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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

A high-performance hybrid algorithm to solve the optimal

coordination of overcurrent relays in radial distribution networks considering several curve shapes

Alexandre A. Kida

^∗

, Luis A. Gallego

DepartmentofElectricalEngineering,LondrinaStateUniversity,Londrina,PR,Brazil

a r t i c l e i n f o

Articlehistory:

Received22October2015

Receivedinrevisedform25May2016 Accepted30May2016

Availableonlinexxx

Keywords:

Evolutionaryalgorithm Overcurrentrelayscoordination Distributionnetworks Optimization

a b s t r a c t

Thecoordinationofovercurrentrelays(OCRs)isessentialforimprovingreliabilityandsecurityindicators inelectricalnetworks.Thispaperproposesahigh-performancehybridalgorithm(HPHA)tosolvethe optimalcoordinationofOCRsinradialdistributionnetworks(RDNs).Theoptimalcoordinationproblem isformulatedasamixed-integernonlinearproblem(MINLP).Weconsideringasdecisionvariables:(1) thepickupcurrent(Ip);(2)timedialsetting(TDS);(3)relaytype;(4)curvetype.TheHPHAiscomposed ofaspecializedgeneticalgorithm(SGA)andanefficientheuristicalgorithm(EHA).Thus,theHPHAfinds theoptimumcombinationofIp,TDS,relaytypesandcurvetypesthatminimizetherelaysoperational times,ensuringtheselectivityforseveralfaultlevels.Theproposedalgorithmconsidersdiscretevalues ofIpandTDS.Thesimulationresultsshowedthattheproposedtechniqueisefficient,fast,reliableand improvesthecoordinationbydecreasingtheoperationaltimesofOCRs.

©2016ElsevierB.V.Allrightsreserved.

1. Introduction

Theelectricalsystemisnotimmunetofailure,anditsprotec- tionhasakeyroleinthepreservationofequipment(generators, switchgears,conductors,capacitors,transformersetc.).Inaddition, it isnecessary toensuremaximumcontinuity of powersupply toconsumers.Interruptionscausedamagebothforusersandfor electricutilities.Forthese,interruptionscanmeanrevenuelosses, damagetothepowerutilityimageandfines[1].Inordertoreduce thenumberofusersaffectedbytheinterruptions,theprotection devicesmust becoordinated.Thisallowsa specificsequenceof operationswhenafaultoccurs.Themaingoalsofthecoordination aresensitivity,selectivity,reliabilityandspeed[2].

Duringafaultcondition,oneofthemainconsequencesforthe electricalsystemistheelevationofcurrentlevels.Thus,itisnatu- raltousetheselevelsasparameterstodetermineifthesystemis faulty.Themostcommondevicesinthiscategoryaretheovercur- rentrelays(OCR),fusesandcircuitbreakers.TheOCRaretypically usedasbackupprotection,butinsomecases,theymaybetheonly typeofprotectionavailable[3].Asmostofthedistributionsystems areradial,theprotectionusingnon-directionalOCRissuitableand iswidelyusedbecauseitissimple,effectiveandcheap.Thispaper

∗Correspondingauthor.Tel.:+5571991001595.

E-mailaddress:alexandrekida@gmail.com(A.A.Kida).

focusesontheinversedefiniteminimumtimeovercurrentrelay (IDMTOCR).Theserelayshavetwoparameters:pickupcurrent(Ip) andtimedialsettings(TDS).However,somerelayscanhavetheir curvetypesmodified(inverse, veryinverseetc.).Wherein,each relaytype(IEEE[4],IEC[5],IAC[6],U.S.[7]etc.)hasadifferentset ofstandardizedcurves.

Thecoordinationproblemcanbeformulatedasamixed-integer nonlinear optimizationproblem. Themain objectiveconsistsin minimizingthetotaloperationaltimesoftheprimaryrelays.These timesrelyonthevaluesofIp,TDS,curvetypesandrelaytypes.Ipis treatedasdiscretebutthevariableTDSisoftenconsideredcontinu- ous.However,someOCRdoesnothavestepsofTDSsmallenoughto betreatedascontinuous,androundingthemtothenearestpossible valuescanleadtomiscoordination[8].

TosolvetheOCRcoordinationproblem,classicaloptimization techniquesareused,suchaslinearprogramming(LP)[2,3,9]and nonlinearprogramming(NLP)[10–12].Aswellas,metaheuristics algorithms(MHAs)havebeenused.TheMHAscanfindhigh-quality solutionsand,insomecases,theglobaloptimalsolutionwithless computationaleffort[13].In[14–16,8,17]geneticalgorithms(GAs) wereusedtosolvetheOCRcoordinationproblem.In[8],theauthor solvedtheproblemwithGAandconsideredIpandTDSasdiscrete variables.In[18,19]theproblemwassolvedusingparticleswarm algorithm.In[20,21]theartificialhoneybeealgorithmswereused tosolvethecoordinationproblem.In[22,23],ahybridalgorithm thatcombinesLPandevolutionaryalgorithmswasusedtosolvethe http://dx.doi.org/10.1016/j.epsr.2016.05.029

0378-7796/©2016ElsevierB.V.Allrightsreserved.

problem.Amethodologythatoptimizeseachparameterindepen- dently,consideringdiscreteIpandTDS,wasproposedin[24].The problemwasalsosolvedusingbinaryintegerprogramming,this wayTDSandIpwereconsidereddiscrete[25].In[26]therewas anadditionofthecurvetypesasdecisionvariables.Theauthors consideredstandardizedandnon-standardizedcurvestypes.

