FP6 4:OO POWED S Y S T E M C O N T R O L WITF ni iST"SSANCE - A C C O M ~ O D A T I O P i

E l e c t r i c a l & Computer Engineer ing ?e?t . i jn ivers i ty o f Alabama in Huntsvil le

Hun t sv i l l e , A1 abama 35899

Abstract

One 3 f the important control problems i n the oper- a t i o n of a n interconnected power system i s l o a d - freQuency control !LCC). I n t t l is paper the LFC problem is inves t iga ted us ing a r e c e n t r e s u l t i n modern control thecry k n o w n as disturbance-accommodation control ( 3 b . C ) . I t i s shohn t h a t the optimal accommodation o f l o a d d i s - turbance c a n l ead to s ign i f icant ly be t te r per formance t h a n t h a t o f convent iona l cont ro l le rs . I t i s f u r t h e r shown that the complete obsot-ption of al l disturbance e f f e c t s i r t h e c l a s s o f power systems considered !s imposs ib le . Yever the less , the d i s turhance e f fec ts in system frequency can be cancelled completely. S i m u - l a t i o r r e s u l t s a r e p r e s e n t e d f o r t h e c a s e o f one area systev ard the d a t a c lear ly demoqstrate the degree o f performance improvement wade possible by a 9,4C con- t r o l l e r .

f *

Af

'i

'e P

P

A

Api

Apg

Uctation

power system nominal frequency

incremental change i n frequent:: i n area i

a r ea i ne r t i a cons t an t

gene ra t ed e l ec t r i c Dower i n MW

power system nominal operating load

in'zremental change i n command signa swed changer

i n MW

1 t o the

inzremental change i n area load

incremental change in power generat ion ' n PLMW

ADtie increvental change in t ie-?ire power i n PuMW

D po'der sys tem se l f - reg t i la t iun coef ic jen t

Tg governor system time constant

T t turbine t ime constant

AXg increvental change ir the governor valve pos i t ion i n PuMW

I - Introduct ion

I n recent yeaps the cortrol o f l a r g e i n t e r - coqnected power systems has received increasing atten- t i o n . I t has always been o f great concern t o ttle power in6us t ry t o be ab le t c m a i n t a i n e l e c t r i c Dower serv ice i n the face of u n k n c w n system disturbances. F o r man:, years the convent ic 'nal control lers employed f o r t h i s . p u r p o s e were a a i n l y based a " t i e - l i n e b i a s control concept [ : I , [2], [:.I, wbere the control law used to reduce t i e a rea cont ro l e r ror was of the

Droportional - plus - in tegra l type . W i t h the devlop- ment o f modern control theory, new concepts have been introduced for the design o f optimal power system c o n t r o l l e r s . Using these new concepts , two m a i n a p - proaches t o the problen o f power system regulation have been considered. The f i r s t acproactl involves the design o f c e n t r a l i z e d c o n t r o l l e r where a l l necessary information must be sen t t o a c e n t r a l c o n t r o l l e r [ A ] , which in tu rn cont ro ls a l l the power generat ing stations within t t le area. This approach i s based on the assurrption t h a t each area can be represented by a s ing le f requency .

The second r r a i n approach. 'Jsirlg modern control theory , has been to design a decen t r a l i zed con t ro l l e r having the property t h a t a t each individual generating s ta t ion wi th in the a rea the loca l sys tem var iab les a re used t o control tile power o u t p u t ? through local con- t r o l l e r s , i n response t c area l o a d a n d frequency vari- a t i o n [ 5 ] , [ G I .

