6824941 Power System Stability Lecture

7/31/2019 6824941 Power System Stability Lecture

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EE4400:PowerEngineering3PowerSystemStability

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EE4400:PowerEngineering36PowerSystemStabilityQuestion:Answer:Whatdowemeanbythetermpowersystemstability?Itisdesirableforallelementsinapowersystemtooperatewithinastablerangeofvalues.Itisalsodesirablethatthesystemreturnstoastablestateafterasystemdisturbancesuchaswhenachangeinloadorgenerationoccurs,orafteracontingencysuchasafaultand/oroutageoccurs.

PowerSystemStabilitycanbeseparatedintotwomaincategories,anglestabilityandvoltagestability:1.AngleStabilitycanbedefinedas“theabilityofinterconnectedsynchronousmachinesofapowersystemtoremaininsynchronism”2.VoltageStabilitycanbebroadlydefinedas“theabilityofasystemtomaintainacceptablevoltagesfollowingasystemcontingencyordisturbance”.Inthischapterofthenoteswewillfocusonthefirstofthesetwomaincategoriesofpowersystemstability,anglestability.

6.1AngleStabilityPowersystemsaregenerallymadeupofalargeinterconnectionofsynchronousmachines.Innormaloperationthesemachinesremaininsynchronismwitheachother,maintainingsteadysynchronisingfrequencyandconstantmachinepowerangledi

fferencesbetweeneachother.Followingadisturbanceinasystemthefrequenciesofsynchronousmachinesundergotransientdeviationsfromthesynchronousfrequencyof50Hzandthemachinepoweranglesundergotransientchange.Thecategoryofanglestabilitycanbeconsideredintermsoftwomainsubcategories:1.Steady-State/Dynamic:Thisformofinstabilityresultsfromtheinabilitytomaintainsynchronismand/ordampenoutsystemtransientsandoscillationscausedbysmallsystemchanges,suchascontinualchangesinloadand/orgeneration.2.Transient:Thisformofinstabilityresultsfromtheinabilitytomaintainsynchronismafterlargedisturbancessuchassystemfaultsand/orequipmentoutages.Thesenoteswillfocusinparticularonthetransientstabilitysubcategoryandonthetechniquesthatcanbeusedtoanalysethetransientstabilityofasystemfollowingadisturbance.Theaimoftransientstabilitystudiesbeingtodetermineifthemachinesinasystemwillreturntoasteadysynchronisedstatefollowi

ngadisturbance.Author:DrCraigAumuller




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6.1.1TheSwingEquationConsiderageneratingunitconsistingofathree-phasesynchronousgeneratorandprimemover,asshowninFigure6-1.

Figure6-1GeneratingUnit

Themotionofthesynchronousgenerator’srotorisdeterminedbynewtonssecondlaw,whichisgivenas:Jαm(t)=Tm(t)−Te(t)=Ta(t)WhereJ(6.1.1.1)

αmTmTeTa

=Totalmomentofinertiaoftherotatingmasses(primemoverandgenerator)(kgm2)=Rotorangularacceleration(rad/s2)=Mechanicaltorquesuppliedbytheprimemoverminustheretardingtorqueduetomechanicallosses(eg.Friction)(Nm)=Electricaltorque,accountingforthetotalthree-phasepoweroutputandlosses(Nm)=Netacceleratingtorque(Nm)

Themachineandelectricaltorques,TmandTe,arepositiveforgeneratoroperation.Therotorangularaccelerationisgivenby

αm(t)=

dωm(t)d2θm(t)=dtdt2dθ(t)ωm(t)=mdt

(6.1.1.2)

(6.1.1.3)

Where

ωm=Rotorangularvelocity(rad/s)θm=Rotorangularpositionwithrespecttoastationaryaxis(rad)

Author:DrCraigAumuller




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Insteadystateconditionsthemechanicaltorqueequalstheelectricaltorqueandtheacceleratingtorqueiszero.Thereisnoaccelerationandtherotorspeedisconstantatthesynchronousvelocity.Whenthemechanicaltorqueismorethantheelectricaltorquethentheaccelerationtorqueispositiveandthespeedoftherotorincreases.Whenthemechanicaltorqueislessthantheelectricaltorquethentheaccelerationtorqueisnegativeandthespeedoftherotordecreases.Sinceweareinterestedintherotorspeedrelativetothesynchronousspeeditisconvenienttomeasuretherotorangularpositionwithrespecttoasynchronouslyrotatingaxisinsteadofastationaryone.Wethereforedefine

θm(t)=ωmsynt+δm(t)Where

(6.1.1.4)

ωmsyn=Synchronousangularvelocityoftherotor,rad/sδm=RotorangularpositionwithrespecttoasynchronouslyrotatingreferenceTounderstandtheconceptofthesynchronouslyrotatingreferenceaxisconsiderthediagraminFigure6-2.Inthisexampletherotorisrotatingathalfthesynchronousspeed,ωmsyn/2,suchthatinthetimeittakesforthereferenceaxis

torotate45°therotoronlyrotates22.5°andtherotorangularpositionwithreferencetotherotatingaxischangesfrom-45°to-67.5°.

