Electrical
and
topological
drivers
of
the
cascading
failure
dynamics
in
power
transmission
networks
Alberto
Azzolin
a,
Leonardo
Dueñas-Osorio
b, ∗,
Francesco
Cadini
d,
Enrico
Zio
a, ca Dipartimento di Energia - Politecnico di Milano, Piazza Leonardo Milano, Italy
b Department of Civil and Environmental Engineering, Rice University, 6100 Main Street, Houston TX 77005, United States
c Chair System Science and the Energy Challenge, Fondation Electricite ’ de France (EDF), Centrale Supélec, Université Paris-Saclay, Grande Voie des Vignes, 92290 Chatenay-Malabry, France d Dipartimento di Meccanica - Politecnico di Milano via La Masa 1 - 20156 Milano, Italy
abstract
Tosystematicallystudykeyfactorsaffectingcascadingfailuresinpowersystems,thispaperadvancesalgorithms forgenerating syntheticpowergrids with realistictopologicalandelectricalfeatures,whilecomputationally
quantifyinghowsuchfactorsinfluencesystemperformanceprobabilistically.Keyparametersaffectinglineout-agesandpowerlosses duringcascadingfailures includelineredundancy,load/generatorlayoutandre-dispatch strategies. Ourstudycombinesasynthetic powergrid generatorwithadirect current(DC) cascadingfailure simulator.Theimpactofeachofthefactorsandtheirinteractions unravelusefulinsightsforinterventionsaimed atreducingtheprobabilities oflargeblackoutsonexistingandfuturepower systems.Moreover,conclusions drawnfroma spectrumofdifferentpowergridtopologiesandelectrical configurationsoffer more generality thantypicallyattainedwhenstudyingspecifictestcases.Lineredundancyanddistributedgenerationappearas themost efficaciousparametersforreducingtheprobabilitiesoflargepowerlossesandmultiplelineoverloads, althoughtheeffectdecayswith networkdensity.Also, re-dispatchstrategiesarecriticalonthedistributionof thecascadingfailuresizeintermsoflinefailures. Theseandrelatedresultsprovidethebasisfor probabilistic analysesandfuturedesignofevolvingpowertransmissionsystemsunder uncertainty.
1. Introduction
Theimpactthatapowertransmissionsystem’stopologyand asso-ciatedelectricalfeatureshaveonoverallsystemreliabilityisstillnot wellunderstood,especiallywhentheirjointeffectisconsideredduring cascadingfailureevents.Andevenwhenconsideringafixednetwork topology,manydifferentelectricalconfigurationsandstatesare possi-ble,asplacementsofgenerationunitsordispatchstrategiesresultin differentpowersystemdynamicsandassociatedreliability considera-tions.
Powersystemreliabilityresearchencompasesthestudyofcascading failures,sincethesefailuredynamicshaveprovenseriousineconomic andsocialterms[1].Standardpracticeinthestudyofcascadingfailures istoapplypowerornetworkflowmodelstoalimitedsetoftestsystems tounderstandevolvingsystembehavior[2–7].Also,thereare topolog-icalstudiesforsimplifiedyetanalyticalexplorationsofsystem proper-ties,typicallybasedonsamplesofrandomlygeneratednetworkswhen theelectricaldataarenotavailable[8–11].However,theprobabilistic analysisofpowersystemssubjectedtocontingenciesisstilllimited,
of-∗Correspondingauthor.
E-mailaddress:leonardo.duenas-osorio@rice.edu(L.Dueñas-Osorio).
tenwithoutthejointeffectsoftopologicalandelectricalconfigurations (e.g.,layout,elementsiting,re-dispatch).Thus,conclusionsdrawnto dateunderprobabilisticmodels,whilevaluable,remainnoteasily gen-eralizable.Ourworkattemptstobridgethisgapbyapplyingacascading failuremodelbasedonanN-2contingencyanalysiswithDirectCurrent (DC) powerflowanalysesonasampleofsyntheticyetrealisticpower networksgeneratedtocapturesomeofthetopologicalaswellas elec-tricalandprobabilisticfeaturesofpowersystems[12].
WhilerelyingonDCpowerflowaddsrealismtothecascadingfailure model,italsorequiresthatthesyntheticnetworksbeadequate electri-cally.Hence,weextendthealgorithmbyWangetal.[12]toproduce syntheticpower gridssuitablefor DCpowerflowcomputations.The numberoffeaturescharacterizingpowergridsishigh,makingthe im-pactanalysisofallofthemprohibitive.Consequently,wegroupkey fea-turesofpowergridsintothreedifferentmacro-areasbasedonknown sensitivityanalyses[2,13]:theunderlyingtopology,theelectrical prop-ertiesof itscomponents,andthecontrolrulesgoverningthesystem. Thisgroupingisalsoconsistentwiththewayeachsyntheticpowergrid isgeneratedinourstudy,asthetopologyisconstructedfirst,thenthe
©2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/ Published Journal Article available at: https://doi.org/10.1016/j.ress.2018.03.011
topologyisenrichedwiththeelectricalparameters(e.g.,impedances, powerdemandandsupplylevels,generationsiting,etc.),andfinally, thedispatch/controlrulesareembeddedintothecascadingfailure sim-ulator.
Atthetopologicallevel,thelineredundancyofthenetworkis cho-sen asinputvariable, sinceadding linesisthe baselineapproach for decreasingsystems’ congestionandhasastraightforwardinterpretation fromaninfrastructuralpointofview.Attheelectricallevel,the genera-tor/loadbuseslayoutisconsideredakeyinputthatwevaryinorderto comparevariousdegreesof centralization,fromclusteredtoamore dis-tributedsitingof loadsandgenerators.Theinterestinpower generationlayout liesintheevolutiontowardssmartgrids[1,14]. Finally,fromtheoperationalpointofview,weconsider two differentpowerre-dispatchstrategies.
Wecarryoutcomputationalexperimentsfromfullfactorialdesigns tostudytheeffect ofkeyinputparametersandoffergeneralinsights [15]. Theinputvariablesinvolvedarefullycrossedwitheach other, allowingustogatherdataoncascadingfailureeffects.Sincethis ap-proachrelies onacatalogueofpowergridsinsteadof aparticular one,theconclusionsdrawnhavemoregeneralappeal.
Asfor system performance, we studythe reliabilityof the power grid,understoodhereastheabilityofthenetworktodeliverpowerto customers(viaapowerlossmetric),whilethephysical infrastructure integrityispreserved(viacountsofoutagesacrosslines)[1]. The com-putationalexperimentsthatfollow,allowusstudyingpowerlossand lineoutagesprobabilistically,particularlyasthetopological,electrical andcontrolparametersofthesyntheticnetworksvary.Thisworkthus complementstwooftheapproachesthatdominateexistingliterature: onegeneralandoftenprobabilistic,butlimitedtotopologicalanalyses [16,17], andtheothercomprehensiveinitselectricalaspectsbut spe-cific to a system model[18,19]. We aimat a middlepoint, where we use DC power flows and also study a variety of system configurations and dispatch strategies, so as to gain generalizable insightsforfutureoperationmanagementandreliability-baseddesign ofevolving powernetworks.
Therestofthearticleisorganizedasfollows:Section2describesour globalstrategytostudyevolvinginfrastructure,particularlyvia compu-tationalexperiments.Section3describesthepowernetworkgeneration procedure andour updatestomake keysystem parameterstunable. Then,Section4describesourcascadingfailuremodel.Section5 pro-videssimulationresults,andSection6discussesoutputsanddraws in-sightsfornetworkoperation.Section7concludesthepaperandprovides ideasforfutureresearch.