The electrical networks encouragedthe industrial, commer- cialandtechnologydevelopment,butmostofthemareoldand nottechnologicallydeveloped.Nowadays,governmentshavebeen encouraging theelectrical utilityto make investmentsin their networkstomakethemmorereliable,safeandtechnological.This way,utilitiesaredevelopingthesmartgrids.Theseareelectrical gridsthatuseadvancedtechnologiestomonitorandactbasedon theinformationaboutthebehaviorofend-usersandgeneration sources.So,thesmartgridcoordinatestheneedsandcapabilities ofallgenerators,gridoperators,consumersandelectricitymarket stakeholders.Withtheobjectiveofoperatingallpartsofthesys- temasefficientlyaspossible,minimizingcostsandenvironmental impactswhilemaximizingsystemreliability,securityandstabil- ity[27].Inthesmartgrids,theenergycontrolcentersneedfastand reliabletechniquesorcomputationalalgorithms,tobeusedinreal- timeapplications.Hence,thereisagreatinterestofelectricutilities tousetechniquesorcomputationalalgorithmstosolveproblems intheirpowergrids.Oneoftheseproblemsistheoptimalcoordi- nationofprotectiondevicesinreal-time.Inthispaperwepropose anefficient,fastandreliablemethodologyforthecoordinationof OCRinradialdistributionnetworks(RDNs).Suchproceduresare essentialfortheoptimaloperationofthenetworks.

TheIp,TDSareusuallyconsideredasdecisionvariables.This paperconsiders theaddition of curve and relay typesas deci- sionvariables. A high-performance hybrid algorithm (HPHA) is proposedtosolvethecoordinationproblem.ThiscombinesaGA withanefficientheuristicalgorithm(EHA).TheGAisresponsible forsolvingthenonlinearoptimizationproblem,whichconsistsin determiningtheIp,relaytypesandcurvetypes.Thiswayispossi- bletofindtheoptimumTDSpossessingthesevariables.IfTDSare treatedascontinuousvariables,theproblemoffindingitsoptimal valuesislinear.Thus,LPtechniquessuchassimplex,dualsimplex, two-phasesimplexandinteriorpointsareusuallyusedtosolveit.

Inthispaper,theEHAwillbeusedinsteadofLPtechniques.Since theTDSarenotencodedinthechromosomeoftheGA,itssearch spaceisgreatlyreduced.TheHPHAalsoensuretheselectivityfor multiplefaultlevels.

Thispaperisorganizedasfollows:Section2,presentsthechar- acteristicsoftheOCRusedinthispaper.Section4,thepresents OCRcoordinationproblemasanoptimizationproblem.Section5.5 presentstheEHAforfindingtheoptimalTDS,inRDN.Section5, presentstheHPHA.Section6,presentsthetestsystemsusedinthis paper.Section7providesacriticalanalysisoftheobtainedresults.

Finally,Section8concludestheworkpresentedinthispaper.

2. Overcurrentrelaycharacteristics

The OCR operates when the input current exceeds a pre- determinedvalue(Ip),sendinga signaltothecircuitbreakerto interruptthecircuit.Thenon-directionalIDMTOCRhasanopera- tiontimethatisinverselyproportionaltotheintensityoftheinput current.Moreover,itdoesnottakeintoaccountthedirectionof thecurrentflow.OneofitsmainapplicationsisintheRDNs,where thedirectionsofthecurrentflowsarealwaysknown[28].These relayshavetheiroperationaltimes(1)determinedbyinternational standards,suchasIEEE(InstituteofElectricalandElectronicsEngi- neers)[4],IEC(InternationalElectrotechnicalCommission)[5],IAC (InverseAlternateCurrent)[6],U.S.(UnitedStates)[7]etc.Theoper- ationaltime ofa relayi(R_i)for eachk faultinside theprimary

Table1

ConstantsofcurvetypesfortheIEEErelay[4].

Curvetype A B P

E.I 28.20 0.122 2.00

V.I 19.61 0.491 2.00

M.I 0.05 0.114 0.02

Table2

ConstantsofcurvetypesfortheIECrelay[5].

Curvetype A N

I 0.14 0.02

V.I 13.50 1.00

E.I 80.00 2.00

L.I 120.00 1.00

Table3

ConstantsofcurvetypesfortheIACrelay[6].

Curvetype A B C D E

E.I 0.004 0.638 0.620 1.787 0.246

V.I 0.090 0.796 0.100 −1.289 7.959

I 0.208 0.863 0.800 −0.418 0.195

S.I 0.043 0.061 0.620 −0.001 0.022

protectionzoneofR_j(Icc^k_j)isshownin(1),anditdependsontheTDS ofR_i(TDS_i)andK_i,j^k.ThetermK_i,j^k varieswiththestandardofR_i,and itrelatedtothevaluesofIpofR_i(Ip_i),Icc^k_j andthecurveconstants relatedtothecurvetypesofRi.FortheIEEE,IEC,IACandU.S.stan- dards,K_i,j^k iscomputedusing(2)–(5),respectively.Tables1–4show theconstantsrelatedtothecurvetypesfortheIEEE,IEC,IACand U.S.standards,respectively.WhereE.I,V.I,M.I,I,L.IandS.Irefer toextremely,very,moderately,standard,longandshortinverse curvetypes,respectively.

