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Electric Power Systems Research
jou rn al h om e p a g e :w w w . e l s e v i e r . c o m / l o c a t e / e p s r
Determining optimal virtual inertia and frequency control parameters to preserve the frequency stability in islanded microgrids with high penetration of renewables
Masoud Hajiakbari Fini, Mohamad Esmail Hamedani Golshan
∗DepartmentofElectricalandComputerEngineering,IsfahanUniversityofTechnology,Isfahan,Iran
a r t i c l e i n f o
Articlehistory:
Received25January2017 Receivedinrevisedform8July2017 Accepted8August2017
Keywords:
Frequencystability Lowinertiamicrogrids Many-objectiveoptimization Renewableenergyresources Virtualinertia
a b s t r a c t
Preservingthefrequencystabilityoflowinertiamicrogrids(MGs)withhighpenetrationofrenewables isaseriouschallenge.Torisetothischallenge,theinertiaconstantofMGswouldbevirtuallyincreased usingenergystorages.However,itisimportanttodeterminethesuitablevalueofinertiaconstantfor thesesystemssuchthatthefrequencystabilityispreservedwithalowercost.Frequencydroopcoefficient ofdistributedenergyresources(DERs)andloadfrequencycontrollers’parameterswouldalsoaffectthe frequencyresponseofMGs.Hence,inthispaper,inertiaconstantistunedtogetherwithfrequencydroop coefficientofDERsandloadfrequencycontrollers’parameters.Determiningtheseparametersismodeled asamulti-objectiveoptimizationproblemand,becausethenumberofobjectivesishigherthanthree, theproblemissolvedbyamany-objectiveoptimizationalgorithm.Comparativesimulationstudieshave beendoneonanMGwithdifferenttypesofDERstoprovethatusingtheproposedstrategyfortuningthe MGparametersnotonlythefrequencydeviationishighlydecreasedbutalsotheamountofloadshedding isconsiderablydiminished.Thiswouldincreasethecustomers’satisfaction.Moreover,consideringthe inertiaconstantasaminimizationobjective,frequencystabilitywouldbepreservedwithalowercost.
©2017ElsevierB.V.Allrightsreserved.
1. Introduction
Duetotheenvironmentalconcerns,thereisanincreasinginter- estinusingrenewableenergysources(RESs)forpowergeneration.
MGswouldprovideasuitableinfrastructureforintegratingRESsto thegridatdistributionlevel.MGscanoperateingrid-connectedor islandedmode.However,inislandedmode,theywouldencounter challengesinfrequencyandvoltagecontrol.Theseproblemswould beexacerbatedinMGswithahighshareofpower-electronically interfacedDERs;duetothelowinertiaofthegrid.Infact,inlow inertiaMGs,powerimbalancesresultinrapidchangesinfrequency thatmightendangerthestabilityofthesystem[1].
Toaddressthesestabilityconcerns,somemethodshavebeen proposedtoincreasetheinertiaofMGs.InRefs.[2,3],usingsyn- chronous condensers have been proposed for this purpose. In additiontocontributingtotheinertiaofthegird,synchronouscon- denserswouldparticipateinreactivepowercontrol.Inverter-based DERswouldalsoemulatetheinertialresponseofsynchronousgen-
∗Correspondingauthor.
E-mailaddresses:m.hajiakbari@ec.iut.ac.ir(M.HajiakbariFini), hgolshan@cc.iut.ac.ir(M.E.HamedaniGolshan).
eratorsbyinjectingapowerproportionaltofrequencyderivative tothegrid.InRef.[4],acontrolmethodhasbeenproposedforcon- verterstoemulatethebehaviorofsynchronousgenerators.Virtual inertiahasbeenimplementedinRef.[5]toincreasethecontribu- tionofdistributedgeneratorstooscillationdamping.Toemulate the inertial behavior of synchronousgenerators, inverter-based DERsrequireatemporarysourceofenergysimilartothekinetic energyoftherotorofsynchronousgenerators.InRef.[6],theenergy storedintherotorofdoublyfedinductiongenerator(DFIG)based windturbinesandalsotheultracapacitor(UC)installedatthedc- linkofconvertersareusedasenergysourcesforinertiaemulation.
InRef.[7],HVDCtransmissionlineiscontrolledtocontributetothe inertiaofthegrid.HVDClinkswouldtransfertheinertialpower generatedbywindfarmstothemaingrid,transferinertialpower fromoneareaofpowersystemtoanotherareaorusetheenergy storedintheDClinktoemulatetheinertialresponse.UCispro- posedinRef.[8]toemulatetheinertialresponseofsynchronous generatorsin anisolatedpowersystem.Althoughsomestudies havebeencarriedoutoncontributionofinverter-basedDERsto theinertiaofMGs,tothebestofauthors’knowledge,asystematic methodfordeterminingthepropervalueforinertiaconstantin islandedMGshasnotyetbeenproposed.
http://dx.doi.org/10.1016/j.epsr.2017.08.007 0378-7796/©2017ElsevierB.V.Allrightsreserved.