2. Globalstrategyforstudyingevolvingpowersystems
Toexplorehowkeyinputsaffectcascadingfailuresinpowersystems, weperformafullfactorialexperimentaldesigntoelicitgeneral conclu-sions.Appropriatevariablesandmodelsforourexperimentaldesignsas discussednext.
2.1. Fullfactorialdesign
Throughoutourwork,wewilldealwithapowergridobjectdefined asfollows:
=(𝑉,𝐸,𝒁,𝑷𝑫,𝑷𝑮,𝑷𝑴𝑨𝑿,𝑪) (1) whereVisthesetofbuses,Eisthesetoflines,and𝐺=(𝑉,𝐸)isthe powergrid topology.VectorZcaptures theimpedancesof thelines, whilePD,PG,PMAXarethepowerdemand,supply,andmaximum supplyof thenodesinV, respectively.VectorCdenotesthecapacity ofthelines.Wedefinethesubset{Z,PD,PG,PMAX}astheelectrical propertiesofthepowergrid.
OurgoalistodeterminewhichparametersaffectinginEq.(1)are bestatreducingcascadingfailures,soastotranslatefindingsintodesign
Table1
Planningmatrixofthefullfactorialexperimentwitheachrow representinga differentfactorandeachcolumna different level.
Factors/Levels 0 1 2
Lineredundancy 𝜁 =−1 𝜁 =0 𝜁 =1 Generatorslayout K=0 K=0.5 K=1 Re-dispatch R:=Proportional R:=OPA
guidelinesgeneralizabletoevolvingpowergrids.Inparticular,we con-siderfactorssuchas:lineredundancy𝜁,theloads/generatorslayoutK, andre-dispatchstrategies.Notethattopologyandelectricalfeaturesare afunctionofthesefactors,suchthatE(𝜁),PD(K),PG(K),andPMAX(K). Inparticular,𝜁 willvaryintherange[−1,1]interpolatingelectric networks generatedbytheRT-nestedSmallWorld(RT-nestedSW) al-gorithm[12],wherearealistictopology correspondsto𝜁 =0,an as-sociatedspanningtreeto,𝜁 =−1,andagreedytriangulation(ordense planarnetwork)to𝜁 =1.Inthisway,weareabletoproducepowergrid structureswithvaryinglevelsoflineredundancy.ParameterKwillvary in[0,1],andisusedtoproducepowergridswithincreased decentral-izedgeneratorlayoutasKincreases.
Inaddition,weanalyzetheimpactoncascadingfailuresof differ-entre-dispatchstrategies.Thetermre-dispatchreferstotheactionof changingthepowersupplyordemandatthepowergrid’snodesinreal time,sothatthetotalsupplyanddemandarebalanced.Wecomparetwo re-dispatchstrategiesinourexperiments.First,aproportionalstrategy wherepowersupplyanddemandatthenodeschangesproportionallyto theirinitialvalues,anddeemedasabaselinestrategy.Second,we con-sidertheOPAmodel[20],whichisasimplificationofhowanoperator mightinterveneinarealisticsystemwhenfacingcomplex contingen-cies.Itconsistsofanoptimizationroutinewhichminimizesthepower losses,subjecttoconstraintsinpowercarryinglimits.TheOPAmodel hasbeenusedinapplicationswithfastdynamicsandlongterm evolu-tionofpowertransmissionsystems[4].
Thecomputerexperimentsaresetasfollows:powergridswith dif-ferentvaluesof𝜁 andKaregeneratedwithourextendedRT-nestedSW algorithm,foralltheirpossiblecombinations.Foreachofthesepower grids,twoseparateN-2contingencyanalysesareperformed,employing oneofthetwore-dispatchstrategieseach.Thisprocedureistantamount toabalancedfullfactorialexperimentaldesign[15], whichunravels the single and joint effects of input factors. Table 1 shows the planningmatrixoftheexperiment.
2.2. Responsevariables
Wecomputethetotalpowernotserved𝑃(𝑙𝑖,𝑙𝑗)
𝑙𝑜𝑠𝑠 andthenumberof linefailuresfromcascades𝑁(𝑙𝑖,𝑙𝑗)
𝑓𝑎𝑖𝑙 ,followingN-2contingencyanalyses onpowergridsin,asassociatedtoapairoffailedlines(li,lj).Hence, evolvesfromitsinitialequilibriumstate0intoanewstate(𝑙𝑖,𝑙𝑗) followingthedynamicswemodeltroughsubsequentAlgorithm4.1.This procedureisthenrepeatedwithreplacementforalldistinctpairsoflines belongingto.WethendefinethetotalN-2contingencyanalysis ef-fectsaspowerlossPtotandlinefailuresNtotasfollows:
𝑃𝑡𝑜𝑡()= ∑ ( 𝑙𝑖,𝑙𝑗)∈ 𝑃(𝑙𝑖,𝑙𝑗) 𝑙𝑜𝑠𝑠 (2) 𝑁𝑡𝑜𝑡()= ∑ ( 𝑙𝑖,𝑙𝑗)∈ 𝑁(𝑙𝑖,𝑙𝑗) 𝑓𝑎𝑖𝑙 . (3)
NotethatPtotandNtotarecomputedforeachpowergrid topologi-calandelectricalconfigurationaswellasre-dispatchstrategy.Knowing howtheseperformanceindicatorsvaryincorrespondencetodifferent
configurationsofinputfactorsallowsustounderstandtheirglobal im-pactoncascadingfailurevulnerabilityandassociatedprobabilitiesof occurrence.
3. Syntheticpowergridsforcomputationalexperiments
Thefollowingsubsectionsreviewtheproceduretogeneratenew net-worktopologiesinagreementwithrealpowergrids.Then,theauthors explaintheroleoflineredundancyandgeneratorlayoutfactors(𝜁,K), whichenablesensitivityanalyses.
3.1. Topologicalpropertiesofpowernetworks
Considerapowergridtopology𝐺=(𝑉,𝐸), with|𝑉| =𝑁 asthe num-berofgenerationsources,aggregateloadsatthesubstationleveland transmissionnodes,and|𝐸| =𝑚asthe numberoflinks or transmis-sionlines.Research hasshown thatcommonlyused syntheticgraph structuressuchassmallworld[21], scalefree[22]andrandom net-worksarenotabletocapturethetopologicalfeaturesofrealpowergrids [12,17,23]. Forexample,Wangetal.[12]foundthatpowergridshave differentconnectivityscalinglawsthanstandardsmallworldgraphs.In fact,theaveragenodaldegree⟨k⟩isconstantanddoesnotscalewith thenetworkdimension,aswouldbethecaseforthesmallworldmodel [21]. Meanwhile,Cotilla-Sanchezetal.[23]noticedthattheaverage pathlength⟨l⟩scalingpropertiesof realpowergridsarein between regulargridsandsmallworldnetworks.Overall,theseandother stud-iescallformodelsthatcapturesystemtopologyaswellasfunctionality consistentwithpracticalpowersystems[12,16,17].
3.2. TheRT-nestedsmallworldmodel
Wangetal.[12]postulatednestingseveralsmall-world(SW) sub-networksintoaregularlatticetoretainelectricalsystemfeatureswhile usingsimpletopologiesthatexhibitrealisticscalingproperties[16,17]. AnotherpowernetworkgeneratorwasusedinPurchalaetal.[24]to testDCpowerflowaccuracyinactivepowerconsiderations.Theyfound boundsforthelinesreactancesandresistanceratios,belowwhichthe DCpowerestimationhasunacceptablehigherrors.TheRT-nestedSW modelhasalsobeenrefinedbyHuetal.andGengeretal.[25,26]in ordertoproducesyntheticDCandACpowerflowtestcases.While syn-theticmodelscontinuedevelopment,weadoptthebasicRT-nestedSW perspectiveandexpandasneededtorealizeourexperimentaldesign. Themethod[12], andassociatedextensionsaresummarizedinTable2.
