A Practical Method for the Direct_1979

8/8/2019 A Practical Method for the Direct_1979

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I E E E 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 - 9 8 , N o . 2 M a r c h / A p r i l 1 9 7 9A P RA CT IC AL METHOD FOR TH E DIRECT AN ALY SIS OF TRANSIENT STABILITY

R . PodmoreMember, IEEE S . VirmaniMemeber, IEEESystems Control, Inc.Palo Alto, California

ABSTRACTThis paper describes th e development and e valua-tion of an analytical method for the direct determina-tion of transient stability. The method developed isbased on th e analysis of transient energy an d accountsfo r th e nature of the system disturbance as well as fo rth e effects o f t ra ns fe r conductances onsystenmbehavior.It has be en evaluated on a 10 generat or 39 bus systeman d on a 2 0 generator 118 bus system. Th e method pre-d ic ts c rit ic al c le ar in g t im es fo r first swing transientstability which agree very closely with the results ofsimulations.Th e main conclusion of the study i s that the basica pp r oa ch d e ve lo p ed is practical, sufficiently accuratean d could be applied to realistic problems in power sys-te m planning an d operation.

INTRODUCTIONThe size an d complexity of modern power systems hasplaced i nc rea sed empha sis on the d eve lo pm en t o f improv-ed analytical techniques. Direct methods fo r analyzing

power system t ran si ent s tab il it y are particularly at-tractive as th ey h av e important applications inthe areaof power system planning, o pe ra ti on , a nd control. Theycan be used, for example, for predictingcritical clear-i ng t im es , fo r real-time security assessment an d in de -velopi ng stra tegi es fo r emergency state control.

Mangnusson [ 1 ] an d later Aylett [ 2 ] developed theoriginal energy based methods fo r stability analysis.In recent years these have b een considered a specialcase of the more general Lyapunov methods - th e energys im pl y b e in g one possible L yap uno v function. Consider-able effort ha s b ee n d ev ot ed to Lyapunov methods fo rpower system stab i li ty analysis in the l as t d ec ad e. Dueto space limitations, discussion of previous work isomitted here but may be found in [ 9 ] ; see also th e sur-veys [ 3 , 4 ] an d th e many references therein. A substan-tial part of th e effort has involved searching fo r bet-ter Lyapunov functions, i.e., Lyapunov functions thateither give larger regions of stability in state spaceor are valid fo r more complex system models. H owever,Ribbens-Pavella ha s shown [ 3 ] that, for the commonlyused system models, th e Lyapunov functions, derived us-in g th e correct state variables, are equivalent to th etransient energy function developed by Aylett [ 2 ] . Th etransient energy remains useful fo r s ta bi li ty a na ly si sand ha s th e advantage that a physical interpretation ofit s properties is available.

The transient energy function contains both kinetican d potential terms. The system kinetic energy, associ-

F 7 8 2 5 1 - 1 . A paper recommended and approved bytheIEEEPowerS y s t e m E n g i n e e r i n g Committee of th e IEEEPowerEngineeringSocietyf o r p r e s e n t a t i o n a t t h e IEEE PES Winter Meeting, NewYork,NY,J a n ua r y 2 9- F eb r ua r y 3, 1 9 7 8 . Manuscript submitted September 1 ,1 9 7 7 ; made a v a i l a b l e f o r printing December 12, 1977.

ated with th e relative motion of machine r ot o rs , isformally independent of the network. The system pote n-tial energy, a ssoc ia ted with th e potential energy ofn et wo r k e le m en ts an d machine rotors, is always definedfo r th e post-fault system, w ho se s ta bi li ty is to be an-alyzed. Th e principal idea of th e direct methods isthat a system's tran si ent stab ility can, fo r a givencontingency, be determineddirectly by comparing thetotal system energy which is ga in ed d ur in g the fault-onperiod, with a c e rt ai n c r it ic al potential energy. Fora two-machine system this critical energy is uniquelydefined and the di rec t a na lysis is equivalent to theequal area criteria.

For a system with three or more machinesth edirectanalysis becomes more difficult. Inthiscase the criti-ca l energy is no t uniquely defined and itsdeterminationbecomes the ke y step in the analysis. It is in thisstep that th e approach proposed her e differs markedlyand significantly from th oseadopted p rev iously which arebased on Lyapunov's second method. According to Lyapun-ov's theorem, th e critical energy is chosen to be thepotential energy at the unstable equilibriumpointclos-est ( i n terms of energy) to t he s tab le equilibriu-npoint.This unstable e qui li br ium p oi nt is called the c l o s e s tone [ 6 ] or t he l owe st saddle point [ 5 ] . This criticalenergy frequen tly yi elds results that are very conserva-tive, especially fo r systems with more than 3 or 4 gen-erators.

This conservativeness has severely l im it ed t heprac tic a l a ppli ca tion of Lyapunov methods to the powersystem transient s ta bi li ty p ro bl em . P re se nt ly , the othermajor limitation of analytical methods is the require-ment of simplified models, i.e., classicallyrepresentedgenerators and constant impedance loads. The latterlimitation is not as restrictive however, since muchuseful information can be obtained from studies withsimplified models. An analytical method can potential-ly provide a broader perspective o f the transient sta-bility problem and v aluably comp lement the step-by-stepsimulation o f i nd iv id ua l cases w it h d e ta il ed m od el s.

In this paper th e reasons for t he c o n s e r v a t i v e n e s sof direct methods are explored in r elation to o bs e r v e dpower system behavior and a new approach is proposedbased on this und er st and ing. T ec hn iq ue s which realizethis modified transient energy m e tho d arediscussed, in-cluding those which account fo r the effects of transferconductances of power system transient behavior. Thesum of these techniques constitute a comp lete alg orith mthat can be used to directly calculate critical clearingtimes without solving any differential equations, andresults which illustrate the practical significance ofthis approach are presented.

MATHEMATICAL FORMULATIONFor th e system m o d e l being c o n s i d e r e d in this in-vestigation the equations o f motion are:

( 1 ). = P. P11 1 ei=I 1where nP . =el j=l3 i

[ C . . si n ( 6 . --6) + D. . cos (6. -6.)ij i 1J 10 0 1 8 - 9 5 1 0 / 7 9 / 0 3 0 0 - 0 5 7 3 $ 0 0 . 7 5 1 9 7 9 IEEE

T . AthayMember, IEEE

5 7 3


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5 7 4P i = P m i - E i G i iC i j = E i E j B i j , D i j = E i E j G i j

a n d , fo r unit i ,P .mlG i iE .1

= mechanical power input= driving point conductance= constant voltage behind thedirect axistransient reactance

u i 6 i = generator rotor s pe ed and a ngl e d e vi a-t i o n s , respectivelyM . = moment o f ine rt ia1B..(G..) = transfer susceptance ( c o n d u c t a r c e ) in1 3 1 3 th e reduced bus admittance matrix.Equations ( 1 ) ar e written w it h r es pe ct to an arbi-trary sync hron ous reference frame. The transformationof these equations into th e c ente r of angle coordinatesno t o nly off ers p hysical insight to the transient sta-bility problem formulation i n general, bu t i n particu-la r provides a concise f ramework fo r th e analysis ofsystems with transfer conductances. Referring to ( 1 )define:

A n An6 1 / M T E Mi .i MT = E M li= 1 i= lthen

nM T UO = I PnI Pi= l i e i . I 1i = 1

n- n A- 2 1E I D.. cos 6. =i=l j=i+l

w0

( 2 )

( 3 )

The dynamics of th e center of angle ( o r center of iner-t i a ) reference are governed by ( 3 ) . By defining newangles an d s pe e ds r e la ti ve to this reference, 0 i A 6 -6A Q ~ ~ ~ ~ ~ ~ 1 i 0and ( L . W- th e system equations of motion become1M.W.P.-P.-M/MP 4111 ei i T CO A (4)

i- i=l...Notice that the center of angle variables s at is fy t heconstraints

n nI M 0.=I M. W. 0i=l i i=l 1 1The transient energy function V , which is always de -fined fo r th e post fault system, can be derived i h sev-eral ways. The expression given in ( 5 ) can be obtained

as in [ 2 ] by first establishing from ( 1 ) the n(n-l)/2relative acceleration equations, multiplying each ofthese by the corresponding relative velocity and inte-grating th e sum of the resulting equa ti ons f r o m a fixedlower limitof the stable equilibrium point (denoted bysuperscript " s " ) to a variable upper limit.