I n t h i s J a p e r , a new type of central jzed o p t i r l a l con t ro l l e r i s p re sen ted fo r a mul t i -a rea power system. using some r e c e n t r e s u l t s fron; distJrbance-accommo- dat ion control theory (04C). I n p a r t i c u l a r , a power sys tem cont ro l le r i s des igned t o ' accomodate" u n - cer ta in external d is turbances using the dis turbance u t i l i z i n g mode of P 4 C . This new type of c o p t r o l l e r makes optimal constructive use o f the dis turbance t o eptlance overall Der%rmance (when t h a t i s p o s s i b l e ) : a n d otherwise minimizes ti-e performance loss conLrib- uted by the d i s turbance ,

11 ~- Modeling Power Systev 3is turbances

:!rider normal cperat ing condi t ions a power system i s con t inua l ly sub jec t ed t o small random-like disturb- ances. Typical example of such disturbances are small changes in the scheduled generation of one machine o r a small i o a d added to the network. The t r a n s i e n t beilavior of a power system following a d i s t u r b a n c e i s in genera l osc i l7a tory . I f t h e s y s t e m i s s t a b l e , t h e s e o s c i l l a t i o n s w i l l be danped o u t t o w a r d a new su iesent opera t ing condi t ion . Those momentary o s c i l l a t i o n s however, are t ransformed into f luctuat ions i n the power f l o w ove r t he t r ansmiss ion l ' ne s . i f a c e r t a i n t i e - l i n e i n a l a rge in te rconnec ted power system undergoes excessive powzr = luc tua t ions i t may t r i o , t h e r e b y d i s - connect ing the tw3 aroups c = rrachines. Ttle degree o f power f l u c t u a t i o n t h a t a systev c a n t o l e r a t e depends pr imari ly on the i n i t i a l oDera t ing cond i t ions .

To deal w i t h these unknown, uncontrollable system i n p u t s e f f e c t i v e l y i t i s e s s e n t i a l t o model, t ' l e i r an t ic ipa ted behavior . One c lass ica l approach cons is t s o f t r e a t i n g t h e d i s t u r b a n c e a s i n i t i a l c o n d i t i o n s on sys tem s ta te var iab les . This approach of fe rs some means t o cope with step or ramp type dis turbances. Another conventional approach i s t o t r ea t d i s tu rbances as a random-type disturbance whicCl are completely

1429 0191-2216 83 0000-1429 S1.00 i 1983 IEEE

e r r a t i c i n n a t u r e h a v i n g n o s i g n i f i c a n t d e g r e e o f r e g - u 1 a r i t . y i n t h e i r w a v e f o r m s .

P o w e r s y s t e m d i s t u r b a n c e s , o n t h e o t h e r h a n d , a r e n o t t o t a l l y i r r e g u l a r b u t i n f a c t h a v e " w a v e f o r m s t r u c t u r e " . T h a t i s , t h e y c a n b e m a t h e m a t i c a l l y mode led as a w e i g h t e d l i n e a r c o m b i n a t i o n o f a s e t o f k n o w n b a s i s f u n c t i o n s

The b a s i s f u n c t i o n f f i ( t ) r e p r e s e n t t h e v a r i o u s w a v e -

f o r m p a t t e r n s t h a t m i g h t b e i d e n t i f i e d i n a c t u a l r e c o r d i n g s o f a r e a d a i l y l o a d and Ci, i = 1 ,2 , . . . , m a r e

u n k n o w n p i e c e w i s e c o n s t a n t p a r a m e t e r s t h a t v a r y i n a n u n k n o w n p a t t e r n . I n p r a c t i c a l a p p l i c a t i o n s , t h e f i ( t )

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

ldh ich has e x p r e s s i o n ( 1 ) as i t ' s g e n e r a l s o l u t i o n [8], where bi, i = 1 ,.. . . ~ a r e k n o w n c o n s t a n t s a n d u ( . t ) i s a

c o m p l e t e l y u n k n o w n i m p u l s i v e f o r c i n g f u n c t i o n . r e p r e - s e n t i n g t h e r a n d o m n e s s o f c o e f f i e i e n t s C i n e q u a t i o n ( 1 ) . E x p r e s s i o n ( 2 ) i s a m o r e c o n v e n i e n i m o d e l f o r DAC d e s i g n p u r p o s e s .