TheinputparametersnecessarytoinitializethealgorithminStep 1{d0, ⟨k⟩,N, andthenumberofsubnetworks},areestimatedfromthe IEEE118-nodesystem[27]asourreferencerealisticnetwork through-outthestudy.Hence,Steps1–4generate powernetworks𝐺=(𝑉,𝐸) withtopologicalpropertiesconsistentwithrealsystems.Then,inStep 5,linereactancesarerandomlygeneratedfromaspecifiedheavy-tailed distributionfittedtotherealpower gridthatonewishestouseasa reference.Inthiswork,aGammadistributionisfoundtofitwellthe impedances ofthe IEEE118-node system[12]. Foreach link of the networkwesampleanimpedancevalue.Then,thesampledvaluesare sortedbymagnitudeinascendingorderandgroupedinto:locallinks, rewirelinks, andlatticeconnection linksaccordingtocorresponding proportions.Linereactancesineachgrouparethenassignedrandomly tothecorrespondinggroupoflinksinthetopology.Wewilldenotewith
ZGthereactancesvectorassociatedtoaparticulartopologyG, wherethe component𝑍𝐺
𝑙 denotesthereactanceoflinel∈E.
WiththepreviousStepandStep7,weextendtheRT-nestedSW algo-rithmandassignelectricalproperties,PD,PG,PMAX,C—necessaryfor DCpowerflowcomputationsandcascadingfailureanalysisasdescribed later.Networksasdescribedconstitutethebaselinesystems correspond-ingtoalineredundancylevelof𝜁 =0(Table1).Notethatchangesin 𝜁 resultinchangestotheparametersd0,⟨k⟩ ofStep1.Weexplainnext
Table2
SummaryoftheRT-nestedSmallWorldmodel(RT-nestedSW)and ex-tensions.
Pseudo-Code RT-nestedsmallworldmodelgeneration Step1 Selectthedesirednumberofsubnetworks,their
properties,andageographicaldistance thresholdd0.
Step2 Constructsubnetworksasfollows:foreachnode iselectlinksatrandom(theirnumberfollows aPoissondistributionwithmean⟨k⟩)from nodesjbelongingto𝑁(𝑖)
𝑑0={𝑗∶𝑑(|𝑗−𝑖|)<𝑑0} (i,jaretheindicesofthenodesconsidered, andd(∙)isadistanceoperator).
Step3 RewirethesubnetworklinksusingaMarkov chainmodel.
Step4 Randomlyconnectthesubnetworkswitheach other,throughlatticeconnections. Step5 Assignthelines’ impedancesusingasuitable
probabilitydistributionmodel.
Step6 Assignpowerdemandandsupplytothenodesin thenetwork.(Extended)
Step7 Assignlinescapacities.(Extended)
howthelineredundancyofthesebaselinenetworksisvariedtoform topologieswithlevels𝜁 =−1and𝜁 =1.
3.3. Boundingmodels
The range in which baseline topologies are allowed to vary is boundedfrombelowandabovebytwolimitingcases:thepowergrid’s spanningtree (ST)which hasa minimumlevelof linestoguarantee connectivity,andthegreedytriangulation(GT)whichapproximatesthe maximumnumberoflinesinatwodimensionalplanarspace.
FromeachbaselinenetworkgeneratedwiththeRT-nestedSW algo-rithm,𝐺𝑖=(𝑉𝐺𝑖,𝐸𝐺𝑖)andassociatedimpedancevector𝒁𝐺𝑖,we con-structthe𝑆𝑇𝑖=(𝑉𝑆𝑇𝑖,𝐸𝑆𝑇𝑖)and𝐺𝑇𝑖=(𝑉𝐺𝑇𝑖,𝐸𝐺𝑇𝑖)networks.Notethat 𝑉𝑆𝑇𝑖=𝑉𝐺𝑖,while𝐸𝑆𝑇𝑖isthesubsetoflinesbelongingtoaspanningtree ofGi,withtheirrespectiveimpedances.Similarly,𝑉𝐺𝑇𝑖=𝑉𝐺𝑖and𝐸𝐺𝑇𝑖 isthesupersetoflinesbelongingtotheplanartriangulationofGi,where theimpedancesofthelinesbelongingto𝐸𝐺𝑇𝑖−𝐸𝐺𝑖aregeneratedasin Section3.2.
Factor𝜁 variesin therange[−1,1]where0indicatesthebaseline topologyGandvaluesin−1≤𝜁 ≤0representthepercentageoflines belongingto𝐸𝐺−𝐸𝑆𝑇uptoobtainingG.Meanwhile,valuesin0≤𝜁 ≤1 indicatethepercentageoflinesbelongingto𝐸𝐺𝑇−𝐸𝐺whichareadded toGuptoformingaGT.Fig.1showstopologiesfor𝜁 =−1,0and1. 3.4. Sitingofloadsandgenerators
PowerdemandvectorPDandapowersupplyvectorPGacrossthe nodes’ setV∈G,helpdeterminethelayoutforgenerators/loadswithin thesyntheticpowergrids,whoseassignmentiscontrolledby parame-terKinTable1.Fromtheloadsperspective,weuseasvectorofpower demandPDasetconsistentwiththeIEEE118-nodesystem[27].We onlyretaintheoriginalcomponentsofthepowersupplyvectorPGand maximumpowersupplyvectorPMAXforthe15majorpower suppli-ers,astheIEEE-118systemhasgeneratorsoverrepresented(i.e., sev-eralserve asboundaryconditionstothelargersystemfromwhichit wasextracted).WecomplementedavailabledataintermsofPMAXfor theIEEE118system,withoptimalpowerflowtestcasedataincluded in Zimmerman et al. [28]. Hence, each component PGi,PMAXi,and 𝑃𝐷𝑖, 𝑖=1,2…𝑁 representsthepowersupplied,themaximumpower thatcouldbesupplied,andthepowerdemandedbybusi,respectively
Fig.1. Sampletopologieswithdifferentredundancylevelsforthesamenodesetlayout:(a)𝜁 =−1(𝑁=118,𝑚=117),(b)𝜁 =0(𝑁=118,𝑚=184),and(c)𝜁 = 1(𝑁=118,𝑚=369).
Fig.2. Samplenetworkswithdifferentgeneratorsetpositionsaccordingto:(a)𝐾=0,(b)𝐾=0.5,and(c)𝐾=1.Nodeswithblackfillaregenerators.
(ifiisaloadnode𝑃𝐺𝑖=𝑃𝑀𝐴𝑋𝑖=0).Therefore,syntheticpowergrids have15generatorbuses,93loadbusesand10transmissionbuses,to maintainrealisticproportions.Weusethefollowingprocedurewhich reliesonaspectralclusteringalgorithm[29]toassignthebuspositions ofthe15generators:
1. Selectextremesinthepowergridasext∈V(theextremesarethepair ofbusesmostdistantfromeachother),andathresholddistanced(K) (intendedasnumberoflinksasmostlineshavesimilarlength). 2. Buildthesubnetwork̃ =( ̃𝑉,𝐸̃)byremovingfrom allthebuses
thatareatadistancemorethandfromext,anddivideitin15bus communities ̃𝑉1,̃𝑉2,…,̃𝑉15usingthespectralclusteringalgorithm. 3. Selectanodeatrandomforeachcommunity ̃𝑉𝑖,andassigntoita
powersupplyandmaximumpowersupplycomponent.
