# Power System Stability By Kimbark 95.pdf

AJOHNWILEY & SONS, INC., PUBLICATIONIEEE Press Power SystemsEngineering SeriesDr. Paul M. Anderson, Series EditorThe Instituteof Electrical and Electronics Engineers, Inc., New York1995 by theInstitute of Electrical and Electronics Engineers, Inc.345 East47thStreet, NewYork, NY10017-23941956 by Edward WilsonKimbarkThisis the IEEE reprinting of a book previously published byJohn Wiley &Sons, Inc. under the titlePower SystemStability,Volume III: SynchronousMachines.All rightsreserved. No part of thisbook may be reproduced in any form,nor may it be stored in aretrieval systemor transmitted in any form,without writtenpermission fromthe publisher.Printed in the United States ofAmerica10 9 8 7 6 5 4 3 2ISBN 0780311353Library ofCongressCataloging-In-Publication DataKimbark, EdwardWilsonPowersystemstabilityI Edward WilsonKimbark.p. em. -(IEEEPresspowersystems engineering series)Originally published:NewYork: Wiley, 1948-1956.Includes bibliographical referencesandindex.Contents: v. L Elements of stability calculations - v. 2.Powercircuitbreakers andprotective relays- v. 3.Synchronousmachines.ISBN0-7803-1135-3(set)1. Electricpowersystemstability. I. Title. II. Series.TKI010.K561995621.319--dc20 94.-42999CIPTomywifeRUTHMERRICKKIMBARKPREFACEThisis thethirdvolume ofathree-volumeworkon power-systemstabilityintended for use by power-systemengineers andby graduatestudents.It grew out of lectures given by the author atNorthwesternUniversityingraduateeveningcourses.Volume I, whichappeared in 1948, covers the elements of thestability problem, theprincipal factors affecting stability, theordinarysimplifiedmethods of making stability calculations,andillustrations of the application of these methods in studieswhichhavebeenmadeonactual powersystems.Volume II,whichappearedin1950, covers power circuitbreakersandprotectiverelays and the influenceof thesedevicesonpower-systemstability.Volume III,thepresent one,dealswiththetheoryof synchronousmachinesandtheirexcitationsystems,anunderstandingof whichisnecessaryfor the justificationof thesimplifying assumptions ordi-narily usedin stability calculationsandforcarrying out calculationsfor theextraordinary cases inwhichgreateraccuracyis desiredthanthataffordedby thesimplifiedmethods. Thisvolume discusses sucheffectsas saliency, damping,saturation, and high-speed excitation.It thusendeavors to givethereader a deeper understanding of power-systemstability than thatafforded by Volume I alone. Steady-statestability,whichwasdiscussed verysketchilyin Volume I, istreatedmorefullyinthe presentvolume.Volume I shouldbeconsideredtobeaprerequisitetothepresentvolume, but Volume IIshouldnot.Iwishto acknowledgemy indebtedness in connectionwith thisvolume,aswell aswiththepreceding ones, tothe.following persons:Tomywife,RuthMerrickKimbark, fortypingtheentire manu-script and for heradviceandinspiration.ToJ. E. Hobson, W. A. Lewis,andE. T. B.Grossfor reviewingthemanuscriptinitsearlierformandformakingmanysuggestionsforitsimprovement. (Theydid not,however, review recent changesviiviii PREFACEandadditionsandthus'cannotbe blamedforanyshortcomings whichthis volumemaynow have.)Tomanufacturers, authors, andpublisherswhosuppliedillustra-tionsorgavepermissionfortheuseofmaterial previouslypublishedelsewhere. Credit for such materialis given at the place whereit appears.EDWARD WILSON KIMBARKSeattle,WaahingtonNovember, 1955CONTENTSCHAPTER PAGEXII SynchronousMachines 1Reactances, Resistances, and Time Constants 2SuddenThree-Phase ShortCircuit 40Mathematical Theory 52Vector Diagrams69Applications to Transient Stability Studies 78Saturation 118XIIIExcitation Systems 137XIVDamper Windings andDamping214XVSteady-State Stability 247INDEX 317ixCHAPTERXIISYNCHRONOUSMACHINESSince the problemof power-system stability is to determinewhetheror not the various synchronousmachineson the systemwillremain insynchronismwith oneanother,thecharacteristicsofthosesynchro-nousmachinesobviously playanimportant part in the problem.o(a)(6)FIG. 