美赛数学建模文章交通建模方面.docx
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美赛数学建模文章交通建模方面.docx
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美赛数学建模文章交通建模方面
Summary
Withtherapidlydevelopingoftraffic,freewaygraduallybecomesthemainstreamwayofshort-distancetravel.Inordertomakethemeansoftransportationbecomemoreperfect,weneedtoimproveinasmanyaspectsaspossible.Tomeasuretheperformanceofafreeway,wemustconsiderthefollowingtwofactors:
trafficflowandsafety.Thesearethemainaspectsthatwemusttakeintoconsiderationtoweighwhetherafreewayisgoodorbad.
Inordertobettersimulatetheactualsituation,weestablishedasimulationmodel.WeadoptedthecoreideasoftheCellularAutomataModel,onwhosebasis,weestablishedanewmodelsuitabletothesimulationoftheperformanceonfreeway.Thekeypointofourmodelisregardingtimeandspacetobediscretewhichisactuallycontinuous.Everyvehiclemustbeincertaindiscreteposition.Inthisproblem,wedividetheroadintomanysame-sizerectangulargrids,thevehiclemustmoveinafixedplace.henumberofgridsstandsforthedistance,thenumberofthegridsthatavehiclemoveperunittimestandsforit’sspeed.Accordingtodifferentrules,differentsmallmodelsarerespectivelyestablishedtostudywhichruleisbetter.InawordthemodelwedesignedhascombinedtheadvantagesoftheCellularAutomataModelandthemostimportantaspectsoftheactualsituationonthehighway.
Tostudytheperformancemoreaccurately,Wehavestudiedunderthefollowingthreeconditions:
1.underverylighttrafficload;
2.underamediumtrafficload(normaltrafficconditions)(mainpart);
3.underaveryheavytrafficload.
Ineachcase,wehaveanalyzedtheperformanceonfreewayanddiscussedthetrafficflowbothintheoryandbysimulation.Wehavealsocalculatedhowthedriversonfreewayguaranteetheirsafetyquantitatively.Afterthat,weexaminedtradeoffsbetweentrafficflowandsafety,andanalyzedthehoweachcaselimitthespeedandovertakingratio.Throughanalysis,wehavegotrelativelyreasonableconclusions.
Differently,incase1,wegaveanactualexampletotestourmodel.Incase2,werespectivelyanalyzedthefollowingandpassingphenomenonindetail.
Safetyonfreewayissoimportantthatwehavestudiedhowmuchthetrafficflowandspeedinfluenceit,wehavecalculatedthetwosafetycorrelationcoefficientsofthetrafficflowandspeedandconcludethatspeedinfluencesafetymost.
Wehavemadecomprehensiveevaluationof“therulethatrequiresdriverstodriveintheright-mostlaneunlesstheyarepassinganothervehicle,inwhichcasetheymoveonelanetotheleft,pass,andreturntotheirformertravellane”,anddesignedanewrulethat“twolaneusedequally”topromotegreatertrafficflowwhileguaranteeingsafe.Thenewrulehasbeentestedbysimulation.
Incountrieswheredrivingvehiclesontheleftisthenorm,wehaveanalyzedtheirperformanceonfreeway,wefoundthatmysolutioncannotbecarriedoverwithasimplechangeoforientation,additionalrequirementsthatthepositionofcabbechangedshouldbeneeded.
Ifvehicletransportationonthesameroadwaywasfullyunderthecontrolofanintelligentsystem,themostobviouschangeisthechangeofovertakingratio(becomesalmost100%),thischangewilldecreasetrafficflowinourearlieranalysis.
Contents
Assumptionandit’sRationality5
1.Model5
1.1Basicmodel5
1.2Feasibilityandrationalityofthemodel6
1.3Howwesettheparametersinthemodel6
1.4Simulationofdifferentsituationthatweuseinthearticle…6
1.4.1Rulesofthesingle-laneCellularAutomatamodel6
1.4.2Thelanechangingmodel7
1.4.2.1Thelanechangingrules7
1.4.2.2Explanationsofthelanechangingrules7
1.4.3LaneChangingModelverification8
2.Differenttrafficdensity8
2.1Underverylighttrafficload9
2.1.1Trafficflowcalculationandsimulation9
2.1.2Safetyguarantee10
2.1.3Speedlimit11
2.1.4Overtakingratiolimit11
2.1.5Anactualexample11
2.2Mediumtrafficload(Normaltrafficconditions)12
2.2.1Threefactorsinfluencingontrafficflowandsimulation12
2.2.2Safetyguarantee16
2.2.2.1Followingphenomenon16
2.2.2.2Overtakingphenomenon17
2.2.3Speedlimit18
2.2.4Overtakingratiolimit18
2.3Veryheavytrafficload18
2.3.1Safetyfactorsanalysis18
2.3.2Influenceontrafficflowandsimulation19
2.3.3Speedlimit(verylowspeed)19
2.3.4Overtakingratiolimit19
2.4.Safetycorrelationcoefficient19
3.Abetterrule20
3.1Description20
3.2Simulation21
4.Forcountriesdrivingontheleft21
5.Intelligentcontrolledtransportationsystem22
Futheranalysisofourmodel22
Conclusion23
Reference24
Assumptionandit’sRationality
Allthelengthisdispersive.Ourmodeldescribesthemovementofeachindividualvehicleaccordingtothestudyfortheirinteractionbytakingvehiclesasdispersiveparticles.CellularAutomatamodeldividesasectionofroadintomanycellsof2metersinlength.
