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Archly ffir Elektrotechnik 72 (1989) 415--426 Ar c h i vf t i r E l e k t r o t e c h n i k9 Springer-Verlag 1989

Excitation controller design of synchronous machine with ou tpu t feedback using highand reduced-order modelsD. P. Papadopoulos, J. R. Smith an d G. Tsourlis, Xa nt hi , Greece

Contents: A practical 8th-order single-input multi ple- outpu t(SIMO) tim e-in vari ant linear model of a synchronousmachine with first order con ventiona l exciter, supplyingpower to the power system through a transformer and atransmission line has been developed. 3?he criterion for theeffective contrib ution of each system state to the total energyresponse at the o utpu t is used to obtain an adequate reduced4th-order model in which the most important states andeigenvalues of the original model are retained. An efficientalgebraic pole--placement method, using output feedbackhas been used to design the 8th and 4th-order models,physically realisable linear excitation controllers (witheasily measurable state variables) for the purpose of sub-stantially enhancing the dynamic stability characteristicsand voltage control of a synchronous machine. The controllerdesigned for the 4th-order modei has also been tested on t he8th order model to demonst rate the val idity of implementin greduced order formulations without degrading the perform-ance of the control. A procedure for implementin g the con-trol strategies is presented in this study, and attention isdrawn to the wider implications of implementing practicaldesigns on large turbogenerat ors current ly in use.

Entwurf des Erregungsreglers einer Synchronmaschinedureh Zuriiekfiihrung auf ein reduziertes M odell[~bersieht: Aus dem Modell einer Synchronmaschine in Formeines linearen zeitinvarianten Systems 8 .0r dnu ng wirdein reduziertes ~:[odell 4. Ordnung abgeleitet, das die wich-tigsten Eigenwerte des Systems beibehglt. Mit Hilfe einesalgebraischen Verfahrens zur Bestimmung der Pole kSnnenlineare Regler der Erregung entworfen werden, die einverbessertes Verhalte n der Maschine bezfiglich Stabil itg tund Spannungsregel ung herbeifiihren. Der ffir das Modell4. Ordnung entworfene tl,egler zeigt bei Betraehtung amSystem 8. Ordnung keine Versehleehterung des Verhaltens.Auf die Bedeutung des Verfahrens ffir praktische Ausf/ih-runge n bei grogen Turbogener atoren wird hingewiesen.

List of principal symbolsd%~d, iqit

load anglefield currentstarer currents in d an d q axis circuits, respectivelymachine terminal current31"

WeVd~ VqVbVtP~ , Q t

T ~ , T mR T , X TRe , XeR~i k ~ , i k q~fXd, XqXkd~ 92kqxmd, xmqRkd, RkqooHKE, T~A V r eVeV e . c . s .c~o8S

field voltagestarer voltages in d an d q axis circuits, respectivelybusbar voltagemachine terminal voltageactive and reactive power delivered at machineterminalsgenerator-shaft mechanical powerairgap electrical torque and generator-shaft me-chanical torquetransformer resistance and reactancetransmissiml line resistance and reactanceexternal system resistance ( R T - t - R z ) and reaet-anee (X T ~- Xz)staf~r resistancedamper circuit currents in d and q axesflux linkagessynchronous reactances in d and q axesfield winding self reactancedamper winding self reaetanees in d an d q axesmagnetizing reaetanees in d and q axesdamper wind ing resistances in d and q axesmachine speed or electrical frequencyinertia, consta ntexciter gain and time constantincrement al (step) voltage reference (input) changeexcitation errorexcitati on controller voltage signaltransformation ratio of transformersubscript, means operating pointLaplace operatortransformation matrix

1 I n t r o d u c t i o nThe enhancement of the dynamic s tabi l i ty charac-teris tics of a practical power sy stem may be achievedby designing a controller to obt ain a closed-loopsystem with pre-assigned dynamic stability charac-teris tics. Many studies aimed at improv ing the dy-namic control lability of turboge nerato rs rely solelyon theoretical considerations and omit t he impor-ta nt and someti mes difficult aspects o f pract icalimplementat ion. A paper s tudy is unl ikely to take

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416 Archly f i i r Elektrotechnik 72 (1989)a c c o u n t o f al l t h e f a c t o r s a ff e c t i n g th e d e g r a d a t i o n o fp e r f o r m a n c e t h a t w i l l i n e v i t a b l y b e f o u n d i n p r a c t i c e .T h e u s e o f a t e s t - b e d m a c h i n e s y s t e m , a l t h o u g h s m a l li n r e l a t i o n t o t h o s e w h i c h t h e a n a l y s is d e s c r i b e d i nt h i s p a p e r i s t o b e d i r e c t e d , a s s i s t s w i t h t h e a s s e s s -m e n t o f t h e p e r f o r m a n c e o f t h e c o n t r o ll e r a n d o fa u x i l i a r y i t e m s s u c h a s t r a n s d u c e r s a n d t h e e f f e c t o ft h e i r t i m e c o n s t a n t s o n t h e o v e r a l l s y s t e m p e r f o r -m a n c e . I n t h e c a s e of t h e l a t t e r , t h e s iz e o f g e n e r a t o ri s o f c o u r se u n i m p o r t a n t .

A p r e r e q u i s it e t o t h e i m p l e m e n t a t i o n o f t h e d e s i g ns t r a t e g y i s t h e f o r m u l a t i o n o f a r e p r e s e n t a t i v e l i n e a rm o d e l o f t h e s y s t e m , t h e c o m p l e x i t y o f w h i c h m a y b er e d u c e d b y r e d u c t i o n t e c h n i q u e s . M a n y r e d u c t i o na n d a p p r o x i m a t i o n m e t h o d s [ 1 - - 6 ] h a v e b e e n i n -t r o d u c e d i n t h e l a s t t w e n t y y e a r s f o r h i g h - o r d e r s t a t e -s p a c e m o d e l s o r h i g h - d e g r e e t r a n s f e r f u n c t i o n m a t r i -c e s o f l a r g e - s c a l e l i n e a r t i m e - i n v a r i a n t m u l t i v a r i a b l es y s te m s . T h e m o d e l r e d u c t i o n m e t h o d s m a y b e a p p l i e dd i r e c t l y t o e i t h e r s t a t e - s p a c e m o d e l f o r m u l a t i o n s o fs y s t e m s o r t o t r a n s f e r f u n c t i o n m o d e l f o r m u l a t i o n s .I n t h i s s t u d y t h e o r d e r r e d u c t i o n m e t h o d o f [ 5 ] w i l lb e u s e d e x c l u s i v e ly , s in c e i t p r o v i d e s f o r r e t a i n i n g i nt h e r e d u c e d o r d e r m o d e l t h e m o s t i m p o r t a n t s t a t e so f t h e o r i g i n a l h ig h o r d e r m o d e l a n d a s a c o n s e q u e n c et h e p h y s i c a l s i g n i f i c a n c e o f t h e s t a t e s i s r e t a i n e da l o n g w i t h t h e i m p o r t a n t e i g e n v a l u e s . F r o m a p r a c -t i c al s t a n d p o i n t t h e s e tw o f e a t u r e s o f t h e m e t h o d a r eo f p a r a m o u n t , i m p o r t a n c e .

T h e m o d e r n c o n t r o l m e t h o d s , w h i c h a r e u s u a l l yu s e d t o d e s i g n a c t u a l c o n t r o l l e rs fo r s y s t e m s o f t h et y p e c o n s i d e r e d h e r e m a y b e c l a s s i f i e d a s e i t h e r" o p t i m u m c o n t r o l s t r a t e g i e s " [ 1 , 7 , 8] o r " a l g e b r a i cc o n t r o l s t r a t e g i e s " [ 9 - 1 2 ] . T h e a l g e b ra i c c o n t r o lm e t h o d s , p a rt i c u l a r ly w h e n t h e y a r e b a se d o n o u t p u t -f e e d b a c k , p r e s e n t c e r t a i n p r a c t i c a l a d v a n t a g e s , f o re x a m p l e e a se o f p r o b l e m f o r m u l a t io n , c o m p u t a t i o n a lr e q u i r e m e n t s a n d i m p l e m e n t a t i o n . T h e p o l e - p l a c e -m e n t m e t h o d o f [ 1 1 ] w i l l b e u s e d i n t h i s s t u d y , s i n c ei t i s e f fi c i e n t a n d r e l a t i v e l y s i m p l e t o a p p l y .