I n g e n e r a l a s y s t e m d i s t u r b a n c e m o d e l l i k e ( 2 ) c a n b e r e p r e s e n t e d i n t h e f o l l o w i n g s t a t e - v a r i a b l e f o r m a t

W ( t ) = L ( t ) Z ( t )

Z ( t ) = D ( t ) Z ( t ) o ( t ) ( 3 ) where L a n d D a r e P x 5 and x 5 k n o w n m a t r i c e s r e s p e c t i v e l y a n d c ( t ) i s a sequence o f c o m p l e t e l y u n - k n o w n , r a n d o m l y a r r i v i n g r a n d o m i n t e n s i t y i m p u l s e s r e p r e s e n t i n g t h e a c t i o n o f W ( t ) i n ( Z ) , F o r m u l t i - a r e a d i s t u r b a n c e s { W l ( t ) , W 2 ( t ) , . . . , W ( t ) ) e a c h W i ( t ) c a n

b e m o d e l e d b y e x p r e s s i o n s 1 i k e ( 1 ) - ( . 2 ) . P

I n t h i s p a p e r t h e s e t o f t h r e e b a s i s f u n c t i o n s

r l , t , t i s c h o s e n t o r e p r e s e n t t y p i c a l l o a d r e c o r d i n g s ; i e . t t e i r r a n d o m c o m b i n a t i o n a p p r o x i m a t e l y r e p r e s e n t s t h e r a n g e o f w a v e f o r m b e h a v i o r w h i c h t h e s y s t e m l o a d d i s t u r b a n c e c a n e x h i b i t . T h e r e f o r e , t h e a r e a l o a d d i s t u r b a n c e W ( t ) will b e m a t h e m a t i c a l l y m o d e l e d b y t h e e x p r e s s i o n

w i t ) = c1 + C 2 t .t c t e - a t 3 (4)

However, f o r m o r e a c c u r a t e r e p r e s e n t a t i o n o f l o a d v a r i a t i o n s , t h e s e t c a n b e e x t e n d e d t o i n c l u d e m o r e r e p r e s e n t a t i v e w a v e f o r m s . I n t e r m s o f ( j ) , e x p r e s s i o n (4) c o r r e s p o n d s t o

W ( t ) = [1 I 0 7 0 , 01 Z ( t ) 0 1 0 0

Z ( t ) = 1; ; ;2 e 1 Z ( t ) + d t )

0 0 -a -2a ( 5 ) 111 - F o r m u l a t i o n o f t h e LFC Prob lem as a P r o b l e m

i n D i s t u r b a n c e - A c c o m m o d a t i o n

The LFC l i n e a r r e g u l a t o r p r o b l e m f o r a m u l t i - a r e a p o w e r s y s t e m m a y b e s t a t e d a s f o l l o w s : A - a r e a p o w e r s y s t e m i s d e f i n e d b y

i ( t ) = A ( t ) X ( t ) + B ( t ) U ( t ) + F ( t ) W ( t )

Y ( t ) = C ( t ) X ( t ) ( 6 )

w h e r e X , U , and W a r e s t a t e , c o n t r o l , a n a d i s t u r b a n c e

v e c t o r s o f d i m e n s i o n s n , r , p r e s p e c t i v e l y a n d t h e m - v e c t o r Y i s t h e s y s t e m o u t p u t m a . E a c h a r e a i n . : - i n t e r c o n n e c t e d s y s t e m i s d e f i n e d b y a s e t o f f i v e s t a t e v a r i a b l e s , o n e c o n t r o l i n p u t ( s p e e d c h a n g e s p o s i t i o n ) a n d o n e d i s t u r b a n c e i n p u t ( a r e a l o a d v a r i a t i o n ) . The c l a s s o f p o w e r s y s t e m s c o n s i d e r e d i n t h i s s t u d y i s g i v e n i n b l o c k d i a g r a m f o r m f o r i - t h a r e a i n F i g u r e 1 .

F i g u r e 1 . B l o c k D i a g r a m o f C o n t r o l A r e a Where T = 2Hi/f*Di

P i

The e q u a t i o n s d e s c r i b i n g t h e d y n a m i c s o f i - t h a r e a i s g i v e n b y 171

iil = - f * / 2 H i Di Xil + TTv (Xi5 - X v 5 ) - Xi2 + Wi

Xi2 = - 1 / T . X + l / T t i Xi3

Xi3 = -l/Tgi Xi3 - 1 / T . RiXil + l/Tgi Ui

t 1 i 2

9'

