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》由会员分享,可在线阅读,更多相关《美赛数学建模比赛论文模板.docx(12页珍藏版)》请在冰豆网上搜索。
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,proportionoflargevehiclesandav