To improve the effectiveness of the protection scheme,the selectivitymustbeguaranteedforarange offaultcurrents.For selectivitypurposes,isconsideredpfaultcurrentlevelsinsideeach primary protectionzone, asshown in (6). For instance,ifp=2, theselectivityisguaranteed,foreachOCR,fortwoexpectedfault levels:theminimumandmaximum.For p>2,morefaultlevels betweentheexpectedminimumandmaximumareconsideredfor theselectivity.Eq.(7)showsthesizeofthediscretizationsteps ofthe selectivityinterval(Icc^k_j), forfaults withintheprimary protectionofR_j.

Forphase OCRs,themaximumand minimumexpectedfault currents are usually the close-in three-phase and the far-end two-phasefaults,respectively.ForgroundOCRs,theclose-inphase- to-ground and the far-end phase-to-ground(through a contact impedance) faults might be used as maximum and minimum expectedfaultcurrents.

T_i,j^k =TDSi·K_i,j^k (1)

K_i,j^k =

Ai Icc^k_j

Ip_i

P_i

−1

+Bi (2)

Table4

ConstantsrelatedtothecurvetypesfortheU.S.relay[7].

Curvetype A B N

M.I 0.023 0.010 0.020

V.I 0.097 3.880 2.000

S.I 0.003 0.003 0.020

I 0.180 5.950 2.000

E.I 0.035 5.670 2.000

Please cite this article in press as: A.A. Kida, L.A. Gallego, A high-performance hybrid algorithm to solve the optimal coordi- R1 R2 R3

F₁ F2 F3

Fig.1.Radialsystemwiththreerelaysandtheirrespectiveprimaryprotection zones.

K_i,j^k =

Ai Icc^k_j

Ip_i

N_i

−1

(3)

K_i,j^k =Ai+ Bi Icc^k_j

Ip_i −C_i

+ Di

Icc^k_j Ip_i −Ci

2+ Ei

Icc^k_j Ip_i −Ci

3 (4)

K_i,j^k =Ai+ B_i

Icc^k_j Ip_i

P_i

−1

(5)

Icc^k_j =Icc^j_min+(k−1)·Icc^k_j (6) Icc^k_j =Icc^jmax−Icc^j_min

p−1 , p>1 (7)

whereT_i,j^k istheoperationtimeofR_iforIcc^k_j.TDS_iistheTDSofR_i. K_i,j^k isrelatedtotherelayoperationaltimeanditvarieswiththe standardofR_i.Icc^k_j istheklevelofshort-circuitwhereR_jactas primaryprotection.A_i,B_i,C_i,D_i,E_i,N_iandP_iareconstantsrelated tothecurvetypesfor,R_i.Icc^k_jisthesizeofthediscretizationstep oftheselectivityinterval.Icc^jmaxandIcc^j_minarethemaximumand minimumshort-circuitcurrentsinsidetheprimaryprotectionzone ofR_j,respectively.

3. Coordinationproblem

InFig.1isshownaRDN.WhereR1,R2andR3refertorelays1,2 and3,respectively.F₁,F₂andF₃correspondtothefaultslocation inthesystem.ForthefaultinF₃,R₃andR₂willactasprimaryand back-upprotection(PPandBP),respectively.ForthefaultinF2,R2

andR₁willactasPPandBP,respectively.Finally,forthefaultinF₁, R₁willactasPPanditistheonlyprotectionavailable.Theareas boundedbydottedlinesaretheprotectionzonesofeachrelay.

AsthefaultissensedbybothBPandPPsimultaneously,the difference betweentheir operational times must be greater or equalthanthecoordinationtimeinterval(T_relay),toguarantee theselectivity.Thisway,thePPwillhavesufficienttimetoclearthe fault,whereintheBPmustactonlyifthePPfailsinclearingthefault.

Usually,isadoptedT_relay=0.4sbecausethecircuitbreakeroper- atingtime,themanufacturingtolerancesandthedesignofsafety timeareapproximately0.13,0.10and0.17s,respectively[1].

4. Mathematicalmodel

Inthissection,thecoordinationproblemofOCRsisformulated asanoptimizationproblem,containinganobjectivefunction(OF), responsibleforminimizingtheoperatingtimesoftherelays,and asetofconstraints.Thisproblemisformulatedasanonlinearand non-convexoptimizationproblem,withahighnumberofrestric- tions[23,29].Inthispaper,Ip,TDS,curvetype,andrelaytypeare consideredasvariables.Ifanyoftheseparametersistreatedasa discrete,theproblembecomesmixed-integer,whichhasagreater

complexity.Theadditionofcurveandrelaytypesasdecisionvari- ables is supportedby thefact that some microprocessor-based OCRsacceptseveralcurvetypesandstandards.Forinstance,the SEPAMSeries20(SchneiderElectric)[30]andthe735/737(Gen- eralElectric)[6]acceptmultiplecurvestandards,suchasIEEE,IEC andIAC.

4.1. Objectivefunction

Inafaultsituation,oneoftheconsequencesfortheelectrical systemisthesuddenincreaseincurrentlevelstodangerousval- ues.Thefasterthefaultisisolated,theloweristhethermaland mechanicaleffortsinthesystem.Intheoptimalcoordinationof OCRs,theobjectiveistoobtainthesettings(Ip,TDS,curvetypes andrelaytypes)thatminimizetheoperationaltimeoftheprimary protectionOCRs[3,31–33].Theoptimalsettingsaredirectlyrelated tothespeedcriteria.TheOFconsistsofminimizingthesumofall operationaltimesoftheOCRs,forallthepfaultlevelsconsidered withinitsprimaryprotectionzone,asshownin(8).

min z= 1 p

p

k=1

m

i=1

T_i,i^k (8)

wherem,kand parethenumberofOCRs,theindexrelated to thefaultlevelanalyzedandthenumberoffaultlevelsconsidered withintheprimaryprotectionzones,respectively.T_i,i^k istheopera- tionaltimeofR_iforafaultinsideitsprimaryprotectionzone,and itcanbecomputedbysettingj=iin(1).