Thisprocedureforcesthegeneratorsandtheloadstobeclustered togetherdependingon:
𝑑=𝑑𝑚𝑖𝑛+𝐾(𝑑𝑚𝑎𝑥−𝑑𝑚𝑖𝑛) (4) wheredministheminimumdistance(innumberoflinks)whereatleast 15busesareatadistancelessthandmin from ext,dmax is the diam-eter of thenetworkandK∈[0,1] isthe inputfactorin our experi-ment.Notethatinthissectionweareinterestedinmodelling differ-entload/generatorgeographiclayouts.Forthisreason,Eq.(4)usesthe numberoflinksasdistance,asthelinklengthdistributionis concen-tratedonasmallrangeof values[30],andthusadequatetocapture geographicpatterns;alternativessuchaselectricaldistances[31],are alsodesirablefordynamicanalyses,butnotnecessarilytoreproduce geographicallayoutpatternsasinFig.2.Thisfigurehasthree exam-plesofgeneratorpositioningfordifferentvalues ofK,so astoassist unravelingtheimpactsofgeneratorsitingoncascadingfailures.
InEq.(4)asKapproaches0,dapproachesdminandonlyafewnodes closetothenetworkextremesareavailabletobecomepowergenerators. Thisresultsinmostpowergeneratorsconcentratedinasmallportionof
thepowergrid,spatiallydistinctwithrespecttotherestofthegrid.As Kapproaches1moreandmorenodesbecomeavailableasgenerators, hencepowercanbesuppliedbybusesalloverthepowergrid,andthe spatialclusteringbetweengeneratorsandloadsblurs.Inparticular,the valuesofKconsideredforsensitivityanalysisare:0,0.5and1(Table1). Thisfactorallowusexploringtheeffectofspatiallydistributedversus centralizedpowergeneration,whichisofinterestfroma“smartgrid” perspective,especiallyasthemainlycentralizedpowergeneration struc-tureofexistinggridsevolvesintoonethatadmitsdecentralizationand distributedgeneration[1]. Asthepositionofpowergeneratingunits hasbeenfoundinPahwaetal.[13]toaffectthefrequencyandvoltage stabilityofpowergrids,italsodeterminesthepathsinDCpowerflows.
3.5. Linecapacityassignments
Toperformcascadingfailuresimulations,itisnecessarytoalsohave transmissionlines’ capacitydataconsistentwithfunctionalsystems.In realpowersystemlines,capacitiesbelongtoasetoffinitediscrete val-ues,whilepowergridsareusually𝑁−1compliant. Toproduce syn-theticpower networkswiththesefeatures, webuilda modelfor the linecapacityallocationthatdiffersfromaproportionalmodelusually foundintheliterature[13,32,5,33].Therefore,tosamplethecapacity Cloflinel,webaseourmodelonatruncatedexponentialdistribution, Eq.(5),withparameter𝜆 estimatedfromthepowerflow-to-capacity ra-tiodataofalargerealsystem(weusedaKolmogorov–Smirnovtestto confirmthattheexponentialmodelwasnotrejected).Specifically,we studiedthePolishgridavailableinZimmermanetal.[28]tosetour ex-ponentialmodel,asitisoneofthemostcompletepowertransmission networkdatasets(notethattheIEEE118 systemdoesnot offersuch
Algorithm4.1
Cascadingfailuresimulator(CFS).
Input:Powernetwork,initialfailuresli,lj∈E
1. 0=,computef 0vectorofpowerflowin0 2. 1=(𝑉,𝐸−{𝑙
𝑖,𝑙𝑗},𝒁 ,𝑷 𝑫 ,𝑷 𝑮 ,𝑷 𝑴 𝑨 𝑿 ,𝑪 )
For:𝑟=1,2…Do
3. AdjustloadandgenerationR:={proportionalorOPA}strategy 4. Compute f rpowerflowvectorin𝑟
5. IdentifyOrasthesetoflinesoutagediniterationr
If|Or|≥1 6. Set𝑟+1=(𝑉,𝐸𝑟−𝑂𝑟,𝒁 ,𝑷 𝑫 𝒓,𝑷 𝑮 𝒓,𝑷 𝑴 𝑨 𝑿 ,𝑪 ) Otherwise:END details): (𝐶𝑙|𝑓𝑙,𝑓𝑙𝑚𝑎𝑥,𝜁)∼ 𝜆||𝑓𝑙||𝑒 −𝜆|𝑓𝑙| 𝑐 𝑐2 ( 𝑒 𝜆|𝑓𝑙| 𝐶𝑚𝑎𝑥𝜁 −𝑒−𝑓𝜆|𝑓𝑙𝑚𝑎𝑥𝑙| )𝐼( 𝑓𝑚𝑎𝑥 𝑙 ,𝐶𝜁𝑚𝑎𝑥 ](𝑐) (5)
whereIistheindicatorfunction,𝜁 istheredundancylevelofthe net-worktowhichlinelbelongsto,𝐶𝑚𝑎𝑥
𝜁 isacapacitylimitdependentonthe lineredundancy,flisthepowerflowingthroughlwhentheDCpower flowiscomputedonthefullnetwork,and𝑓𝑚𝑎𝑥
𝑙 is: 𝑓𝑚𝑎𝑥 𝑙 =max(|||𝑓𝑙|,|𝑓𝑙1|,|𝑓𝑙2|,…,|𝑓𝑙𝑚||| ) 𝑙∈𝐸 (6) with𝑓𝑖
𝑙asthepowerflowingthroughlinelwhenlineiisremovedfrom 𝐸,considering 𝑓𝑙
𝑙=0,and𝑚=|𝐸|. Notethatthesupport ofthe probabilitydensityfunction(pdf)inEq.(5),fordifferentvaluesoffland 𝑓𝑚𝑎𝑥
𝑙 ,coverscapacityvaluesthatarefiniteandguarantee𝑁−1 com-pliance. However,sincewewantour sampletobelongtoa discrete setofcapacities,wediscretizeEq.(5)amongequallyspacedquantiles: {̄𝐶1< ̄𝐶2<…< ̄𝐶𝑛𝜁},wheren𝜁isdependentontheaveragenumberof linesofthenetworkswithredundancy𝜁:
𝑃(𝐶𝑙= ̄𝐶𝑖|𝑓𝑙,𝑓𝑙𝑚𝑎𝑥,𝜁 ) = 𝑒 −𝜆|𝑓̄𝐶𝑖𝑙| −𝑒− 𝜆|𝑓𝑙| ̄𝐶𝑖−1 𝑐2 ( 𝑒 𝜆|𝑓𝑙| 𝐶𝜁𝑚𝑎𝑥 −𝑒 −𝜆|𝑓𝑙| 𝑓𝑙𝑚𝑎𝑥 )𝐼( 𝑓𝑚𝑎𝑥 𝑙 ,𝐶𝜁𝑚𝑎𝑥 ]( ̄𝐶 𝑖). (7)
ThedistributioninEq.(7)reflectsdiscreteandfinitecapacitiesas seeninrealsystems.Ourlinepowerflow-capacityratiosaredistributed inasimilarwaytothePolishgridasareference,andareinitially𝑁−1 compliant.Thedemandingcomputationhereisfor𝑓𝑚𝑎𝑥
𝑙 ,asoneneedsto computethepowerflowforanetworkwithoneofitslinesremoved,for allofitslinesinallthegeneratednetworks.Afterapplyingconsecutively alltheproceduresjustdescribed,acompletesetofsyntheticDCpower grid modelsN isobtained whoseelements ∈ Naredefinedas in Eq.(1).
4. Cascadingfailuremodel
Inthefollowingsectionsweexplainthecascadingfailuremodel em-ployedtostudyallthepowergridconfigurationsinN.