n-i ni = 1 s=i+l 2 Mi

(P M.-P.m.)6.-6 .)-C..(cos 6..-cos 6 ..M T i i M(ii d i j ~ij( dijC d i l6.+6.-26i j 0

Equation ( 5 ) can be algebraically manipulated andwritten in a more convenient form using th e center of

angle variables. Th e resulting exp ression, given i n ( 6 ) ,ca n alternatively be derived from ( 4 ) by applying th eab ov e s te ps to th e n ce nte r o f angle acceleration equa-tions.n -2 nV = 1 / 2 X M i 2 i - P i ( 0 i - 0 i )i = l i=l1n-l n F

- I i+ C i . . ( C o s e . . - co s 0i)0 i + 0 jJL 1- f D. . cos 0 . . d ( 0 + 0 . )S i + e s 1 ) 1 J J I0 . + 0

Th e terms of the transient energy function can bephysi ca lly i nterpreted in th e following way ( a l l changesare with respect to th e s tabl e equilibrium point):

n n 2* 1/2 I M.u. = 1/ 2 X MiW. _ 1/2 MTi= l 1 i= l

Total change in r otor kinetic energy relativeto COA = t ot al c ha nge in rotor kinetic energyminus change in COA kinetic energyn* 1i= l P i i = e i P i i 6 i E 1 06 0Change in rotor potential energy relative toCOA = change in rotor potential energy minuschange in CO A potential energy

* C.. (cos ,. -cos .. )1) 1) 1JChange in magnetic stored energy of branch i j

e i + 6* f D. . cos 0.. d (0+0.)1 1 1o s + e s I ] j 3

i JChange in dissipated energy of branch i j

Th e two expressions fo r th e rotor kineticand poten-tial energies show that th e change in energy associatedwith motion of th e system center of angle is subtractedfrom th e total system energy in order to obtain thetransient energy function. This interpretation of tran-sient energy corresponds with defining a stable systemas one existing in a stable, synchronous equilibriumbut not necessarily in frequency equilibrium [ 5 ] .

Th e synch ronous equilibrium p oints are the s ets ofsystem v ar i ab le s w hi c h satisfy ( 4 ) when the derivativesof speed and angle with respect to the center of angleare zero. These p oi nt s c or re spo nd to extrema o f thetransient energy function and, because the speeds arez e r o , to extrema o f the potential energy component aswell. Within a periodic frame of rotor angles t h e r eexists at most one extremum which is arelative minimumof th e potential energy, th e stable equilibrium point.Th e rest of the extrema are unstable equilibria ( a t mostn-l_ n-1 /ni2 -1 fo r n even, 2 +1/2 1/21-1fo r n odd [ 5 ]which correspond to relative max and saddlepoints ofthe potential energy function.

Th e p hy si ca l s ign if ic an ce of the center o f anglereference in th e transient stability problem formula-tion can be further illustrated b y c o ns id e ri ng in moredetail th e characterization o f equilibria. For systemswithout transfer conductances, the e qui li br ium p oi nt scan be obtained by solving, given app rop riate initialangles, n-l real power equations for an n machine systemwith one reference ( o r swing) machine as in [ 6 ] . Thisworks satisfactorily for such systems b e c a u s e , irrespec-tive of th e operating point, there is no change in load.

( 6 )


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5 7 5Fo r systems with transfer conductances however, th etotal load at an Unstable Equilibrium Point ( U E P ) willdiffer from that at the Stable Equilibrium Point ( S E P )and, when the equations of motion ar e formulated withrespect to a swing machine, th e difference will be al-located to th e swing bu s resulting i n a substantial mis-match. Thus th e s ys te m is clearly no t i n equilibriumas th e swing machine i s accelerating relative to th erest of the system.Equation ( 7 ) , th e re-writtenequilibriumcharacter-ization obtained from ( 4 ) , shows that all absolute ac -celerations ar e equal or that th e relative accelerationsar e zero.( P i P e i ) / M i P C O A / M T i = l , . . , n - l ( 7 )

The sy ste m i s thus in synchronous equilibrium althoughit may no t be i n frequency equilibrium. Th e signifi-cance of defining th e equilibrium condition i n terms of( 7 ) i s that th e relative accelerations are zero an d th eproblem of a large mismatch at the reference bus i stherefore eliminated.THE CONSERVATIVE NATURE OF LYAPUNOV METHODS

The reason for the conservative nature of the Lya-punov m et ho ds h as until recently received little atten-t i o n , probably because this ha s been widely accepted asan inherent characteristic of the method. Fo r th e part-icular case of th e transient energy function, the conser-vativeness can be explained by separating th e transientenergy into kinetic energy an d potential energy compon-ents an d g e n e r a l i z i n g th e well-known mechanical analogyto more than two machines. Th e multi-machine stabilityproblem may be visualized as a ball rolling on th e po -tential energy surface in multi-dimensional space. Th ecoordinates at this space ar e th e rotor a n g l e s an d th epost-fault steady state for ms the minimum point of amulti-dimensional bowl. Depending upon th e total kine-tic plus potential energy of th e ball at th e tine offault clearing, theballcan either escape from th e bowlo v e r a saddle ( i . e . , an unstable c a s e ) or i t can contin-ue to oscillate within th e bowl ( i . e . , astablecase).I f th e motion of th e ball i s u n d a m p e d , i t will continueto oscillate indefinitely an d fo r a conservative e s t i -mate of stability it i s assumed that th e ball may atsome time approach an d escape over th e lowest saddlepoint. ( T h e often repeated argument that the systemwill always, given enough energy, escape over th elowest saddle point i s suspect t h o u g h , as this i m p l i e sthat th e Lyapunov a n a l y s i s y i el d s necessary as well assufficient c o n d i t i o n s . ) However, when this criteriai s applied to p ow er s ys te m stability th e results ar e soconservative that they ar e of little practical value.

Th e reason fo r th e conservative nature of th e pre-dicted s t a b i l i t y r e g i o n i s that even though th e systemt r a j e c t o r y ( i . e . , th e ball i n t h e mechanical a n a l o g y )may ultimately a p p r o a c h th e lowest saddle p o i n t , it mayno t do so in a reasonably finite t i m e . A system thatremains in s y n c h r o n i s m during th e f ir st fe w s w i n g s isgenerally defined as transiently stable an d criticalc l e a r i n g times ar e estimated on this basis. Fo r thosecases i n which th e system t r a j e c t o r y does not a p p r o a c hth e lowest saddle p o i n t d u r i n g th e p e r i o d of interestth e Lyapunov method i s conservative.

Further insight into th e conservativeness problemcan be obtained by r e a l i z i n g that the saddle p o i n t s ofth e potential energy function can be u n i q u e l y associat-ed with th e different p o s s i b l e combinations of unstablegenerators [ 6 ] , an d that th e c o n s i d e r a t i o n of the mini-m um saddle p o i n t , t h e r e f o r e , c o r r e s p o n d s to a s s u m i n gthat th e total transient fault energy contributes toc a u s i n g th e most w e a k l y c o u p l e d group of generators tobe unstable. This a s s u m p t i o n i s c o m p l e t e l y at odds with

observable p ow er s ys te m behavior since it i s well knownthat th e generators which tend to be unstable ar e in -fluenced primarily by the fault location. In particu-l a r , in large systems it i s possible to have v er y w ea k-ly c oupled gen era tors which remain stable because theyar e re mote from the fault location. T he e xi st en ce ofsuch cases explains wh y Lyapunov's results fo r largesystems h av e be en extremely conservative [ 7 ] . On theother hand, i n small 3 or 4 machine studies, reasonableresults ar e o fte n o btained , because there i s a muchgreater chance that the most weakly coupled generatorswill be unstable for any fault location.

