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1306 IEEE Transactions on Energy Conversion,Vol. 14, No. 4, December 1999Simulation of Internal Faults in Synchronous Generators

A . I. Megahed, Student Member, IEEE, and 0. P. Malik, Fellow, IEEEDepartment of Electrical and Computer EngineeringThe University of Calgary, Calgary, Alberta, Canada T2N 1N4

AbstTact- An in te rna l fau l t in the a rm a t u re w i n d -ing pf a s y n c h ro n o u s g e n e ra t o r o ccu r s d u e to thebreakdown of the wind ing insu la t ion . In this p ap e r am e t h o d for s imula t ing in terna l fa u l t s in synchronousgenera tors , u s ing d i rec t phase quan t i t i es , i s descr ibed .S imula t ion resu l t s show ing the f au l t c u r r e n t s , d u r i n g asingle phase to g r o u n d f au l t a n d a t wo p h ase t o g ro u n dfault, are presented.

Keyw ords-Synch ronous Generators, Modelling, StatorWinding Internal Faults.

I INTRODUCTIONAlong with the development of electric power indus-try, the protection of synchronous generators w ith severalparallel paths becomes m ore and m ore important . The in-ternal short circuit current for t he generator may be sev-eral times larger than its terminal short circuit current.Th e strong current could cause severe heat an d mechani-cal damage. Hence, for adequate generator protection, anaccurate method for calculating the internal fault currentsshould be available.Several articles analyzing the in ternal faults of ac m a-chines have been published [l] 121, 131. In 111 121 theSymmetrical Component Method is used and only thefundam ental and the third harm onic components, of timeand spac e, are considered. In the case of internal fault s

in the stator windings of electrical machines, there arestronger space harmonics in th e air gap magnetic field andstronger time harmonics in winding currents. Therefore,a substantial amount of error would be caused by usingthe Symm etrical Comp onent Method. In the Multi-LoopTheory the electrical m achine is considered as formed ofseveral electric circuits, each composed of th e actu al loopsthat are formed by the coils [3]. The inaccuracies in-volved with the calculations of loop inductances and thefocus of the Multi-Loop Theory on hydro-generators withdistributed neu tral ar rangem ent only, prevent the gener-alization of this metho d for different types of synchronousgenerators.PE-1216-EC-0-2-1998 A paper recommended and approved by theIEEE Electric Machinew Committee of the iEEE Power EngineeringSociety for publication in the IEEE Transactions on EnergyConversion. Manuscript submitted August 27, 1997; made availablefar printing Februaly 18,1998.

This paper describes a meth od for sim ulating internalfaults in a synchronous generator, using the direct phasequantities. T he proposed m ethod follows the sam e basiclines followed in [4] or the simulation of externa l faults.The method for calculating the self and mutual induc-tances of the faulted winding of the synchronous machineis based on the analysis presented in [I],[Z] 5], [6], [7]. Inthis paper single phase to gro und faults and two phase toground fault s will only be covered. However, the analysiscan be easily extended t o cover all kinds of internal fau lts.11. INTERNALS I N G L E HASEo GROUND AULTSA schematic representation of a synchronous machinewith two damper coils durin g an internal single phase t oground fault in phase a is shown in Fig. 1. It is assumedthat the armature winding of a synchronous machine con-sistin g of 2 parallel paths per phase is tapped at a cer-tain point of one of the parallel paths of phase a. Thetapped parallel path is divided in two parts, one part isadjacent to the neu tral, which will be referred to as the

rn winding, and the second part adjacent to the machineterminal, which will be referred t o as he n winding. Theremaining, - 1 , parallel paths of phase a are lumpedinto one equivalent winding t ha t w ill be referred to as thep winding. In direct phase quantities, the performance ofa synchronous generator, during an internal single phaset o ground fau lt, connected to a n infinite busbar througha short transmission line, Fig. 2 , can be described byequation s given below.A . Voltage Re lationships

Th e position of the rotor at any instant is specified withreference to the axis of phase a by the angle 0, Fig. 1.In term s of flux linkages, 1, voltage relationships for thestator and rotor circuits, el , are linked with the windingresistances, R I, and instantaneous currents, ~ l s ollows:

