applying pss genconcepts

7/29/2019 Applying PSS GenConcepts

1/30

IEEE T r a n s a c t i o n s o n P o w e r A p p a r a t u s a n d S y s t e m s , V o l . P A S - 1 0 0 , N o . 6 J u n e 1 9 8 1APPLYING P OW ER S YS TE M S TA BI LI ZE RSPART I : G EN ER AL C ON CE PT S

E . V . Larsen ( M e m b e r ) D . A . Swann ( M e m b e r )G e ne ra l E l ec t ri c C o m p a n y , Schenectady, N ew YorkABSTRACT

The general concepts associated w i t h applying p o w e r s y s -t e m stabilizers utilizing shaft s p e e d , ac b us f r e q u e n c y ,a n d electrical power i n p u t s are developed in t hi s firstp a r t o f a three-part paper. T hi s l a y s t h e f o u n d a t i o nf o r discussion of t h e tuning c on c ep t s a n d practicalaspects o f s t a b i li z e r application in P a r t s I I and I I I .T h e characteristics of the " pl an t" t hr ou gh w h i c h thepower s y s t e m stabilizer must operate are d i s c u s s e d a n dt h e implications upon s t a b i li z e r tuning and performanceare n o t e d . A general approach fo r analyzing s t a b i l i z e r sutilizing an arbitrary i n p u t s i gn a l i s d e s c r i b e d a n dapplied t o the frequency a n d e l e c t r i c a l p o w e r i n p u ts i g n a l s .

INTRODUCTIONBeginning i n t h e late 1 9 5 0 ' s an d early 1 9 6 0 ' s mos tof t h e new generating units added to electric utilitys ys t ems were e qu ip pe d w it h c o nt i nu o us l y -a c ti n g v o lt a ger e g u l a t o r s . A s t h e s e units became a larger p e r c e n t a g eo f the generating capacity, i t became a p p a r e n t t h a t t h ev ol ta ge r eg ul at or action had a detrimental impact uponthe dynamic stability ( o r perhaps more properly s t e a d y -s t a t e s t a b i l i t y ) o f t h e power sy stem. Oscillations o fsmall magnitude and low frequency often persisted forl o n g periods o f time and in some cases presented limita-tions on power transfer c a p a b i l i t y . Power s y s t e m s t a b i -lizers were developed t o aid i n damping t he se o sc il la -tions via modulation o f t h e g e n e r a t o r excitation [ 1 , 2 ] .T he a r t a nd s ci en ce of applying power s y s t e m stabilizershas d e ve l op e d c o ns i de r ab l y over t h e p a s t t e n to fifteen

years since t h e first w id es pr ea d a pp l ic at io n t o t h eWestern S y s t e m s o f t h e United S t a t e s . This developmenth a s i n v olv e d t h e use o f various tuning techniques andi n p u t s i g n a l s , and learning to d e a l with practicalproblems such as n oi s e a n d i nt er ac ti on w it h t ur bi ne -ge n e r a t or s haft t o rs i on a l mo de s of vibration.

The desire on t h e p a r t o f t h e a u t h o r s ' Company top ro vi de s ta bi li zi ng e qu ip me nt w h i c h a llows adjustmentf o r g o o d power s y s t e m performance while overcoming t h ep ot en ti al p ro bl ems associated with n oi s e a n d torsionaldestabilization h a s l e d to considerable research i n t h i sarea in r e c e n t years, involving both analytical studiesa n d fie ld tests. I n 1 9 7 5 , a torsional p r ot e ct i ve p ac k-a g e was developed f o r application to stabilizers onl a r g e two-pole units [ 3 ] . The development of similarprotection fo r four-pole u n i t s , w i t h their inherentlylow t ors i on a l frequencies, h a s required much c los e rscrutiny of power s ys t em s t a b i l i z e r applications. T hi spaper de s c r i b e s the r e sult s of this r e s e a r c h , wi t hemphasis on the power s y s t e m performance attainable w i t hs t a b i li z e r s utilizing e a c h o f the t hr e e input s i g n a l sc on s i de r e d most feasible: shaft s p e e d , ac b us fre-quency, and a combination of power and s p e e d . T he s er e s u l t s are presented i n three parts: the firs t part

deals with t h e fundamental a s p e c t s o f applying stabi-lizers with t h e alternative input s i g n a l s , and repre-s e n t s an extension of the c o n c e p t s presented i n thepaper by Concordia a n d deMello [ 2 ] . T he s ec on d par t o ft h i s paper discusses s y s t e m performance criteria, devel-ops tuning c o n c e p t s w h i c h e na bl e a tt ai nme nt o f t h e s ec r i t e r i a , a n d t h e relative performance attainable wi t hpractical stabilizer equipment utilizing the three basicinput s i g n a l s . The t h i r d p a r t discusses t h e practicalconsiderations o f t un in g e qu ip me nt i n the field a n dequipment d e s i g n , including minimizing t h e effects oftorsional destabilization, power s y s t e m noise, a n d , whenusing electrical power as an input s i g n a l , mechanicalpower variations.

BASIC CONCEPTSThe basic function o f a power s y s t e m stabilizer i sto extend stability limits by modulating g e n e r a t o rexcitation t o provide damping to the oscillations o fsynchronous machine r o t o r s relative to one another.

These oscillations o f concern typically occur i n thefrequency range o f approximately 0.2 to 2 . 5 H z , andinsufficient damping o f these oscillations may limit th eability to transmit power. T o provide d a m p i n g , th estabilizer must produce a c ompone n t of e l e c t r i c a l torqueon t h e r o t o r w h i c h i s i n phase with speed variations.T h e i mp l eme n ta t io n d e ta il s d i f f e r , depending upon t h estabilizer input s i g n a l e m p l o y e d . However, fo r anyinput s i g n a l t h e transfer function of the stabilizermu st c omp en sa te f o r t h e gain and phase c h a r a c t e r i s t i c so f the excitation system, the g e n e r a t o r , a nd the powersystem, w h i c h collectively determine the transfer func-tion from t h e stabilizer o u t p u t to t h e c ompone n t ofelectrical t o r q u e w h i c h can be modulated v ia e xc it at io ncontrol. This transfer f u n c t i o n , d e n o t e d G E P ( s ) in thispaper, i s strongly influenced b y v ol ta ge r eg ul at or g a i n ,g e n e r a t o r power l e v e l , and ac s y s t e m s t r e n g t h . T hi ssection discusses t h e general relationship of t h e s ep a r a m e t e r s to G E P ( s ) and h en c e t o stabilizer perform-ance.

The block diagram i n Figure 1 illustrates, in t e rmso f a fe w basic small-signal transfer functions, therelationship b e t w e e n t h e applied t or q ue s on the t ur-bine-generator shaft an d the r e su l ti n g g e ne r at o r r o t o rs p e e d , w an d r o t o r angular displacement, 6 . T hee l e c t r i c a y t o r q u e may b e c on si de re d t o h a v e t w o c o m p o -nents, viz. ( a ) that w h i c h i s prod uced by t h e powers ys t em stabilizer s o l e l y by mo d u l a t i o n of generatorf l u x , T ep and ( b ) t ha t w hi c h r e su l ts fr om a l l o t h e r

A8ATm

80 SM 558-7 A pap er r e c ommende d an d a p p r o v e d b y theIEEE P ow er G e ne ra ti on C ommi tt ee of t he I E E E P o w e rE n gi n ee r in g S oc ie ty fo r p r e s e n t a t i o n a t the IEE E P ESSummer Me e t i n g, M i n n e ap oli s , M in ne so ta , Ju ly 13 -18,1 9 8 0 . M i a n u s c r i p t submitted M a r c h 1 4, 1980; madea v a i l a b l e f o r p r i n t i n g May 7, 1980.

AE I

F I G U R E S T A B I L I Z E R W I T H S P E E D I N P U T -S Y S T E M B L O C K D I A G R A M-1-

3 0 1 7


2/30

3 0 1 8

I r IG E P ( s ) /

F I G U R E 2 S I M P L I F I E D MODEL O F S I N G L E M A C H I N E T O I N F I N I T E BUSsources, including shaft motion, T The func t i on a lr e l a t i o n s h i p between speed and t o r q u e is s h o w n fo r astabilizer employing g e n e r a t o r speed as an i n p u t s i g n a l .T h e c o n t r i b u t i o n o f t o r q u e d u e to t h e s t a b i li z e r p a t h isgiven b y :

AT AP S S ( s ) G E P ( s ) = P ( s ) (1)G

( I n t h i s c a s e s p e e d i n p u t i s u s e d f o r c l a r i t y ; e x t e n s i o nt o o t h e r i n p u t s i g n a l s w i l l b e d i s c u s s e d s u b s e q u e n t l y . )T h e t r a n s f e r f u n c t i o n G E P ( s ) r e p r e s e n t s 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 , t h e e x c i t a t i o n s y s t e m , a n d t h epower s y s t e m . T h i s t r a n s f e r f u n c t i o n c a n b e a p p r o x i -mated with t h e a i d o f F i g u r e 2 , r e p r e s e n t i n g a s i m p l i -fied model o f a m a c h i n e c o n n e c t e d t o a l a r g e p o w e rsystem t h r o u g h a t r a n s m i s s i o n l i n e a s d e r i v e d i n R e f e r -ence 2 . Study o f t h i s f i g u r e r e v e a l s t h a t t h e d y n a m i cc h a r a c t e r i s t i c s o f G E P ( s ) a r e p r o p o r t i o n a l t o t h o s e o ft h e closed l o o p v o l t a g e r e g u l a t o r when t h e g e n e r a t o rs p e e d i s c o n s t a n t ( A W G = 0 ) , i . e . :

KG E P ( s ) - -K 6 a E t3E r e fThis r e l a t i o n s h i p forms t h e basis of s t a b i li z e r t u n i n gprocedures which involve m e a s u r e m e n t o f t h e c los e d-loopvoltage r e g u l a t o r c h a r a c t e r i s t i c t o de t e rmine th e p h a s ecompensation required o f t h e s t a b i l i z e r .

The variation o f G E P ( s ) with e x c i t e r g a i n , gener-a t o r l o a d i n g , a n d ac s y s t e m s t r e n g t h p l a y s a domina n trole i n power s y s t e m stabilizer t u n i n g r e q u i r e m e n t s a n dp e r f o r m a n c e . T h e closed-loop v o l t a g e r e g u l a t o r responsei s primarily a function of the e x c i t e r c ha r a c t e r i s t i c sand t h e ac s y s t e m s t r e n g t h . M o d e r n turbine-generatorstypically ha v e a voltage regulator t r a n s i e n t g a i n ( i . e . ,t h e gain i n t h e frequency range o f i n t e rmac hi n e o s c i l l a -t i o n s , v i z , 0 . 2 to 2 . 5 H z ) o f app roximate ly 2 0 puE f d / p u ' t , which results i n a voltage r e g u l a t o r lo o pcrossover at approximately 1 r a d / s e c w i t h a n om i n a l acs y s t e m [ 2 ] . Experience h a s indicated that this g a i nr e sult s i n satisfactory op e r a t i on over a f a i r l y w i d er a n g e o f s y s t e m c o n d i t i o n s . Howe v e r , t o ob t a i n fullbenefit f r o m h i g h ceiling e x c i t e r s fo r s t a b i l i t y f o l l o w -

i n g a l a r g e d i s t u r b a n c e , h i g h e r g a i n s ma y be r e q u i r e d .S o m e utilities [ 4 ] set t h e i r v o l t a g e r e g u l a t o r s withmuch h i g h e r transient g a i n s , o n t h e order o f 2 0 0 puE f d / p u e t , to e n s u r e maximum p e r f o r m a n c e . With anexcitation s y s t e m h a v i n g r e l a t i v e l y l i t t l e p h a s e l a g ,such h i g h g a i n s w i l l result i n s a t i s f a c t o r y v o l t a g er e g u l a t i o n at l i g h t g e n e r a t o r l o a d s , but will s i g n i f i -c a n t l y d e c r e a s e d a m p i n g o f r ot or o s c i l l a t i o n s a n d ma yc a u s e a n i n s t a b i l i t y which p r e v e n t s f u l l l o a d o p e r a t i o neven f o r r e l a t i v e l y s t r o n g transmission s y s t e m s . T h eremainder o f t h i s paper d e a l s p r i m a r i l y with t h e mor et y p i c a l r e g u l a t o r transient g a i n s i n t h e n e i g h b o r h o o d o f2 0 pu E f d / p u E t , a l t h o u g h many o f t h e c o n c l u s i o n s a l s oa p p l y when e m p l o y i n g h i g h g a i n e x c i t e r s .

