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DESIGN OF POWER SYSTEM STABILIZERBASED ON LQG/LTR FORMULATIONS

G h a d i r RadmanE l e c t r ic a l E n g i n e e ri n g D e p a r t m e n tT e n n e s s e e T e c h n o l og i c a l U n i v e r s i t y

C o o k e v i ll e , T e n n e s s e e 38505

Abstract-T he design of power system stabilizers us-ing L inear Quadra t ic Gauss ian regu la to r methodo logywith Loop Transfer Recovery (LQG/LTR) is considered.A one-mach ine in f in i tebus sys tem is used to test t he ef-fectiveness of the p roposed s tab i l ize r. The s tab i l ize r i sdesigned using a low order l inea r mode l of t h e p o w e r sys-tem. Com pute r s imula t ion is per fo rmed wi th comple tehigh order non-linear model. I t is shown that the closed-loop sys tem wi th the p roposed s tab i l ize r is very robust;i.e. i t remains s tab le fo r a large amount o f d is tu rbanceand /o r pe r tu rba t ion .

I. I N T R O D U C T I O NA reliable interconnected power system must main-

tain constant voltage profile and constant frequency. TheAutomatic Voltage Regulator (AVR) loop is responsiblefor voltage control while the Auto matic Generation Con-t ro l (A GC ) loop regu la tes the f requency by main ta in ingpower balance [1,2]. In case ofdisturbances (load changesor power losses) both the system voltage and the systemfrequency a re d is tu rbed . Th e AVR loop can regu la te thevoltage for a wide range of disturbances. However, th eAGC loop is capable of restoring the frequency only fors m a l l a n d g r a d u a l d i s t u r b a n ce s . T h i s i s d u e t o t h e s l owdynam ics of th e mechanical mod es which have very l i t t ledampin gs. To improve frequency regulation or powerbalance) for a wide range of disturbances, a th i rd con t ro lloop named Power Sys tem S tab i lize r (PSS) i s used . T heopera tion of a PSS can be desc r ibed as follows.

Power imba lance is the difference between the m echan-ical power (P,) a t the genera to r shaf t and the e lec tr ica lpower P,) a t the genera to r te rmina l . To ach ieve powerbalance either P and /o r P should be quickly adjustedupon a d is tu rbance . Quick ad jus tment o f P,,, is not al-ways possible (excep t for th e case of fast valving) nor isi t economical. However, P can be adjusted by adjust-ing genera to r te rm ina l vo l tage th rough i t s f ie ld cu r ren t ,which is rather quick. Using frequency deviation, a PSSprovides the necessary control signal to adjust the f ieldvo ltage. Th is in tu rn ad jus ts P resulting in power bal-ance and frequency regulat, ion.

A power sys tem s tab i l ize r shou ld opera te sa t i s fac to -rily for a wide range of loading conditions; i .e . i t shouldb e r o b u s t. O t h e r a u t h o r s h a v e s h o w n t h a t a P S S d e -sign based on Linear Quadra t ic Regu la to r (LQR ) formu-la t ions i s more robus t and per fo rms be t te r tha n a tra-ditional PSS [3,4]. However, th is PSS requ i res comple temeasurements, which is neither practical nor economi-cal for most cases. Therefore, an estim ator should beadded to th is des ign to es t imate the unmeasured s ta te -variables. T he combined controller an d estimator is nolonger as robust.

Th is paper exp lo res the des ign o f a PSS based onLQG /LTR fo rmula t ion . Here a robus t PSS based onLQR methodology is designed. A Kalman fil ter is usedfor the es t imat ion o f th e unmeasured s ta te va r iab les . Us-ing Loop-Transfe r-Recovery (LTR) m ethod , the Ka lm anf i lte r i s des igned such tha t the robus tness assoc ia ted wi ththe LQ R design is asymptotic ally recovered. Th e de-sign is tested using com puter simulation of a typical one-mach ine in f in i te -bus sys tem. It i s shown tha t a PSSbased on LQ G/LT R is nearly as good as the PSS basedon LQR.