1. Cross sections ofsynchronous machines,(a) round-rotor type, (b) salient-poletype.Synchronous machines are classifiedinto two principaltypes-round-rotor machines .and salient-pole machines (Fig. 1).Generatorsdrivenbysteamturbines (turbogenerators) havecylindrical(round)rotors with slots in which distributed field windingsareplaced. Mostcylindrical rotors are made ofsolidsteel forgings,thoughafewofthem arebuilt upfrom thick steel disks. Thenumber ofpoles is two,four, or six.. Most new machines aretwo-pole.Generators drivenbywaterwheels(hydraulicturbines)havelaminatedsalient-polerotorswithconcentrated field windings and,usually,a large number of poles.Some water-wheel generatorsareprovided withamortisseurwindings12 SYNCHRONOUS MACHINESordamperwindings;others arenot. Synchronousmotorsandcon-densers havesalient-pole rotors withamortisseurwindings.Synchro-nousconvertershaverotatingarmaturesandsalient-polestationaryfieldswithamortisseurs.In the foregoing chapters it wasassumedthateachsynchronousmachine could be representedby a constant reactancein series withaconstantvoltage, and, furthermore, thatthemechanicalangle of therotorof each machine coincided withtheelectrical phaseof that con-stant voltage of themachine. Inview of theimportanceofsynchro-nous machines in the stabilityproblem, these assumptionsrequirefurther investigation, bothto justify theuse oftheassumptions in theordinarycase andtoprovide more rigorousmethodsof calculation inthoseextraordinarycaseswherethe usualassumptions donot giveaccurate enough results.The usual assumptionsmay be in error because of three phenomena,each of which requiresconsideration:1. Saliency.2. Field decrementandvoltage-regulatoraction.3.Saturation.Thewordsaliencyisusedasashort expression forthefactthattherotor ofasynchronousmachinehas differentelectricandmagneticpropertieson two axes 90 elec. deg. apart -thedirectaxis, or axis ofsymmetryof a field pole, and thequadratureaxis, or axis of symmetrymidway between two fieldpoles. Thisdifferencebetween the two axesis present notonly in salient-polemachines but also, to a lesser extent,inround-rotor machines,because of thepresence of thefield windingonthe direct axisonly. Inthis respect bothkinds ofsynchronousmachines differ frominductionmachines. Theeffect of saliency willbeconsideredfirstwhile theeffects of field decrement, voltage-regu-latoraction, and magnetic saturation are disregarded; later, theseothereffects willalsobetakenintoaccount. Aknowledgeofsyn-chronous-machine reactances, resistances,timeconstants,andvectordiagrams is necessary totheunderstanding of all theseeffects.REACTANCES, RESISTANCES, ANDTIME CONSTANTSAs hasalreadybeennoted, asynchronous machine isoftenrepre-sented in a circuitdiagram (the positive-sequencenetwork) by aconstant reactance inseries witha constant voltage. What reactanceshould be used forthat purpose? A greatmanysynchronous-machinereactanceshavebeendefined;for example, direct-and quadrature-axis synchronous, transient, and subtransientreactances. BeforeCONCEPTS OF INDUCTANCE 3proceedingto definethesereactances wemay profitablyreviewthefundamentalconceptsofinductance andof inductive reactance.Fundamental conceptsofinductance. Theconceptsof electriccur-rentandof linesofmagneticflux arefamiliar toallstudentsof elec-tricity.Sincealineofflux isalwaysaclosed curveandanelectriccircuit islikewise aclosedpath, * agivenflux line either does not linka givenelectric circuit at all or it does link it one or more times.ThusFIG. 2. Illustrating flux linkages.Circuit FluxlineFIG. 3.Alineofmagneticflux con-stitutes a positive flux linkagewithanelectriccircuit if it is related to thecircuit according tothe right-hand rule.A diagramsimilartothisone appearson theemblemof theA.I.E.E.[1]inFig. 