For s i t u a t i o n s where t h e v o l t a g e r e g u l a t o r l o o pc r o s s o v e r f r e q u e n c y i s l o w e r t h a n t h e oscillation f r e -quency o f concern, t h e g a i n o f G E P ( s ) a t t h e oscillationf r e q u e n c y c a n b e a p p r o x i m a t e d b y i t s f o r w a r d p a t h , i . e . :G E P ( j w ) - K E X C ( i j w ) I I / W T ( 2 b )

I t i s a l s o assumed t h a t 1 / K T ' i s l e s s t h a n t h e c r o s s -3 d oover f r e q u e n c y . T h e a s s u m p t i o n s behind e q u a t i o n 2 b a r eu s u a l l y valid f o r t y p i c a l modern u n i t s . T h e g a i n i st h e r e f o r e p r o p o r t i o n a l to t h e exciter g a i n a n d i n v e r s e l yp r o p o r t i o n a l t o t h e main g e n e r a t o r o p e n - c i r c u i t f i e l dt i m e c on s t a n t a n d t h e oscillation f r e q u e n c y . T h e g a i ni s a l s o p r o p o r t i o n a l t o t h e p a r a m e t e r K which repre-2sents t h e e f f e c t o f a c h a n g e i n g e n e r a t o r f l u x ( E ' ) o nt o r q u e , a n d : q1 . I n c r e a s e s with g e n e r a t o r l o a d i n g .2 . I n c r e a s e s as t h e a c s y s t e m becomes s t r o n g e r .H e n c e , t h e g a i n i n t h i s p o r t i o n o f t h e l o o p i s h i g h e s twith t h e g e n e r a t o r a t f u l l l o a d a n d operating i n t o t h estrongest ac s y s t e m .

Fo r s i t u a t i o n s where t h e v o l t a g e r e g u l a t o r l o o pc r o s s o v e r f r e q u e n c y i s h i g h e r t h a n t h e oscillation f r e -quency o f concern, as i t i s i n t h e case o f very h i g hr e g u l a t o r g a i n , t h e g a i n o f G E P ( s ) i s no l o n g e r pro-p o r t i o n a l to r e g u l a t o r g a i n b u t i s i n v e r s e l y propor-tional t o t h e parameter K 5 a s i n d i c a t e d by equation 2 a .K r e p r e s e n t s t h e e f f e c t o f E ' on t e r m i n a l v o l t a g e ,w h i c h d e c r e a s e s as t h e ac s y s t e a b e c o m e s s t r o n g e r a n dh e n c e causes t h e g a i n o f G E P ( s ) t o f u r t h e r i n c r e a s e a st h e s y s t e m s t r e n g t h i n c r e a s e s .

S i n c e t h e voltage regulator o p e n - l o o p g a i n i sp r o p o r t i o n a l t o K 6 , t h e crossover frequency d e c r e a s e s a st h e a c system b e c o m e s s t r o n g e r . T h i s i n f l u e n c e s s t a b i -l i z e r p e r f o r m a n c e , a s there will b e more p h a s e l a g too v e r c om e with a strong ac system than with a weak acs y s t e m . T h i s effect i s most pronounced with high v o l t -a g e regulator gain s i n c e t h e c r o s s o v e r f r e q u e n c y i s i nt h e same range a s t h e i n t e r m a c h i n e o s c i l l a t i o n s o fconcern.I n s u m m a r y , t h e power system s t a b i l i z e r must o p e r -ate t h r o u g h t h e " p l a n t " G E P ( s ) which i s d e p e n d e n t upont h e g e n e r a t o r , t h e excitation s y s t e m , a n d t h e powers y s t e m . T h e b a s i c c h a r a c t e r i s t i c s o f t h i s plant whichare significant t o stabilizer applications a r e a s f o l -l o w s :

1 . T h e phase c h a r a c t e r i s t i c s o f G E P ( s ) a r e nearlyidentical t o t h e phase c h a r a c t e r i s t i c s o f t h ec l o s e d l o o p voltage r e g u l a t o r .2 . T h e gain o f G E P ( s ) i n c r e a s e s with generator l o a d .3 . T h e g a i n o f G E P ( s ) i n c r e a s e s a s t h e a c systembecomes s t r o n g e r . T h i s e f f e c t i s a m p l i f i e d w i t hh i g h g a i n v o l t a g e r e g u l a t o r s .

+


3/30

4 . For t y p i c a l voltage r e g u l a t o r transient g a i n s o ft h e o r d e r o f 2 0 p u Efd/pu & t ' t h e gain o f G E P ( s ) a tt h e oscillation frequencies o f concern i s propor-t i o n a l t o t h e r e g u l a t o r g a i n a n d i n v e r s e l y pro-portional t o t h e main generator o p e n - c i r c u i t t i m ec on s t a n t a n d t h e oscillation f r e q u e n c y .5 . T h e p h a s e l a g o f G E P ( s ) increases a s t h e ac systembecomes stronger. This h a s t h e greatest influencewith h i g h g a i n e x c i t e r s , since the v o l t a g e r e g u -l a t o r l o o p crossover f r e q u e n c y a p p r o a c h e s that o ft h e oscillation o f c o n c e r n .

CHARACTERISTICS OF ALTERNATIVE INPUT S I G N A L S -T h e g e n e r a l f r e q u e n c y response characteristics o fs t a b i l i z e r s u t i l i z i n g alternative i n p u t s i g n a l s arediscussed i n t h i s s e c t i o n , t o g e t h e r with t h e basicc o n c e p t s governing t h e t u n i n g aspects and p e r f o r m a n c ec a p a b i l i t i e s . S p e e d i n p u t stabilizers ar e discussedf i r s t , f o l l o w e d b y an extension to f r e q u e n c y a n d poweri n p u t s t a b i l i z e r s .

S p e e d I n p u tA power system s t a b i l i z e r u t i l i z i n g s h a f t s p e e d a san i n p u t must c o m p e n s a t e f o r t h e l a g s i n G E P ( s ) t o p r o -

d u c e a c o m p o n e n t o f torque i n phase with s p e e d c h a n g e ss o a s t o i n c r e a s e damping o f t h e r o t o r o s c i l l a t i o n s , a sd e t a i l e d i n Appendix A . A n i d e a l stabilizer c h a r a c t e r -i s t i c would therefore b e inversely proportional t oG E P ( s ) , i . e . :I d e a l PSS ( s ) = D p 5 5 / G E P ( s ) ( 3 a )

w h e r e D P S r e p r e s e n t s t h e desired damping c o n t r i b u t i o no f t h e s t a b i l i z e r ( d e t a i l e d i n Appendix A ) . Such as t a b i l i z e r c h a r a c t e r i s t i c i s impractical s i n c e perfectcompensation f o r t h e l a g s o f G E P ( s ) r e q u i r e s pure d i f -f e r e n t i a t i o n with i t s as soc ia ted h ig h g a i n a t highf r e q u e n c i e s . A practical s p e e d stabilizer must utilizel e a d / l a g s t a g e s s e t t o c o m p e n s a t e f o r phase l a g s i nG E P ( s ) o v e r t h e frequency r a n g e o f i n t e r e s t . T h e g a i nmust b e attenuated a t h i g h f r e q u e n c i e s t o l i m i t t h ei m p a c t o f n o i s e a n d minimize t o r s i o n a l i n t e r a c t i o n , a n dc o n s e q u e n t l y l o w - p a s s a n d possibly band-reject [ 1 ]f i l t e r s a r e r e q u i r e d . A washout s t a g e i s i n c l u d e d t oprevent s t e a d y - s t a t e v o l t a g e offsets a s system frequencyc h a n g e s .

T sPractical PSS ( s ) = K ws(l+Tws)( l + s T 1 ) ( l + s T 3 )( l + s T 2 ) ( l + s T 4 ) FILT(s)

( 3 b )A s discussed i n t h e previous section t h e stabilizermust o p e r a t e through G E P ( s ) , t h e characteristics ofwhich vary significantly with op e r a t i n g co nd it io ns . Th egain increases as g e n e r a t o r l o a d increases, w h i c h i sdesirable since t h e stability problems for w h i c h t h e

stabilizer i s a p p l i e d a l s o increase with g e n e r a t o r l o a d .H o w e v e r , t h e gain is very high fo r s t r o n g ac s y s t e m swhere t h e stability problem i s m i n i m a l , a nd decreases asthe ac s y s t e m becomes w e a k e r . T h e latter effect causesthe influence o f a speed input stabilizer t o decreasew h e n t h e power s y s t e m requires i t most. I n addition t ot h e fact that gain increases as t h e s y s t e m becomess t r on ge r , t h e phase l a g a l s o i n c r e a s e s . A s a conse-q u e n c e , the stabilizer l o o p o f Figure 1 i s l ea s t s t a b l eunder s t r o n g ac s y s t e m c o n d i t i o n s , a n d t he re fo re t he seconditions establish t h e maximum permissible gain of t h espeed i n p u t stabilizer. Without adaptive ga i n control,the stabilizer gain can n o t b e as high as desired underw e a k ac s ys t em c o n d i t i o n s w h e n t h e s t a b i l i z e r c o n t r i b u-t i on i s n e e de d most.

3 0 1 9Frequency I n p u t

T h e use o f ac b u s f r e q u e n c y a s a stabilizer i n p u tr e s u l t s i n t un i ng p r oc e du r es a n d performance character-i s t i c s s o m e w h a t different f r o m th os e a ss oc ia te d withspeed i n pu t s t ab i li z er s . The primary difference i s thatt h e s e n s i t i v i t y o f t h e f r e q u e n c y s i g n a l t o rotor oscil-l a t i o n s increases a s t h e external transmission s y s t e mbecomes w e a k e r , w h i c h t e n d s to offset t h e reduction i ngain f r o m stabilizer output t o electrical torque,G E P ( s ) , r e s u l t i n g with weaker transmission s y s t e m s .T h i s effect c an b e understood b y u ti li zi ng t h e i n p u tsignal sensitivity factor concept developed i n AppendixB . T h i s factor represents t h e t ra ns fe r f un ct io n f r o mspeed t o t h e stabilizer input s i g n a l , i n t h i s case a cb u s f r e q u e n c y :( 4 )

S ( s ) r a n g e s from zero i n t h e extreme case o f a unitconnected t o an i n f i n i t e l y stiff s y s t e m , t o unity witht h e unit und er open circuit conditions. Between thesee x t r e m e s , o n e approximation of S ( s ) wo u l d be t o equatei t t o a v o l t a g e d i vi si on b e tw e en Fhe internal v o l t a g e o ft h e machine a n d t h e v o l t a g e o f t h e infinite b u s , i . e . :s 1\ 1 X / [ X + XGEN ( s ) ] ( 5 )

T h i s approximation i s valid o n l y f o r manual c o n t r o l ,s i n c e , a s described i n A p p e n d i x B , there exists anadditional path from s p e e d t o t h e i n p u t s i g n a l via t h ev o l t a g e r e g u l a t o r . T h i s effect c a n be included byc a l c u l a t i n g t h e effective internal i m p e d a n c e o f t h egenerator with t h e v o l t a g e r e g u l a t o r a n d i n c l u d i n g t h i si n equation 5 . For t y p i c a l v o l t a g e r e g u l a t o r g a i n s , t h eeffective i m p e d a n c e o f t h e machine i s a p p r o x i m a t e l ye q u a l t o t h e subtransient r e a c t a n c e over t h e f r e q u e n c yr a n g e o f interest t o stabilizer a p p l i c a t i o n .T h e o f f - s e t t i n g effect o f t h i s characteristic i si l l u s t r a t e d i n F i g u r e 3 , which s h o w s t he c h a n g e i n S F OH z ) c a l c u l a t e d f o r a l a r g e f o s s i l unit v e r s u s externalsystem i m p e d a n c e , a n d t h e c o m p o u n d effect o f t h e i n -crease i n s e n s i t i v i t y with decrease i n g a i n t h r o u g hG E P ( s ) ( t h e latter b e i n g p r o p o r t i o n a l to t h e parameterK 2 a s p r e v i o u s l y i n d i c a t e d ) . The g a i n c h a n ge s o f K 2 andS with external system reactance t e n d t o o ff-s et e ac ho t h e r with t h e r e s u l t t h a t t h e net g a i n from generators p e e d ( w G ) t o electrical t o r q u e ( T e p ) i s r e a s o n a b l yconstant f o r a w i d e r a n g e o f system r e a c t a n c e s . Forvery l o w v a l u e s o f system r ea c t a n c e , t he gain i s r e -d u c e d . H e n c e , t h i s a n a l y s i s s u g g e s t s t h a t t h e g a i n o ft h e s t a b i l i z e r ma y b e a d j u s t e d t o o b t a i n t h e b e s t p o s s i -b l e performance under weak a c transmission system c o n d i -t i o n s , w h e r e t h e contribution o f t h e s t a b i l i z e r i sr e q u i r e d m o s t , without concern t h a t t h e gain w i l l b eexcessive a n d cause t h e s t a b i l i z e r t o b e c o m e u n s t a b l eunder s tr on g s y st em c o n d i t i o n s .I n a d d i t i o n , t h e frequency s i g n a l i s mo re s e ns i ti v eto modes o f oscillation between power p l a n t s or l a r g eareas t h a n t o modes i n v o l v i n g o n l y i n d i v i d u a l u n i t s ,i n c l u d i n g t h o s e between units within a power p l a n t .T h i s f o l l o w s since t h e f r e q u e n c y at an a c b u s betweenun i t s i s n e a r a n o d e f o r t h e l a t t e r m o d es of oscilla-t i o n , w h e r e a s i t i s l a r g e l y r e s p o n s i v e t o m o d e s whereint h e units s w i n g c o h e r e n t l y . A s a c o n s e q u e n c e , i t a p -pears t o b e p o s s i b l e t o o b t a i n g r e a t e r d a m p i n g c o n t r i b u -t i on f o r modes o f oscillation between p l a n t s or are asthan would b e o b t a i n a b l e with t h e s p e e d i n p u t .

