5 contingency, the energy definitelydecreases over time.

This obvious decrease in system energy is not soapparent in a larger system. Figure 11 is a plot of theenergy function for the ten-machine system with a *30-34

1

0 .8

~ 0 . 6

0.4

0.2

o ;0 2 0 4 0 6 0 8 1 2 1 4 1 6 1 8

Time sec)

F i g u r e 1 0 . T h r e e - m a c h i n e s y s t e m cr i t i ca l l y c lea red tosh o w d e c r e a se i n sys t e m e n e r g y

Vto t

° . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i . . . . . . . . . - . . . . . . . . . . . . . . . . . . . . . . . ! . . . . . . . . . . . . . . . . . . .

5

~ 4m 3

2

0 .2 0 .4 0 .6 0 .8T i m e (sac)

F i g u r e 1 1 . T e n - m a c h i n e s y s t e m cr i t i ca l l y c lea red to

s h o w a l m o s t c o n s t a n t s y s te m e n e r g y



7 2 Clar i f ica t ions in t rans ient energy fun c t ion method. " M. A . P a i e t a l .

c o n t i n g e n c y . I n t h e l a r g e r s y s t em , t h e p o s t - f a u l t e n e r g y

d o e s r e m a i n r e a s o n a b l y c o n s t a n t , b e c a u s e o f th e d e c r e a s-

i n g i n f l u en c e o f t h e t r a n s f e r c o n d u c t a n c e s a s t h e s y s t e m

i n c r e a s e s i n s iz e . N o t e a l s o t h a t t h e w h i l e t h e f a u l t i s n o t

s e l f -c l e a r i n g , t h e p o t e n t i a l e n e r g y a l s o s t a r t s v e r y c lo s e t o

z e r o .

A s t h e s y s t e m i n c r e a s e s i n s i z e, t h e r e m o v a l o f o n e l i n e

m a k e s l e ss d i f fe r e n c e t o t h e s y s t e m e n e r g y . T h i s p o i n t i s ,

h o w e v e r , i m p o r t a n t i f a n e n e r g y f u n c t i o n i s t o b e u s e d

a f t e r d y n a m i c a l e q u i v a l e n c i n g w h i c h r e s u l t s i n f e w e r

e q u i v a l e n t m a c h i n e s .

V I . C o n c l u s i o n

I n t h i s p a p e r s e v e r a l p o i n t s o f t h e t r a n s i e n t e n e r g y f u n c -

t i o n m e t h o d h a v e b e e n c l a r i f i e d . T h e f i r s t i s a d i s c u s s i o n

o n t h e v a l i d i t y o f t h e s t r a ig h t - li n e a p p r o x i m a t i o n f o r

c o m p u t i n g t h e p a t h d e p e n d e n t i n te g ra l . I t w a s s h o w n

t h a t i t i s m o r e a c c u r a t e t o u s e t h e t r a p e z o i d a l r u l e

w h e n e v e r p o ss i b le . A t h i g h e r l o a d i n g s t h e T R A Pm e t h o d i s m o r e a c c u r a t e t h a n t h e S L m e t h o d . T h e

t r a p e z o i d a l m e t h o d a l s o e x h i b i t s m u c h b e t t e r r e s u l t s

w h e n c a l c u l a t i n g t h e p o s t - f a u l t p o t e n t i a l e n e r g y . T h i s

c o u l d b e v e r y i m p o r t a n t i n v o l t a g e d i p c a l cu l a t i o n s . I n

t h e B C U m e t h o d a f t e r th e c o n t ro l l i n g U E P is c o m p u t e d ,

a s t r a ig h t - l in e a p p r o x i m a t i o n i s n e c e s s a r y to c o m p u t e Vcr.

A n o t h e r c l a r i f i c a t i o n c o n c e r n s t h e p o t e n t i a l e n e r g y

t = 0 . T h e p o i n t w a s m a d e t h a t t h e o n l y w a y t h e p o t e n t i a l

e n e r g y c o u l d b e z e r o a t t h e s t a r t o f th e f a u l t i s i f t h e p r e -

a n d p o s t - f a u l t s y s t e m s a r e i d e n t ic a l . T h i s i s n o t t h e c a s e i f

a n y l i n e s w i t c h i n g o c c u r s ; h e n c e , t h i s s h o u l d b e r e f l e c t e d

i n t h e c o m p u t a t i o n o f t h e e ne r gi e s. I f th e m a x i m u m o f th e

p o t e n t i a l e n e r g y r e f e r e n c e d t o 0 ° i s u s e d t o c o m p u t e g c r ,

t h e n t h e r e i s n o n e e d t o c o m p u t e 0 s i n t h e P E B S m e t h o d .