' i 4 = ' i 1

iCi5 = $ TPV (Xi5 - X v 5 ) ( 7 )

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

Xil = A f i Xi4 = A f i d t

x . = A P a ; Xi5 = APtieidt

xi3 = A X . 1 2 ~

g' (8)

I n t h i s s t u d y we d e p a r t f r o m t r a d i t i o n a l r e p r e s e n t a t i o n o f t h e u n c e r t a i n d i s t u r b a n c e wi i n ( 7 ) a n d u s e t h e waveform mode l ( 3 ) . T h e p r i m a r y o b j e c t i v e o f c o n t r o l U i s t o m a i n t a i n r e g u l a t i o n o f t h e a r e a f r e q u e n c y a n d t i e l i n e p o w e r d e v i a t i o n c l o s e t o z e r o , w h i l e s i r n u l - t a n e o u s l y c o u n t e r a c t i n g t h e s y s t e m d i s t u r b a n c e . To a c h i e v e t h i s o b j e c t i v e a n d i n f o r c e a c o n s t r a i n t o n c o n t r o l e f f o r t o n e may choose t o m i n i m i z e a q u a d r a t i c p e r f o r m a n c e i n d e x o f t h e f o r m

J(U,XO, tO,T) = 1 / 2 X T ( t ) S X ( t ) +

1 / 2 Jt [ X ( t ) Q X ( t ) f UT ( t ) R U ( t ) l d t (9) T T T

0 where S a n d Q a r e s y m m e t r i c n o n n e g a t i v e m a t r i c e s o f a p p r o p r i a t e d i m e n s i o n s a n d R i s p o s i t i v e d e f i n i t e .

1430

The t i ;eory o f d i s t u r b a n c e - accommoda t ion [ e ] can be gsed t o Q p t i m a l l j a c c o m o d a t e t h e d i s t Q r b a n c e s LFC p r o b l e m . I n p a r t i c u l a r , t h e maximum u t i l i z a t i o n mode o f [I,AC [5] can be u s e d t o d e s i g n a LFC c o n t r c ' l e r w h i c h a u t o m a t i c a l l : $ u s e d t S e a c t i o n o f s y s t e m d i s t s r - bances t c , a c h ; e v e m o r e e f f i c i e n t l g z d - f r e q u e n c y c o n t r o l .

VI - L ~ p p l T c a t i o n o f DAC C o n t r o l t o a One-Area ? rob ! em

A s - n g l e c o n t r c l a r e a i s o l a t e d f r o m i t ' s n e i g h - b o r i n g a r e a s c a n b e a p p r o x i m a t e d b y a t h i r d - o r d e r v e r s i o n o f t h e m o d e l (7), w q , e r e t h e s y s t e m s t a t e a n d c o n t r o l v a r i a b : e s a r e X, = .If, X, = 2P X = L X and

IJ = LP . The s y s t e m m a t r i c e s A , 5, C and F i n t h a t g ' 3 9

C c a s e a r e o b t a i n e d f r o m ( 7 ) 2s

I I

i = [ 1,3,0 1

r - f* /?H 1

( 1 0)

D e s i g n o f a Disturbance-Accommodatinq C o n t r o l l e r : By f o r m u l a t i n g t h e LFC p r o b l e m a s a z e r o s e t - p o i n t s t a t e r e g u l a r p o r b l e m , o n e may f o l l o w t h e D A C d e s i g n p r o c e d u r e g i v e n i n [9] t o a b s o r b a1 1 t h e d i s t u r b a n c e e f f e c t s . F o r t h i s p u r p o s e t h e t o t a l c o n t r o l e f f o r t Il(t) i n ( 6 ) i s s p l i t i n t o t w o p a r t s

U ( t ) = u c ( t ) ( 1 1 1

w h e r e U c ( t ) h a s t ' l e t a s k o f c o u n t e r a c t i n g t h e d i s t u r -

bance N ( t ) a n d U r ( t ) i s r e s p o n s i b l e f o r r e g u l a t i o n

X+) + 0 .