Power InputT h e us e o f a c c e l e r a t i n g power a s an i n p u t s i g n a l t ot h e power system s t a b i l i z e r h a s r e c e n t l y received con-

s F ( s ) = a f / a w G


4/30

3 0 2 0

1 . 6

1 . 4

1 . 2

1 . 0

-

z

( D

. 8

. 6

4

. 2

00 . 2 . 4 . 6

TOTAL X E ( P . U . )F I G U R E 3 C O M P E N S A T I O N O F G A I N B Y F R E Q U ES E N S I T I V I T Y F A C T O R

siderable a t t e n t i o n d u e t o i t s i n h e r e n tt or si on al i nt er ac ti on [ 5 ] . P r a c t i c a l d ie l i m i n a t i n g , o r at l e a s t m i n i m i z i n g , t ]mechanical power c h a n g e s appear t o h a v e b e eu t i l i z i n g a h e a v i l y f i l t e r e d s p e e d s i g na p p r o x i m a t e l y corrects f o r mechanical poweT h e f o l l o w i n g a n a l y s i s a p p li e s o n l y to t hc h a r a c t e r i s t i c s o f e l e c t r i c a l power f e e d bi t must b e r e c o g n i z e d t h a t a p r a c t i c a l p o ur e q u i r e s some compensating device fo r me cvariations.T h e most common a p p r o a c h to a n a l y z ii n p u t s t a b i l i z e r i s to t r e a t i t s i n p u t a s to f s p e e d a n d a p p l y t h e same c o n c e p t s u t i l i 2i n g t h e s p e e d i n p u t s t a b i l i z e r . T h i s a p p it h e conclusion t h a t t h e p e r f o r m a n c e c h a r a c tpower i n p u t s t a b i l i z e r are i d e n t i c a l to t h (i n p u t s t a b i l i z e r . T h i s c o n c l u s i o n i s vv a r : . j u s power system m o d e s o f o s c i l l a t is h a f t b e h a v e s a s a r i g i d b o d y ( p r o v i d e d tpower c h a n g e s are c o m p e n s a t e d f o r s o t h a t ti n p u t i s a true m e a s u r e of average a c c e l e r at h e t u r b i n e - g e n e r a t o r ) . For s h a f t t o r s i cv i b r a t i o n , h o w e v e r , the a c c e l e r a t i n g powera t or rotor a l o n e i s c o n s i d e r a b l y differaverage a c c e l e r a t i n g power a c r o s s t h e e n t ih e n c e t r e a t i n g t h e s t a b i l i z e r i n p u t s i g n a lative o f generator s p e e d i s i n v a l i d . A i

approach i s t o utilize t h e input signal sensitivityf a c t o r c o n c e p t , i . e . , t re at e le ct ri ca l p ow er c h a n g e s a sresulting f r o m s p e e d changes v i a t h e el ect ri c po wers y s t e m , rather than causing s p e e d c h a n g e s . This a p -proach yields insig ht i n t o t h e g en er ic d if fe re nc esbetween power input a n d s p e e d input stabilizers, partic-ularly w it h r eg ar d t o torsional i n t e r a c t i o n , a s outlinedi n t h e following discussion.T h e input s i g n a l sensitivity factor f o r power i s :

a PS ( s ) = e =P ww Ga ( T e w G ) a T e

e W G aww b K l e ( s ) + p

s e ow b K l e ( s ( + S / W ) )1 P O w

( 6 a )

( 6 b )

( 6 c )

where T , e X Pe , a n d w G a r e a l l i n per unit on a c o n -sistent b a s e .T h i s input s i g n a l sensitivity factor i s d o m i n a t e d by t h efirst te rm o f equation 6 fo r t h e l o w frequencies o fc o n c e r n , s i n c e t h e lowest value o f Kle i s approximately0 . 5 p.u./radian ( f o r weak transmission s y s t e m s ) a n d with

B U S P e o maximum a t 1 . 0 p . u . t h e br e a k from t h e integral termt o t h e proporational term ( w ) occurs a t approximatelyo n e - h a l f synchronous f r e q u e n c y , i . e . , near 3 0 H z . T h i sI N P U T implies that electrical power and torqu e are equivalentA C T O R S F with respect t o stabilizer performance f o r most subsyn-c h r o n o u s modes o f oscil lation. Sin ce t h e input s i g n a lsensitivity factor f o r power h a s primarily an i n t e g r a lc h a r a c t e r i s t i c , with approximately 9 0 0 o f phase l a g i nt h e frequency r a n g e o f i n t e r e s t , i t would appear t h a t ana d d i t i o n a l 9 0 o f p h a s e l e a d would b e r e q u i r e d i n t h es t a b i l i z e r . However, a non-minimum p h a s e a p p r o a c h canb e u ti li ze d w it h t h e p ow er i np ut s t a b i l i z e r , i . e . , a l a g. 8 1 . 0 c h ar a ct e ri s ti c h a vi n g 2 7 0 o f phase shift c a n b e used

rather than a l e a d characteristic having 9 0 0 o f p h a s es h i f t . T h e advantage o f t h i s approach i s that t h e l a gc h a r a c t e r i s t i c yields a c los ed -l oop i nt er act ion w it h: N C Y I N P U T shaft oscillations w h i c h exhibits a d e c r e a s i n g g a i n withf r e q u e n c y , a s opposed t o t h e increasing gain w ith f r e -q ue nc y a ss oc ia te d with t h e minimum-phase approach u t i -

low level o f l i z i n g a l e a d c h a r a c t e r i s t i c . T h u s , t h e t o r s i o n a lf f i c u l t i e s o f interaction characteristics o f ideal minimum v e r sush e effect o f non-minimum p h a s e stabilizers d i v e r g e at a r a t e o f 4 0. n overcome by d b / d e c a d e a s t h e t o r s i o n a l f r e q u e n c y i n c r e a s e s , withL a l [ 6 ] w h i c h e q u a l interaction at t h e l o c a l mode o f oscillation. r variations. ( a s s u m i n g g a i n s se t for e q u a l l o c a l m o d e p e r f o r m a n c e ) .L e closed l o o p A discussion o f non-minimum p h a s e s t a b i l i z e r s i s i n -a c k ; h o w e v e r , cluded i n Appendix C .w e r s t a b i l i z e rh a n i c a l power The p h a s e l a g o f 2 7 0 c a n be obtained b y i n v e r t i n gt h e e le ct ri ca l p ow er s ig na l a n d designing t h e stabilizersuch that t h e n e t contribution o f t h e stabilizer a n di n g t h e power G E P ( s ) p r o d u c e s 9 0 Y o f p h a s e l a g b y a n i n t e g r a l c h a r a c -t h e derivative t e r i s t i c . With t h i s c r i t e r i a , a n i d e a l s t a b i l i z e r f o rz e d in a n a l y z - p o w e r i s defined b y :r o a c h l e a d s tot e r i s t i c s of aD s e o f a s p e e dr a l i d f o r t h eL o n where t h e- h e m e c h a n i c a lt h e s t a b i l i z e ri t i n g power o no n a l modes o fon t h e g e n e r -e nt than t h ei r e s h a f t , a n da s t h e d e r i v -n alternative

I d e a l P S S p ( s ) = - G E P S SP sGEP ( s ) ( 7 )Note t h a t t h i s r e s u l t i s t h e s a m e a s would be obtainedb y t re at in g p ow er a s t h e derivative o f s p e e d , s i n c einverting P a n d i nt eg ra ti ng w ou ld yield s p e e d , a n dm u l t i p l y i n g % y t h e i d e a l P S S w ( s ) from equation 3 wouldg i v e equation 7 .


5/30

3 0 2 1T he f r e q u e n c y r e s p o n s e o f s u c h a stabilizer i ss h o w n i n Figure 4 . A t l o w f r e qu e n c i e s , l a g must b eintroduced t o compensate f o r t h e l o w p h a s e l a g o fG E P ( s ) , while a t higher frequencies t h e s t a b i l i z e r musta d d p h a s e l e a d t o c o m p e n s a t e f o r t h e secondary l a g s o fboth G E P ( s ) a n d t h e s t a b i l i z e r . Extending t h e l a g a tl o w frequencies a n d t h e lead a t h i g h f r e q u e n c i e s i si m p r a c t i c a l , because o f noise a t t h e h i g h f r e q u e n c i e sa n d because o f p o t e n t i a l l y excessive v o l t a g e offsets f o rl o w frequency p h e n o m e n a on t h e power s y s t e m , i n c l u d i n g

c h a n g e s i n mechanical power i f i t i s not p e r f e c t l ycompensated t o provide a pure a c ce l er at i ng p ow e r s i g n a lf o r t h e s t a b i l i z e r . T h e l o w frequency g a i n i s reducedby using a time constant rather than a pure i n t e g r a l ,e q u i v a l e n t t o t h e w a s h o u t s t a g e o f s p e e d i n p u t s t a b i -l i z e r . T h e combination o f a s m a l l l e a d / l a g s t a g e and al a g a t h i g h frequencies p r o v i d e s a desirable d e c l i n i n ggain w it h fr eq uen cy w h i l e maintaining a d e q u a t e p h a s enear t h e h i g h e s t l o c a l m o d e f r e q u e n c y . A p r a c t i c a lpower i n p u t s t a b i l i z e r would t he refo re h av e t h e f r e -quency response shown b y t h e dashed l i n e i n F i g u r e 4 .

1 0 0

I 0

- cL Jc r -

0 . 1 0 . 1 I 0w 4 R A D / S E C )F I G U R E 4 P O W E R I N P U T S T A B I L I Z E R . I D E A L A N D P R A C T I C A L

I t s h o u l d b e noted t h a t i m p e r f e c t s t e a d y - s t a t ecompensation f o r power c h a n g e s w o u l d necessitate a d d i n ga washout s t a g e , which i s equivalent t o a s e c o n d washouts t a g e i n a s p e e d input s t a b i l i z e r . T h e r es u lt i ng p h as el e a d would b e detrimental t o t h e s t a b i l i z e r performances i n c e , a s described in Appendix A , i t introduces ad e s y n c h r o n i z i n g component o f t o r q u e . I nt er ar ea mo de soften have very l i t t l e synchronizing torque d u e t or e l a t i v e l y w e a k , heavily l o a d e d t i e s , a n d a d d i n g ad e s y n c h r o n i z i n g effect via a power system s t a b i l i z e r c a nc aus e areas t o l o s e synchronism f o l l o w i n g a relativelyminor system d i s t u r b a n c e u n d e r s u c h c o n d i t i o n s .

SUMMARY AND C O N C L U S I O N ST h e g e n e r a l c o n c e p t s associated with a p p l y i n g powersystem stabilizers utilizing s p e ed , f r e qu e n c y , o r poweri n p u t s i g n a l s h a v e been d e s c r i b e d i n t h i s part o f at h r e e - p a r t p a p e r , l a y i n g t h e foundation f o r discussiono f t h e tuning c o n c e p t s a n d p r ac t ic a l a sp ec ts o f s t a b i -

l i z e r application which f o l l o w . T h e characteristics o ft h e " p l a n t " through which t h e power system s t a b i l i z e rmu st op era te i . e . , t h e g e n e r a t o r , e x c i t e r , a n d powers y s t e m , denoted G E P ( s ) , a r e s u c h t h a t t h e g a i n increaseswith g e n e r a t o r loading a n d a c system s t r e n g t h . A l s o ,t h e phase l a g o f t h e " p l a n t " increases a s t h e a c systembecomes s t r o n g e r . F o r s t a b i l i z e r s utilizing s p e e d a n dpower a s i n p u t s , t h es e c h ar ac t er i st ic s i m p l y t h a t t h estabilizer must b e tuned f o r t h e case o f t h e strong a cs y s t e m , a n d t ha t t h e performance will d e c r e a s e a s t h e a csystem becomes w e a k e r . Utilizing a c b u s frequency a s a ni n p u t , h o w e v e r , r e s u l t s i n c h a r a c t e r i s t i c s which dimin-i s h t h e effect o f a c transmission s t r e n g t h on t h e stabi-l i z e r p e r f o r m a n c e , a n d thereby allow tuning f o r t h e weaka c s y s t e m c o n d i t i o n . I n a d d i t i o n , a frequency i n p u tstabilizer i s l e s s sensitive t o m o d e s o f oscillationassociated with i n d i v i d u a l u n i t s , a n d more sensitive t opower s w i n g s between areas than either t h e s p e e d orpower i n p u t stabilizers.

