Dynamic Aggregation of Generator Unit Models

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IEEE Transactions on Power Apparatus and Systems, Vol. PAS-97, no. 4 July/Aug 1978DYNAMIC AGGREGATION OF GENERATING U NI T M OD EL S

A. J . Ge r m o n dMember, I E E E R . PodmoreMember, I E E ES y s t e m s Control, In c .Palo Alto, California

ABSTRACTA new technique i s described for the automaticformation of d yn am ic e qu iv al en ts of g en er at in g u ni tsrepresented by detailed models.The method applies to groups of g en er at in g u ni tsthat are coherent. The parameters of an equivalentmodel of governor, turbine, s yn ch ro no us m ac hi ne , e xc i-tation system and power system stabilizer are identi-fied fo r each group of coherent units, by means of aleast-square fit of their transfer functions. Th etechnique is demonstrated with a large system stabilitydata base.The primary advantage of this procedure i s thatthe reduced model retains a p hy si ca l m e an i ng and thus,

can be used w ith c o nv en t io n al t r an s ie n t stability pro-grams. It also has the capacity for handling th e widerange of models used in practical studies.

INTRODUCTIONThe need to develop fast and cost-efficient sim-plified s ta bi li ty a na ly si s t ec hn iq ue s has motivated re-search in t h e area of reduced-order dynamic models of

power systems. The reduction techniques are based onone of t wo m et ho ds : m o d a l analysis of the linearizedmodel [ 1 ] or coherency recognition [ 2 , 3 ] .

* The modal analysis technique requires thecomputation of eigenvalues, which i s time-consuming, and provides a reduced system ofdifferential equations which c a n n o t be in-terpreted, in general, as r epr esen ti n g m o delsof physical units. Thus, the reduced modelsobtained by modal analysis c a n n o t be usedwithout modifications to conventional stabil-ity programs.

* Re du ct io n b as ed on coherency does n o t pre-sent these disadvantages. An efficientmethod for c oh er en cy a na ly si s has b een d e-veloped and is reported in a companion paper[ 4 ] .

This paper addresses th e problem of grouping gen-erating units that are coherent into an equivalentgenerating u nit model. A method of logarithmic averageha s been proposed [ 3 ] , but applies only when the modelsof the generating units to be grouped are of the sametype.The method presented in this paper does not re-

quire such a restriction. The d at a b as e of the origi-na l s y s t e m can include a variety of models for the~ ~ A t o

F 7 7 1 6 5 - 4 . A paper recommended a n d a p p r o v e d b y t h e IEEE Power S y s t e mE n g i n e e r i n g C o m m i t t e e o f t h e IEEE Power E n g i n e e r i n g S o c i e t y f o r p r e s e n t a -t i o n at t h e IEEE PE S W i n t e r M e e t i n g , New Y o r k , NY, J a n u a r y 3 0 - F e b r u a r y4 , 1 9 7 7 . M a n u s c r i p t s u b m i t t e d A u g u s t 3 0 , 1 9 7 6 ; made a v a i l a b l e f o r p r i n t i n gDecember 2 , 1 9 7 6 .

governors, turbines, s yn ch ro no us m ac hi ne s, excitationsystems and power s y s t e m stabilizers [ 5 , 6 , 7 ] . Thee qu iv al en t g en er at in g units are made of similar modelsand thus are compatible with conventional stabilityprograms.

Th e method considers separately the linear para-m e t e r s and the non-linear limits of the generating unitmodels. First the aggregated transfer functions relat-in g the t ot al mechanical an d electrical power output ofth e coherent generating units to their common speed andterminal voltage are calculated fo r several discretefrequencies. The p a r a m e t e r s of the equivalent transferfunctions are adapted to match the a g g reg a ted t r an s ferfunctions with a minimal error. Then th e equivalentlimits are calculated. The procedure has been imple-mented and experimented with a d ata base representingthe W e s te r n U. S. S y s t e m [ 8 ] . . Equivalents calculatedfor coherent generating units of this system will beused as examples of the method. Finally, the perfor -mance in the time domain of an equivalent mode l will becompared to a full representation of a group of fourgenerating units.

METHODDefinitions

* A coherent group of generating units, for ag iv en p er tu rb at io n, i s a group o f g en er a to r so s ci ll a ti ng w i th the same angular speed, andt er mi na l v olt ag es in a constant complexratio. Thus, th e generating units belongingto a coherent group can be attached to acommon bus, i f necessary through an ideal,complex ratio transformer.

* Th e dynamic equivalent of a coherent group ofgenerating units i s a s i ng le g en er a ti n g unitthat exhibits the same speed, voltage andtotal mechanical and electrical power a s th egroup during any perturbation where thoseu ni ts r em ai n c oh er en t.

Assumptions and Overview of the MethodTh e assumptions are made that the coherent gener-a ti ng u ni ts are on a common bus, with the same terminalvoltage VT , and have the same speed W.

Figure 1 . Generating Unit M o d e l0018-9510/78/0600-1060$00.75 G 1 9 7 8 IEEE

106 0

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The block diagram of Figure 1 r ep re se nt s th e f un c-tio nal r ela ti on s between the mechanical and electricaloutput of an individual_generating unit and its speed uand terminal voltage V , these being considered asi np ut v ar i ab le s.