0885-89 69/99/ 10,00 1998 IEEE

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1307

- Lp Mpm Mpn M p b Mpc Mpf M p k d M p k q -Mmp Lm M m n Mmb MmcMmf M m k d M m k gMnp Mnm Ln Mnb M m Mnf M n k d hfnkqM b p Mbm Mbn b Mbc Mbf M b k d M b k qMep Mcm Mcn Mcb Lc Mcf M e k d M c k qM f p M f m M f n M f b Mfe Lf M f k d 0kdp kdm kdn kdb kde kdf k d 0

L1 =

~ ~ k q p ~ k l m M k l n M k q b M k g c 0 L k q -

Pmncfkd

k q

quad ru ru r rox i s

Fig. 1. Schema tic representation of a synchrono us machine duringan internal single phase to ground fault.

SynchronousGeneratorBusbar

Fig. 2. System representation during an internal single phase toground fault.T he LI matrix can be viewed as having two parts, ahealthy par t and a faulty par t . Th e healthy par t is as-sociated with the self-inductances of the healthy phases(i.e. b c, f d, kq) and the mutual inductances betweenthem. In this part the inductances are the normal values

available in any text book [8]. Th e faulty part is associ-ated w ith th e self and mut ual inductances of the differentparts of the faulty phase a , namely parts p ,m n. T hevalues of th e inductan ces in the fau lty par t of the L1 ma-trix need some modifications from the norm al values, andthese modifications are described in detai l in this section.The fault current, in a short circuited part of the ar-mature winding, produces a mag neti c flux with relativelystronger harmonics than in a case of a whole phase tobe short circuited [6]. Consequently, the differential leak-age is relatively high for a small pa rt of one parallel p athof the ar matu re winding. Therefore, the ratio betweenthe leakage inductances of one pa rt and the whole phaseof the armature winding is not proportional to the ra-tio between the squared effective numbers of turns of thecorresponding windings. However, the ratio between themain inductances is proportio nal to t he ratio between theeffective num bers of tur ns of the corresponding windings,which is: ..where L,,, is the main inductance of the m winding,L,, is the ma in inductance of healthy phase a, Nmis theeffective number of turns of the m winding and N is theeffective number of series turns of phase a. Hence, theself-inductance of the m winding is:

L m = L I ~Lmorno Lmom1 c0s(20m) (14)where L,,, = Lmomo Lmomlc0s(20m), L,o beingthe constant par t and Lmoml being the variable part ofthe main inductance. Th e main inductance is calculatedusing (13), while the leakage inductance JI,, is calcu-lated wit h th e aid of special form s presented in [6]. Due t othe distr ibuted nature of th e winding of the synchronousmachine, the faulted parts of the winding, m,n, have rotordisplacement angles different from that of the p winding,

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1308Pig. 1. Th e displacement angle, E corresponding to thewinding is calcu lated as following:

E =e+s,o 15)In order t o calculate the self-inductance of th e n winding,the sa me procedure described above is followed.

Mutual inductance between any two faulty windingpar ts is proportional to th e ratio of their effective numbe rof turns [5], [6], hence:16)NmM p m = M m p = -Lmmn

As the internal fault does not affect the main inductanceof the faulted ph ase, hence the po rtion of the flux linkinga healthy phase, due to a faulted phase, should be thesame as before fault occurrence.

Based on 17) Mpb = Map = Mob = Mba , Mmb = Mbm =In keeping with t he assum ption of 17), th e flux linkingthe lumped z 1parallel branches, i.e. p winding, shouldremain unchanged. Th is assumption is used to find theself-inductance Lp of the p winding, as follows:

and Mnb = Mbn =w

where L. is the self inductance of healthy phase a andI,,,,,,,, is the main inductance of the n winding. Substi-tuting for L,,, and L in terms of L, 13), andrearranging, yields:Lp = LI. Lm.