4.2. Constraints

The constraint shown in (9) refers tothe T_relay needed to ensuretheselectivitybetweenBPandPP.Theconstraintshownin (10)preventsthattherelaytakesalongtimetooperate,avoiding theequipmentdamageandtheinstabilityofthepowersystem.The constraintsrelatedtotheminimumandmaximumboundariesof TDSandIpareshownin(14)and(11),respectively.Therelaysmust notactduringthenormaloperationofthesystem,asitisshownin (12).R_icannotactasBPiftheminimumshort-circuitcurrentinside itsprotectionzone(Iccprotected,i

min )islessorequalthanIp_i,asshown in(13).

T_BP^k

i,i−T_i,i^k ≥Trelay (9)

T_i,j^k ≤Tmax (10)

Ip^min_i ≤Ip_i≤Ip^max_i (11)

Ipi>LGF·I_loadⁱ (12)

Ipi<Iccprotected,i

min (13)

TDS^min_i ≤TDSi≤TDS^max_i (14)

whereT_BP^k

i,iistheoperationaltimefortheBPofR_i,forIcc^k_i.TDS_i,min andTDS_i,maxaretheminimumandmaximumpossibleTDSforR_i, respectively.TmaxisthemaximumoperationaltimeoftheOCR,for Icc^k_j.Ip^min_i andIp^max_i aretheminimumandmaximumpossibleIpfor R_i,respectively.Iccprotected,i

min istheminimumshort-circuitinsidethe protectionzoneofR_i.LGFistheloadgrowthfactorthatincludes apossibleincreaseonthesystemdemand.Iⁱ_loadisthemaximum expectedloadcurrentinthesystem.

5. Thehighperformancehybridalgorithm

The proposed algorithm is based in the main ideas of the GApresentedin[34],whichwasinitiallydevelopedtosolvethe

Specify the control parameters of the HPHA

(kps and MaxIter)

Create the initial population using HCA-PI

Stop criteria satisfied?

Carry out phases:

- Selection - Recombination - Mutation

Apply Local search no

yes

Acceptation criteria

Stop:

Optimal TDS, Ip, relay and curve types Evaluate the fitness and

unfitness functions

Calculate the optimum TDS using EHA (Pseudocode 3)

Fig.2. FlowchartoftheproposedHPHA.

generalizedassignmentproblem.Thisalgorithmhasseveraldif- ferenceswhencomparedtotheclassicalGA:(1)ineachgeneration cycle,onlyoneoffspringisgeneratedanditcanenterinthecurrent populationifitsatisfiestheacceptationcriteria;(2)theOFiscom- posedbyfitnessandunfitnessfunctions;(3)alocalsearchisusedto improvetheOFoftheoffspring.

SomemodificationsintheGAproposedin[34]weremadeto adaptittothecoordinationproblem.Thecodification,population initialization,OFcomputationandlocalsearchareperformedin adifferentway.TheproposedmethodcombinesEHAwithaGA, inordertoreducethedimensionoftheproblem, makingthis a hybridtechnique.TheflowchartfortheHPHAisshowninFig.2.The parameterskpsandMaxIterreferstothesizeoftheinitialpopulation andthestopcriteria,respectively.TheHPHAstopsiftheincumbent solutiondoesnotchangeafterMaxItergenerationcycles.

5.1. Codification

In this paper,theTDS, Ip,relayand curvetypes are consid- ereddecisionvariables.TheTDSarenotcodedinthechromosome becauseitsoptimalvaluesaredeterminedusingtheEHA.Anexam- pleofachromosomeforasystemcontainingfiveOCRisshownin Fig.3.WhereIpgene,RelaygeneandCurvegeneareIp,relaytypeand curvetypesgenes,respectively.Eachchromosomeisdividedinto threeparts:oneforselectingIp,oneforcurvetypes,andtheother forrelaytypes. Adecimal codification isconsidered.Each gene assignedinIp_genewillcorrespondtoafeasibleIpvalueaccepted bytherespectiveOCR.EachvalueRelaygeneisrelatedtoonerelay type(IEEE,IEC,IACorU.S.),soitvaryfromonetofour.Eachvalueof Curvegeneisrelatedtoonestandardizedcurvetypeoftherespective OCRanditmayvaryfromonetofive.

2 6 1 5 7 2 3 2 1 2 1 1 3 2 2

Fig.3.Exampleofachromosomeoftheproposedalgorithm.

5.2. Initialpopulation

TheinitialpopulationofHPHAcanbecreatedrandomly,butthis canleadtohavingapopulationofpoorquality(veryhighopera- tionaltimesoftherelaysorunfeasibleproposals).Tosolvethis issue,theinitialpopulationcanbecreatedusingaheuristicalgo- rithmthattakessomecharacteristicsoftheproblemtobesolved.

Inthis paper,isproposedaheuristicconstructivealgorithm for thepopulationinitialization(HCA-PI).Thisalgorithmisshownin Pseudocode1,whereIpgenearelimitedby(12)and(13),andthe CurvegenearelimitedbytherespectiveRelaygene.Themainideaof thisalgorithmconsistsintoeliminatesomeinfeasibleproposalsin thepopulationinitialization.