T h e t h i r d c l a r i f i c a t i o n i s r e g a r d i n g t h e p o s t - f a u l t

s y s t e m e n e r g y . A l t h o u g h o f t e n i t i s s h o w n a s c o n s t a n t ,

t h a t i s n o t n e c e s s a r i l y t h e c a s e . I n a s m a l l s y s t e m w i t h

d a m p i n g , i t is e a s y t o s e e t h a t t h e e n e r g y d e c r e a s e s w i t h

t i m e . W i t h l a r g e r s y s te m s , b e c a u s e o f t h e d e c r e a s e d e f fe c t

o f t h e t ra n s f e r c o n d u c t a n c e s , n o t o n l y d o t h e e n e r g i e s

a p p e a r t o s t a r t f r o m z e r o , t h e y a r e a l m o s t c o n s t a n t o r

s l i g h t l y d e c r e a s i n g a f t e r t h e f a u l t i s c l e a r e d .

V II . A c k n o w l e d g e m e n t s

T h e a u t h o r s w o u l d l i k e t o t h a n k t h e N a t i o n a l S c i e nc e

F o u n d a t i o n f o r i ts s u p p o r t t h r o u g h i t s g r a n t N S F E C S

9 1 - 19 4 2 8 . T h e y a l s o w is h t o t h a n k P r o f e s s o r K . R .

P a d i y a r o f I. I. S c ., B a n g a l o r e , I n d i a f o r h i s u se f u l

c o m m e n t s . T h e y a l s o w i s h t o t h a n k t h e r e v i e w e r s f o r

t h e i r u s e f u l s u g g e s t i o n s .

V I I I. R e f e r e n c e s

1 Athay, T, Podmore, R and Virmani, S 'A pr ac t ica l me thod

for direct analys is of t ransient s tabil i ty ' I E E E T r an s . P o w e rA p p a r . S y s t . Vol PAS-98 N o 2 (1979) 573-5 84

2 Uemu ra, K, Ma tsuki , J , Yamada, I and Tsuji , T 'App rox-imation of an e nergy function in transien t s tabili ty analys isof pow er sys tems' E l e c t r . W n g . J p n Vo192 No 6 (1972) 96 -100

3 Padiy ar , K R and Ghosh, K K 'Direct s tabil ity evaluat ion o fpow er systems with detai led gene rator models us ing s truc-ture preserving energy functions ' I n t . J . E l e c t r . P o w e r

E n e r g y S y s t . Vol 11 No 1 (1989) 47 -56

4 Pai , M A P o w e r s y s t e m s t a b il i ty N . H ol land , A ms te r dam(1981)

5 Pa i , M A E n e r g y f u n c t i o n a n a l y s i s f o r p o w e r s y s t e m s t a b i l it y

Kluw er Academ ic Publishers , Boston , M A (1989)

6 Fouad, A A and Vit tal , V P o w e r s y s t e m t r a n s i e n t s t a b i l i t y

u s i n g t h e t ra n s i e n t e n e r g y f u n c t i o n m e t h o d Prentice Hall ,N ew Y o r k ( 1992)

7 Pavella , M and Murthy , P G T r a n s i e n t s t a b i l i t y t h e o r y o f

p o w e r s y s t e m s : f r o m t h e o r y t o p r a c t i c e John Wiley, NewYork (1993)

8 Fouad, A A and Stanton, S E 'Transien t stabil i ty analys is o fa mult i-machine pow er system Par ts I and I I ' I E E E T r a n s .

P o w e r A p p a r . S y s t . Vol PAS-100 N o 8 (1981) 3408-3424

9 Kakim oto, N, Ohsawa, Y and Hayashi , M ' T r ans ien tstability analysis o f electric pow er systems via Lu re type

Lyap unov f unc t ions Par t s I and I I ' T r a n s . l E E J p n Vol 98No 5/6 (1978)

10 Chiang, H D 'Analyt ical results on direct meth ods for powe rsystem transient stability analysis ' C o n t r o l D y n . S y s t . (ed.C T Leond es) Vol. 43 Par t 3 Academ ic Press (1991) pp 185-27 4

11 Anderson, P M and Fou ad, A A P o w e r s y s t e m c o n t r o l a n d

s t a b i l i t y Iowa State Unive rs i ty Press (1977)

12 Little, J N M A T L A B : U s er 's G u id e The Mathw or ks I nc .(1991)

13 Chow , J H and Cheung,K W ' A too l bo x f or pow er systemdynam ics and contro l engineering ed ucation and research'I E E E T r a n s. P o w e r S y s t . Vol 7 No 4 (1992) 1559-1564

14 Chiang, H D, W a, F F an d V ar a iya , P P ' A B CU me thod f ordirect analys is of pow er sys tem transient s tabil i ty ' I E E E

T r a n s . P o w e r S y s t . Vol 9 No 3 (1994) 1194-1208

some clarifications in the transient

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