These concepts ca n be reinforced by means of anillustrative example. Consider the thr ee machine systemshown i n Figure 1 . Th e potential energy surface fo rthis system i s shown i n Figure 2 as ar e th e actual sta-bl e an d unstable system trajectories (determined froma step-by-step simulation) fo r two fault locations,buses 1 an d 2 . T he s tab le cases were cleared withoutline switching at t=0. 20 sec onds and th e unstable casesat t=0.22 seconds. In th e mechanical analogy thesetrajectories correspond to th e tracks of a ball rollingon th e potential surface, where th e effect of th e faulti s t o give th e ball ( t h e system) an initial angle posi-tion ( a n d corresponding potential energy) an d a n initialmomentum vector ( a n d corresponding kinetic e ne rgy ). T hestable equilibrium point, from which th e trajectoriesi n Figure 2 originate, i s a localminimnumof th e poten-tial energy surface where, by construction, th e poten-tial energy i s zero.

1 . 0 8 8 v - I L 1 5 22 . 4 9 _1.5+J.4V . V ' 8 . 1 9J . 4 6 H - 60

j . 0 54 . 2 1

For th e unstable cases it can be seen that the ballescapes t h r o u g h a saddle p o i n t o f th e p o t e n t i a l energyfunction. Th e ball escapes when i t receives j u s t e n o u g henergy fro m the fault condition to reach th e saddle p o i n tthat c o r r e s p o n d s to th e p a r t i c u l a r fault and pass t h r o u g hi t . This i s illustrated i n F i g u r e 3 w h e r e , fo r each of


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5 7 6th e tw o fault cases, th e value of th e total energy gain-ed during th e fault-on period is plotted as a functionof time. Th e value of th e potential energy at each ofth e saddle points (unstable equilibrium points) i s alsoshown. For a particular fault case, th e time at whichth e total energy becomes equal to the potential energyat th e corresponding s ad dl e p o in t ( t h e critical energy)i s the critical clearing time. From Figure 3 the criti-ca l clearing time i s , fo r each of th e two fault cases,approximately 0.21 seconds. In th e Lyapunov method how-ever, th e critical energy i s taken to be th e minimumvalue of unstable equilibrium point energies regardlessof fault location. This corresponds in th e example toU E P ( 1 ) . From Figure 3 it ca n be seen that this results,fo r a fault-on Unit 2 , in the very conservatively pre-dicted critical clearing time of approximately 0.16seconds.

TOETALE R G Y -U N I T 2E N E R G Y IV P E U E P 2) )

E N E R G Y - U N I T 1

T IN ( S E C )

The previous explanation of th e conservativenessof th e Lyapunov method naturally leads to th e conclu-sion that a more realistic criterion fo r determiningtransient stability should be based upon consideringthe unstable equilibrium point corresponding to th egenerators which actually go unstable fo r a particularf a u l t . That i s , th e critical energy should correspondto th e actual boundary of separation r a t h e r than th eweakest one.

Th e modified transient energy method proposed,which i n c o r p o r a t e s this ke y idea, consists of th e fol-lowing steps:Step 1 : Determine th e actual boundary of separa-tion fo r a particular fault location.Step 2 : Calculate th e unstable equilibrium pointcorresponding to the boundary determinedin Step 1 an d calculate th e p o t e n t i a l

energy at that p o i n t . This i s the criti-ca l energy V c .Step 3 : Calculate th e system trajectory during th efault-on period and compute th e total sys-tem energy as a f unction of t i m e .Step 4 : I f th e total system energy at the time offault clearing ( f r o m Step 3 ) i s less thanVc th e system i s considered stable; ifgreater, the s y s t e m i s c o n s i d e t e d unstable.Th e time at which th e total system energybecomes equal to Vc i s th e critical clear-i n g t i m e .While th e analytical approach developed fo r Step 1as well as that fo r incorporating th e effects of trans-

fe r conductances i s discussed in th e following two sec-t i o n s , several intermediate results a re gi ve n here inorder to illustrate th e improved accuracy provided bythis p r o c e d u r e . For these tests transfer conductanceswere neglected an d th e boundaries of separation fo r ap a r t i c u l a r fault location were determined by simulationstudies. Th e 10 unit 3 9 bus system used is describedlater in th e Evaluation section.Th e critical clearing times fo r four fault locations( f a u l t s on generator terminals, cleared without lines w i t c h i n g are shown in Table 1 . In this table th e e n t r y"corresponding UEP" is th e result predicted using th eproposed procedure while the "closest UEP" result wasobtained using th e basic Lyapunov method. Th e criti-cally stable an d unstable clearing times determined bysimulation are also shown. A comparison of these resultsshows that a substantial improvement in th e predictionaccuracy is obtained when th e basic analytical methodis modified to account fo r fault location.

Table 1Critical C l e a r i n g Times fo r 10 Unit 39 Bus S y s t e mCritical C l e a r i n g TimesSimulation Transient Energy

MethodStable Unstable Correspond- ClosestFault Location i n g UE P UE P. 3 .3 .3.2.2Bus 31 .28 .3 0 . 2 8 .2 3Bus 32 .3 0 . 3 2 .2 9 .2 2Bu s 35 .3 4 .3 6 . 3 3 .2 9Bus 38 . 1 8 . 2 0 . 1 8 .1 8ACCOUNTING FOR TRANSFER CONDUCTANCES

Th e major difficulty i n th e analysis of systems withtransfer conductances i s that a closed form expressionfor the total system energy c ann ot b e o bt ai ne d. Manyprevious researchers have neglected transfer conductanceson th e basis that these ar e small. However, th e transferconductances, i.e., the real part of th e off .diagonalelements of the r ed uc ed b us admittance matrix, dependno t only on th e transmission line resistances but alsoon loads modeled as fixed i mp ed anc es . T he re fo re , theseterms can be quite l ar ge an d in general cannot be neg-lected.

Th e integral terms in th e transient energy functionwhich a ri se w he n t ra ns fe r c o nd uc ta nc e s are not neglectedare, from ( 6 ) :e . + en-l n 1 3 A n-l n

i= l i = i + l s s 1 j 1 j i = i j = i + l '1 J

( 8 )

These terms represent energy dissipated in th e transferconductances an d ar e naturally in th e form of an inte-gr al w hi ch is path dependant. T he c or re ct path i s thatof th e actual system trajectory and, when i t i s known,th e terms of ( 8 ) can bevery effectively evaluated num-erically using th e trapezoidal rule [ 9 ] . I n th e modi-fied transient energy method th e only s te p wh er e an act-ual trajectory is ( o r needs to b e ) known is Step 3 , how-ever, where t he f aul t trajectory is used to obtain th etotal energy which the system gains during th e fault-onperiod.

I n order to obtain th e critical energy Vc, which isth e energyat th e UE P corresponding to th e actual bound-ar y of separation, a linear trajectory in th e angle spaceis assumed. This allows the te rm s in ( 8 ) to b e a na lyti-cally evalua ted b etween th e limits O S an d 0 u with th eexpression given in ( 9 ) .


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5 7 7u u s sFe i +0 e i i sD. sin 0 9i 1) e U_0* U_0S+0*s i s j1 1

Figures 4 an d 5 i llustra te typic al errors i n thepotential energy which this approximation introduces.In these figures th e actual ( a s c omputed n umeric a lly ina simulation) an d a pp ro x im at e p ot e nt ia l energy varia-tions as a function of time along th e critical stablea nd u ns ta bl e trajectories are shown fo r two differentfault cases. In th e former case a fi rs t swi ng i ns ta -bility occurs when t he faul t is cleared by line switch-ing at t=0.22 seconds. In th e latter case, althoughsevere intermachine oscillations exist, the approxima-tion is quite ac ceptab le . Inc id entally, because th etotal system energy is constant after fault clearing asshown, this figure dramatically illustrates th e exchangeof system potential and kinetic energy that fundament-a lly c hara cteriz es th e multi-swing behavior.

