Introducing stochastic aspects in macroscopic traffic flow modeling

Introducing stochastic aspects in macroscopic traffic flow modeling Nicolas CHIABAUT LICIT, Laboratoire Ingénierie Circulation Transport (INRETS/ENTPE) Rue Maurice Audin 69518 Vaulx En Velin CEDEX France Abstract The aim of this paper is to present the latest improvements of the wellknown first order macroscopic traffic flow model LWR. For many years the LWR problem was studied as a scalar conservation law and solve with methods coming from hyperbolic differential equation. System s dynamic is macroscopic and resolution is made in Eulerian coordinates. Consequences is that only single pipe modeling is possible.. Moreover, numerical scheme used was non-exact and generated numerical viscosity in solutions. Recently [4] proposes a variational formulation of the LWR model. The author formulates the usual two variables hyperbolic conservation law as an HAMILTON-JACOBI equation. Furthermore, new exact numerical schemes come along with this variational theory (VT) of traffic flow. In the aftermath, [10] added a formulation of the model in Lagrangian coordinates with an exact numerical scheme. The main lacks of the LWR model are now fulfilled and we are going to present these new breakthroughs in the following paper. It will be shown that the LWR model is now multi-classes, multi-pipes and solved microscopically. Complex phenomenon such as capacity drop, relaxation process in lane-changes, impacts of trucks on global flow are captured by the model. 1

1 Introduction Traditionally, traffic flow phenomenon can be represented by both microscopic or macroscopic models. On one hand, microscopic model are based on car-following rules intended to imitate individual behaviors. The studied particle is a single vehicle. Its trajectory x i (t) depends on its speed ẋ i (t) and on the interactions with the motion of its leading vehicle x i 1 (t) and ẋ i 1 (t). x i (t + δt)=f ( x i (t),x i 1 (t),ẋ i (t),ẋ i 1 (t),λ i ) (1) λ i are behavioral parameters without necessarily physical meaning (aggressiveness, experience of the itinerary...). Theses models are difficult to calibrate and validate, and very costly in computing time and memory. In an other hand, the macroscopic model LWR, from the name of the authors LIGHTHILL, WHITHAM and RICHARDS [11, 13], is simple and robust. Based on a scalar conservation law, this model derived from fluid mechanical fields provides a coarse description of traffic flow for a single one-way road using two variables: density k(x, t) and flow q(x, t). x is the space variable and t the time variable. k t + q x = 0 (2) With boundaries conditions and initial conditions this equation can be solved analytically. k and q are known for every t and x in the study set. Traffic is predicted to be stable with transition between stable areas approximated by discontinuous shocks. Those shocks are the solutions of Riemann problems. But as we will see, numerical scheme usually used introduce important numerical viscosity. Moreover, this model have been always formulated in Eulerian coordinates. Individual trajectories can not be build from the solution of the LWR model. Furthermore multi-pipe can not be modeled with the classical LWR statement. However, states of traffic are quite different between two adjacent lanes. This variation have a huge influence in real life: it provokes lane-changing manoeuvres. Thereby local lane-changing phenomenon can not be represented by the LWR model. As we will see later, lane-change have a major impact on traffic conditions specially near entrance and exit. Nevertheless, microscopic models do not seem to us the solution. Actually, dynamic of this approach are based on behavioral parameters impossible to calibrate and validate. For many years, researchers support one of the two approaches and confront each other. Recently, [5] proves that trajectories predicted by a well-known microscopic model match everywhere the results of the LWR model. Controversy is now closed. It seems that every macroscopic model can be computed with microscopic scheme and inversely. In [4] a new formulation of the LWR model is proposed: the variational theory (VT). The author proves that the LWR solution is a set of continuum least-cost 2