C o i r p l e t e c a n c e l l a t i o n o f d i s t u r b a n c e e f f e c t s c a n b e a c h i e v e d i f a n d o n l y i f t h e f o l l o w i n g c o l u m n r a n g e s p a c e c o n d i t i o n i s s a t i s f i e d [ 8 ]

?.[F(t) * H ( t ) ] G X [ B ( t ) ] ; t O s t l T ( 1 2 )

'where ?.[.I d e n o t e s t h e c o l u m n r a n g e s p a c e . T h i s i m p l i e s t h e r a n k c o n d i t i o n

Rank [ B , FL] = Rank [B] ( 1 3 )

A p p l y i n g t h e r a n k c o n d i t i o n ( 1 3 ) f o r o n e - a r e a p r o b l e m ( 5 ) - ( 1 0 )

I O ~ - f * / 2 H 0 0 0 Rank [G(FL ] = Rank 0 0 o j = 2

0 0 0

The r a n k c o n d i t i o n ( 1 3 ) f a i l s f o r o n e - a r e a p r o b l e m . T h u s , c o m p l e t e a b s o r p t i o n g f a l l d i s t r u b a n c e e f f e c t s i n t h e c l a s s o f p o w e r s y s t e m s c o n s i d e r e d i s i m p o s s i b l e .

D e s i g n o f a D i s t u r b a n c e - M i n i m i z i n g C o n t r o l l e r : I t has j u s t b e e n s h o w n t h a t c o m p l e t e c a n c e l l a t i o n o f a l l d i s t u r b a n c e e f f e c t s i n a o n e - a r e a p r o b l e m i s n o t p o s s i b l e . T h e r e f o r e , a s a n a 1 t e r n a t e d e s i g n a p p r o a c h

one L3j' a t t e m p t t o c h o o s e t h e c o n t r o l ! l - ( t ] t o m i n i m i z e

t h e 4 i s t u r b a n c e " r e s i d u a l ' n o r m

~ I B ( t ) U c ( t ) F ( t ) U ( t ) ~ ' ( 1 4 )

The c o n t r o l c o m p o n e n t Y c ( t ) w h i c h m i n i s i z e s (!4j i s n o t

u n i q u e i n g e q e r a : . I f one c11ooses t h e o n e whicc1 i t s e l f has a ninimlJ,x ncrm tb ien i t i s u n i q u e [SI and i s g i v e r ? by

where 5= = :aT 5 j - l i s t h e g e n e r a l i z e d i n v e r s e o f B . F o r t h e s p e c i f i c m o d e l ( 5 ) - ( 1 0 ) e x p r e s s i o n ( 1 5 ) becomes

IJr(t) = - B ' F ( t ) P ( t ) Z(t) ( 1 5 )

- f * !2Y

F i g u r e 2 . C o n t r o l i D i s t u r b a n c e Range Space S i t l u a t i o n f o r a One-Area Power System

T b u s , f o r a n a r b i t r a r y d i s t u r b a n c e ' s t a t e " Z , t h e n o r m ~ B U + F ;4 i s minimum when Llc = 0. I n o t h e r w o r d s ,

t o m i n i m i z e t h e d i s t u r b a n c e r e s i d u a l n o r m ( l a , o n e s i m p l y a p p l i e s n o c o n t r o l ! T h i s T h e o r i t i c a l r e s u l t s u g g e s t s t h a t t h e p u r e m i n i m i z a t i o n o f d i s t u r b a n c e e f f e c t s ( 1 4 ) i n LFC: p r o b l e m i s n o t a n e f f e c t i v e way t o p o s i n g t h e d i s t u r b a n c e - a c c o m m o d a t i o n p r o b l e m . T h e r e - f o r e , o n e s h o u l d t r y some o t h e r f i g l d r e s o f m e r i t f o r d i s t u r b a n c e - m i n i m i z a t i o n . Cine s u c h a l t e r n a t i v e will be e x p l o r e d .

D i s t u r b a n c e A b s o r p t i o n f o r a C r i t i c a l S t a t e V a r i a b l e : I n t h e o n e - a r e a p r o b l e m o n e i s p a r t i c u l a r l y c o n c e r n e d w i t h t h e s y s t e m f r e q u e n c y d e v i a t i o n i n r e s p o n s e t o t h e l o a d d i s t u r b a n c e s . T h u s , r a t h e r t h a n a t t e m p t i n g t o m i n i m i z e ( 1 4 ) o n e may i n s t e a d t r y t o a b s o r b t h e t o t a l e f f e c t o f d i s t u r b a n c e s W(t) o n t h e s y s t e m f r e q u e n c y d e v i a t i o n X .

