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Electricity Blackout and Power Security

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Title:
Electricity Blackout and Power Security Survey and Analysis
Creator:
Xu, Hongsheng
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
Publication Date:
Language:
english
Physical Description:
1 online resource (93 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Industrial and Systems Engineering
Committee Chair:
Pardalos, Panagote M
Committee Members:
Boginski, Vladimir L.
Lan, Guanghui
Hager, William W
Graduation Date:
8/11/2012

Subjects

Subjects / Keywords:
Algorithms ( jstor )
Blackouts ( jstor )
Buses ( jstor )
Economic models ( jstor )
Electric power ( jstor )
Linear programming ( jstor )
Modeling ( jstor )
Power grids ( jstor )
Power lines ( jstor )
Transmission lines ( jstor )
Industrial and Systems Engineering -- Dissertations, Academic -- UF
contingency -- grid -- power -- security
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Industrial and Systems Engineering thesis, Ph.D.

Notes

Abstract:
Electricity power systems are critical to any country’s economy and security. Blackout, the most severe form of power loss to a relatively wide area, posting tremendous societal consequences, could be resulted from different causes from failure fault at a power transmission line to an intentional attack. Therefore, it is of importance to study the behavior of power grid(a complex network) during power blackout with its failure pattern. Based on these findings, we can find the suitable plan for normal daily management/expansion, emergency response and restoration after blackout. However, there are many schools of scholars using different ways to model the Blackout phenomenon, with different conclusions. Since those researches are done over several decades, it is important to organize them in a systematic way so that it could help the future research. Also, several minmax optimization models are proposed with different scenarios to represent the impact of blackout. ( en )
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2012.
Local:
Adviser: Pardalos, Panagote M.
Statement of Responsibility:
by Hongsheng Xu.

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UFRGP
Rights Management:
Copyright Xu, Hongsheng. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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LD1780 2012 ( lcc )

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ELECTRICITYBLACKOUTANDPOWERSECURITY:SURVEYANDANALYSISByHONGSHENGXUADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012HongshengXu 2

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Thisworkisdedicatedtomyparentswhohavesupportedmeinallmyendeavors. 3

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ACKNOWLEDGMENTS First,IwouldliketothankDr.PanosM.Pardalos,thechairmanofmygraduatecommittee,forprovidingmethisopportunitytoworkunderhisadvisingonthisenergytopic.IappreciatehisguidanceandhelpthroughoutmyPh.D.degreestudies.IamgratefultoDr.WilliamW.Hager,Dr.GuanghuiLan,andDr.VladimirBoginski,membersonmygraduatecommittee,fortheirvaluabletimeandattention.IwouldalsoliketothankallmyfriendswhohavebeenprovidingmewithacademichelpandmoralsupportthroughoutmystudiesinDepartmentofIndustrialandSystemsEngineeringatUniversityofFlorida.IextendmyspecialthankstoDr.AmarSapra,whomitisthatinspiredmyenthusiasmofpursuingaPh.D.degreehere.Aboveall,Iwouldliketothankmyfamilyfortheircontinuousfaith,loveandsupportonme.Itistheirbless,careandencouragementthatgivemethemotivationtoachievemygoals. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 10 2BLACKOUTANALYSISSURVEY .......................... 15 2.1Background ................................... 15 2.2ProbabilisticandReliabilityModel ...................... 15 2.2.1CascadingProperty .......................... 15 2.2.2CASCADEModel ............................ 16 2.2.3BranchingProcessModel ....................... 18 2.2.4SimulationandComplexSystemModel ............... 19 2.3OptimizationModel ............................... 24 2.3.1Pre-OptimalPowerFlowModel .................... 25 2.3.2OPFModel ............................... 25 2.3.2.1Linearprogramming(LP)andquadraticprogramming(QP) .............................. 28 2.3.2.2Nonlinearprogramming(NLP) ............... 29 2.3.2.3Integerprogramming(IP)andmixed-integerprogramming(MIP) ............................. 29 2.3.2.4Dynamicprogramming(DP) ................ 30 2.3.3UnitCommitment ............................ 30 2.3.3.1Dynamicprogramming(DP) ................ 32 2.3.3.2Dynamicandlinearprogramming ............. 33 2.3.4ContingencyandInterdiction ..................... 33 2.4BlackoutwithDistributedGeneration ..................... 35 3KEYCONCEPTUSEDINTHISDISSERTATION ................. 37 3.1PowerFlows .................................. 37 3.2DCandACpowerows ............................ 38 3.3PreliminaryModel ............................... 39 3.3.1OptimalPowerFlowModel ...................... 40 3.3.2ContingencyModel ........................... 41 5

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4ECONOMICANALYSISOFTHEN)]TJ /F3 11.955 Tf 9.3 0 Td[(kPOWERGRIDCONTINGENCYSELECTIONANDEVALUATION .................................. 46 4.1Nomenclature .................................. 46 4.2Background ................................... 47 4.3OptimalPowerFlowModelforEconomicAnalysis ............. 50 4.4GraphAlgorithmsforContingencySelection ................. 51 4.4.1SelectionofKBusesbyCriticalNodeMethodandOtherMethods 51 4.4.2SelectionofKLinesbyEdgeBetweenness ............. 53 4.5AN)]TJ /F3 11.955 Tf 11.96 0 Td[(kContingencySelectionbyWorstCaseInterdictionAnalysis ... 54 4.6NumericalExperimentsandComparisons .................. 56 4.7ConclusiveRemarks .............................. 60 5EXTENSIONWORKOFECONOMICANALYSISOFTHEN)]TJ /F3 11.955 Tf 13.22 0 Td[(KPOWERGRIDCONTINGENCYWITHLINESWITCHING ................. 62 5.1Nomenclature .................................. 62 5.2Background ................................... 63 5.3OptimalPowerFlowModelforEconomicAnalysiswithLineSwitching .. 65 5.4NumericalExperimentsandAnalysis ..................... 66 6FUTURERESEARCH ................................ 67 6.1Nomenclature .................................. 68 6.2OptimalPowerFlowwithContingency .................... 69 6.3Analysis ..................................... 70 6.4PreliminaryResult ............................... 71 6.4.1CriticalSets .............................. 71 6.4.2SurvivabilityConstraintsandthePossibleCut ............ 73 REFERENCES ....................................... 81 BIOGRAPHICALSKETCH ................................ 93 6

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LISTOFTABLES Table page 6-1Contingencyselectionwith=0.0005 ....................... 75 6-2Contingencyselectionwith=0.001 ........................ 76 6-3Contingencyselectionwith=0.01 ........................ 77 6-4Contingencyselectionwith=0.1 ......................... 78 6-5Contingencyselectionwith=0.15 ........................ 79 6-6Contingencyselectionwith=0.2 ......................... 80 7

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LISTOFFIGURES Figure page 1-1Visualizingtheu.s.electricgrid,source:Americanelectricpower ........ 11 2-1Cascadingfailurepowersystemmodel ....................... 18 2-2Theloadredistributiontriggeredbyannode-basedattack ............ 22 4-1IEEE-30-bussystem ................................. 57 4-2Loadsheddingcostvs.failedbuses(IEEE-30-BusSystem) ........... 58 4-3Loadsheddingcostvs.failedlines(IEEE-30-BusSystem) ............ 58 4-4Loadsheddingcostvs.failedbuses/lines(IEEE-30-BusSystem) ........ 59 4-5Generatingandloadsheddingcostvs.failedbuses(RTS-96System) ..... 59 4-6Generatingandloadsheddingcostvs.failedlines(RTS-96System) ...... 60 5-1IEEE-30-bussystemwithlineswitching ...................... 66 6-1IEEE-300-bussystem ................................ 72 8

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyELECTRICITYBLACKOUTANDPOWERSECURITY:SURVEYANDANALYSISByHongshengXuAugust2012Chair:PanosM.PardalosMajor:IndustrialandSystemsEngineering Electricitypowersystemsarecriticaltoanycountry'seconomyandsecurity.Blackout,themostsevereformofpowerlosstoarelativelywidearea,postingtremendoussocietalconsequences,couldberesultedfromdifferentcausesfromfailurefaultatapowertransmissionlinetoanintentionalattack.Therefore,itisofimportancetostudythebehaviorofpowergrid(acomplexnetwork)duringpowerblackoutwithitsfailurepattern.Basedonthesendings,wecanndthesuitableplanfornormaldailymanagement/expansion,emergencyresponseandrestorationafterblackout.However,therearemanyschoolsofscholarsusingdifferentwaystomodeltheBlackoutphenomenon,withdifferentconclusions.Sincethoseresearchesaredoneoverseveraldecades,itisimportanttoorganizetheminasystematicwaysothatitcouldhelpthefutureresearch.Also,severalminmaxoptimizationmodelsareproposedwithdifferentscenariostorepresenttheimpactofblackout. 9

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CHAPTER1INTRODUCTION Sincetheindustrialage,human-beinghascreatedmanynetworks,whicharetightlyrelatedtoourdailylife.Thesenetworkscouldbecategorizedasphysicalorengineerednetworks,informationnetworks,biologicalnetworks,cognitiveandsemanticnetworksandsocialnetworks,whichforexampleincludebutnotlimittoelectricalnetwork,computernetwork,biologicalnetwork,articialneuralnetwork,socialnetwork,businessnetworking,radionetwork,telecommunicationsnetwork,televisionnetwork. Astheconceptofnetworkreacheseveryaspectsofthehumansociety,thenewdisciplineofnetworktheoryhasbeendevelopedtostudytheirinterconnectionphenomena,andtoexplorecommonprinciples,algorithmsandtoolsthatgovernnetworkbehavior.Severalparameterssuchasdensity,size,averagedegree,averagepathlength,diameterofanetwork,clusteringcoefcientandconnectedness,whicharerootedfromgraphtheory,havebeenutilizedtoanalyzethepropertiesandcharacteristicsofthenetwork.Basedondifferentparameters,genericnetworkcouldbedescribedbydifferentnetworkmodels,suchasSmallWorldmodel,Scale-Freenetworkmodel,PreferentialAttachmentmodelandSIRmodeletc.TheSmallWorldmodel,Scale-Freenetworkmodel,andPreferentialAttachmentmodelarerandomly-generatedgraphs,whilethersttwoobservethedegree-distributionttedwithapowerlawdistribution.TheSmall-Worldnetworkistheonewhichisstudiedmostlyandwidelysinceithasthesmall-worldpropertieswhicharefoundinmanyreal-worldphenomena,includingroadmaps,foodchains,electricpowergrids,metaboliteprocessingnetworks,networksofbrainneurons,voternetworks,telephonecallgraphs,andsocialinuencenetworks.Withthisproperty,networksarelikelytohavecliques,quasi-cliques,and/orclubs,meaninghighconnectivitybetweenanytwonodeswithinsub-networksduetoahighclusteringcoefcient,andatthemeanwhilemostpairsofnodeswillbeconnectedbyatleastoneshortpathviasomehubnodeswithahighdegree. 10

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Figure1-1. Visualizingtheu.s.electricgrid,source:Americanelectricpower Amongthoseexamplenetworks,electricitynetworkorpowergridhasbeenonthetabledrawinglotsofeyeballs,andonereasonisthatitistheoldestandmosttraditionalofthevariousmegastructures.Theelectricitynetworkhasbeenevolvingfromseveralclose-circuitsystemswithinsomeparticulargeographicareas,whereallenergywasproducedneartheend-userswhichdemandthatenergy.Duringthepastdecadesthepowerinfrastructurehasevolvedintowhatmanyexpertsconsiderthelargestandmostcomplexsystemofthetechnologicalage.Geographically,theNorthAmericanpowergridformsanetworkofover6100generationplants,over365,000mileofhighvoltagelines,over6millionmileoflowvoltagelinesandover45,000transformersthatarecontinuouslyregulatedbysophisticatedcontrolequipments.Consistingofmillionsofmilesoflinesoperatedbyover500companies,theContinentalU.S.powergrid(Fig. 1-1 )isoneoflargestelectricityinfrastructureintheworld,anditisacomplexnetworksystemofpowerplantsandtransmissionlineswhichareindependentlyownedandoperated.However,despiteoftheadvanceintechnologyanddesignoftheelectricalgrid,itspowerdeliveryinfrastructuressufferagingissueacrossthewholeworld.Agedequipmentsandfacilitieshavehigherfailureratesandsubjecttohighermaintenance 11

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andfurtherrepair/restorationcosts;agedareasrequireadditionalsubstationsitesmorethanavailabletomeetthedemandofevergrowingcustomers;problemscausedbyagedequipments&facilities,obsoletesystemlayouts,andmodernderegulatedelectricitydistributionpatternscouldnotbeeffectivelyaddressedbytraditionalconcepts,planning,engineering,operationofpowernetworksystems. TheotherreasonwhyelectricitynetworkissoimportantisthatitrecentlyattractedattentionfromtheHouseForeignAffairsCommitteeaboutitssecurityissue.Deliveryofelectricitypoweriscriticaltoanycountry'seconomyandsecurity.Asaresultoftherecentderegulationofpowergenerationandtransmission,aboutone-halfofalldomesticgenerationisnowsoldoverever-increasingdistancesonthewholesalemarketbeforeitisdeliveredtocustomers,andconsequentlythepowergridiswitnessingpowerowsinunprecedentedmagnitudesanddirections.Thereforesurgesinpowerlinescancausemassivenetworkfailuresandpermanentdamagetomultimilliondollarequipmentinpowergenerationplants. Theelectricalnetworkhasitsowncharacteristicstomakeitselfspecialthanothernetworks.First,ithastoobservethefollowingphysicallaws:(1)Kirchhoff'scurrentlaw:Thesumofallcurrentsenteringanodeisequaltothesumofallcurrentsleavingthenode;(2)Kirchhoff'svoltagelaw:Thedirectedsumoftheelectricalpotentialdifferencesaroundaloopmustbezero;(3)Ohm'slaw:Thevoltageacrossaresistorisequaltotheproductoftheresistanceandthecurrentowingthroughit;(4)Norton'stheorem:Anynetworkofvoltageorcurrentsourcesandresistorsiselectricallyequivalenttoanidealcurrentsourceinparallelwithasingleresistor;and(5)Thevenin'stheorem:Anynetworkofvoltageorcurrentsourcesandresistorsiselectricallyequivalenttoasinglevoltagesourceinserieswithasingleresistor.Second,electricenergyisinstantaneouslyconsumedanditiscurrentlytooexpensivetostore.Third,asthemoderntrendsinforthe21stcentury,theelectricutilityindustryseekstotakeadvantageofnovelapproachestomeetgrowingenergydemandwitheverythinginterconnected. 12

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Withinthiskindofwideareasynchronousgrid,alternatingcurrent(AC)withfrequenciessynchronizedcanbetransmittedthroughoutthewidearea,connectingalargenumberofelectricitygeneratorsandconsumers.However,insuchasynchronousgridallthegeneratorsrunnotonlyatthesamespeedbutalsoatthesamephase,andgenerationandconsumptionmustbebalancedacrosstheentiregrid.Hence,asinglefailureinalocalareacouldcausepowerowtore-routeovertransmissionlinesofinsufcientcapacity,whichmayresultinfurtherfailuresinotherpartsofthegrid,inotherword,thepossibilityofcascadingfailureandwidespreadpoweroutage. Aelectricityblackoutisthesituationwherethereisatotallossofpowertoarelativelywidearea,anditisthemostsevereformofpoweroutage.Therearemanycausesofblackoutinanelectricitynetwork,includingfaultsatpowerplantstations,damagetopowerlines,ashortcircuit,ortheoverloadingofelectricitytransmissionsystems.Blackoutsareespeciallydifculttorecoverquickly,andmaylastfromafewhourstoafewweeksdependingonthenatureoftheblackoutandthecongurationoftheelectricalnetwork.Restoringpowerafterawide-areablackoutneedstobedonewiththehelpofpowerfromothergrid.Intheextremecasewherethereistotalabsenceofgridpower,aso-calledblackstartneedstobeperformedtobootstrapthepowergridintooperation,whichdependsgreatlyonlocalcircumstancesandoperationalpolicies.Sincelocalizedpowerislandsareprogressivelycoupledtogether,inordertomaintainsupplyfrequencieswithintolerablelimitsduringthisprocess,demandmustbereconnectedatthesamepacethatgenerationisrestored,requiringclosecoordinationbetweenpowerstations,transmissionanddistributionorganizations. IntheUnitedStates,thesystem'svulnerabilitytophysicaldisruptionsfromnaturaldisastersandothercauseshaslongbeenstudied.However,thisvulnerabilityhasincreasedinrecentyearsbecauseinfrastructurehasnotexpandedasquicklyasdemandhas,therebyreducingthesystem'stoleranceagainstdeteriorated,failed,orunavailablesystemcomponents.Duringthemostsevereformofpoweroutage, 13