Figure6-2Synchronouslyrotatingreferenceaxis




EE4400:PowerEngineering3PowerSystemStabilityFrom(6.1.1.2)and(6.1.1.4),weseethatequation(6.1.1.1)canbewrittenasd2δm(t)d2θm(t)=J=Tm(t)−Te(t)=Ta(t)Jαm(t)=Jdt2dt2

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(6.1.1.5)

Beingthatweareanalysingapowersystemweareinterestedinvaluesofpowermorethanweareinvaluesoftorque.Itisthereforemoreconvenienttoworkwithexpressionsofpower.Furthermoreitisconvenienttoconsiderthispowerinperunitratherthanactualunits.PowerisequaltotheangularvelocitytimesthetorqueandperunitpowercanbeobtainedbydividingbySrated,sothat:Jωm(t)d2δm(t)ωm(t)Tm(t)−ωm(t)Te(t)pm(t)−pe(t)===pmpu(tpu(t)Srateddt2SratedSratedpmpupepu(6.1.1.6)

=Mechanicalpowersuppliedbytheprimemoverminusmechanicallosses(perunit)=Electricalpoweroutputofgeneratorpluselectricallosses(perunit)

Wedefineaconstantvalueknownasthenormalisedinertiaconstant,or“H”constant.H=storedkineticenergyatsynchronousspeedgeneratorvoltampererating12Jωmsyn=2(joules/VAorperunit−seconds)Srated

(6.1.1.7)Equation(6.1.1.6)becomes

ωm(t)d2δm(t)2H2=pmpu(t)−pepu(t)=papu(t)ωmsyndt2Wherepapu=Acceleratingpower

(6.1.1.8)

Wedefineper-unitrotorangularvelocityas:

ωpu(t)=Equation(6.1.1.8)becomes

ωm(t)ωmsyn

(6.1.1.9)

d2δm(t)=pmpu(t)−pepu(t)=papu(t)ωpu(t)ωmsyndt22H

(6.1.1.10)





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WhenasynchronousgeneratorhasPpoles,thesynchronouselectricalangularvelocity,ωsyn,knownmorecorrectlyasthesynchronouselectricalradianfrequency,canberelatedtothesynchronousmechanicalangularvelocitybythefollowingrelationship.

ωsyn=

Pωmsyn2

(6.1.1.11)

Tounderstandhowthisrelationshiparises,considerthatthenumberofmechanicalradiansinonefullrevolutionoftherotoris2π.If,forinstance,ageneratorhas4poles(2pairs),andthereare2πelectricalradiansbetweenpolesinapairthentheelectricalwaveformwillgothrough2*2π=4πelectricalradianswithinthesamerevolutionoftherotor.Ingeneralthenumberofelectricalradiansinonerevolutionisthenumberofmechanicalradianstimesthenumberofpolepairs(thenumberofpolesdividedbytwo).Therelationshipshownin(6.1.1.11)alsoholdsfortheelectricalangularaccelerationα(t),theelectricalradianfrequencyω(t),andtheelectricalpowerangleδvalues.

α(t)=

Pαm(t)2Pω(t)=ωm(t)2Pδ(t)=δm(t)2

(6.1.1.12)

From(6.1.1.9)weseethat

ωm(t)Pω(t)ω(t)==ωpu(t)=2ωmsynωsynωsynP

2

(6.1.1.13)

Thereforeequation(6.1.1.10)canbewritteninelectricaltermsratherthanmechanical:2H

ωsyn

ωpu(t)

d2δ(t)=pmpu(t)−pepu(t)=papu(t)dt2

(6.1.1.14)

Thisequationisknownasthe“SWING-EQUATION”andisthefundamentalequationindeterminingrotordynamicsintransientstabilitystudies.Theswingequationisnon-linearbecausepepu(t)isanon-linearfunctionofδandbecauseoftheωpu(t)term.Therotorspeed,however,doesnotvaryagreatdealfromthesynchronousspeedduringtransientsandavalueofωpu(t)≈1.0isoftenusedinhandcalculations.







