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- Permanent Link:
- http://ufdc.ufl.edu/UFE0046869/00001
## Material Information- Title:
- Optimal Deployment and Operations of Public Charging Infrastructure for Plug-In Electric Vehicles
- Creator:
- He, Fang
- Place of Publication:
- [Gainesville, Fla.]
- Publisher:
- University of Florida
- Publication Date:
- 2014
- Language:
- english
- Physical Description:
- 1 online resource (15 p.)
## Thesis/Dissertation Information- Degree:
- Doctorate ( Ph.D.)
- Degree Grantor:
- University of Florida
- Degree Disciplines:
- Civil Engineering
Civil and Coastal Engineering - Committee Chair:
- YIN,YAFENG
- Committee Co-Chair:
- ELEFTERIADOU,AGELIKI
- Committee Members:
- WASHBURN,SCOTT STUART
SRINIVASAN,SIVARAMAKRISHNAN LAWPHONGPANICH,SIRIPHONG - Graduation Date:
- 8/9/2014
## Subjects- Subjects / Keywords:
- Algorithms ( jstor )
Batteries ( jstor ) Electric vehicles ( jstor ) Electricity ( jstor ) Energy consumption ( jstor ) Prices ( jstor ) Recharging ( jstor ) Transportation ( jstor ) Travel time ( jstor ) Vehicles ( jstor ) Civil and Coastal Engineering -- Dissertations, Academic -- UF plug-in-electric-vehicles -- price-of-electricity -- public-charging-stations -- regional-level-budget-allocation -- urban-level-location - Genre:
- Electronic Thesis or Dissertation
born-digital ( sobekcm ) Civil Engineering thesis, Ph.D.
## Notes- Abstract:
- Plug-in electric vehicles (PEVs) are vehicles whose battery packs can be recharged from power grids, and the electricity stored on board propels or contributes to propel the vehicles. PEVs include battery electric vehicles (BEVs) and plug-in hybrid electric vehicles (PHEVs). Interests in PEVs have increased dramatically in recent years due to advances in battery technologies, rising prices of petroleum, and growing concern over environment issues. Many governments have incentive policies, such as offering purchase subsidies and deploying public charging infrastructure in convenient locations, to promote the deployment of PEVs. Building public charging infrastructure has a profound impact and is typically associated with a high capital cost. To assist policy makers to optimize the investment, this dissertation is devoted to developing a hierarchical modeling framework where a strategic planning model captures interactions between a regional transportation network and power transmission grids to determine a budget allocation plan for public charging stations among metropolitan areas in the region while a tactic planning model considers the spatial distribution of PEVs and optimizes the locations and operations of charging stations in a metropolitan area. More specifically, at the regional planning level, a static game-theoretic approach is applied to investigate interactions among the availability of public charging stations, destination choices of PEVs, and prices of electricity. The interactions lead to an equilibrium state that can be formulated into a convex mathematical program. We then examine how to allocate the public charging station budget among metropolitan areas in a particular region to maximize social welfare associated with the coupled transportation and power networks. For a particular metropolitan area, given the allocated budget limit, we consider the problem of how to determine the number, locations and types of charging stations within the budget limit. Assuming the locations and types of public charging stations are given, we first develop network equilibrium models with BEVs. Based on the proposed equilibrium models, station location plans are then optimized to maximize social welfare. Lastly, we investigate the operations of public charging stations with a focus on optimizing the prices of electricity at public charging stations. ( 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, 2014.
- Local:
- Adviser: YIN,YAFENG.
- Local:
- Co-adviser: ELEFTERIADOU,AGELIKI.
- Electronic Access:
- RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-08-31
- Statement of Responsibility:
- by Fang He.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Applicable rights reserved.
- Embargo Date:
- 8/31/2015
- Resource Identifier:
- 969976947 ( OCLC )
- Classification:
- LD1780 2014 ( lcc )
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PAGE 1 Designofmoreequitablecongestionpricingandtradablecreditschemes formultimodaltransportationnetworksDiWua,YafengYina ,,SiriphongLawphongpanichb,HaiYangcaDepartmentofCivilandCoastalEngineering,UniversityofFlorida,365WeilHall,Gainesville,FL32611-6580,UnitedStatesbDepartmentofIndustrialandSystemsEngineering,UniversityofFlorida,303WeilHall,Gainesville,FL32611-6595,UnitedStatescDepartmentofCivilandEnvironmentalEngineering,TheHongKongUniversityofScienceandTechnology,ClearWaterBay,Kowloon,HongKong,ChinaarticleinfoArticlehistory: Received1September2011 Receivedinrevisedform7May2012 Accepted8May2012 Keywords: Congestionpricing Tradablecredits Equity MultimodaltransportationnetworksabstractThispaperdevelopsamodelingframeworkthatconsiderstheeffectofincomeontravelersÂ choicesoftripgeneration,modeandrouteonmultimodaltransportationnetworksand explicitlycapturesthedistributionalimpactsofcongestionÂ…mitigationpoliciesondifferent incomeandgeographicgroups.ThemodelingframeworkisappliedtodesignmoreequitableyetefÂ“cientcongestionpricingandtradablecreditschemes.Thedesignmodelsareformulatedasmathematicalprogramswithequilibriumconstraints,andsolvedbyderivativefreesolutionalgorithms.Numericalexamplesarepresentedtodemonstratethemodels andofferinsightonthemechanismsofachievingbetterequityundercongestionpricing ortradablecreditschemes. 2012ElsevierLtd.Allrightsreserved.1.Introduction TrafÂ“ccongestionhasbecomeoneofthemostseveresocialproblemsinmodernsocieties.Foryears,thesolutiontothe risinglevelofcongestionhasbeenaddingadditionalcapacity.However,suchanapproachissubjecttomanyspatialand Â“nancialconstraints.Furthermore,providingmoreroadspacehasbeenproventobeself-defeatingincongestedareasbecausetheincreasedcapacitywillsoonbeoccupiedbyinducedtraveldemands(e.g., GoodwinandNoland,2003;Duranton andTurner,2011 ).Asaresult,market-baseddemandmanagementinstrumentshavegainedincreasingattentionasmore effectiveandcost-efÂ“cientsolutionstothecongestionproblem.Thoseinstrumentsareoftendesignedtomaximizethepeople-movingcapacityofatransportationsystembyaffectingtravelersÂtraveldecisionssuchasmode,departuretimeand route. Ofinterestinthispaperarecongestionpricingandtradablecreditschemes.Theformerhasbeenadvocatedbytransportationeconomistssincetheseminalworkby Pigou(1920) whilethelattercanbetracedbackto Dales(1968) forthepurpose ofmanagingwaterquality.MorespeciÂ“cally,congestionpricingistochargetravelersthemarginalexternalcoststhattheir tripsimposetothesocietytochangetheirtravelbehaviorsinsuchawayastoreducetrafÂ“ccongestionorincreasesocial welfare.Forrecentreviewsonmethodsandtechnologiesforcongestionpricing,see,e.g., YangandHuang(2005),Tsekeris andVoss(2009) ,and dePalmaandLindsey(2011) .Inatradablecreditscheme,creditsaredistributedbythegovernment anddriversarethenrequiredtopayacertainnumberofcreditsinordertoaccesstransportationfacilities.Thecreditscanbe tradedamongtravelersandthepriceisdeterminedbythemarketthroughfreetrading.Bydecidingtheinitialcredit0191-2615/$-seefrontmatter 2012ElsevierLtd.Allrightsreserved. http://dx.doi.org/10.1016/j.trb.2012.05.004 Correspondingauthor.Tel.:+13523929537;fax:+13523923394. E-mailaddress: yafeng@ce.uÂ”.edu (Y.Yin). TransportationResearchPartB46(2012)1273Â…1287 Contentslistsavailableat SciVerseScienceDirectTransportationResearchPartBjournalhomepage:www.elsevier.com/locate/trb PAGE 2 distributionandthesubsequentcreditcharges,thegovernmentcanachieveitspolicygoal.Forrecentinvestigationson tradablecreditschemesformanagingmobility,see,e.g., Verhoefetal.(1997),Viegas(2001),YangandWang(2011),Zhang etal.(2011),Nie(2012) ,and Wangetal.(2012) . Thispaperisconcernedwiththeequityaspectofbothinstruments,whichplaysaprominentroleintheirimplementation.Inpractice,otherwisejustiÂ“edplansareoftenthwartedbytheconcernsanddebatesovertheirequityimpacts.For example,althoughbeingtheoreticallytheÂ“rst-besteconomicsolutiontoreducecongestion,congestionpricinghassofar notreceivedmuchsupportfromthegeneralpublicandevensometransportationprofessionals(e.g. Wachs,2005 ; Jaensirisaketal.,2005 ).Muchoftheoppositioncentersontheperceivedinequalitiesofcongestionpricing( Tayloretal.,2010 ).In particular,somebelievethatcongestionpricingharmsthepoor,whomayhavetopaymoreduetotheirinÂ”exibleschedules orareforcedtoswitchtolessdesirableroutes,departuretimesortransportationmodes.Forarecentexplorationonhowthe equityconcernshavebeenraisedandaddressedinpractice,see Tayloretal.(2010) .Inaddition, Levinson(2010) and Ecola andLight(2009) providecomprehensivereviewsontheequityissuesincongestionpricing. ThispaperattemptstodesignmoreequitablepricingandtradablecreditschemestoalleviatecongestionorimprovesocialbeneÂ“tonmultimodalurbantransportationnetworks.Thisisawell-researchedtopic,particularlyinthecontextofcongestionpricing.Forinstance,some(e.g., YangandZhang,2002;YinandYang,2004 )developedoptimalpricingmodelsfor multiclassnetworkswithsocialorspatialequityconstraints.Others(i.e., Songetal.,2009;LawphongpanichandYin,2010; GuoandYang,2010;NieandLiu,2010 ; Wuetal.,2011 )adoptedPareto-improvingapproachestoensurethatnoneisworse offinthepresenceoftolls.However,noneofthesepricingmodelsisabletocapturethedistributionalimpactsofcongestion pricingondifferentincomegroupsandoffermeaningfuldiscussionontheincome-basedequity.Incontrast,thispaper developsamodelingframeworkthatconsiderstheeffectofincomeontravelersÂchoicesoftripgeneration,modeandroute onmultimodaltransportationnetworksandexplicitlycapturesthedistributionalimpactsofcongestionÂ…mitigationpolicies ondifferentincomeandgeographicgroups.ThemodelingframeworkisappliedtodesignmoreequitableyetefÂ“cientcongestionpricingandtradablecreditschemes.Intheliterature,distributionalimpactsofroadpricinghavebeenexaminedby, e.g., Foster(1974) and Richardson(1974) ,amongothers. FosterandRichardson(1975) emphasizedtheimportanceofexaminingthenetdistributionalimpactsofroadpricingandcalledformeasuresthatminimizeunfavorabledistributionalimpacts onthepoor.Morerecently, Franklin(2006) consideredtheincomeeffectontravelchoicebehaviorsandestimatedthewelfareimpactsoftollingontheSR520BridgeinSeattle,Washington. BureauandGlachant(2008) simulatedandcomparedthe distributionalimpactsofninetollscenariosinParis. KarlstromandFranklin(2009) analyzedtheequityeffectsofcongestion pricinginStockholmusingtravelsurveysconductedin2004and2006.However,allthesestudiesfocusedonevaluationofa givenpricingschemeinsteadofdesigninganoptimalone. Fortheremainder,Section2describestheproblemandpresentsourbasicmodelingconsiderations.Section3formulates thedesignmodelsforequitableandefÂ“cientpricingandcreditschemes,followedbyadiscussionofsolutionalgorithmsin Section4.Section5presentsnumericalexamplesandÂ“nallySection6concludesthepaper. 2.Problemdescriptionandbasicconsiderations Weconsiderageneralmultimodaltransportationnetworkthatincludestwotypesoffacilities,i.e.,transitservicesand highwaylinks.Inthissetting,apricingschemereferstoastrategyfortollingroadsaswellasadjustingfaresontransitlines. Thetollsonhighwaysegmentsarenonnegativewhileadjustmentstotransitfaresareunrestricted.Herein,anegativefare adjustmentmeansthattransitusersaresubsidizedbythetollrevenue.Therefore,acomponentofthepricingschemeinthis papercanbeviewedasrevenuerecycling. Forthetradablecreditscheme,weassumethatthegovernmentwilldistributecreditstoeligibletravelersfreeofcharge. ThecreditscanonlybeusedforaspeciÂ“cperiodoftimeandthusnoonecangainbybankingorstockingcreditsforfuture use.Thecreditsareusedtopayforthecreditchargesonbothhighwayandtransitlinksacrossthenetworkandthecharges mayvaryfromlinktolink.IfthecreditsownedbyanindividualtravelerarelessthanthechargeonaspeciÂ“clink,heorshe cannottravelonthelinkunlessadditionalcreditsarepurchasedfromothertravelers,whomayhaveleftovercreditsorsimplyprefertogiveuptheirtravelrightsinexchangeformoney.Itisassumedthatthegovernmentdoesnotinterferewiththe credittradingmarketbutsolelyactsasamanagertomonitorthesystem.Itisalsoassumedthatthetransactioncostisnegligible.Consequently,thepriceofthecreditissolelydependentonthesupplyanddemandofcreditsonthemarket( Yang andWang,2011 ).Thedesignofacreditschemeinvolvesdeterminationoftheinitialdistributionofcreditsandthecredit chargeoneachlinkofthenetwork.Inourmultimodalnetwork,bothhighwayandtransitlinksarechargednonnegative creditssothattravelerscannotacquirecreditsbytraveling. Wefurtherconsideradiscretesetofusergroupswithdifferentincomesandpreferencesamongthreetravelmodes,i.e., notravel,transitandauto.ItisassumedthatusersÂtraveldecisionscanberepresentedbyanestedlogitmodelwhoseutility functionsarenonlinearinincome.ThenonlinearspeciÂ“cationallowsustomoreaccuratelycapturetheincomeeffectonthe choicebehaviorinthepresenceoftollsorcreditcharges.Accordingto,e.g., Franklin(2006) and BureauandGlachant(2008) , thetraditionalspeciÂ“cationwithconstantmarginalutilityofincomemayleadtoanunderestimateoftheregressivenessofa pricingscheme. TheequivalentvariationisadoptedinthispapertomeasurethenetimpactofacongestionÂ…mitigationpolicy.Itisthe dollaramountthathastobetakenawayfromatravelerintheno-implementationscenariotoleavehimaswelloffashe1274 D.Wuetal./TransportationResearchPartB46(2012)1273Â…1287 PAGE 3 wouldbeintheimplementationscenario(e.g., Nicholson,1998 ).Intuitively,ifthetravelerbeneÂ“tsfromtheimplementation ofapricingorcreditscheme,theequivalentvariationwillbenegative.Theequivalentincome,deÂ“nedastheoriginalincome minustheequivalentvariation,canthusservesasawelfaremeasure.Itisameasureofhowwealthythetravelerfeelsunder theimplementation.Astheutilityofatravelerismonotonicallyincreasingwithincome,anequivalentincomehigherthan theoriginalincomeimpliesthattheimplementationincreasesthetravelerÂsutilitylevel,i.e.,thepricingorcreditschemeis beneÂ“cialtothetraveler.Calculatingequivalentincomeisachallengingtaskintherandomutilityframeworkiftheutilityis anonlinearfunctionofincome.Inthispaper,weadopttheformuladerivedby DagsvikandKarlstrom(2005) tocomputethe expectedequivalentincomeforthewelfaremeasure. TheequityanalysisoftransportationpolicesismultifacetedbecausetherearemultipledeÂ“nitionsoftransportationequityanddifferentwaystocategorizeusers( Litman,2002 ).AlthoughthemodelingframeworkinthispaperisÂ”exibleenough toaccommodateotherdeÂ“nitions,thispaperdeemsapricingorcreditschemetobemoreequitableifitleadstoamore uniformdistributionofwealthacrossdifferentgroupsofpopulationcategorizedbyincomeandgeographiclocations.Such anotionofequitycombinessomeaspectsofbothverticalandspatialequitydiscussedintheliterature.TheGinicoefÂ“cient ( Gini,1912 )isadoptedtoquantifyourworkingdeÂ“nitionofequity.ThecoefÂ“cientmeasurestheinequalityofincomedistributionacrossapopulation,avalueof0expressingcompleteequalityandavalueof1implyingcompleteinequality. Withtheaboveconsideration,weintendtodesignapricingorcreditschemetomaximizethesocialbeneÂ“tandtheincome-basedequitysimultaneously.SincethesetwoobjectivesareoftenconÂ”icting,weseekforabalancebetweenthem. 3.Modelformulations 3.1.Feasibleregion Let( N , L )beadirectedtransportationnetworkwhere N isthesetofnodesand L thesetofdirectedlinks.Asaforementioned,therearetwotypesoffacilitiesinthenetwork,i.e.,highwaylinksandtransitlines.Therearethreetravelmodes,i.e., auto,transitandnotravel,foreachtravelertochoose.Hereinafter,weusethesuperscript H and T todenotehighwaylinks (orautomode)andtransitlines(ortransitmode)andthesuperscript R todenotetheno-traveloption.Eachlink leL hasan associatedtraveltime tlthatdependsonlinkÂ”owvlandlinkcapacity cl.Transitservicesareassumedtosharetheroadway withautomobiles,andthushavethesamein-vehicletraveltimes.However,transitusersmayexperienceadditionalwaiting andtransfertimesattransitstations.Let G bethesetofalltravelergroupsand W denotethesetoforiginÂ…destination(OD) pairs.Thedemandoftravelergroup g forODpair w isdenotedas Dw , gandisassumedtobeÂ“xed. Withtheabovenotations,thesetofallfeasibleÂ”ow-demanddistributionsUforthenetworkcanbedescribedasfollows:Xk 2 Kw ; mfw ; m ; g kÂ¼ dw ; m ; g;8w 2 W ; m 2 M ; g 2 G Ã° 1 Ãž Xm 2 Mdw ; m ; gÂ¼ Dw ; g;8w 2 W ; g 2 G Ã° 2 Ãž fw ; m ; g kP 0 ;8w 2 W ; m 2 M ; g 2 G ; k 2 Kw ; mÃ° 3 Ãžwhere fw ; m ; g kistheÂ”owofusergroup g onpath k ofmode m betweenODpair w , dw , m , gisthedemandofusergroup g between ODpair w usingmode m ,and Kw , misthesetofallpathsconnectingODpair w formode m .Constraints (1)and(2) ensureÂ”ow anddemandbalancesandConstraint (3) requirespathÂ”owvariablestobenonnegative. Forcongestionpricingschemes,thefeasibletollsetWcanberepresentedas:Xg 2 GXw 2 WXm 2 MXk 2 Kw ; gfw ; m ; g kXl 2 LDk ; lsm lP 0 Ã° 4 ÃžsH lP 0 ;8l 2 L Ã° 5 ÃžsR lÂ¼ 0 ;8l 2 L Ã° 6 Ãžwheresm listhetollrateorfareadjustmentformode m onlink l andDk , listhepath-linkincidentmatrix,i.e., Dk ; lÂ¼ 1ifpath k useslink l andDk , l=0otherwise.Constraint (4) requiresthetotaltollrevenuetobenonnegative.Constraints (5)and(6) ensurethatthetollforautomodeisnonnegativeandthetollassociatedwiththeno-traveloptionmustbezero. Fortradablecreditschemes,thefeasiblesetoftradablecreditschemesHcanbedescribedas:pm lP 0 ;8m 2f H ; T g ; l 2 L Ã° 7 ÃžpR lÂ¼ 0 ;8l 2 L Ã° 8 Ãž qw ; gP 0 ;8w 2 W ; g 2 G Ã° 9 ÃžD.Wuetal./TransportationResearchPartB46(2012)1273Â…1287 1275 PAGE 4 Xw 2 WXg 2 Gqw ; gDw ; gÂ¼ Q Ã° 10 Ãžwherepm listhenumberofcreditstobechargedonlink l fortravelmode m ; qw , gisthenumberofcreditsdistributedtothe travelersingroup g ofODpair w and Q isthetotalnumberofcreditsdistributed.Constraint (7) ensuresanonnegativenumberofcreditschargedateachlinkwhileConstraint (8) guaranteestravelerswillnotbechargediftheydecidenottotravel. Constraint (9) impliesthattheinitialdistributionofcreditsisnonnegativeforeachgroupoftravelersandConstraint (10) requiresthetotalcreditsdistributedtobeapredeterminednumber,i.e., Q .Notethatthegovernmentwillalwaysberevenue neutralundertradablecreditsschemesasnomoneywillbecollectedfromthetravelers. 3.2.Multimodaluserequilibrium ItisassumedthattravelersÂchoicesofmodesandroutescanberepresentedbyanestedlogitmodelwheretravelersÂ“rst decidetheirtravelmodesandthenchoosetheroutesamongthoseavailablefortheselectedmodes.Theutilityfunctionfora travelerofgroup g betweenODpair w bymode m onroute k isdeÂ“nedas:uw ; m ; g kÂ¼vw ; m ; g kÃ¾ew ; m ; g kwhere uw ; m ; g kistheutilityofthetraveler;vw ; m ; g kisthedeterministicobservableportionoftheutilityandew ; m ; g kisarandom errorrepresentingunobservablefactors.Asaforementioned,thedeterministicutilityfunctionisassumedtobenonlinear, anditsspeciÂ“cationcanbe,e.g.,theTranslogmodel( Christensenetal.,1971 ):vw ; m ; g kÂ¼ bm ; g 0Ã¾ b1ln Tw ; m kÃ¾ b2ln2Tw ; m kÃ¾ b3ln ygÃ¾ b4ln2Ã° ygÃžÃ¾ b5ln Tw ; m kln ygwhere ygistheincomeorwagerateofatraveleringroup g ; bm ; g 0, b1, b2, b3, b4and b5areparameterstobecalibrated,and Tw ; m kisthetraveltimeofpath k betweenODpair w bymode m ,whichcanbecalculatedasfollows:Tw ; H kÂ¼ Xl 2 LDk ; ltl;8k 2 Kw ; H; w 2 W Tw ; T kÂ¼ Xl 2 LDk ; ltlÃ¾ ~ tw k;8k 2 Kw ; T; w 2 W Tw ; R kÂ¼ 0 ;8k 2 Kw ; R; w 2 Wwhere ~ tw kistheexpectedwaitingandtransfertimefortransitusersonpath k betweenODpair w .Thetransitfrequencyis assumedtobeÂ“xedandknowninthispaper,andthustheexpectedwaitingandtransfertimecanbeestimatedbeforehand foreachtransitpath.Aspreviouslymentioned,traveltime tldependsonlinkÂ”owvl,whichisthenumberoftransitvehicles plusthe(passenger)Â”owofhighwaymodedividedbytheaverageoccupancyofpassengercars.Notethatfortheno-travel mode(mode R ),thedeterministicutilityissettobeaconstantforeachgroup. Inthepresenceoftoll,individualsbearadditionaltravelexpense.TheimpactofthetollamountontheutilityofatravelerbetweenODpair w inmode m bypath k canberepresentedintheaboveutilityfunctionsbyreplacing ygwith ^ yw ; m ; g kdeÂ“nedasfollows:^ yw ; m ; g kÂ¼ yg Xl 2 LDk ; lsm l;8k 2 Kw ; m; w 2 W ; m 2 MUnderatradablecreditscheme,travelersmayselltheunusedcreditstogenerateadditionalincomeorpurchaseextracredits fortravel.Torepresenttheimpactofthecreditschemeontheindividualutility,wereplace ygwith ^ yw ; m ; g kintheaboveutility functionsandassumethat:^ yw ; m ; g kÂ¼ ygÃ¾ p qw ; g Xl 2 LDk ; lpm l ! ;8k 2 Kw ; m; w 2 W ; m 2 Mwhere p isthemarketpriceofcredit. Intheabove,weassumethattheinitialincomesoftravelers,i.e., yg,areindependentoftheirchoicesofmodeorroute. Eveniftravelerschoosenottotravel,theirincomesarenotaffectedduetotelecommuting.However,stayingathomeor telecommutingmayresultinareducedlevelofutility. Asperthenestedlogitmodel,theprobabilityforthegroup g travelerbetweenODpair w tochoosemode m andpath k is givenbythefollowing:Pw ; m ; g kÂ¼ exp vw ; m ; g khw ; m ; g Xj 2 Kw ; mexp vw ; m ; g jhw ; m ; g exp Ã° vw ; m ; gÃž Xm02 Mexp Ã° vw ; m0; gÃžwhere vw ; m ; gistheexpectedutilityforchoosingmode m andcanbecalculatedasfollows:1276 D.Wuetal./TransportationResearchPartB46(2012)1273Â…1287 PAGE 5 vw ; m ; gÂ¼ ln Ã° Xj 2 Kw ; mexp vw ; m ; g jhw ; m ; g ! Ãžhw ; m ; gÃ° 11 Ãžwhere hw , m , gisameasureofthedegreeofindependenceintheerrortermsoftheutilityfunctionfordifferentpathsofmode m forODpair w andusergroup g ( Train,2003 ). Theabovebehavioralconsiderationsleadtoamultimodaluserequilibriumwheretheperceivedutilityofeachtraveleris maximized.ThecorrespondingÂ”owdistributionunderapricingorcreditschemecanbeobtainedbysolvingthevariational inequality(VI)problemsasdescribedin Theorems1and2 respectively. Theorem1. TheÂ”ow-demanddistribution,( f , d )eU,isinthetolleduserequilibriumifitsolvesthefollowingVIproblem:Xg 2 GXw 2 WXm 2 MXk 2 Kw ; mÃ°vw ; m ; g kÃ¾ hw ; m ; gln fw ; m ; g kÃžÃ° ^ fw ; m ; g k fw ; m ; g kÃžÃ¾ Xg 2 GXw 2 WXm 2 MÃ° 1 hw ; m ; gÃž ln dw ; m ; gÃ° ^ dw ; m ; g dw ; m ; gÃž P 0 ;8Ã° ^ f ; ^ d Ãž2UÃ° 12 ÃžProof. TheaboveVIproblemisequivalenttothefollowingconditions:vw ; m ; g kÃ¾ hw ; m ; gln fw ; m ; g kqw ; m ; gÂ¼ 0 ;8k 2 Kw ; m; w 2 W ; m 2 M ; g 2 G Ã° 13 Ãž Ã° 1 hw ; m ; gÃž ln dw ; m ; gÃ¾qw ; m ; g fw ; gÂ¼ 0 ;8w 2 W ; m 2 M ; g 2 G Ã° 14 Ãžwhereqw , m , gand fw , garemultipliersassociatedwithconstraints (1)and(2) . From (13) ,fw ; m ; g kÂ¼ exp qw ; m ; ghw ; m ; g exp vw ; m ; g khw ; m ; g ;8k 2 Kw ; m; w 2 W ; m 2 M ; g 2 G Ã° 15 ÃžSumming (15) forallpathsforthesame w , m and g ,andconsideringconstraint (1) ,wehave:qw ; m ; gÂ¼ hw ; m ; gln dw ; m ; gXk 2 Kw ; mexp Ã° vw ; m ; g khw ; m ; gÃž 0 B B B @ 1 C C C A ;8w 2 W ; m 2 M ; g 2 G Ã° 16 ÃžComparing (11)and(16) yields:qw ; m ; gÂ¼ hw ; m ; gln dw ; m ; g vw ; m ; g;8w 2 W ; m 2 M ; g 2 G Ã° 17 ÃžSubstituting (17) into (14) andapplyingconstraint (2) ,weobtain:fw ; gÂ¼ ln Dw ; g ln Ã° Xm 2 Mexp vw ; m ; gÃž ;8w 2 W ; g 2 GConsequently:dw ; m ; gÂ¼ exp vw ; m ; gXm02 Mexp vw ; m0; g Dw ; g;8w 2 W ; m 2 M ; g 2 G fw ; m ; g kÂ¼ exp vw ; m ; g khw ; m ; g Xj 2 Kw ; mexp vw ; m ; g jhw ; m ; g dw ; m ; g;8k 2 Kw ; m ;; w 2 W ; m 2 M ; g 2 GTheabovetwoequationsdemonstratethatthesolutiontotheVI,theÂ”ow-demanddistribution,followsthenested-logitbaseduserequilibriumconditionsinthepresenceoftoll. h Theorem2. TheÂ”ow-demanddistribution, Ã° f ; d Ãž2U; andthemarketprice, p 2 RÃ¾,isinnetworkandmarketequilibrium underagivenfeasibletradablecreditschemeifitsolvesthefollowingVIproblem:Xg 2 GXw 2 WXm 2 MXk 2 Kw ; mvw ; m ; g kÃ¾ hw ; m ; gln fw ; m ; g k ^ fw ; m ; g k fw ; m ; g k Ã¾ Xg 2 GXw 2 WXm 2 MÃ° 1 hw ; m ; gÃž ln dw ; m ; gÃ° ^ dw ; m ; g dw ; m ; gÃž Ã¾Ã° Q Xg 2 GXw 2 WXm 2 MXk 2 Kw ; mXl 2 LDl ; kpm lfw ; m ; g kÃžÃ° ^ p p Ãž P 0 ;8^ p 2 RÃ¾; Ã° ^ f ; ^ d Ãž2UÃ° 18 ÃžD.Wuetal./TransportationResearchPartB46(2012)1273Â…1287 1277 PAGE 6 Proof. Theproofissimilarto Theorem1 .InadditiontoEquations (13)and(14) ,theequivalentconditionsoftheaboveVI problemalsoincludethefollowing:Q Xg 2 GXw 2 WXm 2 MXk 2 KwXl 2 LDl ; kpm lfw ; m ; g kP 0 Ã° 19 Ãž p Ã° Q Xg 2 GXw 2 WXm 2 MXk 2 KwXl 2 LDl ; kpm lfw ; m ; g kÃžÂ¼ 0 Ã° 20 ÃžEquation (19) ensuresthetotalnumberofcreditschargedislessthanorequaltothenumberofcreditsdistributedbythegovernment.Equation (20) representstheequilibriumconditionofthecreditmarketwherethecreditpricewouldbezeroifthereisan excesssupplyofcredits;ifthepriceispositive,themarketmustclear(e.g., NagurenyandDhanda,2000;YangandWang,2011 ). Thus,thesolutiontotheaboveVIsatisÂ“estheequilibriumconditionsofthemultimodalnetworkandthoseofthecredit marketunderatradablecreditscheme. h TheaboveVIscanbesolvedbyavarietyofexistingalgorithmsintheliterature.Inthispaper,wereformulatethemas regularnonlinearprogrammingusingagapfunctionintroducedby Aghassietal.(2006) .Thenonlinearprogramingmodel forthetolleduserequilibriumisasfollows:minf ; d ;q; fXg 2 GXw 2 WXm 2 MXk 2 Kw ; mÃ°vw ; m ; g kÃ¾ hw ; m ; gln fw ; m ; g kÃž fw ; m ; g kÃ¾ Xg 2 GXw 2 WXm 2 MÃ° 1 hw ; m ; gÃž ln dw ; m ; g dw ; m ; g Xg 2 GXw 2 WDw ; g fw ; gs : t :qw ; m ; g6 vw ; m ; g kÃ¾ hw ; m ; gln fw ; m ; g k;8w 2 W ; m 2 M ; g 2 G ; k 2 Kw ; mqw ; m ; gÃ¾ fw ; g6 Ã° 1 hw ; m ; gÃž ln dw ; m ; g;8w 2 W ; m 2 M ; g 2 G Ã° f ; d Ãž2UIntheobjectivefunction,theÂ“rsttwocomponentscomprisetheobjectivefunctionofa(primal)linearprogramthatthe VIproblemincludeswhilethethirdcomponentistheobjectivefunctionoftheduallinearprogram.Inotherwords,the aboveobjectivefunctionrepresentsthedualitygapofthisprimalÂ…dualpairofsolutions.Iftheaboveprogramcanbesolved withazeroobjectivevalue,partofthesolution,i.e., Ã° f ; d Ãž ,willsolvetheVIproblem.TheVIfortradablecreditschemescanbe solvedusingthesametechnique. 3.3.Welfareandequitymeasures ThissectiondiscussesthemeasuresusedinthispapertoreÂ”ectthechangesintheindividualandsocialwelfaresandthe income-basedequityaftertheimplementationofapricingorcreditscheme. 3.3.1.Individualwelfaremeasure Weemploytheequivalentvariationtorepresentthechangeinindividualwelfareunderapricingscheme,whichisdollar amountthatatravelerwouldbeindifferentaboutacceptinginlieuofthetollcharge.MorespeciÂ“cally,itisthechangeinhisor herwealththatwouldbeequivalenttothetollchargeintermsofitswelfareimpact.Let uw ; gÃ° yg;sÃž betheutilityforatravelerin group g betweenODpairwithincome yginthepresenceoftollrates.Theequivalentvariation,i.e., evw ; gÃ°sÃž ,isdeÂ“nedas:evw ; gÃ°sÃžÂ¼ arg f z : uw ; gÃ° yg z ; 0 ÃžÂ¼ uw ; gÃ° yg;sÃžgBasedontheequivalentvariation,wecancalculatetheequivalentincomethatallowstheindividualtoexperiencethe samelevelofutilitybeforetollingastheoriginalincomedoesaftertolling.Theequivalentincome ew ; gÃ°sÃž fortheindividual isdeÂ“nedas:ew ; gÃ°sÃžÂ¼ arg f z : uw ; gÃ° z ; 0 ÃžÂ¼ uw ; gÃ° yg;sÃžgItfollowsthat evw ; gÃ°sÃžÂ¼ yg ew ; gÃ°sÃž .Theequivalentincomemeasurestheimpactofcongestionpricingoneachindividual travelerinamonetaryterm.Foratraveler,iftheequivalentincomeishigherthanhisorheroriginalincome,thepricing schemeisconsideredtobebeneÂ“cialtohimorher. Similarly,theequivalentvariationandequivalentincomecanbedeÂ“nedforatradablecreditschemeas:evw ; gÃ°p; q ÃžÂ¼ arg f z : uw ; gÃ° yg z ; 0 ; 0 ; 0 ÃžÂ¼ uw ; gÃ° yg;p; q ; p Ãžg ew ; gÃ°p; q ÃžÂ¼ arg f z : uw ; gÃ° z ; 0 ; 0 ; 0 ÃžÂ¼ uw ; gÃ° yg;p; q ; p Ãžgwhere uw ; gÃ° yg;p; q ; p Ãž istheutilityofatraveleringroup g ofODpair w withincome ygundertheinitialcreditdistribution q , thecreditchargeschemepandtheequilibriumpriceofcredit p .Sincethelatterdependsontheformertwo,1wedenotethe equivalentvariationandequivalentincomeunderatradablecreditschemeas evw ; gÃ°p; q Ãž and ew ; gÃ°p; q Ãž . 