美赛数学建模比赛论文模板.docx

美赛数学建模比赛论文模板.docx

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美赛数学建模比赛论文模板

TheKeep-Right-Except-To-PassRule

Summary

Asforthefirstquestion,itprovidesatrafficruleofkeeprightexcepttopass,requiringustoverifyitseffectiveness.Firstly,wedefineonekindoftrafficruledifferentfromtheruleofthekeeprightinordertosolvetheproblemclearly;then,webuildaCellularautomatonmodelandaNaschmodelbycollectingmassivedata;next,wemakefulluseofthenumericalsimulationaccordingtoseveralinfluencefactorsoftrafficflow;Atlast,bylotsofanalysisofgraphweobtain,weindicateaconclusionasfollow:

whenvehicledensityislowerthan0.15,theruleoflanespeedcontrolismoreeffectiveintermsofthefactorofsafeinthelighttraffic;whenvehicledensityisgreaterthan0.15,sotheruleofkeeprightexceptpassingismoreeffectiveIntheheavytraffic.

Asforthesecondquestion,itrequiresustotestifythatwhethertheconclusionweobtaininthefirstquestionisthesameapplytothekeepleftrule.Firstofall,webuildastochasticmulti-lanetrafficmodel;fromtheviewofthevehicleflowstress,weproposethattheprobabilityofmovingtotherightis0.7andtotheleftotherwisebymakingfulluseoftheBernoulliprocessfromtheviewoftheping-pongeffect,theconclusionisthatthechoiceofthechanginglaneisrandom.Onthewhole,thefundamentalreasonistheformationofthedrivinghabit,sotheconclusioniseffectiveundertheruleofkeepleft.

Asforthethirdquestion,itrequiresustodemonstratetheeffectivenessoftheresultadvisedinthefirstquestionundertheintelligentvehiclecontrolsystem.Firstly,takingthespeedlimitsintoconsideration,webuildamicroscopictrafficsimulatormodelfortrafficsimulationpurposes.Then,weimplementaMETANETmodelforpredictionstatewiththeuseoftheMPCtrafficcontroller.Afterwards,wecertifythatthedynamicspeedcontrolmeasurecanimprovethetrafficflow.

Lastlyneglectingthesafefactor,combiningtheruleofkeeprightwiththeruleofdynamicalspeedcontrolisthebestsolutiontoacceleratethetrafficflowoverall.

Keywords：

CellularautomatonmodelBernoulliprocessMicroscopictrafficsimulatormodelTheMPCtrafficcontrol

Content

1.Introduction

Asisknowntoall,it’sessentialforustodriveautomobiles,thusthedrivingrulesiscrucialimportant.InmanycountrieslikeUSA,China,driversobeytheruleswhichcalled“TheKeep-Right-Except-To-Pass（thatis,whendrivingautomobiles,therulerequiresdriverstodriveintheright-mostunlesstheyarepassinganothervehicle）”.

2.Analysisoftheproblem

Forthefirstquestion,wedecidetousetheCellularautomatontobuildmodels,thenanalyzetheperformanceofthisruleinlightandheavytraffic.Firstly,wemainlyusethevehicledensitytodistinguishthelightandheavytraffic;secondly,weconsiderthetrafficflowandsafeastherepresentvariablewhichdenotesthelightorheavytraffic;thirdly,webuildandanalyzeaCellularautomatonmodel;finally,wejudgetherulethroughtwodifferentdrivingrules,andthendrawconclusions.

3.Assumption

Inordertostreamlineourmodelwehavemadeseveralkeyassumptions

●Thehighwayofdoublerowthreelanesthatwestudycanrepresentmulti-lanefreeways.

●Thedatathatwerefertohascertainrepresentativenessanddescriptive

●Operationconditionofthehighwaynotbeinfluencedbyblizzard oraccidentalfactors

●Ignorethedriver'sownabnormalfactors,suchasdrunkdrivingandfatiguedriving

●Theoperationformofhighwayintelligentsystemthatouranalysiscanreflectintelligentsystem

●Intheintelligentvehiclesystem,theresultofthesamplingdatahashighaccuracy.

4.SymbolDefinition

Thenumberofvehicles

Thetime

5.Models

Byanalyzingtheproblem,wedecidedtoproposeasolutionwithbuildingacellularautomatonmodel.

5.1BuildingoftheCellularautomatonmodel

Thankstoitssimplerulesandconvenienceforcomputersimulation,cellularautomatonmodelhasbeenwidelyusedinthestudyoftrafficflowinrecentyears.

Let

bethepositionofvehicle

attime

bethespeedofvehicle

attime

betherandomslowingdownprobability,andRbetheproportionoftrucksandbuses,thedistancebetweenvehicle

andthefrontvehicleattime

is:

ifthefrontvehicleisasmallvehicle.

ifthefrontvehicleisatruckorbus.

5.1.1Verifytheeffectivenessofthekeeprightexcepttopassrule

Inaddition,accordingtothekeeprightexcepttopassrule,wedefineanewrulecalled:

Controlrulesbasedonlanespeed.Theconcreteexplanationofthenewruleasfollow:

Thereisnospecialpassinglaneunderthisrule.Thespeedofthefirstlane（thefarleftlane）is120–100km/h（including100km/h）；thespeedofthesecondlane（themiddlelane）is100–80km8/h（including80km/h）；thespeedofthethirdlane（thefarrightlane）isbelow80km/h.Thespeedsoflanesdecreasefromlefttoright.