F o r t h e o n e - a r e a p r o b l e m we will c o n s i d e r a n i n d i r e c t c o n t r o l a c t i o n t o a b s o r b t h e d i s t u r b a n c e e f f o r t s o n t h e f r e q u e n c y . The d i f f e r e n t i a l e q u a t i o n g o v e r n i n g Y1 = h f c a n h e o b t a i n e d f r o m ( 5 ) - (IO) as

k ( f /2YT T ) li - ! f * /2PT tTg) W - ( f /2HT + f /2HTt) !d

* * t g 4

- f * / 2 U ; ( 1 7 )

E q u a t i o n ( 1 7 ) s h o w s t h a t 11 c a n i n d e e d b e d e s i g n e d t o c o u n t e r a c t t h e d i s t u r b a n c e e f f e c t s i n X, a n d s i m u l t a n - e o u s l y r e g u l a t e X , + 0. The c o n t r o l U ( t ) i n ( 1 1 )

w h i c h a c c o m p l i s h e s c o u n t r a t i o n o f d i s t u r b a n c e e f f e c t i n ( 1 7 ) i s g i v e n b y

C

1431

U (t) = W + ( T t + T ) W + T T W ..

g g t ( 1 8 )

a n d t h e c o n x r o l U r ( t ) w h i c h r e g u l a t e s X1 -+ 0 i n ( 1 7 ) i s

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

U ( t ) = -(K,l X 1 I K 2 X 1 t t3Yl) ( 1 9 )

where K1 , Y 2 and K a r e t o b e d i s i g n e d .

3 y r e p l a c i n q t i e d i s t u r b a n c e t e r m s i n ( 1 8 ) b y e q u i v a l e n t

z t e r m s s g c h a s w=Hz, w=EDz, w=HD z a n d d e r i v a t i v e s o f Xl i n ( 1 5 ) w i t h t h e i r e q u i v a l e n t t e r m s f r o m ( 6 ) - ( 1 0 )

t h e t o t a l c o n t r o l U ( t ) c a n b e e x p r e s s e d b y [lo]

3

2

U ( t j = K Z ( t ) Kx X(t) Z (20)

w h e r e Z ( t ) a n d X ( t ) a r e a c c u r a t e e s t i m a t e s o f Z ( t ) a n d X i t ) w h i c h will b e o b t a i n e d f r o m y ( t ) b y a n o n - l i n e r e a l t i m e s t a t e r e c o n s t r u c t o r . a n d K Z = (KZ1 , K Z 2 , i:

K Z 4 ) and kx = ( K ~ , , xX2, K ) a r e c o n t r o l l e r g a i n

compu ted f rom ( 1 8 ) , ( 1 9 )

z 3 ' x 3

kZ1 l-(f*D/2H) K +(f* D/2H) K 2 1 2

* k = T, A. Tg + (f / 2 H ) K1

2 2

Kz3= T+ r

KZ4' 0

- g

kxl= -K 3 - (ffD/2H) ' K 1 + (f'D/2H) K2

* k = (f D/2H) (f D/2H)K -K + 1 / T T

x2 1 2 t g *

kx3= -if /2H) ic 1

( 2 1 )

Ti;e s i m u l a t i o n r e s u l t s f o r a o n e - a r e a p r o b l e m w i t h a c r i t i c a l s t a t e v a r i a b l e c o n t r o l l e r (io), ( 2 1 ) a n d u s i n g n u m e r i c a l p a r a m e t e r v a l u e s [IO]

Pr = 2000 MU R = 2 . 4 Hz/puMld

Po = 1000 Ml.1 f = 60 r z *

H = 5 . seconds D = 8 . 3 3 X puW!

