POWER SYSTEM DYNAMICS AND STABILITY

1 IPOWERSYSTEMDYNAMICSANDSTABILITYP eter W. Sauerand M. A. PaiDepartment of Electricaland ComputerEngineeringTheUniversity of Illinoisat Urbana-Champaign1406W. ,IL 61801iiCopyrightc 1997(updated2006), theproperty of andis notto be sold,reproduced,or .. Interest..62 ..203 ..614 ..865 .. Stator/NetworkTransients.. (Flux-Decay) Model.. 1176 .. Stator/NetworkTransients.. {Decay Model.. niteBus.. 1527 erential-AlgebraicModel.. statoralgebraicequations.. cationof theTwo-AxisModel.. (FullModel).. : POWER -BalanceForm.. Method .. :Current-BalanceForm.. model.. 2118 .. ModelA.. ModelB.. ects.. ectof loading.. ectofKA.. ectof type of load.. SystemStabilizers.. K6 constants [78,88].. systemstabilizerdesign.. 2709 .. 'sMethod.. (PEBS).. nite-bussystem.. single-machinein .. E( ) anditsusein (BCU)Method .. 313A .. 327 Bibliography333 CONTENTSixxCONTENTSPREFACET heneedforpower systemdynamicanalysishasgrownsigni cantlyinrecent duelargelyto thedesireto utilizetransmissionnetworksformore beenstudiedforyearsin a long-termplanninganddesignenvironment,th ereis a recognizedneedto performthisanalysisin a weeklyor even devotedtodynamicmodelingandsimulationasi t relatestosuch a need,combiningtheoreticalaswellaspractic alinformationforuseas a textforformalinstructionor forreferenceby textforformalinstruction,thisbookassumes a backgroundinelectromechanics,machines,an dpower , thetextwouldnormallybe usedin a graduatecoursein hasbeendesignedforusein a one-semester( fteen-week)}

2 , mosttraditionalmachineandpower systemanalysisbooksandattemptsto follow theindustrystandardsso thata tran-sitionto moredetailandpracticalapplicationis dividedinto two to 6 give anin-troductionto electromagnetictransient analysisanda systematicderivationof rigorousexplanationof modelorigins,development, andsimpli givento thecon-ceptof reduced-ordermodelingusingintegralmanifo ldsas a rmbasisforunderstandingthederivationsand limitationsof a mathematicalintroductionto thistechniqueof to 9 utilizethesedynamicmodelsforsimulationan dstability givento thecalculationof initialconditionsandthealternative analysisis presentedin a sequentialxixiiPREFACE manner,concludingwiththedesignof POWER analysisis formulatedusingenergyfunctionmethods withanem-phasisontheessentialsof thepotentialenergyboundarysurfaceandthec ontrollingunstableequilibriumpoint notclaimto be a completecollectionof allmodelsandsimulationtechniques,butseek sto providea basicunderstandingof moredetailedandaccuratemodelsexistin theliterature,a majorgoalof thisbookis toexplainhow individualcomponent modelsareinterfacedfora systemstudy.

3 Ourobjective is toprovidea rmtheoreticalfoundationforpower systemdynamicanalysistoserve as a startingpoint fordeeper explorationof complexphenomenaandapplicationsin electricpower have so many peopleto acknowledgefortheirassistancein ourcareersandlives thatwe willlimitourlistto sixpeoplewhohave hada directimpacton theUniversity of Illinoispower programandthepreparationof thisbook:StanHelm,forhisdevotionto thepower areaof electricalengineeringforoversixty years;GeorgeSwenson,forhisleadershipin strengtheningthepowerareain thedepartment; MacVanValkenburg,forhisfatherlywisdomand guidance;DavidGrainger,forhis nancialsupportof thepower program;PetarKokotovic,forhisinspiration andenergeticdiscussions;andKarenChitwood , yearsof collaborationat theUniversity of Illinois,we have strivedto maintaina healthy balancebetweeneducationandre-search. We thanktheUniversity , systemshave evolved fromtheoriginalcentralgeneratingstationc on-ceptto a modernhighlyinterconnectedsystemwithimpr ovedtechnologiesa ectingeach partof thesystemseparately.