Pseudocode1. Heuristicconstructivealgorithmforthepopula- tioninitialization(HCA-PI).

1: for(Allindividualsonthepopulation)do 2: for(i=1tonumberofrelays)do

3: Ipgene(i)←randomintegervaluethatcorrespondtoaIpthat satisfiesIⁱ_load·LGF<Ip<Iccprotected,i

min ;

4: Relaygene(i)←randomintegervaluethatvaryfrom1to4;

5: Curvegene(i)←randomintegervaluethatvaryfrom1tothe numberofavailablecurvetypesofRi;

6: endfor

7: endfor

5.3. Objectivefunction

First,itisnecessarytodecodetheinformationofIp,relaytypes andcurvetypesofthechromosome,inordertocomputetheTDS usingtheEHA(moredetailsinSection5.5).Thisway,thefitness functionisevaluatedusing(8).Theunfitnessfunctionispropor- tionaltothenumberofconstraints(11)–(14)violated,multiplied byapenalizationfactor.Thefitnessandunfitnessfunctionsarecom- binedandformtheOF.DuringtheiterativeprocessoftheHPHA, theOFiscomputedseveraltimes.Therefore,theEHAisinteresting duetoitsspeed.

5.4. Localsearch

Afterthephasesofselection,recombinationand mutation,a newoffspringisgeneratedanditcanbefeasibleorinfeasible.In bothcases,alocalsearchisperformedtoimproveitsOFbyperform- inganeighborhoodsearchinIpgeneandCurvegene.Thealgorithmis showninPseudocode2.

Pseudocode2. Localsearchalgorithm.

1: fori=1tonumberofrelaysdo

2: PerformaneighborhoodsearchintheIpgene(i)byincrementingand decrementingoneunit;

3: IftheneighborhoodsearchintheIpgene(i)improvedtheOF,update theIpgene;

4: PerformaneighborhoodsearchintheCurvegene(i)byincrementing anddecrementingoneunit;

5: IftheneighborhoodsearchintheCurvegene(i)improvedtheOF, updatetheCurvegene;

6: endfor

5.5. Efficientheuristicalgorithm(EHA)tocalculatetheoptimum TDS

TheLPisawellknowntechniqueusedtofindtheoptimalTDS [2].Inthispaper,isproposedanalternativemethodentitled“effi- cientheuristicalgorithm(EHA)”tofindtheoptimalTDSinRDN.

ThisalgorithmcanworkwithdiscreteandcontinuousTDS,unlike theLPtechniques.

Please cite this article in press as: A.A. Kida, L.A. Gallego, A high-performance hybrid algorithm to solve the optimal coordi- Theproposedalgorithmisdividedintotwosteps:relaynum-

beringanditerativeprocess.Inthefirststep,thebrancheswhere OCRarelocatedarearrangedinlayers.Inthenextsteptheoptimal TDSiscomputed.

5.5.1. Relaynumbering

In [35] is proposeda methodology for branches numbering basedinlayerstosolvethethree-phasepowerflow.Thesameideas areappliedfortheOCRnumbering.Thisway,theOCRinonelayer isnumberedonlyafterallOCRfromthepreviouslayerwasnum- bered.InFig.4isshownanexampleofasystemwithtenOCRafter thenumbering.InthecaseofR10,R9,...,R1wereanalyzedinthat particularorder,thePPwillalwaysbeanalyzedbeforetheBP.This methodologyallowsasystematicwaytoanalyzetheOCRonlyonce, duringtheEHA.Thisprocedureisperformedonlyonceforafixed RDN.

5.5.2. Iterativeprocess

Thealgorithm is showedin Pseudocode 3.In this step, it is assumedthatalltheOCRarepreviouslynumberedasshownabove.

ThevariablesK_i,i^k andK_BP^k

i,iarecomputedwith(2)–(5),depending onthetypeofR_iandRBP_i,respectively.Moreover,itisnecessaryto checkifthenewTDSdoesnotviolatetheboundofTDS_maxinthe steps10and12.

In this algorithm, each backup OCR hastheminimum TDSs necessarytobecoordinatedwiththeirprimaryprotections.This conditionisachievedwiththestep6ofthePseudocode3.Thepro- cessbeginsintherelaysfurtherfromthesubstationandendsin it.Whenareconsidereddiscretesettings,itisnecessarytoper- form roundingas shown in step 10 of Pseudocode 3 to avoid miscoordination.

Pseudocode3. Efficientheuristicalgorithm(EHA).

1: InitializeallrelayswithTDSmin;

2: fori=mto0do misthenumberofrelays.

3: ifRihasabackuprelaythen 4: RBP_i←backuprelayofRi;

5: fork=1topdo pisthenumberoffaultsinside eachprotectionzone.

6:

Taux←TDSi·K_i,i^k+Trelay;

theminimumoperationaltime requiredofRifortheconsidered faultlevelplusTrelay. 7: TDSaux←Taux/K_BP^k

i,i; TDSBP_irequiredforRBP_itohave anoperationaltimeofTauxforthe consideredfaultlevel.

8: ifTDSaux>TDSBP_ithen

9: ifTDSisconsidereddiscretethen

10: TDSBPi←roundTDSauxtothenextavailablesetting;

11: else forcontinuousTDS.