●Lanechangingrulesbasedlanespeedcontrol

Ifvehicleonthehigh-speedlanemeets

thevehiclewillturnintotheadjacentrightlane,andthespeedofthevehicleafterlanechangingremainsunchanged,where

istheminimumspeedofthecorrespondinglane.

●TheapplicationoftheNaschmodelevolution

Let

bethelanechangingprobability（takingintoaccounttheactualsituationthatsomedriverslikedrivinginacertainlane,andwillnottaketheinitiativetochangelanes）,

indicatesthedistancebetweenthevehicleandthenearestfrontvehicle,

indicatesthedistancebetweenthevehicleandthenearestfollowingvehicle.Inthisarticle,weassumethattheminimumsafedistancegapsafeoflanechangingequalstothemaximumspeedofthefollowingvehicleintheadjacentlanes.

●Lanechangingrulesbasedonkeepingrightexcepttopass

Ingeneral,trafficflowgoingthroughapassingzone（Fig.5.1.1）involvesthreeprocesses:

thedivergingprocess（onetrafficflowdivergingintotwoflows）,interactingprocess（interactingbetweenthetwoflows）,andmergingprocess（thetwoflowsmergingintoone）[4].

Fig.5.1.1Controlplanofovertakingprocess

（1）Ifvehicleonthefirstlane（passinglane）meets

and

thevehiclewillturnintothesecondlane,thespeedofthevehicleafterlanechangingremainsunchanged.

5.1.2Numericalsimulationresultsanddiscussion

Inordertofacilitatethesubsequentdiscussions,wedefinethespaceoccupationrateas

where

indicatesthenumberofsmallvehiclesonthedriveway,

indicatesthenumberoftrucksandbusesonthedriveway,andLindicatesthetotallengthoftheroad.Thevehicleflowvolume

isthenumberofvehiclespassingafixedpointperunittime,

where

isthenumberofvehiclesobservedintimeduration

.Theaveragespeed

isthespeedofvehicle

attime

.Takeovertakingratio

astheevaluationindicatorofthesafetyoftrafficflow,whichistheratioofthetotalnumberofovertakingandthenumberofvehiclesobserved.After20,000evolutionsteps,andaveragingthelast2000stepsbasedontime,wehaveobtainedthefollowingexperimentalresults.Inordertoeliminatetheeffectofrandomicity,wetakethesystemicaverageof20samples[5].

●Overtakingratioofdifferentcontrolruleconditions

Becausedifferentcontrolconditionsofroadwillproducedifferentovertakingratio,sowefirstobserverelationshipsamongvehicledensity,proportionoflargevehiclesandovertakingratiounderdifferentcontrolconditions.

（a）Basedonpassinglanecontrol（b）Basedonspeedcontrol

Fig.5.1.3

Fig.5.1.3Relationshipsamongvehicledensity,proportionoflargevehiclesandovertakingratiounderdifferentcontrolconditions.

ItcanbeseenfromFig.5.1.3:

（1）whenthevehicledensityislessthan0.05,theovertakingratiowillcontinuetorisewiththeincreaseofvehicledensity;whenthevehicledensityislargerthan0.05,theovertakingratiowilldecreasewiththeincreaseofvehicledensity;whendensityisgreaterthan0.12,duetothecrowding,itwillbecomedifficulttoovertake,sotheovertakingratioisalmost0.

（2）whentheproportionoflargevehiclesislessthan0.5,theovertakingratiowillrisewiththeincreaseoflargevehicles;whentheproportionoflargevehiclesisabout0.5,theovertakingratiowillreachitspeakvalue;whentheproportionoflargevehiclesislargerthan0.5,theovertakingratiowilldecreasewiththeincreaseoflargevehicles,especiallyunderlane-basedcontrolconditionsthedeclineisveryclear.

●Concreteimpactofunderdifferentcontrolrulesonovertakingratio

Fig.5.1.4

Fig.5.1.4Relationshipsamongvehicledensity,proportionoflargevehiclesandovertakingratiounderdifferentcontrolconditions.（Figuresinleft-handindicatethepassinglanecontrol,figuresinright-handindicatethespeedcontrol.

istheovertakingratioofsmallvehiclesoverlargevehicles,

istheovertakingratioofsmallvehiclesoversmallvehicles,

istheovertakingratiooflargevehiclesoversmallvehicles,

istheovertakingratiooflargevehiclesoverlargevehicles.）.

ItcanbeseenfromFig.5.1.4:

（1）Theovertakingratioofsmallvehiclesoverlargevehiclesunderpassinglanecontrolismuchhigherthanthatunderspeedcontrolcondition,whichisbecause,underpassinglanecontrolcondition,high-speedsmallvehicleshavetosurpasslow-speedlargevehiclesbythepassinglane,whileunderspeedcontrolcondition,smallvehiclesaredesignedtotravelonthehigh-speedlane,thereisnolow-speedvehicleinfront,thusthereisnoneedtoovertake.

●Impactofdifferentcontrolrulesonvehiclespeed

Fig.5.1.5Relationshipsamongvehicledensity,proportionoflargevehiclesanda