T t = 0 . 3 seconds T = 0 .08 seconds g

F i g u r e 3 , Computed Per fo rmance o f C l o s e d - L o o p System (5)-(10) w i t h C r i t i c a l S t a t e ' i ' a r i a b l e C o n t r o l l e r a n d S t e p L o a d D i s t u r b a n c e

D e s i g n i n g o f a D i s t u r b a n c e - U t i l i z i n g C o n t r o l l e r : P. D A C c o n t r o l l e r w h i c h makes maximum u t i l i z a t i o n o f a n u n c e r t a i n l o a d d i s t u r b a n c e N(t) t o a c h i e v e t h e c o n t r o l o b j e c t i v e s m o r e e f f i c i e n t l y , c a n be d i s i g n e d t o r e g u l a t e s y s t e m f r e q u e n c y d e v i a t i o n X1 t o z e r o w h i l e

k e e p i n g t b e t i m e i n t e g r a l o f f r e q u e n c y d e v i a t i o n ( t i m e d e v i a t i o n ) a s s m a l l a s p o s s i b l e .

F o r D U C c o n t r o l l e r d e s i g n p u r p o s e s t h e p o w e r s y s - tem (6j a n d d i s t u r b a n c e p r o c e s s (3) i s c o m b i n e d t o fori7 a c o m p o s i t e m o d e l

j ? = A S X + S u + i (23)

where X = ( X , Z ) 1 s t h e c o m p o s i t e s y s t e m s t a t e v e c t o r a ne

T .

T h e o p t i m a l D U C c o n t r o l l e r w h i c h a c h i e v e s d i s t u r b a n c e u t i l i z a t i o n w h i l e m i n i m i z i n g a p e r f o r m a n c e i n d e x o f t h e t y p e ( 9 ) s u b j e c t t o ( 6 ) h a s t h e f r o m [ 9 ]

( 2 5 )

where K x i s t h e c o n v e n t i o n a l l i n e a r q u a d r a t i c g a i n a n d

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

0, t y p i c a l DUC c o n t r o l l e r g a i n s a r e g i v e n i n F i g u r e 4 f o r t i e s i n g l e a r e a e x a m p l e (22).

x 2

i s shown i n F i g u r e 3 f o r a s t e p l o a d d i s t u r b a n c e

3.0 3.0 0.0 I n.a T.5 S C C ,

I \Y t

F i g u r e 4 . C o m p u t e d G a i n M a t r i c e s K x ( t ) a n d K x z ( t )

f o r O n e - n r e a C o n t r o l P rob lem w i t h S = d i a q ( 1 0 , l : l ) , 0 = d i a g ( 1 0 . 1 ; 1 ) , e = 1

1432

T h e closed-loop resDonse o f 3ne-area system with D U C c o n t r o l i e r i s cornoared w i t h t h a t of the l inear quadratic ( L C ) cont ro l le r def ined by

The s iwu la t ion r e su l t s fo r d i f f e ren t l oad d i s tu rbances a r e shown i n Figure 5 a n d 6 , assuming ident ica l con- d i t i o n s f o r both types o f c o n t r o l l e r s .

Figure 5 , One-Area Power System Performance with Square !dave Disturbance 0.01 P u Y W ; Yon- Ideal Y"UC a n d L Q C o n t r o l l e r s , S = d i a g ( l @ , l , l ) , 9 = d i a g ( l O , l , l ) , P, = 1

FiGure 6 . 3re-Prea Power System Performance with

Load Disturbance !*! = O.Oi (1-e 1 ; !Jon- Ideal 3UC L Q C o n t r o l l e r s , S = diag ( 1 0 ,

-t

1 , I / r l \ 3 = d i a g ( l O , l , l ) ' 9 = 1

The D U C cont ro l ie r ach ieved a myth f a s t e r r e sponse w i t h substant ia l ly smaller f requency deviat ion for both types of d i s turbances cons idered . The improvement i n performance ildex (9) using D U C cor t ro l l e r w i th r e spec t t c L? c o n t r o l l e r f o r 2 s tep load d i s turbance i s g iven i r Tab?e 1 . ?'le reiative inprovement in performance i rdex de'ined DytT i s remarkable a n d i t i s a b o u t 74'..