I n t h e c o n t e n t o f t h e p r e s e n t w o r k t h e f i r st s t e pi s t o o b t a i n a r e p r e s e n t a t i v e m o d e l f o r t h e s y s t e mu n d e r s t u d y , w h i c h i n p r a c t i c e im p l i e s a r e a s o n a b l ya c c u r a t e b u t n o t n e c e s s a r i l y a n e x c e s s i v e l y c o m p l i -c a t e d f o r m u l a t i o n . O n t h i s b a s i s a n 8 t h - o r d e r S I M Ot i m e - i n v a r i a n t l i n e a r m o d e l o f t h e s y s t e m c o n s i s ti n go f a s y n c h r o n o u s m a c h i n e w i t h c o n v e n t i o n a l e x c i t e rs u p p l y i n g p o w e r t h r o u g h a t ra n s f o r m e r a n d a t r a n s -m i s s i o n l i n e , t o a n e x i s t i n g e l e c t r i c i ty s u p p l y s y s t e m( w h e r e t h e s y n c h r o n o u s m a c h i n e i s r e p r e s e n t e d b y a7 t h - o r d e r m o d e l a n d t h e c o n v e n t i o n a l e x c i t e r b y a1 s t - o rd e r o n e ), is d e v e l o p e d . N e x t , b y u s i n g t h e m o d e lo r d e r r e d u c ti o n m e t h o d o f [ 5 ] a n a d e q u a t e 4 t h - o r d e rr e d u c e d m o d e l d e r i v e d f r o m t h e 8 t h - o r d e r m o d e l i so b t a i n e d . F m - t h e r m o r e , b y a p p l y i n g t h e p o l e - a s s i g n -

m e n t m e t h o d o f [ 1 1 ] t o t h e p r e v i o u s 8 t h - a n d 4 t h -o r d e r m o d e l s a n e x c i t a t i o n c o n t r o l l e r i s d e si g n e d i ne a c h c a se a n d , t h u s , r e sp e c t i v e c l o s e d - l o o p s y s t em sw i t h s i g n i f i c a n t ly i m p r o v e d d y n a m i c s t a b i l i t y c h a -r a c t e r i s t i c s a r e o b t a i n e d . T h e e x c i t a t i o n c o n t r o l l e rd e s i g n e d f o r t h e 4 t h - o r d e r m o d e l w a s t e s t e d o n t h e8 t h - o r d e r s y s t e m m o d e l , i n o r d e r t o v e r i f y i t s e ff e c -t i v e n e s s a n d t h u s g i v e a d d e d j u s t i f i c a t i o n t o t h ep r a c t i c a l m e r i t s o f t h e p r o p o s e d c o n t r o l l e r d e s i g np r o c e d u r e , i . e . v i a t h e r e d u c e d o r d e r m o d e l r o u t e .F i n a l l y , t h e d e s i g n s t r a t e g y w a s a p p l i e d t o a 3 0 M Wt u r b i n e g e n e r a t o r f o r w h i c h t e s t r e s u l t s w e r e a v a i la b l e .

2 M o d e l l in g a n d s i m u l a t i o n o f o p e n - l o o ps y n c h r o n o u s m a c h i n e s y s t e m

M a n y a s p e c t s o f t h e p r e s e n t w o r k , b o t h t h e o r e t i c a la n d p r a c t i c a l , a r e b a s e d o n a n 8 7 .5 k V A m o d e l a l t e r -n a t o r s e t c o n n e c t e d v i a t r a n s m i s s i o n c i r c u i t s t o a ni n f i n i t e b u s a s s h o w n i n F i g . 1 . T h e m o d e l m a c h i n ew a s s e l ec t ed t o h a v e p a r a m e t e r s b r o a d l y t y p i c a lo f t h o s e a s s o c i a te d w i t h l a r g e r m a c h i n e s . S u c hm a c h i n e s h a v e b e e n u s e d in c o n n e c t i o n w i t h p r e v i o u si n v e s t i g a t o r y w o r k [ 13 , 1 4] d i re c t e d t o w a r d s l a r g e rt u r b o - g e n e r a t o r s . T h i s a p p r o a c h f a c i l i t a t e s h a r d w a r ec h a n g e a s r e q u i r e d a n d a l l o w s f o r p r a c t i c a l t e s t s t ob e c a r r i e d o u t w i t h o u t d i s r u p t i o n t o m a i n g e n e r a t i o ns o u r c es . T h e p a r a m e t e r s f o r t h is m a c h i n e , w h i c h w e r eo b t a i n e d f r o m f r e q u e n c y r e sp o n s e t e s t s a n d f o r t h es y s t e m t o w h i c h i t i s c o n n e c t e d a r e g i v e n i n T a b l e 1 ,a n d i t i s r e a d i l y s e e n t h a t w h e r e a s t h e p e r - u n i t r e a c t -a n c e s a r e r e p r e s e n t a t i v e o f t u r b o g e n e r a t o r s i n t h er a n g e o f 2 0 - - 3 0 M W , w i n d i n g r e s i s t a n c e s a r e c o n s i d e r -a b l y h i g h e r . P r e v i o u s a u t h o r s h a v e o c c a s i o n a l l y u s e dt i m e c o n s t a n t r e g u l a t i o n s [ 1 3 ] t o a l l e v i a t e t h i s p r o b l e mb u t f o r t h e p u r p o s e s o f t h e p r e s e n t s t u d y t h i s w a s n o tc o n s i d e r e d n e c e s s a r y s in c e t h e m a c h i n e i s u s e d s o l e l yi n a s s o c i a t i o n w i t h d e t e r m i n i n g t h e f e a s i b i l i t y o fa p p r o a c h .Table 1. Pr incipal System DataSynchronou s Machine87.5 kVA, 415 V, 4-pole x d 1 . 7 5 8 u R a 0.0145 puXq 0.99 pu R~ d 0.00422 pux]d 1.761pu Rkq 0.0126 pu

Xkd 1 . 6 6 4pu Rfd 0.00268 puX~q 0.955 pu H 1. 43 4Xmd 1.658puXmq 0.899pU

Conventional exciter K E 1 v~ 0.472 sTran sform er a.nd R~- 0.0363 pu XT 0.083 8 p utransmission line R 0.03744 pu X L 0.3903 puper unit values on machine rat ing

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D. P. Papadopou los et aI . : Exc itzt ion control ler des ign of synchronous machine 417O r ig i n a l o p e n - l o o p s y s t e m

1 C o n v e n f i o n o t S y n c h r o n o u s i n f i n it e ~ "I e x c i te r g e n e r o to r G e n e r a t o r T r o n s m i s s i o n b u s I I

1 ~ Z T & R~ X L 1 :t i

l v f e v ~ 8 l ii t .