Nomenclature
R Frequencydroopcoefficientofenergyresources H Inertiaconstantofthegrid
Hvir Virtualinertia
Heq Theequivalentinertiaconstantofsynchronousgen- erators
Pref-old Referencepowerofenergyresourceswithoutiner- tiaemulation
Pref-new Referencepowerofenergyresourceswithinertia emulation
f Operationfrequencyofthegrid
E TheenergyrequiredbyUCforinertiaemulation Pinertia Inertialpower
finit Frequencybeforethedisturbance fcrit Criticalfrequency
tcrit Thetimeittakesthefrequencytoreachthecritical frequency
Eavail TheenergyavailablefromUC C CapacitanceofUC
Vinit InitialvoltageofUC
Vmin MinimumpermissiblevoltageofUC Pbase Basepowerofthegrid
Vn NominalvoltageofUC Pn NominalpowerofUC fnadir Frequencyundershoot fmax Frequencyovershoot tst Settlingtimeoffrequency Obji Theithobjectivefunction
t Time
QL0 Initialreactivepowerofload
QL Currentvaluesoftheload’sreactivepower b Coefficientofloadreactivepowerdependencyon
voltage
fn Nominalfrequency tsim Simulationtime
f(t) Deviationofthefrequencyfromthenominalvalue fmin Minimumdesirablefrequency
x Vectorofdecisionvariables
n ThenumberofDERsthatparticipateinfrequency control
xup Theupperlimitvectorforthedecisionvariables xlow Thelowerlimitvectorforthedecisionvariables Objli TheithobjectiveofthelthParetooptimalsolution m Thenumberofobjectives
IdealObj Idealobjectivevector
IdealObji Theithelementoftheidealobjectivevector dl SumofthenormalizedobjectivesofthelthPareto
solution
Tg Governortimeconstant Tt Turbinetimeconstant TBESS BESStimeconstant TUC UCtimeconstant
P˙BESS BESSramprate P˙DEG DEGramprate
PL0 Initialactivepowerofload V0 Initialvoltageofload V Voltageoftheloadbus
PL Currentvaluesoftheloadactivepower
a Coefficientofloadactivepowerdependencyonvolt- age
Kpf Coefficientofloadactivepowerdependencyonfre- quency
Kqf Coefficientofloadreactivepowerdependencyon frequency
Inadditiontoinertiaconstant,loadfrequencycontrollersand frequency droop coefficient of power sources would affectthe frequency response of thegrid. Fine-tuningthe loadfrequency controllerswouldreducethemaximumfrequencydeviationand alsobringbackthefrequencytothenominalvaluefaster.Differ- entmethodshavebeenproposedforloadfrequencycontrol(LFC) inpowersystems.InRef.[9]theperformanceofmodelpredictive controllerforLFCinNordicpowersystemwasinvestigated.Many researchershavefocused ontuningthetraditional proportional integral(PI)/proportionalintegralderivative(PID)controllersusing evolutionaryalgorithms.InRef.[10]bacterialforagingoptimization algorithmhasbeenimplementedtotunetheloadfrequencycon- trollersofanMGwithgenerationrateconstraint(GRC).InRef.[11]
ahierarchicalapproachbasedonfuzzylogichasbeenproposed toimprove the quality of frequency control. Electrical vehicles havebeenimplementedinRef.[12]forfrequencycontrol.InRef.
[13],fractionalorderPIDcontrollershavebeenproposedforLFC inamulti-areapowersystem.Anadaptiveset-pointmodulation techniquehasbeenimplementedinRef.[14]toenhancetheperfor- manceofPIloadfrequencycontrollers.Toimprovetheperformance offrequencycontrollersinathree-areathermalpowersystem,in Ref.[15]governors’frequencydroopcoefficient(R)havebeenopti- mizedtogetherwiththeloadfrequencycontrollers’parameters.
Sincethegridinertiaconstant(H),frequencydroopcoefficientof DERsandparametersofloadfrequencycontrollersallaffectthefre- quencyresponseofMGs,inthispaper,tuningtheseparametershas beensuggestedtoimprovethefrequencystabilityofMGs.Based ondifferentcriteriathatshouldbemettohaveaproperfrequency responseinMGs,asystematicmethodisproposedfortuningallthe parameters,includingR,Handthecontrollersparameters,simulta- neously.Themaingoaloftuningtheseparametersistoimprovethe frequencystabilityofMG.However,thisgoalshouldbeachieved withtheminimumcost.Hence,tuningtheseparametershasbeen modeledasamulti-objectiveoptimizationproblemwhichconsid- ersbothstabilityandeconomicaspects.Consideringthefactthat usualmulti-objectiveoptimizationalgorithms,likenon-dominated sortinggeneticalgorithmII(NSGA-II),wouldnotshowagoodper- formanceinsolvingoptimizationproblemswithmorethanthree objectives[16],a recentlydevelopedmany-objectivekneepoint drivenevolutionaryalgorithm(KnEA)isimplementedforsolving thisproblem.Finally,toselectoneoftheParetooptimalsolutions obtainedbyKnEAasthefinalsolution,astrategybasedonthemini- mumsumofthenormalizedobjectivesissuggested.Also,amethod fordeterminingthecharacteristicsoftheultracapacitorrequired foremulatingthedeterminedinertiaisproposed.
Therestofthispaperisorganizedasfollows:inSection2,the requiredequipmentforincreasingtheinertiaconstantoftheMG isstudied.InSection3,theprocessoftuningtheparametersofMG forimprovingitsfrequencyresponseisexplained.ThestudiedMG isintroducedinSection4.Then,inSection5,bysimulationstudies carriedoutinMatlab/Simulink,theeffectivenessoftheproposed methodisinvestigated.Finally,theconclusionsofthisresearchare presentedinSection6.
2. IncreasingtheinertiaconstantofMG
In this paper, the proper value of inertia constant together withfrequencydroopcoefficientofDERsandloadfrequencycon- trollersparametersaretunedtoimprovethefrequencystability.To contributetotheinertialresponse,thepowerreferenceofinverter- basedDERsshouldbealteredasfollows[17]:
Pref−new=Pref−old−2Hvir.df
dt (1)
wherePref-old,Pref-new,Hvirandfarethereferencepowerwithout inertiaemulation(inpu),thereferencepowerwithinertiaemula-
tion(inpu),thevalueofvirtualinertia(inpus/Hz)andoperation frequencyofthegrid(inHz),respectively.