paths in space-time. Moreover, this problem can be also formulated in Lagrangian coordinates. As a result of these papers, the multi-pipes multi-classes formulation of LWR is now available. Without adding to much parameters (only two), the lane-changing phenomenon can be represented and this whole complexity (decision of changing lane, relaxation of vehicle...) can be captured by a macroscopic model. This paper is organized as follow: section 2 provides some background on macroscopic traffic flow modeling and the state of the art of LWR model. Section 3 presents recent breakthrough of the VT. Section 4 analyses impacts of lane-changing and inter-lane variability on macroscopic measures of traffic flow. Finally section 5 is devoted to outline latest improvements of the LWR model to capture complex phenomenon. 2 Traffic Flow Modeling The section provides basic background of macroscopic traffic flow model as it was taught for many years. As we said before, the LWR model, also called kinematic waves model is derived from fluid mechanical field. Two macroscopic variables are used to describe traffic conditions on the study road: density k and flow q. Both depend on space x and time t. Assuming a concave flux function which links q and k and called the fundamental diagram Q(k,x,t), conservation of vehicles on a highway without entrances or exits leads to the following first order PDE. k t + Q k k x = 0 (3) Density k and flow q can be physically measured. Empirical relationship is exhibited and lead to a approximated theoretical fundamental relationship Q. After many years of debate, most of the traffic theorists community seems to agree on a concave relationship for every kind of road and on triangular shape on highway. This phenomenological law presents a maximum flow q m below which the highway is said uncongested (free flow mode or fluid situation) and above which it is said to be congested. The maximal density is noted k m and the critical density for which the flow is maximum is k c. The slope of the free flow mode is equal to the maximum speed v m and the slop of the congested part is called the wave speed w. 3

q q m v m w k c k m k Figure 1: Fundamental relationship To be solved the LWR PDE needs concave flux function and weak boundary conditions which leads to stable areas. Transition between those stable states can be approximated by discontinuous shocks which are solution of RIEMANN problems. In general, RIEMANN problem are not trivial but in our case the solution is simple. Consider a intial discontinuity of density at location x 1. { ku si x < x k(0,x)= 1 k d otherwise. (4) The solution depends on k u and k d and the entropy condition, to ensure uniqueness of the solution, leads to: acceleration waves, if k u > k d and with the concavity of the flux function, the discontinuity will be dissipated as the time passes, deceleration waves, if k u < k d, the discontinuity propagates at a speed u given by the RANKINE-HUGONIOT formula: v = Q(k u) Q(k d ) k u k d At the same time, [3] and [8] propose a numerical scheme to solve easily quickly the LWR PDE: the GODUNOV scheme which is based on iterative solutions of RIEMANN problems. Briefly, flow between two cells of the scheme is the minimum between upstream cell demand and downstream cell supply. Demand and supply are immediately calculated with the fundamental relationship. This model called Cell Transmission Model (CTM) or STRADA is really useful and used by traffic managers. With a space step x and a time step t, flow and density are calculated as follow: Q(n x,t t + t)= [ ( ) ( )] min (n 1) x n x,t,ω n x (n + 1) x,t (5) 4

K(n x (n + 1) x,t + t)=k(n x (n + 1) x,t) [ ] t + Q(n x,t t + t) Q((n + 1) x,t t + t) (6) x To avoid numerical viscosity, the COURANT-FRIEDRICH-LEVY (CFL) condition must be equalized: x t v m (7) The main lacks of the model are that there is no exact numerical scheme and trajectories are difficult to calculate from the solution. Moreover, its is a single pipe model. Though lane-changing phenomenon impact strongly the traffic condition on highway. Under those conditions, it becomes essential to propose an exact resolution of the LWR PDE in a Lagrangian coordinates, i.e. coordinates fixed to a given fluid particle and moving with it in space-time. 3 Variational Theory of Traffic Flow In this section we will present the variational principle of traffic flow in Lagrangian and Eulerian coordinates. First, we will explain the variational principle with its numerical scheme in Eulerian coordinates. Then, we will give the Lagrangian coordinates statement. 3.1 Variational principle In order to alleviate difficulties of the shocks in the resolution of the LWR PDE and so the problem of discontinuity in x in densities (RIEMANN problems), we will consider the function N(x, t) of cumulative number of vehicles as proposed in [12]. This integrated variable consists of giving a growing number of every vehicle passing some location x 1 at time t 1 since a reference instant t 0. Formally, function N(x, t) of cumulative flow is the following. t1 N(x 1,t 1 )= q(x 1,s)ds (8) t 0 Assuming that N is continuous in time t T and in space x X partial time and space derivative of N are respectively q and k. q(x,t)= N t (x,t), k(x,t)= N (9) x (x,t). On subsistuting 9 into the fundamental relationship we obtain a partial differential equation in N called a HAMILTON-JACOBI equation where N(x,t) is the unknown function. N t ( N ) = Q x (x,t),x,t 5 (10)