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Blackoutwhichisatotallossofpowertoarelativelywidearea,tremendoussocietalconsequencesandsubstantialeconomicallosswouldbeincurred.Thepossiblecausesforblackoutmaybefaultsatpowerstations,damagetopowerlines,ashortcircuit,ortheoverloading,theentireprocedurecouldbesocomplicatedduetothecascadingphenomenonwhichisresultedfromtheself-organizingdynamicalforcesdrivingthesystem.Moreover,thethreatofhumanattacksonthesystemhasbecomemoreserious,too.Therefore,themethodologyandalgorithmsproposedherecouldhaverichapplicationsusedindailyoperationmanagement,emergencystrategyandexpansionplanningforpowergrid. 14

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CHAPTER2BLACKOUTANALYSISSURVEY 2.1Background ThetheU.S.ElectricPowerSystemhasbeenservingthisnationformorethanacenturyeversinceThomasEdisondesignedandbuilttheworld'srstcentralpowerstationinNewYorkCityin1882.Today,however,theagedbutscatteredinfrastructurecombinedwithanincreasingdemandindomesticelectricityconsumption,whichareextremelyvulnerabletoevensmallscaleofunintentionaloutagefailure(saynothingofintentionalcontingency),hasforcedustocriticallyexaminetheconditionandhealthofthenation'selectricalsystemsbeforeitcouldbeoutdatedandunpreparedtodeliverreliableelectricitytotheconsumers.Therefore,thereisalotofacademicresearchwhichaimsatpredictingtheoccurrenceofblackoutandmitigatingtheimpactofblackoutbydesigningandoperatingpowersystemsinthewaythatthepowergridsystemcouldtoleratewithemergencyeventssuchasthelossofpowertransmissionroute,changinginthedemandandgenerationpatterns,etc. 2.2ProbabilisticandReliabilityModel 2.2.1CascadingProperty Insteadofconsideringthepowergridasanintegratedcomplexsysteminamacroscopicway,someresearchershavebeguntouseprobabilisticandreliabilitymodeltodescribethesysteminthemicroscopicway.MostofthepapersonprobabilisticmodelrelatedtopowergridblackoutarecontributedbyIanDobsonandhisco-authors,andtherearemainlythreemodelstheydevelopedand/orlargelyusedtoexploretheaspectsofblackouts.TheCASCADEandBranchingprocessmodelsareusedtorepresentthedynamicfeaturesofthecascadingfailureinatractableway,whiletheOPAmodeldescribesthepowertransmissionsysteminanabstractway. Thecascadingfailureisabuilt-inpropertyofapowergridsysteminwhichthefailureofacomponentcantriggerthefailureofsuccessivecomponentsbyload-shifting. 15

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Cascadingisthekeyfactorforlarge-scalepowerblackout,anditdeservesthecloseattentionpaidfrommanyresearchers. DistributionofIntervalbetweenBlackouts Differentpapershavemadecontraversaryconclusionsaboutthedistributionofthetimebetweenblackouts.Some([ 29 ],[ 58 ],[ 132 ])concludewithexponentialtailorsimilar,whileothersthinkaboutnegativebinomial[ 115 ]orPoissondistribution[ 71 ].Theprobabilitydistributionofthetimebetweenblackoutsisatleastdeterminedbytheprobabilityofthetrigger,andthetimebetweenblackoutsisamixtureofgammadistributions[ 58 ]. DistributionofBlackoutSize Themostquestionpeoplewouldaskwhenhearingofpowerblakcoutisaboutthesizeanditsdistribution.oneofthemostpossibleanswerwouldbetheexponential;analysisofNorthAmericanblackoutstatisticsfromNERC(NorthAmericanElectricReliabilityCouncil)DisturbanceAnalysisWorkingGroup(DAWG)Database( http://www.nerc.com )showthatithasanapproximatepowerlawregionwithanexponentbetween-1and-2([ 9 ],[ 29 ]).Thepowerlawimpliesthatblackoutscanoccurinallsizes,andmostimportantlyatallplaces.Differentpowersystemsindifferentcountries([ 21 ],[ 71 ],[ 131 ])showroughlysimilarformsofblackoutsizedistributioninpowerlawdependence.Dobsonandhiscoworkershavesomepapers([ 42 ],[ 133 ])estimatingtheaveragepropagationoffailuresandthesizeoftheinitialdisturbanceandtopredictthedistributionofblackoutsize. Self-organization elf-organizationistheprocesswhereastructureorpatternappearsinasystemwithoutacentralauthorityorexternalelementimposingitthroughplanning.Manyscholarshavearguedthat,throughcomputersimulationandhistoricaldata,powergridsareself-organizedcriticalsystems,inwhichtheevolvingprocessisquitecomplicated.BasedonasimplemodelofDCloadowandLPdispatchandNERCdataonNorthAmericanblackouts,([ 29 ],[ 28 ]),thedynamicsofblackoutshavesomefeaturesofself-organizedcriticalsystems[ 115 ]. Criticality Astheloadincreases,theaverageblackoutsizeincreasesveryslowly,until,ataloadingcalledthecriticalloading,thereisasharpchangeandaverageblackoutsizestartsincreasingmuchmorequickly([ 27 ],[ 78 ],[ 97 ]). 2.2.2CASCADEModel TheCASCADEmodelconsistsofnitelylargenumberofcomponentswithafailurethreshold,aninitialsystemloading,adisturbanceovercertaincomponent(s),andtheadditionloadingoncomponentcausedbyfailureofothercomponent(s). 16

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IntheCASCADEmodel,therearenidenticalcomponentswithrandominitialloading,andforeachcomponenttheminimumandmaximuminitialloadingisLmin,Lmaxrespectively.ComponentjisassumedtohaveaninitialloadingofL2[Lmin,Lmax]withauniformdistribution,andLj,j=1,2,...,nareidenticallydistributed.Theynormalizedtheinitialloadby`j=Lj)]TJ /F3 11.955 Tf 11.96 0 Td[(Lmin Lmax)]TJ /F3 11.955 Tf 11.95 0 Td[(Lmin,therefore`jisarandomvariablewithastandarduniformdistribution,`j=1beingthefailureload. Tobegincascadeprocess,theyalsoassumethatthereisaninitialdisturbanceDoneachcomponent(Dcouldbe0forcertaincomponents),andfailurescausedbyDonsomecomponentswouldaddanextraloadPonothercomponentsresultinginfurtherfailuresasacascadingprocess.PandDcanbenormalizedasp=P Lmax)]TJ /F3 11.955 Tf 11.96 0 Td[(Lmind=D+Lmax)]TJ /F3 11.955 Tf 11.96 0 Td[(Lfail Lmax)]TJ /F3 11.955 Tf 11.95 0 Td[(Lfail ThedistributionforthetotalnumberSoffailedcomponentsisP[S=r]=nrd(d+rp)r)]TJ /F12 7.97 Tf 6.59 0 Td[(1(1)]TJ /F3 11.955 Tf 11.95 0 Td[(d)]TJ /F3 11.955 Tf 11.96 0 Td[(rp)n)]TJ /F9 7.97 Tf 6.59 0 Td[(r Whenconsideringthetheinteractionmechanismamongindividualcomponents,theyproposedanalgorithmfornormalizedCASCADEwithkinteractions(Fig. 2-1 ): 1. Setallncomponentswithinitialload`1,`2,...,`niidfromstandarduniformdistribution. 2. Samplethencomponentsktimesindependentlyanduniformlywithreplacement,addtheinitialdisturbancedtotheloadofthesampledcomponents,andsetthestageindexitozero. 3. Testeachcomponentswhicharenotmarkedasfailedbeforesamplinginthepreviousstep.Ifcomponentjisnotmarkedasfailedanditsloadlj>1,thenmarkitasfailed.LetMibethenumberoftotalfailedcomponentsinstagei. 17

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4. IncludingthoseMicomponents,samplethencomponentsktimesindependentlyanduniformlywithreplacement,andaddptotheloadofthesampledcomponents. 5. Increasethestageindexby1andgotostep2. Figure2-1. Cascadingfailurepowersystemmodel 2.2.3BranchingProcessModel ByintroducingsomegeneralprocessingmodelssuchasGalton-WatsonbranchingprocesswithgeneralizedPoissondistribution,thesescholarscouldapproximatetheCASCADEmodelwithsimpliedmathematicalmodelsothateachstepofthecascadingfailurepropagationcouldbecalculated.ThebasicideabehindtheGalton-Watsonbranchingprocessisthatthefailureoneachcomponentineachstagewillindependentlyresultinfuturefailuresinthenextstageassociatedwithaprobability 18

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distribution,anditcouldberepresentedinmathematicsasfollowing:Mi+1=M(1)i+1+M(2)i+1++M(Mi)i+1 whereMki+1isidenticallyindependentbyassumptionanddenedasthefailurescausedbykthcomponentfailureinstagei,sothetotalnumberoffailuresisM=1Xk=0Mk Inadditiontoabovemodel,withtheassumptionthateachcomponentfailurewouldcausefurtherfailuresaccordingtoaPoissondistributionofmean,thebranchingprocesscouldbemodeledasatransientdiscretetimeMarkovprocesswithitsbehaviorgovernedbytheparameter.Therefore,thetotalnumberoffailuresbecomesP[M=r]=(r)r)]TJ /F12 7.97 Tf 6.59 0 Td[(1e)]TJ /F9 7.97 Tf 6.59 0 Td[(r r!,01 Theyalsohaveahighlevelversionofprobabilisticmodelofthecascadingprocesswhichutilizesacontinuousstatebranchingprocess[ 133 ].Withallthesemodels,theyproposesomestatisticalestimatorstomeasuretheextenttowhichtheloadsheddingispropagated. However,thisapproximationcanonlyworkinasystemwithmanycomponentsandmanycomponentinteractionssothatseriesoffailurespropagatinginparallelcanbeassumednottointeract,anditcannotreectthemechanismandcomplexitiesofloadingdependentcascadingfailurewhichdoesexistinrealpowernetworksystem. 2.2.4SimulationandComplexSystemModel Inordertodynamicallypredictthebehaviorofthegiganticpowergridundervariouscircumstances,withthehelpofadvancedinformationtechnology,peoplecanusesimulationmodeltonotonlybetterunderstandtheentirepowergriditself,buttoobtainsomeexperimentalresultinatimelymanneraswell.Mostoftheseresearchinvolvescomplexsystemanalysiswithcharacteristicsofcascading,thereforetheyarerelatedto 19

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powerblackoutanalysissincethatpowergridisusuallytreatedasacomplexnetwork,cascadingisauniversalphenomenononpowergrid. Thecomplexsystemornetworkshasalreadydrewalotofattractionfromresearchersinmultipleles,evenbeforethepowersecurityissuehasbeenexposedtopublicattention.Newman[ 98 ],AlbertandBarabasi[ 6 ]reviewtherecentadvancesintheeldofcomplexnetworks,includingmodelingthenetworkandpredictingitsdynamicbehavior.Theydiscussedseveralmainnetworkmodelswithnetworktopologyandnetwork'srobustnessagainstfailuresandattacks.Albert,JeongandBarabasi[ 4 ]especiallyfocusedonthetoleranceagainsterrorsorattacksamongdifferenttypesofcomplexnetworks.WattsandStrogatz[ 129 ]studiesthedynamicsof`small-world'networks,wheretheyexplicitlyclaimedthatthepowergridofthewesternUnitedStatesinsuchcategory. Intheeldofcascadingfailuresimulation,variousmethodsandmodelsareproposedtocapturethecascadingphenomenon,andmostofthemfocusoncertainaspectsofcascadingwhichcouldberepresentedbysuccessivestaticmodels,whilemostofwhicharerelatedtosuchasstaticlineoverloadingatthelevelofDCorACloadow.Thereareseveralpapers([ 11 ],[ 19 ],[ 32 ],[ 33 ],[ 64 ],[ 88 ],[ 109 ],[ 120 ])concentratingonhiddenfailureswithprotectioncontrolandoperatorreaction. Duetothedifcultiesandcomplexitiesofmodelingandtheheavycomputationburden,dynamicanalysissuchasself-organizationduringblackouthasnotbeenwellstudies.However,itiscriticaltostudy,inmoredetail,thetransientstatusoftheevolvingpowergridthatiscontinuallyupgradingtocouplewiththechangingloadandgenerationdemand([ 83 ],[ 122 ]).SimulationsuchastheOPAcouldhelptobetterunderstandtransientreliability([ 28 ],[ 130 ]),andidentifysomehigh-riskcascadesaswell([ 33 ],[ 36 ],[ 108 ],[ 96 ]). Besidesthoseresearches,thereisonewhichcaughtalotofeyeballs,eventheonesfromLarryM.Wortzel,amilitarystrategistandChinaspecialist.OnMarch10, 20

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2009,hepresentedtheU.S.HouseonForeignAffairscommitteethatitshouldbeconcernedonhowtoattackasmallU.S.powergridsub-networkinawaythatwouldcauseacascadingfailureoftheentireU.S.grid[ 134 ],andhisstatementisbasedfromapaperpublished[ 126 ]onthejournalofSafetyScience.Thepurposeoftheresearch,explainedbytheauthorMr.JianweiWangandhiscolleaguesfromDalianUniversityofTechnologyChina,istotrytondwaystoenhancethestabilityofpowergridsbyexploringpotentialvulnerabilities. Inalltheotherstudiescitedbelow,theloadonanode(oranedge)wasgenerallyestimatedbyitsdegreeorbetweennessandtheredistributionloadwereusuallyforwardedfollowingtheshortestpath.Wangproposedanewmeasuretoassigntheinitialloadofanodeandtoredistributeloadamongnodesafterattacking,inordertoreducethecomputingcomplexityusingsomeothermeasuresuchasthebetweenness.TheauthorassumethetheinitialloadLjofanodejinthepowergridisafunctionofitsdegreekjandthedegreesofitsneighbornodeskm(m2)]TJ /F9 7.97 Tf 6.77 -1.79 Td[(j),where)]TJ /F9 7.97 Tf 6.77 -1.79 Td[(jisthesetofallneighboringnodesofthenodej.Theinitialloadofnodejisdenedas:Lj=[kj(Xm2)]TJ /F13 5.978 Tf 4.82 -1.41 Td[(jkm)],andtheredistribution(Fig 2-2 )betweentwoadjacentnodesisdenedas:ij=Li[kj(Pm2)]TJ /F13 5.978 Tf 4.82 -1.41 Td[(jkm)] Pn2)]TJ /F13 5.978 Tf 4.82 -1.41 Td[(i[kn(Pf2)]TJ /F13 5.978 Tf 4.82 -1 Td[(nkf)] TheyalsosetthethecapacityCjofanodejproportionaltoitsinitialload,i.e.,Cj=TLj,andthecascadingprocessbeginswhenLj+ij>Cj.Theyevaluatetheeffectofattackingbythenormalizedavalanchesize(brokennodes)CFattack=Pi2ACFi NA(N)]TJ /F12 7.97 Tf 6.59 0 Td[(1),whereCFiistheavalanchesizeinducedbyremovingnodei.ByadjustingtheparameterandTwhichcaninuencetheinitialloadandnodetolerancerespectively,theynumericallystudiedtheelectricalpowergridofthewesternUnitedStateswith4941nodesand6594edgestoinvestigatethenetworkrobustnessunderattacks. 21

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Figure2-2. theloadredistributiontriggeredbyannode-basedattack WangandChen[ 127 ]proposedacascadingmodelwithalocalweightedowredistributionruleandstudiedonweightedscale-freeandsmall-worldnetworks.Intheirmodel,theyassumethetheweight(ow)ofanedgeijaswij=(kikj),whereistheparameterforthestrengthoftheedgeweight,andkiisthedegreesofnodesi.Theredistributionmodel,whichisthekeytotheirsimulationresearch,isdenedasFim=Fijwim Pa2)]TJ /F13 5.978 Tf 4.82 -1.4 Td[(iwia+Pb2)]TJ /F13 5.978 Tf 4.82 -1.4 Td[(jwjb, where)]TJ /F9 7.97 Tf 6.77 -1.8 Td[(iisthesetofadjacentnodesofi.Inthecasestudy,theauthorsonlyconsideroneedgeijattacking,andobtaintheevaluationresultbynormalizedavalanchesizeSN=Pijsij=Nedge,wheresijisthetheavalanchesizeinducedbycuttingoutedgeij,andNedgeisthetotalnumberinthenetwork.Theyfurthermoreexplorethestatisticalcharacteristicsoftheavalanchesizeofanetwork,thusobtainingapower-lawavalanchesizedistribution. Theconceptofloadentropy,whichcanbeanaveragemeasureofanetworksheterogeneityintheloaddistributionwasintroducedbyBao,Cao,andetc.They 22