Stabilizers utilizing power a s an input ca n b edesigned with a non-minimum p h a s e characteristic, effec-tively utilizing l a g networks rather t h a n t h e l e a dnetworks associated wi t h t h e minimum p h a s e characteris-t i c utilized with s p e e d o r frequency i n p u t . This allowsdamping performance e q u i v a l e n t t o a s p e e d i n p u t s t a b i -l i z e r but with l o w e r gain a t h i g h f r e q u e n c i e s . I nparticular, t h e interaction with t o r s i o n a l m o d e s o fs h a f t vibration e x h i b i t s a d e c l i n i n g g a i n v e r sus f r e -quency c h a r a c t e r i s t i c a s o p p o s e d t o t h e i n c r e a s i n g g a i nw it h f re qu en cy a s s o c i a t e d with s p e e d o r frequency i n p u ts t a b i l i z e r s . While t h i s non-minimum phase characteris-t i c i s l e s s s e n s i t i v e t o high f re qu en cy n oi se a n d t o r -sional i n t e r a c t i o n , i t i s sensitive t o l o w frequencyp h e n o m e n a , s u c h a s c h a n g e s i n mechanical p o w e r , a n dpower i n p u t s ta bi li ze rs mu st t he re fo re b e provided withcompensation f o r t h e s e v a r i a t i o n s . I f t h i s i s d o n e withwashout s t a g e s , phase l e a d i s introduced a t l o w f r e -quencies which i s detrimental t o stability o f interareaswings.

REFERENCES1 ) F . R . S c h l e i f , H . D . Hunkins, G . E . M a r t i n , E . E .

H a t t a n , " E x c i t a t i o n C o n t r o l t o Improve PowerlineS t a b i l i t y " , I E E E T r a n s . V o l . P A S - 8 7 , J u n e, 1 9 6 8 ,p p . 1 4 2 6 - 1 4 3 4 .2 ) C . C o n c o r d i a , F . P . d e M e l l o , " C o n c e p t s o f Synchro-nous Machine Stability a s Affected b y ExcitationC o n t r o l " , I E E E T r a n s . V o l. P A S - 8 8 , April 1 9 6 9 , p p .3 1 6 - 3 2 9 .3 ) R . A . L a w s o n , D . A . S w a n n , G . F . W r i g h t , , " M i n i m i z a t i o no f Power System Stabilizer T o r s i o n a l Interaction o nLarge T u r b i n e - G e n e r a t o r " , I E E E T r a n s . , V o l . P A S - 9 7 ,J a n u a r y / F e b r u a r y 1 9 7 8 , p p . 1 8 3 - 1 9 0 .4 ) P . K u n d u r , D . C . L e e , H . M . Zein E l - D i n , " P o w e rSystem Stabilizers f o r Thermal Units: A n a l y t i c a lT e c h n i q u e s a n d On-Site V a l i d a t i o n " , Paper F 8 0 - 2 2 7 - 9

p r e s e n t e d a t I E E E P E S Winter Meeting, N ew Y o r k ,February 1 9 8 0 .5 ) J . P . B a y n e , D . C . L e e , W . W a t s o n , " A Power S y s t e mS t a b i l i z e r S t a b i l i z i n g S i g n a l f o r T h e r m a l UnitsBased on D e r i v a t i o n o f Accelerating P o w e r " , I E E ETrans. V o l P A S - 9 6 , N o v e m b e r / D e c e m b e r 1 9 7 7 , p p .1 7 7 7 - 1 7 8 3 .6 ) F . P . d e M e l l o , L . N . M a n n e t t , J . M . U n d r i l l , " P r a c -t i c a l A p p r o a c h e s t o S u p p l e m e n t a r y S t a b i l i z i n g fromA c c e l e r a t i n g P o w e r " , I E E E Trans. Vol P A S - 9 7 , S e p -t e m b e r / O c t o b e r 1 9 7 8 , p p . 1 5 1 5 - 1 5 2 2 .


6/30

A P P E NDIX AI n i t i a l Impact o f PSS UponR ot o r O s ci l l at i on Eigenvalues

APPENDIX BA n a l y s i s o f Power S y s t e m Stabilizer A p p l i c a t i o nWith a n Arbitrary I n p u t S i g n a l

I t i s s h o w n h e r e t h a t t h e e i g e n v a l u e nominallya s s o c i a t e d with t h e r o t o r o s c i l l a t i o n m o d e o f interesti n i t i a l l y m o v e s a n amount proportional t o stabilizergain a s t h e g a i n i s i n c r e a s e d from z e r o , a n d i n a direc-tion determined by t h e n e t phase o f t he s t a b i l i z e r ,excitation s y s t e m , g e n e r a t o r a n d power s y s t e m . T h i sc o n c e p t i s u s e f u l i n u n d e r s t a n d i n g t h e relationshipbetween t h e p h a s e c h a r a c t e r i s t i c s o f t h e s t a b i l i z e r pathf r o m s p e e d t o t o r q u e a n d performance o f t h e power systemwith t h e stabilizer l o o p c l o s e d . W e s t a r t with t h ee i g e n v a l u e expressed approximately a s per equation A l ,w h e r e i t i s assumed t h a t t h e damping i s l i g h t .Dv- + J M = i+ j w ( A l )- M i- iA d d i n g t h e power system s t a b i l i z e r r e s u l t s i n effectived a m p i n g a n d s y n c h r o n i z i n g component c o n t r i b u t i o n s :

a TP ( s ) = e p d u e t o a d d i n g P S Sa W G

= D p S S ( w i ) - i W . K p S S ( W i )1

( A 2 a )

( A 2 b )

I n c l u d i n g t h i s c o n t r i b u t i o n i n t h e expression f o r t h ee i g e n v a l u e r e s u l t s i n equation A 3 , a n d A 4 r e p r e s e n t s t h ec h a n g e expressed i n t e r m s o f t h e d a m p i n g a n d synchroniz-i n g c o n t r i b u t i o n s o f t h e s t a b i l i z e r .+ a = - ( D + D P S S ) + j w b ( K o + K P S S )1 L I 2 1 4 1 4

A u . = 1 D p 5 5 ( W j )AWi 2M KpSS(Wi) W

When u s i n g a n i n p u t s i g n a l o t h e r t h a n s h a f t s p e e df o r t h e power system s t a b i l i z e r , t w o a d d i t i o n a l f a c t o r smust b e c o n s i d e r e d i n t h e a n a l y s i s . T h e f i r s t i s t h e" i n p u t s i g n a l s e n s i t i v i t y f a c t o r " which r e p r e s e n t s t h etransfer function from s h a f t s p e e d t o t h e s i g n a l b e i n gused a s a s t a b i l i z e r i n p u t . T h i s transfer function i si n s e r i e s with t h e s t a b i l i z e r l o o p a n d t h e r e b y d i r e c t l yi n f l u e n c e s t h e p e r f o r m a n c e . T h e s e c o n d f a c t o r i s t h e" s t a b i l i z e r i n n e r l o o p " w h i c h i s f o r m e d b y virtue o f t h es t a b i l i z e r o u t p u t h a v i n g a n e f f e c t upon i t s i n p u t s i g n a li n d e p e n d e n t o f s h a f t s p e e d , via f l u x c h a n g e s i n t h eg e n e r a t o r . T h i s i s i l l u s t r a t e d i n F i g u r e B 1 which s h o w st h e s t a b i l i z e r p a t h f r o m s h a f t s p e e d t o e l e c t r i c a ltorque with a n a r b i t r a r y i n p u t s i g n a l X . T h i s can b ee x p r e s s e d m a t h e m a t i c a l l y a s :

A X = S X ( S ) A W G + F B X ( s ) A E p s s= ) A w G + F B x ( s ) PSX(s) A X

X =wE Gps sS X ( S ) =

( B l a )( B l b )

s t a b i l i z e r i n p u t s i g n a lg e n e r a t o r shaft s p e e dstabilizer o u t p u t s i g n a la x GM ,

= " i n p u t s i g n a l s e n s i t i v i t y f a c t o r "FB ( s ) = a xB E p s s

= " i n p u t s i g n a l f e e d b a c k f a c t o r "T h e r e s u l t i n g c o n t r i b u t i o n o f t h e s t a b i l i z e r p a t h i s( A 3 ) t h e r e f o r e :

( A 4 a )( A 4 b )

P ( S ) = A \ T e p / G= S x ( s ) P S S x ( s ) G E P ( s ) / [ l - F B x ( s ) P S S x ( s ) ] ( B 2 )From equation A 4 , i t i s s e e n t h a t t h e c h a n g e i n t h e r e a la n d imaginary c o m p o n e n t s o f t h e e i g e n v a l u e a r e propor-t i o n a l t o t h e r e a l a n d imaginary c o m p o n e n t s o f t h e 'c o n t r i b u t i o n o f t h e s t a b i l i z e r p a t h , a n d h e n c e t h ec h a n g e i n e i g e n v a l u e i s r e l a t e d t o t h i s s t a b i l i z e rc o n t r i b u t i o n :

A. = - P ( j w2 i F P i w ~ ( A 5 )Note t ha t a zero phase characteristic f o r P j w . )will cause t h e r e a l p a r t o f t h e eigenvalue t o increasei n a negative d i r e c t i o n , w h i c h i m p l i e s positive d a m p i n g .Pha s e l a g i n P ( j w . ) r e sult s i n a positive synchronizingcomponent ( a s per e q u a t i o n A 2 c ) and an increase i n f r e -

quency. C o n v e r s e l y , phase l e a d yields a negative syn-c h r o n i z i n g contribution a n d a decrease i n f r e q u e n c y .

F I G U R E B I S T A B I L I Z E R L O O P S W IT H A RB IT RA RY I N P U T XT h e impact of t h e s t a b i l i z e r i n n e r l o o p , w h i c h a p p e a r sin the de n omina t or t e r m of equation B 2 , i s a f u n c t i o n o fthe i n p u t signal ch o se n. F o r t h e case o f s h a f t speedi n p u t , the fee db a c k t e r m is zero a n d h e n c e t h e i n n e rl o o p has no effect. A s an opposite e x t r e m e , utilizingterminal voltage for s t a b i l i z a t i o n w o u l d r e s u l t i n avery l a r g e feedback t e r m a n d a l ow sensitivity factor.Fo r ac bus f r e q u e n c y as a s t a b i l i z e r i n p u t , t h e f e e d b a c kt e r m i s s u f f i c i e n t l y sma ll a n d , a t l e a s t fo r analyzingthe i n p a c t of the s t a b i l i z e r as its gain is i n c r e a s e dfrom z e r o , the fee db a c k t e r m can b e n e g l e c t e d , r e s u l t i n gin the approximation of equation B 3 .

3 0 2 2

A W G


7/30

3 0 2 3P ( s ) ~ P ' ( s ) = S x ( s ) P S S x ( s ) G E P ( s ) ( C 3 )

where P ' ( s ) r e p r e s e n t s t h e forward p a r t o f t h e s t a b i -l i z e r p a t h .I t should b e noted that t h e i n p u t s i g n a l sensitiv-i t y factor i s a function o f t h e v o l t a g e r e g u l a t o r a swell a s power system conditions such as transmissionl i n e s t r e n g t h . T h i s f o l l o w s since t h e stabilizer i n p u tsignal i s sensitive t o c h a n g e s i n t h e v o l t a g e errors i g n a l ( i n t h e same manne r i t i s affected b y stabilizero u t p u t s i g n a l ) , a n d t h e v o l t a g e i s affected b y shafts p e e d . T h i s can b e e x p r e s s e d m a t h e m a t i c a l l y as f o l l o w s :A e = A E +tE -A E ( B 4 )t r e f ps s tB8t/BWG = - S ( B 5 )S x ( s ) = S ( s ) - S E M ( s ) F B ( s ) ( B 6 )

where S M , S E M = i n p u t s i g n a l s e n s i t i v i t y factors f o r Xa n d E t , r e s p e c t i v e l y , under manual control ( i . e . , volt-a g e r e g u l a t o r g a i n = 0 ) .T h e i n p u t s i g n a l feedback factor ( F B x ( s ) ) r e p r e -s e n t s t h e effect o f a c h a n g e i n stabilizer o u t p u t upont h e i n p u t s i g n a l via f l u x c h a n g e s i n t h e g e n e r a t o r , andh e n c e i n c l u d e s t h e c l o s e d - l o o p v o l t a g e r e g u l a t o r charac-t e r i s t i c s .To a n a l y z e t h e c l o s e d - l o o p p e r f o r m a n c e o f t h e powers y s t e m with a s t a b i l i z e r , i t i s useful t o consider t h es t a b i l i z e r a s c l o s i n g a l o o p around a s y s t e m w h i c hi n c l u d e s both t h e i n p u t s i g n a l feedback factor and t h er o t o r . Based on F i g u r e B l , t h i s c a n be e x p r e s s e d a s :G x ( s ) = F B x ( s ) + G E P ( s ) R ( s ) S X ( S )

w h e r eR ( s ) = - s / ( M S + K l e ( s ) w b )

" - 2 2=- S / M ( s _ 2 a . s + W . )3 I L

( B 7 )