1Theequilibriumpriceofcredithereisdeterminedbythetotalnumberofcreditsdistributedandthecreditchargescheme.Itisindependentofhowthe totalnumberofcreditsisinitiallydistributed( YangandWang,2011 ).1278 D.Wuetal./TransportationResearchPartB46(2012)1273Â…1287 PAGE 7 Duetotheexistenceoftherandomerrortermintheutilityfunction,theequivalentincomeisalsorandomforeachindividualtraveler.Inthispaper,weadopttheapproachproposedby DagsvikandKarlstrom(2005) tocomputetheexpected equivalentincome.AlthoughtheformuladerivedbyDagsvikandKarlstromcalculatestheexpectedcompensatingexpenditure,itcanbeeasilymodiÂ“edtocalculatetheexpectedequivalentincome. Letvw ; m ; g kÃ° y ;sÃž bethedeterministicportionoftheutilityforusersingroup g travelingbetweenODpair w onmode m and path k withincomelevel y inthepresenceoftolllevels.Theexpectedequivalentincomeforatravelerinthegroupcanbe calculatedas:E Ã° ew ; gÃ°sÃžÃžÂ¼ Xm02 MXk02 Kw ; m 0Zpw ; m 0 ; g k 0Ã°sÃž 0 Xk 2 Kw ; m 0exp hw ; m 0 ; g kÃ° z ;sÃž hm 0 0 @ 1 Ahm 0 1 exp vw ; m 0 ; g k 0Ã° yg; 0 Ãž hm 0 ! Xm 2 MÃ° Xk 2 Kw ; mexp hw ; m ; g kÃ° z ;sÃž hm Ãžhmdz Ã° 21 Ãžwhere pw ; m0; g k0Ã°sÃž and hw ; m0; g kÃ° z ;sÃž aredeÂ“nedas:vw ; m0; g k0Ã° yg;sÃžÂ¼vw ; m0; g k0Ã° pw ; m0; g k0Ã°sÃž ; 0 Ãž hw ; m0; g kÃ° z ;sÃžÂ¼ max Ã°vw ; m0; g kÃ° yg;sÃž ;vw ; m0; g kÃ° z ; 0 ÃžÃžEquation (21) providesawaytocalculatetheexpectedequivalentincomeforeachindividualtravelerwithoutknowinghis orherexactmodeorpathchoice.ThecalculationoftheexpectedequivalentincomeunderatradablecreditschemeissimilartoEquation (21) withreplacingtheutilityfunctionwiththeoneunderthetradablecreditscheme. 3.3.2.SocialbeneÂ“tmeasure Thesumoftheexpectedequivalentincomeofalltravelerscanserveasawelfaremeasureasitrepresents,atanaggregate level,howwealthytravelersfeelunderapricingorcreditscheme.Indesigningofapricingorcreditscheme,weattemptto maximizethesocialbeneÂ“t,whichconsistsoftwocomponents:theÂ“rstisthetotalexpectedequivalentincomeofalltravelersinthenetworkwhilethesecondisthetotalrevenuecollectedbythegovernment.ThesocialbeneÂ“tassociatedwitha pricingschemescanthusbecalculatedas:SB Ã°sÃžÂ¼ Xg 2 GXw 2 WE Ã° ew ; gÃ°sÃžÃž Dw ; gÃ¾ Xg 2 GXw 2 WXm 2 MXk 2 Kw ; mXl 2 LDk ; lsm lfw ; m kTherevenueunderatradablecreditschemewillalwaysbezero.ThecorrespondingsocialbeneÂ“tcanbethusrepresented as:SB Ã°p; q ÃžÂ¼ Xg 2 GXw 2 WE Ã° ew ; gÃ°p; q ÃžÃž Dw ; gAmeasureofnetbeneÂ“tcanalsobedeÂ“ned,whichdiffersfromtheabovebythetotalinitialincome,aconstant. 3.3.3.Equitymeasure Aspreviouslymentioned,weusetheGinicoefÂ“cienttorepresenttheequityimpactofapricingorcreditscheme.Given theexpectedequivalentincomeoftravelersinthepresenceofapricingscheme,itisstraightforwardtocomputetheGini coefÂ“cientasfollows:GN Ã°sÃžÂ¼ 1 2 Ã° Xg 2 GXw 2 WDw ; gÃž2 E Ã° e Ã°sÃžÃž Xg1; g22 GXw1; w22 WÃ° Dw1; g1 Dw2; g2j E Ã° ew1; g1Ã°sÃžÃž E Ã° ew2; g2Ã°sÃžÃžjÃžwhere E Ã° e Ã°sÃžÃž isthemeanofexpectedequivalentincomesofalltravelers.TheGinicoefÂ“cientunderacreditschemecanbe calculatedsimilarly. TheGinicoefÂ“cientiscalculatedbasedonexpectedequivalentincome.Amoreequitablepricingorcreditschemewill leadtoasmallervalueoftheGinicoefÂ“cient.IftheGinicoefÂ“cientisreducedaftertheimplementation,theschemebeneÂ“ts thepoormorethantherich,ordoeslessharmtothepoorthantherich. Itisworthnotingthatourmodelingframeworkdoesnotimposeanyrestrictionontheequitymeasure.OtherequitymeasuresuchasTheilÂsentropyandAtkinsonindexcanbeusedaswell. 3.4.Designmodels Wenowformulateamathematicalprogramtodetermineapricingschemethatimprovesboththeequityandsocialwelfare.AsthesetwoobjectivesareoftenconÂ”icting,weseekforabalancebetweenthem.Theformulationmaximizesa weightedsumofthenormalizedsocialbeneÂ“tandGinicoefÂ“cientasfollows:D.Wuetal./TransportationResearchPartB46(2012)1273Â…1287 1279 PAGE 8 maxs; d ; fa SB Ã°sÃž SB0Ã° 1 aÃž GN Ã°sÃž GN0s : t : Ã° f ; d Ãž2Uands2WÃ° 22 Ãžwhere SB0and GN0arethesocialbeneÂ“tandGinicoefÂ“cientbeforetheimplementationandaisapositiveweightingparameter.Thedecisionvariablesincludethetollschemeandthecorrespondingtolleduserequilibriumdemand-Â”owdistribution. Thedesignproblemforanoptimalcreditschemecanbesimilarlyformulatedasfollows:maxp; q ; d ; fa SB Ã°p; q Ãž SB0Ã° 1 aÃž GN Ã°p; q Ãž GN0s : t : Ã° f ; d Ãž2Uand Ã°p; q Ãž2HÃ° 23 ÃžThesolutionoftheaboveformulationwillspecifyanoptimalinitialdistributionofthecreditsandthecreditcharging schemeanddeterminetheequilibriumpriceofcreditandtheuserequilibriumdemand-Â”owdistribution. Asformulated,bothmodelsaremathematicalprogramswithequilibriumconstraints(MPEC),aclassofproblemsdifÂ“cult tosolve.CompoundingthedifÂ“cultyarethatbothformulationsarepath-basedandthusrequirepathenumeration(for large-scalenetworks,pathscanbegeneratedasneeded)andthecalculationoftheexpectedequivalentincomeusing (21) involvesnumericalintegration. 4.Solutionalgorithms Becausethecalculationoftheexpectedequivalentincomeinvolvesanumericalintegration,weapplyderivative-free algorithmstosolvetheaboveformulations.Suchanalgorithmsolvesaseriesofmultimodaluserequilibriumproblems, i.e.,theVIformulation (12),(18) ,iterativelybyvaryingthetollorcreditschemesandevaluatestheresultingobjectivevalues inordertoÂ“ndabetterfeasiblesolution.Theprocesscontinuesuntilanoptimumisreached. Wehavetestedseveralexistingderivative-freealgorithms(e.g., MoreandWild,2009 ).Forourproblems,compasssearch algorithm( Koldaetal.,2003 )andtheSID-PSMalgorithmorapatternsearchmethodguidedbysimplexderivatives( CustÃ³dioandVicente,2007;CustÃ³dioetal.,2010 )producegoodsolutionsinaconsistentmanner.Inourimplementation,thealgorithmsarecodedinGAMS( Brookeetal.,2005 )inconjunctionwithMatlab( Wong,2009 )andCONOPT( Drud,1995 )isused asthesolverforthemultimodaluserequilibriumproblems. 5.Numericalexamples 5.1.Networkcharacteristic ThedesignproblemsaresolvedfortheSeattleregionalfreewaynetworkasshownin Fig.1 .Thenetworkconsistsof16 nodes,44linksand30ODpairs.Itisassumedthatthetraveltimefunctionsforthefreewaylinks(solidlinesintheÂ“gure)are inaformoftheBPRfunction.Thecorrespondingfree-Â”owtraveltimeandcapacityofeachfreewaylinkofthenetworkare listedin Table1 ,ascalibratedby Boyles(2009) .Thetraveltimesforalltheotherlinksareassumedtobe5min.Itisfurther assumedthatthereisaprovisionoftransitserviceoneachlink,whichsharesthesameroadwayandthushasthesameinvehicletraveltime.However,extra10minofwaitingandtransfertimeareaddedtoeachtransitpath. TheutilityfunctionisassumedtobeaTranslogfunctionasfollows:vw ; m ; g kÂ¼ bm 0logcygÃ¾ b1ln ^ yw ; m ; g kÃ¾ b2ln Tw ; m kÃ¾ b3ln2Tw ; m kwhere ^ yw ; m ; g kisthenetdailyincomeafterpayingthetollchargeorselling/buyingcreditsforusergroup g travelingonroute k ofmode m betweenODpair w asdescribedinSection3.2,cconvertsdailyincometoannualincome.Theparametersare presentedin Table2 (see Franklin,2006 and Wu,2011 forthecalibrationprocess).Intuitively,theÂ“rsttermintheutility functionrepresentsthetravelersÂmode-speciÂ“cpreference;thesecondtermrepresentstheimpactofincomeandthelast twotermsrepresenttheimpactoftraveltime. Fortheno-traveloption,theutilityisconstantforeachusergroupandODpair.Topreparethoseconstants,weusethe followingequation,whichwascalibratedsothatforeachODpair,approximately20%oftheODdemandwillchooseno-traveloptionintheno-tollscenario:vw ; R ; gÂ¼ bR 0logaygÃ¾ b1ln Ã° ^ yw ; R ; gÃžÃ¾ b2ln e TwÃ¾ b3ln2e Twwhere e Twistheshortestfree-Â”owtraveltimeforODpair w and bR 0issettobe 0.227. Theusersofthenetworkarecategorizedintofourgroupsbasedontheirannualhouseholdincome,i.e.,$20,000,$40,000, $70,000and$120,000,respectively,withgroup1beingthelowest-incomegroupandgroup4thehighest-incomeone.The ODdemandisestimatedandreportedin Wu(2011) . Figs.2and3 presenttheproductionsandattractionsofalltrafÂ“canalysiszones,includingEverett,Seattle,Bellevue,Tacoma,LynnwoodandRenton.1280 D.Wuetal./TransportationResearchPartB46(2012)1273Â…1287 PAGE 9 5.2.Optimalpricingpolicies ThecurrentconditionoftheSeattlenetworkisestimatedbysolvingtheassociatedmultimodaluserequilibriummodel. Currentlythetravelersspendatotalof51,746htravelingwithinthefreewaynetworkwithabout80%choosingtodriveand 4%usingtransitservices.Thetotalannualincomeofthepopulationis$4.8058billionandtheGiniCoefÂ“cientis0.3129. Fig.1. TheSeattleregionalnetworkanditssurroundingregions. Table1 LinkparametersofSeattlenetwork.LinknumberLinkFree-Â”owtraveltimeCapacityLinknumberLinkFree-Â”owtraveltimeCapacity 1(1,2)7686413(5,7)138762 2(2,1)7686414(6,4)35722 3(2,3)14686415(6,5)104944 4(2,4)16589516(6,8)95609 5(3,2)14686417(7,5)137577 6(3,4)10382518(7,8)25994 7(3,5)4686419(7,9)127449 8(4,2)16572220(8,6)95994 9(4,3)10382521(8,7)25609 10(4,6)3589522(8,10)104079 11(5,3)4686423(9,7)127449 12(5,6)10661424(10,8)103950 Table2 ParametersfortheTranslogutilityfunction.ParameterAutoTransit b0Â… 0.227 b19.0149.014 b21.5121.512 b3 0.560 0.560 D.Wuetal./TransportationResearchPartB46(2012)1273Â…1287 1281 PAGE 10 Weassumethat12linkscanbecharged,includinglinks3,4,5,6,8,9,12,13,15,16,17and20,whicharemarkedasred in Fig.1 .Themaximumlinktollrateis$20andthemaximalsubsidyis$20foreachtransitlink. Byvaryingtheweighingfactora,wesolvethetolloptimizationmodeltoobtainaseriesofoptimaltollingschemesforthe network.TheseschemesmayformaParetofrontier,2asshownin Fig.4 .Thecorrespondingtollingschemesaresummarizedin Tables3and4 aspolices1Â…4.Itcanbeobservedthatpolicy1isthemostefÂ“cient,providinganetdailybeneÂ“tof$0.3884million.Atthesametime,itincreasestheGinicoefÂ“cientby4.4%.Infact,thegovernmentistheonlypartythatbeneÂ“tsfromthe policy,collecting$0.7millionintollrevenueeveryday.Allthetravelersexperienceadecreaseintheirequivalentincome.The poorestgroup,i.e.,Group1,suffersthemostlossof7.5%or$1500annually.Incontrast,therichestgroup,i.e.group4,experiencesanegligiblelossof0.06%or$72annually.Itisthussafetoconcludethatpolicy1achievesthemaximalsystemefÂ“ciency atthepriceoflow-incometravelers,therebycompromisingtheequity.Thebasicmechanismofpolicy1istochargeahightoll todiscouragetravelandthusreducetrafÂ“ccongestion.Asthereductionintraveltimeismorevaluabletothehigh-income groups,theyarehardlyharmedbythepolicy.However,thehightollpricesoffasigniÂ“cantportionoflow-incometravelers. Asshownin Table3 ,thetripsmadebythelowest-incomegroupdecreaseby30.0%whereautotripsreduceby44.6%and thetransittripsaremorethantripled.Inthemeanwhile,thetraveldemandofgroup4remainsthesameduetothenegligible changeintheutilitylevelforthegroup.Whentheequityismorefavored,theoptimalpricingpoliciestendtochargelessandsubsidizemore.Aslow-incometravelersconstitutethemajorityoftransitriders,thismechanismactsasameantoprovidemorebeneÂ“ttothepoor.Consequently,thetollrevenueforthegovernmentbecomeslower.Underpolicy2and3,althoughtherichstillsuffersless thanthepoor,thedifferenceinlossissmallerascomparedtopolicy1.Notethattherichestpeopleingroup1actuallybeneÂ“tfromallthesepricingpolicies,asshownin Fig.5 . Fig.2. Compositionsofproductionsofcentroidnodes. Fig.3. Compositionsofattractionsforcentroidnodes. 2Strictlyspeaking,thesolutionsobtainedherearenotguaranteedtobeParetooptimal,becausethetwoobjectivefunctionsarenotconvex(concave) .See GuoandYang(2009) foridentiÂ“cationofbi-criteriaParetosystemoptimalandParetooptimallinktollswithnon-convexobjectivefunctions.1282 D.Wuetal./TransportationResearchPartB46(2012)1273Â…1287 PAGE 11 Withtheequitymeasuregivenenoughweightintheobjectivefunction,policy4exhibitsadifferent,interestingpattern. Thelowest-incometravelersbecometheonesthatenjoythemostbeneÂ“t.Moreover,asshownin Fig.5 ,thelowandhighincometravelersobtainmorebeneÂ“tthanthosemid-incometravelers.TheformerbeneÂ“tsmorefromtransitsubsidies whilethelattervaluesmorethereducedtraveltimes.Allusergroupsarebetteroffcomparedtotheno-tollcondition,suggestingthatpolicy4isParetoimproving. Unfortunately,ourcomputationexperimentsdonotÂ“ndapricingpolicythatreducestheGinicoefÂ“cient.Allthepolicies discussedpreviouslyleadtoamoreunevendistributionofsocialwealth,andnoneofthemisprogressive(beneÂ“tinglowincometravelersmorethanhigh-incomeones).However,ifweallowthetotalrevenuetobenegative,i.e.,thegovernment Fig.4. Paretofrontierofcongestionpricingschemes. Table3 ResultsofoptimalpricingpoliciesforSeattlenetwork.Policy1Policy2Policy3Policy4NotollPolicySa1.000.910.830.500.000.67 NetbeneÂ“t(million$)0.38840.37830.32980.223300.2118 GinicoefÂ“cient0.32660.32350.31850.31340.31280.3119 Totaltraveltime(h)45,51146,33147,86349,00951,74649,442 Tollrevenue($)721,878593,198321,787400 99,753 PercentincreaseinequivalentincomeGroup1 7.54 5.33 1.353.460.004.80 Group2 4.36 3.11 1.190.720.001.11 Group3 1.83 1.12 0.050.960.001.18 Group4 0.060.310.861.430.001.56 AutoTransitAutoTransitAutoTransitAutoTransitAutoTransitAutoTransit GroupdemandGroup1922531609988423910,603602510,206811616,64293999548753 Group211,997150112,257170812,636203812,703252014,43469012,6692700 Group322,194158122,336169022,615184922,764206323,51798422,7412167 Group411,66354211,66756211,71158311,75759911,72441811,747621 Traveldemand55,078678556,249820057,56510,49557,43013,29866,317303057,11114,241 Table4 OptimalpricingpoliciesforSeattlenetwork.LinkPolicy1Policy2Policy3Policy4PolicyS AutoTransitAutoTransitAutoTransitAutoTransitAutoTransit Toll35.001.000.00 1.000.00 5.000.00 15.000.00 20.00 45.001.001.000.000.00 4.250.00 15.000.00 12.25 510.250.008.25 2.003.75 10.002.25 16.001.75 18.00 613.00 1.0013.00 1.0011.50 4.0012.50 1.0011.00 4.00 810.000.508.25 2.503.25 8.000.00 15.502.75 16.00 913.00 0.2512.75 2.2510.75 3.2511.00 0.259.75 3.25 1210.000.0010.000.007.75 3.756.000.006.75 3.75 1320.00 2.0020.00 6.0016.00 12.0012.00 18.008.50 20.00 1511.500.0011.00 1.759.00 2.755.500.009.00 2.75 1615.00 2.0015.00 6.0011.75 10.007.00 18.005.75 18.00 1715.000.2511.25 5.506.75 11.752.00 20.001.00 20.00208.00 0.504.25 4.500.75 10.250.00 16.500.00 18.25 D.Wuetal./TransportationResearchPartB46(2012)1273Â…1287 1283 PAGE 12 hassomeexternalfundingtosupporttheimplementationofcongestionpricing,itremainsfeasibletoimprovebothequity andefÂ“ciency.PolicySin Tables3and4 presentssuchanexample.Thepolicyimprovesthenetworkinallthreeaspects,i.e., theGinicoefÂ“cient,socialwelfareandtotaltraveltime,butrequiresadailysubsidyof$0.1million. 5.3.Optimaltradablecreditschemes Weobtainoptimaltradablecreditschemesforthesamenetworkasshownin Tables5and6 .Intheseschemes,weassumetheinitialcreditsaredistributedaspertravelersÂworklocations,i.e.,allthetravelersfromthesameoriginwillreceive thesamenumberofinitialcredits,whicharedecisionvariablesinourdesignmodel. Fig.6 comparestheoptimalcredit schemes.Itcanbeseenthatalltheschemescanimprovethesocialwelfare,evenforScheme6thatoptimizesonlyequity. Infact,theschemesthatfavorequity,i.e.Schemes3Â…6,improvebothequityandefÂ“ciency. Fig.7 showsthechangesinequivalentincomeforalltravelergroups.ThemostefÂ“cientcreditscheme,Scheme1,beneÂ“ts onlytravelersofhigherincomes.Asmoreweightisgiventoequity,theresultingschemesstarttobeneÂ“tthoselow-income travelers.Infact,Schemes2Â…6areallprogressive.UnderScheme6,theequivalentincomeofthepoorestincreasesdramaticallyby13%.Inthemeantime,thehigher-incometravelersaremadeworseoff.SuchaÂÂRobinHoodÂÂschemeactually chargesmorecreditsforusingbothhighwayandtransitlinks(see Table6 ).Asaresult,asigniÂ“cantnumberoflow-income travelersdecidenottotravelandselltheircreditstoothers,asthisoffersthemthemostvalue.However,thetraveldemand fromhigh-incometravelersisonlyslightlyaffected,whichsuggeststhatthesetravelersarebuyingcreditsfromlow-income travelers,withapricehigherthanthoseunderotherschemes(see Table5 ). Table6 reportsthecreditdistributionandchargesundereachscheme.WiththetotalnumberofcreditsdistributedÂ“xed, amoreequitablecreditschemegenerallychargesmorecreditsonbothhighwayandtransitlinks.Thisisincontrasttocongestionpricingwhereamoreequitableschemeoftenchargesmedium-leveltollsonhighwaylinksbutheavilysubsidizes transitservices.Asmorelow-incometravelersusetransitservices,subsidizingtransitusersfromtollinghighwayisessentiallysubsidizationfromhigh-incometravelerstolow-incomeones,therebyachievingabetterequity.Underatradable creditscheme,thesubsidizationappearsinadifferentbutmoredirectformviacredittrading.Whenacreditschemecharges morecreditsfortravel,itwillincreasethedemandforcredits,andthusthemarketpriceofcredits.Inthiscase,low-income travelersaremorewillingtochooselesspreferableroutesortravelmodestosavetheircreditsandselltheminexchangefor money.Ashigh-incometravelersvaluetheirtravelrightsmore,theywillgenerallypurchaseadditionalcreditsandpaymore tosatisfytheirtravelneeds.Thisleadsatransferofwealthtolow-incomepeopleandyieldsabetterincome-basedequity. Certainly,Scheme6isveryextreme.Inpractice,Schemes4and5maybemorepreferable.Theyarebothprogressiveand Paretoimproving. 5.4.Comparisonofpricingandcreditschemes Fig.8 comparestheperformancesofcongestionpricingandtradablecreditschemes.ItcanbeobservedthatthePareto frontierofthecreditschemestrictlydominatesthatofcongestionpricing.ThesetwoschemesachievethesamelevelofmaximumnetbeneÂ“ts.However,tradablecreditschemescanbeeasilyprogressivewhilecongestionpricingschemesaremostly regressiveforthisnetwork. TheaboveobservationcanbeexplainedbytheadditionalÂ”exibilitythatatradablecreditschemehastooffer.Thegovernmentcandetermineboththeinitialdistributionofcreditsandthecreditchargestoachieveitspolicygoal.Incontrast, withcongestionpricing,itcanonlydeterminehowtochargetravelers,withthewealthdistributiongiven.Inacredit scheme,themarketallowsthosewhovaluetraveltimesavingslesstobedirectlycompensatedbysellingcreditstothose whovaluethemmore.ThismechanismpromisessimplerandfairerdistributionofthebeneÂ“tsfromcongestionreduction. Fig.5. Changesinequivalentincome. 1284 D.Wuetal./TransportationResearchPartB46(2012)1273Â…1287 PAGE 13 Fig.6. Paretofrontieroftradablecreditschemes. Table5 Resultsofoptimaltradablecreditschemes.Scheme1Scheme2Scheme3Scheme4Scheme5Scheme6a1.000.980.910.670.500.00 NetbeneÂ“t(million$)0.40450.39590.38430.37420.31030.1499 GinicoefÂ“cient0.33570.31930.31270.31070.30890.3072 Totaltraveltime(h)45,20145,28845,38045,09744,45842,831 Creditprice($)0.970.991.011.091.301.63 Percentincreasein equivalent income Group1 4.065.445.797.4810.0713.08 Group2 1.132.052.312.332.301.60 Group32.181.941.531.290.64 0.69 Group45.011.331.451.140.31 0.98 AutoTransitAutoTransitAutoTransitAutoTransitAutoTransitAutoTransit GroupdemandGroup1904927919051282597162695909927687721280156971316 Group211,885143311,875144511,958141611,645147810,655163591691084 Group322,103155122,048157021,975157621,720163620,818181119,6971366 Group411,64453911,62854411,58155311,53156711,31761511,161496 Totaldemand54,681631554,602638455,230623953,995644850,511686345,7234261 Table6 Optimalcreditdistributionandcharges.Scheme1Scheme2Scheme3Scheme4Scheme5Scheme6 Initialcreditdistribution Everett0.01.58.215.815.314.4 Seattle0.00.013.19.26.83.4 Bellevue50.320.00.00.00.00.0 Tacoma0.00.07.011.615.621.8 Lynnwood0.05.214.218.821.523.6 Renton0.068.419.613.812.711.9 AutoTransitAutoTransitAutoTransitAutoTransitAutoTransitAutoTransit Creditcharges Link39.41.810.71.95.50.15.80.34.10.66.61.5 Link47.01.48.31.68.80.214.91.319.92.119.914.5 Link511.80.712.10.614.01.612.30.311.50.814.64.1 Link612.90.013.50.018.60.019.91.719.92.519.914.9 Link813.81.113.40.815.31.615.20.219.92.219.914.4 Link917.70.515.50.315.70.216.81.120.03.519.915.8 Link129.70.010.30.016.10.016.62.019.93.119.915.4 Link1320.00.020.00.020.00.020.00.220.00.920.05.0 Link1516.10.513.80.213.40.514.91.218.93.619.915.9 Link1618.10.015.30.012.30.012.50.016.40.017.910.9 Link1715.90.917.70.414.40.913.90.613.41.016.36.4 Link205.80.05.80.04.40.04.50.07.90.020.011.3 D.Wuetal./TransportationResearchPartB46(2012)1273Â…1287 1285 PAGE 14 Itisworthnotingthattheabovecomparisondoesnotconsidermanypracticalissues.Foronething,implementationcosts arenotconsidered.Becauseacreditschemeislikelytobemorecostlytoimplement,thecomparisonisunfairtocongestion pricing.Moreover,theefÂ“ciencyofacreditschememaybeadverselyaffectedbytransactioncostsandspeculationbehaviors inthecreditmarket,whichareignoredinouranalysis. 6.Conclusion WehavepresentedoptimizationmodelstodesignequitableyetefÂ“cientpricingandtradablecreditschemesforgeneral multimodaltransportationnetworks.ThemodelsconsidertheeffectofincomeontravelersÂchoicesoftripgeneration,mode androute,andexplicitlycapturetheimpactsofpricingandcreditschemesondifferentincomeandgeographicgroups.The modelsareformulatedasMPECandsolvedbyderivative-freealgorithms. NumericalexampleswiththeSeattlefreewaynetworkareusedtoillustratethemodel.Intheexamples,weÂ“nditdifÂ“culttodesignamoreequitablepricingpolicyunlessacertainshortfallinthetotaltollrevenueisallowed.Incontrast,a moreequitableandprogressivetradablecreditschemecanbeobtained.Thenumericalexamplesalsohighlightthedifferencebetweenthesetwoinstrumentsinthemechanismofachievingbetterequity. Acknowledgements Theauthorsthanktwoanonymousreviewersfortheirconstructivecommentsandsuggestions.Theresearchispartially supportedbygrantsfromtheNationalScienceFoundation(CNS-0931969)andtheNationalNaturalScienceFoundationof China(61074138).HaiYangÂsresearchispartiallysupportedbyagrantfromtheResearchGrantsCounciloftheHongKong SpecialAdministrativeRegion,China(ProjectNo.HKUST621111). ReferencesAghassi,M.,Bertsimas,D.,Perakis,G.,2006.Solvingasymmetricvariationalinequalitiesviaconvexoptimization.OperationsResearchLetters 34(5),481Â… 490. Fig.7. Changesinequivalentincome. Fig.8. 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D.Wuetal./TransportationResearchPartB46(2012)1273Â…1287 1287 PAGE 1 1 OPTIMAL DEPLOYMENT AND OPERATIONS OF PUBLIC CHARGING INFRASTRUCTURE FOR PLUG IN ELECTRIC VEHICLES By FANG HE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIR EMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 201 4 PAGE 2 2 Â© 201 4 Fang He PAGE 3 3 To my parents PAGE 4 4 ACKNOWLEDGEMENTS Fir st of all, I would like to gratefully and sincerely thank my advisor, Dr. Yafeng Yin, for h is guidance, understanding, patience, support and encouragement. He has been a wonderful advisor, mentor, colleague and friend to me during my Ph . D . study. I feel so fortunate and honored to work with him, and what I learned from him will be beneficial to me for whole life. I would like to thank Dr. Siriphong Lawphongpanich for serving as my committee member, and generously providing numerous valuable guidance and comments on my research. It has always been a pleasant and rewarding experience to discuss res earch with him. I am very grateful to Dr. Lily Elefteriadou, Dr. Scott W ashburn and Dr. Siva Srinivasan for serving as my committee members. Their guidance and support play critical roles in my Ph.D. study. I would like to thank many faculty members from t he Department of Industrial and Systems Engineering and the Department of Economics for their excellent teaching and valuable guidance during my study. I would also like to thank all the fellow students and my colleagues in Transportation Institute for the ir friendship. Lastly but most importantly, I would thank my parents for their selfless support. I would have never made any achievement without them. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGEMENTS ................................ ................................ ............................... 4 LIST OF TABLES ................................ ................................ ................................ ............ 7 LIST OF FIGURES ................................ ................................ ................................ .......... 9 ABSTRACT ................................ ................................ ................................ ................... 10 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ ..... 12 1.1 Background ................................ ................................ ................................ ....... 12 1.2 Problem Statement and Dissertation Obj ective ................................ ................ 16 1.2.1 Optimal Allocation of Public Charging Station Budget at Regional Level 16 1.2.2 Optimal Location of Public Chargin g Stations at Urban Level ................. 16 1.2.3 Optimal Pricing of Electricity at Public Charging Stations for PEVs ......... 17 1.3 Dissertation Outline ................................ ................................ ........................... 18 2 LITERATURE REVIEW ................................ ................................ ........................... 19 2.1 PEV Public Charging Infrastructure Location Problem ................................ ...... 19 2.1.1 Data Driven Approaches for Station Placement ................................ ...... 19 2.1.2 Flow Capturing Location Model ................................ ............................... 23 2.1.3 Flow Refu eling Location Problem ................................ ............................ 25 2.1.4 Clustering Methods for Station Placement ................................ .............. 29 2.1.5 Conceptual Models for Charging Infrastructure Pl anning ........................ 30 2.2 Electricity Pricing for PEV Charging ................................ ................................ .. 33 3 REGIONAL LEVEL ALLOCATION OF PUBLIC CHARGING STATION BUDGET . 35 3.1 Equilibrium of Coupled Transportation and Power Networks ............................ 35 3.1.1 Description of Transportation Network ................................ .................... 35 3.1.2 Description of Power Network ................................ ................................ . 40 3.1.3 Equilibrium of Coupled Transportation and Power Networks .................. 42 3.2 Allocating Public Charging Stations ................................ ................................ .. 48 3.2.1 Model Formulation ................................ ................................ ................... 48 3.2.2 Solution Algorithm ................................ ................................ ................... 49 3.3 Numerical Example ................................ ................................ ........................... 53 3.4 Summary ................................ ................................ ................................ .......... 56 4 URBAN LEVEL LOCATION OF PUBLIC CHARGING STAT IONS ......................... 64 4.1 Base Model ................................ ................................ ................................ ....... 65 PAGE 6 6 4.1.1 Notation ................................ ................................ ................................ ... 65 4.1.2 Definition and Fo rmulation of Network Equilibrium ................................ .. 66 4.1.3 Solution Procedure ................................ ................................ .................. 69 4.2 Equilibrium Model Considering Recharging Time ................................ ............. 71 4.3 Network Equilibrium with Flow Dependent Energy Consumption ..................... 75 4.3.1 Definition of Network Equilibrium ................................ ............................. 75 4.3.2 Model Formulation and Solution Algorithm ................................ .............. 78 4.4 Deploying Public Charging Stations on Urban Road Networks ......................... 81 4.5 Numerical E xamples ................................ ................................ ......................... 83 4.5.1 Examples for NE RT and NE FD ................................ ............................. 83 4.5.2 Examples for CD ................................ ................................ ..................... 85 4.6 Summary ................................ ................................ ................................ .......... 87 5 PRICING OF ELECTRICITY AT PUBLIC CHARGING STATIONS FOR PEVS ..... 97 5.1 Modelli ng Transportation and Power Networks ................................ ................ 98 5.1.1 Description of Transportation Network ................................ .................... 98 5.1.2 Analysis of Power Flows in Distributi on Grid ................................ ......... 101 5.2 Optimal Pricing Models ................................ ................................ ................... 104 5.2.1 Model Formulations ................................ ................................ ............... 104 5.2.2 Solution Algorithm ................................ ................................ ................. 107 5.3 Numerical Examples ................................ ................................ ....................... 111 5.4 Summary ................................ ................................ ................................ ........ 113 6 CONCLUSION ................................ ................................ ................................ ..... 124 LIST OF REFERENCES ................................ ................................ ............................. 127 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 137 PAGE 7 7 LIST OF TA BLES Table page 3 1 Link capacity (100 veh/hr) and free flow travel time (h r) ................................ ....... 57 3 2 Trip production at each origin (100 veh/hr) ................................ ........................... 57 3 3 Input data for transmission lines ................................ ................................ ............ 58 3 4 Input data for generators ................................ ................................ ....................... 58 3 5 Traffic link flow (veh/hr) ................................ ................................ ......................... 59 3 6 Power line flow (M W) ................................ ................................ ............................ 60 3 7 PEV load, power injection and LMP at each bus ................................ .................. 60 3 8 LMPs associated with different allocation plans ($/MWh) ................................ ..... 60 4 1 BEV charger specification ................................ ................................ ..................... 89 4 2 Link capacity (1000 veh/hr) and free flow travel time (min) ................................ ... 89 4 3 Equilibrium link flow (veh/hr) ................................ ................................ ................. 90 4 4 Equilibrium link flow (veh/hr) ................................ ................................ ................. 90 4 5 Path inform ation for the first equilibrium flow pattern ................................ ............ 91 4 6 Path information for the second equilibrium flow pattern ................................ ....... 91 5 1 Link capac ity (veh/hr) and free flow travel time (min) ................................ .......... 115 5 2 O D demands of regular gasoline powered vehicles (veh/hr) ............................. 116 5 3 Trip pr oduction of PEVs at each origin (veh/hr) ................................ ................... 116 5 4 Regular load at each bus ................................ ................................ .................... 116 5 5 Branch series impedance (Ohms) ................................ ................................ ....... 117 5 6 Optimal retail electricity prices at public charging stations ($/KWh) .................... 117 5 7 Optimal retail electricity prices at public charging stations ($/KWh) .................... 117 5 8 Travel cost and real power losses ($) ................................ ................................ . 118 5 9 Link tolls for regular gasoline powered vehicles (min) ................................ ......... 118 PAGE 8 8 5 10 Optimal retail electricity prices at public charging stations ($/KWh) .................. 119 PAGE 9 9 LIST OF FIGURES Figure page 3 1 An illustrative example for equilibrium in coupled networks ................................ . 61 3 2 The coupled transportation and power networks ................................ .................. 62 3 3 Cumulative distribution curve of social welfare ................................ .................... 63 4 1 A toy network with four nodes ................................ ................................ .............. 92 4 2 Sioux Falls n etwork ................................ ................................ .............................. 92 4 3 Charging station utilizations ................................ ................................ ................. 93 4 4 Recharging information ................................ ................................ ........................ 94 4 5 Nguyen Dupius network ................................ ................................ ....................... 95 4 6 Sensitive analyses of budget limits ................................ ................................ ...... 96 5 1 Major components of distribut ion gird ................................ ................................ 120 5 2 A typical feeder ................................ ................................ ................................ .. 120 5 3 The SNR solution procedure ................................ ................................ .............. 121 5 4 The solution framework ................................ ................................ ...................... 122 5 5 The coupled transportation network and test distribution feeder ........................ 123 PAGE 10 10 Abstract of Dissertation Presented to the Graduate School of the University of Florida In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy OPTIMAL DEPLOYMENT AND OPERATIONS OF PUBLIC CHARGING INFRASTRUCTURE FOR PLUG IN ELECTRIC VEHICLES By Fang H e August 2014 Chair: Yafeng Yin Major: Civil Engineering P lug in electric vehicle s (PEV s) are vehicle s whose battery packs can be recharged from power grid s , and the electricity stored on board propels or contributes to propel the vehicle s . PEVs include battery electric vehicles (BEVs) and plug in hybrid electric vehicles (PHEVs). Interests in PEVs have increased dramatically in recent years due to advances in battery technologies, rising prices of petroleum, and growing concern over environment issues. Many governments have incentive policies, su ch as offering purchase subsidies and deploying public charging infrastructure in convenient locations , to promote the deployment of PEVs. Building public charging infrastructure has a profound impact and is typ icall y associated with a high capital cost. To assist policy makers to optimize the investment, this dissertation is devoted to developing a hierarchical modeling framework where a strategic planning model captures interactions between a regional transport ation network and power transmission grid s to determine a budget allocation plan for public charging stations among metropolitan areas in the region while a tactic planning model PAGE 11 11 considers the spatial distribution of PEVs and optimizes the location s and op erations of charging stations in a metropolitan area. More s pecifically, at the regional planning l evel , a static game theoretic approach is applied to investigate interactions among the availability of public charging stations, destination choices of P EVs, and prices of electricity. The interactions lead to an equilibrium state that can be formulated into a convex mathematical program. We then examine how to a llocate the public charging station budget among metropolitan areas in a particular region to maximize social welfare associated with the coupled transportation and power networks. For a particular metropolitan area, g iven the allocated budget limit , we consider the problem of how to determine the number, location s and types of ch a rging stations w ithin the budget limit . Assuming the locations and types of public charging stations are given, w e first develop netwo rk equilibrium models with BEVs. Based on the proposed equilibrium models, station location plans are then optimized to maxim ize social we lfare. Lastly, we investigate the operations of p u blic charging stations with a focus on optimizing the prices of electricity at public charging stations. PAGE 12 12 CHAPTER 1 INTROD U C TION 1.1 Background P lug in electric vehicle s (PEV s ) are vehicle s whose b attery packs can be recharged from power grid s , and the electricity stored on board propels or contributes to propel the vehicle s . PEVs include battery electric vehicles (BEVs) and plug in hybrid electric vehicles (PHEVs). Loosely speaking, the former inco rporates a large on board battery, which can be cha rged while parked via a cord to power grid s . The battery provides energy for an electric motor to propel the vehicle. The latter is also equipped with an internal combustion engine generator that provides electricity to the motor once the initial battery charge is exhausted. Interests in PEVs have increased dramatically in recent years due to advances in battery technologies, rising prices of petroleum, and growing concern over environment issues (e.g., Duv all, 2004 ; Maitra et al., 2010 ). Almost all major vehicle manufactures have their PEV models available in the market, and a fast growing adoption of PEVs is expected. For example, a report predicts that the mark et penetration of PHEVs , will be roughly 1.5 million in 2016 in the United States and the number continues to increase to over 50 million in 2030 with roughly 25% of all newly purchased vehicles being PHEVs ( New York ISO, 2009). Compared to conventional gasoline powered vehicles, PEVs hold several advantages. First, it can reduce the reliance on imported petroleum and thus increase energy security ( US Department of Energy, 2013 ) . Second, PEVs typically achieve better fuel economy. For example, the US Environmental Protection Agency (EPA) rated the selling PEVs, at 115 miles per US gallon gasoline equivalent ( US Department of Energy, 2013 ) . PAGE 13 13 Third, PEVs, operating in the all electric mode, emit no harmful tailpipe pollutants fr om the onboard source of power , wh ereas the extent of emission reduction in the well to wheel assessment depends on the fuel and technology utilized for electricity generation. Last, PEVs provide drivers the opportunity to sell the electricity stored in their ba tteries back to power grid s , thus creating new supplies to meet high demand s in the peak hours of power grid s (e.g., Kempton & Letendre, 1997 ; ) . Despite the above advantages, there are still some significant obstacles to PEV market a cceptance. Considering the driving range of BEVs is limited, the fear of batteries running out of power en route, normally referred to as range anxiety in the literature (see, e.g., Pearre et al., 2011 ), will inevitably prevent the adoption of BEVs. In add ition, the purchase price s of PEVs are so high that gasoline cost saving might not offset them . For instance, Lee & Lovellette (2011) found that according to the purchase and operating costs in 2010 , a PHEV 40 is $5,377 more expensive than an internal comb ustion engine, while a BEV is $4,819 more expensive. M any governments have incentive policies , such as offering purchase subsid i es and deploying public charging infrastruc ture in the convenient locations of urban areas, to promote the deployment of PEVs. For example, the Chinese Ministry of Finance announced a pilot program to provide incentives up to 60, 000 yuan for the private purchase s of new BEVs and 50, 000 yuan for PHEVs in five cities of China ( Motavalli, 2010 ) . California announced to build 200 p ublic fast charging stations and wire for 10,000 plug in units at 1 , 000 locations across the state. British Columbia, Canada, has a plan of building 570 charging stations across the province ( GLOBLE Net, 2012 ). PAGE 14 14 Building public charging infrastructure is ty pically associated with a high capital cost (Dong et al., 2013). The National Research Council (2013) estimated the charging infrastructure investment cost for BEVs to be $3 , 000 per vehicle. To assist policy makers to optimize the investment of public reso urces in building public charging infrastructure, a systematic approach, covering the stages of bo th planning and operation , is needed. In the planning stage, decisions may focus on how to allocate the budget for constructing charging stations among differ ent metropolitan area s in a region , and how to further determine the location s and types of charging stations in a metropolitan area within the allocated budget limit , while price s of electricity are essentially a major decision during the operation stage . Various efforts have been made to optimally locate public charging stations in the literature . Frade et al. (2010) formulated a maximum covering model to locate a certain number of charging stations to maximize the demand covered within a given distanc e. Ip et al. (2010) first applied a hierarchical clustering analysis to identify the demand clusters of PEVs and then formulated simple assignment models to locate charging stations to those identified demand clusters. Nie and Ghamami (2013) developed a conceptual corridor centric optimization model to simultaneously determine battery size s , the number of public charging stations and charging power at each station al ong a corridor to minimize social cost while maintaining a given level of service. Assumin g most PEVs have convenient access to charging opportunities, a variety of electricity pricing strategies have also been evaluated in the literature. Siosh ansi (2012) examined the impact of electricity price s compare d the costs and emissions under different price structures and the ideal case of charging PAGE 15 15 controlled by a system operator. Flath et al. (2013) introduced a n electricity pricing scheme with both temporal and spatial dimensions to improve PEV charging coordi nation based on German mobility data, and found that the spatial price component reflecting local capacity utilization can mitigate load spikes. However, most of the charging station location models in the literature a) ignore the interactions between t ransportati on and power systems, coupled by PEVs (e.g., Galus and Andersson, 2008; Kezunovic et al., 2010); b) rely on the assumption that PEV driver behaviors remain unchanged with respect to public charging station deployment . In addition, the electric ity pricing models in the literature mainly focus on travel profiles, such as route and destination choices, are fixed and independent of electricity price s . This dissertat ion is devoted to overcoming the shortcom ings of the existing literature, and develops a hierarchical modeling framework for both the deployment and operations of public charging infrastructure. More specifically, this dissertation first proposes a strateg ic planning model at regional level that captures the interactions between regional transportation network s and power transmission grid s to determine an optimal budget allocation plan for public charging stations. Then , a tactic planning model is formulate d to optimize the specific number, location s and types of public charging stations in a metropolitan area within the allocated budget limit. Moreover, the tactic planning model recharging their vehicles , and captures the impact of pu blic charging station deployment on their route and recharging decisions . Last ly , the dissertation investigates the impact of the electricity price s at public charging stations on urba n PEV traffic flow distribution and PAGE 16 16 power distribution grid operations, and then optimally design s electricity price s to mitigate the adverse impacts of PEV charging loads on power distribution networks. 1.2 Problem Statement and Dissertation Objective 1.2.1 Optimal Allocation of Pub l ic Charging Station Budget a t Regional Level The policies and measures implemented in the transportation system will change the spatial and temporal distribution of PEVs and thus the pattern of their energy requirement, thereby affecting the operations of the power system. On the oth er hand, the provision of charging infrastructure and the associated charging strategies and expenses will affect the travel patterns of P EVs and consequently the entire transportation system. Using Pennsylvania New Jersey Maryland Interconnection as a cas e study, Wang et al. (2010) demonstrated that, under e xisting charging infrastructure , even a small magnitude of load increase caused by PEV charging activities can have a significant undesirable impact on electricity price s . For the regional level alloc ation of charging station budget , it is thus of importance to consider the interplay between regional transportation and power transmission systems. In this disse rtation, we apply a static game theoretical approach to investigate interactions among the ava ilability of public charging stations, destination choices of PEVs, and prices of electricity. The interactions lead to an equilibrium that can be formulated into a convex mathematical program. We then examine how to a llocate the public charging station budget among metropolitan areas in a particular region to maximize social welfare associated with the coupled networks. 1.2.2 Optimal Location of Public Charging Stations at Urban Level Given the allocated public charging station budget limit for a cit y or urban area, w e consider the problem of how to determine the specific number, locations and types PAGE 17 17 of public charging stations within the city. We envision that the fear of batteries running out of power en route, normally referred to as range anxiety in the literature (see, e.g., Pearre et al., 2011), will inevitably affect the travel choices of BEV drivers, and significantly distinguish their behaviors from those of conventional vehicle drivers. Specifically , w hen traveling between their origins and d estinations, BEV drivers will select routes and decide battery recharging pl ans to minimize their costs while making sure to complete their trips without running out of charge. For optimally locating public charging stations within urban a reas, it is thus of essence to capture the impact of BEV range anxiety and their recharging requirement . In this dissertation, we first develop network equilibrium models with BEVs to predict traffic flow distribution and ven public charging station location plan. Based on the propo sed equilibrium models, station location plans are then optimized to maximize social welfare. 1.2.3 Optimal Pricing of Electricity at Public Charging Stations for PEVs Assuming a significant n umber of public charging stations have already been deployed in the co nvenient location s of urban areas, PEV charging behaviors at these stations will inevitably introduce additional loads to power distributi on networks. The adverse impact of these additio nal loads may result in , among others, loads exceeding the design c apacity of circuit, power loss es increasing and voltage imbalance. This dissertation explores the use of prices of electricity at public charging stations as an instrument, in couple of roa d pricing, to better manage both power distribution and urban transportation networks. PAGE 18 18 1.3 Dissertation Outline This dissertation is organized as follows. Cha pter 2 reviews the related existing studies in the literature, including public charging station deployment and electricity price design models for PEVs. R egional level public charging station budget allocation model is presented in Chapter 3. In Chapter 4, network equilibrium models with BEVs are first discussed. B ased on the proposed equilibrium mod els, a bi level model is then developed to determine the specific number, location s and types of public charging stations in an urban area with an allocated budget limit . Chapter 5 is devoted to optimally design ing the electricity price s at pubic charging stations for PEVs. Chapter 6 provides conclusions with guidelines of future research. PAGE 19 19 CHAP TER 2 LITERATURE REVIEW For PEVs, t here is limited literature on public charging infrastructure planning and electricity pricing for their charging due to their recent emerging popularity. In this chapter, we start from reviewing the existing data driven models for charging station placement . Considering the similarity between locating charging stations for PEVs and placing refueling stations for conventi onal and alternative fueled vehicles, we then briefly introduce the f low refueling location and flow capturing location problems, which are extensively studied in the literature. We then turn our focus to some conceptual models and clustering approaches. F inally, the electricity pricing models dedicated to coordinating PEV charging are reviewed. 2.1 PEV Public Charging Infrastructure Location Problem 2.1.1 Data Driven Approach es for Station Placement The data driven approaches normally develop models using the available travel survey data which is mostly based on conventional vehicles. Therefore, the underlying assumption of these approaches is that t r unchanged after shifting from conventional vehicl e s to PEVs. Dong et al. (2013) proposed an activity based assessment method to evaluate the feasib i lity of BEVs completely fu l fil l ing rogen e ous daily travel needs in the real world driving context. Assumi charger locations and BEV characteristi cs are known, the energy increase in a BEV battery at each intermediate stop, me a sured in miles , is cal culated as follows: PAGE 20 20 where is the energy incre ase of a battery from recharging at the destination of driver th trip on day ; is the range of driver BEV; repres e nts the i s h in g trip and before a possible re c h arging at the destinatio n; denotes the charger power at the destination of driver th trip on day , and is cor responding to different level s of cha r ging stations, which equals zero if no charging station is located ; repres e nts dwell time after driver th trip on day ; lastly, is the ele ctricity consumption rate of driver e relations of battery states of charge for two consecutive trips are written as where denote s the travel distance of driver th trip on day . A negative sta te of charge, i.e., , indicates driver already out of energy before fin i shing trip . Trip and all the following trips are marked as missed trips. A g enetic algorithm is then used to deter m i n e the location s and types of BEV charger s to minim i ze the total number of missed trips of all the BEV drivers on all the t ravel days, given a limited bud g e t. Assuming traveler behavi ors remained unchanged after shifting to BEVs, a case study is conducted using the GPS based travel sur vey data collected in the greater Seat tle metropolitan area and shows that with a small budge t, level 1 chargers are preferred considering they can guarantee necessary network coverage at a low cost. Xi et al. (2013) proposed a simulation optim i zation mod eling framework for locating public charging infrastructure. T he model first determine s a B EV adoption probability of vehi c l e owners living in each sub region, depending on demographic data. Ass uming tour record data for vehi c l e trips is available, Bernoul li trials are then PAGE 21 21 generated to determine B EV flows be t ween the different sub regions , based on the estimated B EV adoption probabilities . For each candidate location, simulations are run to estimate the amount of energy recha r ged in B EV batteries as a func t i on of t he number of installed chargers and char g i ng rate types. Spec i fically, in the simulation, a B EV charges if and only if there is an unoccupied charger upon arrival. Moreover, if deciding to charge, a B EV occupies the charger for entire parking du r a tion. After the simu lation, the problem of determining the location s and types of chargers to maxim ize the recharged energy amount within a budget limit is formulated as a linear integer program. Note that dist r ibution power transfo r mer capacity is also co nsider e d in the model. S ensitive analysis is then conducted by relaxing the assumption that the recharged amount of energy at each location is independent, and inco r porating the co nsideration of more conservative charging behaviors of BEV drivers. Kameda and Mukai (2011) analyzed the utilization of BEV s in an on demand bus system in Japan, where share ride buses transport customers door to door, and pick up and drop off customers as soon as whenever required. The limited range of BEVs and possible long bat tery recharging time may cause problems in the on demand bus system such as rejecting demands due to insufficient charge and increasing customer waiting time because of battery recharging. In order to increase customer acceptance rate and decrease average transport time, an approach is proposed to optimize the placement of charging stations in a service area based on ta xi probe data at Tokyo. The taxi probe data is a historical data of taxis including the latitude and longitude of taxis picking up and drop ping off customers. It is assumed that the demand occurrence of on demand buses and taxis follow s similar spatial and temporal distribution. P otential PAGE 22 22 location s are then ranked according to the amount of outgoing demands, ingoing demands and the summation of the two , respectively. Simulations are run to examine the impacts of charging station placement on the customer acceptance rate and average transport time, where node insertion algorithm is used for vehicle routing. The results show that locating chargi ng stations at the top nodes ranked according to the amount of incoming demands can keep high acceptance rate, probably due to its ability to keep the high levels of BEV battery charge. However, the average transport time is not significantly affected. Ba Chen et al. (2013) proposed a regression model where land use and access attributes are used to predict total parking times per traffic analysis zone. It is shown that parking demand s are most associate d with employment density while parking prices , transit access, student density and network connectivity also appear to be relevant. The o utputs of the parking demand estimation model then serve as in puts to the problem of determining ef ficient charging station location s , which is formulated below: (2 1 ) (2 2) (2 3 ) (2 4 ) (2 5 ) (2 6 ) PAGE 23 23 where parameter is the w alking distance between zones and ; variable denotes the amount of demands in zone choosing to charge at zone ; represents the parking demands, estimated from the proposed regression model; parameter dic t ates the budget; is a big constant number; is a binary variable, which equals one if zone sits a station and zero otherwise; and parameter denotes a distance threshold. The objective function is to minimiz e the total walking distances for PEVs. Equation 2 1 guarantees the demand conservation. Equation 2 2 dictates that PEVs can only choose zones equipped wit h stations to charge. Equation 2 3 s ets the budget limit. Equation 2 4 is to avoid the clustering of charging stations around particular zones. Equations 2 5 and 2 6 guarantee variables are nonnegative and binary respectively. Andrews et al. ( 2012 the Chicago and Seattle travel survey data. A ctivities o f the travelers who fail to complete all their trips using BEVs serve as an input to a charging station location model. The model minimizes the total access distances to charging facilities for all the vehicles. Numerical examples show that both the maximu m and average access distances decrease as the total number of located stations increases. In addition, the percentage of vehicles that recharge exactly on spot, i.e., with zero access distance, also increases as the number of stations increases. 