Table ' 1 . -'le 9re-L.rea Powr Systev w i t h Step h a d '3istiJrbance 3 f ?.Dl PuW; ConDarison o f Performance from "017- Ideal ? Y C a n 3 L O Cont ro l le rs

where the oarameters a re d2f ined as :

JT- <e* .?, - v 2 x:(?: s x t ? l + i ; 2 / ' [ x - l t , : x , : l T u ~ t t ) s u c t : j c t

d :

= :,; ,fT ::.aTt' 2 JtO . .

E - = ' 2.. - ;-?- ;/ 2:: x 120 * -. --I ~ = ' ?JL2 - Jvc !/ x LOC 5

:E = c d t ( 7

't,

----- Conclusions

Qisturbance-accomodating c o n t r o l l e r s f o r power systems can be designed in a systematic way. by a p p l i - ca t ion of the theory of Disturbance-Accommodation Control . I t has been shown t h a t the compiete absorp- t i on o f d i s tu rbance e f f ec t s i n t he c l a s s o C mult i -area power systems considered i s impossi b le . Never the less , t he d i s tu rbance e f f ec t s o n the system *requency can be completely e leminated. i t has been f u r t h e r shown t h a t a power system control ler using the dis turbance- u t i l i z a t i o n mode o f D A C theory makes o p t i m a l construc- t i ve u se o f the uncertain external load dis turbances t o enhance overall performance, when t h a t i s poss ib l e a n d otherwise minimizes the performance loss contri- buted by the l o a d d i s turbance .

Peferences

E l g e r d , n l l e i . , F o s b a , C. E . , ' O p t i m u m Megawatt- Freouency Control of b l t i a r e a E l e c t r i c Energy systems I E E F Trans. P4S-89? P'o. 4, 4pr i l 1 9 7 3 .

Clovic . M . , 'Linear Degulator 3esian for a Load a n d Freouency C o n t r o l , ' !EEE Trans. PAS-91, p p . 2 2 7 1 , 1 9 7 2 ,

Lacarna P o n a l d J , , Johnson, Jot-.nny R . , ' ' D Leaminq Cont ro l le r for t i -e Megawatt Load-"requency Control Problern, I E E E Trans. on system, M a n . a n d Cybernet ics , v o l . SYC-10, Yo. 1 , P O . 4 3 , Jan. 1930.

Elgierd, Q l e I . , Cosha, E.!. , "Ti.e b g a w a t t - "requency Control problem: F new approach v i a optimal Control Theory ' I E E E Trans. PF,S-?9 , Vo. 4 , April 1970.

Tr ipa th i I hl, C., "The Qpt i ra l Cecer t ra i i zed Control of a Large E l e c t r i c Power Sys tem- h tomat ic Gener- a t i o n Cont ro l , ' D h . D . 3 i s se ra t i cn . I Jn ive r s i ty o f Toronto. Toronto, Ontario, Canaja, Jan. 1C7?.

Jamshidi , Y . , "Large Scale S y s t e m s - ~ o ~ e : i n g , C o n t r o l a n d Applicat ions, ' Elsenier Vorth-Folland Book C o , , '!ew York, 1981.

F l g e r $ , n l i e I . , " E l e c t r i c Energy Systeirs Tseo.y: An Tntroductson," 2 n d E d < t i o n , "cGraw-Vill ?oak, L o . , Yew York, 1 9 1 . Johnson, C.3. "Theory o f Cisturbance-Accowmsdation C o n t r o l l e r s , " Contrrj! a n d 9ynamic Systelrs: Advances i n Theory ZY 4ppl i c a t i o n s , v o i , 2 , Acalenic Press, I n c . , Yew Y o r k , 1 , 9 7 6 .

Johnson , C .C. , ' l l t i l i ty of Disturbances i n Dis tur- bance 4ccornmalating Control Problems " Proc. C i f - t een th 4nnual Yeetivg o f the Soc ie ty o f Engineering Sc ience , Inc . , a t Gainesvi l le , Decevber 1978.

Yobadjer, M . , "Power System Control w i t h d i s t u r h n c e -accommodation, ' P 5 . D . Dissera t ion , I J A H , May 1 9 2 2 .

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