12: TDSBP_i←TDSaux;

13: endif

14: endif

15: endfor

16: endif 17: endfor

6. Methods

TheproposedHPHAisappliedintwotestsystems:I(Fig.4)andII (Fig.5).Themaximumcurrentload,currenttransformerratio(CTR), theinitialrelayandcurvetypesandthemaximum/minimumshort- circuitcurrentsinsidetheprotectionzoneofeachOCR,forboth systems,areshowninTables5and6,respectively.Initially,allOCR areIECwithinversecurvetype.Itisassumedthatallrelayandcurve typescanbechanged(eitherbyreplacementorinputparameters).

ItisconsideredTDS_min=0.1,TDS_max=10,T_max=5s,LGF=150%and p=2.TheTDSstepsis0.05and0.01forthetestsystemsIandII, respectively.TheIpvariesfrom50to200%oftheCTR,withsteps

R

¹

R

²

R

³

R

⁴

R

⁵

R

⁶

R

⁷

R

⁸

R

⁹

R

¹⁰

Layer 1

Layer2 Layer 3 Layer4

Layer5

Fig.4.OCRsrenumberinginaRDN(testsystemII).

R

1

R

3

R

²

1

2 3 5

R

⁵

R

4 4 6

Fig.5.TestsystemI.

Table5

OCRdataforthetestsystemI.

R1 R2 R3 R4 R5

Iload(A) 199.5 130.8 68.7 100.7 50.0

CTR 300/5 300/5 100/5 200/5 100/5

Iccmax(A) 3115.0 2010.7 2010.7 1512.5 878.4

Iccmin(A) 1510.5 1046.3 975.1 500.3 325.1

16 16 31 7 7

Ip₁ Ip₂ Ip₃ Ip₄ Ip₅ Fig.6. Chromosomeusedforexample.

of15%.Also,theGAparameterskpsandMaxIteraresetto100and 1500,respectively.

7. Resultsanddiscussions 7.1. AnexampleoftheHPHA

AnexampleoftheprocedureusedforcalculationoftheTDSsand OFoftheproposedalgorithminthetestsystemIisshownasfol- lows.Thisproceduretakesplaceinstagesforthecreationofinitial population,mutation,recombinationandlocalsearch.Forsimplic- ity,onlythevaluesofIparedecisionvariables.Thus,thecurveand relaytypesarefixed,whereallrelaysareIECwithinversecurve type.ThechromosomeoftheGAusedinthisexampleisshown inFig.6.EachgenecorrespondstoIpavailableinthecorrespond- ingrelay.Forexample,asthegenerelatedtoIp1is16,soIp1gets the16thavailableIpavailableatR₁¹(Ip₁←375A).Theremaining decodedIpsareIp₂=375,A,Ip₃=200,A,Ip₄=160,AandIp₅=70,A.

1TheIpsavailableatR1are{150,165,180,195,...,600A}.

Table6

OCRdataforthetestsystemII.

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10

Iload(A) 399.5 120.8 278.7 50.7 70.1 79.2 199.5 199.5 100.7 99.2

CTR 600/5 200/5 500/5 100/5 150/5 600/5 200/5 500/5 100/5 150/5

Iccmax(A) 3115.0 2010.7 2010.7 1512.5 1512.5 1190.0 1190.0 950.4 780.1 780.1

Iccmin(A) 1510.0 1046.3 975.1 630.3 599.1 510.5 699.3 519.1 398.3 372.1

First,allrelaysareinitializedwithTDS_min.Thecountersiand k are initialized withthe number of relays (i←5) and (k←1), respectively.

7.1.1. Step1–coordinationbetweenR₃andR₅

Thecoordinationisperformedfor325.1A(Icc¹5),whichisone oftheconsideredfaultlevelsinsidetheprimaryprotectionzone ofR₅.Inthiscase,R_i←R₅andRBP_i←R3.T_auxiscomputedasthe minimumoperationaltimeofR₃forfaultsinsideitsprotectionzone plusT_relay,asshownin(15).

Taux=TDS5·K_5,5¹ +Trelay=0.1·4.9228+0.4=0.8923s (15) ThevariableT_auxhastheminimumoperationaltimerequired forthebackupprotection(R₃)tobecoordinatedwiththeprimary protection(R5).ThenecessaryTDSforthebackupprotectionhave anoperationaltimeofTauxisTDSaux,anditiscalculatedasshown in(16).

TDSaux= Taux

K_3,5¹ = 0.8923

14.3389=0.0622 (16)

SinceTDSaux<TDS₃,theTDS₃ isnotupdated.Thecounterkis increased(k←2)andthecoordinationisheldfor878.4A(Icc²5),for thesamepairofOCRs,asshownin(17)and(18).

Taux=TDS5·K_5,5² +Trelay=0.1·2.8520+0.4=0.6852s (17) TDSaux= Taux

K_3,5² =0.6852

4.6608=0.1470 (18)

TheTDSaux>TDS3,soTDS3←0.15,because0.1470isnotavail- ableattherelay.²ThecoordinationbetweenR3andR5isfinished, thecounteriisdecremented(i←4)and(k←1).

7.1.2. Step2–coordinationbetweenR₂andR₄

Thecoordinationisrealizeforthefaultlevelof500.3A(Icc¹4), R_i←R₄andRBP_i←R2.ThevariablesT_auxandTDS_auxarecomputed asshownin(19)and(20).

Taux=TDS4·K_4,4¹ +T_relay=0.1·6.0704+0.4=1.0070s (19) TDSaux= Taux

K_2,4¹ = 1.0070

24.2119=0.0416 (20)

As TDSaux<TDS2, theTDS2 is not updated. The counter k is increased(k←2)andthecoordinationisheldfor1512.5A(Icc²4),for thesamepairofOCRs.TheTauxandTDSauxarecomputedasshown in(21)and(22).

