CONTINUOUS DYNAMIC NETWORK LOADING MODELS 1

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1 CONTINUOUS DYNAMIC NETWORK LOADING MODELS 1 Ricrdo Grcí*, Mª Luz López*, Alejndro Niño**, nd Doroeo Versegui* * Deprmeno de Memáics. Universidd de Csill-L Mnch ** Progrm de Invesigción en Tránsio y Trnspore. Universidd Ncionl de Colombi e-mil: Ricrdo.Grci@uclm.es; MriLuz.Lopez@uclm.es; prnsp@ing.unl.edu.co; Doroeo.Versegui@uclm.es 1 INTRODUCTION The dynmic nework equilibrium models hve renewed ineres in Europe, Jpn nd Norh Americ. This is due o he design nd implemenion of Advnced Trnspor Telemic (ATT) sysems require of nework models which my evlue dynmiclly such cions. The min modeling pproches for he dynmic nework equilibrium re: Aggreged mcroscopic models in which rffic flows re regrded in n ggreged wy s fluid nd microscopic pproches in which i is ried o undersnd he behvior of he sysem by modeling he individul vehicles. See Brceló (1998) for overview of some currenly used microscopic rffic simulors. Dgnzo (1995) considers h bsic heory of mcroscopic rffic models for neworks in he conex of dynmic ssignmen would hve o include he les relisic models of: 1. Trffic behvior when he vehicle phs re known. This sub model is known s he dynmic nework loding problem (DNLP) nd consiss of finding emporl rc volumes, rc rvel imes nd ph rvel imes given ime-dependen ph flow res for given ime period. 2. Ph choice when he ime-vrying rc imes re known. 3. Equilibrium o reconcile he predicions iems 1 nd 2. The mcroscopic models for DNLP cn be clssified ccording o severl crierion (see Wu (1998)). In his pper we consider wo cegories defined by wheher rc exi funcions re used or no: 1. Trvel ime funcion models (Asri (1996), Rn nd Boyce (1996), Wu (1998), Xu (1999)). These pproches ssume h he rvel ime from he beginning o he end of n rc of he nework cn be expressed s n incresing funcion of he flow on 1 Reserch suppored under Spnish CICYT projec TRA C2-1/2

2 he rc he ime. These models cn be viewed s n exension of he sic ssignmen model. 2. Kinemic wve heory (Dgnzo (1995), Lebcque nd Khoshyrn (1998)). These models represen he rffic phenomen by pril differenil equions, where he behvior of he rffic given poin in ime-spce is solely ffeced by he se of he sysem in neighborhood of h poin. These pproches rely on he discreizion of heir bsic equions. Any CNDLP mus ddressed hree bsic quesions: () performnce link model, (b) inersecion modeling nd (c) pril flows modeling. The objecive of his pper is o compre compuionlly nd heoreiclly he iem () on he cell rnsmission model of Dgnzo (1994) nd n exension of he rvel ime funcion of Xu (1999). 2 MODEL A: A CONTINUOUS TIME LINK MODEL BASED ON TRAVEL TIME FUNCTION A firs model for he CDNLP is obined from he sic ssignmen models where he rffic volume on link is ime dependen nd he rvel link rversl ime of user rriving link insn is solely deermined by he (ol) rffic volume sill on link he insn. The exi ime he hed of rc of user who eners he il of he rc ime, is equl o τ ()= + s ( v () ) for ll [,T ] where T denoes he les rrivl insn he il node of rc, s (v ()) is rversl ime of link insn nd v () is he rffic volume on link insn. This model considers h he rc dynmics on link obey he equion: where,t v ()dy e ()dy y for ll,t ()= b y [ ] is he sysem period; () nd b () is he rrivl flow re ino link insn,t [ ] e is he exi flow re ino link insn [,T], [ ]. Our formulion ssumes h he model is well posed defining solely he rvel cos funcion nd he enry flow re ino he rc. The resuling behvior does no necessrily sisfies he FIFO condiion. Consider he following uxiliry funcion o give full definiion of his model which indices if user, who enered he insn s, hs rrived he link hed he insn. if τ s () δ = 1 oherwise The ol moun of users h hve gone ou of rc he insn sisfies: e ()= b y ( )dy () δ, y

