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PowerSystemAnalysisK.TomsovicV.VenkatasubramanianSchoolofElectricalEngineeringandComputerScienceWashingtonStateUniversityPullman,WA1.IntroductionTheinterconnectedpowersystemisoftenreferredtoasthelargestandmostcomplexmachineeverbuiltbyhumankind.Thismaybehyperbole,butitdoesemphasizeaninherenttruth:thecomplexinterdependencyofdifferentpartsofthesystem.Thatis,eventsingeographicallydistantpartsofthesystemmayinteractstronglyandinunexpectedways.Powersystemanalysisisconcernedwithunderstandingtheoperationofthesystemasawhole.Generally,thesystemisanalyzedeitherundersteady-stateoperatingconditionsorunderdynamicconditionsduringdisturbances.Electricpowerisprimarilytransmittedasathreephasesignal.Thatis,threeACcurrentcurrentsaresentthatareoutofphaseby120obutofequalmagnitude.Suchbalancedcurrentssumtozeroandthus,obviatetheneedforareturnline.Ifthevoltagesarebalancedaswell,thetotalpowertransmittedwillbeconstantintime,whichisamoreefficientuseofequipmentcapacity.Forlargescalesystemsanalysis,theassumptionisusuallymadethatthesystemisbalanced.Eachphasecanbethenanalyzedindependentlygreatlysimplifyingcomputations.Inthefollowing,theimplicitassumptionisthatthreephasesystemsarebeingused.2.Steady-StateAnalysisInsteadystateanalysis,anytransientsfromdisturbancesareassumedtohavesettleddownandthesystemstateisunchanging.Specifically,systemload,includingtransmissionsystemlosses,arepreciselymatchedwithpowergenerationsothatthesystemfrequencyisconstant,e.g.,60HzinNorthAmerica.Perhaps,theforemostconcernduringsteady-stateiseconomicoperationofthesystembutreliabilityisalsoimportantasthesystemmustbeoperatedtoavoidoutagesshoulddisturbancesoccur.Theprimaryanalysistoolforsteady-stateoperationistheso-calledpowerflowanalysis,wherethevoltagesandpowerflowthroughthesystemisdetermined.Thisanalysisisusedforbothoperationandplanningstudiesandthroughoutthesystematboththehightransmissionvoltagesandthelowerdistributionsystemvoltages.Thepowersystemcanberoughlyseparatedintothreesubcomponents:generation,transmissionanddistribution,andload.Thetransmissionanddistributionnetworkconsistsofpowertransformers,transmissionlines,capacitors,reactorsandprotectiondevices.Thevastmajorityofgenerationisproducedbysynchronousgenerators.Loadsconsistofalargenumberof,andadiverseassortment,ofdevices,fromhomeappliancesandlightingtoheavyindustrialequipmenttosophisticatedelectronics.Assuch,modelingtheaggregateeffectisachallengingprobleminpowersystemanalysis.Inthefollowing,theappropriatemodelsforthesecomponentsinthesteady-stateareintroduced.2.aModeling2.a.1TransformersAtransformerisadeviceusedtoconvertvoltagelevelsinanACcircuit.Theyhavenumeroususesinpowersystems.Tobegin,itismoreefficienttotransmitpowerathighvoltagesandlowcurrentthanlowvoltageandhighcurrent.Conversely,lowervoltagesaresaferandmoreeconomicforenduse.Thus,transformersareusedtostep-upvoltagesfromthegeneratorsandthenusedtostep-downthevoltageforenduse.Anotherwideuseoftransformersisforinstrumentationsothatsensitiveequipmentcanbeisolatedfromthehighvoltagesandcurrentsofthetransmissionsystem.Transformersmayalsobeusedasmeansofcontrollingrealpowerflowbyphase-shifting.Transformersfunctionbythelinkageofmagneticfluxthroughacoreofferromagneticmaterial.Figure2.1aillustratesamagneticcorewithasinglewinding.WhenacurrentIissuppliedtothefirstsetofwindings,calledtheprimarywindings,amagneticfield,H,willdevelopandmagneticflux,φ,willflowinthecore.Ampère’sLawrelatestheenclosedcurrenttothemagneticfieldencounteredonaclosedpath.IfweassumethatHisconstantthroughoutthepaththenNIHl=(2.1)wherelisthepathlengththroughthecoreandNisthenumberofwindingsonthecoresothatNIistheenclosedcurrentbythepathreferredtoasthemagnetomotiveforce(mmf).φφIIFigure2.1a)fluxflowsthroughcorefromfirstwinding,b)fluxislinkedtoasecondsetofwindingsThemagneticfieldisrelatedtothemagneticfluxbythepropertiesofthematerial,specifically,thepermeability.Ifweassumealinearrelationship,i.e.,neglectinghysteresisandsaturationeffects,thenthefluxdensityBorthefluxφislNIHBµµ==orlNIAµφ=(2.2)whereAisthecross-sectionalareaofthecore.Thisrelationshipbetweenthefluxflowinthecoreandthemmfiscalledthereluctance,R,ofthecoresothatNIR=φ(2.3)Now,ifasecondsetofwindings,thesecondarywindings,iswrappedaroundthecoreasshowninFigure2.1b,thetwocurrentswillbelinkedbymagneticinduction.Assumingthatnofluxflowsoutsidethecore,thenthetwowindingswillbeseetheexactsameflux,φ.Since,thetwowindingsalsoseethesamecorereluctance,thetwommf’sareidentical,i.e.,2211ININ=(2.4)Ifthefluxφischangingintime,orequivalentlythecurrentI,thenaccordingtoFaraday’sLaw,avoltagewillbeinduced.Assumingthisidealtransformerhasnolosses,thepowerinputwillbethesameasthepoweroutputso2211IVIV=(2.5)whereV1andV2aretheprimaryandsecondaryvoltages,respectively.Substituting(2.4)andrearrangingshows1212NNVV=(2.6)Thus,thevoltagegaininanidealtransformerissimplytheratioofthenumberofprimaryandsecondarywindings.Apracticaltransformerexperiencesseveralnon-idealeffects.Specifically,theseincludenon-zerowindingresistance,finitepermeabilityofthecore,eddycurrentsthatflowwithinthecore,hysteresis(the