Theenergystoragesusedforprovidinginertialpowershould beabletocontributetotheinertiaofthegriduntilthefrequency reaches its critical value. Hence, the required energy for con- tributingtoinertial responsewithaninertiaconstant ofHvir is determinedas:
E=
tcrit
0
Pinertiadt=
tcrit
0
−2Hvir.df dtdt=
fcrit
finit
−2Hvirdf=2Hvir(finit−fcrit) (2)
wherePinertia,finit,fcrit andtcrit aretheinertialpower(inpu), thefrequencybeforethedisturbance(inHz),thecriticalfrequency (inHz)andthetimeittakesthefrequencytoreachfcrit(ins),respec- tively.Itcanbeassumedthatthefrequencyofthegridbeforethe disturbanceequalstothenominalvalue(fn).
IncaseofusingenergystoragessuchasUCforemulatingthe inertialresponse,theircapacityandmaximumpowershouldbe designed.TheenergystoredinUCdependsonitscapacitanceand nominalvoltage.AstheUCcannotbefullydischargedanditsstored energycanbeextracteduntilthevoltagereachesaspecifiedvalue [18],theenergyavailablefromUCforinertialresponse(inpus)is:
Eavail=C 2
Vinit2 −Vmin2 1Pbase (3)
whereC,Vinit,VminandPbasearethecapacitance(inF),initialvoltage (inV),minimumpermissiblevoltageofUC(inV)andbasepower ofthegrid(inW).ItcanbeassumedthattheinitialvoltageofUC beforethedisturbanceisequaltothenominalvalue(Vn).Basedon Eqs.(2)and(3)therequiredcapacitanceofUCforemulatingan inertiaconstantequaltoHviris:
Creq=4Hvir.
(fn−fcrit)V2n−Vmin2
.Pbase (4)Hence,thenominalstoredenergyofUC(inpus)willbe:
En=1
2CreqVn2 1
Pbase =2Hvir.
(fn−fcrit)Vn2−Vmin2
.Vn2 (5) ThenominalpoweroftheUC(inpu)canbedefinedsuchthat theUCisabletoinjecttherequiredinertialpowertogridincaseof theworstcontingency:Pn=2Hvir.max|df
dt| (6)
wheremax|dfdt|isthehighestrateoffrequencydecay(inHz/s)which happens immediatelyaftertheoccurrence oftheworst contin- gency.
InSection3,amethodfordeterminingtherequiredinertiacon- stanttogetherwithsometunableparametersofMGisproposed.
SubtractingtheinherentinertiaconstantofMGfromtheminimum requiredvalueofinertiaconstant,thevalueofinertiathatshould bevirtuallyemulatediscalculated.
3. ImprovingfrequencyresponseandstabilityofMGsby tuningH,Randloadfrequencycontrollers
IncreasingHand1/Rwoulddecreasethemaximumfrequency deviation.1/Risreferredtoasprimaryfrequencycontrolgain.On theotherhand,whileincreasingHdecreasestheamplitudeoffre- quencyoscillations,increasing1/Rabovea specificvaluewould increasetheamplitudeoftheseoscillations.Also,finetuningthe loadfrequencycontrollers,whichareresponsibleforbringingthe frequencydeviationbacktozero,improvesthefrequencyresponse ofMG. Hence,HandR wouldbetunedtogetherwithloadfre- quencycontrollerstoresultinasmoothfrequencyresponsethat
Fig.1. FrequencyoftheMGafterapowerdeficit.
doesnotviolatethespecifiedlimits.Forthispurpose,determining thepropercriteriaandusinganeffectivemethodfortuningthe parameters,suchthatthesecriteriaaremet,isofgreatimportance.
3.1. Objectivefunctionsandproblemformulation
ThetypicalbehavioroffrequencyofanMGafterapowerdeficit isshowninFig.1.Ascanbeseeninthisfigure,frequencyunder- shoot(fnadir),frequencyovershoot(fmax)andthetimeafterwhich frequencydeviationfromthenominalvaluereturnstothedesired region(tst)areimportantcharacteristicsoffrequencyresponsethat shouldbeminimized.Thesegoalswouldbeachievedbydetermin- ingthepropervaluesofRandHtogetherwiththeparametersofPID loadfrequencycontrollers(KP,KIandKD).Itisimportanttotune theseparameterssuchthatthegoalsareachievedwiththemini- mumcost.Inthefollowing,theobjectivefunctionsthatshouldbe optimizedforfinetuningtheparametersofanMGarediscussed.
ThepurposeofthefrequencycontrolinMG,incaseofapower imbalance,istominimizethemaximumfrequencydeviationwhile bringingthefrequencybacktothedesirableregionassoonaspos- sible.Itis alsoimportanttohavea zerosteadystatefrequency deviation.ThefirstgoalwouldbeachievedbyminimizingObj1and Obj2whicharedefinedasafunctionoffrequencyundershootand overshootinFig.1,respectively.
Obj1=fn−fnadir (7)
Obj2=fmax−fn (8)
wherefnisthenominalfrequency.
Thesecondgoalwouldbeachievedbyconsideringthesettling time,denotedbytstinFig.1,asthethirdobjective:
Obj3=tst (9)
where tst is the time afterwhich thefrequency deviation will remainsmallerthan2%ofthemaximumdeviationforthestudied disturbance.
Although(9)minimizesthesettlingtime,itcannotguarantee thezerosteadystatefrequencyerror;i.e.thefrequencymighthave lowamplitudeoscillationsaroundthenominalvalue.Zerosteady statefrequencydeviationisachievedbyminimizingtheintegralof timemultipliedbyabsoluteerror(ITAE):
Obj4=
t
sim0
t×|f(t)|dt (10)
wheretistimeandf(t)isdeviationofthefrequencyfromthe nominalvalue(f(t)=f(t)−fn)andtsimisthesimulationtime.This objectivefunctionalsowouldresultin areductioninfrequency oscillations.Furthermore,itwouldaffectthesettlingtime.How- ever,sincesomeotherconflictingfactorssuchasthemaximum frequencydeviationareconsideredinEq.(10),tominimizetheset- tlingtime,itisnecessarytoconsiderEq.(9)asoneoftheobjective functions.