4.1. Algorithmforcascadingfailureprocess
ThegeneralstepsareillustratedinAlgorithm4.1(Cascadingfailure simulator,CFS)basedonBienstock[2].Thealgorithmreceivesasinput apowergridobject,andtheindicesoftheinitialcoupleofline fail-ures(li,lj)tostudyN-2reliabilitycompliance.Then,Step1computes thepowerflowattheequilibriumstatebeforeinitialfailures.Step2 ap-pliestheinitialfailuretothepowergridbyremovingthetargetedlines. Then,eachiterationroftheForloopcorrespondstoafailureeventin thecascadesimulation[i.e.,whensomelinesbecomeoverloaded,with 𝑟=1correspondingtotheinitialfailureof(li,lj)].Iftheinitialfailure
doesnotleadtoacascade,thenCFS4.1stopsat𝑟=1.Step3is neces-saryinordertodealwithislandingwhenlinefailuresbreaktheoriginal networkGintomultipleconnectedcomponentswhichmighthavean imbalanceinpowersupplyordemand[34].Step3ishandledbyaDC re-dispatchlogicthatwillbeexplainedinSection4.2.Attheendofthe simulation,systemicmetrics𝑃(𝑙𝑖,𝑙𝑗)
𝑙𝑜𝑠𝑠 and𝑁
(𝑙𝑖,𝑙𝑗)
𝑓𝑎𝑖𝑙 arecomputed,andafter thealgorithmhasrunforallthecouplesoflinesofthenetwork,Ptotand NtotarecomputedasinEqs(2)and(3).
4.2. Re-dispatchlogic
During a cascading failure it is possible that an originally con-nected power network becomes separated in v subnetworks or is-lands: 𝐼𝑠𝑙1,𝐼𝑠𝑙2,…,𝐼𝑠𝑙𝑣 [2].Each island,being asubsetof the orig-inal power grid , is defined consistentlywith Eq. (1) as follows: 𝐼𝑠𝑙𝑖=(𝑉𝐼𝑠𝑙𝑖,𝐸𝐼𝑠𝑙𝑖,𝒁𝐼𝑠𝑙𝑖,𝑷𝑫𝐼𝑠𝑙𝑖,𝑷𝑮𝐼𝑠𝑙𝑖,𝑷𝑴𝑨𝑿𝐼𝑠𝑙𝑖,𝑪𝐼𝑠𝑙𝑖)⊂ .In prin-ciple,islandsdonothavebalancedpowersupplyanddemand,thus re-quiringareadjustmentinvectorsPG,PDthroughanoperationcalled re-dispatch.Re-dispatchisnotapropertyofthepowergridstructureof Eq.(1),andhenceitisembeddedinStep3oftheCFSAlgorithm4.1. Wecouldhavechosenothercontrolactions;however,re-dispatchisstill oneofthemostcommonoperationalactionsusedtoreducecostsand counteractlineoverloads.Meanwhile,otheroperationssuchas trans-missionswitchingaremainlyappliedforplannedoutagemanagement andcostsreduction,buttheirimpactonreliabilityisonlystartingtobe understood[35].Hence,ateachiterationrofanoverloadingevent,the proportionalroutineadjustspowerin𝐼𝑠𝑙1,𝐼𝑠𝑙2,…,𝐼𝑠𝑙𝑣asfollows: ∀𝑖∈𝑉𝐼𝑠𝑙𝑗∶ 𝑃𝐺𝑟
𝑖=𝛼𝑖𝑃𝐺𝑟𝑖−1+𝛾𝑖, 𝑃𝐷𝑟𝑖=𝛽𝑖𝑃𝐷𝑖𝑟−1 (8)
where𝛼i,𝛽iand𝛾itakedifferentvaluesdependingonparticular situa-tions: 𝛼𝑖= 𝑃𝐷𝑟−1 𝑗 𝑃𝐺𝑟−1 𝑗 , 𝛾𝑖=0, 𝛽𝑖=1𝑖𝑓𝑃𝐺𝑟𝐼𝑠−1𝑙𝑗≥𝑃𝐷 𝑟−1 𝐼𝑠𝑙𝑗 (9) 𝛼𝑖=1, 𝛾𝑖= ( 𝑃𝑀𝐴𝑋𝑖−𝑃𝐺𝑟𝑖−1 )( 𝑃𝐷𝑟−1 𝐼𝑠𝑙𝑗−𝑃𝐺 𝑟−1 𝐼𝑠𝑙𝑗 ) 𝑃𝑀𝐴𝑋𝐼𝑠𝑙𝑗−𝑃𝐺𝑟 −1 𝐼𝑠𝑙𝑗 , 𝛽𝑖=1𝑖𝑓𝑃𝑀𝐴𝑋𝐼𝑠𝑙𝑗≥𝑃𝐷𝐼𝑠𝑟−1𝑙 𝑗≥𝑃𝐺 𝑟−1 𝐼𝑠𝑙𝑗 (10) 𝛼𝑖=𝑃𝑀𝐴𝑋𝑖 𝑃𝐺𝑟−1 𝑖 , 𝛾𝑖=0𝛽𝑖= 𝑃𝑀𝐴𝑋𝑟−1 𝐼𝑠𝑙𝑗 𝑃𝐷𝑟−1 𝐼𝑠𝑙𝑗 𝑖𝑓𝑃𝐷𝑟−1 𝐼𝑠𝑙𝑗≥𝑃𝑀𝐴𝑋𝐼𝑠𝑙𝑗 (11) where𝑃𝐺𝑟−1 𝐼𝑠𝑙𝑗 = ∑ 𝑖∈𝑉𝐼𝑠𝑙𝑗𝑃𝐺 𝑟−1 𝑖 ,𝑃𝐷𝑟𝐼𝑠−1𝑙𝑗= ∑ 𝑖∈𝑉𝐼𝑠𝑙𝑗𝑃𝐷 𝑟−1
𝑖 arethetotalpower generatedanddemandedinislandjbeforetheoverloadingeventr.Also, 𝑃𝑀𝐴𝑋𝐼𝑠𝑙𝑗=
∑
𝑖∈𝑉𝐼𝑠𝑙𝑗𝑃𝑀𝐴𝑋𝑖
isthetotalpowercapacityof theisland. Thisproportionallogicissimplebynottakingintoaccountthe capaci-tiesofthesurvivinglinesinthesystemandisclosetoa“no-redispatch simulation”,particularlyasitmaintainspowerbalanceinthenetwork withlimitedoptimizationsteps,interveningonlyincaseofislanding. Proportionalpowergenerationisstillfrequentlyusedinpracticeasin thecaseofgeneratordisconnections[34].Inthecaseoffastcascading failures,therecouldbeinsufficienttimetocarryoutmoresophisticated thanproportionalinterventions.Hence,theproportionallogicoffersa suitablebaselineforcomparisonwithdesiredoptimizationbased strate-gies.
Incontrast,theOPAmodelre-dispatchespowerbysolvingthe fol-lowinglinearprogramforeachisland𝐼𝑠𝑙1,𝐼𝑠𝑙2,…,𝐼𝑠𝑙𝑣:
∀𝑗∈ {1,2,…,𝑣}solve: min Δ𝑝𝑔,Δ𝑝𝑑 ∑ 𝑖∈𝑉𝐼𝑠𝑙𝑗 Δ𝑝𝑔𝑖+100 ∑ 𝑖∈𝑉𝐼𝑠𝑙𝑗 Δ𝑝𝑑𝑖 (12) Subjectto: ∀𝑖∈𝑉𝐼𝑠𝑙𝑗 ∶ −𝑃𝐺𝑟−1 𝑖 ≤Δ𝑝𝑔𝑖≤𝑃𝑀𝐴𝑋𝑖 (13)
P loss P( F D C-1 los s ) N fail N ( F D C-1 fa il )
Fig.3. Probabilisticresponsevariablesasafunctionoftopologyvariationsgovernedby𝜁:(a)log-logplotofthetaildistributionsofPloss,and(b)semi-logplotfor thetaildistributionofNfail.