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considerthatacertaindegreeofinstantaneousnodeoverloadispermissibleinrealitycomplexnetworksandnodebrokageismainlycausedbytheaccumulativeeffectofoverload.Theyalsodisagreethesimpleandunreasonablestrategyofimmediateremovalofaninstantaneouslyoverloadednode.Bytheirstrategyoftheoverloadednoderemoval,itisassumedthatthedensityoftheremovalprobabilityP(L)ofanodeobeysuniformdistributioninthediscreteinterval[0,T].Ateachtimeperiod,theremovalprobabilityiscalculatedas:pi(t)=8>><>>:pi(t)]TJ /F5 11.955 Tf 11.95 0 Td[(1)+P(Li(t))=TifLi(t)>Ci;0ifLi(t)Ci whereLi(t)istheloadofnodeiattimet.Ateachiteration,theloadofallnodesiscomparedwitharandom2(0,1)todeterminedwhetherthecorrespondingnodeshouldberemoved.Whentheloadsofallnodesarenotlargerthantheircorrespondingcapacity,acascadingfailurestops.Theydenestheloadentropytoevaluatetherobustnessofthenetworkunderaninitialremovalonthenodewiththelargestload,buttheydonotmentionabouttheirloadredistributionmodelaftercertainnode'sremoval. Inaddition,anumberofaspectsofcascadingfailureshavebeendiscussedinsomeliteratures,includingthecascadecontrolanddefensestrategy([ 139 ],[ 119 ],[ 94 ],[ 124 ],[ 15 ]),themodelfordescribingcascadephenomena([ 128 ],[ 135 ]),theanalyticalcalculationofcapacityparameter([ 125 ],[ 140 ]),andsoon. Withthehelpofmodernadvancedcomputers,thesimulationofpowergridascomplexnetworkshasbeendevelopedsowellthatitcouldhelpusbetterunderstandthedynamicsofpowernetworkinawaymoreaccurateandlesscostlyintime,thereforeitcouldbeappliedforblackoutanalysisandprediction.However,therearesomeissuesassociatedwiththatapplication.First,althoughtherearealotofproposedmeasurementsforthenodes'degreeoredges'weight,thereisnoscienticproofonthevalidityofthesemeasurements.Second,thevaluesforthetunable 23

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parametersinthemodelarenotwell-recognized,andallthosenumericalstudiesarebasicallyexperimental.Third,inalltheredistributionmodels,theyonlyconsidertheconservationofpowerow(themodiedformofKirchhoff'snodalrule),butpowergridmodeledwithoutapplyingKirchhoff'smeshruleisfarawayfromreality.Forth,allthosemodelonlyconsiderthepowertransmission.Whendealingwithpowerblackoutwithcascading,atleastgeneratorsandconsumershouldbeconsideredintotheintegratedplanning. Duetothoseshortcomingabove,evennumerousrecentpapershaveappliedcomplexnetworkandtopologymethodswithcascadingpropertytostudythestructureanddynamicfunctionofpowergrids,resultsarenotsoidenticalevencontrasttoeachother.HereisapaperfromHines,Cotilla-SanchezandBlumsackshowingthisargument[ 69 ].Theauthorscomparetheanalysisresultsfromavarietyofpowernetworkssubjectedtorandomfailuresanddirectedattackswiththevulnerabilitymeasuresofcharacteristicpathlength,connectivitylossandblackoutsizes.Theychoseseveralcontingencymethodsincludingrandomfailure,degreeattack,maximum/minimum-trafcattackandbetweennessattack.Thenconcludethattopologicalmeasurescanprovidesomegeneralvulnerabilityindication,butitcanalsobemisleadingsinceindividualsimulationsshowonlyamildcorrelation.Mostimportantly,theysuggestthatresultsfromphysics-basedmodelsaremorerealisticandgenerallymoreusefulforinfrastructureriskassessment. 2.3OptimizationModel Mathematicaloptimizationmethodshavebeenusedtosolvemanypowersystemproblemssuchasplanning,operationandcontrolproblemsformanyyears.Inordertoapplyoptimizationmethodsforthepowerproblemsinreality,someassumptionsmustbemadetoderivethemathematicalmodel.However,evenunderthiscircumstances,optimizationoverlarge-scalepowersystemisstillacomputation-intensivetaskwithinthescopeofcontemporaryinformationtechnology.Therearesomanyuncertainfactors 24

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suchasuncontrollablesystemseparation,angleinstabilityandgenerationtrippingintheselarge,complex,andwide-spreadpowersystemthatwouldmaketheabovemodelmorecomplicated,withoutconsideringthenewissuesintroducedbythederegulationofpowerutilities. 2.3.1Pre-OptimalPowerFlowModel BeforethedevelopmentofOptimalPowerFlow(OPF)modelfromitsinceptionin1961andseveralsolutionmethodsinexistencein1978,thereweresomeoptimizationmodelsrelatedtoeconomicpowerowdispatch.Megahedetal.[ 86 ]proposetheconversionofthenonlinearlyconstraineddispatchproblemtoaseriesofconstrainedlinearprogrammingproblems.Systemvoltages,activeandreactivegeneration,andthephaseanglesareconsideredasprototypepartoftheOPFproblem.Thesequantitiesareusedinthelossformula.Accordingtotheauthors,themethodisfastandhasgoodconvergencecharacteristics. 2.3.2OPFModel OptimalPowerFlow(OPF)modelservesasthecenterandcriticalpartinthemathematicaloptimizationappliedblackoutproblemoverpowersystem.ThehistoryofresearchonOPFmodelcoulddatebacktotheearly1960's[ 26 ],anditwasderivedfromthethesolutionoftheeconomicdispatchbytheequalincrementalcostmethod. Economicdispatchisdenedastheprocessofallocatinggenerationlevelstothegeneratingunitsinthemix,sothatthesystemloadmaybesuppliedentirelyandmosteconomically.Researchonoptimaldispatchcouldgoasfarbackastheearly1920',whenengineerswereconcernedwiththeproblemofeconomicallocationofgenerationortheproperdivisionoftheloadamongthegeneratingunitsavailable.Generationdispatchhasbeenwidelystudiedandreportedbyseveralauthorsinbooksonpowersystemanalysis([ 61 ],[ 47 ],[ 39 ],[ 16 ]). AlthoughbothofeconomicdispatchandOPFmodelareoptimizationproblemswiththesameminimumcostobjective,economicdispatchonlyconsidersrealpower 25

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generationandtransmissionwithonlypowerbalanceequationastheconstraint.Ontheotherside,theOPFisastaticnonlinearoptimizationproblemwhichcantakenearlyallelectricalvariables,owbalance,powerowphysics,generatoranddemandnodeboundsandphysicallawsinconsideration,tocomputetheoptimalsettingsinapowernetwork,givensettingsofloadsandsystemparameters.AtypicalOPFmodelis:ming,s,f,Xi2I(higi+risi)X(i,j)2+(i)fij)]TJ /F10 11.955 Tf 23.55 11.36 Td[(X(j,i)2)]TJ /F12 7.97 Tf 6.25 -2.27 Td[((i)fji=8>>>>>><>>>>>>:Pii2C)]TJ /F3 11.955 Tf 9.3 0 Td[(Dii2D0otherwisesin(i)]TJ /F4 11.955 Tf 11.96 0 Td[(j))]TJ /F3 11.955 Tf 11.95 0 Td[(xijfij=08(i,j)jfijjuij8(i,j)PminiPiPmini8i2C0DjDnomj8j2D whereagridisrepresentedbyadirectednetworkG,and: Eachnodecorrespondstoagenerator(i.e.,asupplynode),ortoaload(i.e.,ademandnode),ortoanodethatneithergeneratesnorconsumespower(i.e.,atransmissionordistributionnode).WedenotebyCthesetofgeneratornodes. Ifnodeicorrespondstoagenerator,thentherearevalues0PminiPmaxi.Ifthegeneratorisoperated,thenitsoutputmustbeintherange[Pmini,Pmaxi];ifthegeneratorisnotoperated,thenitsoutputiszero.Ingeneral,weexpectPmini0. Ifnodeicorrespondstoademand,thenthereisavalueDnomi(thenominaldemandvalueatnodei).WewilldenotethesetofdemandsbyD. ThearcsofGrepresentpowerlines.Foreacharc(i,j),wearegivenaparameterxij>0(theresistance)andaparameteruij(thecapacity). GivenasetCoffoperatinggenerators,apowerowisasolutiontothesystemofconstraintsgivenabove.Inthissystem,foreacharc(i,j),weuseavariablefijrepresentthe(power)owon(i,j)(negativeifpoweriseffectivelyowingfromjtoi).Inaddition, 26

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foreachnodeiwewillhaveavariablei(thephaseangleati).Finally,ifiisageneratornode,thenwewillhaveavariablePi,whileifirepresentsademandnode,wewillhaveavariableDi.Givenanodei,werepresentwith+(i)()]TJ /F5 11.955 Tf 7.09 -4.34 Td[((i))thesetofarcsorientedoutof(respectively,into)i. TheaboveconstraintsisfromOhm'sequationindirectcurrent(DC)network.InthecaseofanACnetwork,theycanonlyapproximatesacomplexsystemofnonlinearequations.Theissueofwhethertousethemoreaccuratenonlinearformulation,ortheapproximateDCformulation,isquitenoteasy.Ontheonehand,thelinearizedformulationcertainlyisanapproximationonly.Ontheotherhand,aformulationthatmodelsACpowerowscanproveintractableormayreectdifcultiesinherentwiththeunderlyingreal-lifeproblem. First,ACpowerowmodelstypicallyincludeequationsoftheform:sin(i)]TJ /F4 11.955 Tf 11.96 0 Td[(j))]TJ /F3 11.955 Tf 11.96 0 Td[(xijfij=08(i,j) Here,thefquantitiesdescribeactivepowerowsandthedescribephaseangles.Innormaloperationofatransmissionsystem,onewouldexpectthatijforanyarc(i,j)andthusitcanbelinearized.Hencethelinearizationisonlyvalidifweadditionallyimposethatji)]TJ /F4 11.955 Tf 12.41 0 Td[(jjbeverysmall.However,intheliteratureonesometimesseesthisverysmallconstraintrelaxedwhenthenetworkisnotinanormaloperativemode.Thenonlinearformulationgivesrisetoextremelycomplexmodels,butstudiesthatrequiremultiplepowerowcomputationstendtorelyonthelinearizedformulationtogetsomeusefulandstraight-forwardinformation.Second,nomatterweuseanACorDCpowerowmodel,theresultingproblemshaveafarmorecomplexstructurethantraditionalsingle-ormulti-commodityowmodels,whichwouldleadtoocounter-intuitivebehaviorsimilartoBraess'sParadox. Originally,classicaloptimizationmethodswerecapabletoeffectivelysolveDCOPF,andevenACOPFwithcertainlinearapproximationmethods.Butmorerecently, 27

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duetothewideapplicationofFlexibleA.C.Transmissionsystem(FACTS)devicesandderegulationofpowergrid,itisdifculttodealeffectivelywithmanypowersystemproblemsthroughstrictold-fashionmathematicalformulation.Followingparagraphsbrieydiscussesabouttheimportantmathematicaloptimizationtechniquesusedinpowersystemsproblems: 2.3.2.1Linearprogramming(LP)andquadraticprogramming(QP) Whentheobjectivefunction,constraintsarelinear,anddecisionvariablesarenonnegative,problemscanbeformulatedastheLP([ 117 ],[ 8 ]).T.S.Chungetal.[ 37 ]proposedarecursivelinearprogrammingbasedapproachforminimizinglinelossesandndingtheoptimalcapacitorallocationinadistributionsystem.E.Lobatoetal.[ 81 ]useLPbasedOPFtominimizethetransmissionlossesandGeneratorreactivemarginsoftheSpanishpowersystem.LPcanalsobeusedinvariouspowersystemsapplications,includingreactivepowerplanning[ 101 ],activeandreactivepowerdispatch([ 30 ],[ 31 ]).ProblemsformulatedasLPcanusuallybesolvedbysimplexandInteriorPoint(IP)methods,whilethescaleofproblemscouldbeuptothousandsofvariablesandconstraintsevenusinginexpensivecomputers. BoththesimplexandIPmethodscanbeextendedtoaquadraticobjectivefunctionwhiletheconstraintsmaintainlinear,whicharecalledQP.J.A.Momoh[ 89 ]showedtheextensionofbasicKuhn-TuckerconditionstoemployageneralizedQuadratic-BasedmodelforsolvingOPF,wheretheconditionsforfeasibility,convergenceandoptimalityarediscussed.Thesameauthor[ 91 ]alsopublishedanotherpaperonapplyingInteriorPointmethodstosolvequadraticpowersystemoptimizationproblem.N.Grudinin[ 62 ]proposedareactivepoweroptimizationmodelbasedonSuccessiveQuadraticProgramming(SQP)methods,whichturnedtohavethebestperformancewhilebeingcomparedwithotherveoptimizationmethods.Nanda[ 95 ]developedanewalgorithmforOPFusingFletcher'sQPmethod,andG.P.Granellietal.[ 59 ]proposedSecurity-constrainedeconomicdispatchusingdualsequentialquadraticprogramming, 28

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whichwascomparedwithSQPtodemonstrateabetterresultincomputationtimeandaccuracy. 2.3.2.2Nonlinearprogramming(NLP) Whentheobjectivefunctionand/orconstraintsarenonlinear,problemscanbeformulatedastheNLP.NLPcanbeappliedtovariousareasofpowersystems,suchasoptimalpower([ 90 ],[ 118 ]).J.A.Momohetal.[ 142 ]proposedanonlinearconvexnetworkowprogramming(NLCNFP)modelandalgorithmforsolvingthesecurity-constrainedmulti-areaeconomicdispatchproblem.D.Pudjiantoetal.[ 107 ]usedNLPbasedreactiveOPFfordistributingreactivepoweramongcompetinggeneratorsinaderegulatedmarket. TosolvemostoftheNLP,themostregularwayistostartfromaninitialpointandtoimprovealongacertaindescentdirectioninwhichobjectivefunctiondecreasesincaseofminimizationproblem,andtherearealotofresearchesabouthowtoobtainabetterinitialpointand/ordescentdirectionassociatedwithitssteplength([ 74 ],[ 121 ]).IPmethodsoriginallydevelopedforLPcanbeapplicablehere.SergioGranville[ 60 ]presentedapplicationofanInteriorPointMethodtotheoptimalreactivepowerdispatchproblem.WeiYanetal.[ 137 ]presentedthesolutionoftheOptimalReactivePowerFlow(ORPF)problembythePredictorCorrectorPrimalDualInteriorPointMethod(PCPDIPM). 2.3.2.3Integerprogramming(IP)andmixed-integerprogramming(MIP) Forsomepowersystemrelatedproblems(e.g.generator/transmission-lineONstatus=1,andgenerator/transmission-lineOFFstatus=0),whenallorsomeofthedecisionvariablesarecantakeonlyintegervalues,suchproblemiscalledintegerprogramming,ormixedintegerprogrammingrespectively.Theycanbeappliedtomanyareasofpowersystems,suchasoptimalreactivepowerplanning,powersystemsplanning,unitcommitmentand,generationscheduling([ 12 ],[ 2 ],[ 55 ],[ 41 ],[ 46 ]). 29

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Themostusedapproachetosolveintegerproblemsusingmathematicalprogrammingtechniquesisbranchandbound,andcuttingplanemethods([ 74 ],[ 79 ]).Whenthesizeandcomplexityofmodelsarelargeenough,whilethestructureoftheproblemisnotinspecicform(suchastransportationnetwork),decompositiontechniqueusuallyisapplied([ 7 ],[ 40 ]). 2.3.2.4Dynamicprogramming(DP) DPbasedontheprincipleofoptimalitystatesthatasub-policyofanoptimalpolicymustinitselfbeanoptimalsub-policy.DPcanbeappliedtovariousareasofpowersystems,suchasreactivepowercontrol,transmissionplanning,andunitcommitment([ 70 ],[ 104 ]),butitisveryinefcientduetothecurseofdimensionality. 2.3.3UnitCommitment Sincepowergeneratorscannotinstantlyturnontoproducepower,unitcommitment(UC)requirementmustbefollowedinadvancesothatadequatepowergenerationisalwaysavailabletomeetthedemand,especiallyintheeventthatgeneratorsortransmissionlinesgooutorloaddemandincreases.Unitcommitmenthandlestheunitgenerationscheduleinapowersystemforminimizingoperatingcostandsatisfyingprevailingconstraintssuchasloaddemandoverasetoftimeperiods.Unitcommitmentinpoweroperationplanningconcernstheschedulingofstart-up/shut-downdecisionsandoperationlevelsforpowergenerationunitssuchthatthefuelcostsoversometimehorizonareminimal. UCisnotthesameasdispatching.Dispatchingfocusesonassigningagivensetofpowerplantstoanothercertainsetofelectricdemand,whileUCdeterminesthestart-upandshutdownschedulesofthermalunitstomeetforecasteddemandovercertaintimeperiods.Thedifferencebetweenbothissuesistime.Theusualcommonobjectivesofunitcommitmentscheduleincludeminimizationoftotalproductioncost,minimizationofemissionsandmaximizationofreliabilityandsecurity,andthemostimportantnon-linearconstraintsaretheunit'sminimumup-timeanddown-timerestriction. 30