"L ' - - - - - - - - - - - I I

I I / E x c i t o t i o n I9 v e .c .~ . / ' ~ / - ~ = " c o n t r o l le r :t , - - t 1t ~ i D e s i g n e d :I. . . . . . . . . . . . . . . . . . / c l o s e d - l o o p .L . . . . . . . . . . . . . . . . . . . . . ~ ~ y s l e ~ i

Fig. 1. S implified rep resentat ion of synchronous machine + exciter s~pplying p ower to the electr ic u~il ity sys temthrough ~n int~rconnection network

I n t h e c o n s i d e r a t i o n o f s y n c h r o n o u s m a c h i n es y s t e m s i t i s c u s t o m a r y t o c o m m e n c e w i t h P a r k ' s d -a n d q - ax i s e q u a t i o n s f o r t h e m a c h i n e s n d t h e e q u i v a -l e n t d - a n d q : a x i s e q u a t i o n s f o r t h e e x t e r n a l c i r c u i t sc o n n e c t i n g t h e m a c h i n e t o t h e s u p p l y s y s t e m . I n t h ec o n t e x t o f t h e p r e s e n t w o r k t h e r e s u l t i n g s y s t e m e q u a -t i o n s a r e l i n e a ri s e d t o r e p r e s e n t t h e s y s t e m i n s t a n d -a r d s t a t e - s p a c e d i f f e r e n t i a l f o r m :i = A x + b u ( l a )y = C x ( l b )w h e r e x C R % y ( R m a n d u E R P a re t h e s t a t e , o u t p u ta n d i n p u t v e c t o r s r e s p e c t i v e l y , a n d A , B a n d C a r er e a l c o n s t a n t 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 .

B y a p p l y i n g t h e r e l e v a n t t h e o r y o u t l in e d i nA p p e n d i x A t o t h e s y n c h r o n o u s m a c h i n e , i t s s t a t e -s p a c e r e p r e s e n t a t i o n , i n t h e f o r m o f e q u a t i o n l ( a ),w a s o b t a i n e d , i .e .x T = [ i d i q i l ~ M i kq ~ c ~ ] ( 2 a )uT : [Vm P ~ ] ( 2 b )w i t h t h e e x p l i c i t n u m e r i c a l v a l u e s o f A a n d B , b a s e do n t h e s p e c i f i c o p e r a t i n g p o i n t ,P~ = 0 .8 pu ; Q t == 0 .84 pu; vt : = 1 . 2 9 p u ;vs~ - - 0 .004 3 pu ;,~ = 0.56 Pad ; vb = 0 . 9 7 p ua r e g i v en i n A p p e n d i x B . T h e e i g e n va l u e s o f th e a b o v es y n c h r o n o u s m a c h i n e m o d e l w e r e c o m p u t e d u s i n g a

s p e c i a l s u b r o u t i n e a n d a r e s h o w n i n T a b l e 2 . I t i ss e e n f r o m t h i s t a b l e t h a t t h e s y s t e m i s s t a b l e .

T h e c u r r e n t s i ~ a n d iq a r e n o t m e a s u r a b l e q u a n t i t i e s ,w h i c h p r e s e n t s p r a c t i c a l p r o b l e m s i n t h e a c t u a l d e s i g no f t h e e x c i t a t i o n c o n t r o l l e r, t h u s t h e f o l l o w i n g tr a n s -f o r m e d s y n c h r o n o u s m a c h i n e m o d e l i s o b t a i n e db y u s e o f a n a p p r o p r i a t e t r a n s f o r m a t i o n e x p l a in e d inA p p e n d i x A , i .e .[ x ' ] T = [ v , r i i ~ 4 ~ i ~q ~ ~ ] ( 3 a )[u ' ] T = [u] T = [v/d Pm] (35 )w i t k t h e e x p l i c i t n u m e r i c a l v a l u e s o f t h e m a t r i c e s B ' ,T a n d S g i v e n in A p p e n d i x B . T h e m a t r i x A ' is n o tw r i t t e n s e p a r a t e l y h e r e , s i n c e i t m a y e a s i l y b e i d e n -t i f ie d f r o m t h e 8 t h - o r d e r m o d e l o f t h e s y n c h r o n o u sm a c h i n e a n d t h e c o n v e n t i o n a l e x c i t e r w h i c h w i ll b ed e s c ri b e d n e x t a n d is g iv e n in A p p e n d i x B . T h e c o m -p u t e d e i g en v a l u es o f t h e t r a n s f o r m e d s y n c h r o n o u sm a c h i n e m o d e l a re g i v e n i n T a b l e 2 a n d s h o w c l e a r l yt h a t t h e t w o s y n c h r o n o u s m a c h i n e m o d e l s ( i. e. t h eo r i g in a l a n d t h e t r a n s f o r m e d o n e ) a r e i d e n t i ca l .T h e i n t r o d u c t i o n o f t h e d i f f e re n t i al e q u a t i o nd e s c r ib i n g t h e c o n v e n t i o n a l e x c i t e r t o t h e t r a n s f o r m e d7 t h - o r d e r m o d e l o f t h e s y n c h r o n o u s m a c h i n e ( ac c o r d -i n g t o t h e p r o c e d u r e e x p la i n e d in A p p e n d i x A ) y ie l d st h e 8 t h - o r d e r t r a n s fo r m e d s y n c h r o n o u s m a c h i n e w i t hc o n v e n t i o n a l e x c i t e r m o d e l i n t h e f o r m o f E q . ( l a )a n d ( l b ) , i . e .

[ u ' ] = = [ v s ] ( 4 b )[ y " ] ~ = [ v~ ~:~ ~ v ~ ] ( 4 c )

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418 Archiv fiir Elektrotechnik 72 (1989)Table 2. Simulated csses of open-loop and designed closed-loop systems (C = computed values, D ~ desired values and~t ~ eigenvalues)

s/m model C ~ --0.8006 --1.2724 j14.7439 --7.3045 --21.3114 --46.1202 :k j313.6362f~ Transformeds/m model C A --0.8007 --1.2727 :~ j14.7437 --7.3030 --21.3107 --46.1208 -V j313.6340Transformed C A --0.8008 --1.2729 j14.7437 --7.3033 --21.3112 --46.1208 :~ j313.6367 --2.1200s/m -]- excitermodelo

9 Reduced C A --0.8008 --1.2729 j14.7437 --2.1200order"~09 model9Transformed D ~ --1.1 --1.3 :k j14.7437 . . . . 2.3s/m ~- excitermodel C A --1.0884 --1.2996 j14.7414 --7.9034 --21.2099 --46.1213 j313.6360 --2.3116Reduced D ~ --2.5 --1.7 :k j14.7437 --3.0ordermodel C ~ --2.5 --1.7 ~: j14.7437 --3.0Transformed C A --2.5099 --1.2840 :k j14.7291 --7.3747 --21.4833 --46.1209 -k j313.6366 --2.9580s / m q - excitermodel withcontroller of"~ reduced ordermodel

with the explicit numerical values of A", B" and (7'given in Appendix B. Equation 4(b) presupposes thatthe prime mover maintains Pm constant, which is theusual assumption made for uncoupling the two controlloops of the synchronous machine for carrying-outsmall-input dynamic analysis. The computed eigen-values of the above 8th-order system model are shownin Table 2. The computed time responses of its outputsv t , i t , ~ and vld for the small-input step change A v ~--~ 0.001 pu are shown in Figure 2(a), (b), (c) and (d)respectively. The solution of the 8th-order modelwas obtained by using the l~unge-Kutta 4th-orderform integration technique. F rom Table 2 and Figure 2it is seen that the 8th-order model (original open-loopsystem) is stable but does not necessarily possesssatisfactory damping characteristics and sufficientlyfast response.

3 R e d u c e d o r d e r m o d e l o f o r i g i n a l s y s t e mLastman et al. [5] has shown that a strong relationshipexists between the response of the system states x ~ v tand the out put coefficient matrix C such that E, the itoutput energy criterion, may be found from v/d

n c ~ ~ f dE : ~ d i k f x k ( t ) x ~ ( t ) d t (5) ikdk = l o i k q

when d i , k is an eleme nt of D ~ CT(~. ~o

Thus, if the ith state forms a significant contri-bution to E then that state is deemed appropriatefor retention in the reduced order model. Table 3shows computed results for the par ticipa tion of thevarious system state variables to the total outputenergy of the system subject to a unit impulse input.The results obtained in this way coincide with, andreinforces the initial policy of choosing transformedstate variables to ensure measurability and physicalavailability such that the resulting design could beimplemented without recourse to state estimationtechniques. It also reinforces the policy of basingdesigns on reduced order models.