( B 8 a )( B 8 b )

T h e o p e n - l o o p transfer function o f t h e system i st h e nG H X ( s ) = G x ( s ) P S S X ( S ) ( B 9 a )

= G x C s ) P ' ( s ) / S x s ) G E P ( s ) ( B 9 b )For s p e e d i n p u t , F B i s z e r o a n d S i s u n i t y ,w wh e n c e :G w ( s ) = G E P ( s ) R ( s ) ( B 1 O )

For power i n p u t , F B can b e d e t e r m i n e d b y r e a l i z i n g t h a ti t r e p r e s e n t s t he P e f f e c t o f s t a b i l i z e r output s i g n a lupon power i n d e p e n d e n t o f s h a f t m o t i o n . H e n c e power a n dt o r q u e are e q u i v a l e n t a n d t h e f e e d b a c k factor b e c o m e s :B P a TF B ( s ) e = = G E P ( s ) ( B l l )p B E -pss E p s s

U s i n g t h e power i n p u t s i g n a l s e n s i t i v i t y f a c t oequation 5 , t o g e t h e r with e q u a t i o n s B 7 , B 8 , a nresults i n :G ( s ) = s G E P ( s ) R ( s )

i r f r o m

Note t h a t a n a d d i t i o n a l z e r o a t t h e o r i g i n i s i n t r o d u c e di n t o t h e l o o p when u s i n g power as o p p o s e d t o s p e e di n p u t .A P P E N D I X C

N o n - M i n i m u m P h a s e S t a b i l i z e r sT h e o b j e c t i v e o f a power s y s t e m s t a b i l i z e r i s t oa d d d a m p i n g to r o t o r o s c i l l a t i o n s , which i s a c c o m p l i s h e db y m o d u l a t i n g v o l t a g e r e g u l a t o r setpoint s u c h t h a tr e s u l t i n g t o r q u e c h a n g e s ar e i n p h a s e with s h a f t s p e e d .I n conventional s t a b i l i z e r s l e a d networks a r e p r o v i d e dto c o m p e n s a t e f o r t h e p h a s e l a g s associated with t h eg e n e r a t o r a n d excitation system ( G E P ( s ) o f F i g u r e 1 ) ,r e s u l t i n g i n a " m i n i m u m p h a s e " control l o o p .A n a l t e r n a t i v e a p p r o a c h i s t o a d d a p p r o p r i a t e l a gnetworks s u c h t h a t t h e ne t p h a s e i s - 3 6 0 at t h e f r e -quency o f concern, r e s u l t i n g i n a " n o n - m i n i m u m p h a s e "c o n t r o l l o o p . F i g u r e C l g i v e s a n e x a m p l e where t h e 3 6 0 0p h a s e l a g i s a c c o m p l i s h e d b y :P ( s ) = G E P ( s ) P S S w ( s ) = - K w / s ( C 1 )

I n t h i s f i g u r e , a l l other c o n t r i b u t i o n s t o t o r q u e h a v ebeen l u m p e d i n t o t h e e f f e c t i v e s y n c h r o n i z i n g constantK ( s ) d e f i n e d i n e q u a t i - o n 6 .

A 8A T n

F I G U R E C l N O N - M I N I M U M P H A S E S P E E D I N P U T S T A B I L I Z E RWith a s p e e d input s t a b i l i z e r yielding a pured o u b l e - i n t e g r a l c h a r a c t e r i s t i c , a s pe r e qu at io n C l , a ni n s t a b i l i t y w i l l r e s u l t a s soon a s t h e gain i s increasedf r o m zero. T h i s can b e understood b y examining t h en a tur e o f t h e control l o o p . From Figure C l , t h e forwardpath ca n b e defined a s R ( s ) a s per equation B 8 , with t h ef e e d b a c k p a t h P ( s ) a s p er e qu at io n C l . T h e system h a s ane t o f one open-loop pole a t t he o r i g i n , since o n e o ft h e p o l e s o f t h e stabilizer path i s c a n c e l l e d by t h ezero o f R ( s ) . I n a d d i t i o n , t h e r e e x i s t s a c o m p l e x pairassociated with rotor o s c i l l a t i o n s . Since t h e open l o o pgain i s inverted i n s i g n , when K i s i n c r e a s e d t h ec o m p l e x p o l e s will move further into t h e l e f t - h a l f planea n d b e c o m e more s t a b l e , but t h e p o l e a t t h e origin willmove towards t h e right-half plane and become unstablei m m e d i a t e l y . T h i s i s illustrated i n Figure C 2 a .T h u s , t o s u c c e s s f u l l y i m p l e m e n t a n o n - m i n i m u m phases t a b i l i z e r with speed input i t i s n e c e s s a r y t o u s e at i m e constant a t a l o w frequency r a t h e r t h a n a purei n t e g r a l c h a r a c t e r i s t i c , i . e . :P ( s ) = G E P ( s ) P S S ( s ) = - K / s ( w + s )w ~ w s ( C 3 )

. d R i 1 , T h e p o l e a t t h e origin h a s now been shifted into t h el e f t - h a l f plane b y w . T h i s pole will s t i l l m o v et o w a r d s t h e r i g h t - h a l t plane a n d e v e n t u a l l y become( B 1 2 ) u n s t a b l e , but will p e r m i t some d a m p i n g c o n t r i b u t i o n t orotor o s c i l l a t i o n s a s s h o w n i n Figure C 2 b .


8/30

3 0 2 44

( a )

- K wS 2

K i n s t : Q

- w s

-K W( s ) S

K i n s t >0

( b )F I G U R E C 2 R O O T L O C t W I T H N O N - M I N I M U M - P H A S E S P E E D I N P U T S T A B I L I Z E RT o a n a l y z e t h e characteristics o f t h e power i n p u ts t a b i l i z e r , t h e c o n c e p t s developed i n Appendix B f o r a narbitrary i n p u t s i g n a l a r e u t i l i z e d . F i r s t , a d e v e l o p -me nt p ar al le l t o t h a t used above f o r s p e e d i n p u t i sf o l l o w e d . Utilizing equations B 2 , B l l , a n d B 3 yieldst h e following f o r t he s t a b i l i z e r path with power i n p u t :

P ( s ) = S p ( s ) P S S p ( s ) G E P ( s ) / [ l - G E P ( s ) P S S p ( s ) ] ( C 4 a )= P ' ( s ) / [ l - P ' ( s ) / S p ( s ) ] ( C 4 b )

T o p r o v i d e a n o n - m i n i m u m phase c h a r a c t e r i s t i c f o r l o wv a l u e s o f stabilizer g a i n , t h e forward part o f t h es t a b i l i z e r p a t h , P ( s ) , i s s e t t o t h e double-integralc h a r a c t e r i s t i c o f equation C l , with S ( s ) f r o m e q u a t i o n6 c assuming w = . Including t h e e f f e c t o f t h e s t a b i -l i z e r i n n e r 1 T o o p by s u b s t i t u t i n g i n t o e q u a t i o n C 4 by i e l d s t h e following f o r t h e s t a b i l i z e r c o n t r i b u t i o n :

P ( s ) = - K / s [ ( K / K l e ( s ) t u ) + s ) ] ( C 5 )Note t h a t including t h e stabilizer i n n e r l o o p h a s r e -s u l t e d i n a s t a b i l i z e r c o n t r i b u t i o n w h i c h i s actually a ni n t e g r a l p l u s a t i m e constant s i m i l a r t o e q u a t i o n C 3 .U n l i k e t h e breakpoint w introduced i n t o t h e s t a b i l i z e rw i t h s p e e d i n p u t , h o w e v e r , t h e breakpoint i n equation C 5i s i n h e r e n t l y proportional t o t h e g a i n o f t h e s t a b i l i z e rforward p a t h , K , a n d h e n c e s u g ge s t s t h a t t h e pole a s s o -c i a t e d with t h i s b r e a k p o i n t will n o t become unstable a st h e g a i n i s i n c r e a s e d . T h i s c a n b e verified by d e t e r -mining t h e open-loop transfer function a s per equationB 9 b with the i d e a l characteristic o f equation C l f o rP ' ( s ) , equation B 1 2 f o r t h e s y s t e m , a n d equation 6 f o rt h e i n p u t s i g a n l sensitivity f a c t o r :

G H ( s ) = R ( s ) K / w % K l ( s ) ( 1 + s / w t u ) ( C 6 )T h i s i d e a l open-loop transfer f u n c t i o n h a s l e s s than1 8 0 0 o f p h a s e l a g a t l o w f r e q u e n c y , a n d hence t h e n o n -minimum p h a s e c h a r a c t e r i s t i c c a n b e i m p l e m e n t e d withpower input w i t h o u t having t h e l o w frequency i n s t a b i l i t ywhich i s associated with a s p e e d i n p u t non-minimum phases t a b i l i z e r .

LIST OF S Y M B O L SG E P . ( s ) = T / A E PSS = ' p l a n t " t h r o u g h whichs t a n i l i z e r must o p e r a t e .

= g e n e r a t o r electrical t o r q u e .= c ompone n t of T d u e solely to f l u xc h a n g e s caused C y E p 5 5 .= c ompone n t o f T due t o a l l other

sources ( = T - f ) .e e p= stabilizer o u t p u t s i g n a l .

w t G6E lqE f dEt

EfE r e f tP S S X ( S )E X C ( s )W u bK ST wT1T 2 T 3 s T 4F I L T ( s )x -e

G E Np ep e oK 1 ( s )w- p oA .1

W .

c i .D p 5 5 , K p 5 5D , K0 o 0MF B X ( s )P ( s )G x ( s )R ( s )G H X ( s )

= generator s h a f t s p e e d .= generator s h a f t a n g u l a r d i s p l a c e -- generator i n t e r n a l v o l t a g e .= generator f i e l d v o l t a g e .= generator t e r m i n a l v o l t a g e .= v o l t a g e r e f e r e n c e s i g n a l .

----- error s i g n a l .= s t a b i l i z e r w i t h i n p u t X .= e x c i t a t i o n s y s t e m = A E f d / A & t -

s y s t e m b a s e f r e q u e n c y ( 3 7 7 r a d / s e c @6 0 H z ) .s t a b i l i z e r g a i n .washout time c o n s t a n t .l e a d / l a g t i m e c o n s t a n t s .f i l t e r i n g i n s t a b i l i z e r .-r e a c t a n c e e x t e r n a l t o g e n e r a t o r .e f f e c t i v e r e a c t a n c e o f g e n e r a t o r .e l e c t r i c a l p o w e r .s t e a d y - s t a t e power l e v e l .e f f e c t i v e s y n c h r o n i z i n g c o e f f i c i e n t ,A T / 3 6 , i n c l u d i n g a l l s o u r c e s otherthan s t a b i l i z e r p a t h .w b K l e ( S ) / P e o t he i g e n v a l u e a s s o c i a t e d with i- m o d eo f rotor o s c i l l a t i o n .

t hr o t o r o s c i l l a t i o n f r e q u e n c y o f i -m o d e ." ----- d a m p i n g ------

e f f e c t i v e d a m p i n g a n d s y n c h r o n i z i n gc o e f f i c i e n t s r e s u l t i n g from s t a b i -l i z e r p a t h .d a m p i n g a n d s y n c h r o n i z i n g c o e f f i -c i e n t s .i n e r t i a o f t u r b i n e - g e n e r a t o r s h a f t .i n p u t s i g n a l f e e d b a c k factor =8 X / 8 E p 5 5 .A TT p l / w G n e g l e c t i n g F B X ( s ) .AX/ E P S S i n c l u d i n g r o t o r m o t i o n .

= W G / A Tep= open-loop t r a n s f e r functionstabilizer c o n t r o l l o o p .

of

Th e following are d e f i n e d i n [ 2 ] :K 1 = Te / m 6K 5 = 3 E t / 3 6

K =3 T /WE2 e qK =a E / a E E6 t q

K3 = Z E / R E f d

= mechanical torque. F o r C o m b i n e d d i s c u s s i o n see page 3 0 2 4

T eT e pT e o

E P S ST m


9/30

IEEE T r a n s a c t i o n s o n P o w e r A p p a r a t u s a n d S y s t e m s , V o l . P A S - 1 0 0 , N o . 6 J u n e 1 9 8 1APPLYING POWER SYSTEM STABILIZERS

P A R T I I : P E R F O R M A N C E O B J E C T I V E S AND TUNING CONCEPTSE . V . Larsen ( M e m b e r ) D . A . Swann ( M e m b e r )General Electric C o m p a n y , S c h e n e c t a d y , New York

A B S T R A C T

T h i s p a r t of a three-part p a p e r deals fir s t w i t ht h e performance objectives of p o w e r system s t a b i l i z e r sin te rm s o f the type of oscillations f or w h i c h they a r eintended to provide d a m p i n g , th e operating conditionsf o r w h i c h t h e requirement for s t a b i l i z a t i o n i s g r e a t e s t ,t h e n e e d t o a c c o m m o d a t e multiple modes of o s c i l l a t i o n ,and t h e significance of interplant modes of oscillation.I t n ext t r e a t s stabilizer t u n i n g . G e n e r a l t u n i n g g u i d e -l i n e s are developed as w e l l a s v a r i a t i o n s required f o rdifferent i n p u t s i g n a l s . Th e operating c o n d i t i o n s underw h i c h each type of s t a b i l i z e r s h o u l d be t u n e d are iden-t i f i e d . The r e l a t i o n s h i p b e t w e e n phase compensationtuning a n d root locus analysis i s presented. F i n a l l y ,t h e r e l a t i v e performance characteristics of th e threetypes of stabilizers are examined for b o t h sm a ll p e r -turbations and l a r g e disturbances.