Taux=TDS4·K_4,4² +Trelay=0.1·3.0467+0.4=0.7047s (21) TDSaux= Taux

K_2,4² =0.7047

4.9497=0.1424 (22)

In this case, TDSaux>TDS₂, so TDS₃←0.15. The coordination betweenR2andR4isendedandthecounteriisdecremented(i←3) and(k←1).

2 TheTDSavailableattherelayare0.1,0.15,0.20,0.25,...,10.

7.1.3. Step3–coordinationbetweenR₁andR₃

Thecoordinationis performedfor975.1A(Icc¹3),R_i←R₃ and RBP_i←R1.ThevariablesTauxandTDSauxarecomputedasshown in(23)and(24).

Taux=TDS3·K_3,3¹ +Trelay=0.15·4.3489+0.4=1.0523s (23) TDSaux=Taux

K_1,3¹ =1.0523

7.2554=0.1450 (24)

AsTDSaux>TDS1, so TDS1←0.15. The counter k is increased (k←2)andthecoordinationisperformedfor2010.7A(Icc²3),for thesamepairofOCRs.ThevariablesTauxandTDSauxarecomputed asshownin(25)and(26).

Taux=TDS3·K_3,3² +Trelay=0.15·2.9636+0.4=0.8445s (25) TDSaux=Taux

K_1,3² =0.8445

4.0988=0.2060 (26)

Onceagain,theTDSaux>TDS₁,soTDS₁←0.25.Thecoordination betweenR3 andR1isfinishedandthecounteriisdecremented (i←2)and(k←1).

7.1.4. Step4–coordinationbetweenR₁andR₂

The coordination is realized for 1046.3A (Icc¹2), Ri←R2 and RBP_i←R1.TheTauxandTDSauxarecomputedasshownin(27)and (28).

Taux=TDS2·K_2,2¹ +Trelay=0.15·6.7523+0.4=1.4128s (27) TDSaux=Taux

K_1,2¹ =1.4128

6.7523=0.2092 (28)

As TDS_aux<TDS₁, the TDS₁ is not updated. The counter k is increased(k←2)andthecoordinationisheldfor2010.7A(Icc²2),for thesamepairofOCRs.TheTauxandTDSauxarecomputedasshown in(29)and(30).

Taux=TDS2·K_2,2² +T_relay=0.15·4.0988+0.4=1.0148s (29) TDSaux=Taux

K_1,2² =1.0148

4.0988=0.2476 (30)

DuetoTDSaux<TDS1,theTDS1isnotupdated.Thecoordination betweenR1 andR2isfinishedandthecounteriisdecremented (i←1)and(k←1).AsR_i←R₁anditdoesnothavebackupprotec- tion,theiterativeprocessends.TheoptimumTDSsareTDS₁=0.25, TDS2=0.15,TDS3=0.15,TDS4=0.10andTDS5=0.10.TheOFinthis caseis3.231s.

7.2. ResultsfortestsystemsIandII

Forbothsystems,theOCRcoordinationwillbedoneforthree cases:A,BandC.InthecaseA,TDSandIpareconsidereddecision variables.IncaseB,thereisanadditionofthecurvetypesasdeci- sionvariables.IncaseC,thereisanadditionoftherelaytypesas decisionvariables.TheresultsareshowninTables7and8.

WhentheresultsofcasesAandBareconfronted,theaddition ofcurvetypesasdecisionvariablesimprovedtheOFby25.63and 35.74%forthesystemsIandII,respectively.Whentheresultsof

Please cite this article in press as: A.A. Kida, L.A. Gallego, A high-performance hybrid algorithm to solve the optimal coordi- Table7

ResultsfortestsystemI.

CaseA CaseB CaseC

Ip(A) TDS Relay Curve Ip(A) TDS Relay Curve Ip(A) TDS Relay Curve

R1 375 0.25 IEC I 375 0.20 IEC I 420 6.15 U.S. S.I

R2 375 0.15 IEC I 375 0.10 IEC I 240 2.35 IAC E.I

R3 200 0.15 IEC I 170 0.15 IEC E.I 125 0.25 IEC E.I

R4 160 0.10 IEC I 160 0.10 IEC E.I 160 0.10 IAC S.I

R5 80 0.10 IEC I 80 0.10 IEC E.I 80 0.10 IAC S.I

OF 3.231s 2.403s 1.394s

Table8

ResultsfortestsystemII.

CaseA CaseB CaseC

Ip(A) TDS Relay Curve Ip(A) TDS Relay Curve Ip(A) TDS Relay Curve

R1 440 0.28 IEC I 500 0.17 IEC E.I 500 2.02 U.S. M.I

R2 160 0.21 IEC I 160 0.14 IEC I 160 8.14 IAC S.I

R3 370 0.21 IEC I 310 0.18 IEC I 250 3.17 IAC E.I

R4 80 0.10 IEC I 80 0.10 IEC V.I 80 0.10 IAC S.I

R5 80 0.10 IEC I 80 0.10 IEC E.I 80 0.10 IAC S.I

R6 65 0.10 IEC I 65 0.10 IEC E.I 65 0.10 IAC S.I

R7 250 0.19 IEC I 130 0.47 IEC E.I 160 0.30 IEC E.I

R8 280 0.10 IEC I 130 0.20 IEC E.I 160 1.59 IAC E.I

R9 65 0.10 IEC I 65 0.10 IEC E.I 65 0.10 IAC S.I

R10 65 0.10 IEC I 65 0.10 IEC E.I 65 0.10 IAC S.I

OF 6.539s 4.202s 3.465s

Table9

Relayoperationaltimes(caseC),forthetestsystemI.