Fig.2.Studiedmicrogrid.
Theaboveobjectivefunctionsareminimizedbyoptimaltuning theloadfrequencycontrollers’parameters,frequencydroopcoef- ficientofDERsandinertiaconstantofMG.However,toincreasethe inertiaconstantofMG,energystoragesarerequired.Higheriner- tiaconstantwillresultinahighercostofenergystorages.Hence, minimizingHwouldbeconsideredasthenextobjectiveof this problem:
Obj5=H (11)
whereHisthesumoftheequivalentinertiaconstantoftheonline synchronousgenerators (Heq)and theinertiaconstant which is goingtobevirtuallyemulated(Hvir);i.eH=Heq+Hvir.
Totuneloadfrequencycontrollersparameters,frequencydroop coefficientofDERsandtheequivalentinertiaconstantoftheMG, theoptimizationproblemwouldbeformulatedasfollows:
Minimize(obj1(x),obj2(x),...obj5(x))
x=(KP1,KI1,KD1,...,KPn,KIn,KDn,R1,...,Rn,H) (12) S.t.fnadir>fmin
xlow≤x≤xup
Theconstraintpreventsthefrequencyfromfallingbelowthe minimumdesirablevalue(fmin).xrepresentsthedecisionvariables vector,n isthenumber ofDERsthatparticipateinprimaryfre- quencycontrolandloadfrequencycontrol.Also,xupandxloware theupperandlowerlimitvectorsforthedecisionvariables.
3.2. Solvingtheoptimizationproblem
InSection3.1,severalobjectiveshavebeenintroducedtobe minimizedforimprovingthefrequencyresponseandstabilityof
theMGwiththeminimumcost.Becauseoftheconflictingnatureof theobjectives,fortheproblemsofthistype,generallythereisaset oftrade-offsolutionsknownasParetooptimalsolutions.Thistype ofproblemscanbesolvedbymeansofmulti-objectiveevolutionary algorithmstoobtainasetofParetooptimalsolutions.
However,duetothefactthatthenumberofobjectivesishigher thanthree, ordinaryevolutionaryalgorithmswouldnot showa goodperformanceinsolvingthisproblemandmany-objectiveopti- mizationalgorithmsshouldbeusedtosolveit [16].Kneepoint drivenevolutionaryalgorithm (KnEA) isa many-objectiveopti- mizationalgorithmwhichhasbeennewlyproposedinRef.[16].
Thesuperiorityofthisalgorithminsolvingmany-objectiveopti- mizationproblemsoversomeothermulti-objectivealgorithmshas beenprovedinRef.[16].Hence,KnEAhasbeenimplementedfor solvingthemany-objectiveoptimizationproblemdefinedinthis paper.
3.2.1. Solvingprocess
Principally,KnEAisanelitistPareto-basedmulti-objectiveevo- lutionaryalgorithm.ThemaindifferencebetweenKnEAandother Pareto-basedmulti-objectiveevolutionaryalgorithmslikeNSGA-II isusingkneepointsasasecondaryselectioncriterionbesidesdom- inancerelationship.Addingthiscriteriontotheselectionprocess hasimprovedtheperformanceofKnEAinsolvingmany-objective optimizationproblems.InParetofront,akneepointreferstothe solutioninwhichasmallimprovementinoneobjectiveofsucha solutionresultsinaseveredegradationinatleastoneoftheother objectives.
ThemainstepsofKnEAforsolvingthemany-objectiveproblem oftuningofMGparametersarepresentedbelow.Moredetailson KnEAwouldbefoundinRef.[16].
1.Initialization:Inthissteptheinputparameters,includingthe populationsize(N),numberofiterations,theratioofkneepoints and theupper/lowerlimits ofdecision variables areentered.
Then,theinitialsetsofMGparametersx=(KP1,KI1,KD1,...,KP n, KI n,KD n,R1,...,Rn,H)arerandomlygenerated.
2.Calculatingthevaluesoftheobjectivefunctions:Thedynamic MGmodelsimulatedinMatlab/Simulinkisrunandtheobjec- tivefunctionsincaseofthespecifieddisturbancesarecalculated foreachmemberofMGparametersset.Forthemembersthat resultedinviolationfromtheconstraint,asapenaltymecha- nism,thevalueoftheobjectivesaremultipliedby1000.
3.Binarytournamentselection:Inthebinarytournamentselec- tion, three tournament metrics including the dominance relationship,weighteddistancemeasure,andthekneepointcri- terionareimplemented,tochooseindividualsfromtheparent setsofMGparametersforgeneratingNoffspringsetsbasedon avariation method.Infact,inthebinarytournament mating selection,twosolutionsareselectedrandomlyfromtheparent populationset.Ifoneofthesolutionsisdominatedbytheother one,thenthelatersolutionisselected.Ifnoneofsolutionscan dominatetheotherone,itwillbecheckedwhethertheyareknee points.Ifonlyoneofthesetwosolutionscanbeconsideredasa kneepoint,thenthekneepointisselectedfromtheparentset.
Otherwise,aweighteddistancewillbeusedforcomparingthe twosolutionsandthesolutionwhichhasalargerweighteddis- tancewillwinthetournament.Iftheweighteddistancesofboth solutionsarethesame,oneofthemwillbeselectedrandomly forreproduction.
4.Non-dominatedsortingandidentifyingthekneepoints:Non- dominated sorting is used to form non-dominated fronts.
Different methods have been proposed in literature for this purpose.InKnEA,duetoitscomputationalefficiency,thenon- dominatedsortingmethodproposedinRef.[19],isemployed.
Thenanadaptivestrategy,whichisthoroughlyexplainedinRef.
[16],istakentoidentifythekneepointsofnon-dominatedfronts.