I N T R O D U C T I O NT u n i n g o f s u p p l e m e n t a r y excitation c o n t r o l s f o rs t a b i l i z i n g system modes o f o s c i l l a t i o n h a s b e e n thes u b j e c t of mu c h r e s e a r c h d u r i n g t h e past 1 0 to 1 5 y e a r s .Two ba sic t u n i n g t e c h n i q u e s h a v e b e e n s u c c e s s f u l l yutil ize d wit h p o w e r system s t a b i l i z e r a p p l i c a t i o n s :p h a s e c o m p e n s a t i o n and root locus. P h a s e compensationc o n s i s t s o f a d j u s t i n g th e s t a b i l i z e r t o c o m p e n s a t e f o rt h e p h a s e l a g s t h r o u g h the g e n e r a t o r , e x c i t a t i o n s y s t e m ,and power system s u c h t h a t the s t a b i l i z e r path p r o v i d e st o r q u e c h a n g e s w h i c h are in p h a s e w i t h s p e e d c h a n g e s[ 1 , 2 , 3 , 4 , 5 , 6 ] . T his i s the mo st s t r a i g h t f o r w a r d ap-p r o a c h , e a s i l y understood and i m p l e m e n t e d i n t h e f i e l d ,and t h e m o s t w i d e l y u s e d . S y n t h e s i s by r o o t lo cusinvolves s h i f t i n g th e e i g e n v a l u e s a s s o c i a t e d w i t h th e

power sy stem modes of o s c i l l a t i o n b y a d j u s t i n g th estabilizer pole a nd zero l o c a t i o n s i n t h e s - p l a n e [ 7 , 8 ] .This approach g i v e s additional i n s i g h t t o p e r f o r m a n c e b yw o r k i n g d i r e c t l y with the c l o s e d - l o o p c h a r a c t e r i s t i c s oft h e s y s t e m , a s o p p o s e d to th e o p e n - l o o p n a t u r e o f t hep h a s e compensation t e c h n i q u e , b u t i s more c o m p l i c a t e d t oa p p l y , p a r t i c u l a r l y i n t he f i e ld.

I n d e p e n d e n t o f th e t e c h n i q u e u t i l i z e d i n t u n i n gstabilizer e q u i p m e n t , it i s n e c e s s a r y to r e c o g n i z e t h en o n l i n e a r n a t u r e of p o w e r s y s t e m s a n d t h a t t h e o b j e c t i v eo f a d d i n g p o w e r s y s t e m s t a b i l i z e r s is t o e x t e n d powertransfer limits b y s t a b i l i z i n g s y s t e m o s c i l l a t i o n s ;a d d i n g d a m p i n g i s n o t an e n d i n i t s e l f , b u t a means t oe x t e n d i n g power t r a n s f e r limits. T h i s part o f a t h r e e -part paper a d d r e s s e s the p e r f o r m a n c e c h a r a c t e r i s t i c s ofpower s y s t e m s ta bi l iz er s w i th r e s p e c t t o e x t e n d i n g p o w e rtransfer s t a b i l i t y lim its f o r b o t h r e m o t e generation a n dintertie s itua tio n s. B o t h s m a l l a nd l a r g e d i s t u r b a n c easpects o f p e r f o r m a n c e are i n c l u d e d , r e s u l t i n g i n adefinition of d e s i r e d s t a b i l i z e r performance t o ensure a

8 0 SM 5 5 9 - 5 A paperrecommended and approvedbytheI E E E P o w e r G e n e r a t i o n C o m m i t t e e of the IEEE PowerE n g i n e e r i n g S o c i e t y f or presentation at the IEEE PESS u m m e r M I e e t i n g , M i n n e a p o l i s , M i n n e s o t a , July 13-18,1980.Manuscript s u b m i t t e d M a r c h 1 4 , 1980; m a d ea v a i l a b l e f o r p ri nt in g M a y 7 , 1980.

r o b u s t d e s i g n m e e t i n g t h e system r e q u i r e m e n t s . I na d d i t i o n , a r e l a t i o n s h i p i s established between d e s i r e dp e r f o r m a n c e a n d t h e p h a s e compensation c h a r a c t e r i s t i c s ,l a y i n g t h e g r o u n d w o r k f o r a f a i r l y straightforward f i e l dt u n i n g p r o c e d u r e o u t l i n e d i n Part I I I .P E R F O R M A N C E O B J E C T I V E S

" D y n a m i c " or " S t e a d y - S t a t e " S t a b i l i t y L i m i t sA p p l y i n g power s y s t e m stabilizers can extend powertransfer stability l i m i t s which are characterized byl i g h t l y d a m p e d or spo n t a n e o u s l y g r o w i n g oscillations i nt h e 0 . 2 t o 2 . 5 H z frequency range. This i s a c c o m p l i s h e dv ia excitation control to contribute damping t o t h esystem m o d e s o f oscillation. C o n s e q u e n t l y , i t is t h es t a b i l i z e r ' s a b i l i t y t o e n h a n c e damping under the l e a s ts t a b l e c o n d i t i o n s , i . e . , t h e " p e r f o r m a n c e c o n d i t i o n s " ,which i s important. Additional damping is primarilyr e q u i r e d under conditions of w e a k t r a n s m i s s i o n and heavyl o a d a s occurs, for e x a m p l e , when a t t e m p t i n g t o t r a n s m i t

power over l o n g t r a n s m i s s i o n l i n e s from r e m o t e g e n e r a t -i n g p l a n t s or over r e l a t i v e l y w e a k t i e s b e t w e e n s y s t e m s .C o n t i n g e n c i e s , such as l i n e o u t a g e s or fu el s h o r t a g e s ,often precipitate s u c h conditions. Hence, s y s t e m s whichnormally h a v e adequate damping can o f t e n b e n e f i t f r o ms t a b i l i z e r s d u r i n g s u c h a b n o r m a l c o n d i t i o n s .I t i s i m p o r t a n t to realize t h a t t h e stabilizer i sintended to p ro v id e d am pi n g for small e x c u r s i o n s about as t e a d y - s t a t e operating point, and n o t to enhance t r a n -s i e n t s t a b i l i t y , i . e . , t h e ability to recover from a

severe d i s t u r b a n c e . I n f a c t , t h e stabilizer will oftenhave a d e l e t e r i o u s effect on t r a n s i e n t stability byattempting to pull the g e n e r a t o r field o u t of c e i l i n gt o o early i n response t o a f a u l t . T he s ta bi li ze r o u t p u ti s g e n e r a l l y l i m i t e d to p r e v e n t s er io us i mp ac t on t r a n -s i e n t s t a b i l i t y , but stabilizer t un in g a ls o h a s a s i g -nificant impact upon s y s t e m performance following al a r g e d i s t u r b a n c e , as will b e d i s c u s s e d .S ys t em M od es o f Oscillation

The power s y s t e m oscillations of concern t o s t a -bility occur i n the 0. 2 t o 2. 5 Hz f r e q u e n c y range.These result when t h e r o t o r s of machines, behaving asrigid b o d i e s , oscillate with r e s p e c t t o one a n o t h e rusing t h e electrical t r a n s m i s s i o n path between them t oexchange energy. There are many different modes inw h i c h such oscillations may occur, often s i m u l t a n e o u s l y .Th e first widespread use o f power s y s t e m stabi-l i z e r s occurred when t h e U . S . W es t Coas t utilities

d i s c o v e r e d that they were unable t o fully load their 500k V t r a n s m i s s i o n l i n e s connecting t h e Pacific Northwestand Southwest because o f an oscillatory instability [ 1 ] .Troublesome oscillations r e s u l t e d as a consequence oft h e a g g r e g a t e o f u n i t s a t one end of the i n t e r t i e oscil-l a t i n g against t h e a g g r e g a t e of units a t t h e o t h e r e n d .T h i s has b ec ome kn ow n a s an intertie or interarea modeo f o s c i l l a t i o n , and h a s been experienced in severalsy stems [ 1 , 9 , 1 0 , 1 1 ] . T h e natural frequency of oscilla-tion o f intertie modes i s typically in the range of 0.20t o 0 . 5 H z .The use of power s y s t e m s t a b i l i z e r s was extended t oprovide damping f o r o s c i l l a t i o n s which occur when r e m o t egenerating units are connected to a relatively l a r g e

power s y s t e m through w e a k , essentially radial t r a n s m i s -sion l i n e s [ 1 2 , 1 3 ] . T his has b e c o m e k n o w n as a l o c a l

3 0 2 5


10/30

3 0 2 6mode o f oscillation a n d i t s natural frequency i s t y p i -c a l l y i n t h e r a n g e o f 0 . 8 t o 1 . 8 H z .

Between t h e frequency extremes o f t h e intertie a n dl o c a l modes e xi st o t h e r modes commonly encountered i nweakly connected s y s t e m s [ 1 4 ] . These intrasystem modesr e s u l t f r o m o s c i l l a t i o n s between i n d i v i d u a l u n i t s withina s y s t e m a n d t e n d t o behave s i m i l a r t o l o c a l m o d e s i nt h a t a l a r g e portion o f t h e power oscillation i s t y p i -c a l l y e xp er ie nc ed b y a f e w u n i t s . T h e s e modes w i l l betreated a s l o c a l modes i n t h e discussion which f o l l o w s .F i n a l l y , i t s h o u l d b e mentioned t h a t i f o s c i l l a -t i o n s occur b e t w e e n units i n t h e s a m e plant i t i s aconsequence o f t h e i r controls interacting r a t h e r t h a npower transfer s t a b i l i t y l i m i t s . I t i s generally u n d e -s i r a b l e f o r a s t a b i l i z e r t o r e s p o n d t o t h e s e intraplanto s c i ll a t i o n s , t y p i c a l ly ranging i n frequency f r o m 1 . 5 t o2 . 5 H z , a s t h i s detracts f r o m i t s ability t o e n h a n c et r a n s f e r l i m i t s f r o m t h e power p l a n t . S o m e utilitiesh a v e u s e d a v e r a g e s p e e d derived f r o m multiple u n i t s i n as i n g l e plant a s a n i n p u t t o a l l stabilizers i n t h ep l a n t , t h e r e b y preventing t h e s t a b i l i z e r s f r o m r e s p o n d -i n g t o i nt r ap la nt o sc il l at io ns [ 1 5 ] , a n d summation o fpower h a s a l s o b e e n s u g g e s t e d . A s described i n P a r t I ,a c b u s frequency i s inherently l e s s s e n s i t i v e t o t h e s eintraplant m o d e s than s p e e d o r power i n p u t .Experience suggests t h a t i t i s n o t unusual f o r ag en er at in g u ni t t o participate i n both l o c a l a n d i n t e r -t i e modes o f o s c i l l a t i o n . Power system s t a b i l i z e r s mustt h e r e f o r e b e a b l e t o a c c o m m o d a t e both m o d e s . Since as i n g l e unit o r power plant i s d o m i n a n t i n l o c a l m o d e s ,i t s stabilizer c a n h a v e a very l a r g e i m p a c t o n dampingt h e oscillation. B y c o n t r a s t , a s i n g l e unit experienceso n l y a portion o f t h e t o t a l magnitude o f power o s c i l l a -tion i n t h e intertie m o d e . T h e r e f o r e , a power systems t a b i l i z e r a p p l i e d t o a s i n g l e unit c a n o n l y contributet o t h e d a m p i n g o f a n i n t e r t i e m o d e i n p r o p o r t i o n to t h epower g e n e r a t i o n c a p a c i t y o f t h e u n i t relative t o t h et o t a l capacity o f t h e a r e a o f w h i c h i t i s a p a r t . A s ac o n s e q u e n c e , a stabilizer should b e d e s i g n e d t o p r o v i d ea d e q u a t e l o c a l m o d e d a m p i n g under a l l o p e r a t i n g condi-t i o n s , with p a r t i c u l a r attention to conditions o f h e a v yl o a d a n d w e a k t r a n s m i s s i o n , a n d s i m u l t a n e o u s l y to p r o -vide a h i g h contribution t o d a m p i n g o f i n t e r t i e m o d e s .These c r i t e r i a e n s u r e good p e r f o r m a n c e for a wide r a n g eo f power system c o n t i n g e n c i e s .