L p = La 3 19)where Lo = Ll,+L,. and L1, is the leakage inductance ofhealthy phase a. It should be noted t ha t, if normal opera-tion is simula ted, the above conditions, which are impose don the inductances of the windings of faulty phase a 13-19), would indeed cause th e voltag e ep to be equ al to e,.Also, e, + e, would be equal to e,. The se are necessaryconditions for parallel operation. The formulas of the el-ements of the L1 matrix in 12) are given in AppendixA .C. General

Th e equation of motion of a synchronous generator ca nbe expressed by [4]:

r - - - - - I

SynchronousGeneratorFig. 3.ground fault.System representation during an internal two phase to

where H is the inertia constant, T,,, is the mechanicalinput torque and Ter.is the electrical torque.Equations l , 6 - 11,20) form the complete model fora synchronous machine during an internal single phaseto ground fault. The elements of the L1 matrix in 12)are dependent on rotor position, which varies with time.Hence the performance equations are differential equa-tion s with v ariab le coefficients. As the speed of the gener-ator varies under tran sient ope rating conditions, the per-forman ce equations are nonlinear. Generally, solutions inclosed form cannot be obtained, and numerical solutionshave to be resorted to.111. INTERNAL Two PHASE o GROUND AULTS

Figure 3 shows a synchronous generator having an in-ternal two phase to ground fault in phases a and 6 T h efaulty path of phase is divided in two par ts, one par t isadjacent to the neutral, T winding, and the second partadjacent to the machine terminal, s winding. The remain-ing, 2 - 1, parallel paths of phase 6 are lumped into oneequivalent winding which is the I winding. Th e relation-ship expressed in 1) will still hold durin g an internal twophase fault, as shown in 21) .

dllr2 21)

22)

eZ = - - -Rl 2 i a

ez = [epemen , er~,eoejekdeknlhere= [?/lplDmdnllrrllrr?/lsllrellr/llrl/lkdllr~qlt 23)

Rz = diag[RpRnR,RzR,R,Re -Rj - R k d -Rkq] 24)(25)i z = [ i i i ~ ~ ~ ~ ~ i ~ ~ ~ ~ ~ i k d i k ~

The machine terminal voltages during a n internal twophase t o ground fault are:

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L p Mpm M p n M p a Mpr Mps Mpc Mpf M p k d Mpkq-M n p M n m L n Mnr M n r M m M n c Mnf M n k d M n k qM z p Mzm M m L z Msr Mza Mzc Mzf M z k d M z k qLa= M r p Mrm MPn Mra Lv MFa M m Mrf Mrkd MrkqM a p Msm Msn M ~ Z ~ P s Mse M s / M s k d M ~ k qM c p Mcm M c n Mc, M C PMca Lc Mcf M c k d M c k qMfp M f m M f n M f z MfF Mfs Mfc Lf M f k d 00

M m p Lm MmnMmsMm, Mmr M m c M m fMmkdMmkq

k d p Mkdm M k d n M k d a Mk dr M k d a M k d c M k d f L k dkgpMkqmMkqnMkq MkqrMkqrMkqc 0 0 L k qI

e = O (27)e,, = e p 28)

2Te , =Ebussin(wt ) iz )RTL(29)di di+(-dt - ; i t ) L T L

0I0.05 0.1 0.15 0.2 0.255Lto?

I0.05 0.1 0.15 0.2 0.25-10E511 . I11-

0 0.05 0.1 0.15 0.2 0.25U 5

20

-2200 0 05 0.1 0.15 0.2 0.25

-2 t0 0.05 0.1 0.15 0.2 0.25Time, s

Fig. 4. Computed stat or currents for an internal single phase faultat 6 2 % of one path of phase a.Th e method of stator grounding used in a generator in-

stallation determines the generator's performance duringground fault conditions [9]. Th e high magni tude of faultcurrent which results from solidly grounding a generatoris unacceptable because of the damage it can cause. Asa result, stator windings are grounded in a manner tha twill reduce fault current. Figure 9 shows the sta tor faultcurrents for the single phase fault simulated in Figs. 4, 5but with the ground resistance of the generator included.

V. CONCLUSIONSA synchronous generator during an internal fault canbe represented by a mathematical model in direct phasequantities. Th e mathema tical model regards the faultyphase as composed of three individual windings. The self

0.05 0.1 0.15 0.2 0,25

I0 0.05 0.i 0.15 0.2 0.25-1 120

-2I0.05 0.1 0.15 0.2 0.25Time. D

Fig. 5. Computed rotor currents for an internal single phase faultat 62 % of one path of phase a

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1310A

5

;BI

0 0.05 0.1 0.15 0.2 0.25

Time. eFig. 6.Fault currents for windings p,m,n for an internal two phaseto ground fault at 50 % of one p+th of phases a and 6and mutual inductances of these individual windings arecalculated with the aid of special forms. Th e describedmodel has t he advantage of simulating any kind of fault,while retaining th e generator nonlinear model.