Based on the information of Table 3 the 8th-ordermodel was re-arranged in decreasing order of statevariable participation in the outpu t energy ofthe system, i.e. [x'"]T -~ [ ~ i t v t v l d a ) i / d i k~ i kq ] , [u '"]Table 3. Output energy to system statesState variables Outpu t energy participat ion

7782910392401.060000

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D . P . Papad opou los e t a l. : Exc i t a t i on con t ro l l e r de s ign o f synch ro nous m ach ine 419

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Fi g . ~ a - - e . T im e r e sponses o f t he s t a t e va r i ab l e s (v t , i t, ~ and v f d ) of t he open - loop and c lo sed - loop sys t emi n p u t s t e p c h a n g e A v e = 0 .001 p . u . O pen l oop sys t em : a 8 th -o rde r t r an s fo rm e d sy nchron ous m ~ eh ine ~-. exc i t e rm ode l , b 4 th -o rde r r educed m od e l . D es igned c lo sed - loop sy s t em : c 8 th -o rde r c l o sed - loop m ode l , d 4 th -o rdc r c l o sed - loopm ode l , e 8 th -o rde r c l o sed - loop m ode l w i th t he con t ro l l e r o f t he r educed o rd e r m ode l

= [ u " ] = [ v e ] a n d [ y , " ] T = [ 3 g , V t v f d ] . T h u s , i t i se v i d e n t t h a t t h e a d e q u a t e r e d u c e d o r d e r m o d e l s h o u ldb e o f 4 t h - o r d e r a n d i n t h e f o r m o f e q u a t i o n s l ( a )a n d l ( b ) i t b e c o m e s[xr] 7 = [8 i t s t v i e ] ( 6 a )

[ U r ] = [ u " ] - -- - [ % ] ( 6 b )[ y , ] T = [ x , ] T ( 6 e )

T h e e x p l i c i t n u m e r i c a l v a l u e s o f r e d u c e d m a t r i c e sA t , Br a n d G , w e r e c o m p u t e d w i t h a s p e c i al p r o g r a mb a s e d o n t h e f a c t t h a t t h e f i r s t f o u r d o m i n a n b

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420 Archly fiir Elektrotechnik 72 (1989)eigenvalues of the 8th-order model should be main-rained in the reduced order one; they are given inAppendix B.

As seen from Table 2 the computed eigenvalues ofthe reduced order model agree exactly with thosethat were retained from the 8th-order model. Byassuming zero initial conditions in the 8th-ordermodel the corresponding initial conditions of thereduced order model are given as[ x ~ ( O ) y = [ 2 . 1 4 5 7 0 - 0 . 7 4 1 1 o ] [ v o( O ) ]

The computed responses of the reduced ordermodel ~, is, vt and old for the same input disturbance(zJve = 0.001 pu) are shown in Figure 2 accordingly.It is clear ithat the outputs of the reduced ordermodel follow quite well the corresponding outputsof the 8th-order model.

4 D e s i g n a n d s i m u l a t i o n o f c l o se d - lo o p s y s te m( d e s i g n o f e x c i t a t i o n c o n t r o l l e r s )4.1 Closed-loop system o/ th e trans/ormed 8th-orderopen-loop modelBy applying the relevant theory with output, feed-back outlined in Appendix A to the system ofequations (5a)-(5c) along with appropriate testsfor controllability and algorithms to form theAdj (sI - A") and the coefficients of the character-istic polynomial the following were obtained: thepoly~omiM of the openqoop system P(s) , the poly-nomiM of the closed-loop systems P(s) :-Q( s) . R(s) ,the gain output feedback vector k which is given by

4.2 Close &loo p system el the reduced 4th-orderopen-loop modelBy applying the relevant theory outlined in AppendixA and very much the same procedure followed in theprevious case the following were obtained: the poly-nomiM of the open-loop system P~(s) , the polynomialof the closed-loop system P~(s), the gain ou tpu t feed-back vector k~, i.e.k~ ----- [-0.0 05 6 0.0029 0.0037 1.327] (9)and the ma t r i x )k Of the designed closedqoop systemfrom the relationship

The explicit numerical expressions of Pr(s), Pr(s)and -&r are easily obtainable but are not given herefor the reason mentioned easier. The computedeigenvalues of the above closed-loop system aregiven in Table 2. It is clear that for this case thereis exceptionally good agreement between the desiredeigenvalues and the computed ones from the designedclosed-loop system. The computed time responsesof the outputs ~, it, vt, and old of the designed 4th-order closed-loop sys tem model, for the same inpu tstep change Av ~ = 0.001 pu, are shown accordinglyin Fig. 2. The dynamic stabil ity characteristics ofthis closed-loop sys tem model are by far superior tha nthose of the designed 8th-order closed-loop systemmodel presented in Sect. 4.1.

4.3 Close& loop system o / the trans/ormed 8th-orderopen-loop model w ith the controller o/the reduced4th-order closed-loop modelk T :- [- -0 .0 21 9 0.0167 --0.0412 0.4870] (7)and the matrix A" of the designed closed-loop systemfrom the relationshipA "- - A" - B"k T (8)

The explicit numerical expressions of P(s) , P(s)and A" are easily obtainable but are not given heredue t~) space considerations. The computed eigen-values of the above closed-loop system are given inTable 2. It is seen that in this case there is goodagreement between the desired eigenvalues and thosecomputed by the method. The computed time re-sponses of the outputs v~, i~, ~ and via of the designed8th-order transformed synchronous machine withconventional exciter closed-loop model, for the sameinput disturbance Ave ~ 0.001 pu, are shown in Fig.2a- -d respectively.

I t i s o f p r a c t i c a l i n t e r e s t t o u s e t h e g a i n o u t p u t f e e d -b a c k v e k t o r k r o f e q u a t i o n ( 9) a l o n g w i t h t h e t r a n s -f o r m e d 8 t h - o r d e r o p e n - l o o p s y s t e m m o d e l , d e f i n e d b yE q . ( 4 a - c ) i n o r d e r t o d e s i g n a n e w 8 t h - o r d e r c l o s e d -l o o p s y s t e m m o d e l t o a s c e r t a i n h o w i ts d y n a m i c p e r -f o r m a n c e c o m p a r e s w i t h t h a t o f t h e o t h e r t w oclosed-loop systems, previously designed. The mat rixA'~ of the new 8th-order closed-loop system is obtainedfrom the relationshipAA ' ~ = A " - B " k / ( 1 1 )

The explicit numerical expression of A'~ is easilyobtained but is not given here for reasons mentionedearlier. The computed eigenvMues of this closed-loopsystem are given in Table 2. It is seen tha t this designpresents a considerable improvement over the designoutlined in Section 4.1. The computed time responses ofthe outpu ts v~, it, ~ and old of this 8th-order closed-loop

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p e r f o r m a n c e a n d i n d e e d d e t e r m i n e w h e t h e r t h ec o n t r o l d e s i g n a l g o r i t h m c h o s e n i s a v a l id m e a n s b yw h i c h p r a c t ic a l c o n t ro l l er s m a y b e d e r i v e d . F i g . 3s h o w s t h e p e r c e n t a g e i m p r o v e m e n t t h a t t h e c o n t r o lg i v e s o v e r t h e c o m p l e t e l o c a l r a n g e o f t h e m a c h i n e .T h e r e s u l t s p r e s e n t e d i n F i g . 3 w e r e o b t a i n e d f r o mt h e 8 7 .5 k V A a l t e r n a t o r s e t, b u t i t is c o n t e n d e d t h a tt h e t r e n d i s v a l i d f o r m a c h i n e s o f h i g h e r r a t i n g . I t i si n t e re s t in g t o n o t e t h a t i m p r o v e m e n t , a l t h o u g h a p -p a r e n t t h r o u g h o u t t h e r a n g e o f o p e r a ti o n i s m o r es i g n i f ic a n t a t r a t e d l o a d t h a n a t l o w l o a d c o n d i t io n s .T h i s i s b e c a u s e t h e m o d e l o n w h i c h t h e c o n t r o lw a s b a s e d w a s l i n e a r i s e d i n a p o i n t c o r r e s p o n d i n gt o h i g h l o a d c o n d i t i o n s.