2.1. 2 F low Capturing Location Model Hodgson (1990) first pr oposed a flow capturing location model (FCLM) . Traffic flow on path is captured if and only if there is at least one charging station along . PAGE 24 24 Given a fixed amount of stations, i.e., , the proble m of determining their location s to capture as much flow as possible is formulated as follows: (2 7 ) (2 8 ) where parameter is traffic flow on path ; is a binary variable, which equals one if there is a station at node and zero otherwise; is the set of nodes on path ; lastly, is a binary variable, which equals one if path is intercepted an d zero otherwis e. The objective is to maximize interc epted flow. Equation 2 7 specifies the total number of stations . Equation 2 8 guarantees traffic flow on path is captured if and only if there is at least one charging station along . Berman et a l. (1992) independently studied FCLM and proved that only focusing on junction nodes is enough to capture all flows. Berman et al. (1995) further extended FCLM. One extension considers the beh aviors of flow deviating from existing path s to refuel. In this case, f low on a path is still considered to be captured by a station if the minimum distance between the station and the path is less than a threshold. Hodgson & Berman (1997) relaxed the assumption that there are no additional benefits if a path is captur ed by more than one station s and investigated the billboard location problem where the benefits of displaying an adv ert isement to a driver multiple times are captured . Shukla et al. (2011) proposed a modified flow interc eption facility location model w here a budget limit is given. Th e case study based on the city of Alexandria show s diminishing return on capital PAGE 25 25 investment with the number of refueling stations increasing. Moreover, the flow interception rate corresponding to the current twenty gas stations in the city of Alexandria was calculated as 55%, which is lower than the result, i.e., 57%, of a f l ow interception model with only five facility budget. Building publi c charging stations at the location s capturing as much flow as possible is definitely ben eficial to increasing station utilization. However, simply capturing flow on paths cannot necessarily guarantee the captured PEV flow s are capable of completing their entire trip s. Therefore, results from FCLP may not be optimal for PEVs to recharge batter ies. 2.1. 3 Flow Refueling Location Problem Kuby & Lim (2005) proposed the flow refueling loc ation model (FRLM) that captures the behaviors of alternative fuel ed vehicles stopping at multiple facilities to fully refuel the entire path. It is as sumed that a ll the traffic flow of a OD pair, i.e., , utilize s the shortest path, i.e., , but may have different recharging strategies. A facility combination, i.e., , for path is a set of nodes along , and if a BEV can complete its trip along when the nodes in combination all sit charging stations and the BEV only stops at these nodes to recharge batteries. An algorithm is developed to generate all possible combination that can refuel each path. Given a fixed amount of stations and path flow distribution, the problem of locati ng the stations to refuel as much flow as possible is formulated as a mix integer program. Instead of limiting station candida te sit e s only to the intersection node of any two arcs, Kuby & Lim (2007) extend ed FRLM to add candidate sit e s using three method s. The first method examines each path and add s a node at a location if only recharging at PAGE 26 26 this loc ation can refuel the entire path . The second and third methods are related to the added node dispersion problem (ANDP). By adding a given number of nodes, t he second method, i.e., minmax ANDP, minimizes the maximum sub arc length in the network , while the third method, i.e., maxmin ANDP, maximizes the minimum sub arc length in the network. In numerical examples , FRLM is solved for both original and node added networks. R esults show that adding ANDP sites outperforms the first method and vertices only. Upchurch et al. (2009) introduced a capacit ated FRLM through relaxing the assumption that a refueling station always holds suf ficient spots to serve all the flo w passing through it. Given a fixed total amount of charging stations, i.e., , the problem of optimizing their loc ation s to ma ximize the covered traffic flow can be formulated below : (2 9 ) (2 10 ) (2 11 ) (2 12 ) (2 13 ) where is the traffic volume within OD ; variable denotes the percentage of tr avelers within OD choosing the facility combination to recharge their vehicles; parameter dictates the fraction of trips requiring recharging; parameter represents the number of refueled times at node for travelers of OD stop ping at PAGE 27 27 facility combination ; parameter is the refueling capacity for a station; lastly, is a nonnegative integer representing the number of stations placed at node . Equation 2 9 sets the upper limit for flow sharing of each combinatio n. Eq uation 2 10 specifies the total number of char ging stations. Equations 2 11 2 1 3 are feasible constraints for decision variables. T he use of multiple facilities at a single node and multiple facility combinations for one O D pair is observed in the case s tudy . It is also found that the capa citated FRLM prefers to locate stations in dense flow corridor s to maximize capacity utilization. Finally , the flow requiring fewer refueling stops is also preferred. All the above FRLMs need to first generate possible facility combination s for each path, which may be time con suming. Wang & Lin (2013) formulate d the capacitated multiple recharging station location problem as a mixed integer program without the need of pre generation of facility combination. Specifically , given a set of paths, the problem of locating charging stations to maximize path coverage is formulated as follows: , (2 1 4 ) , (2 15 ) , (2 16 ) , (2 17 ) , (2 18 ) , (2 19 ) , (2 20 ) PAGE 28 28 (2 21 ) , (2 22 ) (2 23 ) where is a binary variable and equals one if BEV can traverse on path without running out of energy; is also a binary variable an d equals one if BEV s use type station at node on path to recharge ; parameter denotes BEV path flow on path and is given; represents the battery state of charge at node on path ; variable is the recharged energ y amount at node on path ; parameter represents the distance of link ; dictates BEV battery range; is predetermined and represents the stay time at node on path ; parameter is the charging power of type s tation; represents the amount of type charging resources at node ; parameter is the vehicle sharing of a type recharging station; parameter denotes the locating capacity at node ; parameter denotes the cos t of locating a type recharging station at node ; lastly, is the budget limit. The objective function is to maximi ze covered path flow. Equation 2 1 4 calculates the battery states of charge at the two ends of each link. Equation 2 15 dictates th e amount of remaining energy plus recharged amount cannot e xceed battery range. Equations 2 16 and 2 17 restrict the amount of recharged energy, i.e., BEV s can only recharge if there are stations, and recharging time is subject to duration time. Equation 2 18 requires there can be at most one type of stations at a node. Equations 2 19 2 21 demonstrate the locating capacity and budget limit for station placement. Finally, Equations 2 22 and 2 23 are binary and nonnegative constraints for variables. PAGE 29 29 The resul ts of case study demonstrate that a mixed station location plan is always better than only locating a single type of recharging station s . Finally , it is worth mentioning that although FRLP takes into account PEV detailed chargi ng behaviors along paths, it still assumes fixed path choice s independent of traffic congestion and public charging station deployment, which is not realistic in practice. 2.1. 4 Clustering Method s for Station Placement Ip et al. (2010) optimized the allocation of recharging statio ns t hrough hierarchical clustering. In the proposed approach , the traffic information from a bounded city area is first collected. Specifically , t he traffic occupancy per segment of 100 meters is measured through techniques such as road pressure sensors an d video surveillance cameras . T he road traffic is then converted to data points in a fixed size grid with t he sizes of points proportional to the traffic occupancy rates at their correspondi n g location s . A hierarchical clustering analysis is applied to pa rtition stations based on their geographical coordinates , which provide s multiple sets of clusters of different sizes and at different levels. Based on any full set of generated clusters, the problem of assigning stations to demand clusters is formul ated a s a simple linear program. Momtazpour et al. (2013 ) proposed coor dinated clustering techniques to aid in placement of charging stations for BEV promotion based on a synthetic population and acti vity data. Before clustering potential location s , their proper ties are first examined. The electricity load for location , i.e., , is computed as: PAGE 30 30 where denotes the number of pepole coming for the purpose of ; is the average square footage per person coming for the purpose of ; represents the electricity consumption of building type . Further m ore, the number of times a trav e ler need to recharge , i.e., , equals where is the total d aily travelling distance and denotes the battery range. A coordinated clu s tering formulation is used to simultaneously cluster location s based on three datasets, i.e., income , geographical location s and B EV charging station attribut es. The last include s three categories, i.e., electricity loads, charging needs and duration time. The set of clu s ters based on it is then used a s a guide to charger placement, with the preference over the c luster having low current electricity load s , high charging needs and long stay duration. 2.1. 5 Conceptual Models for Charging Infrastructure Planning Mak et al. (2013) developed two robust models to determine the location s of battery swapping stations for BEVs. With the assumption that the swap demanding BEVs trave lling al ong a particular path e nter a freeway network following a Poisson process, the minimum number of batteries at a station to guarantee a service level for BEVs is calculated as a function of the arriving rate of the Poisson process . Considering in practice, BEV swapping demands depend on many unpredictable factors, the authors then captured t he uncertainties of battery swapping demands by modeling the arriving rates of BEVs at battery swapping stations as random variables. Two robust op timization models are p roposed. One focuses on minimizing the total construction and maintenance cost s of battery swapping stations while guaranteeing battery PAGE 31 31 inventory is enough under the worst case scenario and with a high probability, the number of batteries at a station does not exceed the maximum allowable number. The other considers a scenario where the budget of deploying stations is from the investment of financial inst itutes. It is thus natural for investors to set some target s on profit rate. The model aims to maximize the probability that the target profit rate is achieved . Tight upper bounds to both models are obtained through solving mixed integer s econd order cone programs. Computational experiment results based on the San Francisco Bay Area freeway network show that the two models generate the solutions of similar qualities, and advancements in the technologies that increase recharging speed s Nie & Ghamami (2013) considered a long corridor with a length of under the assumption that charging stations are uniformly spaced along it, i.e., the distance between any pair of stations is . A conceptual model of simultaneously designing the number of stations, size of BEV batteries and charging power o f stations is formulated as follows: (2 24 ) (2 25 ) where is the function representing the cost of constructing charging facilities with power and spots; denotes the density of BEVs and is the total number of BEVs along the corridor; dictates the cost of manufacturing battery pack s at the capacity ; parameter is the charging efficiency and denotes the time of full y PAGE 32 32 recharging a battery; parameter is referred to as the range tolerance, which means travelers will recharge batteries when 100 % of the battery capacity is consumed; is the lost time due to deceleration and acceleration for recharging; is a fixed time and reflects a certain level of service for BEVs in the corridor; lastly parameter converts energy to travel distance. The objective function is to minimize the total cost s of constructing charging infrastructure and manufacturing batte ry packs. Equation 2 24 guarantees a certain level of service for BEVs along the corridor, i.e., the time wasted on recharging for a BEV traversing along the corridor is less than a given level. Equation 2 25 dictates the battery capacity should be enough for BE Vs to travel between two consecutive stations. Analytical solutions for the above model are derived and primary observations include a higher battery manufacturing cost leads to a smaller battery size and larger charging power; a lower level of ser vice requirement reduces optimal battery size; t he growth in BEV population will make it more preferable to have a smaller battery size and larger charging power; if the average trip frequency is less than 1, a higher trip frequency will increase optimal b attery size and decrease charging power. Similar research was also conducted by Ko et al. (2012) where BEV battery size and charging segment location s are simultaneously determined in a n automated wireless charging electric transportation system. In the i nnovative system, the path of a BEV bus is given and fixed. Different from traditional BEVs which rely on manually plugging in for charging, the BEV bus is charged wirelessly from the p ower transmitters buried under road s . PAGE 33 33 2.2 Electricity Pricing for PEV Char g ing Flath et al. (2013) investigated the temporal and spatial electricity pricing to shift PEV charging activities and mitigate load spikes. Specifically, time based electricity prices are from the hourly wholesale prices while area based electricity prices are deter mined by the following equation: where is the locational surcharge if the current load a t area , i.e., , exceeds the limit, i.e., ; is a parameter related to the shape of price curve. Three PEV charging strategies are examined . The naive charging is to charge batteries whenever possible. The optimal charging is to min imize total charging cost s subject to meeting The h euristic smart charging is to charge batteries only when charging is necessary to fulfill following trips, or electricity price s or battery state s of charge are below some threshold s . Simulation shows that the sole use of time based electricity prices produces high load spikes regardless of charging strategies. However, t hese peaks can be temporally and spatially shifted by introducing a spatial component that reflects local capacity utilization . Sioshansi (2012) inve sti gated the impacts of electricity tariffs on the charging costs and emission of PHEVs. Specifically, PHEV charging under four different patterns of electricity tariffs, i.e., f i xed rates, delay rebate rat es , time of use pricing (TOU) and real time pricing (RTP) , is analyzed. The f ixed rate s means the electricity price is time invariant. The delay rebate rates also assume time invariant electricity prices but PHEV owners are given a rebate if allowing a sys tem operator to control their vehicle charging PAGE 34 34 between the last trip of a day and the first trip of the following morning. TOU represents the electricity tari ff has two different rates during different hours of a day while RTP assumes drivers pay time vari ant price s based on hourly marginal energy cost. Results show RTP performs worst among all of the tariffs, and delay rebate charging is better than TOU, meaning coordinating overnight charging is significantly more important than limiting vehicle mid day c harging. Xi & Sioshansi (2013) investigated the equilib r i um interaction between power s under two price based signals, i.e., marginal pricing and price/quantity based signal s . The marginal pricing scheme charges PEV s based on the most recent marginal cost of providing energy, whereas price/quantity based scheme uses all of the marginal price data collected in all the iterations before achieving the equilibrium. It is proved that the proposed price/quantity based scheme can decentralize a social optimal solution. Numerical examples show that the marginal pricing leads to 6% increase of social costs. Qian et al. (2011) analyzed the impact of BEV battery charging on power distribution systems. The authors considered t hree electricity tariff st ructures, i.e., real time, time of use and fixed rates, and four charging scenarios, i.e., uncontrolled domestic, uncontrolled domestic off peak, smart domestic and uncontrolled public charging. Si mulation results show that for the scenario of uncontrolled domestic charging , a 10% BEV penetration would lead to a 17.9% increase of daily peak demand s . Moreover, the smart charging method is the most beneficial to power distribution systems. PAGE 35 35 CHAP TER 3 REGIONAL LEVE L ALLOCATION OF PUBLIC CHARGING STATION BUDGET This chapter reports our effort s to propose a strategic planning mode l for the public charging station budget allocation problem . More specifically, we adopt a static game theoretical approach t o investigate the interactions among the availability of public charging sta tions, destination choices of P EVs, and prices of electricity. The interactions lead to an equilibrium in the coupled transportation and power networks where prices of electricity , and traffic and power flow distributions can be determined. We formulate the equilibrium conditions into a convex mathematical program. We then examine how to allocate the public charging station budget to maximize social welfare associated with the co upled networks. For the remainder of this chapter, i n Section 3.1, we describe the transportation and power networks and examine the equilibrium state in the coupled networks. A mathematical program is developed to estimate the equilibrium prices of el ectricity and flow distributions in both networks. Incorporating the equilibrium conditions as constraints, Section 3 .2 formulates finding an optimal allocation of public charging stations as a mathematical program with complementarity constraints, and th en proposes its solution algorithm. Section 3.3 presents a numerical example to demonstrate the allo cation model. Lastly, Section 3.4 summarizes this chapter. 3.1 Equilibrium of Coupled Transportation and Power Networks 3.1.1 Description of Transportatio n Network We consider a regional road network where origins and destinations are cities or metropolitan areas. A large portion of vehicles in the network are assumed to be P EVs, and other vehicles are reflected as the background traffic. We attempt to mo del the PAGE 36 36 spatial dis tribution of travel demand of P EVs and their corresponding route choices in the regional network. Previous studies, e.g., Tuttle and Kockelman (2011) and Lin and Greene (2011), have suggested that drivers of PEVs actively seek out charg ing opportunity to avoid gasoline use and thus favor a destination that provides public charging facilities. This explains why hotels and shopping malls have been installing charging stations as their amenities (e.g., Fox News, 2012). Moreover, consideri ng the way how fuel prices affect travel be haviors (e.g., Walsh et al. 2003 ; Weis et al., 2010), it seems reasonable to assume that the charging price will have a similar effect. Kitamura and Sperling (1987) analyzed the refueling behaviors of drivers of gas fueled vehicles and found that price of gasoline is a primary concern of a majority of drivers for selecting fuel stations. Similarly, it is plausible that the charging expense will be one impor tant criterion for drivers of P EVs to select charging sta tions, particularly if the prices of electricity vary substantially among stations. The above behaviors and preferences will likely manifest themselves in shaping the spatial dis tribution of travel demand of P EVs in the region. To reflect their impact, w e assume in this chapter that am ong other factors, drivers of P EVs consider travel time, availability of charging opportunities, and charging expense in selecting destinations. We caution that this is a critical assumption that needs to be verified by fut ure empirical studies. It is further assumed that the above destination choice behavior c an be captured by a multinomial logit model, and drivers select routes in a user optimum manner. Note that multinomial logit models have been applied to predict trav and regional systems (e.g., Southworth, 1981; Kemperman et al., 2002), and the PAGE 37 37 assumption of user optimum route choice has been widely adopted in transportation network modeling (e.g., Sheffi, 1984). Let denote the network of roads, where and are the sets of nodes and links in the network respectively. We denote a link as or the pair of its starting and ending nodes, i.e., . Assume that the travel demands of interest are originated from a set of origin nodes , and denote the demand at each origin as . Next, let denote a set of destinations, where sit public charging stations for P EVs. Note that the sets and are not mutually exclusive. W e use an integer variable to represent the number of charging stations at destination . Let be the node link incidence matrix associated with the network and is a vector with a length of . The vector consists of two non zero components: one has a value of 1 in the component corresponding to origin and the other has a value of 1 in the component corresponding to destination . The origin destination (O D) flows, i.e., ation choices, and thus are decision variables in the network flow model. Let be the link flow at link . Since only a portion of travel demand is of consideration, there exists background traffic in the network. Without losing generality, we her einafter assume the background traffic to be zero, and the travel time for link is a strict ly increasing function of the P EV flow on the link, i.e., . For example, the following form of Bureau of Public Roads (BPR) function can b e used: PAGE 38 38 where is the free flow travel time for link and is the capacity of link . Note that it is straightforward to extend the modeling framework in this chapter to consider multiple user classes if modeling travel choices of the background traffic is of interest. We now model the travel demand distribution of P EVs, given the availability of charging stations and price of electricity at each destination. A combined distribution and ass ignment (CDA) model is proposed to describe P and routes. In the CDA model, the network traffic condition is assumed to be in user equilibrium and the destination choice is represented by a multinomial logit model as follows: where , and are positive coefficients associated with the O D equilibrium trave l time, i.e., , the density of charging stations at the destination, which equals the number of charging stations, i.e., , divided by the area of the destination, i.e., , and the charging expense, which is equal to the unit pric e, i.e., multiplied by the average ho urly energy requirement for a P EV, i.e., . The latter is assumed to be a constant for each OD pair (e.g., Wang et al., 2011). is a location specific constant. For presentation simplicity, we he reinafter substitute for . Note that if , represents the intra zone travel demand. The above logit model implies that the deterministic portion of the utility fun ction of P EV drivers consists of four components: travel t ime, the availability of charging opportunities, charging expense and a constant that encapsulates other factors affecting the attractiveness of a particular destination. With such a consideration of destination choices, the CDA model can be written as fo llows: PAGE 39 39 s.t. ( 3 1) ( 3 2) ( 3 3) ( 3 4) ( 3 5) where is the flow between the O D pair at link . Equation 3 1 calculates the aggregate lin k fl ow. Equation 3 2 ensures the flow balance between origin and destination nodes in the network. Equation 3 3 requires the sum of the demand leaving from any origin to each destination to be equal to the total demand generated at that particular orig in . Equations 3 4 and 3 5 are the non negativity constraints for the link flows and demands. Equations 3 1 3 5 essentially define a set of feasible O D and link flow distributions for the network. Lastly, we emphasize that the coefficients in the abo ve multinomial logit model need to be calibrated using empirical stated or revealed preference data, which may prove some of the parameters to be insignificant. Between the availability of charging opportunities and the price of electricity, the former mo re lik ely affects travel choices of P EV drivers, as suggested in previous studies (e.g., Lin and Greene, 2011). In contrast, the impact of the price of electricity may be less significant because the price may not vary substantially across locations or ev en if it does, it may not yield a meaningful difference in the total charging expense per charge. If neither the availability of public charging opportunities nor the price of electricity has been proved to be strong enough PAGE 40 40 in shaping t he O D demand distr ibution of P EVs, the coefficients, and , will become zero, and the CDA model will consequently reduce to a traditional combined distribution and assignment mode l (e.g., Sheffi, 1984). If another form of the utility function provides a better fit with the data, a corresponding CDA mode l can be developed in a similar fashion. 