∀𝑖∈𝑉𝐼𝑠𝑙𝑗 ∶ 0≤Δ𝑝𝑑𝑖≤𝑃𝐷𝑟−1 𝑖 (14) ∑ 𝑖∈𝑉𝐼𝑠𝑙𝑗 (Δ𝑝𝑔𝑖−Δ𝑝𝑑𝑖)= ∑ 𝑖∈𝑉𝐼𝑠𝑙𝑗 (𝑃𝐺𝑟−1 𝑖 −𝑃𝐷𝑖𝑟−1) (15) ∀𝑙∈𝐸𝐼𝑠𝑙𝑗∶ −𝐶𝑙−𝑓𝑙+𝜀𝑙𝐶𝑙≤𝑩𝑙⋅(𝚫𝒑𝒈−𝚫𝒑𝒅)≤𝐶𝑙−𝑓𝑙+𝜀𝑙𝐶𝑙 (16) whereBl·isthelthrowofthesusceptancematrixBofthenetwork.The objectivefunctioninEq.(12)allowstore-dispatchpowergiving prior-itytogeneratorsbypenalizingpowerdemandmodifications,asavoided wheneverpossibleinpractice.ConstraintsinEq.(13)makesurethatthe powersuppliedbyeachgeneratorisalwaysbelowitsmaximum capac-ity,whileconstraintsinEq.(14)boundthepowerdemandedfromeach loadtoitsinitialvalue(whenthegridisinequilibrium).Constraintsin Eq.(15)ensurethatineachisland,supplyanddemandarebalanced, andconstraintsinEq.(16)forcethere-dispatchactiontorespectthe survivinglines’ capacities.Sinceinrealsituationsthethermalratingof transmissionlinesisneverknownexactlyanddependsonexternal con-ditionsaswellasintrinsicpropertiesofthematerialandshapeofthe conductors,asmallerrornoiseɛlClisintroducedinthenominalcapacity valueCl.ɛl israndomlysampledfromauniformdistributionbetween −0.01and0.01everytimetheoptimizationisperformed. After solv-ingthelinearprogram,thepowersupply/demandofeachbusinIsljis adjusted:
𝑃𝐷𝑟
𝑖=𝑃𝐷𝑟𝑖−1−Δ𝑝𝑑𝑖∀𝑖∈𝑉𝐼𝑠𝑙𝑗 (17)
𝑃𝐺𝑟
𝑖=𝑃𝐺𝑟𝑖−1+Δ𝑝𝑔𝑖∀𝑖∈𝑉𝐼𝑠𝑙𝑗 (18)
TheOPAmodelisa simplificationof howapowergrid operator mightinterveneinre-dispatchingpower.Ithasbeenusedtosolve opti-mizationproblemsrelatedtothepreventionofblackouts[36],aswell astostudythevulnerabilityofinterdependentsystems[37].
5. Computationalexperimentsandresults
5.1. Impactoftopologyoncascadefailurevulnerability
Fig.3showsthelog-logandsemi-logplotsofthetaildistributionof responsevariablesPlossandNfailfortheoriginalsyntheticpowergrids andthetwoboundingtopologiesderivedinSection3.3,representing variouslineredundancyfactors𝜁.
Asexpected, thecomplementary cumulativedistributionfunction (ccdf) for baseline networks (𝜁 =0) is halfway vulnerable between powerdeliveryinterruptionsinthetreenetwork(𝜁 =−1)andthegreedy triangulation(𝜁 =1)[Fig.3(a)].For𝜁 =−1,thepowergridsshowhigh probabilities of power losses, as the probability of Ploss>102MW is above50%.Sincepowerlossesdependonislandingandbus disconnec-tionsinourmodel,suchstrategiesareatoddswhen𝜁 =−1asonly min-imalsetsoflinesnecessarytoensureconnectivityarepresent.Hence, eachlinefailureissufficienttosplitthegridsintomultipleconnected componentsand,dependingonthepositionofgenerators,causepower losses.When𝜁 =0and𝜁 =1,topologiesarelesssensitivetolinefailure givenmorealternativepathstosatisfydemands,andinmostcasesfew failuresdonotaltersystem-levelfunctionalpathways.Infact,the prob-abilityofhavingPloss>102MWafteradoublecontingencyis approxi-mately1%for𝜁 =0,andbelow0.01%for𝜁 =1.However,notethat thetailoftheccdffor𝜁 =0stillreachesvaluescomparabletotheworst scenariosinthetreenetworks,signalingundesirableconfigurations al-thoughatamuchlowerfrequency.Nosuchconfigurationispresentin thegreedytriangulationgiventheshorttail.Themaximumpowerlosses forthethreedifferentnetworkgroupsare:3,879MWfor𝜁 =−1,3,098 MWfor𝜁 =0and598MWfor𝜁 =1onatotalinitialpowerproduction of4,377MW.
TheroleoflineredundancyisdifferentwithrespecttoNtot,where Fig.3(b)showsthatthe𝜁 =0topologieshavehigherprobabilitiesof additionallinefailuresaftertheinitialdoublelinecontingency.Thisis becauselineoverloadsaffectlinesbelongingtoalternativepaths,and sincein𝜁 =−1configurationsonlyonepathexistsbetweeneachcouple ofbuses,nooverloadduetopowerflowre-distributionispossible.In contrast,the𝜁 =0 and𝜁 =1topologiescanbesubjecttolinefailures whenthepowerflowisre-directed.ThetaildistributionofNfailishigher for𝜁 =0than𝜁 =1showingthatNfailisnon-linearandnon-monotonic withrespectto𝜁.The𝜁 =1networkshaveamuchhighernumberof alternativepaths thanthe𝜁 =0 configurations, thusthepowerflow re-directed bytheinitiallinefailuresisdispersedamongalargerset ofpossiblealternativeroutes.Specificresultsshowthatthemaximum numberofconsecutiveoverloadsforthethreedifferentstructuresis4 for𝜁 =−1,28for𝜁 =0and15for𝜁 =1.
5.2. Impactofgeneratorspositions
Regardingtheelectricalstructure,thereisimpactfromdifferent po-sitionsofgeneratorbusesviatheKfactor(spanningfromacentralized
P( F D C-1 los s ) K = 0 K = 0.5 K = 1 N fail N ( F D C-1 fa il ) K = 0 K = 0.5 K = 1
Fig.4. ProbabilisticresponsevariablesasafunctionofsitingfactorK:(a)log-logplotofthetaildistributionsofPloss,and(b)semilogplotforthetaildistribution ofNtot.
P to
t
Nto
t
Fig.5.Multivariatechartofthegroupaverageof(a)Ptot,and(b)Ntotwithrespectto𝜁 andK.
powersupplylayouttoadistributedone).Fig.4showsthelog-logplot oftheempiricaltaildistributionsforPlossandsemi-logofthesame quan-tityforNfail forthedifferentlevelsoffactorK—hereweconsidereda totalof360cases.