T U NI N G C O NC E PT SStabilizers m u s t b e t u n e d to p r o v i d e t h e d e s i r e dsystem performance u n d e r t h e c on di ti on w hi ch r e q u i r e ss t a b i l i z a t i o n , t y p i c a l l y weak s y s t e m s with h e a v y powert r a n s f e r , while at t h e same time b e i n g robust i n t h a tundesirable i n t e r a c t i o n s a re avoided f o r a l l s y s t e mc o n d i t i o n s . A s described i n P a r t I , t h e p l a n t t h r o u g hw h i c h t h e stabilizer mu st operate c o n s i s t s o f t h e g e n e r -a t o r , e x c i t e r , a n d power s y s t e m :G E P ( s ) = A T e p / A E p s ( 1 )

w h e r e G E P ( s ) = th e plant through w h i c h t h e s t a b i -lizer m u s t o p e r a t e .T = component of e l e c t r i c a l torque d u eeP solely to s t a b i l i z e r p a t h .

E p s s = s t a b i l i z e r output s i g n a l .T his plant h as the highest g a i n a n d g re at es t ph as e l a gu nde r c on di ti on s o f f ul l l o a d on t h e u n i t a n d t h es t r o n g e s t t r a n s m i s s i o n s y s t e m . T h e s e c o n d i t i o n s t h e r e -fore r e p r e s e n t the l i m i t i n g case f or a c h i e v a b l e g a i nw i t h a s p e e d or power i n p u t s t a b i l i ze r . W i t h ac b u sfrequency as an i n p u t , th e highest l o o p g a i n occurs w i t h

a moderate t o w e a k a c s y s t e m , which fortunately a l s o h a st h e least phase l a g . Hence, t h e " tu ni n g c on di t io n" f o rspeed a n d power input stabilizers i s w i t h f u l l load andt h e strongest t r a n s m i s s i o n system, b u t with a moderatet o weak system f o r frequency i n p u t . T h e p e r f o r m a n c econdition o c c u r s with a weak transmission s ys t e m wh i c hi s different from t h e t un in g c on di ti on f o r speed andpower input s t a b i l i z e r s . Since t h e gain o f t h e plantdecreases a s t h e system becomes w e a k e r , when using s p e e do r power t h e damping contribution f o r t h e strong s ys t e mshould b e maximized s o a s t o ensure b e s t performancewith a w e a k e n e d s y s t e m .

An e x a m p le w i l l b e used t o illustrate t h e combineduse o f p ha se c o mp en s at i on a n d r o o t l o c u s techniques t omeet t h e a b o v e objective f o r a speed input s t a b i l i z e r .T h e example i s o f a large f o s s i l turbine-generator unitoperated i n t o a very s tr on g s ys te m h a v i n g a total o f 2 0 %external r e a c t a n c e ' , including step-up transformer. Thisrepresents a fairly extreme s it ua ti on , b e ca u se o f t h estrong s y s t e m a n d relatively l i g h t i n e r t i a a s s o c i a t e dwith l a r g e f o s s i l u n i t s . A h i gh i n i t i a l r e s p o n s e exci-t a t i o n system i s a s s u m e d , with a transient gain o f 2 0p . u . E f d / p . u . 8 t A s described i n Part I , t h e speedi n p u t s t a b i l i z e r c o n s i s t s o f a washout s t a g e , a doublelead/lag s t a g e , and a filter t o a tt en ua te h ig h frequencycomponents:T T s ( l + s T 1 ) ( l + s T 3 )PSS ( s ) = K 1 + T s ( l + s T 2 ) ( 1 + s T 4 ) F I L T ( s )w s l + T w s ( 1 + s T Q 2 ( l + s T 4 ) ( 2 )

T h e f i l t e r F I L T ( s ) i s represented with a second orderl a g c h a r a c t e r i s t i c with c o m p l e x r o o t s a t -17.5+j16 r a d /s e c . T h i s representation p ro vi de s ph as e l a g equivalentt o t h a t o f t h e t or si on al b an d reject filter [ 1 6 ] u p t oabout 3 . 5 H z . T o simplify illustration o f t h e basicc o n c e p t s , t h e washout t i m e constant i s s e t a t 1 0 s e c -o n d s , thereby h av in g v ir tu al ly n o i m p a c t upon t h e l o c a lm o d e , a n d t h e l e a d / l a g s t a g e s a r e s e t i de nt ic al ly , e ac hh a v i n g a 1 0 : 1 s p r e a d between t h e l e a d and l a g t i m ec o n s t a n t s . A parameter defined a s t h e compensationcenter f r e q u e n c y , i . e . ,

c = 1 / 2 n J T V T 2 = 4 1 0 / 2 r T ( 3 )i s v a r i e d t o show t h e i m p a c t o f d i f f e r e n t stabilizera d j u s t m e n t s .Phase C o m p e n s a t i o n .

Figure 1 s h o w s t h e variation with l e a d / l a g c e n t e rfrequency on t h e compensated p h a s e , i . e . , t h e phase o ft h e c o m p l e t e s t a b i l i z e r path f r o m s p e ed t o t o r q u e :( 4 )P ( j w ) = G E P ( j w ) P S S ( j w ) = P ( w ) / ( w )

Key p o i n t s t o o b s e r v e from t h i s figure a r e t h e phase a tt h e l o c a l mode frequency o f 1 . 6 H z , 4 , a n d t h e f r e -quency a t which t h e p ha s e p as s es through o o , fo9 0 .

A s shown i n A p p e n d i x A of Part I , t h e initialdirection o f e i g e n v a l u e migration a s stabilizer gain i sincreased from zero i s determined b y t h e p h a s e a t t h el o c a l mode f r e q u e n c y . For perfect c o m p e n s a t i o n , i . e . ,L = 0 , pure p o s i t i v e d a m p i n g will b e a p p l i e d a n d t h eLe i g e n v a l u e w i l l move d i r e c t l y into t h e left h a l f p l a n ewith n o c h a n g e i n f r e q u e n c y . I f phase l a g e x i s t s , t h ef r e q u e n c y will increase in proportion t o t h e amount o fd a m p i n g i n c r e a s e , s p e c i f i c a l l yA W L = - t a n L ' A s ( 5 )


11/30

w h e r e W L = l o c a l m o d e f r e q u e n c y ( r a d / s f c )a L = l o c a l mode d e c a y r a t e ( s e c )A i m p l i e s c h a n g e d u e t o s t a b i l i z e rFor 0 = - 4 5 0 , f r e q u e n c y w i l l increase a t t h e same ratea s d a m p i n g , a n d f o r 0 L = - 9 0 , n o c h a n g e i n d a m p i n g w i l lt a k e p l a c e , b u t f r e q u e n c y w i l l i n c r e a s e . T h i s b a s i cc o n c e p t i s very u s e f u l i n u n d e r s t a n d i n g t h e root l o c u s .

0 . 5 0 . 7F R E Q U E N C Y ( H z )F I R E I C O M P E N S A T E D P H A S E FO R V A R I O U S L E A D / L A G C E N T E RF R E Q U E N C I E S ( f c )

Root L o c u sRoot l o c u s p l o t s are s h o w n i n F i g u r e 2 f o r t h r e es e t s o f s t a b i l i z e r l e a d / l a g a d j u s t m e n t s . T h e s e p l o t sr e p r e s e n t t h e migration o f t h e e i g e n v a l u e s a s s t a b i l i z e rg a i n i s i n c r e a s e d from zero to i n f i n i t y . A l t h o u g hs e v e r a l e i g e n v a l u e s exist f o r t he t o t a l s y s t e m , o n l y t h ed o m i n a n t o n e s a s s o c i a t e d w i t h s t a b i l i z e r o p e n - l o o ptransfer f u n c t i o n G H ( s ) ( d e f i n e d i n Part I , A p p e n d i x B )a r e s h o w n . T h e l i g h t l y d a m p e d e i g e n v a lu e n e a r 1 0r a d / s e c r e p r e s e n t s t h e l o c a l m o d e o f o s c i l l a t i o n ( 1 . 6H z ) . T h e e i g e n v a l u e which s t a r t s a t - 1 7 . 5 + j 1 6 r a d / s e cr e s u l t s f r o m t h e f i l t e r e q u i v a l e n t i n t h e s t a b i l i z e r .I n F i g u r e 2 A , t h e d o u b l e p o l e a t - 2 0 sec a n d t h ed o u b l e z e r o a t - 2 s e c r e p r e s e n t t h e l a g a n d l e a db r e a k s , r e s p e c t i v e l y . O n l y t h e u p p e r h a l f o f t h es - p l a n e i s s h o w n ; t h e l o w e r h a l f i s a mirror i m a g e o ft h e u p p e r h a l f .I n r o o t l o c u s t h e o r y , t h e o p e n l o o p s y s t e m p o l e swill m i g r a t e t o t h e o p e n l o o p s y s t e m zeros a s g a i n i si n c r e a s e d from zero t o i n f i n i t y . Since t h e r e ar e s i xd o m i n a n t p o l e s a n d o n l y two z e r o s , four o f t h e p o l e smust t e n d t o i n f i n i t y a s g a i n i s i n c r e a s e d w h i c h i m p l i e s

j w ( r / s )

3 0 2 7t h a t a n i n s t a b i l i t y w i l l d e v e l o p a t s o m e v a l u e o f g a i n .An o p t i m u m g a i n t h e r e f o r e exists a t w h i c h t h e d a m p i n g i sm a x i m u m . T h e e i g e n v a l u e s c o r r e s p o n d i n g t o t h i s o p t i m u mc o n d i t i o n f o r t h e t h r e e c a s e s p r e s e n t e d are s h o w n on t h er o o t l o c u s c u r v e s by s q u a r e s . For F i g u r e 2 A , t h e o p t i -mum i s c h o s e n where t h e decay r a t e , i . e . , - o = - r e a lpart o f e i g e n v a l u e , o f t he l e a s t d a m p e d mode i s m a x i -m i z e d . For Figure 2 B , t h e optimum i s chosen w h e r e t h ed a m p i n g r a t i o , i . e . , t h e r a t i o o f decay r a t e t o f r e -q u e n c y , a s s o c i a t e d with t h e s o - c a l l e d " e x c i t e r mode"[ 8 ] , which i s becoming l e s s d a m p e d a s gain i n c r e a s e s ,a n d t h e m o d e initially associated with l o c a l m o d e , whichi s becoming m o r e d a m p e d a s gain i n c r e a s e s , a r e e q u a l .M o r e o r l e s s g a i n t h a n t h i s optimum would r e s u l t i n o n er o o t h a v i n g a l o w e r d a m p i n g r a t i o . I n Figure 2 C , d a m p -i n g n e v e r i n c r e a s e s , a n d a gain i s c h o s e n a s t h e maximumb e f o r e s i g n i f i c a n t d e g r a d i n g o f t h e existing dampingo c c u r s .P h a s e C o m p e n s a t i o n , Root L o c u s R e l a t i o n s h i p

O f particular interest i s t h e migration o f t h ee i g e n v a l u e which s t a r t e d a s t h e l i g h t l y d a m p e d l o c a lm o d e f o r F i g u r e s 2 A a n d 2 C . T h e p h a s e c o m p e n s a t i o n a tt h e l o c a l m o d e frequency f o r t h e a d j u s t m e n t s o f 2 A i s- 3 3 0 , b u t - 9 0 f o r c a s e 2 C . A s gain i n c r e a s e s , bothe x p e r i e n c e a n i n c r e a s e i n f r e q u e n c y . T h e direction o fmotion a t a ny po in t a l o n g t h e l o c u s i s governed a p p r o x i -m a t e l y by t h e p h a s e a t t h e frequency which e x i s t s . T h i sc h a r a c t e r i s t i c i s s t r i c t l y t r u e o n l y f o r z e r o d a m p i n g ,but i s approximately c o r r e c t when t h e r o o t i s s i g n i f i -c a n t l y u n d e r d a m p e d , i . e . , u / w < < 1 .

I n Figure 2 C t h e phase l a g i s initially 9 0 0 a n dh e n c e frequency i n c r e a s e s with n o c h a n g e i n d a m p i n g .T h e p h a s e l a g i n c r e a s e s with i n c r e a s i n g frequency whichc h a n g e s t h e s t a b i l i z e r contribution t o a s o m e w h a t n e g a -t i v e d a m p i n g c h a r a c t e r i s t i c a n d h e n c e t h e e i g e n v a l u emoves t o w a r d s t h e r i g h t - h a l f p l a n e a n d eventually b e -comes u n s t a b l e . I n F i g u r e 2 A , t h e l o c u s b e g i n s i np r i m a r i l y a positive d a m p i n g d i r e c t i o n , but eventuallyf r e a u e n c y i n c r e a s e s t o w h e r e t h e ph as e pa ss es through- 9 0 . T h e l o c u s t h e n c o n t i n u e s with a pure i n c r e a s e i nf r e q u e n c y , w h i c h i s a ss oc ia te d w it h a n i n c r e a s e i n t h ep h a s e l a g , t h e r e b y e v e n t u a l l y c a u s i n g a d e c r e a s e i nd a m p i n g t o a p o i n t o f becoming u n s t a b l e .