11. R E V I E W OF L Q G / L T RD E S I G N M E T H O D O L O G YIn this section the design and properties of a compen-

sa to r based on LQ G/L TR des ign methodo logy , as appli-cab le to the PSS in t roduced in th is paper , i s rev iewed .An LQG /LTR based compensa to r i s designed us ing L in -ea r Quadra t ique Gauss ian (LQG ) approach . There fo re ,the LQG approach is reviewed next [5]. Cons ider thesys tem

, = A . ~ + B . ~ + ~ . ~14y = c . x + v (1b)

where i s the s ta te , U i s th e con t ro l s igna l , w and U areuncorrelated white noises with covariances

E{ u w } = WE { v U } = v ( 2 4P b )

a n d A , B , C , and r are matr ices wi th appropr ia ted imens ions . Th e p rob lem is that of f inding a feedback0-7803-0634- 1/92 03.OO OlEEE 1787

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con t rol law th a t m in imizes the pe r fo rmance index1= E { m ( x l . Q x+ R . dtwith Q 2 0, a n d R > 0. Figure (1) shows th e sys tem ofEqua t ions (1) toge ther wi th an LQG compensa to r . Duet o t h e s e p a r a ti o n t h e o r e m IC a n d ICf of Figure (1) m a ybe calculated indep ende nt of each oth er as follows. I(, isthe so lu t ion o f the LQ R prob lem wi th w e ight ing matr icesof Q a n d R, Figure ( 2 ) . K j i s the so lu t ion of th e op t imales t imat ion (o r Ka lman f i l te r ing) p rob lem fo r the sys temof Equations (1) and (2).

T h e r e t u r n r a t i o s fo r n o d e s 1 a n d 2 of Figure (1) aregiven by

H ~ ( s ) K , S. A ) . BH ~ ( s ) K ( s ) G s )

(34( 3 b )

where K ( s ) s the t ransfer func t ion o f the LQG compen-s a t o r . T h e r e t u r n r a t i o H l ( s ) s t h e s a m e as t h e r e t u r nr a t i o a t n o d e 1 of the under ly ing LQR sys tem , F igure (2 ) .

Input nob1II

Ic--------,----------_________1Q G - COMPENSTOR , K s )Figure 1). A linear system together with

i t s LQG-compensa to r .- - - - - - - - - - - - - - - - - - - -L A N TI

-1 x -(SI-A)1a-R I

Optimal stale feedbackFigure 2). A n L Q R s y s t em .

I t i s we ll known th a t a n L QR sys tem possesses a gooddegree of robustness; i .e. in additi on to othe r propertiesi t has good s tab i l i ty marg ins [6]. This i s an exce l len tfea tu re tha t can be app l ied to th e con t rol o f non-l inearsys tems , such a s power sys tems , whose opera t ing po in tsvary within wide ranges.

Th e loop t ransfe r recovery i s a p roc edure t o recoverthe robus tness o f node 1 (hence the robus tness o f theunder ly ing LQR sys tem) at node 2. Th is will enhancet h e r o b u s t n e s s o f t h e L Q G s y s t e m t o t h a t o f t h e u n -der ly ing LQ R sys tem. Th e LTR pro cedure invo lves de -te rmin ing K j s u c h t h a t Hi ) and H 2 ( j ~ )ecomeapprox imate ly equa l fo r t he f requency range o f in te res t:

In th is case the robus tness assoc ia ted wi th the inpu tnodes o f bo th LQG and LQR sys tems wi l l be approx i -mate ly the same. I t shou ld be no ted th a t H l ( j w ) canno tb e m a d e e x a ct l y e q u a l t o H z ( j w ) ; however, if th e trans-fe r funct ion o f the sys tem is s q u a r e a n d m i n i m u m p h a s ethen ICj c a n b e d e t e r m i n e d s u c h t h a t H l ( j w ) becomesarb i t ra r i ly c lose to H z ( ~ u ) .

T h e LTR procedure uses the following facts. Assuming

wi th WO, n e s t i m a t e of th e covar iance o f the no ise w, Ian iden t i ty matr ix , and q a sca la r , the n th e d i f ference be-tween the spec t ra l norms o f H l ( s ) a n d H ~ ( s )pproacheszero as q approaches in fin ity . I t shou ld be no t iced tha tthe nearness of two matr ices i s measured by the i r s ingu la rvalues.

111. T H E P O W E R S Y ST E MA typical one-m achine infinite-bus power system , Fig-

ure 3 ) , i s chosen for th is s tudy . I t i s a ssumed tha t themach ine i s equ ipped wi th a one- t ime cons tan t s ta t ic ex -c i te r wi th bounded ou tpu t [7] a n d a two- t ime cons tan ttu rb ine-governor sys tems [8]. T h e D a n d Q - ax is d a m p e rwind ings a re ignored, a nd t he mach ine is representedwi th th re e dynamic and four a lgebra ic equa t ions as fol-lows [3,9]:

. 1w = Tm T TD)2 H

Th e to rques in the above equa t ions a re in P .u. , and th usnumerically equal to th e correspo nding powers. Fig-ure 4) shows the block diagram representations for the

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ove ral l sys te m. Th e c onne c tions be twe e n the ma c hinea n d the powe r sys te m shows the in te ra c t ions be twe e nthe m. Equa t ion (6 d) shows th a t the sys te m i s nonl ine a r .