1 4E N E R G Y

1 2TOTAL E N E R G Y ( U N S T A B L E C A S E )

ActualI- . . Approate ( U n s t a b l e C a s e )I- - _ - A c t u a l|* _ - *ppoj (Stable Case)| -- Actual T O T A L E N E R C Y ( S T A B L E C A S E )

ability of the post f aul t n et wo r k is suc h that the unitgrouping pattern established during thisperiod is neversignificantly altered. Generally, when this is the case,th e swi tc hi ng time is not a significant factor either.Alternatively, when the post f aul t ne tw or k d oe s have amajor impact on th e ultimate boundary of separation,which i s true in multi-swing cases, then the switchingtime is significant as well, so these two factors tendto go together.Th e approach to determining first swing boundariesis considered first. It relies primarily on a faulttrajectory a pp ro xi ma ti on a nd on several ke y propertiesof th e potential energy function. The estimated systemfault trajectory i s obtained by explicitly integratingtwice a simple a pproxi mati on fo r the unit accelerations.Representing th e center of angle unit accelerating pow-

ers of the faulted system by f i ( i . e . , f i = th e righthand sid e of ( 4 ) ) , th e form of th e approximation is givenin ( 1 0 ) .f i ai bi f l t ( 1 0 )

This form was initially chosen based on careful observa-tion of simula ti on results of th e 10 an d 20 unit testsystems described later in th e evaluation section. Com-putational details involved in determining the unknownconstants ai, b i , i=1,2,...n an d the frequency n areomitted here; basically, two power flow solutions areutilized [ 9 ] . Th e first, at the instant of f au lt a pp li -c at io n, f ix es th e vectors a and b fo r a given frequencyn . The second, along an approximate trajectory shortlyafter t he f aul t, is used to compute r. Angles obtainedfrom this fault traj ec tory approximation are given inTable 2 fo r a particular case on the 1 0 unit system.Also shown fo r comparison are th e actual angles andtho se o bt aine d by assuming a constant a c celerati on dur-in g th e fault. Th e c osi ne a pproxima ti on errors illustra-te d i n Table 2 are among th e w o r s t obtained for numerousfault cases.

Table 2Comparison of F au lt T r aj e ct o ry A ng le s at t=0.4 Sec.(Fault on Bus 1 5 , 10-Unit System)

12ENERGY

10

a

6

-2

DETERMINING BOUNDARIES OF SEPARATIONIn addition to the generation parameters and pre-fault loading condition, which are independant of faultlocation, three fact ors determine the boundary of sep-aration in a particular unstable case: th e f aul t systemtrajectory; th e post fault network; and the switchingtime. Fo r the case of first swing instability, theeffect of the fault system trajectory is dominant. Rough-ly speaking, in these cases the system begins to split

up during t he f au lt -o n period and the synchronizing cap-

Th e potential energy function is a first integralof th e system differential equationswritten withrespectto th e center o f angle. While the approximation intro-duced previously is used to evaluate the energy at aparticular point in the angle space, the potential energygradient i s just the negative of the vector of a c c e l e r a-ting powers calculated using the power flow equations ( 4 ) .Th e power mismatch function F ( e ) defined in ( 1 1 ) cantherefore b e i nt e rp re t ed as th e euclidian norm squaredof this gradient.n 2F(0) f2 ( 0 ) (11)i=1

T he s cal ar function F(0) is a measure o f closeness toanequilibrium point. Indeed, this property is t he basisof mi ni miz a tion techn iques fo r calculating equilibriumpoints [ 8 ] ; in this project a variation of the Davidon-

ConstantCosine AccelerationUnit Actual Approximation Approximation1 -38.0 -38.1 -40.82 55.7 55.4 64.13 63.2 63.2 73.14 98.5 98.8 114.75 91.2 91.0 85.76 95.0 95.7 98.77 100.7 101.2 111.38 43.9 42.4 55.69 71 .7 71 .5 75.310 8.2 9.4 1.7


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57 8Fletcher-Powell ( D F P ) method ha s been used fo r t h l i spurpose.

Second order curvature i n f o r m a t i o n of the potentialsurface i s contained i n ( t h e negative o f ) th e Jacobianmatrix. I n th e region containing the SEP th e surface i sconvex an d i s referred to as th e principle region; itsclosed boundary i s the principle singular surface [ 5 ] .This i s the boundary of th e region of dynamic stabilitythat corresponds i n a tw o machine system to th e peak ofthe power transfer curve. A load flow algorithm basedon the minimization of F ( 0 ) will, initialized insideth e principle singular s u r f a c e , converge to th e S E P .Initialized outside, it will ( a l m o s t always) convergeto a UEP.

The basic procedure fo r determining first swingboundaries of separation, which utilizes these proper-ties of th e potential e ne rgy a nd th e fault trajectoryapproximation, consists of th e following steps:Step 1 : Fo r th e given f a u l t , determine th e faulttrajectory approximation parameters a i ,b i , i = 1,2,...n an d n l .Step 2 : Calculate th e post fault reduced admit-

tance matrix an d compute the post faultstable equilibrium point e S E P 2 .Step 3 : Using th e approximation of Step 1 , findth e t i m e at which the ( p o s t - f a u l t ) powermismatch function F ( e ) i s a maximum alongthe fault t r a j e c t o r y . Th e angles at thist i m e ar e e S s : t h i s i s ( v e r y close t o )th e i n t e r s e c t i o n point of th e fault tra-jectory with th e p r i n c i p l e s i n g u l a r sur-f a c e .Step 4 : Construct th e vector e S S - 6 S E P 2 an d nor-malized it to form th e direction vectorh.Step 5 : Solve th e on e d i m e n s i o n a l minimizationproblem

min F ( e O ( X ) ) AF = ( e ( X * ) )where 0 ( X ) e SS + X * h

A ""an d 0 ( X * ) ouStep 6 : With 0 u as a s t a r t i n g p o i n t , us e th e DFPminimization load flow t e c h n i q u e to ob-tain the unstable e q u i l i b r i u m p o i n t 0 u .O u characterizes the first s w i n g boundaryof separation an d th e p o t e n t i a l energy atthat point i s th e critical energy fo r theparticular fault case.F i g u r e s 6 an d 7 illustrate this p r o c e d u r e . InFigure 6 F ( 0 ) i s plotted versus time a l o n g an actual( s i m u l a t i o n ) c r i t i c a l l y unstable t r a j e c t o r y fo r a caseon the 1 0 unit s y s t e m . Th e relative maximum shown cor-responds to O S in th e p r o c e d u r e , while the relativeminimum occurs with the t r a j e c t o r y p a s s i n g b y the UE Pof i n t e r e s t . A contour p l o t , b a s i c a l l y a s l i c e out ofthe n-l dimensional s u r f a c e , of F ( 0 ) i s shown i n F i g u r e7 fo r th e p o s t - f a u l t network of t h i s fault case. Th eactual unstable t r a j e c t o r y , cleared at O S S f o r i l l u s t r a -t i v e purposes ( s l i g h t l y but i n s i g n i f i c a n t l y b e y o n d th ecritical c l e a r i n g p o i n t in this c a s e ) , i s also shown.T h e fault t r a j e c t o r y a p p r o x i m a t i o n i s q u i t e accur-at e an d its intersection with th e p r i n c i p l e s i n g u l a rsurface O S S , determined i n S t e p 3 , i s shown i n F i g u r e 7 .The basic idea of this s t e p , which t y p i c a l l y r e q u i r e sseveral cubic i n t e r p o l a t i o n s an d c o r r e s p o n d i n g power

flow s o l u t i o n s , is to le t the fault trajectory ru n longe n o u g h fo r th e effects of the faulted system to become

* e q I

1800 2 ( d e g r e e s )

established. ( A power flow solution i s a single eval-uation of th e RH S of Equation ( 4 ) with a given se t ofa n g l e s . ) Th e direction vector formed i n Step 4 i s alsoi l l u s t r a t e d , as i s the point of minimum F(0) along i to u , which is determined in Step 5 . Again, on e or tw ocubic interpolations ( t w o power flow solutions e a c h )ar e the major computations here. Finally, although itappears that O u is very near a UEP other than ou dueto th e slice effect, th e UE P calculation of Step 6 doesi n fact converge to O u as represented by the dottedline in Figure 7 .

Determining the boundary of separation, that i sth e se t of unstable generators, i s more difficult whenth e synchronizing intermachine oscillations of th epost-fault system ar e significant which is the case fo rmultiple swing instabilities. A simple an d promisingapproach to this problem has be en developed that i sbased on th e predictor/corrector type of integrationscheme.