2 flux line1 linkscircuit C once, line 2links it fourtimes, andline 3 does not link it at all. Thefluxlinkage1/Jof acircuit is definedasthealgebraicsumof thenumbersof linkagesof allthelines of fluxlinkingthe'circuit,thepositivedirectionoflinkage' beingtakenac-cording to Fig. 3. Thusin Fig.2, if there were no flux lines linking thecircuit exceptthoseshown, andifeachlinerepresentedone weberofflux, the fluxlinkageof thecircuit would be -1 +4 +0 = 3 weber-turns. Ingeneral,theflux shouldbedividedinto infinitesimal partsd(j),eachrepresented bya lineor tube linkingthe circuit ntimes; thefluxlinkageof the circuit is then givenby1/1 =f ndcP weber-turns*Theflux linkage of apart of a circuit, such as a winding of a machine,which isnot aclosed path, can be defined asthelinkage of theclosedpath formedbythewinding anda short linefrom one terminal of thewinding to theother.4 SYNCHRONOUS MACHINESAccordingtoFaraday'slawof electromagnetic induction, anychangeinthefluxlinkageof acircuit inducesanelectromotive force e, thesign andmagnitude of which are given byd1/le =- - voltsdt[2]where t is timein seconds. A positive e.m.f. tends to set up a positivecurrent(that is, a current in thedirectionof the circuit, Fig. 3).Theappliede.m.f. requiredtoovercome theinducede.m.f. isofoppositesignfromthe latter. Faraday's lawis perfectlygeneral: itholdswhether thechange of fluxis caused by a change of current inthecir-cuitconsidered, by a change of currentin anothercircuit, byadefor-mationof thecircuit, byrelativemotionof one circuitwithrespectto another, or by relative motion of magneticmaterialsor permanentmagnets with respect tothe circuit.Therearesome circuits theflux linkages of which aresubstantiallyproportional to thecurrent:1/1 =Li weber-turns[3]where L is theself-inductance of thecircuit inhenrys and iisthecur-rent of thecircuitinamperes. Theself-inductanceofsuchacircuitmay be regarded as the flux linkage per ampere:1/1L =-; henrys

## Power System Stability By Kimbark 95.pdf

ec-e 2.5 :::J"0C" 2.030 60 90 120 150 180Angle of directaxisfromaxis of phase a(elec. deg.)Fro. 28. Variation ofself-inductance of armature phase a with position of thefield.Thetest points, taken on a 4-pole 15-kva. 220-volt1,800-rpm.synchronous motor,comparefavorablywith the curveshowingthe theoretical variationof an idealmachine: Laa=L, +Lm cos28 = 2.36 +0.796 cos 28mho (FromRef. 34 bypermission.)INDUCTANCESVB. ROTOR POSITION 55[96]inductanceissimilar,exceptthatthemaximumvalueoccurs wherethedirect axis coincides withtheaxis ofphase b, that is, at 8= 120or -60.. Hence,theproperequationforphase b isobtainedbyre-placing 8 in eq. 94 by 8- 120.L bb =L8 +L m cos 2(8- 120)=L8 +i; cos (28 +120)[95]Similarly, for theself-inductance of phase c,Lee= L8 +Lmcos 2(8+120)=L8 +Lm cos (28 - 120)Themutualinductance between twoarmaturephases also varies withtheposition of thefield. It is alwaysnegative, anditsgreatest abso-lutevalueoccurs where thedirectaxislies midwaybetween theaxis180' 30 60 90 120 150Angle of directaxisfromaxisof phase a (elec. deg.)...-:-s:-1.5Q-----f-----r-.----+-----,t'r---+----oQ)- Realor-directa_xisSubstituting eqs. 173 and174 intoeq. 177,V =E +rI +jXdld - xql q[179]The voltages, currents, and link-ages in eqs. 172to 179areconstantsequal to the crest values of thecorrespondingpositive-sequencequantities, as indicated in eqs. 169 for currentI. However, the corre-spondingeffectivevaluescanbeusedequallywellinanyexcept theequations involvingfield current If.Thevectordiagram correspondingtoeq. 179 is shown inFig. 30. ItisdrawnforpositiveIf, la" Iq, and{j. Theseconditionscorrespondthesteady-stateinternal voltage.VECTORDIAGRA1\.IS 71to motor operation withlagging (magnetizing) current, as may beseen fromthe followingconsiderations:If IfandI d areboth positive,eq. 172a shows thattheir effects arecumulative; henceId is magnetiz-Thefieldandarmaturepoles areasshowninFig. Nowif(b) Iq:FIG. 31. Locationoffieldand armaturepolesfor positiveI Jand (a)positiveIonly, (b)positiveI q only.I q is assumed tobepositive and I d tobezero,then,sincethequadra-tureaxiswastaken900aheadof thedirectaxis, thepoleswill be lo-catedasshowninFig. 