Fig.4(a)and(b)showthattheprobabilityofbothhighpowerlosses andadditionallinefailuresdecreasesasKincreases(i.e.,aswemove fromacentralizedstructuretoadistributedone).Infact,centralized powergenerationstructureshaveimportanttielineswhichconnectthe powersupplyagglomerateswiththepowerdemandsites.Failingthese tielinescausemorewidespreadpowerinterruptionsasreflectedinthe highervaluesofexpectedPloss.Incontrast,thedistributedgeneration cases(𝐾=1)aremorerobustsincethesupplyanddemandbusesare spreadthroughthenetworkandnot separablebyafewlinefailures. Thesegeneralconsiderationsdonotapplyequallytoeverytopological type,asonecanseefromthemultivariatechartforthe𝜁 × Kgroups inFig.5.Asexpected,𝜁 =−1networksaregreatlyinfluenced bythe sitingofgenerators.Infact,incentralizedgeneratorconfigurationsthe numberoftielinesisminimalforthistypeofnetwork.Eventheinitial doublecontingency,insomeconfigurations,candisconnectentirelythe supplynodesfromthedemandnodes.Theothernetworktopologiesare notsosensitive,thankstotheirN-1compliantdesign,asevenwhenall
generatorsareclusteredtogether,thereisenoughredundancyinthetie linestopreventthecompletepowersupply/demanddisconnection(see Fig.2forclusteringexamples).
ThelayoutofloadandgeneratorsseemtoaffectthelineoutagesNtot inallconfigurations,especiallythe𝜁 =0case.For𝜁 =−1thedistributed generatorsslightlyincreasethequantityoflineoutagesinthenetworks Ntot—anoppositetrendwithrespectto𝜁 ={0, 1}.Overall,onenotices thattheroleof𝜁,atlowtomediumlevelsthatcapturemostrealistic systems,iscriticalwithrespecttoKindeterminingthebehaviorofboth powerlossesandlinefailures.
5.3. Impactofre-dispatchpolicy
Inadditiontothetopologicalstructureandelectricalfeaturesofa powergrid,there-dispatchofpowergenerationisoneofthemost fre-quentlyusedcontrolactionstakenbyoperatorstodecreaseoperational costs andincrease reliability(thus essential in thecase of islanding [34]).Whencontingenciesstrike,re-dispatchisalsoappliedasa mit-igationactiontoavoidadditionalcascadefailures;however,insome realfast-evolvingcases,wrongre-dispatchcanactuallyworsenthe sit-uation[38]. Fig.6shows thelog-logplotof thetaildistributionfor
Fig.6. Probabilisticresponsevariablesasafunctionofre-dispatchstrategy:(a)log-logplotofthetaildistributionsofPloss,and(b)semi-logplotforthetail
distributionofNtot.
Fig.7. Multivariatechartforidentifyinginteractionsbetween(a)re-dispatchandtopology,and(b)re-dispatchandgeneratorsitingK.
Plossandthesemi-logplotofthetaildistributionforNfail,acrosscases whichsharethesamere-dispatchprocedure.Inparticular,weconsider aproportionalprocedureandtheOPAmodel.Theimpactofthemore sophisticatedOPAcontrolstrategyinpreventingoverloadsisclearin Fig.6(b).Infact,differentfromtheproportionalmodelwhichtriesto accommodatethedemandinagreedyway,theOPAmodelsacrifices powerdemandinordertopreventlinefailureswheneverthelinear pro-gram(12)–(18)isfeasible,thuspreventingcascadingfailurestospread. RegardingPloss,OPAseemstosucceedinreducingpowerlossesinthe mostextremescenarios,butdoesnothavesignificantimpactonsmall andmediumpoweroutagesasevidencedinFig.6(a).Thisisbecause theOPApolicyfocusesonoptimizingpowerlossesundertheconstraint ofnoadditionaloverloads(Eq.(16)),thusavoidingextendedcascades whichrepresentstheextremescenarioswherelargeamountsofpower arelost.
Fig.7showsthemultivariatechartforNtot,comparingthemean ef-fectofthecombinationsofthere-dispatchfactorwithtopologyand gen-eratorposition.TheOPAre-dispatchsuccessfullypreventslinefailures inalldifferenttopologiesandalsowithrespecttoallgenerator config-urationsinK.Overall,thebestimprovementisobtainedinthe interme-diatelineredundancylevel𝜁 =0whichwasidentifiedasmore vulner-abletolineoverloads(andtheclosesttoredundancylevelsinpractice).
Meanwhile,Ptotoffersnodiscernibleinteractionpattern,mainlybecause asnotedinFig.6,themaineffectofthere-dispatchfactoronPtotisitself weak,influencingtheoutcomesofonlyrarescenarios.
6. Insightsforpowergriddesignandoperation
Thisworkshowsthatlineredundancyandgenerator/loadlayout fac-torshaveasignificantimpactontherobustnessofpowergridstopower lossandcascadinglinefailures.Fromthelineredundancyperspective, aminimumamountofredundanttransmissionlinesisclearlynecessary toavoidpowerinterruptionseverytimealinefails.AsshowninFig.3, iftopologiesaredenserthanthe𝜁 =−1case(approachingthe𝜁 =0 lay-out),powerlossesdecrease,althoughitisstillpossibletooverloadlines due topowerflowre-distribution,andleadtowidespread blackouts. Atanextreme,ifaconsiderablenumberofredundantlinesareadded (approachingthe𝜁 =1layout),itbecomespossibletodoboth: signif-icantlydecreasetheoverloadfrequencyofthetransmissionlinesand avoidlargeblackouts.Inpractice,however,buildingnewtransmission istooexpensivetopursuea𝜁 =1 powergrid.Therefore,weidentifya baselinecombinationofthetopologicalfactorsofredundancyand gen-eratorlayoutwhich,whilebeingattainableinpractice,stillachievea goodlevelofprotectiontocascadingfailures.Tothisend,weperform
Fig.8. Confidenceintervalsofthegroupmeanof(a)Ptotand(b)Ntotfordifferentlineredundancyvalues.
Pto
t
Nto
t
Fig.9. Confidenceintervalsofthegroupmeansof(a)Ptotand(b)Ntotfordifferentlineredundancyvaluesandgeneratorlayouts.
additionalsimulationswithpowernetworkswithintermediate redun-dancylevels,in-between𝜁 ={−1,0, 1}.Sincethedesignofthecontrol strategycanbecarriedoutindependentlyfromthechoicesof𝜁 andK, weperformsimulationswiththeproportionalre-dispatchonly.Inthis wayweobtainlowerboundsonthepowernetworkrobustnessthatcan thenbefurtherimprovedbyapplyingamoresophisticatedcontrol strat-egy,suchasOPA(Figs.6and7).Fig.8(a)and(b)showtheconfidence intervalsforthemeanofPtotandNtotaswevarytheredundancylevel ofthebaselinecase(0onthex-axis)toward−1or1withproportional re-dispatch.
Fig.8(a)showsthataddinglinesalwayshelpssavingpowerlosses asPtotismonotonicallydecreasing.ThebehaviorofNtotismore com-plexandnotmonotonicasalsoevidencedinFig.3.Havingonlyafew redundantlinesabovetheSTconfigurationleadstohighlyvulnerable topologies,whilecontinuingtoaddmorelinesafterthepeak,slowly improvesthesituation(Fig.8(b)).Also,after25%ofthelineswhich separateGTfromGareaddedtothelatter,significantimprovementis achieved,butbeyondthatpointimprovementismarginal.Hence, build-ingnewtransmissionlineswouldnotbethemosteconomicandtime efficientstrategytoreducethepowergridriskofcascadingfailuresfor densertopologies,someasuresrelyingoncontrolstrategiesand/or dis-tributedgeneratorsshouldalsobeincluded.
Consideringthelayout of loadsandgeneratorswe show they af-fectmainlyNtot.Onaverage,moredistributedconfigurationsrenderless
powerlossandnoticeablysmallercascadingfailures.Themagnitudeof theimprovementsdependsontheredundancylevelofthenetwork.In particular,thelowerthenumberofredundantlinesinthegridbeforee becoming trees,themoresensitivetheresponsetogeneratorlayout. Fig.9showstheconfidenceintervalsforthemeanofPtotandNtotaswe varytheredundancylevelofthebaselinetopologyforthethree differ-entgeneratorslayouts.LookingattheconfidencebandsinFigs.8(b)and 9(b),wenoticehowforNtottheintervalsarewiderforsparsetopologies andclusteredgeneratorslayouts.
