美国数学建模C题英语版Word文档格式.docx

美国数学建模C题英语版Word文档格式.docx

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BradFinney

Summary

WeadaptaaLotka-Volterraecologicalcompetitionmodeltodescribethecar（andlighttruck）market.Weassumethatgasoline-poweredinternalcombustionenginevehicles（ICE）,plug-inhybridvehicles（PHEV）,andbattery-electricvehicles（BEVs）performlikeorganismscompetingforasharedbutlimitedresource.Fororganisms,thisresourcemightbeafoodsupply;

inthecarmarket,manufacturerscompeteforconsumerdollars.Theequationsdescribetheratesofchangeofthreedependentvariables,thepopulationsofcarsofeachtype.Themodelparametersdescribegrowthrates,interspeciescompetition,andcarryingcapacities,whichindirectlyrelatetoconsumerpreferences,economicconditions,governmentinfluences,andimprovementsinautomotivetechnologies.VariablesandparametersusedinthemodelarelistedinTable1

Weassumethatintrinsicgrowthratesareconstant,butrefinementstothemodelcoulddescribethemasfunctionsoftime,marketforces,orstochasticvariables.Weassumethatthecarryingcapacitygrowsat1%,consistentwiththehumanpopulationgrowthrateoftheU.S.[TheWorldBankGroup

2011].Welumptogethermodelparameters,withdeterministicvariablesreflectingallaspectsinfluencingconsumerchoice.

Weinvestigatefivescenariosofchangesinconditionsaffectingtheautomarket.Abasescenariousescurrentyearlygrowthratesandcurrentpopulations;

othersinvestigateeffectsofhighoilprices,increasedbatteryperformance,governmentinvestment,andhighelectricityrates.

TheUMAPJournal32

（2）（2011）165–178.cCopyright2011byCOMAP,Inc.Allrightsreserved.

Permissiontomakedigitalorhardcopiesofpartorallofthisworkforpersonalorclassroomuseisgrantedwithoutfeeprovidedthatcopiesarenotmadeordistributedforprofitorcommercialadvantageandthatcopiesbearthisnotice.Abstractingwithcreditispermitted,butcopyrightsforcomponentsofthisworkownedbyothersthanCOMAPmustbehonored.Tocopyotherwise,torepublish,topostonservers,ortoredistributetolistsrequirespriorpermissionfromCOMAP.

Wecomparethepresentvalueoftwocarmodelscurrentlyavailable,toexaminethecompetitivenessofthesecars.Withoutcurrentgovernmentsubsidies,theNissanLeafhaslowerpresentvaluethantheHondaCivicandsoisatadisadvantageagainsttheCivic.TheLeafwouldbecompetitivewithoutsubsidieswithalinearriseingaspricesto$5/gallon,increasedBEVefficiency（inkWh/miledriven）,andhigherresalevalues.

Table1

Symboltable

VariableDescriptionValue/UnitsGrowthModel

G（t）numberofICEvehicles-

E（t）numberofBEVs-

G（t）numberofPHEVs-

rG,rE,rHintrinsicgrowthrates

1.52yr_1

KG,KE,KHmaximumpopulations300⇥106

!

GeffectofICEonBEV0.7

"

GeffectofICEonPHEV1.2

↵EeffectofBEVonICE1.2

EeffectofBEVonPHEV0.8

↵HeffectofPHEVonICE1.2

HeffectofPHEVonBEV1.0

EconomicModel

C0initialcapitalinvestmentUS$

LresalevalueatendofownershipUS$

BtneteconomicbenefitinyeartUS$

Rtrepair/maintenancecostinyeartUS$

nvehicleownershiplifeyr

Inominalinterestrate%

kcompoundingperiodsperyear12

ieffectiveannualinterestrate%

Theprimaryweaknessofourmodelislackofdataforcalibrationandtesting.

AllscenariospredictaneventualshifttoBEVsfromICEs,butthetimingdependsongrowthratesandthevaluesofthecompetitivenessparameters.

WithasignificantinitialinvestmenttoincreasethenumbersofBEVs,theequilibriumpointbetweenICEsandBEVsoccursin2030,comparedtoabasecaseof2035.Ifcoalisexpensiveandbatterypriceshigh,thispointdoesnotoccuruntil2043;

ifoilpricesriserapidly,increasingthecompetitivenesscoefficientsofBEVs,thispointoccursin2028.Whenallfactorscombine–oilpricesrise,batterytechnologyimproves,andelectricityremainsinexpensive—thenBEVs,ICEs,andPHEVswouldbepresentinequalnumbersin2027.

ElectricCars167

Introduction

WeadaptanecologicalcompetitionmodeltodescribethepopulationdynamicsamongICEcars,BEVs,andPHEVs.Weinterpretmodelresultsusingpresent-dayfiguresforownershipcosts,tailpipeandpower-plantemissions,electricalgenerationcapacity,andoilconsumption.Thisanalysisallowsaroughestimateofthegreenhousegas（GHG）emissions,extraelectricitydemandduetoelectricvehicleuse,andspecificmeasuresformanufacturersandgovernmentstospeedelectricvehicleadoption.