3.1.2 Description of Power Network We consider a competitive wholesale power market, consistent with the one proposed by the U.S. Federal Energy Regulatory Commission (FERC, 2003). In the market, an independent sys tem operator (ISO) undertakes the daily operations of the transmission grid using locational marginal pricing. More specifically, the ISO accepts supply and demand bids submitted by market participants, i.e., buyers and generators, and is responsible for determining the power commitments (supplies) to meet the demands, with an objective of maximizing social welfare while ensuring the system security. In this case, the price of electricity at each location equals the marginal cost of providing electricity at that location, i.e., the locational marginal price (LMP). LMP reflects the market clearing price at each location. Based on their LMPs, the buyers pay the ISO for the dispatched power (e.g., Sun and Tesfatsion, 2010). We assume in this chapter that t he retail el ectricity prices faced by the P EV drivers at different public charging stations in the same destination are the same and equal to the LMP at the specific location. We further assume that there are a variety of active participants in the wholes ale power market, leading LMP to vary substantially by location (e.g., Lewis, 2010 and Feldman, 201 2 ). The spatial variability of LMP will influence travel patterns of P EVs, which in return has significant impacts on LMPs and the operations of the power n etwork. PAGE 41 41 The clustered P EV loads at public charging stations will have attendant effects on LMPs of electricity, which can be estimated by solving an optimal power flow problem. All current market designs utilize the DC power flow models for LMP calculat ion, which are linear approximations to the AC models (e.g., Litvinov, et al., 2004). Although the approximations lead to some loss of accuracy, the results match fairly closely with the full AC solutions (Overbye et al., 2004). This chapter adopts the st andard DC power flow model to compute the LMP at each bus, including the destinations with public charging stations. Without losing generality, we assume that there are only one generator and one buyer at each bus. Public charging stations for PEVs will create additional electricity load to the bus at their particular locations. Let denote the power network, where and are the sets of buses and transmission lines in the network respectively. For the generator at bus , is the real power injection, and and represent the lower and upper r eal power limit respectively. We further denote a transmission line or branch as the pair of its starting and ending buses, i.e., . For each branch, let represent the real power flowing in the branch and represent the therma l limit of real power flow. The standard DC power flow model is written as follows: s.t. ( 3 6) ( 3 7) PAGE 42 42 ( 3 8) ( 3 9) ( 3 10) ( 3 11) ( 3 12) where is the total cost of generating amount o f electricity at bus , which is assumed to be a strictly convex function with . is the regular real power load at bus , which is given. If the regular real power load is not given and needs to be treated as a decision variable, its corr esponding (inverse) demand function should be specified. Consequently, the objective function of the DC power flow model will be changed to maximizing social benefit. is t he additional load created by P EVs, defined as follows: , where denotes the set of the pairs of travel destinations and their serving buses. is the inverse of the pu reactance and is the multiplication of the base apparent power and voltage angel (in radians). In the above, Equation 3 6 is the nodal power balance constraint, whose associated Lagrangian multiplier is the LMP at the node. Equation 3 7 is the linear real power branch flow equation. Equations 3 8 3 11 ensure the feasibility of the real power flo w and power injection. Lastly, Equation 3 12 sets the voltage angle at the reference node 1 to be zero . 3 .1.3 Equilibrium of Coupled Transportation and Power Networks The interactions b etween destination choices of P EVs and price of electricity will lead to an equilibrium state in the coupled networks. The state implies a market equilibrium where the supply of electricity matches its demand so that the price of PAGE 43 43 electricity and the power flow can consequently be determined. The state als o encapsulates a user equilibrium in the tr ansportation network where no P EV is able to improve its perceived utility by unilaterally changing its route or destination. Combining the KKT conditions of the CDA and DC power flow models yields a mathematical definition of th e equilibrium state in the coupled networks. The definition is a nonlinear complementarity system as follows: ( 3 1) ( 3 12) ( 3 13) ( 3 14) ( 3 15) ( 3 16) ( 3 17) ( 3 18) ( 3 19) ( 3 20) ( 3 21) ( 3 22) ( 3 23) ( 3 24) ( 3 25) PAGE 44 44 where is the Lagrangian multiplier associated with Equation 3 2 , with Equation 3 3 , with Equation 3 6 , with Equation 3 7 , with Equation 3 8 , with Equation 3 9 , with Equation 3 10 , w ith Equation 3 11 and with Equation 3 12 . Moreover, is the marginal generation cost at bus . Below we present an example with a three node road network and a three bus power network to illustrate the equilibrium concept. As show n in Figure 3 1 , the road network consists o f nodes 1, 2 and 3 where 5000 P EVs generated at node 1 are destined to the other two nodes. There are three and five public charging stations at nodes 2 and 3 respectively, whose serving buses compose the power grid with another bus, i.e., 4. There is no electricity load at bus 4, and the regular load at buses 3 and 5 are both 100 MW. It is assumed that the total generation cost functions at buses 4 and 5 are linear and the unit costs are $10 and $15 per MWh re spectively. We use the BPR function to calculate the travel time of links 1 2 and 1 3, whose free flow travel times and capacities are both 60 minutes and 4000 veh/hr respectively. Other parameters are assumed as follows: ; ; ; ; and . We solved Equations 1 25 for the illustrative network to obtain the equilibrium solution. At equilibrium, 3377 P EVs leave from n ode 1 to 3, generating additional electricity load of 27MW and facing a LMP o f $10 per MWh. The other 1623 P EVs leaving for node 2 create electricity load of 13MW at a price of $15 per MWh. The power injections at buses 4 and 5 are 227 MW and 13 MW respect ively. Instead of solving the nonlinear complementarity system Equations 1 25 , this section presents a convex mathematical program whose solution is the equilibrium O D PAGE 45 45 demands and link flows (or simply traffic flow distribution) at the transportation netw ork, and the power injection and branch flow (or simply power flow distribution) at the power network. We hereinafter refer to it as the network equilibrium problem, i.e., NEP, and the formulation is as follows: s.t. ( 3 1) ( 3 12) The equivalence of NEP to the equilibrium definition ( 3 1) ( 3 25) can be easily establis hed by examining the KKT condition of NEP as follows: ( 3 1) ( 3 12) PAGE 46 46 where is the Lagrangian multiplier associated with Equation 3 2 in NEP, with Equation 3 3 , with Equation 3 6 , with Equation 3 7 , with Equation 3 8 , with Equation 3 9 , with Equation 3 10 , with Equation 3 11 and with Equation 3 12 . Comparing the above KKT conditions to the definition of equilibrium, i.e., Equations 3 1 3 25 , one can find that these two systems are the same except that and are times of and in Equations 3 1 3 25 . More specifically, the solutions to NEP, i.e., the traffic and power flow distributions, satisfy the equilibrium definitions Equations 3 1 3 25 and the equilibrium prices of electricity are the Lagrangian multiplier s associated with Equation 3 6 in NEP. Theo rem 3 1. Given the availability of public charging stations, the equilibrium state always exists. The equilibrium traffic link flows, O D demand distributions and real power injections are unique. Proof. As the objective function of NEP is continuous and its feasible region is there must exist a solution to NEP. In other words, equilibrium always exists in the coupled networks. Define the convex feasible region of NEP as , and thus its solution will satisfy the following first order optimality condition: PAGE 47 47 ( 3 26) Suppose that there are two optimal solutions, and denote them as and . Substituting them into Equation 3 26 yields the following: ( 3 27) ( 3 28) Summing Equation s 3 27 and 3 28 leads to: ( 3 29) Because the BPR, logarithm and the marginal generation cost functions are all strictly increasing, Equation 3 29 implies that , and . In other words, must be unique. PAGE 48 48 3 .2 Allocating Public Charging Stations 3 .2.1 Model Formulation With the proposed framework of equilibrium analysis of the coupled networks, we now investigate how to allocate a given number of public charging stations to a set of potential locations, a strategic decision in the deployment planning of public charging infrastructure. It is assumed that the government agency attempts to maximize social welfare associated with both the transportation and power networks. We propose a macroscopic planning model that determines the optimal number of charging stations allocated to each metropolitan area. Being strategic, the model does not optimize exact locations and capacities of the allocated charging stations, which are expected to b e decided by another tactic planning model proposed in Chapter 4 . Defining a nonnegative integer variable to represent the number of additional charging stations allocated to node , we formulate the public charging station allocation problem (PC SA) as follows: s.t. ( 3 1) ( 3 14) and ( 3 16) ( 3 25) ( 3 30) ( 3 31) ( 3 32) PAGE 49 49 where is the given total number of public charging stations, and denotes the set of potential locations. In the above, the objective is to maximize social welfare of the coupled networks for an average hour. I t consists of four components. The first one represents the total expected utility of P EV drivers where is the equilibrium travel time between the O D pair r s. Because the utility encapsulates the charging expense paid by P EV d rivers, which, however, should be viewed as a transfer, the second components, i.e., the charging expense, is thus added. The last two components represent the total generation cost of electricity and the total construction cost of charging stations. is the construction cost of a charging station at destination . In the constraints, Equations 3 1 3 14, 3 16 3 25 and 3 30 ensure that the traffic and power flow distributions and prices of electricity are in equilibrium. The others require the to tal number of located charging stations to be less than and the charging stations can only be allocated to the potential locations. Note that by solving PCSA, not only can we obtain the allocation plan but also the equilibrium traffic distribution, whic h can be potentially used as inputs to the tactic planning model to determine the locations and capacities of allocated charging stations. 3.2.2 Solution Algorithm PCSA is a mathematical program with complementarity constraints (MPCC), a class of problem s difficult to solve (see, e.g., Luo et al., 1996). The problem is non convex and standard stationarity conditions such as the KKT conditions may not hold for it (Scheel and Scholtes, 2000). Compounding the difficulty is that PCSA also contains integer d ecision variables, which, however, can be replaced by another set of PAGE 50 50 complementarity constraints (e.g., Zhang et al., 2009). Thus, our reformulation of PCSA is a regular MPCC with no integer variables. More specifically, let denote the smallest integ er number such that , and thus the number of charging stations allocated to node can be represented as where is a binary variable for . However, our reformulation of PCSA treats as a continuous decision variable and introduces instead another complementarity constraint of the form: . When combined with , the previous equation forces to be binary. Therefore, our reformulation of PCSA is a regular MPCC, which can be solved by many existing algorithms (see, e.g., Luo et al., 1996, and references cited therein). However, some of these algorithms only work well for small and medium problems while others, especially those based on solving equivalent nonlinear programs (e.g., Fletcher and Leyffer, 2004 and Lawphongpanich and Yin, 2010), can handle larger problems. If the number of potential allocation plans is limited, PCSA can be solved by simply solving NEP associated with each pl an and then comparing their resulting social welfare. However, if the number of plans is prohibitively large, an effective solution algorithm is needed. Note that PCSA has a similar mathematical structure as a discrete equilibrium network design problem, which can be solved using an active set algorithm (Zhang et al., 2009; Lou et al., 2009). When applied to the allocation model, the algorithm will iteratively solve NEP associated with a given allocation plan, and then construct a binary knapsack problem to update the plan. To tailor the active set algorithm to PCSA, we define three new sets, , and with PAGE 51 51 ; . Based on the sets, a restricted version of PCSA, i.e., RPCSA, can be formulated as follows: s.t. ( 3 1) ( 3 14) and ( 3 16) ( 3 25) ( 3 33) ( 3 34) ( 3 35) ( 3 36) ( 3 37) Note that RPCSA is another m athematical program with complementarity constraints, but its optimal solution can be easily obtained by solving NEP with an allocation plan compatible with . The procedure of the active set algorithm is described below: Step 0: Set , , and . Solve NEP with a plan vector compatible with the pair . Step 1: Using the results from NEP, construct a solution to RPCSA with ( . Then, determine and , the smallest and largest values of Lagrangian multipliers PAGE 52 52 associated with Equations 3 36 and 3 37 . Set as the social welfare associated with the current plan, and go to Step 2. Step 2: Set and adjust the active sets by performing the following steps: a) Let solve the following knapsack problem for plan adjustment: s.t. If the optimal o bjective value of the knapsac k problem is zero, stop and the current solution is optimal. Otherwise, go to Step 2b. b) Set c) Solve NEP with a plan vector compatible with and calculate its social welfare denoted as . If , go to Step 2d because the pair ( leads to an increase in social welfare. Otherwise, set , where is sufficiently small, and return to Step 2a. d) Set , , and . Go to Step 1. For simplicity, the initial active pair in Step 0 corresponds to allocating no charging station. The pair corresponding to another feasible plan would certainly work. In Step 1, finding the largest and sma llest Lagrangian multipliers is time consuming. In PAGE 53 53 our numerical example reported in the next section, we used commercial nonlinear solver such as CONOPT (Drud, 1994) to directly solve RPCSA with the solution constructed from solving NEP being the initial solution. The multipliers generated by CONOPT worked very well. The objective of the plan adjustment problem at Step 2a is to maximize the total estimated increase to social welfare by adjusting the current allocation plan, where means shifting from to and analogously, means the opposite. If the optimal objective value is zero, then no adjustment leads to a higher social welfare, a criterion used to terminate the algorithm above. O therwise, an updated plan compatible with is obtained in Step 2b. Because the Lagrangian multipliers are only estimates of the changes, may not lead to an actual increase in social welfare. Step 2c is to verify it by solving N EP with . If does not lead to an adjustment problem in Step 2a with . Zhang et al. (2009) proved that the above active set algorithm terminates after a finite number of iterations. The solution is strongly stationary solution if some assumptions are satisfied. 3.3 Numerical Example The allocation problem is solved for a coupled network that we created based on the topology of the Sioux Falls road network and a subset of the IEEE 118 bus system ( http://motor.ece.iit.edu/Data ), as shown in Figure 3 2. The transportation netw ork consists of 76 directed links and 24 nodes, 12 of which are origins and destinations. Note that although the network has the same topology as the Sioux Falls network, it has PAGE 54 54 been scaled up to represent a regional network whose free flow travel time an d capacity of each link are reported in Table 3 1. Table 3 2 presents the trip production at each origin. The power network is composed of 16 transmission lines (undirected links) and 12 buses (nodes), each of which is connected to a destination node. F or the DC power flow model, the base apparent power is 100 MVA. Table 3 3 lists the upper limit on magnitude of real power flow in each branch and its B value, i.e., the inverse of the pu reactance. The total generation cost function at each bus is assum ed as follows: where , , and are coefficients, which are given in Table 3 4 as well as the lower and upper limits on real power production of each generator. Other parameters include , , and in the logit based destination choice model and , the average hourly ener gy requirement for charging a P EV, equals kWh. Moreover, in allocati ng the charging stations, we did not consider their construction costs. In this numerical example, we assumed that there are three existing charging stations at each city, and then allocated additional 20 charging stations to five potential nodes in the c oupled network, i.e., nodes 1, 2, 4, 5 and 10. We assumed that at most seven charging stations can be allocated to each node, and applied the active set algorithm to solve the allocation model. With allocating no charging stations being the initial solut ion, we obtained a solution of allocating seven stations at node 4, six at node 5 and seven at node10, which yields a social welfare of 317,650. The corresponding traffic and power link flows are reported in Table s 3 5 and 3 6 respectively. Table 3 7 lis ts the LMP at each bus together with its real power injection and the power load PAGE 55 55 generated by P EVs. It can be observed from the table t hat the power load created by P EV is 16% of the total power injection in the power network. In order to verify the effect iveness of the active set algorithm, we enumerated all possible allocation plans, i.e., 2 , 226, and then solved the corresponding NEP for each of them and found the optimal allocation plan with the largest social welfare. Indeed, the best solution we obtain ed from using the active set algorithm is the global optimal allocation plan. In addition, we observed in our numerical experiments that the active set algorithm was very efficient and consistently produced good results with different initial solutions. T he observation is consistent with Zhang et al. (2009), demonstrating the potential of applying the active set algorithm to large scale networks where the number of allocation plans is prohibitively large and the enumeration method fails. However, we did n ot prepare such a large scale example in this chapter because the focus here is not on the computational aspect. Figure 3 3 plots the cumulative distribution curve of the social welfare of those 2 , 226 possible allocation plans. It can be observed that th e social welfare varies approximately from 210,000 and 320,000, and many plans fall into the range between 230,000 and 290,000. Such a dispersed distribution highlights the importance of making a wise allocation decision and the need for an optimal alloca tion model. In Table 3 8, we further compare the LMPs corresponding to the best allocation plan obtained from using the active set algorithm with those corresponding to two randomly selected plans, and the situation without newly allocated charging statio ns. The first randomly selected plan allocates four charging stations at node 1, seven at node 2, four at node 4 and five at node 5, and the other allocates one at node 1, five at node 2, four PAGE 56 56 at node 4, five at node 5 and five at node 10. It can be obse rved that the P EV loads have substantial impacts on the prices of electricity and the impacts are very sensitive to the allocation decisions. Finally, it is easy to observe that the best obtained plan allocates public charging stations in three cities whe reas two randomly selected plans allocate stations in four or five cities. One may favor the latters because they serve more cities. If it is the intent to find a plan to ensure certain coverage for equity or social justice, additional constraints can be introduced to the formulation to guarantee that, e.g., the number of charging stations at each city should be more than a certain threshold or the number of cities served should be larger than a threshold. 3.4 Summary We have adopted a game theoretical approach to investigate the interactions among availability of public charging opportunism, destination and route choices of P EVs and price of electricity in coupled transportation and power networks. The interactions lead to an equilibrium where equilib rium prices of electricity, and traffic and power flow distributions can be determined. A convex mathematical program is formulated to describe the equilibrium state. Built upon the proposed equilibrium analysis framework, the problem of optimally alloca ting public charging station budget is formulated as a mathematical program with complementarity constraints, and is solved by an active set algorithm. It is observed from the numerical exampl e that the charging load from P EVs has substantial impact on th e operations of the power network and the price of electricity. Consequently, it is important to consider this impact when allocating public charging stations. The proposed location model will be of help to government agencies for deployment planning of public charging infrastructure. The active set algorithm has been demonstrated to be effective and efficient. PAGE 57 57 Table 3 1 . Link capacity (100 veh/hr) and free flow travel time (hr) Link Capacity Link Capacity Link Capacity 1 2 1.8 6.02 10 11 1.5 20 17 16 0.6 10.46 1 3 1.2 9.01 10 15 1.8 27.02 17 19 0.6 9.65 2 1 1.8 12.02 10 16 1.5 10.27 18 7 0.6 46.81 2 6 1.5 15.92 10 17 2.1 9.99 18 16 0.9 39.36 3 1 1.2 46.81 11 4 1.8 9.82 18 20 1.2 8.11 3 4 1.2 34.22 11 10 1.5 20 19 15 1.2 4.42 3 12 1.2 46.81 11 12 1.8 9.82 19 17 0.6 9.65 4 3 1.2 25.82 11 14 1.2 9.75 19 20 1.2 10.01 4 5 0.6 28.25 12 3 1.2 46.81 20 18 1.2 8.11 4 11 1.8 9.04 12 11 1.8 9.82 20 19 1.2 6.05 5 4 0.6 46.85 12 13 0.9 51.80 20 21 1.8 10.12 5 6 1.2 13.86 13 12 0.9 51.80 20 22 1.5 10.15 5 9 1.5 10.52 13 24 1.2 10.18 21 20 1.8 10.12 6 2 1.5 9.92 14 11 1.2 9.75 21 22 0.6 10.46 6 5 1.2 9.90 14 15 1.5 10.26 21 24 0.9 9.77 6 8 0.6 21.62 14 23 1.2 9.85 22 15 1.2 20.63 7 8 0.9 15.68 15 1 0 1.8 27.02 22 20 1.5 10.15 7 18 0.6 46.81 15 14 1.5 10.26 22 21 0.6 10.46 8 6 0.6 9.80 15 19 1.2 9.64 22 23 1.2 10 8 7 0.9 15.68 15 22 1.2 20.63 23 14 1.2 9.85 8 9 1 10.10 16 8 1.5 10.09 23 22 1.2 10 8 16 1.5 10.09 16 10 1.5 10.27 23 24 0.6 10.1 6 9 5 1.5 20 16 17 0.6 10.46 24 13 1.2 11.38 9 8 1 10.10 16 18 0.9 39.36 24 21 0.9 9.77 9 10 0.9 27.83 17 10 2.1 9.99 24 23 0.6 10.16 10 9 0.9 27.83 Table 3 2 . Trip production at each origin (100 veh/hr) Origin 1 2 4 5 10 11 13 Production 12.92 12.71 12.18 12.51 13.51 13.41 11.46 Origin 14 15 19 20 21 Production 13.20 14.11 13.31 11.70 11.93 PAGE 58 58 Table 3 3 . Input data for transmission lines Line Capacity (MW) Line Capacity (M W) 1 2 175 66.23 4 13 500 25.91 2 4 175 3.98 13 14 500 50 1 4 175 4.63 10 14 500 37.31 4 5 175 6.90 11 15 175 4.59 4 10 175 6.67 11 19 175 8.55 5 10 500 74.07 15 19 175 9.85 5 11 175 17.83 19 21 175 3.60 10 11 175 26.60 20 21 500 27.03 Table 3 4 . Input data for generators at Node Load (MW) Lower limit (MW) Upper limit (MW) (10 2 ) 1 63 25 100 1.28 17.82 10.15 2 84 25 100 1.28 17.82 10.15 4 277 50 200 1.39 13.29 39 5 78 0 0 0 0 0 10 0 50 200 1. 