Anintroductiontopowersystemstability Stabilitystudiesevaluatetheimpactofdisturbancesontheelectromechanicaldynamicbehaviourofthe powersystem. Maximumpowertransfer: Themaximumpowertransferredbetween2nodeswilloccurwhentheimpedancebetweenthenodesis equaltotheconjugateoftheloadimpedanceatthereceivingnodei.e.node2.Thismeansthathalfof thetotalpowertransferrediswastedinthetransmissionlineimpedance.ForaPowerSystemEngineer, amaximumof15%transmissionlosswouldbeacceptable.Furthermore,thePowerSystemEngineeris facedwithavariableloadrangingfromaminimumtoamaximumvalueatornearaconstantvoltage level.Ifthevoltagevariationismorethanaspecifiedvalue,theperformanceoftheequipmentsuffers andthelifeofmostoftheconnected&energizeddevicesaresacrificed.Forexample,theoutputofa lampisreduceddrasticallywhenitoperatesbelowacertainvoltagelevel.Theinductionmotordraws morecurrentforthesametorquewhenoperatedatlowerthanratedvoltagesandunderextreme conditionsoflowvoltage,themotormaystall.Ontheotherhand,synchronousgenerators(onthe sourceside),whenareforcedtotransferpowermorethancertainlevels(knownassteadystatestability limits)mayfalloutofstepandsupplytocustomersmaybeaffectedintheformofbrownoutorblack outconditions.Themagnitudeofpowerthatcanbetransmittedfromasourcetoasynchronousloads (likeheaters,lighting,SCIM)dependsontherangeofvoltagethatisavailablefromthesourceand maybetoleratedattheloadcentersaswellasthecurrentcarryingcapacityofthevariouscomponents ofthegrid(network).When2ormoresynchronousmachinesarerunningonthesamepowersystem,it isfoundthatapowertransferlimitexistseventhoughthevoltagesatbothnodes(buses)atthespecified valuesThereactivepowerflowbetween2nodesisfunctionofthevoltagelevelsonthe2buseswhile theactivepowerflowisfunctionofthepahseanglesatthe2buses.Ifasynchronousmotoris connectedtoasynchronousgenerator,lossofsynchronismwillcausethemotortostall.When2 generatorsareconnectedtothepowernetwork,lossofsynchronismwillresultinwildfluctuationsin thetransmissionnetworkcurrents&voltages.Thepowertransferbetweenthesourcesisalternatively positive&negativewithaveragetransferequalto0.Separationoftheoutofsynchronismmachine throughtheopenningofthepertinentcircuitbreakeristheonlysolutionforsuchacondition. Formsofinstabilityinpowersystems: Thereare2formsofinstabilityinpowersystems,thestallingofasynchronousloadsandlossof synchronismofthesynchronousmachines.Thesynchronousstabilitycanbefurthersubdividrdinto steadystateandtransientstate.Theformeristhesystemstabilityunderconditionsofgradualor relativelyslowchangeosload.Itisassumedthattheloadisappliedataratewhichisslowwhen comparedtoeithernaturalfrequencyofoscillationofthemajorpartsofthesystemortotherateof changeofthefieldfluxintherotatingmachine(inresponsetochangeinloadcurrent).Thelatterrefers tothemaximumflowofpowerthroughapointwhensudden&largechangestakeplaceinthenetwork withoutloosingstability(lossofsynchronismofconnectedgenerators).Thesechangesmaybebrought bywhenfaultsoccurorwhenadditionorsheddingoflargeincrementsofloadstakeplace.Dynamic stabilityisatermusedwhensynchronousmachinesareoperatedalongwithfastactingvoltage

regulators,thestabilitylimitsofthesystemarehigherthanwhenrathersloweractingregulatorsare used.Dynamicstabilityismadepossiblebytheactionofthefastactingvoltageregulatorswhichare capableofchangingthefluxataratefasterthanthatcausingthesystemtofalloutofstep.Insteady statestability,theregulatoractsslowlyinordertoadjusttheterminalvoltagetothepresetvalue. Stabilitystudiesusuallysolvethefollowingproblems:findingtheinertiaconstantoftherotating componentsofthemachine(H)fromthemomentofinertia(IorJ),findingthefrequencyandperiodof oscillationofthemachineunderslighttemporarydisturbancecondition,findingtheequivalentHfor2 unitsoperatinginparallel,calculatingthecrticalclearingangle&criticalclearingtimeunder3phase faultcondition,findingtheloadangleequationundernormal&faultconditions,calculatingtheswing equationformultimachinesystemduring&postfault(afterremovingfaultyportionofnetwork) conditions.Ingeneral,thedisturbancesthatmayaffectthestabilityofasystemare:shortcircuit conditions,lossofatiecircuit,switchingoperations,lossofaportionofanonsitegeneration,impact motorloadingandstartingofalargemotorascomparedtothegeneratingcapacity. Synchronousgenerators: Thestatoraswellastherotorareconnectedtoapowersource.Therotorisconnectedtoad.c.source. Thestatorisconnectedtoana.c.one.Thesynchronousspeedis120timesthefrequencyofthesystem dividedbythenumberofthepolesinthemachine.Thisisthespeedoftherotorofthemachine.Its frequency/speedcharacteristicsisaverticalline.henasynchronousgeneratorisoperatingasasingle unit,itisnecessaryfortherotorspeedtoremainconstantandindependentofthetorqueinorderthat thefrequencytobemaitainedconstant.Thesynchronousmachinecanbeviewedasaspecialcaseofa transformer(currenttransformer).Whenthecurrenttransformerisoperatedwithaconstantprimary current(inthesynchronousmachineitisthedcfieldcurrent),ithasalargefluxvariationbetweenno load&fullload.Atnoload,itsfluxisfixedbytheMMF(magnetomotiveforce)oftheprimary currentalone,whileatloadthetheresultantfluxisdeterminedbyboththeprimary&secondary MMFs.Insynchronousmachines,thesecondarycurrentisthatofthearmature(stator).Thegenerator operationcanbefullydescribedbythefollowingcharacteristics:noload&airgap,shortcircuit(Potier triangle),load,external,regulationcurve,shortcircuitratio&directaxissynchronousreactance.The angledbetweenV&Efisthebasicvariableofthesynchronousmachine,itisknownasthepower angleortorqueangle.Visthemachineterminalvoltage&EfistheEMFinducedinthearmature winding(bythefieldflux).Foraconstantfieldcurrent(i.e.constantEf),theelectromagneticpower& torqueofthesynchronousmachinedependsolelyupontheangled.Iftheregulatoroftheprimemover (aturbineforexample)allowstheprimemovertoreceiveaddedinput(steam),thegeneratorwilltend tobedrivenatanincreasedspeed.ButsincetheRPMofthemachineisfixedbythenumberofpolesof themachineandtheline(busornode)frequency(n=120f/rpmP),thentheincreasedinputwill ewsultinanadvanceofthepolestructurei.e.Efwillbemovedaheadofthelinevoltagetoanewangle d.Undersuchcondition,additionalcurentinthearmature(Ia)willbedeliveredtotheline(and consequentlytotheconnectedloads&loadcenters).IaisalmostinphasewithV,thismeansthatthe outputpowerisanactiveone.Thus,iftheoutputofasynchronousgenerator,whichisoperatingin parallelwithothergenerators,istobeincreased,itsprimemovermustbeacceleratedbysupplyingit withmoremechanicalpower(throughtheuseofsteamforexample)andviceversa.Ontheotherhand,

avariationinthefieldcurrentatafixedloadandvoltagelevelwillcausethereactivecurrent(power) injectedintothelinetochange.