Thetimeintervalisonesecond.Aslengthisdispersive,thetimeisdispersive.Wemaketheminimumtimeintervalisonesecond.Onesecondisshortenoughtodescribethemotionofthecar.
Thelengthofthecaristhesame.Insomefreewaythebigtruckisprohibited.Thenumberofsmallcarsdominates.Wejuststudythesituationthatsmallcardrivingonthefreeway.Thelengthofthecaris4metersorso.Wetakeitasaaverage2celllength.
Thenumberofthecarsonaselectedsectionoftheroadisaconstant.Weonlystudyasectionoftheroad.Wedesigneditasaclosedloopwhichmeansonecargetsoutandonecarenters.Sothenumberofthecarsonaselectedsectionoftheroadisaconstant.Inthisway,thedensityofthecarsontheroadisaconstant.
Weignorethefactorofweatherandseason.Differentweathermayleadtodifferenttraffic.Thesituationiscomplexthatwehavetoignorethesefactors.
Thesteeringwheelisontherightsideofthecar.Itisacommonthatncountrieswheredrivingvehiclesontheleftisthenormthesteeringwheelisontherightsideofthecar.ItisalsoafactinUSandChina.
Passingisnotallowedtosingleroad.Atthesametimeacellcanbeoccupiedbyonlyonecar.Sothecarcannotpassanothercarinfrontonthesameroad.
Analysisoftheproblem
1.Model
1.1Basicmodel
Thekeypointofourmodelisregardingtimeandspaceasdiscretewhichisactuallycontinuous.Everyvehiclemustbeincertaindiscreteposition.Inthisproblem,wedividetheroadintomanyofthesamesizerectangulargrids,thevehiclemustmoveinafixedplace.
Thenumberofgridsstandsforthedistance,thenumberofthegridsthatavehiclemoveperunittimestandsforit’sspeed.
1.2Feasibilityandrationalityofthemodel
Whenweanalyzetheproblem,thedistanceweconsiderislongenough,andtimeisalsolongenough,dividingtimeandspaceintomanysmallpartswillnotinfluencetheresultsofanalysisandsimulationsomuch.Onthecontrary,thewaywemaketimeandspacediscretewillsimplifytheanalysisandcalculationprocesstoagreatextent,itcanalsomakesimulationmuchmoreeasy.
1.3Howwesettheparametersinthemodel
Consideringthevariousaspectsoffactors,theBasicparameterdefinitenessisasfollows:
Inthismodel,thelengthofeachcellis2meters,per2successivecellscontainonevehicleandthese2successivecellsareinthesamestateatmomentt,i.e.thespeedofvehiclecontained.Maximumspeedofvehicleis120km/h(33m/s).Minimumspeedofvehicleis80km/h(22m/s).Thusinthismodel,maximumspeed(vm)is16celllength/second,minimumspeed(vmin)is11celllength/second.Speedvaluerangeisvmin~vmandrenewaltimeintervalis1second.
1.4Simulationofdifferentsituationthatweuseinthearticle
1.4.1Rulesofthesingle-laneCellularAutomatamodel
VariablesymbolsusedinthisModelaredefinedasfollows.
xn(t):
thepositionofthevehicleatmomentt;
vn(t):
thespeedofthevehicleatmomentt;
an(t):
theaccelerationofthevehicleatmomentt~t+1;
gn(t):
thenumberoffreesitesaheadofthevehicle,i.e.
gn(t)=xn-1(t)-xn(t)-2.
Thestatesofallvehiclesonroadconductsynchronousrenewalaccordingtothefollowingrules.
AccelerationRule:
ifvn(t)≤gn(t),thevehiclewillaccelerate.
Ifgn(t)-vn(t)<2,thenan(t)=gn(t)-vn(t).
Ifgn(t)-vn(t)≥2,thenan(t)=2.
Ifvn(t)=vm,thenan(t)=0.
DecelerationRule:
Ifvn(t)>gn(t),thevehiclewilldecelerate.
Ifgn(t)-vn(t)>-2,thenan(t)=gn(t)-vn(t).
Ifgn(t)-vn(t)≤-2,thenan(t)=-2.
CorrectionRule:
Iftheaccelerationofthevehicleisan(t)atmomentt,ontheassumptionthattheforwardvehicleisdeceleratedatmaximumdeceleration,thenatmomentt+1.
Ifvn(t+1)≤gn(t+1),thentheaccelerationofthevehicleisan(t).
Ifvn(t+1)>gn(t+1),thentheaccelerationofthevehicleisan(t)-1,andrecalculatethevn(t+1)andgn(t+1),untilvn(t+1)≤gn(t+1).
Thus,theactualaccelerationofthevehicleisacorrectedvalue.
1.4.2Thelanechangingmodel
Lanechangingistheemphasisanddifficultyofmulti-laneroadtrafficflowsimulation.Alanechangedecisionprocessisassumedtohavethefollowingthreesteps:
productionoflanechangingdesire,feasibilityanalysisonlanechangingactivityandimplementationoflanechangingactivity(Zou,2002).
Basedonthesingle-laneNSmodel,K.Nagelhasputforwardthemulti-lanetrafficsimulationmodel,inwhich,thevehiclesmovingineachlaneshallconformtotheNSruleandsatisfytheLane-changingrules(Nagel,1998/Wagner,1997)whenchanginglanes.Thisarticleputforwardakindoflane-changingmodelthatissuitableforvehiclemovementinorderontheurbanroadsundertheunobstructedcondition,whichisshowntomatchtherealvehi
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