39 13.29 39.00 11 77 25 100 1.28 17.82 10.15 13 0 0 0 0 0 0 14 0 0 0 0 0 0 15 39 100 420 1.36 8.34 64.16 19 28 0 0 0 0 0 20 0 0 0 0 0 0 21 0 80 300 1.09 12.89 6.78 PAGE 59 59 Table 3 5 . Traffic link flow (veh/hr) Link Flow L ink Flow Link Flow 1 2 180.67 10 11 178.48 17 16 292.76 1 3 1119.94 10 15 113.74 17 19 48.41 2 1 153.64 10 16 27.79 18 7 168.10 2 6 1158.08 10 17 25.81 18 16 421.39 3 1 69.69 11 4 969.50 18 20 26.94 3 4 1866.17 11 10 1375.07 19 15 335.17 3 12 46.42 11 12 26.99 19 17 830.17 4 3 33.11 11.14 136.92 19 20 282.69 4 5 1911.90 12 3 829.23 20 18 589.49 4 11 57.18 12 11 262.19 20 19 376.36 5 4 2071.02 12 13 56.98 20 21 225.37 5 6 20.16 13 12 1074.98 20 22 20.21 5 9 313.81 13 24 185.32 21 20 195.73 6 2 43.79 14 11 1095.64 21 22 705.63 6 5 1116.93 14 15 202.70 21 24 361.92 6 8 313.40 14 23 91.71 22 15 705.63 7 8 168.10 15 10 1550.92 22 20 13.97 7 18 11.44 15 14 156.49 22 21 85.13 8 6 295.88 15 19 250.04 22 23 20.21 8 7 11.44 15 22 85.13 23 14 149.81 8 9 291.66 16 8 116.36 23 22 13.97 8 16 10.30 16 10 597.79 23 24 77.74 9 5 1111.21 16 17 22.60 24 13 305.16 9 8 11.42 16 18 15.49 24 21 190.21 9 10 605.47 17 10 537.41 24 23 129.60 10 9 1122.63 PAGE 60 60 Table 3 6 . Powe r line flow (MW) Line Flow Line Flow 1 2 13.44 4 13 123.12 2 4 47.08 13 14 125.17 1 4 53.21 10 14 128.25 4 5 68.37 11 15 140.81 4 10 71.66 11 19 175 5 10 58.27 15 19 98.23 5 11 112.74 19 21 109.31 10 11 147.56 20 21 3.94 Table 3 7 . P EV load, power injection and LMP at each bus at Node P (MW) Power Injection (MW) LMP ($/MWh) 1 1.77 25 17.4 2 1.52 25 17.4 4 34.01 148.14 17.4 5 24.63 0 17.4 10 37.53 148.14 17.4 11 3.51 25 17.4 13 2.04 0 17.4 14 3.08 0 17.4 15 5.99 284.03 16.06 19 4.60 0 15.43 20 3.94 0 15.43 21 3.55 116.87 15.43 Table 3 8 . LMPs associated with different allocation plans ($/MWh) at node Without P EVs Best plan obtained Randomly selected plan 1 Randomly selected plan 2 1 15.96 17.40 17.20 17.30 2 15.96 17.40 17.20 17.30 4 15.96 17.40 17.20 17.30 5 15.96 17.40 17.20 17.30 10 15.96 17.40 17.20 17.30 11 15.96 17.40 17.20 17.30 13 15.96 17.40 17.20 17.30 14 15.96 17.40 17.20 17.30 15 15 .55 16.06 16.15 16.11 19 15.36 15.43 15.66 15.55 20 15.36 15.43 15.66 15.55 21 15.36 15.43 15.66 15.55 Social welfare 317,650 232,436 253,317 PAGE 61 61 Figure 3 1 . An illustrative example for equilibrium in coupled networks PAGE 62 62 Figure 3 2 . The coupled transportation and power networks PAGE 63 63 Figure 3 3 . Cumulative distribution curve of social welfare 0% 20% 40% 60% 80% 100% 21 22 23 24 25 26 27 28 29 30 31 32 Cumulative Probability Social Welfare x 10000 PAGE 64 64 CHAPTER 4 URBAN LEVEL LO CATION OF PUBLIC CHARGING STATION S In this chapter, given the allocated budget limit for a city or urban area, we consider how to determine the number, locations and types of charging stations to build within the city. Specifically, assuming the locations and types of charging station is given, we first propose network equilibrium models incorporating the impact of limit range and charging requirement behavior s . Based on the proposed equilibrium models, w e then optimally determine station deployment plans. Note that t he notations used in this chapter are different from those in Chapter 3, and are redefined wherever appropriate. The literature of alternative fuel vehicles has considered the potential need o f recharging those vehicles to reach their destinations. For example, Wang and Lin (2009) formulated the refueling logic of vehicles to be a system of linear equations, assuming travelers between an origin destination (O D) pair choose the shortest path, which is given and fixed. For electric vehicles, activity based approaches have been proposed to investigate individual vehicle routing and scheduling problem with recharging in the literature (e.g., Kang and Recker, 2009, 2012; Schneider et al., 2012). Adler et al. (2013) defined a BEV shortest walk problem that finds the route between an O D pair with minimum detouring, and proved the problem to be polynomially solvable. Note that the obtained shortest walk may include cycles for detouring to recharge batteries. In contrast to these previous studies, this chapter proposes network equilibrium model s that predict how the limited driving range and recharging flow dist ribution on regional or large metropolitan road networks where charging stations PAGE 65 65 are few and far between. Among others, the most relevant studies in the literature include Jiang et al. (2012), and Jiang and Xie (2013). In the former, a network equilibriu m model is formulated upon paths whose lengths are within the driving ranges of BEVs. The so called path constrained traffic assignment model can be solved efficiently by a solution algorithm proposed in the latter. Both studies do not consider rechargin g behaviors of BEV drivers and assume the energy consumption of BEVs is independent of traffic congestion. For the remainder of this chapter , assuming that the energy consumption of a BEV is not affected by traffic congestion, i.e., the consumption is flo w independent, Section 4.1 first formulate s a network equilibrium model that abides by the driving range of BEVs and accommodates their recharging decisions. An iterative solution procedure is proposed to solve the model efficiently. Section 4.2 further extend s the model to consider the time required for recharging, which can be substantial, depending on the amount of recharged energy and the type of charging stations . Moreover , considering the potential impact of traffic congestion on the fuel economy o f BEVs, Section 4.3 investigate s a novel network equilibrium model with flow dependent energy consumption. Section 4.4 formulates the problem of optimizing the locations and types of charging stations within a given budget limit as a bi level model, which is solved by a genetic based algorithm. Numerical examples are provided in Section 4.5. Lastly, Section 4.6 concludes the chapter. 4.1 Base Model 4.1.1 Notation We consider a regional or metropolitan road network. Let denote the network where and are the sets of nodes and links in the network respectively. We PAGE 66 66 denote a link as or its starting and ending nodes i.e., . Travel demands are between a set of O D pairs, i.e., . Let and be the travel demand and the set of paths between O D pair respectively. In addition, represents the traffic flow on path of O D pair . We further denote as the origin node of O D pair , and is the path link incidence, which equals 1 if path traverses link and 0 otherwise. Let and be the traffic flow and distance of link . The travel time of link is a strictly increasing function of the flow on the link, i.e., . For exampl e, the following Bureau of Public Roads (BPR) function can be used: where is the free flow travel time of link , and represents the capacity of link . 4.1.2 Definition and Formulation of Network Equilibrium It is assumed in this chapter that all vehicles in the network are BEVs. This assumption is not necessarily restrictive as the models proposed below can be easily extended to accommodate both electric and regular vehicles. It is fu rther assumed that a limited number of charging stations are located at certain nodes of the network, and thus vehicles travelling along a path may not pass by a charging station. We thus have the following definition: Definition 4 1 . A path is usable if a BEV is able to complete the path without or with recharging. The distance of a usable path must be within the driving range of a BEV, if none charging station exists along the path. Otherwise, it is usable as long as the vehicle can recharge its batte ry at charging stations along the path to avoid running out of charge PAGE 67 67 b efore reaching its destination. Figure 4 1 shows a simple example to further illustrate the definition. The O D pair 1 2 is connected by three paths, i.e., 1 2, 1 3 2 and 1 4 2. A char ging station is located at both nodes 3 and 4. It is assumed that the battery size of the BEV is 24 KWh; its initial state of charge is 4 KWh and the energy consumption rate is 0.3 KWh/mi. It is easy to see that path 1 2 is not a usable path. Along the other two, the BEV can reach node 3 or 4 without running out of charge. Because the vehicle can recharge at node 3 or 4 to achieve a driving range up to 80 miles if fully charged, it can reach the destination successfully. Therefore, both path 1 3 2 and path 1 4 2 are usable. When traveling between their origins and destinations, it is reasonable to assume that BEV drivers select routes to minimize their travel costs, which may include electricity cost and travel time cost. Note that the former is much smaller than the latter. For example, considering an electricity price of $0.12 per kWh, a value of travel time of $20 per hour and a travel speed of 50 miles per hour, the electricity cost for a Nissan Leaf 2013 is 8.7% of its travel time cost (US Depart ment of Energy, 2013). Therefore, we hereinafter simply adopt time minimization as the decision criterion for route choices. More specifically, we assume that travelers choose the paths with the least travel time among all the usable paths. In our base model, we consider homogenous vehicles with the same battery size and initial state of charge, another assumption that can be easily relaxed. Moreover, it is assumed in Sections 4.1 and 4.2 that traffic congestion does not affect the fuel economy of BEVs. The energy consumption of a BEV thus depends PAGE 68 68 on travel distance rather than travel time. The above assumptions and considerations yield the following network equilibrium: Definition 4 2 . At equilibrium, all the utilized paths are usable and the travel t imes of all the utilized paths of one O D pair are the same, which are less than or equal to that of any unutilized usable paths of the same O D pair. We are now ready to construct our base model to describe the above network equilibrium. Because the ene rgy consumption of a BEV is flow in dependent, the usability of a path is also independent of traffic flow and can be pre determined. Denote the set of all usable paths between O D pair as and we have the following formulation for network equi librium (NE) with BEVs: s.t. (4 1) (4 2) where is the path flow vector. In the above, the constraints ensure flow balance between each O D pair and the nonnegativity of path flows respectively. As compared to the classical formulation of Beckmann et al. (1956), NE requires the path flow of each unusable path to be zero. It i s straightforward to verify the equivalency of NE to Definition 4 2 and that the equilibrium link flow is unique with the strictly increasing travel time function previously assumed. Note that the p ath constrained traffic assignment problem in Jiang et al . (2012) can be also formulated as NE, if includes the paths no longer than the driving range of BEVs. PAGE 69 69 4.1.3 Solution Procedure As formulated, NE is a convex program with linear constraints. If all the usable paths are enumerated beforehand, i t can be solved easily by commercial nonlinear solvers such as CONOPT (Drud, 1994). Considering path enumeration is time consuming, this section presents an iterative solution procedure. The procedure starts with a subset of , , and solves a restricted version of NE defined upon the subset. Another sub problem is then solved to determine whether the solution to the restricted NE solves the original formulation. If not, a new usable path will be generated and added to the subset and the iter ation proceeds until termination. A few new variables are introduced for formulating the sub problem. For each node , we use to represent the upper limit of electricity that the node can provide for recharging. The variable equals 0 if there is no charging station at node ; otherwise, it is a sufficiently large constant because the number of chargers at a charging station is assumed to be sufficient. Let and be the battery size and initial state of charge. For a BEV travelling between O D pair , the recharging amount of electricity at node is and the state of charg e at node after recharging is . Let be the node link incidence matrix associated with the network and is a vector with a length of | |. The vector consists of two nonzero components: one has a value of 1 in the component correspo nding to the origin of and the other has a value of 1 in the component corresponding to the destination of . Given the current link flow solution to the restricted NE, the sub problem, shortest usable path finding or SP, can be formu lated as follows for each O D pair : PAGE 70 70 s.t. (4 3) (4 4) (4 5) (4 6) (4 7) (4 8) (4 9) (4 10) where and are sufficiently large constants; is the energy consumption rate of BEVs; is a binary variable, which equals 1 if link is utilized and 0 otherwise; is a variable that equals 0 if link is utilized and is unrestricted otherwise. In the above, the objective function is to minimize the total travel time. Equation 4 3 e nsures flow balance. Equations 4 4 and 4 6 specify the relation between the states of charge of BEV batteries at the starting and ending nodes of any utilized link. Equation 4 5 ensures that BEVs do not run out of charge on any utilized li nk. Equation 4 7 suggests that BEVs can only recharge at nodes with a c harging station while Equation 4 8 sets the upper and lower bounds of the states of char ge of BEV batteries. Equation 4 9 specifies the ini tial state of charge. Finally, Equation 4 10 requires to be binary. SP is a mixed integer linear program, and can be easily solved by commercial solvers such as CPLEX 12.2 for small or medium sized problems. In addition, with a PAGE 71 71 simple twist, the algorithm proposed by Jiang and Xie (2013) can be utilized to solve SP. More specifically, during the labeling process, if a node with a recharging station is selected to calculate the new labe ls, zero will be used as its distance label instead of its original distance label. Similar algorithms can also be found in Laporte and Pascoal (2011) and Adler et al. (2013) among others. For each O D pair , the optimal solution to SP, denoted as , can be used to construct a shortest usable path, i.e., . The iterative procedure of solving NE can thus be written as follows. As the number of usable paths in a network is finite, following procedure terminates in a finite number of steps. Step 0 : For each O D pair , solve SP with . Construct . Step 1 : Solve the restricted NE upon . Denote and as the optimal solutions and multipliers associated with Equation 4 1 . Step 2 : For each O D pair , solve SP. For , if , add into . If for all O D pairs, stop and is the equilibrium link flow distribution; Otherwise, go to Step 1. 4.2 Equilibrium Model Considering Recharging Time The base model does not take into account the recharging ti me, which can be substantial, depending on the recharging amount and the power of the charger. For example, for a BEV with a 24 KWh battery, it may take 20 hours to replenish a depleted battery at 1.2 KW power level. At 60KW power level, 24 minutes are s till needed (ETEC, PAGE 72 72 2010 a ). See Table 4 1 for the charging powers of BEV chargers that are currently deployed (Morrow, 2008). 4.1 .2. In Figure 4 1, both paths 1 3 2 and 1 4 2 are usable and travelers thus choose between them. Assuming that their travel times are 25 and 20 minutes respectively due to different speed limits, travelers would prefer path 1 4 2, if the recharging time is n ot considered. If travelers consider the recharging time and aim to reach their destinations as quickly as possible, they do not necessarily fully recharge their vehicles at nodes 3 or 4, where the remaining level of charge of the battery is one KWh. To complete their trips, travelers only need to recharge 0.5 KWh at node 3 or two KWh at node 4. Suppose that it takes 10 minutes for the charging stations at nodes 3 and 4 to recharge one KWh of electricity, and the recharging time will be 5 and 20 minutes respectively. Travelers would thus prefer path 1 3 2 to 1 4 2, as the total time to complete the former is 30 minutes while 40 minutes for the latter. The route choice is thus different with or without considering charging time. Motivated by the above e xample, this section extends the base model to consider the impact of recharging times on choices of route and recharging amount. It is assumed that BEV drivers attempt to minimize their trip times, which include travel times (more explicitly, driving tim es) and recharging times. Besides route choices, they also decide where and how much to recharge their vehicles. Consequently, at network equilibrium, all the utilized paths of each O D pair will be usable and yield the same trip times, which are less th an or equal to that of any unutilized usable path of the same O D pair. PAGE 73 73 To describe the equilibrium, let denote the time it takes for a BEV to recharge amount of electricity at node . For simplicity, we assume a linear charging time function, i.e., , where the first component represents the fixed time for the recharging activity; the second component is the variable time and depends on the type of char gers as shown in Table 4 1. Without loss of generality, we assume that there is at most one charging station at each node with the same type of chargers. If there is no charging station at node , both and are zero. Note that by add ing another cost component to the charging function, one can consider the impact of different prices for recharging at different types of charging stations. In this chapter , we assume the same unit price of electricity at all charging stations. Let represent the minimal time that a BEV of O D pair needs to spend on recharging activities when traversing a path . Because of the flow independence of the energy consumption of BEVs, it is straightforward to examine each usable pa th to determine optimal charging locations and amounts to compute . As the charging time function is linear, can be uniquely specified. The network equilibrium with recharging time (NE RT) can thus be formulated as follows: s.t. ( 4 1) ( 4 2) The above formulation is another convex program with linear cons traints and commercial nonlinear solvers such as CONOPT can solve it globally. Similarly, it is PAGE 74 74 straightforward to verify the equivalency of the formulation and that the equilibrium link flow is unique. To avoid path enumeration, an iterative solution pro cedure similar to that in Section 4.1 can be adopted to generate usable paths and their charging times as needed. The sub problem, shortest usable path finding with recharging time or SP RT, can be formulated as follows for each O D pair : s.t. (4 3) (4 10) (4 11) (4 12) where and indicates whether BEVs recharge at node . In the above, the objective function is to minimize the total trip time that includes the driving time, i.e., , and the recharging time, i.e., . Equations 4 11 and 4 12 ensure that equals 1 if BEVs recharge at node and 0 otherwise, as the objective function is to minimize . Similarly, SP RT is another mixed integer linear p rogram, and can be easily solved by commercial solvers such as CPLEX 12.2 for small or medium instances. However, because it encapsulates both route and recharging decisions, the labeling process to solve SP discussed in Section 4.1 is not readily applica ble to solve SP RT. Let denote the optimal solution to SP RT for O D pair . Based on , we can easily construct the usable path with minimal trip time, i.e., . In addition, and dictate the optimal recharging plan along the path, and PAGE 75 75 the minimal recharging time is . For completeness, we present the following solution procedure to NE RT: Step 0 : For each O D pai r , solve SP RT with . Construct and compute . Step 1 : Solve the restricted NE RT upon . Denote and as the optimal solutions and multipliers ass ociated with Equation 4 1 . Step 2 : For each O D pair , solve SP RT. For , if , add into . If for all O D pairs, stop and is the equilibrium link flow distribution; Otherwise, go to Step 1. 4.3 Network Equilibrium with Flow Dependent Energy Consumption In the above models, the energy consumption of BEVs depends only on the distance travelled. Bigazzi et al. (2011) investigated the effect of traffic congestion on the fuel economy of BEVs and found that BEVs may become more fuel efficient as the average speed in creases, particularly at local arterials. Although it is difficult to foresee how future developments in the battery and vehicular technologies may enhance the fuel economy of BEVs at various traffic conditions, we assume in the section that the energy co nsumption for a BEV to traverse a link increases as the travel time increases. This implies that the fuel economy of BEVs becomes flow dependent. So is the usability of a path between an O D pair. 4.3.1 Definition of Network Equilibrium With the flow de pendent usability of paths, the network equilibrium described in Definition 4 2, does not necessarily exist. To see this, consider a single O D network PAGE 76 76 connected by two parallel links. The O D travel demand is 5.5, and the travel time functions for two l inks are and while their lengths are 1 and 0.5 respectively. Suppose that the initial state of charge of BEV batteries is 1.6 and the energy consumption at each link is the function of distance and travel time, i.e., . Although there is no charging station in the network, both paths are usable when they are in free flow conditions. Consequently, three possible scenarios emerge at equilibrium, i.e., only link 1 or 2 is utilized or both links are utilized. It is easy to verify that if only either link 1 or 2 is utilized, the increased link travel time will cause the link unusable. If both links are utilized, according to Definition 2, they will have the same travel times. We thus obtain , , , and . Apparently, link 1 becomes unusable because . This contradicts to Definition 2, suggesting that the defined equilibrium does not exist in this network. This implies that with flow dependent energy consumption, the utilized usable paths do not necessarily have the same travel times when the network achieves equilibrium. We then claim that and is an equilibrium flow pattern, under which travel times and energy consumptions are , , and . One can verify that under such a flow pattern, no drivers have incentives to switch paths unilaterally. Specifically, drivers of path 1 have no interest to change to path 2 for a longer travel time. Reversely, drivers of path 2 have no interest to switch to path 1 either, because doing so would make path 1 unusable and thus compromise the chance of completing their trips. The above example motivates us to investigate a more general definition of ne twork equilibrium with flow dependent path usability. Several new variables are PAGE 77 77 introduced for this purpose. For each path , we divide it into sub paths, denoted as , , based on the locations of charging stations along the path. Each sub path starts and ends at a charging station, origin or destination, and there is no additional charging stati on along a sub path. The link energy consumption is assumed to be strictly increasing with the distance and the travel time . For model simplicity, we do not consider recharging time in this section. It is further assumed that each BEV starts its trip with a fully charged battery. Note that these two assumptions are not restrictive and can be relaxed. Below we introduce a concept of charging depleting paths to facilitate the presentation of our idea: Definition 4 3. A charge depleting path is a usable path with the energy consumption of a BEV reaching exactly its full battery capacity between two adjacent charging stations along the path. Recall that represents the battery size. The above definition suggests that along a charge depleting path, there exists at least a sub path where the energy consumption, i.e., , is equal to . Definition 4 4. A flow pattern is in user equilibrium if all the utilized paths are usable; the travel times of utilized regular or non charge depleting paths are equal and no larger than that of any unutilized usable path; the travel times of ut ilized charge depleting paths may be less than or equal to that of a utilized non charge depleting path. Mathematically, the user equilibrium conditions can be written as follows: If , then (4 13) If , and where , then (4 14) PAGE 78 78 If , and , , then (4 15) If , and , , then (4 16) where is the travel time of path . Equation 4 13 ensures that all the utilized paths are usable. Equations 4 14 and 4 15 specify that the travel time of each utilized path of O D pair equals , except for the charge depletin g paths whose travel times can be less than or equal to . Equation 4 16 dictates that the travel time of any unutilized usable paths is greater than or equal to . Note that if we denote as the equilibrium travel cost, for any utiliz ed charge depleting path , according to Equation 4 14 , there must exist a variable, say, such that . In a way, the variable may be interpreted as the cost of range anxiety that arises when the energy consumption reaches the limit. When the energy consumption is flow dependent, route choices of travelers affect not only the path travel time but also the path usability. When choosing paths, travelers would consider both effects, and the variable may captur e their sense of risk for being running out of charge along the path. However, we caution that such an interpretation depleting paths. 4.3.2 Model Formulation and Solution Algorithm The problem of finding a user equilibrium flow pattern as defined in Definition 4 4 can be formulated as the following nonlinear complementarity problem (NCP), which we call network equilibrium with flow dependent energy consumption or NE FD: (4 17) PAGE 79 79 (4 18) (4 19) (4 20) (4 21) (4 22) (4 23) We now briefly show that the solution to NE FD satisfies Equations 4 13 4 16 by examining both the utilized and unuti lized paths. For any utilized path, i.e., , Equation 4 19 leads to Equation 4 13, and Equation 4 21 implies that , . Consequently, Equation 4 23 reduces to . For a charge depleting path, together with Equation 4 20, Equation 4 23 implies that , which is corresponding to Equation 4 14 . For a regular or non charge depleting path where , , from Equation 4 21 , . Equation 4 23 thus implies that , which is corresponding to Equation 4 15 . For any unutilized path, i.e., , Equations 4 19, 4 21 and 4 23 are satisfied, and Equation 4 22 essentially reduces to . For any usable path, from Equations 4 20 and 4 22 , we have , which is PAGE 80 80 corresponding to Equation 4 16 . Note that for an unusable path, there exists a sub path where . Considering can be any nonnegative number, Equation 4 22 does not impose any re striction on the travel time of the unusable path, which is reasonable because travelers would have no incentive to switch to it. On the other hand, it can be verified that for any flow d istribution satisfying Equations 4 13 4 16 , a corresponding vector o f can be found such that NE FD is solved. Define a set . The NCP is equivalent to finding that solves the following variational inequality (VI): The NCP or VI may admit multiple link and path flow solutions. An example is provided in Section 4. 5 to demonstrate such a non uniqueness property. Solution a lgorithms proposed for solving VI or NCP in the literature can be applied to solve NE FD. Among others, one approach is to covert the formulation into a nonlinear optimization problem via a gap function (e.g., Lo and Chen, 2000; Wu et al., 2011) and then solve it accordingly. Further computational efficiency may be obtained by applying linear approximation to the functions of link travel time and energy consumption and then solving a series of simplified programs (e.g., Wang and Lo, 2010). PAGE 81 81 If all the pat hs can be enumerated, solving NE FD is no more difficult than, e.g., solving asymmetric path based traffic assignment models. It is thus beyond the scope of this dissertation to develop an efficient solution algorithm to solve the NCP or VI formulation. See, e.g., Facchinei and Pang (2003a, b), for recent developments. To avoid path enumeration, an iterative solution procedure similar to that in Section 4.1 can be adopted to generate paths as the procedure proceeds. The path generation sub problem is si milar to SP. The only difference is to add to SP additional cuts that correspond to charge depleting paths under the current flow solution because travel times of those paths are likely to be less than the equilibrium cost of the O D pair. More specifica lly, if we denote as one of the charge depleting paths under the current flow solution, one valid cut can be written as . 