1 0 6 1Since all the machines of a coherent g r ou p h av e,b y d efi n it i on , th e same speed deviation W , equations

( 1 ) are summed-up for all the machines of a group toform the m ech a ni c al Eq uat i on ( 2 ) of the equivalent gen-erator-turbine:A similar block diagram i s used to model an equiv-alent generating u n i t , with the individual mechanicalpower replaced by the total mechanical power and th e

electrical power by the t ot a l elec t ri c al p ow er o ut pu tfor the group.Th e objective of the method i s to specify t h ec h a r a c t e r i s t i c s of t h i s e q u i v a l e n t model, given t h emodel of each individual unit. This will be done byconsidering separately th e r o t o r dynamics, the governorand turbine model, the synchronous machine model, th eexcitation system model and th e power system stabilizermodel.Th e assumption i s n o w made that t h e linear andnonlinear c h a r a c t e r i s t i c s of th e e q u i v a l e n t models ca nbe identified separately.Th e linear parameters of each e q u i v a l e n t model arenumerically adjusted to obtain a minimal error between

i ts t ra ns fe r function and the sum of t he t ra ns fe r func-t i o n s of the individual u n i t s . The error to be mini-mized i s th e sum of the squares of the magnitude of therelative d i f f e r e n c e , fo r s p e c i f i e d discrete frequen-c i e s .Th e transfer functions to be approximated areindicated i n Table I . The s p e c i f i c problems and theequivalent parameters f o r th e non-linear character-i s t i c s of ea ch m od el are discussed i n th e followingsections of th e paper.

Table I Open-Loop Transfer FunctionsTo Be A p p r o x i m a t e d By The E q u i v a l e n t ModelsROTOR DYNAMICS

The mechanical equation fo r one machine i s :2 H K = P - P - D . u .j dt mj e j I j

withu p.u. s p e e d deviation

mj e- (E D w ( 2 )M j 3

MVA: All p ar ame te rs being referred to the same base

* Th e equivalent inertia constant i s th e sum ofthe individual inertia constants.* Th e e qui va le nt d am pi ng factor i s th e sum ofthe individual damping factors.

EQUIVAL ENT T U RB INE - GOV E RNOR SYSTE MAggregated Transfer Function

The Western System data base includes severalmodels o f g ov er no rs , turbines and also comprehensivemodels of g ov er no r a nd turbine( s) . Bl oc k diagrams ofth e major g ov er no r a nd turbine models are representedi n [ 5 ] .

Assuming a speed variation of small a m p l i t u d e , th evalve an d v al ve rate limits are neglected and a lineartransfer functionG . ( s ) = P m j ( s ) / A u ( s ) ( 3 )

where Au = ( F - u ) denotes the generator speed error,can be calculaed fo r each generating u n i t . Since th ei n p u t speed variation i s the same fo r each generatingunit of a coherent group, the variation of mechanicalpower for the coherent group i sP M ( s ) = I M j ( 5 )M j ~ ~ ~ ( 4 )

T h e transfer f u n c t i o n s G . ( s ) of E q u a t i o n ( 3 )consist of two elements i n s e r i e a , th e governor trans-fe r f u n c t i o nG ( s ) = P G ( s ) and th e turbine transfer functionG A W ( s )G ( s ) = PM(s)T P G V ( s )For each governor and t u r b i n e , th e transfer func-t i o n s GG an d G T are evaluated numerically fo r dis-crete values of the variable s along th e i m a g i n a r ya x i s . Th e aggregated transfer function P M ( s ) / A u ( s ) i s

( 1 ) obtained by summation.Th e aggregated governor-turbine transfer functioncalculated fo r one group of th e Western system i srepresented i n th e Bode d i a g r a m of F i g u r e 2 .

i n er t ia c o ns t an t (generator + turbine)in MWsMV Am e c h a n i c a l power i n p .u .

electromagnetic power in p . u .D damping f a c t o r

Equivalent ModelDifficulties were e n c o u n t e r e d in t he choice o f asingle equivalent m o d e l t h a t w o u l d accurately fi t thefrequency response of c o h e r e n t g r o u p s constituted ofboth steam and h y dr o g e ne r at o rs . Therefore, when t h i scircumstance occurs, th e s tea m un it s are grouped t oform an equivalent s t e a m unit a n d t h e hydro units toform an e q ui v al e nt h y dr o unit w h i c h are both attachedto the same bus.

Open-Loop Transfer Function E q u i v a l e n t ModelW / ( E P m e ) Rotor Dynamics

) p / A W Governor + Turbinem) i T / V T Excitation System +Synchronous MachineV s o / U P ow er S ys te m Stabilizers o ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

H

pmp e

2H dw _:i i dt

I G ( s ) - A w ( s )i3



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1062Equivalent

Magnitude Aggregated --r p . u . o f 100 MV A sp.u. o f synchronous s p e e d I \

_1 0 0 . h a g

= > , p a s e \

- 1 0 .

1 .. 0 1 . i 1 frequency [ H z ]Figure 2 . Bode plots of th e aggregated and equivalentgovernor-turbine transfer function fo r a coherent groupof the Wes te rn System.

The transfer function of an equivalent hydro gov-ernor i s specified as

* Th e error function E j w i ) - G 2 )i G ( j w i ) li s calculated.

