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Macroscopic Simulation of Widely Scattered Synchronized Traffic States Martin Treiber1 and Dirk Helbing1,2 1II. Institute of Theoretical Physics, University of Stuttgart, Pfaffenwaldring 57, D-70550 Stuttgart, Germany 2Department of Biological Physics, Eötvös University, Budapest, Puskin u 5–7, H–1088 Hungary (February 1, 2008) 9 9 Recently,aphasetransitiontosynchronizedcongestedtraffichasbeenobservedinempiricalhighway 9 data [B. S. Kerner and H. Rehborn, Phys. Rev. Lett. 79, 4030 (1997)]. This hysteretic transition 1 hasbeendescribedbyanon-local,gas-kinetic-basedtrafficmodel[D.HelbingandM.Treiber,Phys. n Rev. Lett. 81, 3042 (1998)] that, however, did not display the wide scattering of synchronized a states. Here, it is shown that the latter can be reproduced by a mixture of different vehicle types J like cars and trucks. The simulation results are in good agreement with Dutch highway data. 3 1 05.70.Fh,05.60.+w,47.55.-t,89.40.+k ] Recent publications stressed the fact that, whereas in are extracted from real traffic data by determining the h free traffic the observed flow-density diagram is well de- proportion of long vehicles (”trucks”). This method al- c e scribed by a unique one-dimensional flow-density rela- lows to simulate a uni-directional multi-lane freeway by m tion, in congested traffic the empirical data points are an effective one-lane model for one car species with av- ratherdistributedoveratwo-dimensionalregion[1,2]. In erage, but varying parameter values. Our simulations - t ordertoaccountforthisfact,Krauß[3]hasrecentlypro- are carried out with the empirically measured boundary a t posed that the driver behavior is changed in congested conditions, and the results are quite realistic. s traffic compared to free traffic. Instead of this, Lenz et For the simulations, we use the macroscopic, gas- . at al.[4]andSchreckenberg[5]havesuggestedthatthewide kinetic-based traffic model (GKT model) [6,7], which m scattering of data is caused by an anticipation effect of showsa realisticinstability diagramand the characteris- drivers who not only react to the respective vehicle in ticpropertiesoftrafficflows[7]demandedbyKernerand - d front but also to the traffic dynamics further ahead. Rehborn[1]. More importantly, this model is able to de- n In contrast to these ”microscopic” approaches which scribe the hysteretic phase transitionto congestedstates o simulate the interactions of individual vehicles, macro- with high traffic flow [8] (called ”synchronized traffic” c scopic models describe the evolution of the macroscopic [1]) which typically occurs behind on-ramps, gradients, [ velocity V(x,t) = v and the vehicle density ρ(x,t) = or other bottlenecks of busy freeways [6]. α 1 1/s , whicharelhocailaveragesofthe ”microscopic”ve- According to the gas-kinetic-based model, the evolu- α v h i locities v of the vehicles α and their center-to-center tion of the vehicle density ρ(x,t) in dependence of the 9 α distancess . Allfluctuatingquantitieslikeindividualve- time t and the position x along the freeway is given by 1 α locityvariationsordistance distributionsareeliminated. the continuity equation 1 1 This means that, in deterministic macro-simulations, all ∂ρ ∂(ρV) Q 0 self-organized structures (like stop-and-go waves or con- + = rmp . (1) 9 gested traffic) are smooth. ∂t ∂x nL 9 Therefore, some researchers believe that, while the Here,V(x,t)denotestheaveragevelocityofthevehicles. / t wide scattering of the congested flow-density data may At on-ramps (or off-ramps), the source term Qrmp/(nL) a m be reproducedby microsopic trafficmodels, macroscopic is given by the actually observed inflow Qrmp > 0 from ones will fail for principal reasons. However, motivated (oroutflowQrmp <0to)the ramp,dividedbythe merg- - by the circumstance that the scattering has been ob- ing length L and by the number n of lanes. Otherwise d n served in aggregated rather than single-vehicle data, we it is zero, reflecting the conservation of the number of o areconfidentthata macroscopicsimulationofthis effect vehicles. The averagevelocity obeys the equation c should be possible. ∂V ∂V 1∂(ρθ) 1 : v Inthefollowing,wewillshowthatthescatteringcanbe + V = + (U V) . (2) ∂t ∂x −ρ ∂x τ − i explained by the fluctuations caused by a heterogeneous X traffic population, which enter the macroscopic