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6.1.2ElectricpowerequationIntheswingequationthemechanicalpowerfromtheprimemover,pmpu,isconsideredtobeconstant.Thisisareasonableassumptionastheconditionsintheelectricalnetworkcanbeexpectedtochangebeforethesloweractingcontrolgovernorcancausetheturbinetoreact.Theelectricalpower,pepu,willthereforedeterminewethertherotoraccelerates,deceleratesorremainsataconstantsynchronousspeed.Electricalnetworkdisturbancesresultingfromextremechangesinsystemloading,networkfaultsandcircuitbreakeroperationwillcausethegeneratoroutputtochangerapidlyandtransientswillexist.ThesynchronousmachineisrepresentedintransientstabilitystudiesbyatransientinternalvoltageE inserieswithitstransientreactanceX dasshowninFigure6-3.

Figure6-3Simplifiedsynchronousmachinemodelfortransientstabilitystudies

Generatorsarenormallyconnectedtosystemscomposedoftransmissionlines,transformersandothermachines.Whensystemsarelargeenough,astheymostoftenare,an“infinitebus”behindasystemreactancecanrepresentthem.Aninfinitebusisanidealvoltagesourcethatmaintainsconstantvoltagemagnitude,phaseandfrequency.Figure6-4illustratestheconnectionarrangementofthesynchronousgeneratortotheequivalentsystem.

Figure6-4Synchronousgeneratorconnectedtoasystemequivalent

Therealpowerdeliveredfromthegeneratortotheinfinitebus(andthereforethesystem)istherefore:E Vbuspe=sinδ=pmaxsinδ(6.1.2.1)XeqWhereXeq=X d+X





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6.1.3TheequalareacriterionAmethodknownastheequalareacriterioncanbeusedforaquickpredictionofstability.Thismethoddeterminesifamachinemaintainsitsstabilityafteradisturbancebygraphicalinterpretationoftheenergystoredintherotatingmass.Itis,however,restrictedtoeitheronemachinesystemsconnectedtoaninfinitebusortotwomachinesystems.Morecomplexandaccuratenumericalsolutionofthenon-linearswingequationcanbeperformedbycomputerandisespeciallyapplicabletotheanalysisofmulti-machinesystems.Inthesenoteswewillnotfocusonthisnumericalsolutionmethod,preferringtolookatthesimplerequalareamethod.Figure6-5showsplotsofelectricalpowerpeandmechanicalpowerpmversusthepowerangleδ.ThegeneratingunitillustratedinFigure6-5isinitiallyoperatinginasteadystatepe=pm=pm0andδ=δ0.Ifastepchangeinthemechanicalpoweroccurssothatitincreasestopm=pm1attimeequalszero.Therotorhasinertiaandassuchtherotorpositioncannotchangeinstantaneously,δm(0−)=δm(0+).Astheelectricalpowerangleisrelatedtotherotorpositionandelectricalpowerisrelatedtotheelectricalpoweranglethentheelectricalpowerdoesnotchangeinstantaneously,δ(0−)=δ(0+)=δ0andpe(0−)=pe).Themechanicalandelectricalpowerswillbeunbalancedandtheacceleratingpowerwillacttoincreasetherotorspeedandδwillincrease.Whentheangleδreachesthedesiredvalueofδ1thentheacceleration,d2δ/dt2,willbezerobut

asthevelocityisabovesynchronoustheangleδwillcontinuetoincreaseandovershootthetarget.Oncepastδ1theelectricalpowerbecomesgreaterthanthemechanicalpowerandtherotordecelerates.Afterreachingamaximumvalueitbeginstoswingbacktowardsδ1.Iftherewerenodampingpresentthentheangleδwouldcontinuetooscillateabouttheδ1point.Damping,howeverispresentduetomechanicalandelectricallosses,andδeventuallysettlesdowntoitsfinalsteadystatevalueδ1.

Figure6-5ElectricalandMechanicalpowerversusδ




EE4400:PowerEngineering3PowerSystemStabilityIfweconsidernowtheswingequationandassumethatωpu(t)≈1.0then:2Hd2δ=pmpu−pepuωsyndt2Ifwetiplybothsidesbydδ/dtandusetheidentityddδdδdt=2dtdt)becomes2Hd2δωsyndt2dδdtdδHddδ=(pmpu−pepu)=dtωsyndtdt222dδ2dt

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(6.1.3.1)

(6.1.3.2)

Multiplyingequation(6.1.3.2)bydtandintegratefromδ0toδweobtainthefollowingexpression:

dδd∫dt=δ∫(pmpu−pepu)dδωsynδ00HOr

δ

2

δ

(6.1.3.3)

Hdδωsyndt

2δ

=∫(pmpu−pepu)dδδ0

δ

(6.1.3.4)

δ0

Notethattheaboveintegrationbeginsatδ0andendsatsomearbitraryangleδ.Thevalueofdδ/dtiszeroatδ0asthemachineisinsteadystate.Thevalueofdδ/dtisalsozeroatδequaltoδ2,astherotorchangesdirectionbacktowardsδ1.Thelefthandsideof(6.1.3.4)equalszeroforδ=δ2andtherefore:

∫(pδ0

δ

m u

−pepu)dδ=0

(6.1.3.5)

Ifweseparate(6.1.3.5)intoacceleratinganddeceleratingareasweobtainthefollowingequation:



−p)d−p)d∫(p44244δ+δ∫(p44244δ=01313δm uepumpuepu0

δ1

δ1

(6.1.3.6)

re A1

0

re A2





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Notehowthetwoelementsof(6.1.3.6)equatetotheareasA1andA2showninFigure6-5andinorderforthetwoequationstobesatisfiedthetwoareasmustbeequal.Thisiswhywecallthisthe“equalarea”criterion.Inpractice,suddenchangesinmechanicalpowerdonotoccurasthetimeconstantsassociatedwiththeprimemoverdynamicsareintheorderofseconds.However,stabilityphenomenasimilartothatdescribedabovecanalsooccurfromsuddenchangesinelectricalpowerduetosystemchangessuchassystemfaults.Thefollowingthreeexamplesillustratehowtheequalareacriterioncanbeusedtodetermineifasystemwillbeunstableafterathree-phasefault.Thedeterminationofthecriticalclearingtime(CCTortcr),whichisthelongestfaultdurationthatcanbeallowedforstabilitytobemaintained,willalsodiscussed.





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6.1.4DesignmethodsforimprovingtransientstabilityThereareanumberofdesignmeasuresthatcanbeimplementedtoimprovethetransientstabilityofapowersystem:1.ImprovethemaximumpowertransfercapabilityImprovingthemaximumpowertransfercapabilityofasystemmeansthatpowercanbetransferredthroughalternativeun-faultedportionsofthenetworkwhenafaultoccurs.Theeffectofafaultonthesystemwillnotbeasextreme.Themaximumtransfercapabilityofasystemcanbeimprovedbythefollowingmethods:a.Implementandusehighersystemvoltagelevels(systemlosseswilldecreaseascurrentflowswillbelower,especiallyimportantincaseswherelinedistancesarelarge)b.Installadditionaltransmissionlines.c.Installlinesandtransformerswithsmallerreactancevaluesd.Installseriescapacitivetransmissionlinecompensationtoreducetheoverallreactanceoflinese.InstallstaticVARcompensatorsandflexibleACtransmissionsystems(FACTS)2.ImplementhighspeedfaultclearingItisvitaltoclearfaultsbeforethecriticalclearingtimeisreachedsothequickerafaultisclearedthebetter.3.Implementhighspeedre-closureofcircuitbreakersAsthemajorityoftransmissionlineshortcircuitsaretemporary,re-closurepostfaultcanbebeneficialinprovidingbetterpowertransfercapability.Caremustbetakeninthiscasetoensurethatthere-closingonapermanentfaultandanysubsequentreopeningwillnotadverselyaffectthestabilityofthesystem.4.ImplementsinglepoleswitchingThemajo

rityofshortcircuitsaresinglelinetogroundandtheindependentswitchoutofonlythefaultedphasemeansthatsomepowerflowcancontinueacrossthefaultedline.Studieshaveshownthatsinglelinetogroundfaultsareself-clearingevenwhenonlythefaultedphaseisde-energised.5.UsegeneratorswithlargermachineinertiaandwithlowertransientreactanceAlargersynchronousmachineinertiaconstant(H)resultsinareductioninangularaccelerationandthereforeaslowingdownofangularswings.Thecriticalclearingtimeisincreased.Reducingthemachinetransientreactanceincreasesthepowertransfercapabilityduringfaultsandintheperiodspostfault.Author:DrCraigAumuller



EE4400:PowerEngineering3PowerSystemStability6.Usefastresponding,highgainexciters

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Modernexcitationsystemscanbedesignedtoactquicklyandwithhighgainintheeventofsensingalowterminalvoltageduringfaults.Theeffectistoincreasethegeneratoroutputduringthefaultandpostfaultperiods.Criticalclearingtimesareincreased.7.ImplementfastvalvingSomesteamturbinesareequippedwithfastvalvingtodivertsteamflowsandrapidlydecreasethemechanicaloutput.Whenafaultoccursneartothegeneratortheelectricalpoweroutputisreducedandthefastvalvingactstobalancethemechanicalandelectricalpowers.Thisprovidesreducedaccelerationandlongercriticalclearingtimes.8.BreakingResistorsInpowersystems,areasofgenerationcanbetemporarilyseparatedfromtheloadareas.Whentheseparationoccursthebreakingresistercanbeinsertedintothegenerationareaforasecondortwoinordertoslowtheacceleration.




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6824941 Power System Stability Lecture