Consider th e situation at th e completion of Step 5where th e UEP estimate e u has be en determined. Th e as-sumption in searching along a straight line fr om th e prin-ciple singular surface i s that intermachine synchronizingtorques due to the post fault network will no t signifi-cantly affect th e grouping pattern which t he fa ul t tra-jectory h a s . e s t a b l i s h e d . T he b as ic idea now is to treatthe angles O u as if they were th e result of th e predic-tion step in an integration step an d to compute a cor-rection st ep b as ed on th e post-fault power flow e q u a t b n s .Assuming that the f aul t iscleared at th e principle sin-gular surface, t he i ni ti al unit accelerating powers andtheir time derivatives c an b e obtained from a power flowsolution at that point and the speeds as obtained fromthe fault trajectory approximation. Using th e acceler-ating powers computed by a power flow at d u an d an esti-


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mate for the time step obtained by assuming a linearv ar ia ti on o f th e norm of machine angles over a shorttime interval, a cubic polynomia l i n time can be fittedfor th e accelerating powers on each unit between th epoints 6 S S an d 0 . This can then be a na lytic a lly i nte-grated i n order to complete th e correction step [ 9 ] .Base d on this c or re ctio n step, a new directionvector i s formed and Step 5 of the b as ic p r oc ed ur e i sr epe ate d. Wh ile the step taken ca n be quite large, in

which case th e predicted angles may be in absoluteerror with respect to time, th e procedure effectivelyd ete ct s r elat ive motion, i.e., th e relative amount ofsynchronizing energy which each ma chin e exchanges withth e network. In the case of a first swing instability,th e predictor/corrector step makes v er y l it tl e changei n th e initial direction v ec to r ass um ed and th e effecti s to confirm th e f ir st UE P estimate.DETERMIN ING S Y STEM ENERGY DURING FAULT

D ir ec t m e th od s of stability analysis ar e limitedto stationary systems. Fo r th e power system transientStability problem the y are applicable only to th e post-fault network an d i t i s still necessary to obtain th etotal energy which the system gains during th e fault-on period. This has previously b ee n a cc om pl is he d byperforming a step-by-step integration of th e faultedsystem equations an d simultaneously calculating th eappropriate energy or Lyapunov functions.While initially developed to ai d b ound ar y o f sep-aration determination, th e fault trajectory approximationintroduced i n th e previous section ca n be used to obtainthis t ot al e ne rgy very effectively. This i s illustratedi n Figure 8 wher e the actual (froma simulation) an d th eapproximate energy gained during th e fault i s plottedfo r a typical case on th e 10 unit system. Th e approxi-mation i s valid well beyond th e critical clearing time.As a result th e modified transient energymethod report-ed here i s truly a direct method in the sense that th estability assessment which it provides can be obtainedwithout solving an y differential equations.

6 0 1T O T A L E U E R OD U R D N G h A I I L T A C T U A L5 0 -

20 /

1 0 ,

0 . 2 0 0.N0 S R C O R D S 0 . 6 0t c

EVALUATIONTh e basic approach to evaluation which ha s beenadopted i s to perform a reasonably c o m p r e h e n s i v e serieso f c la ss ic al s t a b i l i t y analyses using th e accepted met-ho d of simulation an d to compare th e results with thoseobtained from th e m od if ie d tr an sie nt energy method.While these two t e c h n i q u e s differ m a r k e d l y , both ca n beused to calculate critical c l e a r i n g t i m e s , so t h i s fami-

liar transient s t a b i li t y p e r f o r ma n c e measure provides

5 7 9an explicit basis of comparison.

Two test systems have been used. Th e f ir st isa10 unit 39 bus syste m which i s representative of theNew England s yst em and was stud ie d in [ 7 ] . A line dia-gram, generator parameters a nd i ni ti al conditions ar eprovided i n th e Appendix. Th e second i s a. 20 unit 118bus IEEE test system which wa s also studied i n [ 7 ] .Generator parameters and initial conditions fo r this sys-te m ar e i n th e Appendix. Complete data fo r both systemsi s i n [ 9 ] .A range of faults were tested i n c l u d i n g faults onb ot h ge ne r at or and lo ad buses, cleared with an d withoutline switching. Considerable insight ha s been gained onth e different characteristics of these systems, andmanyof the faults were chosen so as to create situationsthat critically test th e method. Under these f a u l t s ,th e systems exhibited a wide range of s t a b i l i t y behaviorincluding single an d multi-unit as well as first s w i n gan d multiswing instability.Table 3 shows critical c l e a r i n g times calculateddirectly along with those obtained vi a simulation fo rth e 10 unit system. Th e first d ir ec t r es ul t c o r r e s p o n d sto using th e first swing boundary determination procedure

i n Step 1 of th e proposed method while th e second resultcorresponds to u s i n g , i n addition, thepredictor/correct-or step. Table 4 , s im il ar ly a rr ange d, shows criticalclearing times calculated for th e 20 unit system.Table 3Direct Calculation of Critical C l e a r i n g Times for10 Unit System

"Line Tripped ' S I M U L A T I O N DIRECT C A L C U L A T I O N S* - f a u l t e d P r e d i c t o r /b u s s t a b l e unstable First S w i n g Corrector2 * - 3 0 . 2 4 0 . 2 6 0 . 2 3 5 0 . 2 2 54 * - 1 4 0 . 2 2 0 . 2 4 0 . 2 1 5 0 . 2 1 55 * - 1 1 ( 0 . 2 0 0.22 0.21 0.211 5 * - 1 6 0 . 2 2 0.24 0.23 0.23_ 2 3 - 2 4 * 0 . 1 8 0 . 2 0 0.175 0.175

2 5 * - 2 6 0 . 1 8 0 . 2 0 0.19 0.222 8 - 2 9 * 0 . 0 4 0 . 0 6 0.06 0.063 1 * 0 . 2 2 0 . 2 4 0.22 0 . 2 23 5 * 0 . 2 4 0 . 2 6 0 . 2 5 0 . 2 53 7 * 0. 2 2 0 . 2 4 0 . 2 3 0 . 2 3Table 4Direct Calculation of Critical Clearing Times fo r20 Unit System

SIMULATION DIRECT CALCULATIONSLine T r i p p e d*- f a u l t e d P r e d i c t o r /b u s s t a b l e u n s t a b l e F i r s t S w i n g Corrector2 0 . 1 8 0 . 2 0 0 . 1 8 0 . 1 83 0 . 4 6 0 . 4 8 0 . 5 0 0 . 3 94 0 . 3 2 0 . 3 4 0 . 3 1 0 . 3 35 0 . 3 8 0 . 4 0 0 . 3 6 0 . 3 65 1 0 . 4 6 0 . 4 8 0 . 4 6 0 . 0 01 3 0 . 3 4 0 . 3 6 0 . 3 3 0 . 3 51 8 0 . 3 2 0 . 3 4 0 . 3 3 0 . 3 5

Details of individual cases are p r o v i d e d in [ 9 ] .Th e general conclusion drawn from these results i s t h a t ,fo r an i n s t a b i l i t y that i s e s s e n t i a l l y first s w i n g , th ebasic procedure y i e l d s results o f p r a c t i c a l s i g n i f i c a n c e .Th e p r e d i c t o r / c o r r e c t o r i s effective i n d e t e c t i n g th epresence of major post fault s y n c h r o n i z i n g oscillationsan d determining th e relative unit motions that thereforeoccur. In some cases this additional s t e p i s sufficientfor obtaining th e correct m u l t i s w i n g b o u n d a r y whereas inother cases i t i s not.


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58 0CONCLUSION

This paper has contributed to the c le ar understand-ing a nd e ff ec ti ve removal of s ev er al m aj o r d if fi cu lt ie sth at h ave se rio us ly im pe de d th e a pp li ca ti on o f directmethods to th e power system transient stability-problemfo r many years. By accounting fo r th e effects o f faultlocation a nd t ra ns fe r conductanceson th e system's tran-s ie nt b eha vi or , the approach to transient energy stabilityanalysis developed ha s provided results which have practi-ca l meaning a nd i mp or ta nc e. Th e method developedis ableto e ff ic ie nt ly and d ir ec tl y c al cul at e c rit ic al c le ar ingtimes associated with first swing transient instabilityw it ho ut e x pl ic it ly s ol vi ng any differential equations.Th e r es ul ts o bt aine d are practical in th e sense thatthey are sufficiently accurate an d they are derived us-ing power system models which, although simplified, arestill widely used by th e industry.