* A numerical grad ient technique i s used forcorrecting th e u n kn o wn p a ra m et e rs to minimizethis error. In addition, it i s required thatth e error be zero for w=O. T he p ro ce ss isinterrupted when a technically small error i sobtained. Th e parameters so found ar e th oseof the equivalent model. The transfer func-tion of the equivalent m od el f ou nd by thismethod i s represented as well in Figure 2 .

Valve and Valve Rate LimitsThe valve limit for the equivalent model i s cal-culated as the sum of the individual gate limits.No valve rate limit i s calculated for the equiv-alent, since this parameter i s not represented in sev-eral of th e WSCC models.

EQUIVALENT SYNCHRONOUS MACHINE AND E XCI TATION MODEL SI n Table I , the transfer function i T / V T was des-ignated as excitation system a nd s yn ch ro no us machine.In Figure 3 these entities ar e s e pa r at e ly r e pr e-sented. The key assumption that the machines are onthe same bus permits one to consider th e transfer func-t i o n between the terminal current and the terminalvoltage, which i s l i n e a r . Th e block diagram of Figure3 applies to both th e individual and th e equivalentmachine.

( l + S T 2 )G G ( s ) = K 2( l + s T ) ( l + s T 3 )with unknown parameters K , T 1 , T 2 , T 3 . Th e transferfunction of an e q u i v a l e n t hydro unit is s p e c i f i e d as

* _ l ~ - s T wG ( s )G T ( l+sTw/2hydrowith u n kn o wn p a ra m et er Tw. The transfer function of a nequivalent s t e a m governor and t u r b i n e i s s p e c i f i e d as

* l + s T 2 1G ( s) - _ _ _ _ _ _ _GT ( l + s T ) ( l + s T ) l + s T 4steamwith unknown parameters T 1 , T 2 , T 3 , T 4 .

The last transfer f u n c t i o n represents a p p r o x i -mately a s t e a m governor with a non-reheat turbine andwas chosen as th e m o s t a p p r o p r i a t e to represent suc-c e s s f u l l y the e q u i v a l e n t transfer f u n c t i o n of a steamunit with the minimum number of parameters. The de-tails of the process of identification are e x p l a i n e dwith th e example of F i g u r e 2 :* Fo r discrete complex f r e q u e n c i e s , j w i , thetransfer function of the individual governor-turbines are calculated and summed-up. Theresult i s a curve on th e Bode plot referredto as the " a g g r e g a t e d transfer function"G ( j w i ) .* S t an d ar d v a lu es a re a s s i g n e d to the unknownparameters of th e e q u i v a l e n t governor andt W r b i n e and the e q u i v a l e n t transfer f u n c t i o nG ( j u i ) i s calculated.

( t h e symbol * indicates e q u i v a l e n t variablesand p a r a m e t e r s )

S Y S T E MS T A B I L I Z E R J

Figure 3 . Synchronous machine and excitationsystem model.The equivalent i s formed i n three steps. F i r s t ,th e transfer function i s aggregated without the excita-tion i n p u t . This p r o v i d e s the parameters of the e q u i v -alent synchronous machine.

Then th e equivalent of th e excitation system i s formed.Since the output of the individual excitation systemsare applied to synchronous machines that have differentc h a r a c t e r i s t i c s , it i s necessary to w e i g h those out-puts to fo rm the equivalent field v o l t a g e . F i n a l l y ,t h e equivalent p ow er s ys te m stabilizer i s determined.The coherent groups may be constituted of combinationsof the following models:Synchronous M a c h i n e s : Th e two-axes model with o n efield w i n d i n g i n the direct a x i s and one d a m p i n g wind-i n g in th e quadrature axis i s used fo r each individualmachine ( F i g u r e 1 5 i n A p p e n d i x ) . The round r o t o r ma-



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chine and " cl as si ca l m od el " a re considered a s specialcases. Th e transients of the stator are neglected.Excitation Systems: Seven different models or rotatingand static excitation systems are represented [ 7 , 8 ] .Power System Stabilizers: Three models are repre-sented, with either speed deviation, shaft s l i p oraccelerating power as input signal. Only power systems t ab i li z er s w i th the same input are aggregated.Formation of the Equivalent Synchronous Machine

Assuming coherency:0 The differences of rotor angles between ma-c h i n e s of a c oh er en t g ro up remain constant.* The terminal voltage i s the same for eachmachine of t he g ro up .These assumptions ar e used to d em on st ra te i n theAppendix that a dynamic equivalent can also be repre-sented as a two-axes model. Th e justification i s th eequivalence of th e e l e c t r o m a g n e t i c power output.T he t ot al p.u. electromagnetic power output P ofth e coherent group i s :

=e ( V q j i q j + V d j i d j )With th e terminal voltage and t h e stator current ex -pressed for each machine i n i t s own reference axes.

Since the terminal voltage i s common, th e totalelectric power i s expressed as :

e Q E l Q j +VD E D j ( 5 )where D and Q r ep re se nt c om po ne nt s o n an arbitrarypair of orthogonal axes.