5.Environmentalselection:ToselectNindividualsastheparent setsofMGparametersofthenextgeneration,environmental selectionisperformed.Beforedoingenvironmentalselection, KnEAperformsnon-dominatedsortingasexplainedinRef.[19]
toformNF non-dominatedfronts.Then,itstartschoosingthe non-dominated solutions fromthefirst non-dominated front (F1).WhenthenumberofsolutionsinF1ishigherthanthepopu- lationsizeN,thekneepointslocatedinF1arechosenasparents forthenextpopulation.Ifthenumberofkneepoints(NP1)is alsolargerthanN,thenNkneepointswithalargerdistancefrom thehyperplanearechosen.Otherwise,NP1kneepointstogether with(N−NP1)othersolutionsinF1withalargerdistancefrom thehyperplaneofF1areselected.Whenthenumberofsolutions inF1issmallerthanN,KnEAselects(N−|F1|)parentsolutions fromthesecondnon-dominatedfrontbasedonaproceduresim- ilartowhatexplainedforF1.ThisprocessisrepeateduntilN parentpopulationforthenextgenerationisfound.
Theaboveprocedurerepeatsuntilthemaximumnumberofiter- ationsisreached.Intheend,thealgorithmproducesanumberof Paretooptimalsolutionsamongwhichonehastobeselectedasthe finalsolution.TheprocedureofselectingoneoftheParetooptimal solutionsasthefinalsolutionisexplainedinSection3.2.2.
3.2.2. Selectingthefinalsolution
Whenthestopcriterionismet,theoptimizationalgorithmwill giveusanumberofParetooptimalsolutionsamongwhichonehas tobechosenasthefinalsolution.Here,ameasurebasedonthesum
ofthenormalizedobjectiveshasbeenproposedtoselectoneofthe NParetooptimalsolutionsasthefinalsolution:
dl=
mi=1
ObjliIdealObji
(13)
wheretheObjliistheithobjectiveofthelthParetooptimalsolution, misthenumberofobjectivesandIdealObjiistheithelementof theidealobjectivevectorwhichisdefinedasfollows:
IdealObj=
min
l=1→Nobjl1, min
l=1→Nobjl2,..., min
l=1→Nobjml
Theithelementofthisvectorequalstotheminimumnon-zero valueoftheithobjectiveamongalltheobtainedParetooptimal solutions.
TheParetooptimalsolutionswouldberankedbasedonthed calculatedforeachsolutionandifthesolutionwiththeminimum disnotsatisfactoryforthedecisionmaker,theothersolutionscan beconsideredaccordingtotheirrank.
4. StudiedMG
4.1. Configurationofthestudiedmicrogrid
To illustrate the effectiveness of the proposed method for improvingthefrequencystabilityofMGs,amicrogridwithvarious DERsincludingwindturbinegenerator(WTG),photovoltaic(PV), dieselenginegenerator(DEG)andbatteryenergystoragesystem (BESS)hasbeenstudied.Inthissystem,PVandWTGarecontrolled toinjectthemaximumpowertothegridandarenotdispatchable.
But,DEGandBESSparticipateinfrequencycontrol.
ThestudiedMGwhichisshowninFig.2isamodifiedIEEEstan- darddistributionsystem[20,21].Aneighthordermodelisused forsynchronousgenerator.Dieselengineandgovernorarerepre- sentedbythefollowingtransferfunction[22]:
GDEG(s)= 1
(1+sTg)(1+sTt) (14)
whereTgandTtaregovernorandturbinetimeconstants,respec- tively.
ThemodelusedforBESSis[22]:
GBESS(s)= KBESS
(1+sTBESS) (15)
UCwhichisimplementedforinertiaemulationismodeledby thefollowingtransferfunction[8]:
GUC(s)= 1
(1+sTUC) (16)
Consideringthefactthattherateofchangeofoutputpowerof eachDERislimitedandthiswouldhighlyaffecttheirfrequency response,therampupandrampdownrateoftheoutputpowerof theseDERshasbeenconsideredas[23]:
P˙BESS<20%/s, P˙DEG<40%/min
Theeffectoffrequencyandvoltagedependencyofloadsonthe activeandreactivepowersismodeledasfollows[24]:
PL=PL0(V V0)
˛
(1+Kpff) QL=QL0(V
V0)
b
(1+Kqff)
(17)
wherePL,PL0,V,V0,a,andKpfarecurrentloadactivepower,initial activepoweroftheload,loadvoltage,initialvoltageofload,voltage dependencycoefficientofactivepowerandfrequencydependency coefficientofactivepower,respectively.Also,QL,QL0,b,andKqf
Fig.3.FrequencyresponsemodelofthestudiedMG.
arecurrentloadreactivepower,initialreactivepoweroftheload, voltagedependencycoefficientofreactivepowerand frequency dependencycoefficientofreactivepower,respectively.
ThevalueofparametersofDERsandloadsaretakenfromRefs.
[8,20,22,24]andgiveninAppendixA.
4.2. Frequencyresponsemodel
Todecreasethecomputationalcostofsolvingtheoptimization problem,itisdesirabletosimplifythemodelusedforfrequency responsestudies.Inisolatedpowersystems,frequencydynamics aremainlyaffectedbyrotorandgovernor-turbinesystemdynam- ics.Exciterandgeneratortransientsaremuchfaster andcanbe neglected[8].Hence,thefrequencyresponsemodelshowninFig.3 isaccurateenoughforinvestigatingthefrequencyresponseofMGs incaseofpowerimbalances[8,22].Inthismodel,Heq andDare theequivalentinertiaconstantofthesynchronousgeneratorsand loaddampingfactor.Loaddampingfactormodelsthedependency ofloadspowerinfrequency.ReqBESSandReqDEGaretheequivalent frequencydroopcoefficientofBESSandtheequivalentfrequency droopcoefficientofDEGs,respectively.Heq(inpus/Hz)canbecal- culatedasfollows[24]:
Heq=
ni=1
HiSi
fn.Sbase (18)
where Hi, Si, Sbase and n are theinertia constant of theith synchronousgenerator(ins),thenominalcapacityoftheithsyn- chronousgenerator(inMVA),thebasepowerofMG(inMVA)and thenumberofthesynchronousgenerators,respectively.
