A Distributed Algorithm for Distribution Network Reconfiguration

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2018ChinaInternationalConferenceonElectricityDistributionTianjin,17-19Sep.2018ADistributedAlgorithmforDistributionNetworkReconfigurationYuanqiGao,StudentMember,IEEE,PengWang,Member,IEEE,andNanpengYu,SeniorMember,IEEEactuationofloadsandswitcheswhicharecriticaltotheAbstract—Distributionnetworkreconfigurationalgorithmsimplementationofdistributionnetworkreconfiguration.changethestatusofsectionalizingandtieswitchestoreduceThedistributionnetworkreconfigurationoutcomesneednetworklinelosses,relievenetworkoverloads,minimizelossofsatisfytwosetofoperationalconstraints.First,thesystemload,orincreasehostingcapacityfordistributedenergyoperatinglimitssuchasthelineflowlimitmustnotbeviolated.resources.MostoftheexistingworkadoptcentralizedorhierarchicalapproachestosolvethenetworkreconfigurationSecond,theradialityconstraintmustbeenforcedonproblem.Thispaperproposesadistributedschemefornetworkdistributionnetwork,thatis,noloopshouldbeformed.reconfiguration.Inthedistributedapproach,theswitchesareTheexistingliteratureondistributionnetworkreconfig-representedbyedgecomputingagentswhocommunicateandurationcanbedividedintotwogroupsbasedonthesolutionworkcollectivelywiththeirneighborstofindtheoptimalmethodology.Thefirstgroupadoptsheuristicmethodswithinreconfigurationsolution.Thisschemebreaksthecomplexwhichtherearetwoapproaches.Thefirstapproachstartswithacomputationtasksrequiredbycentralizedalgorithmsintomuchsmallerones.Italsorelievesthecommunicationanddatasharingmeshednetworkandthenopentheswitchthatwillcontributeburdenvianeighbor-to-neighborcommunication.Simulationthemosttotheobjectivefunction[3,4,7,8].Theprocedureresultsona16-busdistributiontestfeederdemonstratethatthecontinuesuntilaradialnetworkisachieved.Thesecondqualityofthedistributedsolutioniscomparabletothatoftheapproachstartsfromaradialnetworkandselectsapairofcentralizedapproach.closedandopenswitchesandexchangetheirstatus[5,6].SelectingsuchapairrequiresanaccurateestimationoflossIndexTerms—Distributionnetworkreconfiguration,reductionduetotheexchange.distributedoptimization,mixed-integerprogramming.ThesecondgroupofliteratureformulatesthedistributionI.INTRODUCTIONnetworkreconfigurationproblemasamixed-integerprogramoracombinatorialoptimizationproblem.TheoptimizationYchangingthestatusofswitches,distributionnetworkBproblemissolvedbyeithergeneral-purposemetaheuristicreconfigurationalgorithmsimprovethesystemalgorithmsordeterministicones.Metaheuristicalgorithmsperformanceunderbothnormalandabnormaloperatingsuchassimulatedannealing[9],geneticalgorithm[10],andantconditions.Theobjectivesofdistributionnetworkcolonyalgorithm[11]havebeenusedtosolvethenetworkreconfigurationrangefromloss,overload,andoutagereductionreconfigurationproblem.Thedeterministicalgorithmsworkbytohostingcapacityenhancement.Thedistributionnetworklinearizingorconvexifyingtheoriginalproblemandconvertingreconfigurationfunctionalityisespeciallyvaluablewhenaittoamixed-integerlinearorconvexoptimizationproblem.largernumberofdistributedrenewableresources(DERs)haveThenmixed-integerlinearormix-integerconvexoptimizationbeenorareexpectedtobeinstalled[1].BothFederalsponsoredalgorithmsareadoptedtosolvetheproblem[1,12,13,14].Theprogramsandmarketforcesarefacilitatingthewide-spreaddeterministicapproachhasanumberofadvantagessuchastheadoptionofsmartgridtechnologiessuchastheadvancedrepeatabilityofsolutions,guaranteesofglobaloptimality,andmeteringinfrastructureandremotecontrollableswitches[2].easeofimplementationthankstotheoptimizationsolvers.ThesetwotechnologiesenabledremotedatacollectionandMostoftheexistingmethodsfollowedthecentralizedcontrolframeworkwithinwhichallnetworkdataarecollectedThisworkwassupportedinpartbyCaliforniaEnergyCommissionunderandsentbacktothecontrolcentertodeterminetheagreementEPC-15-090.Y.GaoandN.YuarewiththeDepartmentofElectricalandComputerreconfigurationsolution.TheswitchcontrolsignalsarethenEngineering,UniversityofCalifornia,Riverside,CA,92521USA(e-mail:sentfromthecontrolcenterviathecommunicationnetworkforygao024@ucr.edu;nyu@ece.ucr.edu).switchactuation.AlthoughcentralizedapproacheshaveshownP.WangiswithPacificNorthwestNationalLaboratory,Richland,WA99352USA.(e-mail:peng.wang@pnnl.gov).goodnumericalperformanceonsomedistributiontestfeeders,CICED2018PaperNo.201805280000165Page1/51730

12018ChinaInternationalConferenceonElectricityDistributionTianjin,17-19Sep.2018theyusuallyresultinhighlatencyandcommunicationformulation,namelythenodalvoltagemagnitudelimit(2)andbottleneckinthesystem.Thedistributedapproachesonthebranchflowlimit(3):otherhandhavegreatpotentialinreducingthecommunication?2???≤?2≤?2???∀?∈?\?0(2)?burden,improvingcybersecurityandpreservingtheprivacyof?2≤??2???∀??∈?(3)??ℓsmartmeterdata[15].where?2isthesquarednodalvoltagemagnitudeofnode?;?Inthispaper,weproposeadistributedalgorithmto?ℓ∈{0,1}isabinaryvariablerepresentingtheclose(?ℓ=1)overcomethecommunicationbottleneckproblembyandopen(?ℓ=0)statusofeachswitch;?denotesthesetofdistributingthecomputationtaskamongthenetworkswitches0allnodesinthedistributionnetwork;?isthesetofsubstation(agents)withonlyneighbor-to-neighborcommunications.Ournodes(referencenodes).contributionsareasfollows.First,anoveldecomposedformulationofthedistributionnetworkreconfigurationB.PowerFlowModelproblemisdeveloped.Second,analternatingdirectionmethodTheDistFlowequations[14]areadoptedtocapturetheofmultipliers(ADMM)releaseandfixalgorithmisadoptedtopowerflowconstraints.solvetheprobleminadistributedmanner.Third,weintroduce22??=∑???−∑(???−??????)+????∀?∈?(4)adistributedapproximatedNewton’smethodtospeedupthe??∈???∈?distributedoptimizationalgorithm.?=∑?−∑(?−??2)+??2∀?∈?(5)Therestofthepaperisorganizedasfollows.Section2???????????formulatesthedistributionnetworkreconfigurationproblem.??∈???∈??2=?2−2(??+??)+(?2+?2)?2∀??∈?(6)Section3presentsthedistributedalgorithm.Section4shows????????????????thesimulationresults.Section5providestheconclusion.?2+?22???????=2∀??∈?(7)??II.PROBLEMFORMULATION2???20??=?∀?∈?(8)A.Overviewwhere??+???isthecomplexnetpowerinjectionatnode?;Oneofthemostcommonlyusedobjectivesofthenetwork???+????isthecomplexbranchpowerflowofline??;???+reconfigurationproblemistheminimizationoflinelosses.The????istheimpedanceofline??;??