It i s shown in Appendix that j i D and j i Qrelated to VD and V Q b y the expression: are

1063The position of th e equivalent axes and th e para-meters of the e q ui v a le n t m o d el are calculated to mini -miz e the difference between the electric power outputP of the equivalent and Pe eTh e aggregation of the synchronous machine pro-ceeds as follows: ( s e e Appendix)* First th e position of the axes 0 i s calcu-lated, such that YDD' Y , , and Y Q F i nEquation ( 6 ) are made negligibly small i n thefrequency range.* Then th e pa ra met er s o f the equivalent-modelare adjusted separately for each axis byfitting the g p e r a t i o n a l admittance Y * withY and Y with Y D QD Q QD Q D _This i s obtained by adjusting th e parameters ofth e equivalent to fi t i t s o pe ra ti on al a dm it ta nc es toth e operational admittances of the aggregated machinesFigure 4 represents an example of t h i s procedure.

Magnitude[ p . u . ] magnitude

. - 1 0 0 .

p h a s e1 0 .-90-

phase EquivalentAggregated

-180_. 0 1 . 1 1 ' . f r e q u e n c y [ H z I

i D ( S ) / Y D D ( S ) Y D Q ( s ) VD Y D FaiD(s) + ~ ~ ~ ~ ~ ~ ~ e F D6 )

iQ( s) y Q D ( s ) Y Q Q ( s ) /VQ y QFwhere Y D D ( s ) , Y D Q ( s ) , Y Q D ( S ) and Y Q Q ( s ) d e p e n d o nth e parameters of th e individual m a c h i n e s and on th eposition of th e axes D and Q .

The e q u i v a l e n t machine b e i n g a two-axes model aswell, it s electric power output i s :* * *P V= + V ie Q Q D D

withD ( s ) 1 0

i ( s ) Y Q D ( s )

( 7 )

DQ(s (eDF0V QI ~ + eFD (8)

Figure 4 . Bode plot of the aggregated and equivalentadmittance i n the direct a x i s , fo r acoherent group of t he We st er n System.Aggregation of th e E x c i t a t i o n System Models [ 6 , 7 ]

Each i n d i v i d u a l e x c i t a t i o n system i s representedby a single-input single-output block d i a g r a m . In thecase of t he We st er n System data b a s e , the excitation i sone of th e models of the Western System C o o r d i n a t i n gCouncil [ 8 ] .

Ignoring the regulator l i m i t s , th e linear transferf u n c t i o n of on e e x c i t e r i s GE j ( s ) .The terminal voltage V T i s th e c o m m o n i n p u t toeach e x c i t e r . A s s u m i n g this input small e n o u g h thatnone of the regulators reaches i t s l i m i t , the outputefd i s expressed fo r each e x c i t e r as:e fd ( s ) = G ( s ) AV ( s ) ( 9 )

where A V T ( V R E F + V s o - V T ) i s t h e terminal voltageerror.



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1064The field voltages e , applied to the indivi-dual synchronous machine moSels result in a contribu-tion to the total current denoted as A i D ( s e e Appen-d i x ) .

I i s = I D( s fd (s) A V . ~ i (s)e s) AV(s)) T(Y d f ( s ) GE A V T ( s ) ) ( 1 0 )

The transfer f u n c t i o nA i D ( s )I = ( Y d f . ( s ) C O s G j ( s ) )A V T ( s ) )j

has to be identified with the transfer f u n c t i o n betweenthe same variables i n t h e e q u i v a l e n t model: Figure 6 . IEEE type 1 excitation system model.A i ( s ) e ( s )D - Y ( s ) * FDA V T ( s ) A V T ( s ) ( 1 1 )It i s convenient to rewrite t h i s condition as

G * ()= e F D =E (s) A T Id f

COs4 i GE (s) (12)y D (s) Iand consider th e e x p r e s s i o n between brackets as a" w w e i g h t - f a c t o r " W . ( s ) that accounts fo r the p a r a m e t e r sof th e s y n c h r o n o u s ] machines to which th e excitationsystems are connected. This i s r e p r e s e n t e d in F i g u r e5 .

V T

Magnitude Equivalent--ggregated10 0

1 0 .0 Phase

-4 5

- 9 0. 0 1 . 1 l f r e q u e n c y l H z ]

I

Figure 5 . Formation of the e q u i v a l e n t field v o l t a g e b yweighting th e output of the i n d i v i d u a l exci-tation systems.At this stage, Y D ( s ) i s known from th e p a r a m e t e r sof the e q u i v a l e n t s y n - c r o n o u s m a c h i n e . Th e r i g h t - h a n dside of E q u a t i o n ( 1 2 ) c a n thus b e c al cu la te d fo r dis-c r e t e f r e q u e n c i e s .

E q u i v a l e n t ModelA W o d e l of Type I ( F i g u r e 6 ) with a trahsfer func-tion G E ( s ) represents the e q u i v a l e n t . I g n o r i n g th enonlinear l i m i t s , the parameters o f t h i s model area d j u s t e d , u s i n g a g r a d i e n t s e a ; c h to minimize t h e meansquare of the e r r o r between G E ( s ) and the r i g h t - h a n ds i d e of E q u a t i o n ( 1 2 ) . F i g u r e 7 r e p r e s e n t s th e resultof t h i s i d e n t i f i c a t i o n fo r a group of the Western sys-tem.