TocalculateloaddampingfactorfromthedetailedmodelofMG, itisassumedthatDERsdonotcontributetothefrequencyresponse.
IfapowerdeficitwiththemagnitudeofPD(inpu)occursinsuch acase,bymeasuringthesteadystatefrequencydeviation(fssin Hz),loaddampingfactor(inpu/Hz)wouldbecalculatedasfollows:
D= PD
fss (19)
Also,foronlineestimationoftheequivalentinertiaconstantof powersystemsandloaddampingfactor,somemethodshavebeen proposedinRefs.[25,26].
Fig.4whichcomparesthefrequencyobtainedusingthedetailed modelofMGandthesimplifiedfrequencyresponsemodeldepicted inFig.3,showsthat thefrequency responsemodel hasa good
Fig.4. Comparisonbetweentheresponseofthedetailedmodelandthesimplified frequencyresponsemodel:(a)incaseofa0.86MWpowerdeficitand(b)incaseof a4.5MWpowerdeficit.
accuracyincaseofbothsmallandlargedisturbances.Theopti- malparameterspresentedinthesecondrowofTable2,havebeen usedinthesesimulations.
Duetothefactthatthefrequencyresponsemodelhasagood accuracyandcalculationoffrequencyusingthismodelconsider- ablyreducesthecomputationalcost,thismodelhasbeenusedfor theoptimizationprocess.Also,thesimulationresultspresentedin thenextsectionareobtainedusingthefrequencyresponsemodel.
The upperand lower limitsof the decision variables of the optimizationproblemareconsideredas:0.0149<H<0.2pus/Hz, 0<1/R1<5pu/Hz, 0<1/R2<5pu/Hz, −5<KP1<5, −5<KI1<5,
−5<KD1<5,−5<KP2<5,−5<KI2<5,−5<KD2<5.
5. Resultsandanalysis
Inthissection,thenecessityofincreasingtheinertiaconstant ofalowinertiaMGforachievinganacceptablefrequencyresponse hasbeeninvestigatedanddifferentstrategiesfortuningtheparam- etersofMGhavebeencompared.
Astheemulatedinertiaconstantisdependentonthecapacityof theinstalledUC,increasingitwillrequireanincreaseinthecapacity ofUC.Hence,avaluefortheemulatedinertiaconstantshouldbe determinedthatisenoughforkeepingthefrequencystabilityofthe MG,incaseoftheworstcontingency,inalloperatingconditions.
Theworstoperatingconditionfromthefrequencystabilitypointof viewiswhenthesystemhastheminimumload,whichresultsin thelowestloaddampingfactor,andtherenewablesproducetheir maximumpower,whichresultsinthelowestinertiaconstant.Such anoperatingconditionforthestudiedMG,includingloadspower andrenewablesgeneration,isshowninFig.2.Theinertiaconstant, frequencydroopcoefficientofDERsandloadfrequencycontrollers parametersaretunedsuchthatthefrequencyiskeptabovefminif theworstcontingencyhappenswhenthesystemhasthelowest valueofHeqandD.Thetunedparameterswouldbefixedforallthe operatingconditionsoftheMG.
WhilelowervaluesofRresultinlowerfrequencydeviations in case of small disturbances, they might result in oscillatory frequencyresponseincaseoflargedisturbances.Hence,theopti- mizationproblemissimultaneouslysolvedforasmallandlarge disturbancetoguaranteetheacceptableoperationofMGincase ofboth small and largedisturbances.In this section,theworst contingencywhich is in thescopeof loadfrequency control(a 4.5MWpowerdeficit),isconsideredasthelargecontingencyand theminimumallowedfrequencyoftheMGbeforeactivatingthe
Fig.5.FrequencyoftheMGincaseofa0.86MWpowerdeficit.
Fig.6.FrequencyoftheMGincaseofa4.5MWpowerdeficit.
underfrequencyloadsheddingrelays(48Hz)isconsideredasfmin forthiscase.Ontheotherhand,a0.86MWdisturbanceisconsid- eredasthesmalldisturbanceforwhichtheproblemissolved.The valueoffminforthiscaseisconsideredtobe49.5Hz.tsimusedfor optimizationequals1000s.
5.1. Comparisonbetweentheperformanceofdifferentstrategies usedfortuningtheparameters
ToinvestigatethenecessityoftuningH,firstly,onlyloadfre- quencycontrollersand frequencydroop coefficientof DERsare tuned(theinertiaconstanthasnotbeentuned);thisstrategyis calledOriginalH.Then,Hhasbeentunedtogetherwithcontrollers andfrequencydroopcoefficientofDERsandthisstrategyiscalled TunedH#1.Finally,toshowtheeffectofconsideringHasoneofthe objectives,Hhasbeenconsideredasoneofthedecisionvariables tobetunedtogetherwithcontrollersandfrequencydroopcoef- ficientofDERsbutObj5 isomitted;thisstrategyiscalledTuned H#2.Aftertheterminationofoptimizationprocess,MOalgorithm achievesasetofParetooptimalsolutions.AnumberofthePareto optimalsolutionsofTunedH#1strategyaregiveninTable1.The objectiveswithsuperscript1and2arerelatedtothelargeandsmall disturbance,respectively.Obj5istheinertiaconstantoftheMGand isthesameforbothdisturbances.Therearesolutionswithhigher inertiawhichresultinlowerfrequencydeviation.However,these solutionswillimposeahighercostofenergystorages.
Here,theParetooptimalsolutionsarerankedaccordingtotheir respectivedcalculatedusingEq.(13)andthenoneischosenas thefinalsolutionbasedonitsdandthedecisionmakerpreference.