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Althoughtheplanninghorizonforunitcommitmentinprincipleshouldbecontinuousintime,theunitcommitmentmodelstypicallyareindiscretetimeduetotheavailabilityofdata,theexecutiontimeforschedulingdecisionsandcomputationlimitationonthecomplexMIPincontinuoustime.HerewedemonstrateaverybasicUCformulation: minpti,stj,wtjTXt=1IXi=1Ci(pti,uti)+TXt=1IXi=1Sti(uti)ut)]TJ /F12 7.97 Tf 6.59 0 Td[(1i)]TJ /F3 11.955 Tf 11.95 0 Td[(uti1)]TJ /F3 11.955 Tf 11.95 0 Td[(uli,i=1,2,...,I;t=2,3,...,T)]TJ /F5 11.955 Tf 11.95 0 Td[(1l=t+1,t+2,...,minft+i)]TJ /F5 11.955 Tf 11.96 0 Td[(1,Tgltj=lt)]TJ /F12 7.97 Tf 6.59 0 Td[(1j)]TJ /F5 11.955 Tf 11.95 0 Td[((stj)]TJ /F4 11.955 Tf 11.95 0 Td[(jwtj),j=1,2,...,Jl0j=linj,lTj=lendj,t=1,2,...,TIXi=1(utipamxit)]TJ /F3 11.955 Tf 11.96 0 Td[(p)it)Rt,t=1,2,...,TIXi=1pti+JXj=1(stj)]TJ /F3 11.955 Tf 11.96 0 Td[(wtj)Dt,t=1,2,...,Tpminitutiptipmaxituti,i=1,2,...,I,t=1,2,...,T0stjsmaxjt,j=1,2,...,J,t=1,2,...,T0wtjwmaxjt,j=1,2,...,J,t=1,2,...,T0ltjsmaxj,j=1,2,...,J,t=1,2,...,T Here,TdenotethenumberofsubintervalsoftheoptimizationhorizonandsupposethereareIthermalaswellasJpumped-storagehydrounits.TheThevariableuti2f0,1g,i=1,2,...,I,t=1,2,...,Tindicateswhetherthethermalunitiisinoperationattimet.Variablesptj,stj,wtj,j=1,2,...,J,t=1,2,...,Taretheoutputlevelsforthethermalunits,thehydrounitsingenerationandinpumpingmodes,respectively.Thevariablesltjdenotethell(inenergy)oftheupperdamofthehydrounitjattheendofintervalt,j=1,2,...,J,t=1,2,...,T.Theobjectiveisthesumofthefuelcostandstart-upcostwithparameterCi,Stirespectively,andtheconstraints 31

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includethepoweroutputboundsofunitsandthelloftheupperdam,loadcoverage,reservemanagementofthethermalunits,balancesforhepumped-storageplants,andminimumdowntimesforthermalstressesinthecoalredblocks. Themoststraight-forwardwaytosolvetheUCeconomicoptimizationisbruteforce,whichenumeratesallpossiblecombinations,eliminatesthepossibilitiesthatdonotmeettheobligationsset,andnallychoosesthebestofalltheremainingpossibilities.Eventhoughthisalgorithmiscomputationallyintensive,mostofcurrentmethodologiesareavariationonthebruteforce,inwhichsomeproceduresareaddedtoreducethenumberofpossibilitiesenumerated. Theunitcommitmentproblembelongstotheclassofcomplexcombinationaloptimizationproblems.Severalmathematicalprogrammingtechniqueshavebeenproposedtosolvethistime-dependentproblem. 2.3.3.1Dynamicprogramming(DP) DPsearchesthesolutionspacethatconsistsoftheunits'statusforanoptimalsolution[ 116 ].Thesearchcanbecarriedoutinaforwardorbackwarddirection.Thetimeperiodsofthestudyhorizonareknownasthestagesoftheproblem.Thecombinationsofunitswithinatimeperiodareknownasthestates([ 72 ],[ 63 ]).Lowery[ 70 ]startsfromapreviously-determinedoptimalUCplanningandgraduallyaddspowerplantstoobtainoptimalsolutionsforhigherdemands.Hobbsetal.[ 82 ]initializetheirapproachwithoptionscalculatedforprecedingperiods.CohenandYoshimura[ 38 ]proposedabranch-and-boundmodelwhichstartsfromapreviouslyobtainedoptimum.TheUCproblemmayalsobedecomposedintosmallersubproblemsthatareeasilymanagedandsolvedwithDP,wherethemasterproblemisoptimized,linkingthesub-problemsbyLagrangemultipliers.VandenBoschandHonderd[ 24 ]decomposedthemainproblemintoseveralsub-problemsthatareeasiertosolve.ThedecompositionproposedbySnyderetal.[ 116 ]consistsofgroupingpowerplantsfromthesametype. 32

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TheadvantageofDPisitsabilitytomaintainsolutionfeasibility.DPbuildsandevaluatesthecompletedecisiontreetooptimizetheproblemathand.Butitsuffersfromthecurseofdimensionalitybecausetheproblemsize(numberofstates)increasesrapidlywiththenumberofgeneratingunitstobecommitted,whichresultsinanunacceptablesolutiontime.Toreducethedimension,searchspaceandexecutiontime,severalapproacheshavebeendeveloped,includingDP-SC(dynamicprogramming-sequentialcombination)[ 103 ],DP-TC(dynamicprogramming-truncatedconbination)[ 102 ],DP-STC(whichisacombinationoftheDP-SCandDP-TCapproaches)[ 76 ]andDP-VW(variablewindowtruncateddynamicprogramming)[ 102 ].Thevariationofwindowsizeaccordingtoloaddemandincrementindicatesasubstantialsavingincomputationtimewithoutsacricingthequalityofthesolution,andthesolutionofalloftheseDPmethodsissub-optimal. 2.3.3.2Dynamicandlinearprogramming TheUCproblemcanbesolvedbyusingregulardynamicprogramming(DP)orDPwithsuccessiveapproximationofthesolutionspace.Linearprogramming(LP)solvestheeconomicdispatchwithinUCforoptimalallocationoffuel/generation.Dantzig-Wolfedecomposition,whenused,partitionsthelinearprogramintosmaller,easilymanageableLPsubproblems([ 85 ],[ 51 ]).TheprimarydisadvantageofLPsolutionsisthenumerousvariablesneededtorepresentthepiece-wiselinearinput-outputcurves. 2.3.4ContingencyandInterdiction ContingencyandInterdictionanalysis,whichassessestheabilityofthepowergridtosustainvariouscombinationsofpowergridcomponentfailuresbasedonstateestimates,isacriticalpartoftheenergymanagementsystem.Here,thecontingencymeansasetofunexpectedeventshappeningwithinashortduration.Theunexpectedeventscanbefailuresofbuses(generators,substations,etc)ortransmissionanddistributionlines.optimizationisusedtomaximizetheblackoutsizeduetocontingenciescausedbybyattackedwithlimitedresources([ 13 ],[ 45 ],[ 111 ]). 33

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Inthepast,duetotheheavycomputationinvolved,thecontingencyanalysiscanbeonlyanalyzedforonlyaselectsetofN)]TJ /F5 11.955 Tf 12.65 0 Td[(1contingency,orN)]TJ /F5 11.955 Tf 12.65 0 Td[(1reliabilitycase,whichisthefailureofonecomponent(abusoraline)hasbeenanactiveresearcharea.Milanoetal.usesN)]TJ /F5 11.955 Tf 12.09 0 Td[(1contingencycriterionasaninitialoptimaloperatingconditiontoestimatethesystem-wideavailabletransfercapability[ 87 ].Hedmanetal.analyzetheN)]TJ /F5 11.955 Tf 12.33 0 Td[(1reliableDCoptimaldispatchwithtransmissionswitchingbymodifyingeconomicdispatchoptimizationproblemstoincorporatetheexibilityoftransmissionassets'states[ 67 ].WhileensuringN)]TJ /F5 11.955 Tf 12.44 0 Td[(1reliability,thesameauthorsalsopresentaco-optimizationformulationoftransmissionswitchingproblemandthegenerationunitcommitment[ 66 ]. However,aselectricitydemandcontinuestogrowandrenewableenergyincreasesitspenetrationinthepowergrid,analysisoftheN)]TJ /F5 11.955 Tf 12.92 0 Td[(1reliabilityisnotsufcientformanyrealapplicationswithmultiplefailurestodiscoverthevulnerabilitiesofpowergrids.Althoughthecombinatorialnumberofcontingencystatesimposesasubstantialcomputationalburdenforanalysis,theN)]TJ /F3 11.955 Tf 12.03 0 Td[(kcontingencyanalysisforfailuresofmultiplecomponents(totallykbusesandlines)canreectalargervariationofvulnerabilitiesofapowersystemandattractalotofresearchfocus.Salmeron,Wood,andBaldickappliedalinearizedpowerowmodelandusedabi-leveloptimizationframeworkalongwithmixed-integerprogrammingtoanalyzethesecurityoftheelectricgridandtoobtaintheworstcontingencyselection,wheretheinterdictionmodelistoidentifycriticalsetsofapowergrid'scomponents,e.g.,generators,transmissionlines,andtransformers,byidentifyingmaximallydisruptive,coordinatedattacksonagrid,whichaterroristgroupmightundertake[ 111 ].Pinaretal.modeledthepowergridvulnerabilityanalysisasamixedintegernonlinearprogramming(MINLP)problem,andusedaspecialstructureintheformulationtoavoidnonlinearityandapproximatetheoriginalproblemasapurecombinatorialproblem[ 106 ].Bienstockusedthetwoapproachesoftheintegerprogrammingandanew,continuousnonlinearprogrammingformulationforcomparisononvulnerabilityevaluationoverlarge-scalepowergridfailures([ 22 ],[ 23 ],[ 123 ]). 34

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InthepaperfromFanetal.[ 50 ],thecriticalnodemethod,originallyusedforresearchingeneralgraphs[ 14 ],isinthersttime,tobeappliedtostudyonpowergrids,whichisoneofthemaincontributionofthispaper.Moreimportantly,thispaperisamongoftherstfewpaperstoevaluatedifferentN)]TJ /F3 11.955 Tf 12.12 0 Td[(kcontingencyselectionbygraphalgorithmsandinterdictionmethodsthroughtheeconomicobjective(generatingandloadsheddingcost)ofthosecontingencystates.Inthispaper,severalgraphalgorithmsandinterdictionmethodsforcontingencyselectionaresurveyedandcomparedwithournewmethodrepresentedinformulation.Theirnewmodelcanselectthecontingencystateincludingbothbusesandlines,whichisabigplusthantheothers.Alsotheevaluationmeasurementoneconomicalemphasisisquitenew,andleadtosomeinterestingconclusions. 2.4BlackoutwithDistributedGeneration Traditionally,electricityindustriesgeneratemostoftheirpowerinlargeandcentralizedfacilities,fromsuchasfossilfuel,nuclear,largesolarorhydropowersources.Thiskindofbusinessstrategieshasexcellenteconomiesofscaleandsomeotherconsiderations,suchasofhealth&safety,logistics,environment,geographyandgeology,butusuallyelectricitytransmissionoverlongdistanceswouldnegativelyaffecttheenvironment.Theotherapproachisviadistributedgeneration,whichcanreducetheamountofenergylostintransmissionandthesizeoftransmissioninfrastructureduetoclosenessbetweensupplyanddemand.Averygoodapplicationexampleofdistributedgenerationismicrogrid,whichisalocalizedgroupingofelectricitygeneration,energystorage,andloads.Undernormaloperatingcondition,itisconnectedtoatraditionalcentralizedgrid,howeveritcouldbedisconnectedfromtherefunctioningautonomously.Fromthepointofviewoftheelectricitynetworkoperator,amicrogridcouldbetreatedasoneentity,butinsteadofsimplyreceivingelectricitysupply,itcouldsustainitselfforalongtimewithoutexternalelectricitysourceorevensometimesoutputelectricityforothers. 35

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Theblackoutisalwaysprecededbyasequenceofcascadingfailureswhichbreaksdowntransmissionlinesandgenerators,andthusleadstolargevariationinpowerow,itsroutingandbusvoltageduethemechanismofloadbalanceinamassivelyinterconnectednetworksystem.Withtheemergingofdistributedgeneration,thephenomenaofislandingmustbetakeninconsiderationduringblackoutanalysis,especiallyforreal-timedecisionmakingintheearlystageofpossibleblackout.Islandingisthesituationwhereadistributedgeneratorcontinuestopoweraneighborhoodwhileelectricalgridpowerfromtheelectricutilityisnotavailable. Bypropermanagementofintentionalislanding,theoperatordisconnectsthoseislands(localizedgroupingofelectricitygeneration,energystorage,andloads)fromthegrid,andforcesthedistributedgeneratortosupplypowerforthelocaldemand.Strategylikethiscouldlargelyreducetheburdenofsubstationsandgeneratorswhicharealreadycrumblingduetoemergentpowerreroutingtomeettheremotepowerdemandduringpoweroutage,andrelievetheoverloadingtransmissionlinesduringthecriticalrestorationphase[ 65 ].Distributedgenerationcouldincreasethereliabilityandsecurityofpowersupply[ 10 ]byprovidingelectricitytothemedium-voltageandlow-voltagenetworkswhereitismostneeded,incaseofhighervoltagenetworkfailures.Alotofconceptualbutpioneerpapershavebeenpublishedonhowtoapplyislandingoperationtomitigatethespreadofpoweroutateandpreventpossibleblackout([ 48 ],[ 43 ],[ 80 ],[ 3 ],[ 49 ]),howeverduetothecomplexityofinteractingmechanismwithintheislandandinter-islands,therearequiteafewpreliminaryresearchesonreal-timealgorismtoquicklydetectthepossibleoptimalislandingstrategywhenpowersystemvulnerabilityisapproachingtoanextremeemergencystate([ 138 ],[ 141 ],[ 105 ],[ 92 ]). 36

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CHAPTER3KEYCONCEPTUSEDINTHISDISSERTATION Inthischapter,somebasicbutkeyconceptsareintroducedhere. 3.1PowerFlows AbriefintroductionaboutDCpowerowmodelispresentedhere.Forthepurposesofourproblem,agridisrepresentedbyadirectednetworkG,where: Eachnodecorrespondstoagenerator(i.e.,asupplynode),ortoaload(i.e.,ademandnode),ortoanodethatneithergeneratesnorconsumespower(i.e.,atransmissionordistributionnode).WedenotebyPthesetofgeneratornodes. Ifnodeicorrespondstoagenerator,thentherearevalues0PminiPmaxi.Ifthegeneratorisoperated,thenitsoutputmustbeintherange[Pmini,Pmaxi];ifthegeneratorisnotoperated,thenitsoutputiszero.Ingeneral,weexpectPmini0. Ifnodeicorrespondstoademand,thenthereisavalueDnomi(thenominaldemandvalueatnodei).WewilldenotethesetofdemandsbyD. ThearcsofGrepresentpowerlines.Foreacharc(i,j),wearegivenaparameterxij>0(theresistance)andaparameteruij(thecapacity). GivenasetCoffoperatinggenerators,apowerowisasolutiontothesystemofconstraintsgivennext.Inthissystem,foreacharc(i,j),weuseavariablefijrepresentthe(power)owon(i,j)(negativeifpoweriseffectivelyowingfromjtoi).Inaddition,foreachnodeiwewillhaveavariablei(thephaseangleati).Finally,ifiisageneratornode,thenwewillhaveavariablePi,whileifirepresentsademandnode,wewillhaveavariableDi.Givenanodei,werepresentwith+(i)()]TJ /F5 11.955 Tf 7.09 -4.34 Td[((i))thesetofarcsorientedoutof(respectively,into)i. Inthesystemgivenbelow,constraintsaretypicalfornetworkowmodelsrespectively:owbalance,powerowphysics,generatoranddemandnodebounds. 37