simula- TransportTerm PressureTerm RelaxationTerm r tions via the boundary conditions. We will distinguish a According t|o{tzhi}s, the|tem{pzora}l cha|nge{ozf th}e average cars and trucks characterizedby different sets of param- velocityisgivenbyatransportterm(causedbyapropa- eter values. These define two equilibrium flow-density gationofthevelocityprofilewithV),aso-calledpressure relations of pure car traffic and pure truck traffic, re- term (that reflects dispersion effects due to the finite ve- spectively, which are close to each other at small vehicle locity variance θ of the vehicles), and a relaxation term densities,butconsiderablydifferentinthecongestedden- (describing the adaptation to a dynamic equilibrium ve- sity regime. For mixed traffic, we interpolate between locity U with a certain relaxation time τ). In our gas- both parameter sets and, hence, between both equilib- kinetic-basedmodel, the analyticallyderivedformula for rium relations, using a weighted average. The weights the dynamical equilibrium velocity is 1 Treiber/Helbing: Macroscopic Simulation of Widely Scattered Synchronized Traffic States 2 θ+θ′ ρ′T 2 sourcesoffluctuations than observedones. A reasonable U =V 1 B(δ ) , (3) 0" − 2A(ρmax)(cid:18)1−ρ′/ρmax(cid:19) V # abgyredeismtienngtuwishitihngemtwpoirviceahlicdleastatycpaens aolnrleya,dsyhobret vreeahcichleeds (”cars”)andlongones(”trucks”,withalengthofatleast where V is the desired (maximum) velocity, T the aver- 0 7m). Eachtypeischaracterizedbyitsownparamterset. age time headway at large densities, and ρ the max- max ForthecarsweassumeadesiredvelocityV =112km/h, imum vehicle density. A prime indicates that the corre- 0 anaveragetimeheadwayT =1.0satlargedensities,and sponding variable is taken at the advanced ”interaction point”x′ =x+γ(1/ρ +TV)ratherthanattheactual a maximum density ρmax =110vehicles/km. Trucks are max described by the parameters V =90km/h, T =5s, and position x. This accounts for the anisotropic anticipa- 0 ρ = 100vehicles/km. The remaining model parame- tion behavior of drivers. The monotonically increasing max ters are the same for both types: τ = 25s and γ = 1.6. ”Boltzmann factor” The parameters in the constitutive relation (5) for the e−δV2/2 δV e−y2/2 variance have also been chosen identical. B(δV)=2"δV √2π +(1+δV2) −∞dy √2π # (4) Accordingtothephilosophyofmacroscopicmodels,we Z nowdefine time-dependent ”effective”modelparameters X(t)asweightedaveragesoftherespectivecarandtruck canbederivedfromgas-kineticformulas[7]anddescribes parameters X and X : the dependence of the braking interaction on the dimen- car truck sionless velocity difference δ = (V V′)/√θ+θ′. Fi- X(t)=p (t)X +[1 p (t)]X . (9) V truck truck truck car − − nally, the dynamics ofthe variance canbe approximated Here, p (t) is the proportion of trucks averaged over by the constitutive relation truck a time interval ∆t around t [Figure 2(a)]. Although the approximations behind the resulting ”effective” macro- ρ(x,t) ρ θ(x,t)= A0+∆Atanh − c V2(x,t), (5) scopic simulationmodel are rather crude, it yields a sur- ∆ρ (cid:20) (cid:18) (cid:19)(cid:21) prisinglygoodagreementwithempiricaldata. Evenbet- ter results are expected for macroscopic models which where the coefficients A = 0.008, ∆A = 0.015, ρ = 0 c explicitly take into account different vehicle types and 0.28ρ , and ∆ρ = 0.1ρ have been obtained from max max lane-changing interactions among the freeway lanes [12]. single-vehicle data [7]. WesimulatedtrafficflowonasectionoftheDutchtwo- Thevelocity-densityrelationresultingforthismodelin lane motorway A9 from Haarlem to Amsterdam (Fig- spatially homogeneous and stationary equilibrium reads ure 1) from the detector cross-section D1 (0km) to D6 (5.7km). For this purpose, the measured single-vehicle V˜2 4V2 V (ρ)= 1+ 1+ 0 (6) data were aggregated to 1-minute averages of the veloc- e 2V0 − s V˜2  ity,trafficflow,andtruckproportion(Figure2). Between 7:30 am (450 min) and 9:30 am (570 min) in the morn-   with ing of November 2, 1994, we find transitions from a low- densityregimetoahigh-densityregimecorrespondingto 1 1 1 A(ρ ) transitions between free and congested traffic. Figure 3 V˜(ρ)= max . (7) illustrates that the congested state at D2 is connected T (cid:18)ρ − ρmax(cid:19)s A(ρ) with a considerable velocity drop, while the flow is decreased only by about 10%, both in the empirical data