+???istheshuntadmittanceconstraintsoftheoptimizationproblemincludetheoperatingfrombus?toground.Ithasbeenshown[16]thatforpracticallimitssuchasthelineflowlimits,thepowerflowconstraints,radialnetworks,thesystemofequationsEq.(4-8)hasauniqueandthenetworkradialityconstraint.Thepowerflowsolutionneartheflatvoltagesolution.Therefore,theyareconstraintsensurethatthesteady-stateoperatingconditionsaresufficeforthereconfigurationapplication.consistentwiththeelectricloads,distributedgenerationsandSinceequation(7)definesanon-convexfeasibleset,thethephysicsofthedistributionnetwork.Thenetworkradialityrelaxationistypicallyapplied[14]:constraintsrequirethateveryprimaryfeederofthedistribution?2+?22????networkhavearadialtopology.Thegoalofnetwork???≥2∀??∈?(9)??reconfigurationistofindon/offstatusforallswitchesthatNotethatEq.(9)definesaquadraticconeandcanbehandledbyminimizethenetworklosseswhilesatisfyingalloperatingmanyoptimizationsolvers.constraints.Inthiswork,thedistributionnetworkisassumedtobereasonablybalancedsothatthesingle-phaserepresentationC.RadialityConstraintofthethree-phasenetworkisacceptable.ItisalsoassumedthatToenforcethenetworkradialityinthereconfigurationeachlinesegmenthasaswitchinstalledwhichcanberemotelyproblem,themethodproposedin[1,12]isadopted:controlledfornetworkreconfiguration.???+???=?ℓ∀ℓ∈?(10)Theobjectivefunctionoflinelossminimizationisgivenby:∑?=1∀?∈?\?0(11)??min∑??2(1)?????∈?(?)??∈??=0∀?∈?0(12)??2where???istheresistanceofline??.???denotesthesquared?∈{0,1}∀?∈?\?0,?∈?(?)(13)??magnitudeofcurrentflowingonline??.?isthesetofalllines0≤?ℓ≤1∀ℓ∈?(14)inthenetwork.Notethateachline??hasareferencedirectionwhere???,???,?ℓarevariablesassociatedwitheachline??;?→?associatedwithit.Throughoutthepaperweuse??to?(?)isthesetofneighbornodesof?.Ithasbeenshown[12]denotealineifthereferencedirectionisneeded;otherwisewethatEq.(10-14)aresufficientfortheradialityforeachgraphwillsimplyuseℓinplaceof??.componentthatisconnectedtooneofthereferencenodes.TwooperatinglimitswillbeconsideredintheproblemThereforeEq.(10-14)togetherwithEq.(4-8)definearadialCICED2018PaperNo.201805280000165Page2/51731

22018ChinaInternationalConferenceonElectricityDistributionTianjin,17-19Sep.2018networkforpracticalloads.canbepartitionedinto???(1)={2},?????(1)={3},???(1)=???Wesummarizethefinaloptimizationproblemas:{4},and?????(1)={5}.?minNetworkloss:Eq.(1)s.t.Operatinglimits:Eq.(2,3)B.ADMMRelease-and-Fix(OP1)Powerflow:Eq.(4,5,6,8,9)ThissubsectiondescribesthedistributedsolutiontoproblemNetworkradiality:Eq.(10,11,12,13,14)(OP1).Thestatevectorofagent??isdefinedas???=22(??)2(??)?Thedecisionvariablesare???,???,???,???,???,?ℓ∀??∈?;[?,?,?2,?,?,?,?,?].Superscripts(??)are????????????ℓ?2∀?∈?.Problem(OP1)isamixed-integerconic?introducedonthevoltagevariables.Thisisbecausethesameprogrammingproblemandcanbesolvedbyexistingsolversinvoltagevariable?2issharedbyallagentsthatareincidentto?acentralizedmanner.node?andmustbedistinguished.Asaresult,agentsmustagreeonthevalueofsharedvoltagevariables:III.DISTRIBUTEDSOLUTIONMETHODOLOGY?2(??)=?2(??)∀??∈?,∀??∈?????(??)(15?)???Inthissection,weproposeadistributedsolutiontoproblem?2(??)=?2(??)∀??∈?,∀??∈???(??)(15?)???(OP1).First,(OP1)willbedecomposedintoacollectionof2(??)2(??)?????=?∀??∈?,∀??∈?(??)(15?)coupledsub-problems.Second,eachofthesub-problemsis???2(??)2(??)