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X(i,j)2+(i)fij)]TJ /F10 11.955 Tf 23.55 11.36 Td[(X(j,i)2)]TJ /F12 7.97 Tf 6.25 -2.27 Td[((i)fji=8>>>>>><>>>>>>:Pii2C)]TJ /F3 11.955 Tf 9.3 0 Td[(Dii2D0otherwisei)]TJ /F4 11.955 Tf 11.96 0 Td[(j)]TJ /F3 11.955 Tf 11.96 0 Td[(xijfij=08(i,j)jfijjuij8(i,j)PminiPiPmini8i2C0DjDnomj8j2D 3.2DCandACpowerows TheaboveconstraintsisfromOhm'sequationindirectcurrent(DC)network.InthecaseofanACnetwork,theycanonlyapproximatesacomplexsystemofnonlinearequations.Theissueofwhethertousethemoreaccuratenonlinearformulation,ortheapproximateDCformulation,isquitenoteasy.Ontheonehand,thelinearizedformulationcertainlyisanapproximationonly.Ontheotherhand,aformulationthatmodelsACpowerowscanproveintractableormayreectdifcultiesinherentwiththeunderlyingreal-lifeproblem. First,ACpowerowmodelstypicallyincludeequationsoftheform:sin(i)]TJ /F4 11.955 Tf 11.96 0 Td[(j))]TJ /F3 11.955 Tf 11.96 0 Td[(xijfij=08(i,j) Here,thefquantitiesdescribeactivepowerowsandthedescribephaseangles.Innormaloperationofatransmissionsystem,onewouldexpectthatijforanyarc(i,j)andthusitcanbelinearized.Hencethelinearizationisonlyvalidifweadditionallyimposethatji)]TJ /F4 11.955 Tf 12.41 0 Td[(jjbeverysmall.However,intheliteratureonesometimesseesthisverysmallconstraintrelaxedwhenthenetworkisnotinanormaloperativemode.Thenonlinearformulationgivesrisetoextremelycomplexmodels,butstudiesthatrequire 38

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multiplepowerowcomputationstendtorelyonthelinearizedformulationtogetsomeusefulandstraight-forwardinformation. Second,nomatterweuseanACorDCpowerowmodel,theresultingproblemshaveafarmorecomplexstructurethantraditionalsingle-ormulti-commodityowmodels,whichwouldleadtoocounter-intuitivebehaviorsimilartoBraess'sParadox. 3.3PreliminaryModel Theapproachtoidentifyingcriticalsystemcomponentsrstdevelopsanetworkvulnerabilitymodeltorepresenttheoptimal-attackproblemthataterroristgroupmightface(whichisthesimplest).Thegeneralmodelisamax)]TJ /F3 11.955 Tf 11.96 0 Td[(minproblem: max2minpc|psubjecttog(p,)b,p0. Aninterdictionplanisrepresentedbythebinaryvector,whosekthentryis1ifcomponentkofthesystemisattackedandis0otherwise.Foragivenplan,theinnerproblemisanoptimalpower-owmodelthatminimizesgenerationcostsplusthepenaltyassociatedwithunmetdemand,togetherdenotedbyc|p.Here,prepresentspowerows,generationoutputs,phaseanglesandunsatiseddemand;crepresentslinearizedgenerationcosts,andthecostsofunsatiseddemand.Theoutermaximizationchoosesthemostdisruptive,resource-constrainedinterdictionplan2,whereisadiscretesetrepresentingattacksthataterroristgroupmightbeabletoexecute.Inthismodel,gcorrespondstoasetoffunctionsthatarenonlinearin(p,).Theinnerprobleminvolvesasimpliedoptimalpower-owmodel,withconstraintg(p,)functionsthatarelinearinforaxed=^. 39

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3.3.1OptimalPowerFlowModel WeapproximateactivepowerowswithaDCmodel,whichneglectsreactivepowereffectsandnonlinearlosses.Thisapproximationisnormallyacceptableinthecontextoflong-term. IndexSet i2I nodes g2G generators l2L transmissionlines c2C consumernodes s2S substations l2LNodei linesconnectedtonodei l2LSubs linesconnectedtosubstations Parameters di originalloadofconsumernodei PLinel transmissioncapacityforlinel PGeng maximumoutputfromgeneratorg xl,rl resistance,reactanceoflinel(xlrl;seriessusceptanceisBl=xl=(r2l+x2l)) hg generationcostforunitg fi load-sheddingcostforconsumernodei DecisionVariable PLinel powerowonlineforlinel PGeng generationfromgeneratorg i phaseangleatnodei Si load-shedcostforconsumernodei 40

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FormulationofOptimalPower-Flowmodel:minPLine,PGen,,SXghgPGeng+XifiSisubjecttoPLinel=Bl((i))]TJ /F4 11.955 Tf 11.96 0 Td[((j))8l(i,j)XgPGeng)]TJ /F10 11.955 Tf 12.13 11.35 Td[(Xl(i,j)PLinel+Xl(j,i)PLinel=Xi(di)]TJ /F3 11.955 Tf 11.95 0 Td[(Si)8i)]TJ /F5 11.955 Tf 13.24 2.66 Td[(PLinelPLinelPLinel8l0PGengPGeng8g0Sidi8i Theobjectiveistominimizegeneratingplussheddingcostsmeasured.Firstconstraintapproximatesactivepowerowsonthelines.Secondconstraintmaintainspowerbalanceatthenodes.Thirdandfourthconstraintsetmaximumlinepowerowsandgenerating-unitoutputs.Minimumpoweroutputsaresettozeroforallgeneratingunitsforsimplicityhere,butextensionstononzerominimaarestraight-forward.Fifthconstraintstatesthatloadsheddingcannotexceeddemand.Thiswillbeaasubproblemoftheattackingmodeldescribednext. 3.3.2ContingencyModel Inthecontingencymodel,agroupofterrorists,willmakeacoordinatedsetofresource-constrainedattacksonthepowergrid.Wemakethefollowingassumptionsontheeffectofeachattack.However,wewouldextendthemeaningofattackingtoabroaderscopesuchthatacomponentfailure,impactofdailymaintenanceorplanningexpansionwouldbeincludedinthisformtouniversalapplicationpurpose. Lineattack:linesunderanattackareopened. Generatorattack:Thegeneratorisdisconnectedfromthegrid. Nodeattack:Alllines,generation,andloadconnectedtothenodearedisconnected. Substationattack:Allconsumernodesconnectedtothesubstationaredisconnected.(especiallyimportantforblackoutcausedbycomponentfailures) 41

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Terroristresourceconstraintscanaccommodateinformationfromintelligencesources,herewemodelthisfeaturethroughasimpleknapsackconstraint. AdditionalSetsandParameters I2I,G2G,L2L,S2S attackablenodes,generators,linesandsubstations,respectively. MNodei,MGeng,MLinel,MSubs resourcerequiredtoattacknodei,generatorg,linelandsubstations,respectively. M totalattackresourceavailable DecisionVariablesforAttacking I2I,G2G,L2L,S2S attackablenodes,generators,linesandsubstations,respectively. Nodei,Geng,Linel,Subs binaryvariablesthattakethevalue1ifnodei,generatorg,linelandsubstationsareattacked,respectively;0otherwise. FormulationofContingencymodel:minNodei,Geng,Linel,SubsF(Nodei,Geng,Linel,Subs)subjecttoXi2IMNodeiNodei+Xg2GMGengGeng+Xl2LMLinelLinel+Xs2SMSubsSubsMk=0,ifk2 (I[G[L[S) andwhereF(Nodei,Geng,Linel,Subs)=minPLine,PGen,,SXghgPGeng+XifiSi 42

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subjecttoPLinel=Bl((i))]TJ /F4 11.955 Tf 11.95 0 Td[((j))(1)]TJ /F4 11.955 Tf 11.96 0 Td[(Linel)(1)]TJ /F4 11.955 Tf 11.96 0 Td[(Nodei)(1)]TJ /F4 11.955 Tf 11.95 0 Td[(Nodej)Ysjl2LSubs(1)]TJ /F4 11.955 Tf 11.96 0 Td[(Subs)8l(i,j)XgPGeng)]TJ /F10 11.955 Tf 12.13 11.36 Td[(Xl(i,j)PLinel+Xl(j,i)PLinel=Xi(di)]TJ /F3 11.955 Tf 11.96 0 Td[(Si)8i)]TJ /F5 11.955 Tf 13.24 2.66 Td[(PLinel(1)]TJ /F4 11.955 Tf 11.95 0 Td[(Linel)(1)]TJ /F4 11.955 Tf 11.95 0 Td[(Nodei)(1)]TJ /F4 11.955 Tf 11.96 0 Td[(Nodej)Ysjl2LSubs(1)]TJ /F4 11.955 Tf 11.95 0 Td[(Subs)PLinelPLinel(1)]TJ /F4 11.955 Tf 11.95 0 Td[(Linel)(1)]TJ /F4 11.955 Tf 11.96 0 Td[(Nodei)(1)]TJ /F4 11.955 Tf 11.96 0 Td[(Nodej)Ysjl2LSubs(1)]TJ /F4 11.955 Tf 11.96 0 Td[(Subs)8l0PGengPGeng(1)]TJ /F4 11.955 Tf 11.95 0 Td[(Geng)8g0Sidi8i Thismodelmaximizesgenerationcostsplusload-sheddingcosts,whichisevaluatedthroughtheinnerminimizationproblemthatconsistsoftheoptimalpower-owmodelwithattackedcomponentsremoved.Attheouterlevel,therstinnerconstraintreectstheterrorists'optionstoattackdifferentcombinationsofcomponentsinthegridwithoutexceedingtheirresources.Thesecondinnerconstraintdenesterroristactionsasbinaryvariablesandensurenon-attackingofcertaingridcomponents. Theconstraintsoftheouterproblemaresimilarasthepureoptimalpower-owone,withtakingintoconsiderationthecomponentsthathavebeenattackedthroughthebinarystatusvariables.Thecomputationalchallengeisfromthemax-minstructureoftheproblem.Theoptimalobjectivevalueofthelinearizedversionoftheinnerminimization,asafunctionofcontinuous,isconvex.Hencetheentireprogramisthemaximizationofaconvexfunction,whichisusuallyadifculttask,whileitisjustthesimplestscenarioconsidered.However,wecanstilldosomelinerizationtomakethenalcalculationeasier. 43

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Fortheconstraintsoftheouterproblemwith(1)]TJ /F4 11.955 Tf 10.21 0 Td[(Linel),(1)]TJ /F4 11.955 Tf 10.22 0 Td[(Nodei),(1)]TJ /F4 11.955 Tf 10.21 0 Td[(Geng)and(1)]TJ /F4 11.955 Tf -458.7 -23.9 Td[(Subs),sincethedecisionvariablearebinary,theycouldbelinearized.Hereisasimpliedproof. GivenX,Yasbinaryvariables,Wasacontinuousvariable,casaparameter,wehavethatthesetwoconstraints(1)and(2)areequivalent: WcXY; (3) 8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:letZ=XY,thenWcZ;ZX;ZY;ZX+Y)]TJ /F5 11.955 Tf 11.95 0 Td[(1;Z0; (3) Therefore,wecanclaimthatthesetwoconstraints(3)and(4)areequivalent.Here,X,Y,Zarebinaryvariables,Wasacontinuousvariable,casaparameter. WcXYZ; (3) 8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:WcV;VXandVYandVZ;VX+Y+Z)]TJ /F5 11.955 Tf 11.96 0 Td[(2;VX)]TJ /F5 11.955 Tf 11.96 0 Td[(1andVY)]TJ /F5 11.955 Tf 11.95 0 Td[(1andVZ)]TJ /F5 11.955 Tf 11.95 0 Td[(1;V0; (3) 44

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Hereisaproof:Obviously,usingthefactfrom(1)and(2),wecanhave: 8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:letU=XY,thenWcUZ;UX;UY;UX+Y)]TJ /F5 11.955 Tf 11.96 0 Td[(1;U0; (3) Continuingthesimilarprocedure,wecanhave: 8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:letV=UZ,thenWcV;VU)VXandVY;VZ;VU+Z)]TJ /F5 11.955 Tf 11.96 0 Td[(1)VZ)]TJ /F5 11.955 Tf 11.95 0 Td[(1andVX+Y+Z)]TJ /F5 11.955 Tf 11.95 0 Td[(2;V0; (3) Byinduction,thoseconstraintsfortheouterproblemwith(1)]TJ /F4 11.955 Tf 11.65 0 Td[(Linel),(1)]TJ /F4 11.955 Tf 11.65 0 Td[(Nodei),(1)]TJ /F4 11.955 Tf -458.7 -23.91 Td[(Geng)and(1)]TJ /F4 11.955 Tf 11.96 0 Td[(Subs)arelinearizable. 45

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CHAPTER4ECONOMICANALYSISOFTHEN)]TJ /F3 11.955 Tf 11.96 0 Td[(KPOWERGRIDCONTINGENCYSELECTIONANDEVALUATION Contingencyanalysisisimportantforprovidinginformationaboutthevulnerabilityofpowergrids.Manymethodshavebeenpurposedtousetopologicalstructuresofpowergridsforanalyzingcontingencystates.Consideringfailuresofbusesandlines,wepresentandcompareseveralgraphmethodsforselectingcontingenciesinthispaper.Anewmethod,calledcriticalnodedetection,isintroducedforselectingcontingenciesconsistingoffailuresonbuses.Besidesthesemethods,weincludeaninterdictionmodelwhichprovidestheworstcasecontingencyselection.Ourmeasurementforcontingencyevaluationistomaximizethesocialbenet,ortominimizethegeneratingandloadsheddingcost.Comparingwithothermeasurementsforcontingencyselection,ourmodelisbasedoneconomicanalysisandisreasonableforevaluatingtheselectedcontingencystate.Additionally,acontingencyconsistingofbothbusesandlinesisalsostudied.Thischapterisbasedonapublishedpapter[ 50 ]. 4.1Nomenclature IndicesandIndexSets I:setofbuses,includingsubstations,generatorsandloadconsumers Let1gifori2Ibetheindicatorsuchthat1gi=1ifiisagenerator,and1gi=0otherwise;andlet1dibetheindicatorsuchthat1di=1ifiisaloadconsumer,and1di=0otherwise (i,j)2L:transmissionlines,i,j2Iandi
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bij:susceptanceofline(i,j);bijiscomputedfromtheresistanceandthereactanceofline(i,j)2L hi:generationcostatbusi2I ri:penaltycostforload-sheddingati2I k:thenumberofcomponents(busesorlines)tobefailed Decisionvariables gi:powergenerationatnodei2I fij:powerowonline(i,j)2L si:loadshedatnodei2I i:phaseangleatnodei2I i:i=1ifthebusiisselectedasoneofkcomponentsforfailures;i=0otherwise ij:ij=1iftheline(i,j)isselectedasoneofkcomponentsforfailures;ij=0otherwise 4.2Background ContingencyanalysisisakeyfunctionintheEnergyManagementSystem(EMS),whichassessestheabilityofthepowergridtosustainvariouscombinationsofpowergridcomponentfailuresbasedonstateestimates.Thecontingencymeansasetofunexpectedeventshappeningwithinashortduration.Theunexpectedeventscanbefailuresofbuses(generators,substations,etc)ortransmissionanddistributionlines.Inthepast,theN)]TJ /F5 11.955 Tf 13 0 Td[(1contingency,orN)]TJ /F5 11.955 Tf 12.99 0 Td[(1reliability,analyzingthefailureofonecomponent(abusoraline)hasbeenanactiveresearcharea[ 66 100 ].However,analysisoftheN)]TJ /F5 11.955 Tf 12.43 0 Td[(1reliabilityisnotsufcientformanyrealapplicationswithmultiplefailurestodiscoverthevulnerabilitiesofpowergrids. TheN)]TJ /F3 11.955 Tf 12.17 0 Td[(kcontingencyanalysisforfailuresofmultiplecomponents(totallykbusesandlines)canreectalargervariationofvulnerabilitiesofapowersystem.However,combinatorialnumberofcontingencystatesimposesasubstantialcomputational 47

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burdenforanalysis.Generally,contingencyanalysisconsistsoftwosteps:contingencyselectionandevaluation[ 93 ].Contingencyselectionistoselectasetofpotentialcontingencies,andtoreducethesetofcontingenciesefciently;Contingencyevaluationexaminesseverecontingenciesindetails.Sometimes,vulnerabilityanalysis,whichaimstondasetofsmallgroupoflineswhoselosscancauseasevereblackout,isalsoused. Methodsforcontingencyselectionincludeautomaticmethods,onlinesecurityanalysis,rankingbyapproximatepowerowsolutions,concentricselection,sparsevectormethods,geneticmethods,hybridmethodsandetc.Thesemethodsarebrieyreviewedin[ 35 93 ].Recently,probabilityanalysis[ 33 ],powerowtrafc[ 5 126 ],limitationsongenerationandlinecapacity[ 27 ],andseveralothermethodsbystudyingthetopologystructuresofpowergrids[ 68 ]areusedforvulnerabilityanalysis.In[ 68 ],Hinesetal.reviewedandcomparedthreevulnerabilitymeasuresforcontingencyevaluation:characteristicpathlength,connectivity,blackoutsize.Additionally,theyalsocomparedthecontingencyselectionmethods,suchasrandomfailures,degreeattack,maximum-trafcattack,minimum-trafcattackandbetweennessattack,underthreemeasures.Theanalysisconsideredthefailuresofbusesonly.In[ 68 ],theyconcludedthatunderthemeasureofcharacteristicpathlength,thosebusesselectedbythenodebetweennessmethodaremostcrucialforthepowergrid;underthemeasureofconnectivityloss,thosebusesselectedbydegreebasedmethodaremostcrucial;andunderthemeasureofblackoutsize,thosebusesselectedbymaximum-trafcmethodaremostcrucial. Inthispaper,besidesmethodscomparedin[ 68 ],weintroducethecriticalnodedetectionmethod[ 14 ]tondbuseswhoplaysimportantrolesinapowergrid.Forfailureonbuses,weconsiderthesegraphalgorithms:random,degreebased,maximum-trafc,minimum-trafc,nodebetweenness,andcriticalnodedetectionmethods;Forfailuresonlines,thegraphalgorithmconsiderstheedgebetweenness. 48