3 In his work he model rvel ime funcion includes explicily he cpciy consrin on he link s flows nd he rffic ligh conrol sysem. The model kes ino ccoun he consrin v () u for ll where u is he upper bound on he flow on he link. This consrin imposes when i is cive n over surion dely he begin of he rc. A funcion Γ is inroduced in order o ckle he rffic signls. The ol ime of rversing n rc he insn is given by formul: Γ ( () ) τ 3 MODEL B: THE LWR MODEL FOR ONE LINK The firs order fluid pproximion of rffic flow dynmics proposed by Lighhill nd Whihm (1955) nd Richrds (1956) (he LWR model), provides descripion of rffic behvior for link using hree vribles h vry in ime nd spce: flow, q, densiy, k nd speed υ. All rod link wihou enrnces or exis, sisfies he following conservion principles x 2 k ( x,)dx + q ( x 2,) q ( x 1,)= x 1 for ny locions x 1 nd x 2 of he link nd for ny ime. Under he ssumpion of k is differenible, he previous equion cn be expressed s he pril differenil equion k ( x,) + q ( x,) = x The bsic LWR model ssumes he relion beween flow nd densiy observed under sedy se condiion for homogenous rod, where Q is differenible nonnegive, funcion h is zero for k = nd k =kj. q ( x,)= Q ( k ( x,) ) The hird vrible, υ is, by definiion, υ ( x,)= q ( x,) k ( x,), hen he relionship beween speed nd densiy s υ ( k )= Q ( k ) k. Model A do no ble o gurnee he respec o FIFO rule. Codin nd Brceló (1996) show h he LWR model nd firs order implici regressive finie difference pproximion hold FIFO properies. Dgnzo (1994) inroduces numericl procedure, he cell-rnsmission model, in order o pproxime he LWR equions. This mehod converges o he proper soluion under he generl condiions h rise in rffic problems nd i hs been used in his work.

4 4 NUMERICAL RESULTS From heoreicl poin of view, i would be in esing o find weker condiions on he rversl ime funcions h ensure he FIFO condiion (Xu (1999)). We hve illusred h i is impossible for simple cse. 4.1 Comprison of models The firs compuionlly es h i is developed rises he comprison of he models described before, subjec o generl condiions h llow o mke hem comprble. This numericl experimen hs been crried ou using b ()= 4 e 1 nd he following models: A. I esblishes he ime of rip hrough he rc is given by rversl ime funcion. The considered funcion is of he ype s ( v )= L υ + β ( v ) α, where L is he lengh of he rod link, υ is he free-flow speed, v is he number of vehicles in he rc, nd α nd β hey re prmeers of he funcion. A'. I consiues he previous model bu king ino ccouns he cpciy consrin. B. I is he LWR model nd i is solved using he cells rnsmission lgorihm rised by Dgnzo (1995), considering rpezoidl relion beween he inensiy of flow (q) nd he densiy (k), wih which he prmeers o consider re he free flow speed (υ ), he mximum flow inensiy (q mx ), he speed of congesion disurbnce propgion (w) nd he possible mximum densiy or (jm densiy) in he rc (k mx ). The vlue of w is seled like 25 kilomeers per hour. B'. Bsed on model B, bu insed of using rpezoidl q-k relionship, considers funcionl form derived from he model A. The obined resuls re shown in Figure 1. I is observed he loss of FIFO propery in he model A once sops he rrivl of vehicles o he rc nd he drined one begins, siuion h does no ke plce in he oher models. Figure 1. Tes 1