As a result of this progress further research isplanned in three major areas. Firstly, continued devel-opment of techniques fo r analyzing multiple swi ng i nsta-bility an d fo r unstable equilibrium point classificationi s planned that will increase the reliability of thebasic approach. Secondly it is intended that the algor-ithms be adapted to exploit network sparsity in order tof ur th er i nc r ea se their computational efficiency. Third-l y , an investigation will be made of the implicationsan d effects of modeling assumptions on transient energystab i li ty a na lysi s i nc ludi ng those of variations in fieldan d int er na l m ac hin e v ol ta qe s, d am ping, and n on -l in ea rloads.

With further development, the transient energymethod i s likely to yield new an d powerful c omputation -al tools fo r stability analysis that have potentiallyva luable a pplic ati on s in system planning and operation.Fo r system planning it could be employed as a screeningtool which would be useful fo r preliminary and/or longrange studies b y a ll ow in g a large number of alternativeplans to be studied in a fast and approximate manner.Fo r syst e m o p er ati on , m et ho ds b as ed on transient energycan be used fo r o n-l in e a ss es sm en t and enhancement oftransient stability.

ACKNOWLEDGEMENTThis research effort was supported by the EnergyResearch and Development Administration un de r c o nt r ac tnumber E(49-18) 207 6. T he d ir ec ti on a n d a s s i st a n c e pro-vided by Lester H. Fink and Kjell Carlsen of ERDA isgratefully acknowledged.

REFERENCES[ 1 ] P.C. Magnusson, "The Transient - Energy M e t l h o d ofCalculating Stability", A I EE T r an sa c ti on s Vol. 6 6 ,1947, pp. 7 47-755.[ 2 ] P . D . Aylott, "The Energy-Integral Criterion o f

Transient Stability Limits of Power Systems", pr o-ceedings IE E (London), Vol. 105(C), July 1958,pp. 527-536.

[ 3 ] M . Ribbens-Pavella, "Critical Survey of TransientStability Studies of Multimachine Power Systems byLiapunov's Direct Method", Proceedings of the 9thA nn ua l A ll e rt o n C on fe r en c e on Circuits and SystemsTheory, October 1971, pp. 751-7 6 7 .[ 4 ] A . A. Fouad, "Stability Theory-Criteria for Trans-ient Stability", Engineering Foundation ConferenceSyst ems Enginee ring for P ower, Henniker, New Hamp-shire, 1975.[ 5 ] C . J. Tavora and 0. J. M. Smith, "Stability Analy-si s of Power Systems", IEEE Transactions on Power

8 2 p a r a t u s an d S y s t e m s , Vol. P A S - 9 1 , M a y / J u n e 1 9 7 2 ,pp, 113 8-1145.[ 6 ] F. S . Prabhakara an d A. H. El-Abiad, " A SimplifiedDetermination of Transient Stability Regions forLiapunov Methods", IEEE Transactions on P owe r A pp -aratus and Systems, Vol. PAS-94, March/April 1975,

pp . 672-689.[ 7 ] G. W. Bills, et.al., " On-Line Stability AnalysisStudy" RP90-1 Report for the Edison Electric Insti-tute, October 12, 19 7 0.[ 8 ] A. M. Sasson, C. Trevino and F. Aboyte s, " Im pr ove dNewton ' s Load F l ow T hr o ug h a Minimization Technique",IEEE Transactions, Vol. PAS-90, 5 , pp 19 7 4-19 8 1,.September-October, 19 7 1.[ 9 ] T . Athay, R. Podmore and S. Virmani, "TransientEner gy S tab il ity A nal ysis ". Engineering FoundationC o nf e re n ce -S y st e m E ng in e er i ng for Power, Henniker,New Hampshire, August 21-26, 1 9 7 7 .

APPENDIX

Table 5Generator P a r a m e t e r s and Initial Conditions fo r1 0 Unit 3 9 Bus System (100 MW Base)

Unit 'P u A n g l eN o . H p u X d pu E pu ( r a d i a n s )1 5 0 0 . 0.006 1.036 8 -0.13442 30.30 0.06 97 1.196 6 0.34073 35.80 0.0531 1.1491 0.34174 38.60 0.0436 1.0808 0.29855 26.00 0.132 1.397 1 0 .50886 34.80 0.05 1.1910 0.33767 26.40 0.049 1.1394 0.34998 24.30 0.057 1.0709 0.30 709 34.50 0.057 1.136 8 0.533510 42.0 0.031 1.0929 -0.0087

Table 6Generator Parameters an d Initial Conditionsfo r 20 U n i t , 118 Bus System ( 1 0 0 MW Base)Unit H pu Xd pu E pu Angle( B u s ) (radians)No .

1 8. 0 0.0875 0.9875 -0.24952 22.0 0.0636 1.0955 0.36933 8.0 0 .1750 1.1808 -0.16884 14.0 0 .1 000 1.1280 0 .1 6 6 05 26.0 0 .0538 1.0525 0.16976 8.0 0.0875 0.9778 -0.24927 8. 0 0.0875 1.0005 -0.28058 8.0 0.0875 1.0 025 -0.42519 8.0 0.0875 1.0283 -0.413910 12.0 0.1167 1.2072 0.046311 10.0 0.1400 1.1351 0.02051 2 12.0 0.1167 0.9793 0.100 61 3 20.0 0.0700 1.1491 0.20911 4 20.0 0.0700 1.0852 0.21081 5 30.0 0.0467 1.0342 0.23671 6 28.0 0.0500 1.1266 0.19671 7 32.0 0.0438 1.0 423 0.43371 8 8.0 0.0875 1.0425 -0.00691 9 1 6 . 0 0.0875 1.1511 0.164120 15.0 0.0467 0.9957 -0.2730


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Unit 39 Bus N e w England Test System

Thomas M. A t h a y h a s o b t a i n e d t h r e e d e g r e e s i nelectrical e n g i n e e r i n g ; the B.S. in 1974 fromM i c h i g a n T e c h n o l o g i c a l U n i v e r s i t y , and t h eM . S . a n d E n g i n e e r ' s d e g r e e s b o t h i n 1 9 7 6 f r o mMassachusetts Institute of T e c h n o l o g y .W h i l e a t MIT h e s e r v e d as b o t h a T e a c h i n gAssistant and a R es ea rc h A s s i s t a n t , th e latterassignment b e i n g with the Electronics SystemsL a b o r a t o r y h e l p i n g i n t h e d e v e l o p m e n t o fdecentralized strategies for the control of inter-c o n n e c t e d p ow e r s y s t e m s .S i n c e j o i n i n g S C I i n 1 9 7 6 , M r . A t h a y h a s b e e n w o r k i n g w i t h t h eS y s t e m s A n a l y s i s D i v i s i o n . He h a s b e e n w o r k i n g p r i m a r i l y onh e d e v e l o p m e n t o f d i r e c t m e t h o d s f o r t r a n s i e n t s t a b i l i t y a n a l y s i s a n dt h e d e v e l o p m e n t o f a d v a n c e d a u t o m a t i c g e n e r a t i o n c o n t r o la n d a l g o r i t h m s . T h e s e two p r o j e c t s a r e b e i n g c o n d u c t e d f o r