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Toevaluateselectedcontingencies,economicanalysisofcascadingfailuresshouldbeincluded.Afterselectingcontingenciesbythesegraphalgorithms,weevaluatetheeconomicconsequenceofthesecontingencyselectionbyusingthedirectcurrentoptimalpowerow(OPF).TheOPFminimizesthegeneratingandloadsheddingcostormaximizesthesocialbenetinfact.Inthepast,theeconomicanalysisforreliabilityorcontingencyanalysishavebeenstudiedinsomeproblems,forexample,N)]TJ /F5 11.955 Tf 13.05 0 Td[(1reliabilityin[ 100 ],andunitcommitment[ 136 ].Inthispaper,basedtheOPFmodelwhoseobjectiveisassumedtominimizethegeneratingandloadsheddingcost,weusethisobjectiveaseconomicanalysistoevaluateselectedcontingencystates.Bythisanalysis,weassumethatunderfailuresonsomebusesorlines,thepowersystemisstilloperatedinanoptimizedway.Especially,hereweconsidertheloadsheddingcost,whichcanmeasuretheunsatisfactorydemandincaseoffailuresinasystem. Additionally,theworstcontingencyselectioncanbeobtainedandevaluatedbyinterdictionmethods.Weadapttheinterdictionmodelintroducedby[ 111 112 ]toselectkcomponentsforfailuresandalsoevaluatethembyeconomicanalysisinonemodel.Theinterdictionmodelistoidentifycriticalsetsofapowergrid'scomponents,e.g.,generators,transmissionlines,andtransformers,byidentifyingmaximallydisruptive,coordinatedattacksonagrid,whichaterroristgroupmightundertake[ 111 ].Bymeansofmaximallydisruptive,themodelistheworstcaseeconomicanalysisofthecontingency. Thecriticalnodedetectionmethodisformulatedbyamixedintegerlinearprogram(MILP),andalsotheinterdictionmethodbyamixedintegernonlinearprogram.Someheuristicmethodforcriticalnodedetection,nodeandedgebetweennessarereviewed,whilelinearizationtechniquesareproposedfornonlinearprograms.AlldiscussedgraphalgorithmsandinterdictionmethodsarecomparedontwoIEEEtestsystems. 49

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4.3OptimalPowerFlowModelforEconomicAnalysis Asdiscussedinprevioussection,aftercontingencyselectionbygraphalgorithms,whichdecidethevaluesof=(1,,jIj),=(ij:i=1,,jIj,j=i+1,,jIj),theOPFmodelisusedforevaluatingtheselectedcontingencies.ForagivenIEEEtestsystemandgivenvalueofk(usuallyrangesfrom2to10),eachgraphalgorithmspresentsacontingency,.Thusinthefollowingmodel( 6a )-( 4 ),,aretheknownfailedcomponentsinapowergrid.Tomaximizethesocialbenet,weminimizethegeneratingandloadsheddingcostPi2I(higi+risi)intheOPFmodelinthepowergridswithfailedbusesorlines.Theeconomicanalysismodelforpowergridswithfailedcomponentscanbeformulatedasfollows: z(,)=ming,s,f,Xi2I(higi+risi) (4)s.t.fij=bij(i)]TJ /F4 11.955 Tf 11.96 0 Td[(j)(1)]TJ /F4 11.955 Tf 11.96 0 Td[(i)(1)]TJ /F4 11.955 Tf 11.95 0 Td[(j)(1)]TJ /F4 11.955 Tf 11.96 0 Td[(ij),8(i,j)2L (4))]TJ /F5 11.955 Tf 13.17 2.66 Td[(FijfijFij,8(i,j)2L (4)Xjfji+gi=Xjfij+(Di)]TJ /F3 11.955 Tf 11.95 0 Td[(si),8i2I (4)0giGi(1)]TJ /F4 11.955 Tf 11.95 0 Td[(i),8i2I (4)0siDi,8i2I (4) Forconstraint( 4 )online(i,j),itisanapproximateactivepowerowontransmissionline.Ifoneendingbusofthislineisfailed(i.e.,i=1orj=1),orthelineisfailed(i.e.,ij=1),thepowerowonthislineis0.Theconstraint( 4 )ensuresthemaximumpowerowoneachline.Theconstraint( 4 )ensuresthethepowerbalanceateachbus,wheretheserveddemandatbusiis(Di)]TJ /F3 11.955 Tf 12.34 0 Td[(si).Theconstraint( 4 )limitsthemaximumgeneratingoutputGi(1)]TJ /F4 11.955 Tf 12.04 0 Td[(i).Ifthegeneratoratbusiisfailed(i.e.,i=1),nopowerisprovidedbythisgenerator.Theconstraint( 4 )statesthelimitationofloadsheddingbythemaximumload. 50

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Thisformulationislinearprogramwithgivenvaluesof,,andcanbesolvedefcientlybysimplexmethodinCPLEX. 4.4GraphAlgorithmsforContingencySelection Inthissection,graphalgorithmsareusedforcontingencyselectionofbusesorlines,respectively. 4.4.1SelectionofKBusesbyCriticalNodeMethodandOtherMethods Usingthecontingencyselectionmethodsin[ 68 ],weaddanewmethod,calledcriticalnodedetection,forcontingencyselectionofbuses.Forcontingencyselectiononbuses,othermethods,suchaspowergenerationlimitsin[ 27 ]andrankingindex[ 93 ],arealsousedforanalysis. Inthissection,weassumeallfailures,orattackshappenonbuses.IntheN)]TJ /F3 11.955 Tf 12.59 0 Td[(kcontingencyanalysismodel,thenumberoffailedbusesisk,i.e.,Pi2Ii=k. Therandomfailure[ 68 ],selectskbusesforfailurewithequalprobability.Thus,kbusesfromIarerandomlyselectedandxedi=1.Allotherisaresettobe0.Thedegreebasedmethodistoselectkbusesstartingwiththehighestdegreebus,tillthekthhighestdegree,andseti=1forthesecorrespondingkbuses. Themaximum-trafcandminimum-trafc[ 5 126 ]methodsarebasedonthepowerowofeachbus:Ti=jgi)]TJ /F5 11.955 Tf 11.95 0 Td[((Di)]TJ /F3 11.955 Tf 11.96 0 Td[(si)j+Xj:j2Ijbij(i)]TJ /F4 11.955 Tf 11.96 0 Td[(j)j,wherethetermjgi)]TJ /F5 11.955 Tf 12.3 0 Td[((Di)]TJ /F3 11.955 Tf 12.31 0 Td[(si)jistheabsolutevalueofnetpowerinjectionintobusibygeneratorsandloadconsumers,andthesummationpartisthesumofpowerowsinandoutofbusi.Themaximum-trafcmethodselectskbuseswithhighestTis,whiletheminimum-trafcmethodselectskbuseswithlowestTis.Inthispaper,weusetheOPFmodel(seesectionbelow)atnormalstatetondthetrafcTioneachbus. 51

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Thenodebetweennessmethod[ 5 35 ]istondkbuseswithhighestnodebetweenness.Thenodebetweenness[ 54 ]forbusiisdenedasCB(i)=Xs6=i6=t2Inst(i) nst,wherenstisthenumberofshortestpathsfromstot,andnst(i)isthenumberofshortestpathsfromstotthatpassthroughabusi.Tocalculatethebetweennessofallbusesinapowergridneedscalculatingtheshortestpathsbetweenallpairsofbusesinapowergrid.BymodiedFloydWarshallalgorithm,andJohnson'salgorithmforsparsegraphs,thecomputationalcomplexitycanbeO(jIj3)andO(jIj2logjIj+jIjjLj),respectively.ByusingBrandes'algorithm[ 25 ],itcanbemoreefcientwithcomplexityO(jIjEj). Thecriticalnodedetectionproblem(CNP),introducedin[ 14 ],istodetectasetofverticesinagraphwhosedeletionresultsinthegraphhavingtheminimumpairwiseconnectivitybetweentheremainingvertices.Assumekbusesarefailedwithi=1,andi=0forallotherbuses.InCNP,thesetfi:i=1gofbusesistobeselected.Letxij=1denotethatbusiandjareinthesamecomponentafterdeletionofkbuseswithincidentlines,andxij=0otherwise.TheconstraintsofCNPincludethatthenumberofbusesinthedeletedsubsetisk,i.e.,Pi:i2Ii=k,andallthreelinesinLhavetherelationthatiftwolinesareintheresultedgraph,anotherlineisalsointheresultedgraph,i.e.,maxfxij+xjt)]TJ /F3 11.955 Tf 12.4 0 Td[(xit,xij)]TJ /F3 11.955 Tf 12.4 0 Td[(xjt+xit,)]TJ /F3 11.955 Tf 9.3 0 Td[(xij+xjt+xitg1.Therelationbetweenxijandicanbeexpressedasxij+i+j1undertheobjectiveofminimization.TheobjectiveofCNPistominimizethepairwiseconnectivitybetweentheremainingvertices.Therefore,theCNPforselectingkbusesinapowergridN=(I,L)can 52

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formulatedasfollows:minXi,j:vi,vj2Vxij (4)s.t.xij+i+j1,8(i,j)2L (4)xij+xjt)]TJ /F3 11.955 Tf 11.95 0 Td[(xit1,8i,j,t2I (4)xij)]TJ /F3 11.955 Tf 11.95 0 Td[(xjt+xit1,8i,j,t2I (4))]TJ /F3 11.955 Tf 11.95 0 Td[(xij+xjt+xit1,8i,j,t2I (4)Xi:i2Ii=k (4)i,xij2f0,1g,8i,j2I (4) ThisproblemisNP-hardandtheformulation( 4 )-( 4 )forCNPisamixedintegerlinearprogram,whichcanbesolvedbyCPLEX.In[ 14 ],heuristicmethodsbasedmaximumindependentsetproblemisproposedwithcomputationalcomplexityO(jIj2jLj).Inthispaper,forsmallIEEEtestsystem,MILPbyCPLEXisusedandforlargesystems,heuristicmethodsareused.In( 4 ),ifiisrequiredtosatisfyi=0fori2fi2I:1gi=0g,allkfailuresarehappeningongenerators. 4.4.2SelectionofKLinesbyEdgeBetweenness Forfailuresonlines,weuseedgebetweenness[ 35 ]forcontingencyselection.Theedgebetweennessisadaptedfromnodebetweenness,anditcanbeexpressedasC(i,j)B=Xs,t2Inst(i,j) nst, wherenstisthenumberofshortestpathsfrombusstobust,andnst(i,j)isthenumberofshortestpathsfrombusstotthatpassthroughline(i,j).Thealgorithmsforndingthenodebetweennesscanalsobeusedtondtheedgebetweenforeachlineofapowergrid.Afterobtainingklineswithhighestedgebetweenness,wesetij=1fortheselinesandsetij=0forallotherlines. 53

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Recently,studiesofvulnerabilityforlinesinpowergridcanbefoundin[ 23 44 106 ],whereoptimizationmethodsareusedtondsmallgroupsoflineswhosefailurecancausesevereblackout.Additionally,contingencyselectionoflinesisalsostudiedbytnessfunctionoflinecapacity[ 77 ],linecapacitymethod[ 27 ]andetc. 4.5AN)]TJ /F3 11.955 Tf 11.96 0 Td[(kContingencySelectionbyWorstCaseInterdictionAnalysis Inprevioussection,bygraphalgorithms,westudythecontingencyselectionforbothbusesandlines.However,exceptthemaximumandminimumtrafcmethodsforselectingbuses,allmethodsarefromtheprospectiveofgraphs,ortopologystructuresofpowergrids.Inthefollowing,theinterdictionmodelisusedfortheworstcaseanalysis. Theinterdictionmodelforpowergridwasproposedin[ 111 112 ],whereacontingencyofmaximallydisruptiontoapowergridcanbederived.Theinterdictionmodelhasbeenusedtomodelintelligentadversaries,e.g.,terrorists.Iffi:i=1gandf(i,j):ij=1gdenotethesetsofdestroyedbusesandlines,respectively,theconstraintPi2Ii+P(i,j)2Lij=kcanbeinterpretedasaresourceconstraintonterrorists.Thedecisionvariablestomaximallydestroyapowergridinclude,.Afterthedisruptive,thesystemoperatorsolvesOPFbyreadjustingpowergenerations,demandsandpowerowstominimizetheconsequence.Theinterdictionmodelisformulatedasfollows: max,ming,s,f,Xi2I(higi+risi) (4)s.t.Constraints( 4 )-( 4 ) (4)Xi2Ii+X(i,j)2Lij=k (4)i,ij2f0,1g,8i2I,(i,j)2L (4) Iftheconstraintij=0forall(i,j)2Lisaddedtothismodel,allinterdictionscanonlyhappenonbuses.Similarly,iftheconstrainti=0isadded,allinterdictionscan 54

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onlyhappenonlines.Additionally,ifweaddPi2Ii=k1andP(i,j)2Lij=k2,wherek1+k2=k,thenumberofinterdictedbuses(orlines)arelimited. Inthisinterdictionanalysismodel,forxed,,theminpartin( 5 )withconstraints( 5 )constructtheOPFmodel.Assumetheassociateddualvariablesforconstraint( 4 )isij,andij,ijfor( 4 ),ifor( 4 ),ifor( 4 ),ifor( 4 ).Takingthedualityofthispartandcombiningunknownvariables,,wecanobtainthefollowingequivalentformulationfortheproblem( 5 )-( 5 ):max,,,,,,,X(i,j)2LFij()]TJ /F4 11.955 Tf 9.3 0 Td[(ij+ij)+Xi2IDi(i+i)+Xi2IGi(1)]TJ /F4 11.955 Tf 11.95 0 Td[(i)i (4)s.t.i+ihi gi (4)i+iri si (4)ij+ij+ij)]TJ /F4 11.955 Tf 11.96 0 Td[(i+j=0 fij (4))]TJ /F10 11.955 Tf 12.1 11.36 Td[(Xj:j>ibij(1)]TJ /F4 11.955 Tf 11.96 0 Td[(i)(1)]TJ /F4 11.955 Tf 11.96 0 Td[(j)(1)]TJ /F4 11.955 Tf 11.96 0 Td[(ij)ij+Xj:j
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thetermvij=ijuijbyvijij)]TJ /F3 11.955 Tf 11.95 0 Td[(Ub(1)]TJ /F3 11.955 Tf 11.96 0 Td[(uij),vijLbuij,vijij)]TJ /F3 11.955 Tf 11.96 0 Td[(Lb(1)]TJ /F3 11.955 Tf 11.96 0 Td[(uij),vijUbuij, (4) andthetermwi=(1)]TJ /F4 11.955 Tf 11.95 0 Td[(i)ibywii)]TJ /F3 11.955 Tf 11.96 0 Td[(Ubi,wiLb(1)]TJ /F4 11.955 Tf 11.95 0 Td[(i),wii)]TJ /F3 11.955 Tf 11.95 0 Td[(Lbi,wiUb(1)]TJ /F4 11.955 Tf 11.96 0 Td[(i), (4) whereLb,Ubarelowerandupperboundsforij,iandcanbechosenassmallenoughorlargeenoughconstants,respectively. Bysolvingtheproblem( 4 )-( 4 )withlinearizations( 4 ),( 4 )and( 4 ),whichconstructamixedintegerlinearprogram,wecanequivalentlyobtainsolutionstotheoriginalproblem( 5 )-( 5 ).Inthispaper,weuseCPLEXtosolvetheequivalentlinearprogramformulation. 4.6NumericalExperimentsandComparisons ThelinearprogramformulationsforOPFandinterdictionmodelsareimplementedinC++andusingCPLEX11.0viaILOGConcertTechnology2.5.Inthissection,wetestallproposedalgorithmsandmethodsforcontingencyselectionandevaluationontwobussystems:IEEE-30-BussystemtestcaseandRTS-96testsystem[ 1 ]. Inthissection,werstconsiderN)]TJ /F3 11.955 Tf 12.1 0 Td[(kcontingencywithkvaryingfrom1to10.Busandlinefailuresareinvestigatedseparatelyforcomparingcosts.Finally,weconsiderfailuresofbusesandlinesinonecontingencybyinterdictionmodel. ForexperimentsonIEEE-30-BusSystem,thegeneratingcostisnotconsidered,i.e.,settinghi=0fori2I.TherelationshipbetweenloadsheddingcostandthenumberoffailedbusesunderthesegraphalgorithmsandtheinterdictionmethodisshowninFig. 4-2 .Fromthisgure,wecansee,underthemeasurebyminimizingloadshedding,theinterdictionmethodpresentsmuchworsepowergridperformancethanothercontingencies.Thedegreebased,nodebetweenness,maximum-trafcandcriticalnodedetection(CNP)methodsallselectthecrucialbusesforthissystem,whilethe 56