4 Thetechniquesforanalysisofpower systemshave beena ectedmostdrasticallyby thematurity of otherdisciplineswithinelectricalengineer ing,thefoundationsof theanalysisareoftenhiddenin assumptionsandmeth-ods thathave resultedfromyearsof ,we have a hostof techniquesandmodelsmixedwiththeartof powerengineeringand,at theotherextreme,we have sophisticatedcontrolsystemsrequiringrigo roussystemtheory. It is necessaryto strike a balancebetweenthesetwo extremesso thisbookis to seeksuch a middlegroundin theareaof modelingandsimulationliesintheneedto capture(withminimalsizeandcomplexity) the\phenomenaofinterest."Thesephenomenam ustbe understood beforee ective simulationcanbe POWER systemdynamicsandstability is clearlyanex-tremelybroadtopicwitha longhistoryandvolumesof ways to divideandcategorizethissubjectforbothedu cationandresearch. Whilea substantialamount of informationaboutthedynamicbehaviorof POWER systemscanbe gainedthroughexperienceworkingwithandtes tingindividualpiecesof equipment, thecomplexproblemsandoper-atingpractices of largeinterconnectedsystemscanbe betterunderstood if12 CHAPTER1.

5 INTRODUCTION thisexperienceis coupledwitha as transient networkanalyzershave a valuein providinga physicalfeel-ingforthedynamicresponseof POWER systems,buttheyarelimitedto smallsizesandarenot exibleenoughto a placein thestudyof systemdynamics,capability and exibility have thestudyof POWER systemdynam-icsandstability [1]. deMelloclassi eddynamicprocessesinto threecategories:1. Electricalmachineandsystemdynamics2. Systemgoverningandgenerationcontrol3. Prime-mover energysupplydynamicsandcontrolIn thesamereference,C. ConcordiaandR. P. Schulzclassifydynamicstudiesaccordingto fourconcepts:1. Thetimeof thesystemcondition:past,present, or future2. Thetimerangeof thestudy:microsecondthroughhourlyrespons e3. Thenatureof thesystemunderstudy:newstation,newline, Thetechnicalscope of thestudy:faultanalysis,loadshedding,sub- synchronousresonance, theseclassi cationssharea commonthread:Theyemphasizethatthesystemi s notin steadystateandthatmany modelsforvariouscom-ponents mustbe usedin varyingdegreesof detailtoallow e cient rsthalfof thisbookis thus devotedto thesubjectof modeling,andthesecondhalfis devotedtotheuseof anexhaustivetreatment.

6 Rather,fundamentalconceptsarepresentedas a foundationforprobingdeeper into thevastnumber of important andinterestingdynamicphenomenain POWER of a POWER systemcanbe representedin a block-diagramformat,asshownin , iFuelSourceFurnace&BoilerTurbineGenerato rNetworkLoadsEnergyControlCenterP, :Systemdynamicstructureresentationdoes notshow allof thecomplexdynamicinteractionbetweencompo nents andtheircontrols,it serves to broadlydescribe , therehasbeena majordivisioninto themechanicalandelectricalsubsystemsas notabsolute,however,sincetheelectricalsi declearlycontainscomponents withmechan-icaldynamics(tap-changing-und er-load(TCUL)transformers,motorloads,etc .)andthemechanicalsideclearlycontainscom ponents withelectricaldy-namics(auxiliarymotordr ives,processcontrols,etc.).Furthermore,b othsidesarecoupledthroughthemonitoringan dcontrolfunctionsof classi cationof dynamicphenomenais theirnaturaltimerangeof typicalclassi cationis shownin similarconceptis presentedin [6].Thistime-rangeclassi cationis important becauseof itsimpactoncomponent shouldbeintuitivelyobviousthatit is notnecessaryto solve thecomplextransmissionlinewave equationsto investigatetheimpactof a changein minda statement madeearlierthat\thesystemis notin steadystate.

7 "Evidently, dependingonthenatureof thedynamic4 CHAPTER1. INTRODUCTIONL ightning PropagationSwitching SurgesStator Transients andSubsynchronous ResonanceTransientStabilityGovernor and LoadFrequency ControlBoiler and Long-Term Dynamics10 710 510 (sec) :Timerangesof dynamicphenomenadisturbance,portionsof thepower systemcanbe consideredin \quasi-steadystate."Thisratherambiguoust ermwillbe explainedfullyin thecontextoftime-scalemodeling[2]. cationsof dynamicphenomenaillustratethepotentialco mplexity of even smallor POWER systemdynamicsandstability arecompoundedim-menselyby thecurrent sizeof generalsystemstructureis shownin notnecessarilycom-monto interconnectedsystemsthroughouttheworld, it represents a typicalNorthAmericansystemandserves to illustratetheconceptof a \large-scalesystem."If we speculateaboutthepossiblesizeof a singleinterconnectedsystemcontainingnine coordinatingcouncils,fourpoolsper 20 GeneratorsPool2 10 CompaniesCoordinatingCouncil 3 5 PoolsInternational9 :Systemorganizationalstructurecouncil,si xcompaniesper pool,andtengeneratorsper company, thetotalpossiblenumber of (138{765kV)thentypicallyconsistsof over 10, ,thecurrent demandin theninecoordinatingcouncilswithintheNort hAmericanElectricReliability Council(NERC)exceeds500,000MW[3].}