??(??)(15?)solvedbyanagent(switch)vialocalcomputationand??=??∀??∈?,∀??∈??neighbor-to-neighborcommunication.Inthefollowing,wefirstUsingthedefinitionof???,problem(OP1)canbewrittenas:definetheagentsandtheircommunicationgraph,thenwe?min∑??????presentthetwo-stepdistributedalgorithm.???,??∈???∈?A.DefinitionofAgentsandCommunicationGraphs.t.???∈???∀??∈?(OP2)Weassumeeachswitchhascomputingcapabilityandcan∑??????=??∀?∈?communicatewithitsneighbors.Theagentsaredefinedasthe?∈?(?)switchesinthenetwork.Itisassumethateachlinehasaswitch.where???isalocalmixed-integersetwhosecontinuousHence,wedonotdistinguishtheconceptofswitch,line,andrelaxationisconvex;?(?)isthesetofneighbornodesof?;theagentandrefertothemasagentintherestofthepaper.Wedefinetheneighbors?(??)ofeachagent??astheagentsmatrices???andvectors??areidentifiedthroughproblem(OP1)aswellasEq.(15).Wereferthefirstsetofconstraintsinthathaveanodeincommonwithagent??.??itselfisnotin(OP2)aslocalconstraintsandthesecondthecoupling?(??).Inotherwords,let?=(?,?)bethegraphrepresentingconstraints.Tosolveproblem(OP2)inadistributedmanner,thedistributionnetwork,thenthecommunicationgraphis?wederivetheaugmentedLagrangianfunctionbyabsorbingthe?=(?,?)whereifℓ∈?and?∈?,thenℓ?∈?ifℓand?couplingconstraintsintotheobjectivefunction:?areincidentin?.Ingraph-theoreticterms,?iscalledthelinegraphof?.min?=∑???+????????,??∈???∈?∑??(∑??−?)+??∈?(?)??????∈?(OP3)2?∑‖∑?∈?(?)??????−??‖2?∈?2s.t.???∈???∀??∈?where?>0iscalledthepenaltyparameter.Problem(OP3)Fig.1.ExampleofagentsandtheircommunicationsmaybesolvedinadistributedmannerbythealternatingAnagent??’ssetofneighbors?(??)ispartitionedintofourdirectionmethodofmultipliers(ADMM)[17,18,19].subsetsbasedontheirandagent??’sreferencedirections.TheADMMalgorithmcanbeusedtosolveconvexseparable????Denote?(??)astheneighborsthatconnecttonode?with?problems.However,thepresenceofbinaryvariables???,???intheir“from”nodesbeing?.???(??),?????(??),and???(??)are???(OP3)destroystheconvexity.Aheuristicremedytohandledefinedinasimilarmanner.Fig.1showsanexamplebinaryvariableswasintroducedin[17]andtheresultingillustratingtheconceptsmentionedinthissection.TherearemodifiedalgorithmiscalledADMMRelease-and-Fix.ThefiveswitchagentsinFig.1(a).Thearrowsdenotethereferencemodifiedalgorithmproceedsbyiteratingbetweentwostages.directions.Fig.1(b)showsthecommunicationgraphoftheThefirststage(ADMM-Release)isidenticaltoconventionalnetwork.Accordingtothereferencedirection,?(1)={2,3,4,5}ADMMwiththeexceptionofthepresenceofbinaryvariables.ThegoalofADMM-Releaseistosearchforfeasiblebinarysolutions,whichare“stable”acrossmultiplerunsofCICED2018PaperNo.201805280000165Page3/51732

32018ChinaInternationalConferenceonElectricityDistributionTianjin,17-19Sep.2018ADMM-Release.InordertoencourageexplorationofnewNowwecanapproximate?−1asfollows[20]:binarysolutions,thepenaltyparameter?willbegradually?−1=(?+?)−1decreasedafterafeasiblesolutionisfound.Inordertoforce111−11−−−−convergencetoa“stable”solution?willgraduallyincrease.=?2(?+?2??2)?2Afteranewstablebinarysolutionisfound,thesecondstage−1−1−1−1≈?2(?−?2??2)?2(ADMM-Fix)fixesthebinarysolutionfromADMM-Releaseandsolvesthesimplifiedoptimizationproblemwithonly=?−1−?−1??