Figure 7 . Bode plot of the aggregated and e q u i v a l e n texcitation transfer function fo r o n e coher-ent group of th e W e s te r n System.Regulator Limits

The equivalent limits are calculated a s s u m i n g thata step i n p u t equal to th e regulator limit i s a p p l i e dsimultaneously to each exciter system. Fo r such a stepi n p u t , th e output i s ( F i g u r e 8 ) .

e (s ) =FD s 11i

v R M A X / ( K E + S E ) W . ( s )+ T E / ( E + ( 1 3 )

[.. -II



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V R M A X-a

Figure 8 . Simplified e xc it er m od el for la rg e s tepinput.Using th e initial and f in al v al u e theorem [ 6 ] :

u = w , A f or accelerating powerFigure 9 . Block diagram of an excitation system withpower system s t a b i l i z e r .

E * W.(s=o)j E j i

V S L I M I( 1 4 )

Fo r th e equivalent model, th e same limit i s :V *RMAXT E *Since the parameter T E * i s already kn ow n fr omfitting the linea r t ran sfer function, this relationdetermines VRMAX*.L i me ( t ) =t 4 o

I (XEV+MS * W . ( 0 ) ( 1 5 )iMAXjiFo r t he e q ui v al en t model, th e same limit i s :

VRMAX *E( K +SEMAX)* FDMAX*

Since VRA is already known, this relationdetermines

~ \ *( KE + SEMAX) and EFD MA

Exci ter S at ura ti o n CoefficientTwo discrete values of th e s at ur at io n f un ct io nS ( E f d ) are g i v e n fo r each e x c i t e r , at E f d MA X and. 5 5 E d MAX' An e x p o n e n t i a l relation i s assumed fo rt h i s sun E on . The e q u i v a l e n t function i s * c a l c u l a t e dat two p o i n t s ( i n i t i a l o p e r a t i n g p o i n t an d E f d M A X ) . I t

i s assumed that t h i s function i s e x p o n e n t i a l as w e l l .POWER SYSTEM STABILIZERS

A power system stabilizer introduces a correctionto th e reference voltage of an excitation system. T h i scorrection i s f u n c t i o n a l l y related to an i n p u t s i g n a las described on th e block d i a g r a m of F i g u r e 9 .The models i n th e WSCC data b as e h av e either th efrequency d e v i a t i o n , th e shaft s l i p o r t h e a c c e l e r a t i n gpower o f the g e n e r a t i n g unit as i n p u t . T h e r e are threetypes of power system s t a b i l i z e r s , d e p e n d i n g on thei n p u t s i g n a l . We found that o n l y s t a b i l i z e r s with th esame i n p u t s i g n a l could be a g g r e g a t e d .

VR EF

Figure 1 0 . Block diagram of the equivalent excitersystem and power system stabilizer.

T he p a ra me te r s of th e equivalent excitation s y s t e mof Figure 1 0 are already known at this stage (they haveb e en d et er mi ne d wi th o ut s t ab i li zer signal). A n equiva-lent stabilizer remains to be added in order that thedependance o f the field voltage on the stabilizer inputsignal be represented.Since this input signal u ( s ) i s the same fore v e r y stabilizer of a coherent group, the relation i s :*

Ae ( s )us) Wis) GE s ' s.s

I( 1 6 )

Whereas, fo r th e equivalent,

Ae FD()u ( s)

* *= G (s) * G (s)E S ( 1 7 )

*The parameters of G ( s ) are adjusted to minimizeth e mean square p.u. error between t h e transfer func-tions o f E qu at io ns ( 1 6 ) and ( 1 7 ) .

These aggregated and equivalent transfer functionsare shown on Figure 11 for a group of the Western Sys-tem.

Th e equivalent limit of the power system stabiliz-er is such that

* *VS G v G ( 0 ) W.(0) (18)SLIM GE()SLIM.Ej

1 0 6 5

d e F DL i m dt =t + + 0

I V R E F



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106 6

Figure 1 1 . Bode plot of the aggregated and equivalentA e f d power s y s t e m stabilizer t r an s fe r f un c ti o nfor one group of th e Western System.INTERFACE OF T HE D YN AM IC AGGREGATION PROGRAMWITH COHERENCY A ND S TA BI LI TY PROGRAMS

The dynamic aggregation program i s designed to usethe same set of data and th e same i n p u t formats as astability program. In a d d i t i o n , it r e q u i r e s th e l i s tof coherent generating units i n each group.The output i s a reduced set of data fo r a stabil-i t y program.The dynamic aggregation program has b een designedto interface through files with th e other programs.

APPLICATIONAGGREGATION OF THE COHERENT GROUP S OF THEWESTERN SYSTEM

PROGRAM PERFORMANCEOn e coherent gro up of the We st er n S ys te m has beenused as an example of the d yn am ic a gg re ga ti on tech-nique.This group includes 8 un it s w it h hydro-mechanicalgovernors. Two of t hem h ave a rotating exciter, 5 havea static exciter and a power system stabilizer with

a c ce le ra t in g p o we r input, one has a discontinuous exci-tation system. The B od e d ia gr am s of the aggregate ande q ui v al en t t r an s fe r functions for this group are plot-ted in Figure 2 , 4 , 7 and 1 1 .The aggregation of the g o ve rn o rs , m a ch i ne s, exci-tation systems and power system stabilizers was per-formed for all the c o he r en t g r ou ps of the Western Sys-tem.The total time fo r coherency analysis, networkreduction and formation of a reduced system i s esti-mated to be a fraction of th e time required fo r a sta-bility run of the f ul l s ys te m.