Robin Podmore wasborn in NewZealand onJuly 20 , 1947. He received a Bachelor ofEngineering degree with first c lass honors, fromthe University of Canterbury, Christchurch,NewZealand, in 1968 and aPh.D. from thesame u n i v e r s i t y i n 1 9 7 3 .From 1973 to 1974 he was a Post-DoctoralFellow with the Power System Research Groupa t t h e U n i v e r s i t y o f S a s k a t c h e w a n , C a n a d a . I nMay 1974 he joined Systems Control, Inc. He i sp r e s e n t l y M a n a g e r o f t h e P o w e r S y s t e m sD i v i s i o n a n d i s r e s p o n s i b l e f o r s u p e r v i s i o n a n d management o f&D a c t i v i t i e s i n t h e areas o f c o m p u t e r a p p l i c a t i o n s f o r power s y s t e mo p e r a t i o n a n d c o n t r o l . He h a s m a n a g e d a n d w o r k e d as p r i n -i n v e s t i g a t o r on s e v e r a l m a j o r r e s e a r c h a n d d e v e l o p m e n t p r o j e c t sp o n s o r e d b y E l e c t r i c P o w e r R e s e a r c h I n s t i t u t e a n d U . S . D e p a r t m e n tf E n e r g y . H i s p r i n c i p l e p r o j e c t a s s i g n m e n t s h a v e i n c lu de d d ev e l o p-

ment o f d y n a m i c e q u i v a l e n t s a n d a n a l y t i c a l t e c h n i q u e s f o r use i n t r a ns i e n t s t a b i l i t y s t u d i e s , d e v e l o p m e n t o f new t e c h n i q u e s f o r o n- l i nm o n i t o r i n g a n d r a t i n g o f u n d e r g r o u n d c a b l e s , a n d d e v e l o p m e n t a n d i mp l e m e n t a t i o n o f new m e t h o d s f o r a u t o m a t i c g e n e r a t i o n c o n t r o l .D r . P o d m o r e i s a member o f t h e PES S y s t e m D y n a m i c P e r f o rmance S u b c o m mi t t ee a n d i s a R e g i s t e r e d P r o f e s s i o n a l E n g i n e e r i n t hS t a t e o f C a l i f o r n i a . He h as a u t h o r e d or c o - a u t h o r e d a p p r o x i m a t e l y 2p u b l i c a t i o n s i n t h e areas o f n e t w o r k a n a l y s i s , t r a n s i e n t s t a b i l i t ye c o n o m i c d i s p a t c h , a u t o m a t i c g e n e r a t i o n c o n t r o l a n d s e c u r i t y a s s e s sm e n t .

S u d h i r V i r m a n i o b t a i n e d h i s B . T e c h ( H o n s .d e g r e e f r o m t h e I n d i a n I n s t i t u t e o f T e c h n o l o ga n d t h e M . S . a n d P h . D . d e g r e e s f r o m t hU n i v e r s i t y o f W i s c o n s i n .He i s c u r r e n t l y S e n i o r P r o j e c t Manager aS S S y s t e m s C o n t r o l , I n c . w h e r e h e i s w o r k i n g t hareas of power s y s t e m d y n a m i c s a n d c o n t r o lA R ;; : A ; ; ; ; P r i o r t o j o i n i n g S y s t e m s C o n t r o l , I n c . , D r . V i rm an i w or ke d a t American E l e c t r i c Power S e rv i c e C o r p o r a t i o n a n d S t a g g S y s t e m s , I n c . iNew Y o r k .

D i s c u s s i o nR . P . S o o d ( D ep a r t me n t o f E l e c t r i c a l E n g i n e e r i n g , R e g i o n a l E n g i n e e ri n g C o l l e g e , K u r u k s h e t r a , I n d i a ) : The a u t h o r s ar e to b e h i g h l y com- m e n d e d f o r t h e i r e f f o r t s t o i m p r o v e upon e x i s t i n g methods o f e v a l u at i o n o f t r a n s i e n t s t a b i l i t y o f l a r g e m u l t i m a c h i n e power systems u s i nd i r e c t m e t h o d s .T h e a u t h o r s h a v e e x p e r i m e n t a l l y d e m o n s t r a t e d a b e t t e r accuracy ir e s u l t s b y a c c o u n t i n g f o r t h e e f f e c t s o f f a u l t l o c a t i o n and t r a n s f e r cond u c t a n c e s on t h e s y s t e m ' s t r a n s i e n t b e h a v i o u r f o r e v a lu a ti n g t h e c r i t i c ac l e a r i n g t i m e a s s o c i a t e d w i t h t h e f i r s t s w i n g t r a n s i e n t s t a b i l i t y . Woul

5 8 1

3 9

Figure 9 1 0


10/12

5 8 2t h e a u t h o r s p l e a s e express t h e i r o p i n i o n f o r a s s e s s i n g t h e i n f l ue n ce o f i n -c l u s i o n o f o t h e r m a c h i n e parameters s u c h as p o l e s a l i e n c y , d a m p i n g orf a s t governor a c t i o n a n d n o n - l i n e a r l o a d s t o t h i s p r o b l e m b y p r o p o s e da l g o r i t h m a n d t h e e f f e c t o n t h e o v e r a l l s y s t e m p e r f o r m a n c e ? Would t h ei n c l u s i o n o f t h e s e parameters l e a d t o c o m p l i c a t e d p r o c e d u r e i n d i g i t a lc o m p u t a t i o n s a n d c o s t ?R e c e n t l y B i l l i n t o n a n d Kuruganty h a v e proposed a p r o b a b i l i s t i ci n d e x f o r t r a n s i e n t s t a b i l i t y . Th e d i s c u s s o r f e e l s t h a t i n c o r p o r a t i n g , ex -p l o i t a t i o n o f n et wo r k s p a r s i t y , s t u d y o f i m p l i c a t i o n s o f m o d e l i n ga s s u m p t i o n a n d t h e p r o b a b i l i t y i n d e x [ A ] i n t h e p r o p o s e d a l g o r i t h mc o u l d l e a d t o a p o w e r f u l c o m p u t a t i o n a l t o o l f o r t r a n s i e n t s t a b i l i t y .

REFERENCE[ A l R . B i l l i n t o n , P . R . S . K u r u g a n t y " A p r o b a b i l i s t i c i n d e x f o r t r a n -s i e n t s t a b i l i t y " paper N o . A 7 8 2 3 1 - 3 , paper p r e s e n t e d a t IEEEP E S , W i n t e r m e e t i n g , New Y o r k , J a n . 2 9 - F e b . 3 , 1 9 7 8 .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 21 , 1 9 7 8 .

J . S a b a t e l a n d M. R i b b e n s - P a v e l l a ( U n i v e r s i t y o f L i e g e , L i e g e ,B e l g i u m ) : The a u t h o r s ar e t o b e commended f o r t h e i r pa pe r ; i t i s an i n -t e r e s t i n g w o r k on t h e p r a c t i c a l us e o f d i r e c t m e t h o d s f o r t r a n s i e n ts t a b i l i t y a n a l y s i s o f power systems. We f e e l t h a t a f e w p o i n t s ar e w o r t ht o b e b r o u g h t out i n c o n n e c t i o n w i t h t h e two main q u e s t i o n s t r e a t e d i nt h e paper.A . W i t h r e s pe ct t o t h e d e t e r m i n a t i o n o f t h e "boundary o f separa-t i o n " , we w o u l d l i k e t o make t h e f o l l o w i n g comments.1 . C o n t r a r y t o t h e a u th o rs ' b e l i e f , we t h i n k t h a t t h e argument ac-c o r d i n g t o w h i c h " t h e ( m u l t i m a c h i n e ) system w i l l a l w a y s , g i v e n e n o u g henergy, escape over t h e l o w e s t s a d dl e p o i n t" , h a s s e l d o m b e e n u s e d ; i t i s