5 P r a c t i c a l i m p l e m e n t a t i o n o f t h e e x c i ta t i o n c o n t r o l l e r

s y s t e m m o d e l , f o r t h e s a m e i n p u t d i s t u r b a n c e Ave= 0 .0 0 1 p u , a r e s h o w n i n F i g . 2 a - e r e s p e c t i v e l y . T h ed e g r e e o f e n h a n c e m e n t a c h i e v e d i n t h e d y n a m i cs t a b i l i t y c h a r a c t e r i s ti c s i n t h i s e a se i s p r a c t i c a l l y t h es a m e as t h a t a c h i e v e d w h e n u s in g th e 4 t h - o r d e rc l o s ed - l o o p s y s t e m m o d e l .

T h i s i s a s a t i s f a c t o r y r e s u l t , s i n e e i n a p r a c t i c a ls i t u a t i o n , w h e r e a s y s t e m o f la r g e d i m e n s i o n s i s in -v o l v e d , t h e s y s t e m d e s i g n e r w o u l d u n d o u b t e d l yp r e f e r t o s y n t h e s i s e a c o n t r o l b a s e d o n r e d u c e d o r d e rf o r m u l a t i o n s i f a n a s s u r a n c e o f s a t i s f a c t o r y s y s t e mb e h a v i o u r c a n b e g u a r a n t e e d . T h e d i f f e r e n c e s i n p e r -f o r m a n c e b e t w e e n d e s i g n s b a s e d o n h i g h e r - o r d e r a n dr e d u c e d - o r d e r m o d e l s a r e n o t s i g n i f i c a n t f r o m a p r a c -t i c a l p o i n t o f v i e w a s i n d i c a t e d b y c o m p a r i s o n o fr e s u l t s i n p r e v i o u s s e c t i o n s . N e v e r t h e l e s s , u n t i l s u c hc o m p a r i s o n s h a v e d u l y b e e n c a r r i e d o u t , s u c h a c o n -c l u s i o n w o u l d b e m e r e l y s p e c u l a t i v e a n d n o t g i v e t h es o u n d a s s u r a n c e s t h a t a r e r e q u i r e d . F i n a l l y , i t s h o u l db e s t a te d t h a t w h e n o u t p u t f e e d b a c k i s u s e d f o r p o l e-a s s i g n m e n t i n h i g h - o r d e r s y s t e m s , s o m e p r o b l e m s c on -e a r ni n g t h e e f f ec t iv e n e s s o f t h e m e t h o d m a y a p p e a r .T h i s f e a t u r e o f t h e a l g o r i t h m l e n d s f u r t h e r w e i g h t t ot h e u s e o f r e d u c e d - o r d e r m o d e l s. F u r t h e r m o r e , b yc o m p u t a t i o n o f A d j ( s I - A ) i n s y s t e m s o f h i g hd i m e n s i o n s , e s p e c i a l l y w h e n t h e m a t r i x A i s n o ts p a r s e , a c c u m u l a t i v e e r r o r s d u e t o r o u n d i n g a p p e a r .A n e x c i t a t i o n c o n t r o l l e r , d e r i v e d f r o m e i t h e r e o n -v e n t i o n M o r m o d e r n a l g e b r a i c c o n t r o l s tr a t e g i e s, i sg e n e r a l ly b a se d o n a m a t h e m a t i c a l m o d e l o f t h e s y s t e ml i n e a r i s e d i n a p a r t i c u l a r o p e r a t i n g p o i n t . H o w e v e rs i n ce t h e c h a r a c t e r i s t i c s o f t h e g e n e r a t o r v a r y w i t hd i f f e r in g o p e r a t i n g c o n d i t i o n s d u e t o n o n - l i n e a r i ti e si n h e r e n t i n t h e m a c h i n e , t h e p e r f o r m a n c e o f t h ed e r i v e d c o n tr o l m u s t b e e v a l u a t e d a t o p e r a t i n g c o n -d i t i o n s o t h e r t h a n t h o s e u s e d a s a b a s i s o f d e s ig n . I nt h i s w a y i t i s p o s s i b l e t o a s s e s s t h e d e g r a d a t i o n i n

F o r t h e d e s i g n e d e x c i t a t i o n c o n t r o l l e r t o b e i n t e -g r a t e d i n t o a p r a c t i c a l s y s t e m ( r e p r e s e n t e d i n F i g . 1 ) i tm u s t b e t r a n s fo r m e d i n t o a c t u a l h a r d w a r e . T h e b l o c kd i a g r a m r e p r e s e n t a t i o n o f t h e e x c i t a t i o n c o n t r o l l e ri s s h o w n i n F i g u r e 1 a n d w h e n t a k e n t o g e t h e r w i t ht h e r e s u l t s o f E q s . (7 ) a n d ( 9) i t d e f i n e s t h e f o r m o fd e s i g n e d e x c i t a t i o n c o n t r o l l e r s f o r t h e p o w e r s y s t e mu n d e r c o n s i d e r a t i o n .

T h e s c h e m a t i c d i a g r a m w h i c h s h o w s o n e m e t h o do f i m p l e m e n t i n g t h e d e s i g n e d e x c i t a t i o n c o n t r o l l e r i ss h o w n i n F i g . 4 . T h e g a i n s o f t h e o p e r a t i o n a l a m p l i f i e r sm a y n o w b e d e t e r m i n e d b y f o ll o w in g s ta n d a r d p r a c -t i c e s . F o r e x a m p l e , s t a r t i n g w i t h t h e s u m m a t i o na m p l i f i e r a n d t a k i n g i n t o a c c o u n t t h e o p e r a t i n g c o n -d i t i o n v a l u e s o f t h e o u t p u t v a r i a b l e s ( i. e. v t, it , c$ a n dv /d ) a n d t h e a s s o c i a t e d c o m p u t e d f e e d b a c k g a i n skT = [/Cv~ #4 /c~ /% ~ ] , one m ay de te r m in e th e v a lue so f t h e e x c i t a t i o n c o n t r o l s i g n a l v o l t a g e V e.c .s, t h ea s s o c i a t e d i n p u t s a n d t h e g a i n A ~ . W i t h t h e v a l u e s o ft h e i n p u t s t o t h e s u m m a t i o n a m p l i f ie r k n o w n t h eg a i n s o f t h e o t h e r o p e r a t i o n a l a m p l i f i e r s ( i . e . A 1 , A 2 ,A ~ a n d A 4 ) m a y b e d e t e r m i n e d s i n c e a l l i n f o r m a t i o nr e q u i r e d i s a v a i l a b l e . H o w e v e r , c a r e m u s t b e t a k e ni n c o n v e r t i n g t h e p . u . i n f o r m a t i o n i n to t h e c o r r e -s p o n d i n g p h y s i c a l q u a n t i t i e s . I t m u s t b e p o i n t e d o u tt h a t , s i n c e t h e m o d e l u s e d f o r t h e s i m u l a t i o n t a k e sn o a c c o u n t o f t h e i n h e r e n t d e l a y s i n t h e v a r i o u st r a n s d u c e r s e t c . o f t h e f e e d b a c k s i gn a ls , t h e i m p r o v e -m e n t o f t h e d y n a m i c s t a b i l i t y c h a r a c t e r i s t i c s p r e -d i c t e d b y t h e s i m u l a t i o n s o f t h e d e s i g n e d c l o s e d -l o o ps y s t e m w i l l b e h i g h e r t h a n t h o s e o b t a i n e d f r o m a c t u a lt e s t s .T o i l l u s t r a t e t h e e f f e c t i v e n e s s o f a f o r m a l l y d e r i v e dc o n t r o l s t r a t e g y c o m p a r a t i v e s i m u l a t i o n s w e r e c o n -d u c t e d u s i ng a 3 0 M W s y n c h r o n o u s g e n e r a t o r f o rw h i c h b o t h d a t a a n d t e s t r e s u l t s w e r e a v a i l a b l e . T h ed a t a f o r t h e m a c h i n e a r e g i v e n i n T a b l e 4 .