PP BP Iccmin Iccmax

TPP TBP TBP−TPP Icc(A) TPP TBP TBP−TPP Icc(A)

R1 – 0.827 – – 1510.5 0.531 – – 3115.0

R2 R1 0.722 1.158 0.436 1046.3 0.274 0.677 0.404 2010.7

R3 R1 0.334 1.254 0.920 975.1 0.078 0.677 0.600 2010.7

R4 R2 0.007 3.175 3.168 500.3 0.005 0.406 0.401 1512.5

R5 R3 0.006 3.470 3.464 325.1 0.005 0.413 0.409 878.4

casesBandCareconfronted,theadditionofrelaytypesasdecision variablesimprovedtheOFby41.99and17.54%forthesystemsI andII,respectively.Agreaterimprovementhappenedwhencon- frontedcasesAwithC,whichtheadditionofrelayandcurvetypes asdecisionvariablesimprovedtheOFby56.86and47.01%forthe systemsIandII,respectively.Theresultsshowedthattheaddition ofmoredecisionvariablesintotheproblemgreatlyreducedthe OCRoperationaltimes.Theconsiderationoftherelaysandcurves typesasdecisionvariables,resultinawiderangeofcurvesshapes

thatcanbechosen.Thisflexibilityimprovedthecoordinationof OCR.

ForthecaseC,thecoordinationgraphicsfortheOCRforthetest systemIandIIareshowninFig.7.Also,theoperationaltimesfor bothPPandBPareshowninTables9and10.Theselectivitywas achievedfortheconsideredfaultlevels,becauseofthedifferences betweenT_BPandT_PParegreaterthanT_relay,forallOCRspairs.Also, theOCRdidnottaketoolongtooperate,sincealltheTBPandTPP

arelessthanT_max.

Table10

Relayoperationaltimes(caseC),forthetestsystemII.

PP BP Iccmin Iccmax

TPP TBP TBP−TPP Icc(A) TPP TBP TBP−TPP Icc(A)

R1 R0 0.985 – – 1510.5 0.609 – – 3115.0

R2 R1 0.432 1.457 1.024 1046.3 0.389 0.789 0.400 2010.7

R3 R1 1.177 1.607 0.430 975.1 0.389 0.789 0.400 2010.7

R4 R2 0.005 0.501 0.496 630.3 0.004 0.404 0.400 1512.5

R5 R2 0.005 0.512 0.506 599.1 0.004 0.404 0.400 1512.5

R6 R3 0.005 4.507 4.502 510.5 0.004 0.842 0.838 1190.0

R7 R3 1.325 2.212 0.886 699.3 0.441 0.842 0.400 1190.0

R8 R7 0.827 2.519 1.692 519.1 0.300 0.700 0.400 950.4

R9 R8 0.005 1.421 1.416 398.3 0.004 0.406 0.401 780.1

R10 R8 0.005 1.656 1.651 372.1 0.004 0.406 0.401 780.1

Fig.7.CoordinationgraphicsforthecaseC.

AperformancebenchmarkbetweenEHAandLPtechniqueswas realized,thus,theTDSweretreatedascontinuousvalues.TwoLP solverswereutilized:thestate-of-the-art CPLEX^® andtheopti- mizationtoolofMatlab^®.ThetestsweredoneinaIntel^®Core^TM i5-3317UCPU@1.7GHz, and theresultsare shown inTable 11.

TheEHAandtheLPsolversprovidedthesameanswers,butthey demandeddifferentcomputationaltimes.In theworstcasesce- nario(testsystemII),theEHAwas2.69timesfasterthanthefastest LPsolver(CPLEX^®).TheMatlab^®solverwastheslowestalternative tosolvethisproblem.Inaddition,theEHAcanhandlediscreteTDS differentlyfromLPtechniques.Thisway,theEHAshowedtobea superiortechniquetofindtheoptimumTDSinRDN(Fig.8).

Table11

TimecomparisonforfindingtheoptimumTDSforthecaseA(consideringcontinu- ousTDS).

Testsystem EHA(ms) PL(CPLEX^®)(ms) PL(Matlab^®)(ms)

I 0.39 1.21 15.25

II 0.58 1.56 15.87

Fig.8.ConvergenceoftheHPHA.

8. Conclusion

This paper presented a high-performance hybrid algorithm (HPHA)tosolvetheproblemofovercurrentrelays(OCRs)coordi- nationinradialdistributionnetworks(RDNs).TheHPHAcombines ageneticalgorithm(GA)withanefficientheuristicalgorithm(EHA) todecreasethesearchspaceoftheGA,astheoptimumtimedial settings(TDS)arecomputedwiththeEHA.Severaladvantagesare obtainedusingtheEHA:(1) itcanhandlediscreteandcontinu- ousTDS;(2)itdoesnotdependoncommercialsolvers;(3)itis fasterthanthelinearprogramming(LP)techniquesand(4)ithasa simpleimplementation.TheEHAshowedtobeamoreinteresting approachthanLPtechniquesforRDN.

Severaldecision variablesareconsideredintheoptimization problem:TDS, pickupcurrent (Ip),curvetypes andrelaytypes.

Foreachdecisionvariablesaddedtotheproblem,theOCRoper- ationaltimeswerereduced.Therefore,itisinterestingtoconsider intheoptimizationproblemtheadjuststhatdonotimplyaddi- tionalcosts,like TDS,Ip and curvetypes.The additionof relay