w e l l k n o w n , i n s t e a d , t h a t t h e c o n s i d e r a t i o n o f t h i s s a d d l e p o i n t - w h i c hp r o v i d e s a t h e o r e t i c a l e s t i m a t e o f t h e s t a b i l i t y d o m a i n f o r t h e e n e r g y -t y p e L i a p u n o v f u n c t i o n - h a s o f t e n b e e n a c k n o w l e d g e d as t h e m a i nreason f o r t h e c o ns e rv a t is m o f t h e m e t h o d ( s e e f o r i n s t a n c e p . 1 7 6 o fr e f . [ A ] ) . T r i a l s t o p r e c i s e l y g e t r i d o f t h i s d r a w b a c k h a v e l e d us t o pro-p o s e a m e t h o d d e s c r i b e d i n [ B ] .We h a v e c o m p a r e d ou r r e s u l t s w i t h t h o s e p r o v i d e d b y t h e p r o -c e d u r e s e t f o r t h i n t h e p r e s e n t p a p e r . T h e y ar e e x p r e s s e d i n t e r m s o fc r i t i c a l c l e a r i n g t i m e s ( t Q ) o f s e v e r a l s i m u l a t e d t h r e e - p h a s e s h o r t c i r c u i t sa p p l i e d t o s i x d i f f e r e n t power s y s t e m s . T h e t a b l e b e l o w c o n t a i n s a l lt h e s e r e s u l t s ; e x c e p t f o r t h o s e o f c o l u m n 2 , t h e y are a l l o b t a i n e d b y u s -i n g t h e e n e r g y - t y p e L i a p u n o v f u n c t i o n ( V ) w h i c h h a s t h e samea n a l y t i c a l e x p r e s s i o n w i t h t h e " t r a n s i e n t e n e r g y " f u n c t i o n : c o l u m n 3g i v e s t h e t , s f o u n d w h e n d e t e r m i n i n g t h e " b o u n d a r i e s o f s e p a r a t i o n "a c c o r d i n g t o t h e m i n i m i z a t i o n p r o c e d u r e o f t h e p r e s e n t p a p e r ; c o l u m n 4g i v e s t h e t , s f o u n d v i a t h e p r o c e d u r e o f r e f . [ 1 B ] ; c o l u m n 5 g i v e s t h e t c sp r o v i d e d b y t h e s t a n d a r d L i a p u n o v c r i t e r i o n , t h a t i s w h e n r e l a t i n g t h es t a b i l i t y d o m a i n t o t h e " c l o s e s t " u n s t a b l e e q u i l i b r i u m p o i n t . Weo b s e r v e t h a t , a l t h o u g h o b t a i n e d i n a d i f f e r e n t way, t h e r e s u l t s o f c o l -umns 3 a n d 4 h a v e a l m o s t t h e same d e g r e e o f a p p r o x i m a t i o n : g e n e r a l l yb e t t e r t h a n t h o s e o f c o l u m n 5 , t h e y may l e a d h o w e v e r t o l a r g e e r r o r s ,even i n some cases o f " f i r s t s w i n g i n s t a b i l i t y " .I n t h i s same t a b l e , c o l u m n 6 g i v e s t h e t h e t c s o b t a i n e d w h e n co m-p u t i n g t h e l i m i t v a l u e o f t h e V - f u n c t i o n b y r e l a t i n g i t t o t h e m a c h i n e ( s )w h i c h t h e f i r s t g o e s ( g o ) o u t o f s t e p ; t h i s d e t e r m i n a t i o n h a s b e e n p r o -p o s e d i m p l i c i t l y or e x p l i c i t l y i n [ 9 ] a n d [ C ] . We s e e t h a t t h e s e r e s u l t s a r ea l m o s t a l w a y s i n g o o d a g r e e m e n t w i t h t h o s e o f t h e s t e p - b y - s t e p m e t h o da n d are a l w a y s r e l i a b l e . N e v e r t h e l e s s , t h e p r a c t i c a l use o f t h i s p r o c e d u r ei m p l i e s t h e f a s t d e t e r m i n a t i o n o f t h e a b o v e " i n t e r e s t i n g " m a c h i n e .T r i a l s t o r e l a t e i t t o t h e one p r o v i d e d b y t h e m i n i m i z a t i o n p r o c e d u r e o ft h e p r e s e n t p a p e r d o n ' t seem s u c c e s s f u l ; t h r o u g h o u t ou r s i m u l a t i o n s ,we h a v e n o t s y s t e m a t i c a l l y e x p e r i e n c e d t h i s c o i n c i d e n c e . We w o u l d a p -p r e c i a t e t h e a u t h o r ' s comments on t h i s p o i n t .

BLE[ 1 v 2 3 4 5 6 7

C) tn04 : a ) Q OOYX C4 ;QOkk p r W C 0 0 - 00 0 C- H C R1 4 4 n 4 0 D 40 0 C4 ) v C IO _ 4 0 4) 6 '0.1 a d u a I 00~ ~ ~ c o 1 -1 0 00 0 Cl -o )4 ) . C 4)0. 4)4)- 4) 4C- *HO-0 ) 4 ) U U . mU) hU)>U) OB C U 4ooc-O 0 t ) C - to0c3 U. HCne24: 323 4443I3I3 0-HO10yse -004-0 3 - 0 4)ClA 34 41 4 4 ) - 140C43 8 3 8 2300A4-P0)

w 0140 0> - H 7 40-H 9CO 04 5 - E 0 .-~ 0 1 4 0 O C zn1B 3 34 N 1 4 U) 0 14 4 4 u3 0I d cU 0 . 0 om0 0-. - 0 Id 0 000syst4eB 4 ) 4 ) 4 )3-machi- 3 3 18-19 20 1 7 1 7 21 20ne 2 ) C - 2 32-33 3 4 3 4 3 1 3 3 3 1system j 1 1 37-38 >60 3 9 3 7 3 6 3 8

1 1A 3 41-42 3 8 3 8 2 9 3 8 3 35-machi- 1A 1 42-43 4 2 ( 1 ) 4 3 2 7 3 4 4 0ne 3 1 B 3 38-39 50 4 3 28 3 4 3 4system S B 3 18-19 2 1 1 8 1 3 1 8 1 53 A , - - 3 28-29 4 5 4 5 2 1 2 6 201 1 35-36 3 5 3 5 . 2 6 3 5 3 4

7 - n q a c h i - 2 2 41-42 4 2 4 2 3 4 3 6 3 6ne 3 3 39-40 3 9 3 9 28 3 9 3 8system 4 - , - 5 50-51 5 2 50 3 0 4 7 4 25 5 35-36 3 6 36 26 36 3 66 6 52-53 5 3 5 3 3 9 50 5 09-machi- 3 3 26-27 27 27 21 27 2 7ne 1 1 : : 1 46-47 N C ( 1 ) 5 6 39 3 9 3 9 4 0system 2 4 7 73-74 7 5 7 5 5 3 67 7 1

15-machi- Q 1 : : 01 41-42 4 5 3 8 2 8 3 8 37ne C l C l 41-42 4 6 4 6 3 8 3 8 4 3system M l M l 36-37 3 8 3 8 3 3 3 8 3 440-machi- C001 cool 32-33 3 1 . 1 8 1 5 2 4 >43ne AWIR3 AWIR3 24-25 >35 12 10 2 4 >30system MERC1 DOELlA 25-26 NC (1 ) 10 8 20 >37

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


11/12


12/12

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numerous c a s e s , a n d t h e c o s i n e a p p r o x i m a t i o n was f o u n d t o b e s i m p l e ra n d more a c c u r a t e .Th e d i s c u s s o r s q u e s t i o n t h e l i n e a r t r a j e c t o r y a p p r o x i m a t i o n u s e d i na c c o u n t i n g f o r t ra ns f er c o n d u c t a n c e s . T h i s i s a s i m p l e a p p r o x i m a t i o n ,a l t h o u g h t h e errors i n t r o d u c e d h a v e b e e n t o l e r a b l e f o r v e r y n o n l i n e a rt r a j e c t o r i e s , e . g . , see F i g u r e 5 . Th e t r a n s f e r c o n d u c t a n c e s h a v e a s i g n i f i -c a n t a f f e c t on s y s t e m b e h a v i o r , and t h a t a p p r o x i m a t i o n i s much mores a t i s f a c t o r y t h a n i g n o r i n g them c o m p l e t e l y . T h e y a l s o h a v e a s t r o n g i n -f l u e n c e on t h e l o c a t i o n and n a t u r e o f e q u i l i b r i u m p o i n t s , w h i c h i s a

s e p a r a t e i s s u e from t h e e v a l u a t i o n o f t h e p o t e n t i a l energy a t t h e UEP.F o r e x a m p l e , t h e a p p r o x i m a t i o n u s e d b y t h e d i s c u s s o r s f o r d e t e r m i n i n gi n t e r e s t i n g UEPs can b e q u i t e i n a c c u r a t e when t h e t r a n s f e r c o n d u c -t a n c e s are i n c l u d e d .I n c l o s i n g we w o u l d l i k e t o a g a i n t h a n k t h e d i sc us s or s f o r t h e i r e f -f o r t s i n p r e p a r i n g t h e i r comments.M a n u s c r i p t r e c e i v e d May 18 , 1 9 7 8 .

ANGLE ( C O A coordinates, degrees)GENERATOR Critical UEP Closest I n t e r e s t i n g UEPVP E = 9 . 3 1 ( t = . 2 1 s e c ) VP E = 5.93(t = . 1 6 s e c ). ~ ~ ~ ~ ~ v

1 - 4 4 . 7 -25.72 138.8 135.43 8 9 . 2 4 7 . 04 7 5 . 3 35.65 8 6 . 7 4 6 . 86 7 7 . 2 37.67 7 8 . 3 38.58 60.1 27.99 84.2 44.9

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A Practical Method for the Direct_1979

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