- 50 ' ( s e c - 1 )( B ) f c - 3 . 4 H z ; T I / T 2z T 3 / T 4 - . 1 5 / . 0 1 5

j w ( r / s )

a ( s e c ' 1 )( C ) f c 1 0 H z ; T 1 / T 2 : T 3 / T 4 - . 0 5 / . 0 0 5

F I G U R E 2 ROOT L O C I F OR TUNING E X A M P L E S P E E D I N P U T , S T R O N G SYSTEM, FUL L LOAD

a , ( s e c - ' )( A ) f I . O H z ; T , / T 2 : T 3 / T 4 . 5 / . 0 5


12/30

3 0 2 8Summary o f Tuning E x a m p l e

T h e significant p a r a m e t e r s a s s o c i a t e d with eachl e a d / l a g s e t t i n g a r e consolidated i n Table I . Thesei n c l u d e p h a s e c o m p e n s a t i o n a t t h e l o c a l m o d e f r e q u e n c y ,$ 6 t h e f r e q u e n c y . a t which t h e p h a s e l a g p a s s e s through9 h , f 9 o , t h e optimum g a i n , K O P T ' t h e decay r a t ea s s o c i a t e d with t h e most l i g h t l y d a m p e d s y s t e m mode witht h e optimum stabilizer g a i n , G O P T ' t h e g a i n a n d f r e -quency a t which a n i n s t a b i l i t y o c c u r s , K I N S , f N S T 'r e s p e c t i v e l y , a n d t h e g a i n o f t h e stabilizer a a y p i -c a l i n t e r t i e frequency o f 0 . 4 H z w i t h t h e optimum s t a b i -l i z e r g a i n , K I .

Table ISUMMARY O F TUNING EXAMPLE

f c O P T P- (( H z ) ( s e c1 . 02 . 03 . 45 . 01 0

- 3 . 5- 4 . 0- 4 . 8- 3 . 6- 0 . 8

K O P T K I K L f - 9 0 0( d e g ) ( H z )2 . 5 3 - 3 3 08 1 1 - 2 9 02 0 2 3 - 4 2 04 0 4 2 - 5 9 03 5 3 6 - 9 0

2 . 7 53 . 3 03 . 5 53 . 5 01 . 6 0

K I N S T f I N S T( H z )7 . 5 2 . 92 5 3 . 36 5 3 . 71 2 0 3 . 71 3 0 2 . 6

I t i s seen from t h i s t a b l e t h a t v e r y g o o d l o c a l m o d ed a m p i n g ca n b e o b t a i n e d with a w i d e r a n g e o f l e a d / l a gs e t t i n g s , b u t decreases r a p i d l y a s t h e c o m p e n s a t i o ncenter frequency b e c o m e s g r e a t e r t h a n 5 H z . For com-pensation center f r e q u e n c i e s u p t h r o u g h 5 H z , t h e bestl o c a l m o d e d a m p i n g occu rs w i t h a c e n t e r f r e q u e n c y n e a r3 . 4 H z . T h e h i g h e s t g a i n at intertie f r e q u e n c i e s ,h o w e v e r , occurs with t h e h i g h e r c o m p e n s a t i o n c e n t e rf r e q u e n c y o f 5 H z . I n g e n e r a l , t h e h i g h e s t c o m p e n s a t i o ncenter frequency which p r o v i d e s a d e q u a t e l o c a l moded a m p i n g w i l l y i e l d t h e greatest contribution to intertiem o d e s o f o s c i l l a t i o n . T h e l a s t f o u r p a r a m e t e r s o f T a b l eI suggest t h e f o l l o w i n g g u i d e l i n e s f o r s e t t i n g t h el e a d / l a g s t a g e s t o a c h i e v e a d e q u a t e l o c a l mode d a m p i n gwith maximum c o n t r i b u t i o n t o intertie modes o f o s c i l l a -t i o n . T w o basic criteria i n terms o f p h a s e c o m p e n s a t i o na r e :1 . I t i s most i m p o r t a n t to maximize t h e b a n d w i d t hwithin w h i c h t h e p h a s e l a g remains l e s s than 9 0 0 .This i s tru e e v e n t h o u g h l e s s than p e r f e c t p h a s ecompensation r e s u l t s a t t h e l o c a l mode f r e q u e n c y .

TUNING WITH ALTERNATIVE I N P U T SIGNALSSpeed I n p u t Stabilizers

T h e system u s e d i n t h i s a n d t h e f o l l o w i n g s e c t i o n st o establish t h e c o n c e p t s o f t u n i n g t h e t h r e e basict y p e s o f s t a b i l i z e r s d i f f e r s f r o m t h e previous e x a m p l ec a s e i n t h a t a f o u r - p o l e t u r b i n e - g e n e r a t o r i s u s e d . A sw i t h t h e p r e v i o u s e x a m p l e , t h e t u n i n g i s e x a m i n e d f o rt h e c a s e with a s t r o n g transmission system o f 20% r e a c -t a n c e a n d f u l l l o a d o n t h e g e n e r a t o r s i n c e t h i s r e p r e -s e n t s t h e m o s t r e s t r i c t i v e c a s e f o r s p e e d i n p u i t s t a b i -l i z e r s . T h e lead/lag s p r e a d o f t h e stabilizer t i m ec o n s t a n t s w a s i n i t i a l l y s e t a t 1 0 : 1 , t h e s a m e a s used i nt h e previous a n a l y s i s with t h e t w o - p o l e m a c h i n e , a n dt h e l e a d t i m e c o n s t a n t s a t 0 . 2 s e c o n d s ( f = 2 . 5 H z ) .T h e l o c u s o f d o m i n a n t r o o t s a s a f u n c t i o n o f s t a b i l i z e rg a i n a r e s h o w n i n Figure 3 a . T h e l o c a l mode e i g e n v a l u emoves t o t h e l e f t a n d i n c r e a s e s i n d a m p i n g while t h er o o t associated with t h e f i l t e r i n g i n t h e s t a b i l i z e rbecomes l e s s s t a b l e a n d e v e n t u a l l y g o e s u n s t a b l e . F o rt h i s c a s e , t h e optimum g a i n i s a b o u t 3 0 a n d t h e i n s t a -bility gain i s about 9 0 .

T h e w i d e s p r e a d between t h e l o c i o f t h e e x c i t e rmode a n d l o c a l m o d e i s i n d i c a t i v e o f e x c e s s s t a b i l i z e rl e a d . S i n c e t h e i n e r t i a a n d r e a c t a n c e o f a f o u r - p o l emachine a r e g r e a t e r t h a n f o r a t w o - p o l e m a c h i n e , t h el o c a l m o d e f r e q u e n c y i s l o w e r , o t h e r c o n d i t i o n s beinge q u a l . C o n s e q u e n t l y , t h e r e q u i r e d phase l e a d a t l o c a lmode f r e q u e n c y i s l e s s t h a n with a two-pole m a c h i n e , a n dt h e l e a d / l a g r a t i o c a n t h e r e f o r e b e r e d u c e d . F i g u r e s 3 ba n d 3 c s h o w t h e resulting r o o t l o c i f o r different l e a dtime c o n s t a n t s , maintaining 6 : 1 s p r e a d s o n both s t a g e s .T h e r e d u c e d phase l e a d o f c a s e 3 b i s c l o s e t o optimumf o r l o c a l m o d e . Placing t h e l e a d t i m e c o n s t a n t s a th i g h e r f r e q u e n c i e s r e s u l t s i n even l e s s phase l e a d a n dt h e l o c a l mode g o e s unstable r a t h e r than t h e e x c i t e rmode a s shown i n Figure 3 c .

I t i s instructive t o c o m p a r e t h e s e l e c t e d g a i n s f o rt h e th ree cases shown i n Figure 3 . Tabulated i n T a b l eP T ' K NST K H O K P T T l T 3 / T 2 T 4 r ep r e s e n t i n g t h egain a t ' h i g h t r e q d e n c i e s , a n d K t h e gain a t 0. 4 H zIr e p r e s e n t i n g t h e g a i n a t a n i n t e r t i e f r e q u e n c y . T h e

t a b l e a l s o s h o w s t h e l o c a l m o d e e i g e n v a l u e s , \ - O C A L 'c o r r e s p o n d i n g t o K O P T . 'Table I I

COMPARISON O F SPEED I N P U T STABILIZER TUNING

2 . The p h a s e l a g at t h e l o c a l m o d e f r e q u e n c y should b el e s s t h a n about 4 5 . T h i s ca n b e i m p r o v e d somewhatb y decreasing t h e washout time c o n s t a n t , b ut t ool o w o f a washout time c o n s t a n t will add p h a s e l e a da n d an a s s o c i a t e d d e s y n c h r o n i z i n g effect to t h eintertie o s c i l l a t i o n s . I n g e n e r a l , i t i s best tok e e p t h e washout time c o n s t a n t g r e a t e r than on es e c o n d .T h e g a i n a n d f r e q u e n c y a t w h i c h a n i n s t a b i l i t y o c c u r sa l s o p r o v i d e a n indication o f a p p r o p r i a t e l e a d / l a gs e t t i n g s . The r e l a t i o n s h i p o f these p a r a m e t e r s top e r f o r m a n c e ar e u s e f u l i n root l o c u s a n a l y s i s and i nf i e l d t e s t i n g .3 . T h e f r e q u e n c y a t w h i c h a n i n s t a b i l i t y o c c u r s i sh i g h e s t f o r t h e b e s t l e a d / l a g s e t t i n g s . T h i s i sr e l a t e d t o m a x i m i z i n g t h e bandwidth within whicht h e p h a s e l a g remains l e s s than 9 0 0 .4 . The o p t i m u m g a i n f o r a p a r t i c u l a r l e a d / l a g s e t t i n gi s c o n s i s t e n t l y about o n e - t h i r d o f t h e i n s t a b i l i t yg a i n .

F i g T 1 / T 2 , T 3 / T 4 " L O C A L OPT K I N S T K H K I3 a . 2 / . 0 2 , . 2 / . 0 2 - 3 . 0 + j 8 . 03 b . 2 / . 0 3 3 , . 3 / . 0 5 - 3 . 5 + j 8 . 53 c . 1 2 / . 0 2 , . 3 / . 0 3 3 - 3 . 5 + j l l . 5

3 02 05 0

9 0 3 0 0 0 3 76 0 7 2 0 2 81 5 0 1 8 0 0 5 8

A l l d e s i g n s g i v e g o o d d a m p i n g f o r l o c a l m o d e , b u t t h e6 : 1 l e a d / l a g s p r e a d y i e l d s l a r g e r gain a t i n t e r t i ef r e q u e n c y p e r gain a t high f r e q u e n c y , i . e . , K /KH i sl a r g e r . Optimum t un in g m ig ht l i e between c a s e s M b , a n d3 c with a trade-off r e q u i r e d between high f r e q u e n c y g a i na n d intertie damping c o n t r i b u t i o n s . C a s e 3 b w i l l b eused f o r s u b s e q u e n t c o m p a r i s o n with t h e p e r f o r m a n c e o fother t y p e s o f s t a b i l i z e r s . Note t h a t f o r a l l c a s e s t h eoptimum gain i s about one-third o f t h e instability g a i n ,c o m p a r a b l e t o t h e p r e v i o u s e x a m p l e with a t w o - p o l e u n i t .Power Input Stabilizer

A s indicated i n Part I , t h e interest i n usingaccelerating power a s a s t a b i l i z e r i n p u t s i g n a l r e s u l t sf r o m t h e i n h e r e n t l y l o w l e v e l o f torsional i n t e r a c t i o nd u e t o i t s n o n - m i n im u m p h a s e c h a r a c t e r i s t i c . A s a


13/30

3 0 2 9j w ( r / s )

( a ) T 1 / T 2T 3 / T 4

l 7

L Lu L J U)T-C )

- k g r- 5

j 2 0

( - 1 7 . 5 + 1 6 )j ' 5

I l O

j 5

[ - /

L JC - ) :C/ ' c

- 2 . 5 0 2 . 5a ( s e c )

F I G U R E 3 R O O T L O C I W I T H S P E E D I N P U T , V A R I O U S S E T T I N G Sc o n s e q u e n c e , a s i n g l e t i m e c o n s t a n t l a g a t 0 . 0 6 s e c o n di s a s s u m e d t o provide sufficient torsional attenuationf o r t h i s e x a m p l e . T h e g e n e r a l f r e q u e n c y c h a r a c t e r i s t i c so f a p r a c t i c a l power i n p u t s t a b i l i z e r a r e d e v e l o p e d i nP a r t I . Appropriate s e t t i n g s f o r t h i s e x a m p l e w e r ed e t e r m i n e d t o b e :

P S S p = K ( 1 + . 2 5 s ) ( l + . 1 5 s ) ( 1 + . 0 6 s ) ( 6 )T h e l a g / l e a d s t a g e o f . 5 / . 2 5 seconds contributes t h ep h a s e l a g r e q u i r e d a t l o w l o c a l mode f r e q u e n c i e s ( w e a ktransmission s y s t e m s ) a n d t h e l e a d / l a g s t a ge o f . 1 5 / . 0 5s e c o n d s p r o v i d e s t h e p h a s e l

applying pss genconcepts

Documents