4.4 Deploying Public Charging Stations on Urban Road Networks With the proposed BEV network equilibriu m model s , we now investigate how to deploy public charging stations to a set of candidate locations. Specifically, g iven a budget limit, we propose a planning model that determines the locations and types of public charging stations to maximize social welf are. C onsidering NE FD may yield multiple equilibria , and it is challenging to solve NE FD for large size networks, we adopt NE optimizing station location plan s . Define a binary variable to represent whether a public charging station of type is built at node . Moreover, we use and to highlight the impact of station location plans on the sets of usable paths and optimal recharging plans. Given the PAGE 82 82 above notation, t he public charging station deployment problem (CD) is formulated as follows: s.t. ( 4 24 ) (4 25 ) , ( 4 26 ) ( 4 27 ) s.t. (4 2) , , where is the inconvenience cost s to travelers caused by failing to complete trip ; ; denotes the cost s of building a type charging station at node ; represents the budget limit; lastly, is the set of potential locations for charging stations. In the above, the objective function represents total so cial costs, including the total driving and recharging time, and the inconvenience cost s caused by missed trips. Equation 4 24 specifies the budget limit for st ation deployment. Equation 4 25 dictates at most one station can be built at a po tential locatio n. Equation 4 26 requires to be binary. Finally, Equation 4 27 states the traffic flow distribution is predicted by NE RT . PAGE 83 83 To solve CD, we utilize a genetic algorithm based procedure (Goldberg 1989; Yin, 2000). A genetic algorithm (G A ) is a search and optimization heuris tic, which is inspired by the process of natural selection, and has been applied to solve many real world eng ineering and science problems. 4.5 Numerical E xamples 4.5.1 Examples for NE RT and NE FD In this section, we present numerical examples to demonst rate the proposed ne twork equilibrium models. We first solve NE RT for the Sioux Falls network (see Figure 4 2 ), which consists of 24 nodes, 76 links and 552 O D pairs. Table 4 2 reports the free flow travel time and capacity of each link. The O D demand s are downloaded from Bar Gera (2013). The link distances are assumed to be 2.5 times of the link free flow travel times. The battery capacity, i.e., , is set as 24 KWh, consistent with the battery size of a Nissan Leaf (Nissan USA, 2013). Th e energy consumption rate is 0.29 KWh/ mi, as per the EPA fuel economy of Nissan Leaf 2013 (US Department of Energy, 2013). Other parameters include min, , and min /KWh. There exist five charging stations in the network, i.e., two level 1 stations at nodes 11 and 15, two level 2 at nodes 5 and 16, and one level 3 at Node 12. We adopt CONOPT and CPLEX 12.2 to solve NE RT and SP RT respectively, and the equilibrium l ink flows are reported in Table 4 3. In order to analyze the impact of the initial state of charge of battery on the equilibrium and charging station utilization, we also solve NE RT with another two initial states of and , in addition to . Figure 4 3 compares the average utilizations, i.e., average numbers of recharging trips, of three types of charging PAGE 84 84 stations in these three cases. It can be observed that as the i nitial state of charge increases, the utilizations of all three types of charging stations decrease. In addition, level 3 station is always the most popular, followed by level 2 and then level 1, which makes intuitive sense as higher power stations offer less amount of recharging time and are thus more favored by travelers. The average recharging frequency, amount and time in all three cases are depicted in Figure 4 4 . It can be observed that the highest average recharging frequency is 0.32 when , suggesting that on average a BEV recharges 0.32 times to complete its trip. Correspondingly, the recharging amount is 0.97 KWh, which takes 7 minutes on average. As the initial state of charge increases, the average recharging frequ ency, amount and time all decrease. Particularly, when the initial state equals , both the recharging amount and time are almost zero. All these observations suggest that the current driving ranges of BEVs are sufficient in the example ne twork, if their batteries are reasonably charged. Considering flow dependent energy consumption, we solve NE FD for the Nguyen Dupius network shown in Figure 4 5 . This network consists of 13 nodes, 19 links, and four O D pairs. The link characteristics and O D demands can be found in Nguyen and Dupius (1984). We assume that the link distance is 1.5 times of the link free flow travel time. The initial battery state and battery capacity are both equal to 16.56 KWh. The link energy consumption function i s . Two charging stations are located at nodes 6 and 11, respectively. During the solution iterations, we adopt PATH (Ferris and Munson, 1999) to solve NE FD upon a subset of paths, which are generated b y solving SP with cuts using CPLEX 12.2. With different PAGE 85 85 initial solutions, two equilibrium link flow patterns are obtained and reported in Table 4 4. It can be observed from Table 4 4 that these two flow patterns are substantially different. For instanc e, the flow on link 17 changes from 1.99 to 13.81. The non uniqueness and possible substantial variation of equilibria will inevitably create challenges for predicting BEV traffic flow distribution in the case of flow dependent energy consumption. Depend ing on specific policy goals, different strategies may be applied. For example, from the perspective of robust policy making, one may examine and design against the worst case equilibrium flow pattern (e.g., Lou et al, 2010). Tables 4 5 and 4 6 present th e detailed path information of these two flow patterns, including path travel time, flow and sub path energy consumption. It can be observed from Table 4 5 that under the first flow pattern, there are six utilized paths in total, of which only path 6 is c harge depleting. However, its travel time is the same as path 4 5 and thus the cost of range anxiety is zero. In contrast, under the second flow pattern in Table 4 6, there are two different path travel times for O D pair 4 2, and three for O D pair 4 3. Indeed, paths 5, 7 and 8 are all charge depleting and incur costs of range anxiety, as reported in the table. 4.5.2 Examples for CD In this section, we first solve CD for the Nguyen Dupius network, and compare the optimal station location plans , total s under three different budget limits , i.e., $900/veh, $1,100/veh and $1,500/veh . Compared to Section 4.5.1 , d ifferent parameter values are used . Specifically, the link distance is 0.8 times of the link free f low travel time; the energy consumption rate equals 0.24 KWh/ mi; . The costs of building a leve 1 station, i.e., , PAGE 86 86 equals $7,000; level 2, i.e., , equals $25,000; level 3, i.e., , equals $50,000 (ETEC, 2010 b ; U.S. Department of Energy , 2012 ) . T he inconvenience cost from a missed trip, i.e., , is assumed to be equivalent to a 500 minute delay. The set of potential locations for charging stations, i.e., , include s all the 13 nodes of the network. All the other parameters have the same val ues as those in Section 4.5.1. We utilize the GA solver in MATLAB R201 3a to solve CD with NE RT solved by CPLEX 12.2 within each iteration of the GA procedure to eva luate total social costs associated with different location plans. The r esults report that all the three budget limits can eliminate missed trips for all O D pairs. Moreover, Figure 4 6 (A ) shows the optimal station location plans under different budget limits, and Figure 4 6 (B ) compares their corresponding total social costs and average rechargi ng time for drivers. It can be observed that as the budget limit increases, more level 3 stations are deployed, and both the total social verage recharging time decrease , which makes sense because high power stations can substantially r total social costs. In order to demonstrate the potential of CD model for solving more realistic problems, we solve CD for the Sioux Falls network . We divide the OD demands and link capacities of Section 4. 5.1 by 100 , and use the results as inputs for this example . . The set of potential locations for charging stations, i.e., , include s nodes 4, 5, 11, 10 and 15 . The budget limit equals $120 ,000. The obtained optimal plan locates a level 1 station at node 10, two level 3 at node s 4 and 11 respectively . The corresponding total social costs are 30900 with no missing trips for any O D pair. Instead of consuming the entire budget limit, it can be observed that the obtained PAGE 87 87 optimal loca ti on plan only utilizes 89% of it, with the remaining budget still capable of affording several extra level 1 stations. The explanation might be that drivers always favor high power stations and thus locating extra charging stations, especially level 1 one s, may not necessarily lead drivers to use them. 4.6 Summary We have investigated the network equilibrium problems with battery electric vehicles. Considering the limited driving range and the recharging need of electric vehicles, we assume that drivers o f electric vehicles select paths to minimize their driving times while ensuring not running out of charge. We then define the network equilibrium conditions and formulate them into a mathematical program. An iterative procedure is proposed to solve the p rogram to find the equilibrium flow pattern. We further extend the model to consider the recharging time and flow dependent energy recharging decisions and captures the i choices. With the energy consumption being flow dependent, the second extension and usa bility of paths. Based upon the p roposed battery electric vehicle network equilibrium model s , we formulate the problem of determining the number, locations and types of charging stations within a budget limit on urban road networks as a bi level model , which is then solved by a genetic al gorithm. We analyze in the numerical examples the utilization for different types of charging station s , and the average recharging frequency, amount and time under various initial states of charge. It is observed that regardless of initial state of charg e, travelers always favor higher power charging stations, and the current BEV battery size PAGE 88 88 may be sufficient for trips across the example network. Moreover , the equilibrium flow pattern is shown to be non unique if the energy consumption is flow dependent . Lastly, it is demonstrated that optimal charging station deployment plans may not fully use the entire budget limit because locating extra low power stations does not necessarily lead drivers to use them. PAGE 89 89 Table 9 4 1. BEV charger specification Charging level Level 1 Level 2 Level 3 Power(KW) 1.44 6 90 Charging circuit 120V, 15A 240V, 30A 50V, 200A Table 10 4 2. Link capacity (1000 veh/hr) and free flow travel time (min) Li nk Capacity Link Capacity Link Capacity 1 2 3.6 6.02 10 11 3 20 17 16 1.2 10.46 1 3 2.4 9.01 10 15 3.6 27.02 17 19 1.2 9.65 2 1 3.6 12.02 10 16 3 10.27 18 7 1.2 46.81 2 6 3 15.92 10 17 4.2 9.99 18 16 1.8 39.36 3 1 2.4 46.81 11 4 3. 6 9.82 18 20 2.4 8.11 3 4 2.4 34.22 11 10 3 20 19 15 2.4 4.42 3 12 2.4 46.81 11 12 3.6 9.82 19 17 1.2 9.65 4 3 2.4 25.82 11 14 2.4 9.75 19 20 2.4 10.01 4 5 1.2 28.25 12 3 2.4 46.81 20 18 2.4 8.11 4 11 3.6 9.04 12 11 3.6 9.82 20 19 2.4 6.05 5 4 1.2 46 .85 12 13 1.8 51.80 20 21 3.6 10.12 5 6 2.4 13.86 13 12 1.8 51.80 20 22 3 10.15 5 9 3 10.52 13 24 2.4 10.18 21 20 3.6 10.12 6 2 3 9.92 14 11 2.4 9.75 21 22 1.2 10.46 6 5 2.4 9.90 14 15 3 10.26 21 24 1.8 9.77 6 8 1.2 21.62 14 23 2.4 9.85 22 15 2.4 20.6 3 7 8 1.8 15.68 15 10 3.6 27.02 22 20 3 10.15 7 18 1.2 46.81 15 14 3 10.26 22 21 1.2 10.46 8 6 1.2 9.80 15 19 2.4 9.64 22 23 2.4 10 8 7 1.8 15.68 15 22 2.4 20.63 23 14 2.4 9.85 8 9 2 10.10 16 8 3 10.09 23 22 2.4 10 8 16 3 10.09 16 10 3 10.27 23 24 1. 2 10.16 9 5 3 20 16 17 1.2 10.46 24 13 2.4 11.38 9 8 2 10.10 16 18 1.8 39.36 24 21 1.8 9.77 9 10 1.8 27.83 17 10 4.2 9.99 24 23 1.2 10.16 10 9 1.8 27.83 PAGE 90 90 Table 11 4 3. Equilibrium link flow (veh/hr) Link Flow Link Flow Link Flow 1 2 2100 10 11 23462 17 16 21238 1 3 8500 10 15 13632 17 19 16672 2 1 2500 10 16 14934 18 7 11285 2 6 4100 10 17 7872 18 16 15862 3 1 8100 11 4 4100 18 20 11835 3 4 20958 11 10 21842 19 15 8680 3 12 23000 11 12 21265 19 17 19638 4 3 19500 11 14 10796 19 20 8382 4 5 24658 12 3 24058 20 18 12062 4 11 4700 12 11 18142 20 19 6972 5 4 23900 12 13 25065 20 21 4435 5 6 17400 13 12 23100 20 22 9120 5 9 10458 13 24 19400 21 20 4800 6 2 4500 14 11 11800 21 22 9688 6 5 18500 14 15 6900 21 24 10535 6 8 20100 14 23 5600 22 15 16728 7 8 12685 15 10 11600 22 20 7472 7 18 9100 15 14 7904 22 21 8400 8 6 21600 15 19 13056 22 23 6200 8 7 10500 15 22 13480 23 14 5600 8 9 9300 16 8 14515 23 22 6512 8 16 14166 16 10 16965 23 24 8900 9 5 8600 16 17 17000 24 13 17335 9 8 8266 16 18 17720 24 21 12188 9 10 19893 17 10 6600 24 23 9212 10 9 17100 Table 12 4 4 . Equilibrium link flow (veh/hr) Link Flow #1 Flow #2 Link Flow #1 Flow #2 1 48.57 47.62 11 48.01 47.74 2 2 1.43 22.38 12 0 0 3 30 26.34 13 20.56 38.59 4 0 3.66 14 21.43 3.66 5 28.01 31.41 15 28.01 16.19 6 20.56 19.87 16 20 31.56 7 20.56 38.59 17 1.99 13.81 8 21.43 3.66 18 38.57 56.34 9 10 10 19 38.01 26.19 10 20 16.34 PAGE 91 91 Tab le 13 4 5. Path information for the first equilibrium flow pattern O D Path ID Node sequence Path flow Path travel time Sub path Energy consumption 1 2 1 1 12 8 2 10 45.92 1 12 8 2 13.68 1 3 2 1 12 6 7 11 3 20 52.86 1 12 6 7.26 6 7 11 5.88 11 3 2.12 4 2 3 4 5 6 7 8 2 28.01 72.67 4 5 6 8.32 6 7 8 2 8.2 4 4 5 9 10 11 2 1.99 72.67 4 5 9 10 11 16.26 11 2 3.39 4 3 5 4 5 9 10 11 3 18.57 70.72 4 5 9 10 11 16.26 11 3 2.12 6 4 9 13 3 21.43 70.72 4 9 13 3 16.56 Table 14 4 6. Path information for the second equilibrium flow pattern O D Path ID Node sequence Path flow Path travel time Cost of range anxiety Sub path Energy consumptio n 1 2 1 1 12 8 2 10 39 0 1 12 8 2 12.88 1 3 2 1 12 6 7 11 3 16.34 57.07 0 1 1 2 6 6.76 6 7 11 6.03 11 3 2.97 3 1 5 9 13 3 3.66 57.07 0 1 5 9 13 3 16.02 4 2 4 4 5 6 7 8 2 16.19 85.06 0 4 5 6 10.14 6 7 8 2 7.82 5 4 5 9 10 11 2 13.81 75.32 9.74 4 5 9 10 11 16.56 11 2 3.40 4 3 6 4 5 6 7 11 3 15.22 95.19 0 4 5 6 10.14 6 7 11 6.03 11 3 2.97 7 4 9 10 11 3 22.38 94.09 1.1 4 9 10 11 16.56 11 3 2.97 8 4 5 9 10 11 3 2.4 80.59 14.6 4 5 9 10 11 16.56 11 3 2.97 PAGE 92 92 Figure 4 4 1. A t oy network with four nodes Figure 5 4 2 . Sioux Falls network PAGE 93 93 Figure 6 4 3 . Charging station utilizations PAGE 94 94 Figure 7 4 4 . Recharging in formation . A) Average recharging frequency. B) Average recharging amount. C) Average recharging time. PAGE 95 95 Figure 8 4 5 . Nguyen Dupius network PAGE 96 96 Figure 9 4 6 . Se nsitive analyses of budget limits . A) Optimal station plans. B) Average recharging time and total costs. PAGE 97 97 CHAPTER 5 PRICING OF ELECTRICITY AT PUBLIC CHARGING STATION S FOR PEVS PEVs introduce additional load s to the power system. As Hadle y and Tsvekova (2009) projected, it may roughly lead to a 15% increase of the base load on the peak day load curve in 2030. The impact of this additional load on power distribution grid s have been extensively investigated. For example, Taylor et al. (2009) concluded that PEV loads clustered in certain areas and time intervals could result in loads exceeding the current design capacity of circuit and suggested utilities to conduct distribution feeder level analyses to ascertain that these new demands can be met. Karnama (2009) investigated the impacts of PEVs on the distribution grids of three distinct areas in Stockholm and found that accommodating PEV loads is more problematic in residential areas with high population density. The author also concluded that power losses can be reduced by regulated charging. Putrus et al. (2009) analyzed three charging scenarios of PEVs, i.e., uncontrolled, off peak, and scheduled and phased charging. The authors showed that a large deployment of PEVs could cause power qualit y problems and voltage imbalance, which can be mitigated by reasonably distributing PEV loads among three phases. Even though different types of demand management schemes have been proposed in previous studies to mitigate the adverse impacts of PEV loads, few have charging behaviors to better distribute their charging loads. A large number of PEVs connecting in one area might cause deterioration of power quality and security of supply since the power distribution networks are not designed for such mobile loads (Green et al., 2011). This chapter attempts to adjust the prices of electricity at public charging stations to influence the spatial distribution of PEV PAGE 98 98 chargin g loads to mitigate their impacts. It is assumed that a significant number of public charging stations have been deployed in urban areas. Their locations and prices of electricity will affect the spatial distribution of PEVs and thus the pattern of their e nergy requirements, thereby affecting the operations of power distribution systems. We propose a modeling framework to optimize the prices of electricity at public charging stations to minimize real power losses in the power distribution grid, and develop the i ntegrated pricing of electricity and roads to minimize both real power losses and total travel cost in the coupled power and transportation networks. For the remainder of this chapter, Section 5.1 describes the modeling of urban transportation networ k and radial power distribution grid. Based on the models developed in Section 5.1 , Section 5.2 formulates the pricing model of electricity and roads as a mathematical problem with complementarity constraints and proposes its solution algorithms. Section 5 .3 presents a numerical example to demonstrate the effectiveness of pricing of electricity and roads in managing power distribution grids and urban transportation networks. Lastly, Section 5 .4 concludes this chapter . 5. 1 Modelling Transportation and Power Networks 5. 1 .1 Description of Transportation Network We consider an urban transportation network where there are two classes of vehicles, i.e., PEVs and regular gasoline fueled vehicles. We assume that the origin destination (O D) demands of the latter ar e given while PEVs have flexibility in choosing their travel destinations to conduct activities such as shopping and recreation. We are thus interested in modeling the spatial distribution of PEVs and link flow distributions of PEVs and regular vehicles in the urban transportation network. PAGE 99 99 To model the transportation network, we follow the notation in Section 3 .1.1 , with a slight difference when describing charging station existence and multi class travelers. Without losing generality, we assume that there is at most one public charging station at each node and then use a binary variable to represent the existence of such a charging station, which is one if a charging station is located at node and zero otherwise. is assumed to be given an d fixed in this chapter . Let represent the O D flows of regular vehicles and is the trip production of PEVs at each origin, both of which are given. The O D flows of PEVs, choic es, and thus are decision variables in the network flow model presented later in this chapter. Similarly with Chapter 3 , we assume that the above destination choice behavior of PEVs can be captured by a multinomial logit model as follows: Compared to Section 3.1, the coefficient in the above logit model is associated with the dummy variab le representing the existence of a charging station, i.e., , instead of the c harging station density, which is attributed to our assumption of each destination in a urban transportation network sit ting at most one charging station. It is further ass umed that when travelling between their origins and destinations, both PEVs and regular vehicles choose the paths with minimum travel times ( e.g., Sheffi, 1984). Consequently, the network will achieve multi class user equilibrium. Given the availability of charging stations and their corresponding electricity prices, the equilibrium O D flows of PEVs and link flow distributions of PEVs and regular vehicles PAGE 100 100 can be estimated by solving the following multi class combined distribution and assignment ( M CDA) mod el: s.t. ( 5 1 ) ( 5 2) ( 5 3) ( 5 4) ( 5 5) ( 5 6) ( 5 7) where and are flows of PEVs and regular vehicles on link between O D pair , respectively. In the above, Equation 5 1 calculates the aggregate link flow. Equations 5 2 and 5 3 ensure flow balance at each node for demands of PEVs and regular vehicles between each O D pair in the network (e.g., Hearn and Ramana, 1988). Equation 5 4 requires the sum of the demand of PEVs leaving from any origin to each destination to be equal to the total demand of PEVs generated at that particular origin . Equations 5 5, 5 6 and 5 7 are the non negativity constraints. As formulated, M CDA is a convex program with all linear constraints. Since the KKT conditions are both sufficient and necessary to a convex program (e.g., Bazaraa et al., 2006), the equivalence of CDA to the assumed network equilibrium conditions can be established easily by examining its PAGE 101 101 KKT conditions. M CDA can be also efficiently solved by commercial nonlinear sol vers such as CONOPT (Drud, 1994). 5.1.2 Analysis of Power Flows in Distribution Grid Urban areas are served by power distribution grids, which consist of distribution substations and feeders as shown in Figure 5 1. Typically, a distribution substation is fed by a sub transmission line and serves one or multiple feeders. Feeders deliver electric power from a substation to different end users. With rare exception, they are radial (Kersting, 2002). In a feeder, there are buses that connect electrical loads fr om end users. Buses are connected by branches. A feeder may consist of a main line and several laterals. For the modeling purpose, the feeder can be divided into multiple layers with the main line in one layer and a lateral being another layer, as shown in Figure 5 2. Without loss of gen erality, we assume in this chapter that the urban transportation network of interest is served by a power distribution grid that consists of a distribution substation and a radial feeder. In a feeder, AC power flow has multi ple phases and its distribution follows electrical load at each bus and the base voltage at the distribution substation, a system of nonlinear equations can be established to estimate power flows, both active and reactive, bus voltages and power losses. Many studies have been conducted to construct accurate systems of nonlinear equations, i.e., power flow models, and developed efficient algorithms to solve them (e.g., Tinney an d Hart, 1967; Cheng and Shirmohammadi, 1995; Ciric and Ochoa, 2003). See, e.g., Kersting (2002), for an overview of power flow analysis in distribution grids. PAGE 102 102 This chapter adopts the power flow model proposed by Baran and Wu (1989), which assumed a balance d three phase and transposed radial distribution feeder. Although the assumption of a balanced and transposed feeder leads to inaccuracy to some extent, Mwakabuta and Sekar (2007) conducted a comparative study and concluded that the power flow model yields similar voltage profiles as compared to the empirical data for the IEEE 34 node test feeder. For the power flow analysis, we adopt totally different notations from Section 3.1.2. Specifically, l et DS ( ) denote the distribution feeder, where , and are the sets of buses, branches and layers in the feeder respectively. For bus and layer , and are the active and reactive powers that flow into the sending end of the branch located in layer and emanated from bus , and represents the bus voltage magnitude and is considered to be given and fixed at the distribution substation. We further denote a branch as the pair of its starting and ending buses, i. e., . For each branch, is the series impedance, where is the resistance and is the reactance. We also use to denote the set of end nodes of each layer, and and respresent the sets of the pairs of branches and their locating layers, and the pairs of buses and their layers respectively. Finally, is the set of the pairs of destination nodes and their serving buses. The power flow equations are written as follows: ( 5 8) ( 5 9) PAGE 103 103 ( 5 10) ( 5 11) ( 5 12) ( 5 13) ( 5 14) where is regular constant electrical loads, both active and reactive, at bus , which are given. is the additional (active) electrical load created by charging PEVs, which is equal to ; Moreover, represents the shunt capacitor and is the magnitude of the base voltage. In th e above, Equations 5 8 5 10 , referred to as branch flow equations, represent the relationship between the magnitudes of active or reactive powers and voltages at two ends of each branch. Equations 5 11 5 12 are nonnegative constraints. Equation 5 13 repres ents a boundary condition, requiring the active and reactive powers flowing out of the end bus of each layer to be zero. Equation 5 14 sets the magnitude of the voltage at the substation (labeled as 0) as the base voltage. To solve the above nonlinear sys tem, Baran and Wu (1989) proposed a simplified Newton Raphson method, taking advantage of the radial structure of the feeder and numerical proper ties of the Jacobian matrix of Equations 5 8 5 14 to reduce the number of decision variables and simplify the c alculation of the Jacobian matrix. More specifically, the variables are reduced to the active and reactive powers only emanated from substation and all the buses that join different layers. Hereinafter, we PAGE 104 104 represent these powers as vector . Once is determined, the active and reactive powers emanated from other buses and the voltage magnitudes of all buses can be calculated as per Equations 5 8 5 10 and 5 14 . Therefore, solving the power flow equations essentially reduces to finding a v ector of to ensure that Equation 5 13, the boundary condition , is satisfied. To facilitate the presentation, we represent the boundary conditions as , and thus the solution procedure is to seek for such that . Figure 5 3 illustrates the pro cedure, which we refer as the simplified Netwton Raphson or SNR procedure. In Figure 5 3, when constructing the system Jacobian matrix, branch flow equations, i.e., Equations 5 8 5 10 , and the chain rule for computing derivatives are utilized. Based on th e numerical properties and special structure of the Jacobian matrix, all the off diagonal components can be dropped except those in the last row. Moreover, since the matrix is almost constant, it needs to be constructed only once and then can be used in al l iterations. 