ProblemFormulation

BecausemostelectricitygeneratedintheU.S.comesfromcoal,BEVsdonotrepresentawholesalemovementawayfromfossilfuels.Theenvironmentalimpactofcoal-producedelectricityvariesbylocation,dependingoninfrastructureandemissionsregulations.Pricefluctuationsforoilandcoalareunavoidableanddifficulttopredict,varyingbothregionallyandtemporally.Additionally,initialcostsofnewtechnologyarehigh;

butwithincreasedproduction,pricescanbeexpectedtodecrease.Governmentsupportintheearlystagesofdevelopmentofnewtechnologiesiscriticalbutcannotbeundertakenwithoutresearchintotheviabilityandmarketabilityofthetechnology.

Electriccarshavelimitations:

shortrangesandlittleinfrastructureforrecharginginpubliclocations.Bothfactorscanbeexpectedtoimproveasmoreelectriccarsaresold.Chargingtimeisalsoaconsideration;

afullchargetakesapproximatelyfivehours[PerujoandCuiffo2010],andrapidchargingatpublicstationswouldplacehugedemandsontheelectricalgrid.Alternatively,abatteryexchangeprogramwouldallowchargingataslowerrateandoffpeak;

however,higherinitialcapitalwouldberequiredtostore,maintain,andstockpilebatteries.IncentivetobuyBEVswoulddependstronglyonsuchinfrastructurealreadybeinplaceorplanned.

ModelGoals

Wedevelopamodeltoestimatemarketpenetrationofelectriccarsundervaryingconditions,includingthepricesandavailabilityofoilandcoal,investmentininfrastructure,andtherateofdevelopmentofbatterytechnology.

WeusetheLotka-VolterraequationsdescribinginterspeciescompetitiontosimulatethecompetitionbetweenstandardICEs,BEVs,andPHEVs,andweusetheresultingestimatesofmarketsharetocalculatechangesinCO2production,oilconsumption,andelectricitydemand.WealsoperformaseparatebutrelatedeconomicanalysiscomparingownershipcostsoftheNissanLeafandtheHondaCivicSIoveraneight-yearlife.

，

WesolvethesystemviatheRunge-Kutta-Fehlbergmethod,usingcoefficientsderivedbyCash-Karp.Thismethodgivesefficientsolutionswithoutintroducingexcessiveround-offerror[ChapraandCanale2002].

Limiteddatamakereliablecalibrationofthemodelimpossible.Themodelisthereforebestusedforrunningvariousscenariosandcomparingoutcomes.

EconomicEffects

Thecostoffuel（gasolineandelectricity）canberepresentedinthegrowthmodelviathecompetitionparameters.Ifgasolinepricesrise,electriccarsbecomemorecompetitive;

ifthecostofelectricityrises,gasolinecarsbecomemorecompetitive.

Alargerinvestmentinchargingorbatteryswapstations,chargingportsinpublicparkingfacilities,andresearchtoimprovetechnologyshouldresultinagreaterinfluxofBEVs.

Batterytechnologyhasbeenimprovingrapidly.Prices,lifecycles,range,andefficienciesofbatterieswilldictatethestrengthoftheBEVpopulation.

PresentValueModel

TheeconomicviabilityoftheBEVandthePHEVcanbequantifiedintermsofownershipcostoverthelifeofavehicle.Weconvertforecastsofgasolinepricesandelectricityratestocostperyearbasedondrivinganaverageof11,720mi/yr[U.S.EnergyInformationAgency2010].Weaddaverageannualmaintenanceandrepaircosts（Rt）of$500/year[Automotive.com2011].WithdatafortheNissanLeafandtheHondaCivicSIfromCallaway[2011]andPenn[2011],wecomparethecostofpurchasingandoperatingeachcarforeightyears[WillisandFinney2004].Themodelcaneasilybeadjustedforchangesininterestrate,resourcecosts,maintenancecosts,drivingdistance,batterytechnology,andresalevalue:

NetOwnershipCost

where

•NetOwnershipCost（n）isthepresentvalueoftheownershipcostover

ann-yearhorizon,

•C0istheinitialcost,

•Listheistheresalevalueafternyears,

•B（t）andR（t）arethebenefitsandtherepair/maintenancecosts,

•Iisthenominalinflationrate,andtheeffectiveannualinflationrateiis

.

（Thiscalculationdoesnottakeintoaccountthecostofbuyingthecaroncreditandpayingforitoveranumberofyears.）

EffectsontheEnvironment

Thepotentialpowerdemandof100,000electriccarschargingsimultaneouslywouldbe440GW[PerujoandCuiffo2010].Ifchargingcoincidedwithpeakdemand,additionalcapacitywouldbeneeded.At236MWpertypicalcoal-firedplant[U.S.EnergyInformationAgency2011a],these

100,000carswouldrequiretheequivalentof1,865morepowerplants.

EffectsonHumanHealth

Inthemodel,potentialhealthrisksaredeterminedfromtheamountsoftailpipeemissionsandemissionsfromcoal-firedpowerplants.HealthrisksperICEhavebeenestimatedtocost$103pervehicleannuallyfromparticulatematter[Guoetal.2010],byassigningmonetaryvaluestoprematuredeathsandincreasedillnessrates.

Results

Asrepresentedin

（1）–（3）,thebehaviorofG,E,andHismostlycontrolledbythevalueoftheinterspecificcompetitioncoefficients.WematchedmodelparameterstothebestavailabledataforgrowthratesofPHEVs,BEVsandICEs,theirr