31b,correspondingtomotoroperation(unlikepolesattract).NegativeIdwouldbedemagnetizing, andnegativeIqwouldrepresentgenerator action.Note thattheoriginal circuitequations (91) wereset upwith suchconventions that positive power means power consumed(that is,motor operation). As a consequence, eq.179, which wasderived fromtheoriginal equations,bearsthesamesignconventionsandisoftheformV=E+ZI[180]In dealingwithgenerators the conventions on signsareusually takenso that eq. 180becomesorV=E-ZIE=V+ZI[181][182]requiring that either the signof Iorthe signs of V andEbe changed.If the sign of I (that is, of both Idand Iq) is changed without changingthesignof If, thenpositive Idisdemagnetizingandpositive Iqrep-resents generator action.Thischangedsign of l sand Iq(includinginstantaneou8 values id andiq) is ineffect inthe rest of this book andisingeneraluseelsewhere.The vector diagramfor generator actionwithlaggingcurrent is72 SY'NCHRONOUS MACHINESshown in Fig. 32. Inthisdiagramthequadrature axis is drawn hori-zontally, as is customary,and resistance is neglected.In both vector diagrams theexcitationvoltage Eliesalongthepositive quadrature axis.Ifthevector jxqI - that is, the total arma-turecurrent multipliedbythequadrature-axissynchronous reactance- is addedtotheterminal-voltage vectorV, a vector Eqd is obtainedwhich alsolies onthequadratureaxis. If thegeneratorisoperatedIIIII ",,"1 /d------ 90. ;'I I \1IIIIId - _I, 1\ 90' -: , \ /t \/DirectaxisFIG.33. Vector diagram of synchronous generator in thetransient state.VectorE' remainsconstant duringarapidchangeofarmaturecurrent I, butVand Echange.the amount of the transient-impedance drop,whichconsistsofre-sistance and transient-reactance components. Itis less than theexcitationvoltage by the difference of thesynchronous-impedanceand transient-impedance drops.A vector diagramcorresponding toeqs. 204, 206, 208, 209, and210is given in Fig.33.In the steadystateprecedinga changeofarmaturecurrent,theexcitation-voltage vector E lies on the quadrature axis; thatis, Ed = O.Thecorresponding valuesof Eiand E/maybecomputed,respec-tively, byaddinglq(xq - x/) to0andbysubtractingld(xd-xi)fromEq Then duringa changeof armature current E'remainsconstant; Evariesandmaydepart fromthequadratureaxisunless,xq= xqSuppose that components I d and Iq of the armaturecurrentundergochanges ofIIIdandIIIq, respectively. Thecorrespondingchanges inother quantities are as follows:IlVtf = 0IlVto = 0IlVtd =- Li tJ.ldIlVtq = -L/III qTRANSIENT STATE 773MI La - La'611=- - Ma=Ma [216]2 LII M13 MaL,- L/st, =2L si, M si, [217)aa af:1Eq= (xa-xi) f:1ia[218]f:1Ea = - (xq- x/) f:1iq[219]f:1Ei = 0 [220]l:1E/ = 0[221]f:1Vq= -r f:1Iq- xi f:1Id[222]f:1Va = -r Md+x/Mq[223]Salient-polemachine. Sinceasalient-polemachinehasnoquadra-ture-axisfield circuit, I a =0andEa = 0, andtheexcitationvoltageE=jEqalwayslies onthequadratureaxisinthe transient stateasf---------Eq =Ek------- Eqd---j------,JE'FIG. 34. Vector diagram of salient-polesynchronousmachine(xq' = xq) . VoltagevectorsE, Eqd, andE' alllieonthequadratureaxis, bothinthetransient stateandin thesteadystate.well as in the steady state. Furthermore, since x/= Xq,Ei=Ed=0,andthevoltagebehindtransient reactance, E' =jE/,also lieson thequadratureaxis,as does Eqa, thevoltagebehindXq The vectordia-gramis shown inFig. 34.Turbogenerator. Intheturbogenerator,which has a solid cylindricalrotor, xi andx/ are very nearly equal,and Xa and Xqare almostequal. Consequently, thexldropsneednot besplit intodirect-and78 SYNCHRONOUS MACHINESquadrature-axiscomponents. The vector diagramfor the transientstate is shown inFig. 