AscanbeseenfromTable1,solutionnumber7hasthelowestd andithasbeenselectedasthefinalsolution.Theparametersof theMGtunedbydifferentstrategiesaregiveninTable2.Fig.5 showsthatincaseofthesmalldisturbance(0.86MW),OriginalH couldpreservefrequencystability.However,thefrequencydevia- tionishigherthanTunedH#1andTunedH#2strategies.AsFig.6 shows,usingtheparameterstunedbyOriginalHstrategy,thefre- quencyviolatestheminimumpermissiblelimitincaseoftheworst contingencywhichisinthescopeofloadfrequencycontrollers.In fact,thisfigureshowsthatwhenHisnottuned,theoptimization
Fig.7. FrequencyoftheMG:(a)incaseof0.43MWpowerdeficitand(b)incaseof a3.22MWpowersurplus.
algorithmcouldnotachieveanysolutionthatdoesnotviolatethe minimumpermissiblelimit.Thiscertifiesthenecessityoftuning Hforachievingasolutionwhichdoesnotviolatetheoptimization constraint.
Having determined thevalueof virtual inertia,therequired powerandenergyspecificationofUCisdeterminedbasedonthe methodexplainedin Section2.Toemulatetheinertiaconstant determinedbyTunedH#1strategy,aUCwithnominalpowerof 5.87MWandstoredenergyof23.92MW.sisrequired.However, iftheparametersdeterminedbyTunedH#2strategyaregoingto beused,aUCwithnominalpowerof6.01MWandstoredenergy of31.83MW.swillberequired.FromFigs.5and6itcanbefound thatusingtheparameterstunedbytheTunedH#1andTunedH#2 strategies,thefrequencyofMGismaintainedwithinthepermissi- blelimitsincaseofbothlargeandsmalldisturbances.Usingthe parameters obtainedby TunedH#2strategy, MGexperiences a smallerfrequencydeviationincaseofbothlargeandsmalldistur- bances.However,TunedH#2strategyhasledtoaUCwithahigher capacity.Consideringa200D/kWcostofinstallationforpowerand a10kD/kWhcostofinstallationforenergy[8],theUCinstallation costwillbe1240kD and1290kD fortheTunedH#1andTuned H#2strategies,respectively.
Itisalsoworthmentioningthatacomputerwitha24GBRAM anda2.93GHzCPUwith24coreshasbeenusedforsolvingthe optimizationproblems.Theoptimizationprocesselapsed832,965 and1385sfortheTunedH#1,TunedH#2andOriginalHmethods, respectively.
5.2. PerformanceofTunedH#1andOriginalHstrategiesfor differentpowerimbalances
InthestudiedMG,loadfrequencycontrollersshouldkeepthe frequencyabove48Hzincaseofpowerdeficitslessthan4.5MW andloadsheddingmightbetriggeredforthelargerdisturbancesto keeptheminimumfrequencyabove47Hz[8,27].Here,theunder- frequencyloadsheddingmethodproposedinRef.[21],hasbeen implemented.However,accordingtotherequirementsofthestud- iedMG,thethresholdsforactivatingtheloadsheddingstepshave beenalteredtothefollowingvalues:48,47.8,47.6and47.4Hz.In thissection,somecasestudiesarecarriedouttoinvestigatethe frequencyresponseofMGwiththedesignedparameters.
5.2.1. CaseI:a0.43MWstepincreaseindemand
Fig.7(a)showsthefrequencyresponseofthesystemincaseofa 0.43MWstepincreaseindemand.ItisclearthatbothoftheTuned H#1andOriginalHstrategiescouldmaintainthefrequencywithin
Table1
ThevalueofobjectivesforanumberofparetooptimalsolutionsobtainedbyKnEA(superscript1and2refertothelargeandsmalldisturbance,respectively).
Obj11 Obj12 Obj21 Obj22 Obj31 Obj32 Obj41 Obj42 Obj5 d
1 1.36 0.05 0.008 0.00 37.7 20.4 589 2.8428 0.1888 32
2 1.51 0.06 0.011 0.00 35.8 16.0 706 3.5431 0.1688 50
3 1.68 0.06 0.014 0.000 37.1 16.9 766 4.0333 0.1499 51
4 1.74 0.07 0.007 0.003 36.9 35.1 599 4.7951 0.1441 73
5 1.90 0.07 0.022 0.000 25.3 15.1 1701 1.6329 0.1303 62
6 1.38 0.05 0.000 0.026 41.3 44.5 506 10.61 0.1858 290
7 1.63 0.06 0.000 0.00 39.6 43.7 602 8.8980 0.1540 28
8 1.61 0.06 0.0589 0.005 97.1 40 2719 3.040 0.1566 207
IdealObj 1.34 0.05 0.0005 0.0001 25.3 9.4 390 0.5120 0.1272 –
Table2
ParametersofthestudiedMGtunedbasedondifferentstrategies.
1 Req DEG
pu/Hz 1Req BESS
pu/HzH
pus/Hz
KP1 KI1 KD1 KP2 KI2 KD2
TunedH#1 0.5682 0.7058 0.1540 −2.0719 −0.0721 −1.7581 −0.1249 −0.0030 −0.5424
TunedH#2 1 0.9924 0.2000 −0.1530 −0.2496 −0.7481 −1.9500 −0.0039 −1.9280
OriginalH 1 0.9168 – −1.3220 −0.0362 −0.7673 0.3896 −0.0017 −0.1215
Fig.8.FrequencyoftheMGincaseofthelargestPVunitoutage.
theallowedlimits.However,usingTunedH#1,MGwillexperience asmallerfrequencydeviation.
5.2.2. CaseII:a3.2MWstepdecreaseindemand
ItcanbefoundfromFig.7(b)thatasaresultofa3.2MWstep increaseinthedemandofMG,theperformanceofOriginalHstrat- egyhas beensignificantly deteriorated and thefrequency goes considerablyabovethepermissiblelimit.However,usingTuned H#1strategythefrequencystabilityofMGincaseofthispower surplushasbeenpreserved.