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Figure4-1. IEEE-30-bussystem(From:PowerSystemsTestCaseArchive) minimum-trafccannotpresentcrucialselection.Excepttherandomfailures,allcostsproposedbyothermethodsaremonotonouslyincreasingaswellasthenumberkincreases.Additionally,whenk6fortheworstcaseinterdictionmethod,thewholepowersystemiscompletelyfailed(nogenerationandnosatiseddemand). InFig. 4-3 forIEEE-30-BusSystem,assumethecontingencyconsistingonlyoflines,andassumeloadsheddingistheonlyconsideredcost.Similarly,theinterdictionmethodpresentstheworstcaseselectionoflinesasweexpect.AsthenumberKoffailuresisincreasing,theloadsheddingcostincreasesaswell. Asdiscussedinprevioussection,theinterdictionmethodcanselectthecontingencyconsistingfailuresonbothbusesandlines.InFig. 4-4 ,weshowthecostoffailuresonbothbusesandlines.Forgivenvalueofk,byaddingPi2Ii=k1andP(i,j)2Lij=k2in 57

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Figure4-2. Loadsheddingcostvs.failedbuses(IEEE-30-BusSystem) Figure4-3. Loadsheddingcostvs.failedlines(IEEE-30-BusSystem) theinterdictionmodel,wherek1+k2=k,weassumetherearek1=dk=2efailedbusesandk2=bk=2cfailedlines.Inthisgure,wecanndthatthecontingencyconsistingofonlybuseshasthemostinuenceforthesystem.Whenkislargeenough(e.g.,k6forcasesofbusesonlyandbusesandlines,k8forthecaseoflinesonly),thewholesystemiscompletelyfailed.Analyzingthereasonbehindthatbusesareimportantnodes,onepossibleexplanationisthatourdatasetsassumethatalllineshavehighcapacitycomparingthosegenerationsonbuses. ToimproveefciencyofthecomputationforMILPproblemsinlargesystems,forexample,theinterdictionmethodforRTS-96System,thefailedbusesorlineswithhighappearanceswhenKissmall,canbexedfordecreasingthecomputational 58

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Figure4-4. Loadsheddingcostvs.failedbuses/lines(IEEE-30-BusSystem) timewhenKislarge.ForexperimentsonRTS-96System,bothgeneratingandloadsheddingcostsareconsidered.InFig. 4-5 ,therelationshipbetweengeneratingandloadsheddingcostandthenumberoffailedbusesundergraphalgorithmsandtheinterdictionmethodispresented.Fromthisgure,wecansee,underthemeasurebyminimizinggeneratingandloadsheddingcost,theinterdictionmethodpresentstheworstcaseselectionofcontingencies.Themaximum-trafcmethodprovidesthesecondworstcontingenciesconsistently,whiletheCNPmethodprovidesthethirdworstcontingenciesinmostofvaluesfork.Similarly,excepttherandomfailures,allcostsproposedbyothermethodsaremonotonouslyincreasingaswellasthenumberkincreases. Figure4-5. Generatingandloadsheddingcostvs.failedbuses(RTS-96System) 59

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InFig. 4-6 forRTS-96System,assumethecontingencyconsistingonlyoflines.Similarly,theinterdictionmethodpresentstheworstcaseselectionoflinesasweexpect.AsthenumberKoffailuresisincreasing,thegeneratingandloadsheddingcostincreasesaswell. Figure4-6. Generatingandloadsheddingcostvs.failedlines(RTS-96System) 4.7ConclusiveRemarks WestudytheN)]TJ /F3 11.955 Tf 12.72 0 Td[(kcontingencyselectionandevaluation,bywhichthefailuresofmultiplecomponentscanbeselectedandevaluatedtoreectvulnerabilitiesofapowersystem.Afterreviewingsomegraphalgorithms,weintroducedthecriticalnodedetectionmethod.Thesemethodscanndthecontingenciesconsistingoffailuresonlyonbuses,likegenerators,substations,etc.Forcontingencyconsistingoffailuresontransmissionordistributionlines,theedgebetweennessmethodisused.Bymaximizingthesocialbenet,equivalentlyminimizingthegeneratingandloadsheddingcost,weuseeconomicanalysisforevaluatingselectedcontingencies.Fortheworstcasecontingencyanalysis,theinterdictionmethodwithlimitedkcomponents'failuresisusedforbothcontingencyselectionandevaluation.Comparingwithgraphalgorithmsforcontingencyselection,theinterdictionmodelcanselectcontingenciesconsistingbothbusesandlines.BynumericalexperimentsontwoIEEEpowergridtestcases,wendthatinterdictionalwaysselectthemostcrucialcomponents,whilethemaximum-trafc 60

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methodandcriticalnodedetectionmethodselectthesecondmostcrucialbuses.Forallcases,theminimum-trafcmethodndtheleastcrucialones. FutureresearchdirectionsincludeconsideringthesecurityconstraintsintheOPFmodel,usingalternatingcurrentpowermodels,etc.Inaddition,usingcontingencyanalysisforndingvulnerabilitiesofpowersystemsshouldprovidesuggestionsforsystemplannerandoperator.Todesignarecoveryplanforapowergrid,ortodesignawarningsystemforsystemsreachingakindleveloffailuresisalsoimportant. 61

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CHAPTER5EXTENSIONWORKOFECONOMICANALYSISOFTHEN)]TJ /F3 11.955 Tf 11.96 0 Td[(KPOWERGRIDCONTINGENCYWITHLINESWITCHING Withthearisingawarenessofthesecurityissuesinpowersystems,contingencyanalysisforunexpectedfailureshasbeenanimportantprobleminrecentyears.Inordertoleveragegridcontrollability,thepowergridoperatorscanapplythetechniqueoftransmissionlineswitchingtomitigatetheimpactcausedfromtheattacking.However,regardingtheexponentialnumberofcontingenciesandswitchingchoicesoflines,itbringssomecomputationalissueofnonconvexdiscreteoptimizationproblem,whichdoesnothavewell-developednon-exponentialalgorithm.Inthissection,resultsareshowntoprovidesomeinsightaboutapplyinglineswitchingwithN)]TJ /F3 11.955 Tf 12.89 0 Td[(kpowergridcontingencymodel. 5.1Nomenclature IndicesandIndexSets I:setofbuses,includingsubstations,generatorsandloadconsumers Let1gifori2Ibetheindicatorsuchthat1gi=1ifiisagenerator,and1gi=0otherwise;andlet1dibetheindicatorsuchthat1di=1ifiisaloadconsumer,and1di=0otherwise (i,j)2L:transmissionlines,i,j2Iandi
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ri:penaltycostforload-sheddingati2I k:thenumberofcomponents(busesorlines)tobefailed Decisionvariables gi:powergenerationatnodei2I fij:powerowonline(i,j)2L si:loadshedatnodei2I i:phaseangleatnodei2I i:i=1ifthebusiisselectedasoneofkcomponentsforfailures;i=0otherwise ij:ij=1iftheline(i,j)isselectedasoneofkcomponentsforfailures;ij=0otherwise ij:ij=1iftheline(i,j)isswitchedonbyoperator;ij=0otherwise 5.2Background Traditionally,thesystemoperatortreatstransmissionassets(linesortransformers)asstaticassetswithinOptimalPowerFlow(OPF)problems,whicharethenetworkowproblemfortheelectricalgrid.Therefore,powergridismodeledasinterdictionmodelwithcontingency,overwhichthesystemoperatordispatchesgeneratorstominimizecost.Inaddition,systemoperatorscanchangethetopologyofsystemstoimprovegridcontrollabilityandincreasetransmissioncapacity.Inthemeanwhile,thealreadybuilt-inlinecapacityredundanceoftheelectricgridensuringthemandatoryreliabilitystandard,canprovidethepossibilitytoimprovetheefciencyofthenetworkagainstworst-casescenarios. Glavitsch[ 56 ]introduceshowtousethetransmissionswitchinginresponsetocontingencyconditionsasacorrectivemechanism.Hemodelsthisproblemwithamathematicalformulationandcomparessomesearchingtechniquessolvetheproblem.Bakitzisetal.[ 20 ]usesaMixedIntegerProgrammingmrthod(acontinuousvariableformulationfortheswitchingdecisionaswellasdiscretecontrolvariables)tomodelthe 63

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thetransmissionswitchingproblems.Bacheretal.[ 17 ]havearesearchonACpowernetworkwithtransmissionlineswitchinginordertomitigatelineoverloadingissue.However,theyassumethatthegenerationdispatchisalreadydeterminedandxedwhichdonottaketheadvantageofoptimizationoverthenetworktopology.Rolimetal.[ 110 ]haveareviewpaperforthetransmissionswitchingmethodsandthesolutiontechniques.Mazietal.[ 84 ]proposeamethodofaheuristictechniquetosolvetheproblemofalleviatingtransmissionlineoverloadingduringcontingencybytheuseoftransmissionswitching. TheinitialconceptofthedispatchableOptimalPowerFlow(OPF)networkmodelwasrstproposedbyO'Neilletal.[ 99 ].Fisheretal.[ 52 ],furtherintroducedtheconceptofincorporatingtheswitchingoftransmissionlineintodispatchoptimizationformulations.Gorenstinetal.[ 57 ]studyasimilarproblemwithtransmissionswitching,andtheyusealinearintegerapproximateOPFformulationtosolvetheproblembasedonbranchandbound.Shaoetal.([ 113 ],[ 114 ])workontheuseoftransmissionswitchingtosolvetheproblemsoflineoverloadingandvoltageviolationsbased.Theyproposeanewsolutionusingasparseinversetechniqueandafastdecoupledpowerowtondthebestswitchingactionswithlesscomputationtime. Recently,transmissionlineswitchinghasbeenanalyticallystudiedinordertoreducedispatchcostinpowersystemscheduling,andupto25%dispatchcostcanbesaved[ 52 66 67 75 ].InBacheretal.[ 18 ],theydemonstratesthatcomparedwithtraditionaltechnique,itcouldreduceelectricallossesinthenetworkbytemporarilyopeningatransmissionline.Fliscounakisetal.[ 53 ]proposedamixedintegerlinearprogramtotominimizelosseswithoptimaltransmissionlinetopology.Traditionally,modeldosearchfortheoptimaltopologybutdonotco-optimizethegenerationtomaximizethemarketsurplus.Incontrasttotheseapproaches,theiroptimaltransmissionswitchingmodelmaximizesthemarketsurplusbyco-optimizingthetransmissiontopologyalongwithgeneration. 64

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SpecialProtectionSchemes(SPSs),arebecomingamainstreamprotocolinelectricgridoperationssincetheycanbeusedtosolveavarietyofissuesfrommaintainingvoltagestabilitytoacorrectiveactionthatistakenonceaspeciccontingencyoccurs.Giventhefactthatthetopologyofthepowergridwillbechangedifoneormoretransmissionlinesareattackedordisconnected,transmissionlineswitchingscanalsobeincorporatedintosystemoperatorspost-disruptiondecisionforabettermitigationeffect.Gridoperatorsidentifyspecicgridconditionswhereitcanbeadvantageoustoimplementanautomatic,predeterminedcorrectiveactioninresponsetospecicabnormalgridoperations.Forexample,inPJM,SpecialProtectionSchemeshaveincludedtransmissionlineswitchingasoneoperationduringcontingencies[ 73 ]. 5.3OptimalPowerFlowModelforEconomicAnalysiswithLineSwitching max,ming,s,f,,Xi2I(higi+risi) (5)s.t.Constraints( 4 )-( 4 ) (5)Xi2Ii+X(i,j)2Lij=k (5)bij(i)]TJ /F4 11.955 Tf 11.96 0 Td[(j))]TJ /F3 11.955 Tf 11.96 0 Td[(fij+(1)]TJ /F4 11.955 Tf 11.96 0 Td[(ij)Mij00 (5)bij(i)]TJ /F4 11.955 Tf 11.96 0 Td[(j))]TJ /F3 11.955 Tf 11.96 0 Td[(fij)]TJ /F5 11.955 Tf 11.96 0 Td[((1)]TJ /F4 11.955 Tf 11.96 0 Td[(ij)Mij0 (5)X(i,j)2Lij=m8(i,j)2L (5)i,ij,ij2f0,1g,8i2I,(i,j)2L (5) Thisisabi-levelproblem,andtheinnerlevelcouldnotbedualizedasbeforebecauseoftheintegerdecisionvariableij.Hereinthismodel,welimitthetotalnumberoflinesthatcouldbeswitchedtomasaresourceconstraint. 65

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5.4NumericalExperimentsandAnalysis ThelinearprogramformulationsforOPFandinterdictionmodelswithlineswitchingareimplementedinC++andusingCPLEX11.0viaILOGConcertTechnology2.5.Inthissection,wetestallproposedalgorithmsandmethodsforcontingencyselectionandevaluationonthebussystem:IEEE-30-Bussystemtestcase,andwehavesomepreliminaryresulthere. Figure5-1. IEEE-30-bussystemwithlineswitching Comparedwithgure 4-3 withk=2andk=6,wecanseethatwhenthereisnolineswitchingallowed(m=0),theobjectivevalueisthesameasinlastchapter.However,whenthereislineswitchingallowed,wecanseesomeimprovementovertheobjectivevalueforthecaseofk=2.TheeffectoflineswitchingcanNOTbeobservedforthecaseofk=6,sincethewholepowergridisalmostdownandthereislittleroomfortheoperatortomanipulate. 66

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CHAPTER6FUTURERESEARCH Withthearisingawarenessofthesecurityissuesinpowersystems,contingencyanalysisforunexpectedfailureshasbeenanimportantprobleminrecentyears.However,regardingtheexponentialnumberofcontingencies,thesecuritycheckingforthefeasiblepowerowsunderdifferentsituationsiscomputationallyexpensiveandalmostintractableinshorttimeperiods.Inthisresearch,weplantousesomeoptimizationapproachestoclassifythelargecontingencysetintosmalleronesbasedontheoptimalpowerowmodel.Wewanttoshowthatcheckingsmallestsubsetsofcontingenciesbytherobustoptimizationmethodisenoughforcheckingthewholeset'sfeasibility.Bytheseapproaches,thecomputationalcomplexityforlargepowersystemswillbereducedsignicantly. AccordingtothestandardsoftheNorthAmericanElectricReliabilityCorporation(NERC),powersystemsarerequiredtoperformnecessaryadjustmentsundernormalandcontingencyconditionstoensuresystemreliability.Ifonlyasingleelementislost(N)]TJ /F5 11.955 Tf 12.25 0 Td[(1contingency),thesystemmustbestableandallthermalandvoltagelimitsmustremainwithinapplicablerating.Theloss-of-loadisnotallowedforN)]TJ /F5 11.955 Tf 12.47 0 Td[(1contingency.Inthecaseofmultiplesimultaneousfailures(N)]TJ /F3 11.955 Tf 12.48 0 Td[(kcontingency),thesystemstillhastomeetthestable,thermalandvoltagelimits,butplannedorcontrolledloss-of-loadisallowed,toalimiteddegree[ 34 ].Inthismodel,weintroducetheconceptoftoensurethepowergridsurvivabilitycriterion,whichrequiresatleast(1-)fractionoftheinitialtotaldemandmustbemetwhenthereareuptokcomponentsfailedwithinthegrid.Itmeansthatforno-contingencystateandcontingencystateswithk=1,noloss-of-loadisallowedand=0;forcontingencystateswithk2,asmallfractionoftotalloaddemandcanbeshedand0<1. 67

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6.1Nomenclature IndexSetsandIndices I:thesetofbuses(indexedbyi,j) G:thesetofgeneratingunits(indexedbyg) Gi:thesetofgeneratingunitslocatedatbusi E:thesetofdirectedtransmissionlinesconnectingbuses(indexedbye) E.i:thesetoftransmissionlinesorientedintobusi Ei.:thesetoftransmissionlinesorientedoutofbusi e=(ie,je):thetailbusieandheadbusjeoftransmissionlinee C(k):thesetofallN-kcontingencieswithexactlykfailedelements c:indexforcontingencyandc2C Parameters Be,Fe:susceptanceandthermalcapacityoftransmissionlinee2E Di:loaddemandatbusi P g, Pg:lowerandupperboundsofgenerationlevelforunitg2G N:thenumberoftotalsystemelementsconsistingofgeneratingunitsandtransmissionlines(i.e,N=jGj+jEj) k:thenumberoffailedelementsasappearinginN-kcontingency,andusuallypositiveintegers(i.e.,k=1,2,3,) cg2f0,1g:cg=1meansthatthegeneratingunitgisfailedincontingencyc,andcg=0otherwise ce2f0,1g:cte=1meansthatthetransmissionlineeisfailedincontingencyc,andce=0otherwise DecisionVariables pg:thegenerationlevelofunitg fe:thepowerowontransmissionlinee 68