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42 2Tab le 4 . Da t a fo r 3 0 M W sy n ch ro n o u s g en era to r sy s t emgath ~g: 37.5 MV A, 11.8 kV, 0.8 pf(tag), 3000 rpmxa 1.816 pu X q 1.648 puX m d 1.676 p~ X m q 1.508 pUxifl 1.816 p u x d 0.249 pux a 0.141 pu rid 0.00107 10ur a 0.002 pu r k d = r k q 0.003 t8 puH 5 , 3 sTra.nsmission Circui~:Trans form er XT 0 .1328 pu

R T 0.00505 puLin e X/~ 0.047 5 Io~R~ 0 .017 24 pu

S t e p f a l l i n s y s t e m v o l t a g eT h e p u r p o s e o f th i s s i m u l a t i o n i s t o c o m p a r e r a t e s o ft e r m i n a l - v o l t a g e r e c o v e r y fo r a g e n e r a t o r w i t h c o n -v e n t i o n a l e x c i t a t i o n c o n t r o l a n d a c o n t r o l l e r b a s e d o nt h e f o r m a l l y d e r iv e d s t r a t e g y w h e n t h e m a c h i n e i ss u b j e c t e d t o s u c c e s s i v e f a ll s in s y s t e m v o l t a g e v b. T h ec u r v e s F i g . 5 a f o r t h e c o n v e n t i o n a l c o n t r o l , a n d F i g .5 b f o r t h e s y n t h e s i s e d m o d e r n c o n t r o l , w e r e p r o d u c e db y r a i s i n g v b f r o m I p u t o v a r i o u s v a l u e s d e p e n d i n go n t h e t e s t c o n s i d e r e d t o g i v e t h e i n i t ia l c o n d i t i o n s ,a n d t h e n s t e p p i n g vb b a c k t o I p u . T a b l e 5 g i v e s d e t a i l so f t h e s t e p - c h a n g e c o n d i t io n s u s e d i n e a c h s i m u l a t i o n .T h e s i m u l a t i o n r e p r e s e n t s s t e p r e d u c t i o n s o f s y s t e mv o l t a g e o n t h e g e n e r a t o r , a n d i t is s e e n t h a t c o n v e n -t i o n a l c o n t r o l i s l e s s e f f e c t i v e a t c o n t r o l l i n g v o l t a g ew i t h e a c h s u c c e e d i n g s t e p o f vb . I t i s a l s o a p p a r e n tt h a t t h e f o r m a l l y d e r i v e d s t r a t e g y e f f e c ts a f a s t e rv o l t a g e re c o v e r y a n d m a i n t a i n s s t a b i l i t y b e y o n dc u r v e C a f t e r w h i c h t h e c o n v e n t i o n a l c o n t r o l is i n -

Archiv f/Jr Elcktrot,echnik 72 (1989)e f f e c t iv e . I t s h o u l d b e e m p h a s i s e d t h a t b o t h c o n t r o l le r su s e d t h e s a m e f o r w a r d a m p l i f i c a t i o n s t a g e s , t h e o n l yd i f f e r e n c e b e i n g in t h e f e e d b a c k q u a n t i t i e s a p p l i e d ,t h a t i s, t e r m i n a l v o l t a g e o n l y i n t h e e a s e o f t h e c o n -v e n t i o n a l c o n t r o l a n d t h e s t a t e s v t , i t , v l e a n d ~ i n t h ec a s e o f t h e n e w m o d e r n c o n t r o l.T h r e e - p h a s e f a u l t c o n d i t io nI n t h i s s i t u a t i o n i t is t o b e e x p e c t e d t h a t a n e x c i t a t i o nc o n t r o l d e r i v e d u s i n g s t a t e f e e d b a c k w i l l s h o w s o m ei m p r o v e m e n t i n p o s t -f a u l t v o l t a g e re e o v er ~ a n d w i ll b em u c h l e s s e f f e c ti v e i n e x t e n d i n g t h e l i m i t s o f t r a n s i e n ts t a b i l i t y . F i g u r e 6 sh o w s t h e s i m u l a t i o n o f t h e p o s t - f a u l tv o l t a g e re c o v e r y t r a n s i e n t f o r t h e 3 0 M W s y n c h r o n o u sg e n e r a t o r c o m p a r e d w i t h t h e r e c o v e r y r ec o r d e d d u r in gt e s t c o n d i t i o n s . T h e f a u l t w a s a p p l i e d a t t h e h i g h -v o l t a g e te r m i n a l s o f t h e g e n e r a t o r t r a n s f o r m e r . A ni m p o r t a n t f e a t u r e o f t h i s s i m u l a t i o n r e l a t e s t o t h er a t e o f r i se o f v o l t a g e a n d i t s e f f e c t o n a s y n c h r o n o u sm o t o r lo a d c o n n e c t e d t o t h e s y s t e m .

Table 5 . In i t ia l condi tions fo r s i rau lat ion of s tep reduct ionsin sy s t em v o l t ag eFinal read ing v b v t P Q1,0 1 .1 0 . 8 - 0 . 6Ste p initial (a) 1,08 1.1113 0.8 0.128conditions (b) 1, 11 2 1.1 0.8 --0.11,9for (c) 1,1 51 1.1 0.8 --0.358(d) 1 ,1 .8 9 t . I 0 .8 -0 . 5 9(e) 1 .2 18 1 .102 0 .8 -0 .75 7(f) 1.257 1.104 0.8 --0.9 86(g) 1 .2 9 6 t .1 0 4 2 0 .8 - -1 .2 2

o : Iv t ~ ~ A . C .

Vuttogermnsfo rmero: 1

A . C .

Perfectrect i f icat ion

Cur ren tTransformer

Opticalsensor

l~ ig . 4 . Schemat, ie d ia gram of ex ci tat ion con t ro l ler imple me ntat ion

To exci tat ionsummat ionpo in t

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D, P. Papad opoulo s et a l . : Exc itat ion control ler des ign of synchronous machine 4231 . I 5p . ~ .1 . 1 0

1 . 0 5

i . 0 00 . 9 5

O . g O

0 . 8 5

0 . 8 01 . 1 5~U .1 . 1 0

1 . 0 5

i O 0~ r 0 , 9 5

0 . 9 0

0 . 8 5

0 . 8 0

&

o : 5 1 : o l : s z : o z . sf

Fi g . 5 a a nd b . Step-fal l in sys tem voltage, a conventionalcontrol, b designed design control1. 4p . u .1 .21 .0

0 . 8i~ . - . 60 4

0 2

- - - j F o u t t c l e a r e d3 8 0 m s

1

I /~ J r ' - - / R e fe r e n c e

1

' " , - ' i e v e l

J- / - - C o n v en tio n a l e x c i t a ti o n. . . . N e w d e s i g n o f e x d t a t i o n