VALIDATION T E S T FOR A 4-MACHINE SYSTEMThe objective of this test was to verify that theassumptions and th e criteria of fitting transfer func-t i o n s were meaningful, and to check th e prototype pro-grams for the aggregation of the synchronous machinesand exciter systems.T he c oh er en cy analysis program was no t yet avail-a b l e , thus a group of four supposedly coherent ma-c h i n e s , connected to an infinite bus through a trans-mission line ( F i g u r e 1 2 ) was selected for a stabilitysimulation. The disturbance was a voltage d i p of 50%a p p l i e d at th e infinite bus for . l s .

V T

Load Bu s

Figure 1 2 : Test case of the equivalent synchronousmachine and excitation system in thetime domain.Two cases were c o n s i d e r e d :

ObjectiveThe purpose of th e a g g r e g a t i o n of the coherentgroups of th e Western System wa s to a ss es s the d i f f i -culties encountered in a p p l y i n g th e method to a l a r g e -scale system fo r o b t a i n i n g a reduced m o d e l , and toevaluate th e performance of the a g g r e g a t i o n program.

ProcedureThe Western Systems Coordinating C o u n c i l ( W S C C )power-flow and s t a b i l i t y data base p r o v i d e d t h e datafo r a coherency a n a l y s i s p e r f o r m e d with a method des-cribed i n a c o m p a n i o n paper [ 4 ] . A l i s t of 8 7 coherentgroups was obtained out of the i n d i v i d u a l 3 3 7 m a c h i n e s .44 groups had o n l y o n e generator. T h i s l i s t wa s pro-vided as i n p u t to the d y n a m i c a g g r e g a t i o n program inaddition to the WSCC s t a b i l i t y data base.

* In Case 1 , th e same fault i s simulated withth e individual machines and individual exci-ter systems modelled and a g a i n with th eequivalent machine and e q u i v a l e n t exciters y st e m m od el .The t e r m i n a l voltage, and total electricpower are plotted i n F i g u r e s 1 3 and 1 4 .Th e t e r m i n a l voltage a nd t ot al electric powerobtained with the e q u i v a l e n t machine modelare plotted o n the same f i g u r e s .There i s a g o o d agreement between the termi-na l voltages an d to tal electric p ow er o ut pu tsobtained with the individual models and withth e e q u i v a l e n t . This i s remarkable since thecoherency between th e individual generatorsha d a tolerance of almost 20 d e g r e e s .



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Total Electric PowerP e [ p . u . of 10 0 MVA]

8 .e q u i v a l e n t model

f u l l system.time

1 . 1 . 5Figure 1 3 . Comparison, i n the t i m e d o m a i n , of t h etotal electric p ow er o ut pu t of th e fullmodel and t h e reduced model.

Terminal Voltage1.04

1 . 0 0

. 9 2

.8 8

.8 4

1067CONCLUSIONS

The following work ha s been performed:* A c on si st en t method has been developed toaggregate th e r es pe ct iv e g en er a ti n g u n it com-ponents ( m a c h i n e s , excitation system, tur-bine-governors a nd p ow er s ys te m s ta bi li ze rs )on the basis o f c oh erenc y, to form the com-ponents of an equival ent generating u n i t .* T he p ar am et er s of t h e equivalent models arei d e n t i f i e d by prescribing a criteria whichc o r r e s p o n d s to f i t t i n g transfer functions i nth e complex plane.* Th e individual and equivalent models are com-p a t i b l e with exis ti ng s ta bi li ty programs.* The method has been applied to a wide rangeof generating unit dynamic models includingal l those in the WS CC d at a base.The key conclusions which can be drawn from thisstudy can be stated a s :* Th e feasibility of the method, demonstrated

by the formation of equivalents fo r the Wes-t er n S ys te m.* Th e capability of th e program to handle largescale transient s ta bi li ty g en er at in g unitd at a b as es in a reasonable t i m e .Th e proposed method i s a practical approach toreducing th e dynamic order of a p ow er system represen-t a t i o n .Th e program developed will form one of th e ke yelemehts i n a comprehensive package of programs fo r

forming dynamic equivalents.

Further tests in the t im e d om ai n on th e WSCC Sys-t em will be performed t o evaluate the accuracy of thereduced models. Satisfactory results were already ob-tained in th e preliminary test with four machines.

APPENDIXSYNCHRONOUS MACHINE MO DE L A ND'OPERATIONAL ADMITTANCE MATRIXi r d X Q Rfd fd q XZI~ ~ Refd ~ ~ ~ ~ ~ ~ l qa d x V q I aq

equivalent modelfull system d -a x i s q a x i s

. 5 . 6 1. 1 . 5Figure 1 4 . Comparison, in the time domain, o f theterminal voltage of the full m od el a ndthe reduced model.