This also determines the equilibrium traffic flow by and in the simulation. In addition, the congested traffic Q (ρ)=ρV (ρ), (8) state relaxes to free traffic downstream of the on-ramps e e [Figures 4(c) and (d)]. A comparison with Figures 1(b) which, for a given parameter set, is a one-dimensional and 3(c) of Ref. [8] suggests that the congestion in the curve. However, as will be shown in the following, the investigated data corresponds to synchronized traffic. empiricallyobservedtwo-dimensionalregionof”synchro- As inflow and outflow boundary conditions, we used nized” congestedstates canbe reproducedby simulating thedataofthecross-sectionsD1andD6,respectively,as a mixture of different vehicle types. Although it has shown in Figure 2(b). There are two on-ramps and one not been stressed clear enough, it is known from mi- off-rampinthe consideredsection. Forallramps,weuse crosimulations that heterogeneous traffic produces con- the empirical data of the traffic flow Q , divided by rmp siderablefluctuations ofthe aggregatequantities likethe the number n = 2 of lanes [6] [Figure 2(c)], and assume vehicle density and the average velocity [9–11]. Never- a merging length L=200m. theless, we do not need to carry out microsimulations to In the simulation, congested traffic first sets in at account for the two-dimensional scattering of synchro- t 450min near the on-ramp at D3a, which agrees well ≈ nized traffic states. It is sufficient to simulate traffic in with the empirical findings. We started the simulation a macroscopic way with empirically obtained boundary 50 min earlier to eliminate any effects of initial condi- conditions, including the varying proportion of long ve- tions, and to show the spontaneous nature of the tran- hicles (”trucks”). Thus, we do not need to assume other sition [Figure 3(a)]. In Figure 4(b), the free traffic flow Treiber/Helbing: Macroscopic Simulation of Widely Scattered Synchronized Traffic States 3 before the breakdown (t < 450min) and after the re- They may serve as prototype for introducing stochastic- covery (t > 570min) is delineated by the points at the ityintomacroscopicequationsinacontrolledandempir- low-density(left) side ofthe diagram,whichmoreorless ically justified manner. define a one-dimensional curve. In contrast, the congested traffic state is represented by the points at the high-density (right) side, which are distributed over a Acknowledgments two-dimensional region. Some minutes later, the front of the congested state crosses the on-ramp at D2, which Theauthorswanttothankforfinancialsupportbythe causes congested traffic upstream of it. In accordance BMBF (research project SANDY, grant No. 13N7092) with the mechanism of the formation of synchronized and by the DFG (Heisenberg scholarship He 2789/1-1). traffic proposed in [6], the congested state upstream of TheyarealsogratefultoHenkTaaleandtheDutchMin- D2 has a lower flow and a higher density [Figure 4(a)] istryofTransport,PublicWorksandWaterManagement than that between D2 and D3a. The congested states for supplying the freeway data. aresustainedfornearlytwohours,untiltheinflowsfrom both the main road and the on-ramps are considerably decreased,whichshowsthehystereticnatureofthetran- sition. Summarizing our results, one can say that the macroscopic, gas-kinetic-basedtraffic model allows to simulate synchronized traffic, including the associated scattering [1] KernerBSandRehbornH1996Phys.Rev.E53,R4275. of the flow-density data in the congested regime. Sim- [2] KernerBS1998 inProc. Third Int. Symp. Highway Ca- ulations of this model with only one vehicle type [6,13] pacity,Vol.II,edRysgaardR(RoadDirectorate,Copen- suggested that the phenomenon of synchronized traffic hagen, Denmark). as such (i.e., high traffic flows at low velocities) does [3] Krauß S 1998 Microscopic Modelling of Traffic Flow not depend on the existence of different types of vehi- (PhD thesis, DLR,Cologne), FB 98-08. cles. However,asisoftenthecaseforself-organizednon- [4] Lenz H, Wagner C K, and Sollacher R 1998 European chaotic patterns resulting from deterministic dynamics, Physical Journal, in print. [5] SchreckenbergM 1998 To be published. the flow-density diagram is essentially one-dimensional. [6] HelbingDandTreiberM1998Phys.Rev.Lett.81,3042. In this Letter, we showed that a realistic scattering in [7] Treiber M, HenneckeA,and HelbingD 1999 