I~i~'. 6. Po st-faulg rec ov ery vo ltag e

6 C o n c l u s i o n s a n d d i s c u s s io nA p r a c t i c a l p r o c e d u r e , a p p r o p r i a t e f o r t h e d e s i g n o fe x c i t a t i o n c o n t r o l le r s o f s y n c h r o n o u s m a c h i n e s c o n -n e c t e d t o i s o l a t e d o r i n t e r c o n n e c t e d n e t w o r k s , h a sb e e n p re s e n t e d . T h e m a t h e m a t i c a l b a c k g r o u n d r e -q u i r e d fo r t h e d y n a m i c m o d e l l i n g o f a s y n c h r o n o u sm a c h i n e w i t h e x c i t e r h a s b e e n p r e s e n t e d i n c o n c is ef o r m . T h e a b o v e m e t h o d s h a v e b e e n a p p l i e d s u cc e ss -f u l l y t o t h e d e s i g n o f a n e x e it a t, o n c o n t r o l l e r f o r a n8 7 .5 k V A s y n c h r o n o u s m a c h i n e s u p p l y i n g p o w e r t oa n e l e c t r i c it y s u p p l y s y s t e m t h r o u g h a t r a n s f o r m e ra n d a t r a n s m i s s i o n li n e , u s i n g a n o r i g in a l S IM O t i m ei n v a r i a n t l i n e a r 8 t h - o r d e r s y s t e m m o d e l a n d a na d e q u a t e r e d u c e d 4 t h - o r d e r s y s t e m m o d e l . F r o m t h et h r e e c l o s e d - l o o p s y s t e m s d e s i g n e d ( i . e . b a s e d o n t h e8 t h - o r d e r m o d e l , t h e 4 t h - o r d e r r e d u c e d m o d e l a n dt h e 8 t h - o r d e r m o d e l w i t h t h e c o n t r o l l e r o f t h e 4 t h -o r d e r r e d u c e d m o d e l ) , t h e b e s t e n h a n c e m e n t i n t h es y s t e m d y n a m i c s t a b i l i t y c h a r a c t e r i s t i c s w a s a c h i e v e db y t h e t w o l a t t e r c a s e s w h i c h v a l i d a t e s t h e u s e o f4 t h - o r d e r m o d e l s a s t h e b a s is o f d e s ig n . P r a c t i c a l t e s t sc a r r i e d o u t o n t h e a l t e r n a t o r s y s t e m s h o w t h a t i m -p r o v e m e n t i n r e s p o n s e i s a t t a i n a b l e t h r o u g h o u t t h ew o r k i n g r a n g e o f t h e m a c h i n e w h i c h i n e s s e n c e e l i m i -n a t e s t h e d i r e c t n e e d t o u p d a t e a c o n t r o l l e r i n a na d a p t i v e m a n n e r w i th c h a n g i n g l o a d c o n d i t i o n s . T h ec o n t r o l l e r t o p r o d u c e t h e r e s u l t s g i v e n i n F i g . 3w a s c o n s t r u c t e d u s i n g a n a l o g u e t e c h n i q u e s a n d in c l u -d e d t h e e f fe c t, o f t h e t r a n s d u c e r d e l a y s . F u r t h e r i m -p r o v e m e n t s s h o u l d b e o b t a i n e d b y u s in g d i g it a lt e c h n i q u e s , a n d w o r k i s c o n t i n u i n g i n t h i s d i r e c t i o n .

A p p e n d i x AOver v iew o / s ynchr onous gener a tor s y s t em mode l l ing [1 ]T h e g e n e r a l f o r m o f t h e v o l t a g e e q u a t i o n s f o r a s y n -c h r o n o u s g e n e r a t o r s y s t e m i n P a r k ' s f r a m e o f r e f e r -e n c e a r e :[v] = [i ] p[i] + ([G,~] + [G~]) co,[i.] + [R] [i] (B .I)a n d t h e e q u a t i o n o f m o t i o n o f t h e g e n e r a t o r s h a f t isg i v e n b y(H/7~]) p2~ = T,~ - - [i]T [Gm] [i] (B.2 )w h e r e t h e w i n d i n g c u r r e n t s a r e g i v e n b yt i l t = ( id i ~ i zd i k ~ i ~ ) ( B . 3 )a n d t h e v o l t ag e v e c t o r b y[v ]y = (vb si n (3 vb co s (3 vzd 0 0) (B .4)[ L ] i s t h e m a c h i n e i n d u c t a n c e m a t r i x , [ G m ] a n d [ G Jm o d i f i e d i n d u c t a n c e m a t r i c e s a n d [ R ] t h e d i a g o n a lm a t r i x o f w i n d i n g r e s is t a n c e s .

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42 4 Archly f i i r Elektrotechnik 72 (1989)F o r s m a l l p e r t u r b a t io n s a b o u t a g i v e n o p e r a ti n g

p o i n t , f r o m e q u a t i o n ( B . 1 )[ A v ] = [L ] p [ A i ] - F ([G~ ] + [O,])Ao~r[io]

+ ([a,~] + [ r ~ o [ ~ ] + [R] [~ i ] (B .5)a n d f r o m e q u a t i o n (B .2 )p A e , r = ( = I / H ) ( A T , , - - ( [ / e l T ( [ am]

+ [G,~]T)) [A,] (B.6)pA(~ = AoJr

C o l l e c t i n g t e r m s f r o m e q u a t i o n s ( B . 5 ) a n d ( B . 6 )t h e c o m p r e h e n s i v e f i rs t - o r d e r d i f f e r e n t ia l f o r m

w h e r e A , U . V a n d C a r e t h e m a t r i c e s o f e i g e n v a lu e s ,e i g e n v e ct o r s a n d c o n t r o l o u t p u t r e s p e c t iv e l y p a rt i -t i o n e d a p p r o p r i a t e l y a s ~ i s g i v e n b y

/~ LU~ ~J ~T h e i n i t i a l c o n d i t i o n s o f t h e r e d u c e d o r d e r m o d e l a r eg i v e n f r o m t h e f o l l o w i n g e x p r e s s io nxr(O = V n [ U l t x ~ ( O ) + U ~x~( 0) ] - - t ~ A ~ - I / ~ u ( 0 ) ( B .1 2 )

I t i s t o b e n o t e d t h a t t h e s o l u t i o n o f e q u a t i o n s( B . 1 0 a ) , ( B . 1 0 b ) a n d ( B . 1 2 ) w i l l y i e l d y~( t ) w h i c hw i ll b e a c l o s e a p p r o x i m a t i o n o f t h a t o f t h e o ri g i n a ls y s t e m , i .e . o f y ( t ) .m ] [ [ L ] -1 {[~] -~- ~%[G]}

--~//~[4 ]T {[a~] + [a~]m}1 0- + o 0

o - " o l/Z (B.7)m a y b e o b t a i n e d .

I f i t is a s s u m e d t h a t t h e s y n c h r o n o u s m a c h i n e h a sa f i r s t - o r d e r c o n v e n t i o n a l e x c i t e r d e s c r i b e d b y t h ee q u a t i o np v f ~ = ( - 1 . 1 T E ) r i d + K ~ I ~ , ( ~ G o ~ - v , ) (B.8)t h e n t h e o v e r a ll m o d e l fo r t h e m a c h i n e a n d e x c i t e ri s o b t a i n e d u s i n g E q s . ( B . 7 ) a n d ( B . 8 ) .

F o r t h e p u r p o s e o f t h e p r e s e n t w o r k t h e s t a t e s a r er e q u i r e d t o b e i n t e r m s o f p h y s i c a l q u a n t i t i e s. T ot r a n s f o r m t h e s t a t e s o f E q s . ( B . 7 ) a n d ( B .8 ) t o a n e ws e t, in t h e p r e s e n t c o n t e x t i d a n d iq a r e t r a n s f e r r e d t ov t a n d i t b y t h e m e t h o d g i v e n i n l% e f. [ 1] .