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i:thephaseangleonbusi qi:loadshedding(i.e.,unsatiseddemand)atbusi Q(c):theminimumamountofloadsheddingincontingencyc g,e:unknownvariablescorrespondingtoce2f0,1gforidentifyingacontingency 6.2OptimalPowerFlowwithContingency Duringthecontingencyc2C(k),thefollowingprogram( 6 )isusedforcontrolledoperationstondtheoptimalpowerow.Controlledoperationsincludeloadshedding,andisusedforcontrollingtheamountofloadshedding. OPF(c,k,):Q(c,k,)=minf,p,,qXi2Iqi (6a)s.t.Xe2E.ife+Xg2Gipg=Xe2Ei.fe+(Di)]TJ /F3 11.955 Tf 11.96 0 Td[(qi),8i2I (6b)Be(ie)]TJ /F4 11.955 Tf 11.96 0 Td[(je)(1)]TJ /F4 11.955 Tf 11.95 0 Td[(ce))]TJ /F3 11.955 Tf 11.96 0 Td[(fe=0,8e2E (6c))]TJ /F3 11.955 Tf 11.96 0 Td[(Fe(1)]TJ /F4 11.955 Tf 11.95 0 Td[(ce)feFe(1)]TJ /F4 11.955 Tf 11.96 0 Td[(ce),8e2E (6d)P g(1)]TJ /F4 11.955 Tf 11.95 0 Td[(cg)pg Pg(1)]TJ /F4 11.955 Tf 11.96 0 Td[(cg),8g2G (6e)0qiDi,8i2I (6f)Xi2IqiXi2IDi (6g) Theobjective( 6a )istominimizetheamountofloadsheddingintheprocesstondafeasiblepowerowamongthesystem.Theconstraint( 6b )foreverybusiistoensuretheowbalanceofi,wherethesatiseddemandatiisDi)]TJ /F3 11.955 Tf 12.32 0 Td[(qi.Theconstraint( 6c )computestheowamountonlinee,dependingthedifferenceofphaseanglesofitstwoendingbuses.Ifeisinthecontingencyc(i.e.,ce=1),theowiszero,aswellaslimitedby( 6d ).Theowoneachlineeislimitedbythelinecapacityshownin( 6d ).Theconstraint( 6e )forgeneratingunitgistoensurethatthegenerationlevelislimitedwithinthelowerandupperbounds.Whenthisgeneratingunitgisinthecontingency 69

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c(i.e,cg=1),thegenerationlevelis0.Theconstraint( 6f )ensuresthattheloadsheddingamountateachbusi,whilethelastconstraint( 6g )presentsathresholdforthetotalamountofloadshedding. Notewhenk=0,i.e.,c2C(0),theformulation( 6 )istheoptimalpowerowforeconomicdispatch,wherenoloadsheddingisallowedandtheobjectiveistominimizethegeneratingcost. 6.3Analysis Intheformulation( 6 ),thevectorsc,cisknownanddenedbycontingencyc.However,amongthelargenumberofcontingencieswithinC(k),whichoneistheworst-caseleadingtolargeamountofloadsheddingisnotknownuntilyoucheckallcontingencies.Here,weuserobustoptimizationmethodinthenetworkinterdictionmodeltondtheworst-casecontingencywithrespecttotheloadshedding.Thenetworkinterdictionmodelforpowergridswasproposedin[ 112 ].Followingtheirmodel,weassumethatexactlykfailedcomponentsongeneratingunitsand/ortransmissionlines,andaddtheconstraintsforcontrolledloadsheddingbythethreshold.Fordifferentvaluesofkand,themodelforworst-casecontingencyidenticationcanbeformulated 70

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asfollows, Q(k,)=max,minf,p,,qXi2Iqi (6a)s.t.Xe2E.ife+Xg2Gipg=Xe2Ei.fe+(Di)]TJ /F3 11.955 Tf 11.95 0 Td[(qi),8i2I (6b)Be(ie)]TJ /F4 11.955 Tf 11.96 0 Td[(je))]TJ /F3 11.955 Tf 11.95 0 Td[(fe)]TJ /F3 11.955 Tf 11.95 0 Td[(Me0,8e2E (6c)Be(ie)]TJ /F4 11.955 Tf 11.96 0 Td[(je))]TJ /F3 11.955 Tf 11.95 0 Td[(fe+Me0,8e2E (6d))]TJ /F3 11.955 Tf 11.96 0 Td[(Fe(1)]TJ /F4 11.955 Tf 11.95 0 Td[(e)feFe(1)]TJ /F4 11.955 Tf 11.95 0 Td[(e),8e2E (6e)P g(1)]TJ /F4 11.955 Tf 11.96 0 Td[(g)pg Pg(1)]TJ /F4 11.955 Tf 11.95 0 Td[(g),8g2G (6f)0qiDi,8i2I (6g)Xi2IqiXi2IDi (6h)Xg2Gg+Xe2Ee=k (6i)g,e2f0,1g,8g2G,e2E (6j) Theobjective( 6a )hastwolevels,wheretheouterleveldenotestheidenticationtheworst-casecontingencies,whiletheinneronedenotestheoptimaloperationsinthecorrespondingcontingency.Theconstraint( 6i )ensuresthattherecanexactlykfailedcomponentsinthesystem.Constraints( 6c ),( 6d ),( 6e )ensurethattheowfeisBe(ie)]TJ /F4 11.955 Tf 12.28 0 Td[(je)andwithinthecapacitylimitsifthelineisnotinthecontingency,and0ifinthecontingency,whereMisalarge-enoughconstant. Assumethatoptimizedvalueforthisprogramisc=(,),whichidentiesaworst-casecontingencywithrespecttotheamountofloadshedding. 6.4PreliminaryResult 6.4.1CriticalSets Inordertohavesomebasicideaaboutthiscriticalcontingencysubset,theoptimalcontingencyselection(transmissionlineonly)isobtainedbysolvingthepreviousmodel 71

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usingCPLEXonIEEE-300-Bussystemtestcase.Theresultsareshowninthefollowingtableswithdifferentkand. Figure6-1. IEEE-300-bussystem(From:PowerSystemsTestCaseArchive) Fromthetablesbelow 6-1 6-2 6-3 6-4 6-5 6-6 ,wecanseethatsametransmissionline(s)areinthesolutionfordifferentkwithinthesamescenarioof.Inthescenarioofepsilon=0.0005,line(60,238)and(32,35)areshowninallthecases,(39,62)and(41,92)areinmostofthecases;Inthescenarioofepsilon=0.001,line(60,238)isshowninallthecases,(32,35),(121,154)and(40,68)areinmostofthecases;Inthescenarioofepsilon=0.01,line(60,238)isshowninallthecases,(35,36),(39,62)and(41,92)areinmostofthecases;Inthescenarioofepsilon=0.1,line(60,238)and(41,92)areshowninallthecases,(32,35)isinmostofthecases;Inthescenarioofepsilon=0.15,line(60,238)isshowninallthecases,(32,35),(40,68)and(41,92) 72

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isinmostofthecases;Inthescenarioofepsilon=0.2,line(60,238)isshowninallthecases,(35,36)and(40,68)areinmostofthecases.Line(60,328)appearsinallthesolutionswithdifferentandk,soitisdenitelythemostcriticalcomponentinthispowergrid. 6.4.2SurvivabilityConstraintsandthePossibleCut Similarworkhasfocusedonidentifyingasmallsetofarcswhoseremoval(tomodelcompletefailure)willresultinanetworkunabletodeliveraminimumamountofdemand.Aproblemofthistypecanbesolvedusingmixed-integerprogrammingtechniquestechnique[ 23 ].Anexampleofthisapproachisusedin[ 23 ],whereasanapproximationtotheN)]TJ /F3 11.955 Tf 12.35 0 Td[(kproblemwithACpowerows,alinearmixed-integerprogramtosolvethefollowingcombinatorialproblem:removeaminimumnumberofarcs,suchthatintheresultingnetworkthereisapartitionofthenodesintotwosets,N1andN2,suchthatD(N1)+G(N2)+cap(N1,N2)Qmin HereD(N1)isthetotaldemandinN1,G(N2))isthetotalgenerationcapacityinN2,cap(N1,N2))isthetotalcapacityinthe(non-removed)arcsbetweenN1andN2,andQminisaminimumamountofdemandthatneedstobesatised.Thequantityintheleft-handsideintheaboveexpressionisanupper-boundonthetotalamountofdemandthatcanbesatised,theupper-boundcanbestrictbecauseunderpowerowlawsitmaynotbeattained. Typically,thiskindofexactmethodcouldnothandellargescaleproblems,becausetheexponentialnumberofcontingenciescouldmaketheproblemintractable.However,combinedwithsurvivabilityconstraint,itmeansthatifbyanyselectionofremovalofktransmissionlinesfromthepowergrid,theconstraintabovecouldbesatised,thatcontingencyisnotfeasible,andwecanusethistogeneratepossiblecuttingplaneand/ortoeliminatesomepossiblebranches. 73

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BysubstitutetheconstraintofQmin=Xi2IDi intoD(N1)+G(N2)+cap(N1,N2)Qmin wecanget(Xi2N1Di)]TJ /F10 11.955 Tf 12.4 11.35 Td[(Xi2N1qi)+cap(N1,N2)(1)]TJ /F4 11.955 Tf 11.96 0 Td[()Xi2N2Di)]TJ /F10 11.955 Tf 12.4 11.35 Td[(Xi2N2Gi (Pi2N1Di)]TJ /F10 11.955 Tf 12.91 8.97 Td[(Pi2N1qi)isdifferencebetweentheactualloadsheddingandtheaveragedloadsheddinginsetN1,and(1)]TJ /F4 11.955 Tf 12.84 0 Td[()Pi2N2Di)]TJ /F10 11.955 Tf 12.85 8.96 Td[(Pi2N2GiisthedifferencebetweenaveragedtotaldemandandactualgenerationinsetN2.ThisconstraintmeansthattheloadsheddingcouldnotbetoomuchunbalancedbetweensetsN1andN2. 74

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Table6-1. ContingencySelectionwith=0.0005 kLines 1[60][238].2[32][35][60][238].3[32][35][39][62][60][238].4[32][35][41][92][60][238][124][159].5[32][35][41][92][60][238][93][186][107][112].6[32][35][39][62][40][68][41][92][60][238][93][186].7[32][35][39][62][40][68][60][238][103][139][106][147][121][154].8[32][35][39][62][41][92][60][238][103][139][106][147][107][112][121][154].9[32][35][35][36][39][62][41][92][60][238][103][139][106][147][107][112][121][154].10[32][35][35][36][40][68][41][92][60][238][95][103][103][139][106][147][107][112][121][154].11[32][35][35][36][40][68][60][238][103][139][105][106][106][147][107][112][114][115][121][154][124][159].12[32][35][39][62][40][68][41][92][60][238][95][103][103][139][105][106][106][147][107][112][114][115][124][159]. 75

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Table6-2. ContingencySelectionwith=0.001 kLines 1[60][238].2[40][68][60][238].3[40][68][60][238][121][154].4[32][35][60][238][93][186][121][154].5[32][35][40][68][60][238][93][186][121][154].6[32][35][40][68][60][238][93][186][107][112][121][154].7[32][35][40][68][41][92][60][238][103][139][105][106][121][154].8[32][35][40][68][60][238][93][186][103][139][107][112][114][115][121][154].9[32][35][35][36][40][68][41][92][60][238][95][103][103][139][105][106][107][112].10[32][35][35][36][40][68][41][61][41][92][60][238][103][139][106][147][107][112][121][154].11[32][35][35][36][39][62][40][68][41][61][41][92][60][238][103][139][106][147][107][112][121][154].12[32][35][35][36][39][62][40][68][41][61][60][238][93][186][103][139][106][147][107][112][114][115][121][154]. 76

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Table6-3. ContingencySelectionwith=0.01 kLines 1[60][238].2[41][92][60][238].3[39][62][41][61][60][238].4[39][62][40][68][41][61][60][238].5[39][62][40][68][41][61][41][92][60][238].6[35][36][39][62][41][61][41][92][60][238][106][147].7[35][36][39][62][40][68][41][61][60][238][103][139][106][147].8[32][35][35][36][39][62][40][68][41][61][41][92][60][238][103][139].9[32][35][35][36][39][62][40][68][41][61][41][92][60][238][103][139][106][147].10[32][35][35][36][40][68][41][61][41][92][60][238][103][139][106][147][107][112][146][148].11[32][35][35][36][40][68][41][61][41][92][60][238][103][139][107][112][121][154][137][138][146][148].12[32][35][35][36][40][68][41][61][41][92][60][238][103][139][105][106][106][147][107][112][124][159][146][148]. 77

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Table6-4. ContingencySelectionwith=0.1 kLines 1[60][238].2[41][92][60][238].3[32][35][41][92][60][238].4[41][92][60][238][105][106][114][115].5[32][35][41][92][60][238][103][139][106][147].6[35][36][39][62][41][61][41][92][60][238][114][115].7[32][35][41][61][41][92][60][238][103][139][114][115][121][154].8[32][35][39][62][41][61][41][92][60][238][103][139][114][115][124][159].9[32][35][35][36][39][62][41][61][41][92][60][238][103][139][105][106][124][159].10[32][35][35][36][40][68][41][61][41][92][60][238][95][103][103][139][107][112][114][115].11[35][36][39][62][40][68][41][61][60][238][95][103][103][139][107][112][114][115][124][159][146][148].12[32][35][35][36][39][62][40][68][41][61][41][92][60][238][95][103][103][139][105][106][106][147][124][159]. 78

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Table6-5. ContingencySelectionwith=0.15 kLines 1[60][238].2[60][238][114][115].3[41][92][60][238][106][147].4[40][68][60][238][103][139][106][147].5[32][35][41][92][60][238][103][139][106][147].6[32][35][40][68][41][92][60][238][106][147][114][115].7[32][35][40][68][41][92][60][238][103][139][106][147][114][115].8[32][35][40][68][41][92][60][238][95][103][106][147][114][115][124][159].9[35][36][40][68][41][92][60][238][95][103][106][147][107][112][114][115][124][159].10[32][35][35][36][39][62][41][61][41][92][60][238][103][139][106][147][107][112][124][159].11[32][35][35][36][39][62][41][61][41][92][60][238][103][139][106][147][114][115][121][154][124][159].12[32][35][35][36][40][68][41][61][41][92][60][238][103][139][106][147][107][112][114][115][121][154][124][159]. 79

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Table6-6. ContingencySelectionwith=0.2 kLines 1[60][238].2[60][238][105][106].3[40][68][60][238][107][112].4[41][61][60][238][103][139][107][112].5[35][36][40][68][60][238][103][139][107][112].6[35][36][39][62][40][68][60][238][106][147][107][112].7[35][36][39][62][60][238][93][186][103][139][106][147][107][112].8[35][36][39][62][40][68][60][238][103][139][105][106][106][147][107][112].9[35][36][39][62][40][68][41][92][60][238][103][139][105][106][106][147][107][112].10[32][35][35][36][39][62][41][92][60][238][103][139][106][147][107][112][114][115][124][159].11[32][35][35][36][40][68][41][92][60][238][95][103][103][139][106][147][107][112][114][115][124][159].12[32][35][35][36][39][62][41][92][60][238][95][103][103][139][106][147][107][112][114][115][124][159][146][148]. 80

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BIOGRAPHICALSKETCH HongshengXuwasbornandraisedinChengdulocatedinsouthwesternpartofChina,andhespenthisrst18yearsthere.HongshengreceivedhisBachelorofSciencedegreeincivilengineeringfromTsinghuaUniversity.HeenrolledinthePh.D.programofDepartmentofCivilEngineeringin2006topursuehisdegreeintransportationengineering.In2008,withaMasterofSciencedegreefromcivilengineering,hestartedhisPh.D.newprogramintheDepartmentofIndustrial&SystemsEngineering.Hespenttherstoneandahalfyearsworkingonoptimizationrelatedcomputationproblems,andgainedalotofexperienceofprogramming.Then,HongshengjoinedtheCenterforAppliedOptimization(CAO)inthespringof2010,andfocusedonoptimizationoverpowersecurityissues.HereceivedhisPh.D.fromtheUniversityofFloridainthesummerof2012.Hisresearchinterestsincludepowernetworkoptimizationanditsapplication,globaloptimizationandapplications,designandanalysisofcomputeralgorithms,parallelcomputinginmathematicalprogramming,softwaredesignanddevelopment. 93