Theseresultssuggestthatdispersingpowergeneratingunitsacross thepowergridhelpstoavoidlineoverloadswheneverthenumberof redundantlinesisnotsufficient.Thistypeofinterventionishappening ad-hocviasmartgridtechnologiesanddistributedgeneration(DG) to-day.Mostlikely,combinationsoftransmissionlinebuildupandDG, which requirenovelcontrols,offera foreseeablesolutiontomanage powerlossesandcascadesinevolvingpowersystems.
Regardingre-dispatch,wefindithasadeterminantroleinavoiding theoverloadofadditionaltransmissionlinesafterinitialcontingencies materialize.Moreover,correctivere-dispatchactionsinteractina syn-ergisticwaywiththepowergridtopologies,bringingweaker configu-rationsatapproximatelythesamelevelofthestrongerones.Inpower gridsvulnerabletooverload,theoptimizedOPAmodelcouldgenerally avoidlargescalecascadesandassociatedblackouts,relativetosimpler proportionalstrategies.
7. Summaryandconclusions
Thisstudydevelopsnewstrategiestoevaluatecascadingfailure dy-namicsthroughabroadsetofrealisticpowergridtopologiesoperated withdifferentpowerre-dispatchstrategies.Cascadingfailuresare sen-sitivetotheinitialpowergridtopology,supply/demandstates,andthe controlactionsappliedwhilecascadesevolve.Mostexistingworkis spe-cifictocasestudysystemswithsettopology,andthusconclusionsare typicallynotgeneralizable.Theapproachtakeninthisworkisinstead toexplorekeytopological,electricalandcontrolinputsacrossrealistic powergridsinordertofindparametersettingstosafeguardgridsfrom cascadingfailures.Thequantitativeresultsdrawnfromthisexploration aretranslatedintohigh-levelguidelinesforreliability-basedpowergrid designthataregeneralizable,sincetheglobal-to-localcomputational strategyemployedhereisprobabilisticandaccommodatesanensemble oftopologicalandelectricalsystemconfigurations.
Resultsshowthatforreducingtheprobabilityoflargepowerlosses, improvements atthestructure/layout levels arenecessary,including strategiesthatincreaselineredundancy,decentralizegenerators,oruse combinationsofthem(atlevelsnottoo distantfromrealisticsystem configurations).Meanwhile,ifmajorrisksderivefromtoofrequent over-loads,itisbettertoinspectthecontrolpoliciescurrentlyemployedin thesystemandassessifitisnecessarytoimproveoroptimizethem. Inparticular,powerre-dispatchshowsthatlineoverloadcontainment throughoptimizedloadsheddingandpowergenerationre-scheduling iseffectiveinpreventthepropagationoffailuresinallthepowergrid configurationsconsidered.
Overallresultsshowthatthebestpracticalsolutiontotheproblemof minimizingtheprobabilityofpowerlossesandmultiplelineoverloads istohaveatopologyslightlymoreredundantthantheaveragelevel forrealisticsystems0<𝜁 <0.25,coupledwithdistributedgenerators layoutK>0.50.Inadditiontothistopologicalsolution,anoptimized re-dispatchstrategywouldbringevengreaterbenefitsbycurtailingthefew cascadingfailuresthatstillcanbreacharobustpowernetworkdesign. Also,increasedlineredundancyandgeneratordecentralizationlevels reducethevariabilityofperformancemeasures,suchasthenumberof linesoverloaded,furtheringthemanageabilityofcomplexpower sys-tems.
Futureresearchincludesextendingthecomputationalexperiments presentedinthisworkbyconsideringACpowerflow[34]andother formsofcontrolactionssuchasthosethatmodifythetopologyofthe grid[35]. Moreover,insteadofconsideringsingleordouble contingen-cies,theinitialfailureeventscouldbegeneratedwithhazard-consistent simulationapproaches,soastorepresentriskduetoextremeeventsand quantifytheassociateduncertaintyinthecascadingdynamicsofpower gridsystems.Andriskanalysesthatexploitthehierarchicalstructureof networksalsoofferuntappedopportunitiesforpowersystems[39].
Notationlist:
Factors
𝜁: lineredundancyfactor K: generatorlayoutfactor R: re-dispatchfactor
Sets
N: setofpowergridobjects
E: setofarcsofthegraphrepresentingthepowergridtopology V: setofverticesofthegraphrepresentingthepowergrid
topol-ogy
Objects:
: powergridobject
0: powergridobjectatinitialstate
𝑟: powergridconfigurationaftertherthlinefailureevent
Powergridelectricalparameters:
Z: vectorcontainingtheimpedancesofthearcs
PD: vectorcontainingthepowerdemandsforbusesinthepower grid
PDi: powerdemandforbusi
PG: vectorcontainingthepowersuppliesforbusesinthepower grid
PGi: powersupplyforbusi
PMAX: vectorcontainingthepowersupplieslimitsforbusesinthe powergrid
PMAXi: powersupplylimitforbusi
C: vectorcontainingthecapacitiesforthearcsinthepowergrid
Powerflowandcapacityallocationparametersandvariables:
l: lineofthepowergriddefinedasanarcwithimpedanceand capacity(e,Z,C)
fl: powerflowingthroughlinel 𝑓𝑖
𝑙: powerflowingthroughlinelwhenonlylineiisfailed 𝑓𝑚𝑎𝑥
𝑙 ∶ maximumpowerflowingthroughlobtainedfrom𝑁−1 con-tigencyanalysis
fr: powerflowvectorassociatedwith𝑟
̄𝐶𝑖: ithsmallercapacityvaluethatcanbesampledfromthe ca-pacitydistribution
𝐶𝑚𝑎𝑥
𝜁 ∶ maximumcapacity valueallowedfor agivenvalueofline redundancy𝜁
Re-dispatchparametersandvariables:
𝛼i: coefficientforproportionalpowergeneratoradjustmentfor busi(proportionalre-dispatch)
𝛾i: constantcoefficientforpowergeneratoradjustmentforbusi (proportionalre-dispatch)
𝛽i: coefficientforproportionalpowerdemandadjustmentforbus i(proportionalre-dispatch)
𝚫pg: powergenerationadjustmentvector(OPAre-dispatch)
𝚫pd: powerdemandadjustmentvector(OPAre-dispatch) Δpgi: powergenerationadjustmentforbusi(OPAre-dispatch) Δpdi: powerdemandadjustmentforbusi(OPAre-dispatch)
B: susceptancematrix
ɛl: noiseinthecapacitynominalvalueforlinel
Responsemeasures:
𝑃(𝑙𝑖,𝑙𝑗)
𝑙𝑜𝑠𝑠 ∶ amountofpowernotsuppliedatthefinalstatewhenli,ljare selectedasinitialfailures
𝑁(𝑙𝑖,𝑙𝑗)
𝑓𝑎𝑖𝑙 : amountoflinesfailedatthefinalstatewhenli,ljareselected asinitialfailures
Ptot: totalpowerlossacrossN-2contingencies Ntot: totalcascadinglinesacrossN-2contingencies
Acknowledgments
Theauthorswouldliketoacknowledgethesupportofthisresearchby thePolitecnicodiMilano,RiceUniversity,theU.S.NationalScience Foundation (GrantsCMMI-1436844andCMMI-1541033),theU.S. DepartmentofDefense throughitsArmyResearch Office(Grant W911NF- 13-1-0340),andtheDataAnalysisandVisualization Cyberinfrastructure fundedbyNSF undergrantOCI-0959097.
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