5 I is worh he rouble o mke noice h he models A' nd B' hve very similr performnce hroughou he simulion, since in boh occsions he cpciy is resriced nd he rvel speed is given by n equivlen funcion. Finlly i is possible o wrie down h he difference beween models B nd B' kes plce he momen which model B reches he siuion reproduced by he model A' siuion in which he simulion of model B' kes cre of he vriions of he rvel speed wheres model B sys simuling he sme iniil speed. As conclusion, model A requires he cpciy consrin for congesed neworks in order o pllie he loss of FIFO condiion nd for obining ccure predicions of rvel ime. 4.2 Empiric es This es looks for o compre he rel behvior of wy secion wih he esimions mde by ech one of he sudied models. I ws developed on he d of rffic sudy mde by he Trnsi nd Trnspor Reserch Progrm (PIT) of he Universidd Ncionl de Colombi in Bogoá (Colombi), sudy where ow groups disnced 3.96 kilomeers, regisered he mriculions of he vehicles were observed h circuled round he Luis Crlos Glán Avenue. I is imporn o menion h he Luis Crlos Glán Avenue is 1 meers wide chnnel of rod, wih wo-wy senses of circulion, h is loced o he Wes of Bogoá nd is chrcerized by non-congesed flow condiions. The obined informion corresponds o wo dys (Thursdy, Februry 11 of 1999 nd Mondy, Februry 15 of 1999) in he Es - Souh circulion sense (leving he ciy) nd is consolided in 15 minues inervls during period of wo hours (1: o 12:), in where i ws reled he inervl of nlysis, he number of coinciden vehicles (vehicles observed in boh guging poins) nd he verge ime used by hese o cross he 3.96 kilomeers. Addiionlly he percenge h represened he volume observed in poins respec o he ol of observed vehicles. Figure 2. Tes 2

6 The funcion h ppers in he Figure 2 represen he min simulion resul, including he esimions done by he models A nd B wih he rel performnce of he rc. Becuse he Luis Crlos Glán Avenue, objec of he simulion, is no congesed, he models A' nd B' such produces he sme resuls h he models A nd B, hus re excluded from he represenion. In greemen wih he resuls observed in he grph, in noncongesed condiions, s much he model A s he B suibly represens he performnce of he rffic in rel rc, wih which he goodness of boh models cquires knowledge. REFERENCES Asri, V. (1996). A coninuous ime link model for dynmic nework loding bsed on rvel ime funcion. In 13h Inernionl Symposium on Trnsporion nd Trffic Theory (Lyon, Frnce), pp Brceló, J., L. Ferrer, D. Grcí, M. Florin, nd E. Sux. (1998). Pssenger ssignmen in congesed rnsi neworks: A hisoricl perspecive. In P. Mrcoe nd S. Nguyen (Eds.), Advnced Trnsporion Modeling, pp Msschuses: Kluwer Acdemic Publishers. Csillo, J., P. Pindo, nd F. Beniez. (1993). A formulion for he recion ime of rffic flow models. In Trnsporion nd rffic heory, proceeding of he 12h Inernionl Symposium on Trnsporion nd Trffic Theory (Berkeley, USA), pp New York, Elsevier. Codin, E., nd J. Brceló. (1996). A sysem opiml rffic ssignmen model wih disribued prmeers. In Binco nd P. Toh (Eds.) Advnced Mehods in Trnsporion Anlysis, pp Berlin: Springer-Verlg. Dgnzo, C. F. (1994). The cell rnsmission model I: A dynmic represenion of highwy rffic consisen wih he hydrodynmic heory. Trnsporion Reserch 26B, pp Dgnzo, C. F. (1995). A finie difference pproximion of he kinemics wve model of rffic flow. Trnsporion Reserch 29B, pp Lebcque, J. P., nd M. Khoshyrn. (1998). Firs order mcroscopic rffic flow models for neworks in he conex of dynmic ssignmen. In Proceeding of he 6h EURO Trnsporion Meeing (Gohenburg, Sweden). Lighhill, M. H., nd G. B. Whihm. (1955). On kinemics wves II: A heory of rffic on long crowded rods. Proceeding of Royl Soc. A 229, pp Rn, B., nd D. Boyce. (1996). Modeling dynmic rnsporion neworks. Springer-Verlg. Richrds, R. I. (1956). Shock-wves on he highwy. Opns. Res. 4, pp Wu, J. H., Y. Chen, nd M. Florin. (1998). The coninuous dynmic nework loding problem: mhemicl formulion nd soluion mehod. Trnsporion Reserch 32B, pp Xu, Y. W., J. H. Wu, M. Florin, P. Mrcoe, nd D. L. Zhu. (1999). Advnces in he coninuous dynmic nework loding problem. Trnsporion Science 33, pp