8 At anaverage250 MWper generator,thisroughlycon rmstheestimateof over 2000generatorsin systemsrangingin sizefromthesmallestcompany to wellas regionalpower viewof thepotentialsize,dynamicstudiesmustbe capa-bleof su cientlyaccuraterepresentationwithoutproh ibitive systemengineeringproblemsinherent in such a complextaskwas emphasizedin two benchmarkreportsby Departmentof Energy(DOE)andtheElectricPower Research Institute(EPRI)[4, 5].Thesereportsresultedin a meetingof internationalleadersto identifydi-6 CHAPTER1. INTRODUCTION rectionsforthefutureof thistechnology. Thesereportssetthestagefora wholeneweraof POWER cant impactof thesereportswas thestimulationof newideasthatgrewinto student InterestThedynamicperformanceof POWER systemsis important to boththesystemorganizations,fromaneconomi cviewpoint, andsociety in general,fromareliability viewpoint. Theanalysisof POWER systemdynamicsandstabilityis increasingdailyin termsof number andfrequencyof studies,as wellasin complexity beendiscussedaccordingto basicfunction,time-scaleproperties, theessenceof thechallengesof e ective simulationof POWER ,modelingandsimulationcapturethephenomen aof rststepin thisprocessis understandingthephenomenaof solidphysicalandmathematicalunderstandin gcanthemodelingandsimulationproperlyre mathematicalmodelsmustbe understood, andtheirpurposemustbe well de accomplished,theminimalcostis achievedby modelreductionandsimpli cationwithoutsigni cantlossin thetime-scaleclassi cationof POWER systemdynamics,thefastesttran-sients aregenerallyconsideredto be orientedtowardsystemanalysisratherthanco mponent design,thesetransients arediscussedin thecontextof theirpropagationinto otherareasof as conductortemperature,motion,andchemicalr eactionareimportant aspectsof such high-speedtransients,we focusmainlyon a circuitview.

9 Wherevoltageandcurrent areof insulationbreakdown,arcing,andlightningp ropagationrarelylendthemselves to incorporationinto standardcircuitanalysis[7], somesimula-tionsoftwaredoes includea portionof thesetransients [8, 9]. Froma systemviewpoint, thetransmissionlineis themaincomponent thatprovidestheinterconnectionto programs(EMTP)describedin [8] and[9] areuniquefortheirtreatment of switchingphenomenaof valueto designers,theyincludethecapability to studythepropagationof transients theEMTP programa systemanalyst'stool as wellasa designer' usedin theseprogramsarediscussedin networkanalysisareusuallycat-egorizedby linelengths(long,medium,short)[10]{[12]. Thislinelengthconceptis interesting,andpresents a majorchallengein thesystematicfor-mulationof example,moststudentsandengineershave beenintroducedto theargument thatshunt capacitanceneednotbe includedin short-linemodelsbecauseit hasa negligiblee ecton\theaccuracy."Thus,a shortlinecanbe modeledusingonlyseriesre-sistanceandindu ctance,resultingin a single(fora singleline)di erentialequationin thecurrent ,therewouldalsobe a di modelfromtwo or threedi erentialequationsto onlyoneisa processthathasto be justi edmathematicallyas wellas physically.}

10 Aswillbe shown,the\long-line"modelinvolves partialdi erentialequations,which in somesenserepresent anin nitenumber of ordinarydi nity to oneis, indeed,a majorreductionanddeserves ,it is important to be analysisunlesstheyaretheresultof a reductiontechniquesuchas thevarioustraditionaltransmis-sionlinemo delsillustratesthispoint the\long-line"modelbeginswiththeconstruc tionof anin nitesimalsegment of length xin assumedtobe :Transmissionlinesegmentenoughthatmagnet icandelectric elde ectscanbe consideredseparately,resultingin per-unitlengthlineparametersR0;L0;G0;C0. Thesedistributedparametershave theunitsof ohms/mi,henries/mi,etc., , initself,anapproximationof theexactinteractionof theelectricandmagnetic , of course,is thatas theincrementalsegment approacheszerolength,theresultingmodelgi ves a good approximationof Maxwell' certainspecialcases,it canbe shownthatsuch anapproachis indeedvalid([13], {397).Thelinehasvoltagesandcurrents at itssendingend(k) andreceivingend(m).}