* In Case 2 , no exciter system was modelled,but the field voltages were kept c o n s t a n tinstead. The same fault was simulated withth e individual machine models, and again w i t hthe equivalent machine model. The conclu-sions were the same as in Case 1 .

timei d c o m p o n e n t in direct axis of p.u. s ta to r c ur re nti component in quadrature axis of p.u. stator c u r r e n t

2 . Svd component in direct axis o f p.u. terminal voltagev component in quadrature axis of p.u. terminalq voltage

Figure 1 5 . T wo axes model of the synchronous machine.Referring to Figure 1 5 and to the usual definition ofthe per unit reactances and time-constants, t h e Laplaceequation of the two-axes synchronous m a c h i n e r e a d s as:

4



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1 0 6 8

~ d q ~/ v d e t Y f f d (19)f'q\ ( : q d 0, \0/ewhere th e position of the reference axis q with respectto the direction of th e terminal voltage is the r oto rinternal angle 6 . ( F i g u r e 1 6 )

andy d qy q d

y d f ,

- 1 + s T ' d o )xd + s x ' T vX d +sd do

1 + s T ' )q ox + s x ' T 'q q qo

-Kx + s x ' T 'd q do

- 4 - - ~ 1 - - V Tq ~ . = 6 . j - 0-~~~ J

D d .Figure 1 6 . Definition of reference a x e s.

Th e e q u a t i o n of every m a c h i n e of a c o h e r e n t g r o u pi s p r o j e c t e d o n a common p a i r of axes, D an d Q . T h edirection of th e axis Q with respect to the terminalvoltage b e i n g denoted a s 0 , and c . = ( 6 . j - 9 ) , the totalcurrent becomes:i D DD y D Q DFl

I J ~ + eFDYQQ vQ YF

~ ~ ~ ~ ( 2 0 )

with YDD = - (dqj Y q d j )

DQ I

S i n f C o s 4j j

dq j C o s 2 - Yd. i n 2 .qdj IQD = dq j S i n 2 . + Yd. Cos.i i~~~ qjY = -YQQ DD

Y D F e F D Y d f j Cos f * e f d jF FD j j ~ ~ ~Y * e I Y d f j S i n 4. e f d j

Fo r Equation ( 2 0 ) to r e p r e s e n t a t w o - a x e s s y n c h r o n o u smachine, it i s n e c e s s a r y to have an angle e such thatYQ F and the diagonal elements Y D D and Y Q Q vanish.Th e diagonal elements do not vanish fo r an arbitrarydirection G of t h e axes of t h e equivalent machine.They do so i f they are rotated by the a m o u n t A E ) , with:

t an 2 AO = -Q DDDQ QD ( 2 1 )Since th e admittances Y D D , Y , Y and Y O D a refrequency dependant, t h i s c o n d i t i o n aRd the conaitionY Q F = 0 c a n only be a p p r o x i m a t e d . A good a p p r o x i m a t i o nwas found when Equation ( 2 1 ) i s satisfied for s + j - .

ACKNOWLEDGEMENTST h i s work was sponsored by th e Electric PowerResearch Institute under Project RP- 76 3 a nd managed fo rEPRI by Timothy Yau. The Pacific Ga s and ElectricCompany contributed by providing system data and trans-i e n t stability cases. The individual assistance ofClifford C . Young, Ji m F . Luini and John Rostoni ofPacific Ga s and Electric Company i s gratefully appre-c i a t e d .

REFERENCES[ 1 ] J . M . U n d r i l l , J. A. Casazza, E. M . Gulachenski,L . K. Kirchmayer, " E l e c t r o m e c h a n i c a l Equivalentsfo r Us e in Power System Stability Studies," I E E E

Trans., Vo l. P AS -9 0, September/October 1 9 7 1 , pp.2060-2071.[ 2 1 Systems Control, Inc., "Coherency-Based DynamicEquivalents fo r Transient Stability Studies,"Final Report on EPRI Project RP -9 0- 4, P ha se II,January 1 9 7 5 .[ 3 ] R . W. deMello, R . Podmore, K. N. Stanton, "Coher-ency-Based Dynamic Equivalents: Applications i nTransient Stability Studies," 1975 PICA ConferenceProceedings, pp . 2 3-31.[ 4 ] R . Podmore, "Identification of Coherent Generatorsfo r D y n a m i c E q u i v a l e n ts , " p a p e r F771555-5. IEEEWinter Power Meeting, Ne w York, January 1977.[ 5 ] " D y n a m i c Models fo r Steam and Hydro Turbines inPower System S t u d i e s , " I E E E Committee Report,IEEE Trans., V ol . P AS -9 2, November/December 1973,pp . 1904-1915.[ 6 ] "Excitation System D y n a m i c C h a r a c t e r i s t i c s , " I E E ECommittee Report, I E E E Trans., Vol. P A S - 9 2 ,J a n u a r y / F e b r u a r y 1 9 7 3 , pp . 64-75.[ 7 ] IEEE Committee Report, " C o m p u t e r R e p r e s e n t a t i o n ofExcitation S y s t e m s , " I E E E Trans., Vol. P A S - 8 7 ,June 1 9 6 8 , pp. 1460-1464.[ 8 ] Dynamic Equivalents fo r Transient StabilityS t u d i e s " , Final Report prepared fo r EPRI P r o j e c tRP-763 b y S ys te ms C o n t r o l , I n c . , A p r i l 1977.