. v, Z=R+ix V-

d-axisFigure 3). A one-machine inf ini te-bus

sys te m.

overnor turbine1( l+Srp) l+S?J

1 I I ocat ioab, I

Figure 4). A block-diagram representa-t ion of the power system.

The loa ding c ondi t ion (ope ra t ing poin t ) a nd the pa -ra me te r s of the sys te m a r e a ssume d as follows:

P i u t o R + j X G + j B1.0 1.05 1.0 0.06+j0.6 0.25 - 0 . lH X d X L X q TA D4.63 0.973 0.19 0.55 7.76 0

K A T A Vmax Vmin K g Ti50 0.05 6.0 -6.0 0.027 0.3 1 .0 .I V. D E S IG N O F PSS

T h e PSS is designed using a low order l inear designmode l obta ine d as fo llows. T he dyn a mic s of the turb ine -gove rnor a re ignore d by a s suming P, = Prefd u r i n ga t r a ns ie nt pe r iod . T he r e su l t ing sys te m is the n lin-e a riz ed a b out th e ope ra t ing poin t. F igure (5) h o w s t h el inear ized model together with t he desired compensato r .Not ic e t ha t th e inc re me nta l va lue s of t he s igna l s a re use din th i s mode l . A lso , not i ce tha t Pref n d Vref d o n o ta ppe a r s inc e the i r inc re me nta l va lue s a re a s sume d z e -ros . Moreover , a der ivat ive block with transfer functionIC, . s / 1+ r, * s) is a d d e d t o t h e p a t h o f U in orde rt o g u a r an t e e t h a t t h e PSS i s turne d off dur ing s te a dy

ope ra t ion . A reasonable numerica l va lue was chosen for7 e.g. T = 1.5) t o g u a r a n te e t h a t U, is a smo oth we llbe ha ve d s igna l , a nd K , i s c ons ide re d as a design param-e te r . A l though the de r iva t ive b loc k i s pa r t of the PSS,t e mpora r i ly i t is c o m b i n e d w i t h t h e p l a n t t o g i v e t h ede s ign mode l for LQG/L TR c o mpe nsa tor . Th e de s ignmodel is expressed in s ta te-space form:

whe re x = [ A ~ , A E F D , A ~ ; , A W , Z ]s t h e s t a t e , U ist h e i n p u t t o t h e d e r i v a ti v e b l oc k , y i s the inc re me nta lfrequency, and A , B, a n d C r e t h e s y s t e m m a t ri c e sof a ppropr ia te d ime ns ions . W i th th e a bove nume r ic a lvalues , the matr ices are :

0.0 0.0 0.0 377 0.0-3.596 -20.001 -576.39 0.0 -1.00.129 -.342 0.0 0.0-.172 0.0 -.158 0.0 0.00.0 0.0 0.0 -.667( 8 4

B = [O .O 6.667 0.0 0.0 0.0041 ( 8 b )c = [ O 0. 0. 1.0 0.01. ( 8 c )

Th e e igenva lue s for the sys te m wi thout PSS a r e-0.0606 8.0 316 , -0.6667, -5.2968, -14.9382

which shows the very poor d a mp ing a s soc iate d to theme c hanic al mo de s of the sys te m. Th e obje ct ive i s toprovide a s tabil iz ing s ignal U t h r o u g h PSS t o i m p r o v et h e d a m p i n g . T o d e si g n a n L Q G / L T R b a s ed PSS, suc h asthe one shown in F igure l ) ,we first find I

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Th is sys te m wa s de s igne d a s suming the we ight ing ma t r i -ces of R = 1, a n d Q = dia g(0 q 2 0 0 0). Variousvalues of q2 a n d I< , we re t r i e d a n d i t wa s found th a t forz = le4 a n d I< , = 0.01 he r e su l t ing LQR sys te m wa s

one with acceptable response . Th e result ing gain h , wasf o u n d t o b eK , = [77.899 0.619 109.312 2736.521 0.001 (9)

which gives an LQR-system with e igenvalues:-1.6932fj8.5421, -0.4701, -6.4140, -14.8824 .This shows a s igni f ic a nt improve me nt in the da mping ofthe sys te m.