5. 2 Optimal Pricing Models 5.2.1 Model Formulations With the proposed CDA model and power flow equations, we now investigate the design of prices of electricity at public charging stations. We assume that the power distribution feeder is ma naged by a distribution company (DisCo), who purchases power from the wholesale electricity market with a contract price and is responsible for utilizing the feeder to deliver electricity from the distribution substation to end users (LÃ³pez Lezama et al., 2011). We consider different authoritarian regimes of DisCo. In the simplest setting, DisCo is able to determine the retail price of electricity at each public charging station to impact electrical loads generated by PEVs to minimize power losses PAGE 105 105 over the distribution feeder. As DisCo can be a public entity who will be concerned with social welfare, it can optimize the prices of electricity at public charging stations to minimize both power losses in the feeder and total travel time across the transportatio n network. Lastly, we consider that DisCo, in cooperation with a traffic agency, determines both prices of electricity and road tolls for regular vehicles to minimize both power losses and total system travel time. Such cooperation may not exist today. How ever, The general pricing model can be formulated as follows: s.t. ( 5 1) ( 5 14) ( 5 15) ( 5 16) ( 5 17) PAGE 106 106 ( 5 18) ( 5 19) ( 5 20) ( 5 21) where, in the objective function, represents the contract price at which DisCo purchases electricity from the wholesale power market; is the active power flowing from the substation, which is equal to the amount of energy DisCo purchases from the wholesale power market, and is the value of travel time. In the constraints, with Equation 5 4 ; is the toll at link for regular vehicles; is the retail price at which DisCo sells electricity to users at all places other than public charging stations, and is assumed to be given and fixed; and denotes the retail price of el ectricity at public charging station, to be determined by DisCo and lastly, denotes the upper limit for electricity price at public charging stations . In the above, the objective function is to minimize real power losses in the distribution gr id and total travel cost in the urban transportation network. Equations 5 1 5 7 and 5 15 5 19 are the KKT conditions of the CDA model in the presence of tolls and ensure that the traffic flow pattern is in multi class tolled user equilibrium. More specific ally, Equations 5 15 5 18 imply that travel costs of all utilized paths between an O D pair are the same, which are less than or equal to those of any unutilized path of the same O D pair. Equation 5 19 implies the multinomial logit model discussed in Sect ion 5 .1. 1 . As aforementioned, Equations 5 8 5 14 are power flow equations. PAGE 107 107 Equation 5 20 ensures nonnegative revenue for DisCo, i.e., the total revenue from selling electricity should be no less than the expense of purchasing it from the wh olesale power ma rket. Equation 5 21 sets the upper limit for electricity prices at public charging stations. Note that the capacity of a distribution line is usually sufficient for real operations, and we thus do not incorporate line capacity constraints. The model can be easily extended to include those constraints if line capacities are a concern for a particular network (e.g., Momoh and Wang, 1997). The above formulation is general and can reduce to different problems under different authoritarian regimes of DisCo. To facilitate the discussion, we refer to the above formulation as ERP PLTC (electricity and road pricing for power losses and travel cost minimization). If DisCo does not cooperate with the traffic agency and no tolls will be charged, i.e., , the problem reduces to EP PLTC ( electricity pricing for power losses and travel cost minimization). Lastly, if DisCo is only concerned about real power losses in the power distribution gird, the problem essentially becomes designing electricity p rices at public charging stations to minimize real power losses, which we refer as EP PL. 5.2.2 Solution Algorithm The above pricing model is a mathematical program with complementarity constraints (MPCC), a class of problems difficult to solve (see, e.g. , Luo et al., 1996). The problem is non convex and standard stationarity conditions such as the KKT conditions may not hold for it (Scheel and Scholtes, 2000). Many (see, e.g., Luo et al., 1996, and references cited therein) have proposed special algorithm s to solve them. Some of these algorithms only work well for small and medium problems while others, PAGE 108 108 especially those based on solving equivalent nonlinear programs (e.g., Fletcher and Leyffer, 2004 and Lawphongpanich and Yin, 2010), can handle larger pro blems. In this chapter, we first implement the manifold suboptimization algorithm proposed by Lawphongpanich and Yin (2010). More specifically, we specify a pair of active sets for each O D, based on which each complementarity constraint is split into two regular constraints. The model thus becomes a regular nonlinear program that is a restricted version of the original MPCC and can be directly solved by a commercial nonlinear solver, e.g., CONOPT. Using the Lagrangian multipliers generated by CONOPT, the pairs of active sets are updated and the procedure continues iteratively until a strongly stationary point is reached. Lawphongpanich and Yin (2010) proved that the procedure ensures the feasibility of the solution at each iteration and terminates in a fin ite number of iteration. Note that the manifold suboptimization algorithm yields a strongly stationary solution, whose quality may not be good because strongly stationary conditions are one of the weaker optimality conditions (e.g., Scheel and Scholtes, 2 000; Fletcher and Leyffer, 2004; Lou et al., 2010). To explore and hopefully obtain a better solution, we reformulate the pricing model as an equivalent bi level programming model, and then develop a derivative free solution algorithm to solve it. Such typ e of algorithms may be particularly effective if the number of pricing decision variables is not prohibitively large, e.g., in the problems of EP PL or EP PLTC. For the derivative free solution algorithm, consider the following bi level program model equiv alent to EP PLTC: PAGE 109 109 s.t. s.t. (5 1) (5 7) where is sufficient large constant number. In t he objective function, the last component is a penalty to ensure that Equation 5 20 is satisfied. represents the simplified power flow equations as discussed in Section 5.1.2. Here we add , the O D demands of PEVs, to highlight the impact s of PEV electric loads on the operations of the power distribution grid. In a derivative free algorithm, to evaluate a given price design, M CDA can be solved to estimate the transportation network performance and the resulting O D demands of PEVs are th en fed into the power flow equations, which is then solved by the SNR procedure as shown in Figure 5 3 to estimate power losses. Such an evaluation procedure is efficient because both subproblems can be solved efficiently. A derivative free algorithm typic ally utilizes the information from evaluating multiple price designs to update and find a better design. The process continues until an optimum design is reached. PAGE 110 110 In this chapter , we choose the SID PSM algorithm, a pattern search method guided by simplex derivatives, to solve the bi level model (CustÃ³dio and Vicente, 2007; CustÃ³dio et al., 2010), after reviewing and comparing a variety of derivative free algorithms including metaheuristics (e.g., Yin, 2000; Kolda et al., 2003; More and Wild, 2009). SID PSM identifies a subset of points with some desirable geometrical properties as a sample set in the beginning of each iteration and computes the corresponding simplex derivatives based on this sample set. The gaps between the corresponding simplex derivatives and exact derivatives of the objective function are determined by the size and the geometrical properties of the sample set. Using the simplex derivatives, a descent indicator, e.g., a negative simplex gradient, can be computed. In SID PSM, an improved so lution is found by search and poll steps. The search step aims to find an improved feasible mesh point by evaluating a finite number of points lying on the mesh. The iteration is declared as successful if such a mesh point is found. Otherwise, the poll ste p will be conducted. SID PSM orders the polling direction set with respect to the angle between a polling vector and the descent indicator. The objective function is evaluated at each polling point following the determined order. If an improved polling poi nt is found, declare the iteration as successful and otherwise unsuccessful. The mesh size is increased or remains unchanged after each successful iteration and is decreased if the iteration is unsuccessful. The global convergence of this algorithm is esta blished with the mesh size going to zero. In this chapter , the SID PSM algorithm and the SNR subroutine are implemented in MATLAB R2011 while M CDA is solved using the CONOPT solver in GAMS 23.6 (Brooke et al., 2005). Figure 5 4 shows the procedure. PAGE 111 111 5.3 N umerical Examples The pricing models were solved for a coupled network that we created based on the topology of Sioux Falls road network and IEEE 34 bus test feeder (Kersting, 2000), as shown in Figure 5 5. The transportation network consists of 76 direct ed links and 24 nodes, 12 of which are origins and destinations, and are connected to buses in the distribution feeder (See, e.g., Table 5 6 for the pairs of destination nodes and their serving buses). The free flow travel time and capacity of each link in the transportation network are reported in Table 5 1. The distribution feeder is composed of 26 buses, a simplification from the IEEE 34 bus test feeder as suggested by Mwakabuta and Sekar (2007). The O D demands of regular vehicles and trip productions o f PEVs are reported in Tables 5 2 and 5 3 respectively. Table 5 4 reports the regular electrical load at each bus. The series impedance of each branch in the distribution feeder is presented in Table 5 5. Other parameters were specified as follows: , and in the logit based destination choice model; , the hourly average energy requirement for charging a PEV, equals KW; , the contract price DisCo pays to purchase energy from the wholesale electricity market is $100/MWh and , the electricity price users at all places other than public charging stations pay, is $300/MWh; , the maximum electricity price at public charging stations, equals $650/MWh; the shunt capacitors at buses 844 and 848 are 300 and 48 KVar respectively. From the input data, it can be observed that the market penetration of PEVs is 10% and their electrical load is around 15% of the total power injection in the feeder. Finally, bus 800 is the distribution substation and its voltag e magnitude equals 24.9 KV. PAGE 112 112 In all numerical examples, we assumed that there is one public charging station at each destination node. Using an initial design that the same price of electricity is charged at public charging stations as the retail electricit y price at other areas, i.e., $0.3/KWh, we first implemented the manifold suboptimization and SID PSM algorithms to solve EP FL. The latter yielded a better solution that reduces real power losses from 1341.9 KW of the initial design to 1195.1 KW, a 10.9% reduction. We also observed that although it may terminate at a stationary point of low quality, the manifold suboptimization algorithm was much faster than SID PSM. The optimal prices of electricity at public charging stations are presented in Table 5 6. One can easily observe that the optimal design essentially sets the prices to be 0 at four public charging stations (buses) close to the distribution substation, i.e., buses 806, 812, 824 and 828, and the others to be the upper limit. Intuitively, if more PEV loads are attracted to buses close to the distribution substation, real power losses will be reduced due to shorter power delivery distances. However, the total travel time under such a price design is 62,088 minutes, a slight increase from 62,022 min utes, the travel time under the initial design, implying that the transportation system is made slightly worse off. We further solved EP PLTC with the value of travel time being $20/hr. The two algorithms yielded solutions of essentially the same quality. The optimal electricity prices from the manifold suboptimizaton algorithm are reported in Table 5 7. Table 5 8 compares the resulting travel cost and real power losses with those associated with the initial design. PAGE 113 113 It can be observed from Table 5 8 that al though the optimal price design can reduce both the system travel cost and real power losses, the reduction in their sum is as small as 0.2%. This is hardly surprising, because the prices of electricity can only affect a small portion of vehicles in the tr ansportation network and thus may have limited effects in managing the network. In order to be more effective, electricity pricing can be complemented by road pricing. We further solved EPRP PLTC by the manifold suboptimization algorithm to explore the pot ential of integrated pricing of electricity and roads. The resulting optimal tolls and prices of electricity are reported in Tables 5 9 and 5 10 respectively. The integrated pricing strategy was found to achieve more meaningful reduction in the sum of tra vel cost and power losses. The best obtained objective value is $20,272, a 2.6% reduction as compared to the initial design. Moreover, the travel cost and real power losses are reduced by 2.6% and 4.5% respectively. More advanced pricing strategies, such a s differentiated road pricing of PEVs and regular vehicles, are expected to be more effective for integrated management and operations of urban transportation and power distribution networks. This, however, is beyond the scope of this chapter. 5.4 Summary In this chapter, we have investigated optimal designs of prices of electricity at public charging stations to mitigate the adverse impacts of charging loads of PEVs on the power distribution grid in an urban area, and, in couple of road pricing, to reduc e system travel time on the transportation network. Our numerical examples have demonstrated that electricity pricing can be an effective tool for addressing the challenges that PEV charging loads impose on power distribution grids, and integrated PAGE 114 114 pricing of electricity and roads has potential for better managing and operating the coupled transportation and power networks. PAGE 115 115 Table 15 5 1. Link capacity (veh/hr) and free flow travel time (min) Link Capacity Link Capacity Link Capacity 1 2 3.60 181 10 11 3.00 600 17 16 1.20 314 1 3 2.40 270 10 15 3.60 811 17 19 1.20 290 2 1 3.60 361 10 16 3.00 308 18 7 1.20 1404 2 6 3.00 478 10 17 4.20 300 18 16 1.80 1181 3 1 2.40 1404 11 4 3.60 295 18 20 2.40 243 3 4 2.40 1027 11 10 3.00 600 19 15 2.40 133 3 12 2.40 1404 11 12 3.60 295 19 17 1.20 290 4 3 2.40 775 11 14 2.40 293 19 20 2.40 300 4 5 1.20 848 12 3 2.40 1404 20 18 2.40 243 4 11 3.60 271 12 11 3.60 295 20 19 2.40 182 5 4 1. 20 1406 12 13 1.80 1554 20 21 3.60 304 5 6 2.40 416 13 12 1.80 1554 20 22 3.00 305 5 9 3.00 316 13 24 2.40 305 21 20 3.60 304 6 2 3.00 298 14 11 2.40 293 21 22 1.20 314 6 5 2.40 297 14 15 3.00 308 21 24 1.80 293 6 8 1.20 649 14 23 2.40 296 22 15 2.40 619 7 8 1.80 470 15 10 3.60 811 22 20 3.00 305 7 18 1.20 1404 15 14 3.00 308 22 21 1.20 314 8 6 1.20 294 15 19 2.40 289 22 23 2.40 300 8 7 1.80 470 15 22 2.40 619 23 14 2.40 296 8 9 2.00 303 16 8 3.00 303 23 22 2.40 300 8 16 3.00 303 16 10 3.00 308 2 3 24 1.20 305 9 5 3.00 600 16 17 1.20 314 24 13 2.40 341 9 8 2.00 303 16 18 1.80 1181 24 21 1.80 293 9 10 1.80 835 17 10 4.20 300 24 23 1.20 305 10 9 1.80 835 PAGE 116 116 Table 16 5 2. O D demands of regular gasoline powered vehic les (veh/hr) 1 2 4 5 10 11 13 14 15 19 20 21 1 0 71 71 71 58 59 68 53 51 49 32 32 2 71 0 68 70 59 60 49 51 51 70 32 37 4 71 68 0 71 58 58 51 49 45 43 33 35 5 71 70 71 0 61 52 49 48 44 39 43 44 10 58 59 58 61 0 72 49 53 71 63 51 49 11 59 60 58 52 72 0 51 71 60 51 40 33 13 68 49 51 49 49 51 0 47 46 37 32 33 14 53 51 49 48 53 71 47 0 71 61 51 47 15 51 51 45 44 71 60 46 71 0 71 69 62 19 49 70 43 39 63 51 37 61 71 0 71 60 20 32 32 33 43 51 40 32 53 69 71 0 71 21 32 37 35 44 49 33 33 47 62 60 71 0 Table 17 5 3. Trip production of PEVs at each origin (veh/hr) Origin 1 2 4 5 10 11 13 Production 78 76 73 75 81 80 69 Origin 14 15 19 20 21 Production 79 85 80 70 72 Table 18 5 4. Regular load at ea ch bus Nodes Active power(KW) Reactive power(KVar) Nodes Active power(KW) Reactive power(KVar) 802 55 29 858 34 18 808 16 8 848 60 48 816 174 89 834 146 73 840 27 21 860 142 91 890 450 225 836 40 20 824 44 22 862 28 14 830 45 20 842 9 5 828 7 3 844 450 338 854 4 2 PAGE 117 117 Table 19 5 5. Branch series impedance (Ohms) Branch Resistance Reactance Branch Resistance Reactance 800 802 0.547 0.407 834 842 0.090 0.045 802 806 0.367 0.273 836 840 0.275 0.137 806 808 6.837 5.066 836 862 0.090 0.045 808 812 7.955 5.918 842 844 0.432 0.215 812 814 6.307 4.692 844 846 1.165 0.580 814 850 0.003 0.002 846 848 0.170 0.084 816 824 3.268 1.627 850 816 0.099 0.049 824 828 0.269 0.133 852 832 0.003 0.002 828 830 6.543 3.256 854 852 1 1.789 5.867 830 854 0.166 0.083 858 834 1.866 0.929 832 858 1.568 0.781 860 836 0.858 0.427 832 888 0.000 0.000 888 890 2.240 1.667 834 860 0.647 0.322 Table 20 5 6. Optimal retail electricity prices at public charging stati ons ($/KWh) Destination node Serving bus Optimal retail price Destination node Serving bus Optimal retail price 1 806 0 13 812 0 2 848 0.65 14 888 0.65 4 842 0.65 15 890 0.65 5 846 0.65 19 834 0.65 10 844 0.65 20 828 0 11 858 0.65 21 824 0 Table 21 5 7. Optimal retail electricity prices at public charging stations ($/KWh) Destination node Serving bus Optimal retail price Destination node Serving bus Optimal retail price 1 806 0 13 812 0.65 2 848 0 14 888 0.65 4 842 0.65 15 890 0.58 5 846 0.65 19 834 0.13 10 844 0.41 20 828 0 11 858 0.60 21 824 0.49 PAGE 118 118 Table 22 5 8. Travel cost and real power losses ($) Scenario System travel cost Power losses Total travel cost plus power losses Initial design 20 ,674 134 20,808 Best obtained design 20,639 129 20,768 Table 23 5 9. Link tolls for regular gasoline powered vehicles (min) Link Toll Link Toll Link Toll 1 2 12.05 10 11 0.42 17 16 0.04 1 3 15.37 10 15 0.39 17 19 1.15 2 1 0.0 6 10 16 0.00 18 7 0.00 2 6 4.91 10 17 0.00 18 16 0.00 3 1 0.06 11 4 1.68 18 20 1.84 3 4 0.02 11 10 0.30 19 15 3.23 3 12 0.00 11 12 0.96 19 17 0.99 4 3 0.22 11 14 1.73 19 20 0.40 4 5 0.36 12 3 0.00 20 18 1.12 4 11 1.34 12 11 1.38 20 19 0.64 5 4 0.08 12 13 0.00 20 21 0.78 5 6 0.76 13 12 0.00 20 22 0.18 5 9 1.66 13 24 1.66 21 20 0.44 6 2 11.47 14 11 1.79 21 22 0.20 6 5 0.91 14 15 0.86 21 24 0.59 6 8 0.46 14 23 0.25 22 15 0.19 7 8 0.01 15 10 0.30 22 20 0.05 7 18 0.00 15 14 0.81 22 21 0.12 8 6 1. 46 15 19 0.70 22 23 0.09 8 7 0.00 15 22 0.07 23 14 0.36 8 9 0.39 16 8 0.00 23 22 0.09 8 16 0.34 16 10 0.18 23 24 0.15 9 5 0.86 16 17 0.12 24 13 1.04 9 8 0.05 16 18 0.00 24 21 0.54 9 10 0.12 17 10 0.22 24 23 0.17 10 9 0.37 PAGE 119 119 Table 24 5 10. Optimal retail electricity prices at public charging stations ($/KWh) Destination node Serving bus Optimal retail price Destination node Serving bus Optimal retail price 1 806 0 13 812 0.65 2 848 0 14 888 0.65 4 842 0.65 15 890 0. 65 5 846 0.65 19 834 0.28 10 844 0.44 20 828 0 11 858 0.32 21 824 0.50 PAGE 120 120 Figure 10 5 1. Major components of distribution gird Figure 11 5 2. A typical feeder PAGE 121 121 Figure 12 5 3. The SNR solution procedure PAGE 122 122 Figure 13 5 4. The solution framework PAGE 123 123 Figure 14 5 5. The coupled transportation network and test distribution feeder PAGE 124 124 CHAPTER 6 CONCLUSION This dissertation develops a hierarchical modelling framework to enhance decision making in public PEV charging infrastructure deployment and operations. The proposed framework compr ehensively considers the impact of PEV charging on power transmission and distribution grids, predict s and recharging behaviors in response to the pubic char ging infra structure deployment, and will thus be of help to government agencies for optimizing their investment in deploying pubic charging infrastructure for PEVs. At regional planning level, a game theoretical approach is used to capture the interactions between regional transportation networks and power transmission grids , coupled by PEVs. The interactions lead to an equilibrium where prices of electricity, and traffic and power flow distributions can be determined. Built upon the proposed equilibrium analysis f ramework, we formulate the problem of allocating public charging station budget among different cities as a mathematical program with complementarity constraints. It is observed from the numerical example that the charging load from PEVs has a substantial impact on the operations of the power network and the price of electricity. Note that t he proposed model is based on a critical assumption that availability of public charging stations and prices of electricity will affect the destination choices of PEVs. This assumption needs to be verified by future empirical studies. At urban planning level, considering their recharging needs, we assume that drivers of PEVs simultaneously select paths and decide recharging plans to minimize their trip time while ensu ring not running out of charge. We then define the corresponding net work equilibrium conditions, and propose their mathematical PAGE 125 125 formulation and solution procedure. Based upon the proposed network equilibrium models, we formulate the problem of optimally d etermining the number, locations and types of public charging stations within a budget limit on urban road networks as a bi level model, which is then solved by a genetic algorithm. I t is demonstrated in the numerical examples that optimal charging station deployment plans may not fully use the entire budget limit because locating extra low power stations does not necessarily lead drivers to use them. Note that the proposed network equilibrium model is trip based and overlooks potential connections among di fferent trips conducted by the same traveler. One way to this limitation is to incorporate activity based analyses (e.g., Recker, 1995, 2001; Lam and Yin, 2001; Gan and Recker, 2008; Kang and Recker, 2013), which, however, will impose computational challen ges. Our future study will attempt to find a right balance between the computational tractability and model realism. We further note that the fuel economy of electric vehicles is also affected by random factors such as ambient temperature (Lohse Busch et al., 2013). Moreover, Franke et al. (2012) conducted a field study to analyze the comfortable range for different BEV drivers, which is defined as the lowest remaining battery state of charge that a user would experience comfortable. It is observed that the comfortable range is affected by factors such as the internal control belief, ambiguity tolerance and daily range practice of drivers. In other words, travelers, especially risk averse ones, likely reserve a safety margin to hedge against the variati ons of energy consumption. To model this, we can treat the link energy consumption as a random variable and assume travelers to adopt a percentile energy consumption to judge the usability of each path (see, e.g., Lo et al., 2006 ; Xu et al., 2011; Nie, 201 1). The resulting network equilibrium is left for further PAGE 126 126 investigation. Lastly, in the numerical examples, we only solve the network equilibrium problem with flow dependent energy consumption on a small network. The computational difficulty lies in solvi ng the NCP or VI formulation efficiently at each iteration. Our future study will investigate the possibility of applying new techniques, such as linear approximation (see, e.g., Lawphongpanich et al., 2014), to further enhance computational efficiency. A ssuming a significant number of public charging stations have been located in urban areas, we have inv estigated the integrated design of the electricity prices at public charging stations and road pricing to minimize real power losses and total travel time in the coupled transportation and power distribution networks. Our numerical examples have demonstrated that the integrated pricing of electricity and roads has potential for better man aging and operating the coupled networks. The proposed models are stat ic. It is necessary and useful to extend the models to consider dynamic travel demands and electricity loads, and capture the changes of charging behaviors of PEVs in response to time varying prices of electricity. Another extension is to enhance the accur acy of estimating real power losses in distribution grids by considering components such as voltage regulators and transformers, which, however, will inevitably complicate the power flow analysis and may result in the utilization of software packages and p latforms such as WindMil (Milsoft Integrated Solutions, 2005) and OpenDSS (Dugan, 2010). Once again, the challenge will then be to balance computable tractability and modeling realism. PAGE 127 127 LIST OF REFERENCES Adler, J., Mirchandani, P.B., Xue, G., Xia, M., 2013. The electric vehicle shortest walk problem. Compendium of Papers DVD of TRB 92nd Annual Meeting , Transportation Research Board. Andrews, M. et al., 2012. Modeling and optimization for electric vehicle charging. 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PAGE 137 137 BIOGRAPHICAL SKETCH Fang He received his Bachelor of Science degree in civil engineering from the Tsinghua University in 2010 in China. After that, Fang started to pursue his Ph.D. degree in the Department of Civil and Coastal Engineering at the University of Florida under the supervision of Dr. Yafeng Yin. Fang is transportation systems modeling and optimization. During his Ph.D. study, Fang received interdisciplinary academic training in transpor tation engineering, industrial and systems engineering and economics, co authored eight papers and made nine presentations at various conferences. |