Phys. Rev. theflow-densityplanecanbesimulatedbydistinguishing E 59, in print. several vehicle types with different parameter sets, the [8] Kerner B S and Rehborn H 1997 Phys. Rev. Lett. 49, measuredproportionsofwhicharethe weightsfordeter- 4030. mining the time-dependent “effective” parameter set. A [9] Nagel K, Wolf D E, Wagner P, and Simon P 1998 Phys. reasonable agreement with empirical data from Dutch Rev. E58, 1425. highways is already obtained for two different vehicle [10] HelbingDandSchreckenbergM1998“Cellularautomata types, cars and trucks. Our results also indicate that, simulatingexperimentalpropertiesoftrafficflows”,Phys. when studying dynamical phenomena in empirical traf- Rev., submitted. fic data, itis highly recommendedto thoroughlyanalyze [11] HelbingDandHubermanBA1998“Movinglikeasolid the proportion of trucks, which shows surprisingly large block”, Nature,in print. variations [Figure 2(a)]. [12] Shvetsov V and Helbing D 1998 “Macroscopic dynamics Noticethattheassumedparametervariationsduetoa of multi-lane traffic”, Phys. Rev.E, submitted. changing truck fraction can explain both the relatively [13] Helbing D and Treiber M 1999 “Numerical simulation low scattering of flow-density data in the low-density of macroscopic traffic equations”, Computing in Science regimeandthewidescatteringintheregimeofcongested and Engineering, in print. traffic. Whilethesmallamountofscatteringforfreetraf- fic at low densities is caused mainly by variations of the individual desired velocity (with a standard deviation of about 10% of the mean value), the main reason for the considerable scattering for congested traffic at densities above30vehicles/kmarethevariationsofthetimehead- way (which are of the order of 100%). There are other effects that influence scattering of congestedtraffic (e.g., lane changes), but they will only increase the scattering of the flow-density data. Finally, we mention that the equations of the gas- kinetic-based traffic model together with relation (9) for the stochastic quantities V , T, and ρ , represent e max stochastic partial equations with multiplicative noise. Treiber/Helbing: Macroscopic Simulation of Widely Scattered Synchronized Traffic States 4 120 Rottepolderplein S17 Badhoevedorp -- km/h) 18000 SimEumlaptiiroicna rl edsautlat y ( 60 D1 D2 D3D3aD4 D5 D6 D7 D8 D9 Velocit 2400 (a) FIG. 1. Overview of the evaluated stretch of the Dutch 0 Highway A9from Haarlem to Amsterdam. 420 480 540 Time (min) 3000 2500 (b) Simulation result 0.8 h) Empirical data of trucks 00..46 (a) D2 ow (veh/ 112050000000 on Fl 500 acti 0.2 0 Fr 420 480 540 0 420 480 540 Time (min) Time (min) FIG.3. (a)Velocity,and(b)trafficflowatD2accordingto 3000 themodel,incomparisonwiththeempiricalone-minutedata. Inflow (D1) s/h) Outflow (D6) The breakdown of velocity is a result of a dynamical transi- cle 2000 tion, since neither the initial conditions, nor the boundary hi conditions, ortherampflowsusedin thesimulations contain e w (v 1000 any significant peaks. Flo (b) 0 420 480 540 Time (min) 3000 3000 (a) (b) h) D2 D3 1500 On-Ramp after D2 es/ 2000 2000 hicles/h) 1500000 (c) Off-ORna-mRpa mbepf oart eD D33a w (vehicl 1000 1000 w (ve 0 Flo o Fl 0 0 0 25 50 75 100 0 25 50 75 100 420 480 540 Time (min) 3000 3000 (c) (d) h) D4 D5 FIG. 2. (a) Proportion of trucks, from one-minute av- cles/ 2000 2000 erages. (b) Upstream and downstream boundary conditions hi e for the flow, taken from measured one-minute data at the w (v 1000 1000 cross-sectionsD1andD6,respectively. (c)Flowsofthethree o ramps in the considered section. The off-ramp at detector Fl D1 was left out. It leads only to changes of the traffic situa- 0 0 tion upstreamof thecross-section D1for whichnodatawere 0 25 50 75 100 0 25 50 75 100 available. Density (vehicles/km) Density (vehicles/km) FIG.4. The displayed points in density-flow space corre- spond to empirical 1-minute data (dark crosses) and related simulation results (grey boxes), separately for the cross sec- tions D2, D3, D4, and D5. The simulations manage to re- produce both the quasi-linear flow-density relation at small densities and thescattering overa two-dimensional region at highdensities. Forcomparison,wehavedisplayedtheequilib- rium flow-density relations for traffic consisting of 100% cars (—),and 100% trucks(- - -). Rottepolderplein S17 Badhoevedorp - - D1 D2 D3 D3a D4 D5 D6 D7 D8 D9