B. Loop Transfer RecoveryTh e sec ond s te p for de s igning LQG/LTR ba se d c om-

pe nsa tor i s t o f ind ICf using LTR procedure . I

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Time (Second)i m e (Second)In N

v,3 5

a,s;0 1 2 3 4 5

T i m e (Second)n 1

0 1 2 5 4 5ITime (Second)/p/---* ....................... n lF:cda s

r

T i m e (Second)In

I0 1 2 3 4 5Time (Second)

Inr?c

u z L ;n .....- - - - _ _ _ _- - - ____9

T i m e (Second)

T i m e (Second)In

In9T i m e (Second)

I0 1 2 3 4 5Time (Second)

v

0 1 2 3 4 5T h e (Second)ime (Second) -Figure 7). Simula t i on r esu l t s a ssuming one of t h e Figure 8 ) . Simulat ion resul ts assuming a three-phase

sus t a ined sho r t c i r cu i t i n t he m idd le ofo n e of the l ines._-- Sys t em wi thou t PSS. . . . .Sys t em wi th L QG/L T R- based PSSl ines is t r ipped.. . . . . Sys t em wi th L QG/L T R- based PSS- - - _ System with LQR-based PSS

Sys t em wi thou t PSS

- - - - Sys t em wi th L QR- based PSS1791

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exciter (governor) s ta t ic ga inc ont ro l ( f i lt e r ) ga in m a t r ixgain of the der ivat ive blocktransfer function of a compensatore lec tr ica l (mechanical) powerreference powermachine terminal power before any dis tur-ba nc e ss ta te weighting ma t r ixsecond diagonal e lement of Qc ont ro l we ight ing ma t r ixt ie- l ine impedanceme c ha nic a l ( e le ct r i ca l ) da mping torqueme c ha nic a l torque at ma c hine sha f tcontrol vectorexcita t ion control s ignal genera ted by PSSme a sure me nt noi se ( in te ns i ty)d q) -c ompone nt of ma c hine t e rmina lvoltage

W ma c hine f r e que nc yTA exciter t ime-constantd o d-axis t ransi ent open-c ircuit

t ime -c ons ta ntT g ( 4 gove rnor ( tu rb ine ) t ime -c ons ta ntTa t ime-cons tant of th e der ivat ive block

R E F E R E N C E S[I] 0. I. Elgerd, Electric Energy Systems Theory: An

Introduction, McGraw-Hill, New York, 1982.[2] Charles A. Gross, Power System Analysis, J o h n

Wiley Sons, New York, 1986.[3] Yao-Nan Yu, Electric Power System Dynamics,

Academ ic Press , New York, 1983.[4] Hamd y A. M. Mou ssa , Yao-Nan Yu, Opt imal power

sys te m s ta b il i z at ion through e xc i t a t ion a n d/or gove r-nor control , l E E E Transactions on Power Appa-ratus and Systems, Vol. PAS-91, No. 3, pp. 1166-

Vma x(Vmin) ma xim um (minimum ) outpu t of the e xc it e rVref reference voltage [ 5 ] J . M. Macie jowski , Multivariable Feedback Design,V ~ O V ~ ) ma c hine - te rmina l 00 -bu s) voltage prior Addison-Wesley, New York, 1989.

[6] Brian D. 0 Ande rson, John B. Moore , Optimalvt ma c hine t e rm ina l vol t a ge Control: Linear Quadratic Methods, Pren tice Hall ,

Englewood Cliffs, New Jersey, 1990.[7] IEEE Co mmit te e Re por t , E xc i ta t ion Sys te m Mod-WO els for Power System St abil i ty Stu dies , IEEE Trans-

X d ( X L ) d-a xis ( t r a ns ie nt ) r e a c ta nc e actions on Power Apparatus and Systems, Vol.PAS-100, No. 2, pp. 494-509, Feb ruary 1981.x qY s y s t em o u t p u t [8] P. M. Ande rson a nd A . A . Fouad, Power SystemA pref ix deno ting an incremen ta l va lue 1977.6 ma c hine a ngula r pos i t ionr inpu t noise ma t r i x Ne w York , 1956.

1174, Ma y/Jun e 1972.

to a ny d i s turba nc e sinput noise ( intensi ty)es t imate of the input noise intensi ty(e s t ima te of ) s t a te ve c torq-axis reactance

W W>x(*>

t s ta te in t rod uc e d by the de r iva t ive b lock Control and Stability, Sc ienc e Pres s , Ep hra t a , PA,[9] Edward Wilson Kimbark, Power System Stability:

Synchronous Machines, Dover Publica t ions , Inc . ,he i- th s ingular va lue of matr i x A

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