M o d e l r e d u c t i on [5 ]A r r a n g i n g t h e l i n e a r i s e d s y s t e m e q u a t i o n s i n t h eo r d e r o f d e c r e a s i n g e n e r g y p a r t i c i p a t i o n [ 5 ] o f i t ss t a t e s o r b y a r r a n g i n g t h e o r d e r in t e r m s o f t h e p h y s i -c a l s ig n i f ic a n c e a n d e a s e o f i m p l e m e n t a t i o n o f s t a t e s ,w h i c h i n t h i s c a s e a r e t h e s a m e , t h e s y s t e m e q u a t i o nm a y b e w r i t t e n

---- u (B .9 )~ [ A 21 A2~J x~ B~T h e r e d u c e d o r d e r m o d e l i s g i v e n b y ,

~ = [ A l l - A 1 2U T ~ IU 2 1 ] x ~ + [ B 1 - - A I ~ U ~ I A ~ I B ~ ] u(B. lOa)

y ~ - - [C ~ -- C ~ U ~ U 2 ~ ] x ~ - - C 2 U ~ 1 A 2 1 B ~ u ( B . 1 0 b )

P o l e - p l a c i n g t e c h n i q u e [11]I f t h e l i n e a r S I M O m o d e l , E q s . ( l a ) a n d ( l b ) , i sc o n t r o l l a b l e w i t h d i s t i n c t r e a l o r c o m p l e x e i g e n v a Iu e st h e c o n t r o l l a w m a y b e f o u n d f r o mu = - -k Tx + ~o (B.13)w h e r e k T i s t h e 1 n g a i n s t a t e f e e d b a c k v e c t o r a n d coi s t h e n e w s c a l a r c o n t r o l i n p u t , s u c h t h a t t h e s o u g h tc l o s ed - l o o p s y s t e m i s o b t a i n e d f r o m ,

= (A - - B k +) x + Be~ -~ Ax ~ B~o (B .1 4a )y = C x ( B . 1 4 b )

I f i t i s a s s u m e d t h a t l s y s t e m s t a t e s a r e m e a s u r a b l ea n d t h e s y s t e m s t a t e v e c t o r is a r r a n g e d i n o r d e r o fm e a s u r a b i l i t y , t h e f e e d b a c k c o n t r o l l a w b e c o m e su = - - [ k T 0] x @ co = - k [ y + o~ ( B.15)w h e r e k i s t h e u n k n o w n l 1 g a i n o u t p u t f e e d b a c kv e c t o r . I n t h i s c a s e l p o l e s o f t h e c l o s e d - l o o p s y s t e mm a y b e p re a s si g n e d w h er e a s t h e r e m a i n in g n - lp o l e s w i l l b e f r e e .

T h e c h a r a c t e r i s ti c p o l y n o m i a l o f t h e s o u g h t c l os e d-l o o p s y s t e m P ( s ) , i n te r m s o f t h a t o f t h e o p e n - l o o ps y s t e m P ( s ) e r e , b e c o m e sP(s): [ s I - - A i = [ s I - - A + B k V I : [ s I - A !

+ kT[ A dj ( s I - - A ) ] B- -- - P ( s ) + kTM(s) B (B.16 )

E q u a t i o n ( 13 .1 6 ) m a y b e r e w r i t t e n i n t h e f o r mk TM (s) B - - 1 0 ( 8 ) - P ( s ) , a n d a f t e r e q u a t i n g e o e f f i -

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D. P. Papadopoulos et al. : Excita tion controller design of synchronous machine 425cients of equal powers of s one obtains the followingmatrix equation in compact formRk = ~ (B.17)where the elements of k are the unknowns.

If t = n and R, is invertible, t hen equ ation (B.17)yields the gain state f eedback vecto r k = R-I ~. Onthe oth er hand, if l < n equation (B.17) ma y be writ-ten in partitioned form as

M atr ix B ' o / t rans formed synchronous machine open-loop model and transformation mairices T and S

- 16.3489 0.2669-11.1719 0

2873.2964 0--2845.3904 0

--8.5153 00 04,2841 110

T =

---1. 1580 0.2239 1.1967 1.1958 --0.2272 --0.07490.8968 0.4425 0 0 0 00 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 10 0 0 0 0 0

--0.0O24-000001

which, when Rll is invertible, yields the gain outputfeedback vector kI = R~]1~I subj ect to the condi tionR21R~1~h = ~]2.

A p p e n d i x B

Matrice s A and B o f synchronous mach ine open-loopmodel

S : ,=

A =

-0.0142000000

B =

0-000000

- --47.7 738 797.7444 --0.0 845 --2.1 433 --486.8357 --446.22 20--1 118 .21 48 --44.1 950 830.5969 830.5969 --5.9 260 257.4357

--2. 7847 46.4997 --7.6 912 11.9467 --28.37 72 --26.00 98--44.8161 748.3560 7.5775 --14,8 322 --456,6 956 --418. 5964

--105 2.313 6 --41.5 904 781.6462 781.6462 --9.7 105 242.26390 0 0 0 0 033.0181 --244.00 48 --72.1 264 --72, 1264 79.2618 0

- 31.5652 0-0 0

2874.1499 0--2831.6542 0

0 00 00 110

1.0082-1.36980.05880.94571,289110

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42 6 A rch ly fi i r E l ek t ro t echn ik 72 (1989)Matrices A", B'" and C" o/ trans]ormed synchronousmachine with conventional exciter open-loop model

A ,

4 . ~ I i ckh l , J . ; S inha , N . K . : M ode l r educ t i on fo r l i nea rm ul t i - va r i ab l e sys t em s . IEEE Trans . , A C -25 (1980)1121- -1127

A , , _

- - 3 0 2 . 1 4 7 7 - - 3 5 3 . 3 4 3 4

4 0 .2 8 90 3 7 .7 4 23 - 4 0 . 0 4 8 6 4 0 .9 2 3 4 - 6 . 0 7 4 9 - 9 . 3 2 6 1 - 0 . 1 2 5 0 i1 2 0 8. 50 3 1 9 6 1 .0 2 5 0 - 1 0 7 8 . 7 6 0 8 - 1 0 7 9 . 4 8 9 5 - 1 6 4 . 6 1 0 2 - 1 9 5 . 7 7 8 7 - - 1 . 4 2 2 2 1

6 0 .1 9 6 3 7 4 .6 2 5 9 - - 7 9 .7 2 7 6 - 6 0 . 0 3 4 0 - 1 4 . 6 9 9 4 - 2 1 . 5 0 3 3 - 0 . 0 8 7 3 19 6 8 . 7 8 4 9 1 2 0 1 . 0 1 2 8 - - 1 1 5 1 .7 5 9 4 - 1 1 7 3 .2 7 3 1 - 2 3 6 . 5 6 8 3 - - 34 6 . 0 68 3 - 1 . 4 05 0 16 0 0 .5 6 9 2 - 3 9 7 . 9 2 4 1 6 2 . 95 0 0 6 3 . 5 0 54 1 2 6. 7 50 9 2 7 8 . 2 25 5 - - 0 . 1 6 8 2 1

0 0 0 0 0 0 1 l2 8 9 . 4 5 1 2 2 8 9 . 1 7 1 8 1 0 . 6 0 7 8 - - 2 2 . 6 2 0 3 0 . 7 3 3 2 I

0 0 0 0 0 0 0

1 6 . 3 4 8 9 -1 1 . 1 7 1 9

2 8 7 3 . 2 9 6 4- - 2 8 4 5 . 3 9 0 4

- - 8 . 5 1 5 304 . 2 8 4 1

- - 2 . 1 2 0 0

~00

I

I OioL2 . 12_~

(y , =

- 11

0

0 0 00

10

Matrices A r, Br and C~ o/ the reduced order open-loopmodel

t- - 1 0 2 . 8 8 9 7

A r = - 1 1 8 . 0 1 3 9t . 7 1 3 60

r - 3 2 3 . 5 5 8 1 ]_ / - 4 1 2 . 5 1 0 8 [,

B ~ / - 2 . 1 8 1 1 [L 2.12d

e ~ = 1 00 1

1

8 6 . 3 3 7 4 - - 2 0 9 . 3 6 7 6 ]9 6 . 8 2 7 6 - - 2 3 9 . 6 5 2 4 1 ,

- - 1 . 4 5 1 0 0 ~ . 7 1 5 5 j

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