35.Vector diagram for thesubtransientstate. Intheprecedingtreat-ment ofthetransientstate, amortisseurwindings or theirequivalentwereassumedabsent, because the equations that wereused asastarting point werederived under that assumption. If,however,amortisseurs are present, their effectfollowing asuddenchangeincircuit conditionsdies awayinaperiodof afew cycles,after whichtherelations just derived are valid.f+---------Eq- --- -- -FIG. 35. Vectordiagramofsynchronousmachinewithsolid, roundrotorint hetransientstate. Xd = Xq and xi = Xq'. Inthesteady state,Ed=0andE = Eq,but the same is not truein thetransient state.Attheinstant ofasuddenchange incircuit conditions, however,thefluxlinkagesof theamortisseur windings, aswell asof thefieldwindings,are substantially constant. Under these conditions,thefictitiousvoltage, E"= E/' +jE/', known asthevoltage behindthesubtransient impedance, or the subtransient internal voltage,likewiseremainsconstant. Equationssimilar tothosederivedfor thetran-sient state (205, 206,208, 209, 210, and 212through 223)canbewritten for thesubtransient statebyreplacing E/ by E/',E/byE/', E' by E", x/by x/' , and x/by x/'. Thevectordiagramforthesubtransient state is similar to Fig. 33, with the changes justnoted.APPLICATIONS TOTRANSmNT STABILITYSTUDmsThe periodofmechanical oscillationof a machine during adis-.turbanceisoftheorder of1 sec.,andthe behavior ofthemachineduring thefirst quarter-periodoften suffices toshowwhetherthesys-TRANSIENT STABILITY STUDIES 79temisstableor unstable.Thesubtransient timeconstant ofama-chineis only-about 0.03 to0.04sec.,whichis short compared with theperiodof mechanicaloscillation. Therefore, in stabilitystudies thesubtransientphenomena are disregarded. The armature time con-stant is about 0.1to0.3sec. forashort circuit at the armature termi-nals, but it isgreatlydiminished by external resistance. Thereforethe d-e,component of armaturecurrent is ordinarily disregarded also,thoughitseffectmaybeappreciableincaseofafaultator neartheterminals of themachine.The only electrical transient whichneedbe consideredinastabilitystudy isthe "transient component." Its timeconstant isfromabout0.5 to10 sec.,usually being longer than the period ofmechanical oscil-lation. Therefore the transient component cannotbe ignored althoughitsdecrement is frequently ignored.Thetransientcomponent canbetaken into account inanumber ofways whichdiffer as to the accuracy of the assumptions and theamountofcalculationrequired. Thefollowingdifferent assumptionscan be maderegarding aSalient-polemachine:1. The voltage behind direct-axistransient reactance may beassumed constant (constant Fig. 34); or2.The flux linkage of the field winding may be assumedconstant(constant Eq' , Fig. 34);or3.Fielddecrementandvoltage-regulator action(ifany)maybecalculated(varying Eq' ) .Theseassumptionsarelistedintheir orderofincreasingaccuracyanddecreasingsimplicityofcalculation. Method1wasemployed inearlier chapters of this bookand is commonly usedinpractice. Meth-ods 2 and 3 will be explained, illustrated, andcompared with Method 1inlater sections of thischapter.Amachinewithasolidcylindricalrotorhasrotorcircuitsonbothdirect and quadrature axesconcerningwhich assumptions must bemade. Consequently, thefollowing methods canbe usedforaRound-rotor machine:1.ThevoltageEl' (Fig. 33or 35) behinddirect-axis transientreactancemay beassumed constant; or2. Thefluxlinkages of therotor circuitsonbothaxes maybeassumed constant (that is,constant Eel and Eq' ,Fig. 33); or3. Theeffect of decrement andvoltage-regulatoractionontheflux linkage of the direct-axisrotor circuit (that is,onEq' ) maybe80 SYNCHRONOUS l\'IACHINEScalculated, while thequadrature-axis circuit IS assumed open(Ed' = 0);or4.Thedecrementoftherotorflux linkagesof bothaxesandtheeffectofvoltage-regulatoraction, if any, onthedirect-axislinkagemaybecalculated.WewillnowconsiderMethod2forasalient-polemachine. Todoso we needthe transientpower-anglecurveof the machine.Power-anglecurvesof asalient-polemachine.Steady state.Theelectricpoweroutputofasalient-polegeneratormaybeexpressedintermsof the terminal voltage V, the excitation voltage Eq, theanglebetweenthesetwovoltages, andthecomponentsofthesynchronousimpedance r,Xd, andxq All these quantities areshowninthevectordiagram, Fig. 34.Ifeqs, 194 and195aresolvedfor IdandIq, withEd set equal tozero, the result is:I- VdT +(Eq - Vq)xqd= 2T +XdXqI =(Eq - Vq)r + VdXdq r2+XdXqThe power outputis[224][225][226][227]Substitution of the values of Iqand I d intothis expression yields:P_ VIl(- VdT +Eqxq - VqX