−1(19)continuousvariables.Thesetwostageswillalternateuntilthewherethethirdequationwiththeapproximationsignisstoppingcriteriaismet.analogoustothefirstorderTaylorseriesexpansion1≈1−TheADMM-Fixstepconvergesslowly.Tospeedupthe1+??near?=0.NotethanEq.(19)enablesthecomputationofdistributedcomputationweproposeanapproximatedNewton’s−1?andtheupdateoflocalvariablestobecarriedoutmethodtoreplacetheADMM-Fixstep.(approximately)locallyasfollows:C.ApproximatedNewton’sMethod??−1−1−1???←???−?ℓ∇ℓ?+?ℓ∑?ℓ???∇??(20)AfterafeasiblebinarysolutionisfoundbyADMM-Release,?∈?(??)⋃??theADMM-FixproblembecomesthesameassolvingtheSince?ℓ?=?if?≠ℓ,?∉?(ℓ),thecomputationofeachDistFlowequationsEq.(4-8)inadistributedmannerwithaterminEq.(20)requiresonlytheinformationofagent??anditsgivennetworkconfiguration.Tosolvetheproblem,wefirstneighbors.linearizeEq.(7)oftheDistFlowequationsas:Insummary,theapproximatedNewton’smethodhastwo2???̃+2???̃−?2??̃2−?2??̃2??????????????levelsofiterations.Intheouteriteration,problem(OP4)is=?2?+?2?−?2??2?∀??∈?(16)formedbyobtaining??,??,and??frompreviousiteration;?????????????where?istheiterationnumberandvariableswithatildẽintheinneriteration,variablesareupdatedusingEq.(20).denotestheincrement,e.g.,?̃=??−?.Theresulting??????systemoflinearequations,namelyEq.(4,5,6,8,16),isdenotedIV.SIMULATIONRESULTSas??=?.Next,weproposeadistributedalgorithmwhichThissectionpresentsasimulationstudytovalidatethesolvethislinearsysteminaniterativemanner.proposeddistributedalgorithmfornetworkreconfiguration.?22(??)2(??)?WefirstdescribethetestsystemandthendiscusstheresultsWedefineanewvector???=[???,???,???,??,??]withfrombothcentralizedandtheproposeddistributedalgorithm.continuousvariablesonly.Solving??=?isequivalenttosolvingthefollowingunconstrainedoptimizationproblem[20]:Inparticular,thecomputationspeedoftheADMMalgorithm1???2andourproposedapproximatedNewton’smethodiscompared.??min,??∈??=2∑‖??????−???‖2(OP4)Thedistributiontestfeederdescribedin[21]isusedinthe????∈?where??and??areidentifiedfrom?and?byrearrangingsimulation.ThedistributiontestfeederisshowninFig.2????equationsandvariablesaccordinglyandappendingEq.(15)toenforcevoltageconstraints.Notethat??isaconstantmatrix??while??dependslinearlyon??forallneighbors,?,ofagent?????(excluding??itself).Wewouldliketosolveproblem(OP4)usingNewton’siterationwherethegradientandHessianmatrixcanbederivedasfollows:∇?=???(????−??)∀ℓ∈?ℓℓℓℓℓ??=∇?=?????∀ℓ∈?(17)ℓℓ???ℓℓℓℓ???=∇?=−?????∀ℓ,?∈?,?∈?(ℓ)ℓ????ℓℓ???ℓ???ℓ?=?∀ℓ,?∈?,?∉?(ℓ)Fig.2.16-bustestfeederHowever,itischallengingtoinverttheHessianmatrix?ofthewherethedotsrepresentloadbuses;thesolidlinesrepresentobjectivefunctioninadistributedmanner.Therefore,ansectionalizingswitchesanddashedlinesrepresenttieswitches.methodtoapproximatetheinverseHessianisneeded.Todoso,Agentsarerepresentedbyaredbox.Theedgesofthelet’sdefinetwonewmatrices:?=????(?1,?2,…,?|?|)communicationgrapharerepresentedbyreddashedlines.where?ℓ=??ℓℓand?whereInitially,allsectionalizingswitchesareclosedandalltie(1−?)?ℓℓ??ℓ=?switchesareopen.Theglobaloptimumsolutionfoundbythe?ℓ?={(18)?ℓ???ℎ??????centralizedalgorithmwasreportedin[21].ThenetworkCICED2018PaperNo.201805280000165Page4/51733

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