REFERENCES[l] V.A. Kinitsky, Calculation of interna l fault curren ts in syn-chronous machines , I E E E Trona. on P A S , vol 84 o. 5 pp.381-389 May 1965.[Z] V.A. Kinitsky, Digital computer calculation of interna l faultcurrentsin a synchronousmachine , I E E E nons. on P A S , vol .87 o. 8 p. 1675-1679 Aug. 1968./3]X H Wang, L.Z. Zhang, W.J. Wang, and Z.H. u, Researchand application of protection relay Bchemes for internal faultsin stator windings of B large hydro-generator with multi branchand distributed arrangement , I E E 5 t h Int. Conf. on Develap-menta in P o w e r System Protection, , no. 368 p. 51-55 1993.

0I

0 05 0.1 0.15 0.2 0.252

I0 05 0.1 0.15 0.2 0.25

JU 0.05 0.1 0.15 0.2 0.25

20

-20 05 0.1 0.15 0.2 0.25Time, 6

4

J0.05 0.1 0.15 0.2 0 25

ao I0 0.05 0.1 0.15 0 2 0.25-112,

0.05 0.1 0.15 0.2 0.25-20 Time,Fig. 8.Computed rotor currents for an interna l two phase to groundfault at 50 % of one path of phases a and b .141 P. Subramaniam and O.P. Malik, Digital simulation of a syn-

chronous generator in direct-phase quantities , Proc. IEE, vol.118 o. 1 pp. 153-160 an. 1971.[5]V.A. Kinitsky, Mutual inductances of synchronous machineswith damper windings , I E E E Dana. on P A S , vol. 83 o. 10pp. 997-1001 Oct. 1964.l6l V.A. Kinitsky, Inductaneesof a Dortion of the arm atu re wind-ing of synchronous machines , I E E E Trons on PAS , vol. 84no. 5 pp. 389-396 May 1965.V.A.Kinitsky, Calculation of synchronousmachine constantsby a digital computer , Proe. I E E E P I C A Conference, pp.74-90 1965.Paul C. Krause, A n a l p i t of Electr ic Machinery, McGraw-HiliInc, 1986.C.J. Mazina (Coordinator), I E E E Tutorial a n the Protection ofSynchronous Generators , IEEE Tutorial Course, IEEE PowerEngineering Society Special Publ . no. 95 TP 102 995.

0 0 5 0 1 0 15 0 2 25

J

0 0.05 0.1 0.15 0.2 0.25Time, sFig. 7. Fault curre ntsfor windingsz, , 8 , c far an internal two phaseto ground fault at 50 % of one pat h of phases and b . Fig. 9. Computed stat or curren ts for an internal single phase faults t 62 % of one pat h of phase a with the ground resistance included.

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A P P E N D I XA

The components of the L1 matrix are given by [l] 4],[6J, PI:

1311LCMet = Mj .

= L.0 Lmolcos(20 - 9=Mat cos(8 F)

M d = Mhdc =Me/ COS(8 9Mckq= Mkpe =-M an sin(0 9Lf =c ons t a n tMfkd = Mkd, =co nsta ntL k d =c ons t a n tLkq =constant

whereL,o, Lmal,Mao = inductance coefficients of healthy ar-mature windings.LOO= + LmaoLmo = L I ~LrnamoL ~ o L I ~ Lrnono8 , , = 8 _ m e i , -6M.j = Mutual inductance between armature and fieldwinding.Mnq= Mutu al inductance between armature and q-axisdamper winding.

A P P E N D I X

not shared with the L1 matr ix are:Formulas of the components of the Lz matr ix, that are

whereNp N , = effective number of turns of the r ,s windings.Om = 8 +&,, b,, = displacement angle of the T windingfrom the original axis of the b winding due to the fault.Om = 8 +&,, b,, = displacement angle of the s windingfrom the original axis of the b winding due to the fault.

B I O G R A P H I E SMegahed A.I ., obtained his BS c. and MS c. in EEfrom Alexandria University in 1991 and 1994. He is cur-rently working towards his Ph.D . in power systems at t he

University of Calgary.Malik O.P., graduated in EE in 1952, obtained M.E.degree in 1962 and Ph.D. degree and D.I.C., London, in1965. From 1952 to 1961 he worked w ith electrical utilitie sin India. He is at present a professor at the University ofCalgary.