5.2.3. CaseIII:largestPVunitoutage
TheoutputpowerofPVunitsmightchangeasaresultofchanges inthetemperatureandsunirradiance[28].Fig.8showsthefre- quencyresponse ofthestudiedMGwhen thepowergenerated bythelargestPVunitfallsfromthemaximumvalue(2.6MW)to zero.UsingtheparametersobtainedbyOriginalHstrategy,fre- quencygoesconsiderablybelow48Hzandunderfrequencyload shedding shouldbecarried outin this case topreventthefre- quencyfromfallingbelowtheminimumacceptablevalue.Onthe otherhand,usingtheparameterstunedbytheTunedH#1strat- egy,thefrequencyismaintainedwithinthepermissiblelimitsand underfrequencyloadsheddingisnotrequired.
5.3. Loadshedding
Inthissection,thefrequencyresponseofthestudiedMGwith theparameterstunedusingtheproposedTunedH#1strategyis investigated,fordifferentcontingencieswhichrequireloadshed- ding,andcomparedwiththeOriginalHstrategy.
Table3demonstratesthatincaseoflargepowerdeficits,using theparameters obtainedby both strategies, the load shedding schemewassuccessfulinstoppingthefrequencydecline.However, usingtheparameterstunedbytheTunedH#1strategy,loadshed-
Fig.9.FrequencyoftheMGincaseofa4.5MWpowerdeficitwhentheloadhas increasedby30%.
dingisavoidedinmanycasesandintheothercasestheamount ofloadsheddingisconsiderablylowerthanwhentheparameters tunedbyOriginalHstrategyareused.Inthisway,abetterqualityof servicewouldbeprovidedforthecustomersandtheirsatisfaction levelisincreased.
5.4. RobustnessofMGusingtheparameterstunedwiththe proposedmethod
Inanypowersystem,loadchangeswiththetimeandthiswould changetheoperatingpointofthesystem.But,astheemulatediner- tiaconstantdependsontheinstalledcapacityoftheUC,itcannotbe increasedtothevalueshigherthanthedesignedvalue.Fig.9shows thattheparametersdesignedfortheminimumloadconditions,are abletomaintainthefrequencystabilityincaseoftheworstcon- tingency(a4.5MWpowerdeficit)whentheloadofsystemhas increasedby30%.Tosupplytheloadinthiscase,another8MVA DEGhasbeenconnectedtothe69KVbusofthesystem.Itshouldbe mentionedthatforabetteroperationoftheMG,theparametersof thesystemexceptHwouldberetunediftheoperationconditions changedconsiderably.
Also,itispossiblethatthereisaninaccuracyintheparameters estimatedformodelingthesystem.Eventheparametersofthesys- temmightchangebytime[29].Hence,itisimportanttoinvestigate theperformanceoftheMGincaseofparametersvariations.Fig.10 showsthat,usingTunedH#1strategy,MGhasanacceptablefre- quencyresponseincaseofa4.5MWpowerdeficitwhenthetotal HandDoftheMGare10%higher/lowerthanthenominalvalues.
Furthermore,asshowninFig.11,10%decrease/increaseinthetime constantsofDERs(TBESS,Tg,TtandTUC)doesnothaveaconsiderable effectonthefrequencyresponseofMG.
Table3
PerformanceoftheMGincaseofseverpowerdeficits.
Pd(pu) 0.10 0.15 0.20 0.25 0.30
OriginalHstrategy Pshed(pu) 0.0198 0.0809 0.1490 0.2053 0.2593
fnadir(Hz) 47.51 47.41 47.40 47.39 47.39
TunedH#1strategy Pshed(pu) 0 0 0 0.0152 0.0829
fnadir(Hz) 49.61 49.14 48.48 47.40 47.54
Fig.10.TheeffectofchangeinHandDonthefrequencyresponseofMGincaseof a4.5MWpowerdeficit.
Fig.11. TheeffectofchangeinthetimeconstantsofDERsonthefrequencyresponse ofMGincaseofa1.5MWpowerdeficit.
6. Conclusion
Inthis paper,improvingthefrequency stability oflow iner- tiaMGswasinvestigated.Ithasbeenshownthatfine-tuningthe loadfrequencycontrollers’parametersandfrequencydroopcoef- ficientofDERs,couldnotsolelyguaranteethefrequencystability oflow inertiaMGs.Hence,tuningtheinertiaconstanttogether withloadfrequencycontrollers’parametersandfrequencydroop coefficientofDERshasbeenproposedin thispaperforimprov- ingthefrequencystabilityoflowinertiaMGs.Infact,tuningthese parametershasbeenmodeledasamulti-objectiveoptimization problemwhichconsidersbothtechnicalandeconomicconcerns.
Becausethenumber ofobjectivesishigherthanthree,a many- objectiveoptimizationalgorithmhasbeenimplementedforsolving thisproblem.Additionalfindingfromthestudiescarriedoutinthe paperare:
1.KnEAhasagoodperformanceinsolvingtheoptimizationprob- lemdeterminedinthispaper.
2.TuningtheparametersoftheMGwiththeproposedMOmethod hashighlyimproveditsfrequencyresponse.
3.ComparingtheproposedTunedH#1strategywiththeOriginal Hmethodcertifiesthenecessityofdeterminingthepropervalue ofinertiaconstantinlowinertiaMGs.
4.Usingtheparameterstunedbytheproposedmethod,MGhasa goodperformanceincaseofchangesintheoperatingconditions ofMGanditsparameters.
AppendixA.
TheparametersofthestudiedMG:
KBESS= 1,D= 0.012pu/Hz,H= 0.0149pus/Hz,TBESS= 0.1s,
Tg=0.08s,Tt= 0.4s,TUC= 0.2s,Pbase= 21.5MW,Kpf
= 0.012pu/Hz,Kqf= 0.010pu/Hz,a= 1b= 2.
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