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1 0 6 9D i s c u s s i o n

R . H . P a r k ( F a s t L o a d C o n t r o l I n c . , B r e w s t e r , M A ) : A p p a r e n t l y i n t h eW e s t e r n S y s t e m s i t i s c u s t o m a r y t o c a r r y o u t s t u d i e s o f p o w e r s y s t e mb eha vi or du r i ng d i s t u r b a n c e s w i t h a l l i n t e r c o n n e c t e d m a c h i n e sr e p r e s e n t e d . W h e r e t h i s i s n o t d o n e , a s may a p p l y , f o r t h e m o s t p a r t ,e a s t o f t h e R o c k i e s , t h e e f f e c t c a n o n l y b e t o c o n s i s t e n t l y i n t r o d u c e e r -r o r s i n t h e r e s u l t s o f c o m p u t a t i o n .A c c o r d i n g l y , i t w o u l d b e u s e f u l , i f , i n a f u t u r e p a p e r , i n f o r m a t i o nw e r e t o b e p r e s e n t e d t h a t w o u l d e v i d e n c e t h e e x t e n t o f e r r o r t h a tt y p i c a l l y c a n d e v e l o p i n d e p e n d e n c e o n t h e t y p e o f d i s t u r b i n g e v e n t a n dt h e p e r c e n t a g e r e p r e s e n t a t i o n o f t h e t o t a l i n t e r c o n n e c t e d s y s t e m .W h i l e i t i s h i g h l y d e s i r a b l e t o h a v e a v a i l a b l e e f f e c t i v e w a y s o fd e v e l o p i n g e q u i v a l e n t s f o r g r o u p s o f s y s t e m g e n e r a t o r s , i t c o u l d a l s o b ew o r t h g i v i n g a t l e a s t l i m i t e d c o n s i d e r a t i o n t o t h e d e v e l o p m e n t o fe q u i v a l e n t s f o r m o t o r l o a d s .P o s s i b l y t h e r e w o u l d b e s t b e o n e e q u i v a l e n t f o r s m a l l m o t o r s ,w h i c h w o u l d i n c l u d e m o t o r s u s e d i n a i r c o n d i t i o n e r s , o n e f o r m o t o r s o fi n t e r m e d i a t e s i z e , a n d w h e n a p p r o p r i a t e , o n e f o r m o t o r s o f l a r g e s i z e .

M a n u s c r i p t r e c e i v e d F e b r u a r y 2 2 , 1 9 7 7 .

A . J . Germond a n d R . P o d m o r e : T h e a u t h o r s t ha nk M r . P a r k f o r h i sc o m m e n t s a n d a r e i n a g r e e m e n t w i t h h i m . I n r e l a t i o n t o h i s f i r s t c o m -m e n t , t h e c o h e re nc y b a se d d y n a m i c e q u i v a l e n c i n g p r o g r a m h a s b e e n e x -p a n d e d t o h a n d l e u l t r a - l a r g e data b a s e s i n t h e r a n g e o f 1 0 , 0 00 b u ss esa n d 2 , 0 0 0 m a c h in es . T h e e x p a n d e d p r o g r a m i s b e i n g u s e d b y t h e ECARC o h e r e n c y S t u d i e s T a s k F o r c e t o c a l c u l a t e d y n a m i c e q u i v a l e n t s f o r t h edata b a s e s w h i c h a r e c o m p i l e d f o r t h e e n t i r e E a st er n U .S . i n t e r c o n n e c -t i o n b y t h e NERC M u l t i - r e g i o n a l M o d e l i n g G r o u p . T h e r e s u l t a n td y n a m i c e q u i v a l e n t s w i l l b e a m u c h m o r e m a n a g e a b l e s i z e f o r t r a n s i e n ts t a b i l i t y s t u d i e s . T h e y w i l l a l s o a l l o w a n a s s e s s m e n t o f how t h e m o r ec o m p l e t e r e p r e s e n t a t i o n o f t h e E a s t e r n U . S . i n t e r c o n n e c t i o n e f f e c t s t het r a n s i e n t s t a b i l i t y r e s u l t s a s s u g g e s t e d b y M r . P a r k .We a g r e e t h a t t h e d e v e l o p m e n t o f d y n a m i c e q u i v a l e n t s f o r m o t o rl o a d s i s a n i m p o r t a n t p r o b l e m w h i c h d e s e r v e s f u r t h e r s t u d y . An ap -p ro ac h w h i c h a g g r e g a t e s m o t o r s w h i c h h a v e s i m i l a r s i z e a n dc h a r a c t e r i s t i c s a p p e a r s t o b e a l o g i c a l o n e . T h e c o n c e p t s a n d t e c h n i q u e sw h i c h h a v e b e e n u s e d f o r a g g r e g a t i n g t h e s y n c h r o n o u s g e n e r a t o rm o d e l s ( e . g . , i d e n t i f y i n g e q u i v a l e n t p a r a m e t e r s b y f i t t i n g b o d ed i a g r a m s ) s h o u l d a l s o b e u s e f u l f o r a g g r e g a t i n g m o t o r m o d e l s .

M a n u s c r i p t r e c e i v e d D e c e m b e r 2 2 , 1 9 7 7 .

Dynamic Aggregation of Generator Unit Models

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