In [4], the author decides to solve this equation by using what we call in traffic flow theory the NEWELL s postulate. In [12], it has been shown that with boundary conditions D (conditions on frontier of X and T ) the value of function N at a point P is: N P = min{b P + W ; W W P P } (11) where P P is the set of all paths from D to P, W is the set of all directed wave paths emanating from D (which mean that their slope is always equal at one of the slope of the fundamental relationship), W is a space-time path, B P is the value of N at the beginning of the path W and W is the cost of the path. The cost of the path depends on slopes but also on the fundamental relationship. DAGANZO proposes the following cost for every path W = x(t) from a point B D to P. ( ) R(ẋ,x,t)=sup Q(x,t,k) k.ẋ (12) k Furthermore, it has been proven that this cost guarantee unicity of the solution of 11. Without going in details, notice that 12 is the physical cost of the path only if ẇ = Q k. Otherwise, the cost is bigger. Function N can be calculated at every point P =(x,t) by determining the least-cost path emanating from D. The LWR model is reduced to a short-path algorithm. Thereof DAGANZO proves that optimal path in (x, t) space are optimal wavepaths without angle. Wave-path have their slope equal to one of the slope of the fundamental diagram, i.e. ẋ = Q k. Without angle means that optimal path are straight line. Thus, it seriously bounds the possible values for N. For example, with a triangular shape relationship, only two slopes are allowed: once positive ẋ = v m and a second negative ẋ = w. To conclude this section, traffic condition, i.e. flow and density, can be determine by choosing the minimum value coming from straight line emanating from set D of boundary condition with slope equal to u 3.2 Numerical resolution { Q k With a triangular shape relationship simple numerical scheme can be used to determine N. Only two path are allowed: one with a positive slope ẋ = v m and one with a negative slope ẋ = w. The costs of these paths are respectively nul and k x x. Time step is still t and space step x. A rectangular scheme in the (x,t) is exact at every point if and even if v x /w N assuming that the discretisation is consistent with the CFL condition, i.e. x/ t = v x. Function N can be calculated iteratively for every point of the scheme. ( ) N(x, t)=min N(x,t t),n(x+,t n t)+k x x (13) In this case, the numerical scheme is exact at every point of the grid. If v x /w / N, error is still bounded. }. 6

3.3 Lagrangian approach We will present briefly the numerical scheme used for the Lagrangian resolution of the LWR model. Recently, [10] proposes to solve the LWR model in Lagrangian coordinates inspired by the variational principle explained in [4]. The LWR solution is X(n, t) the solution of a HAMILTON-JACOBI equation (14). t X = V ( n X,n,t) (14) where V is speed-spacing relationship depending on spacing s, i.e. n X = s = 1/k, n the vehicle number and t the time. v(n,t)= t X is the speed of the vehicle n. The conservation equation associated is the usual LWR hyperbolic equation express in s and v = V (s,n,t) variables: t s(n,t)+ n V (s,n,t)=0 (15) q v q m v m v m u 1 =0 w u 2 =wk m k c k m k s m =1/k m s c =1/k c s Figure 2: (a) Fundamental diagram, (b) speed-spacing relationship. Numerically one can compute the model on a GODUNOV scheme with parameters t and n. The discrete solution becomes for a triangular fundamental diagram Q [10]: ( X(n,t + t)=min X(n,t)+v m t, ) (1 α)x(n,t)+αx(n n,t) w t (16) where: v m is the free speed of vehicle n, α = wk m t/ n where w (respectively v m ) is the wave speed (respectively the free flow speed) of the vehicle n n. 7

As in the usual GODUNOV resolution of the LWR model, a CFL condition is needed to avoid numerical diffusion. n max ( ) s V(s,n,t) t (17) s,n,t 4 Drawbacks of a single pipe modeling For years first and second order macroscopic model have been focused on single pipe modeling. Finding arguments to model a five lane road by a single pipe was not always necessarily simple. Recently, it was shown that complex phenomenon can be captured by the multi-pipe approach. 4.1 Capacity drop Studies of freeway bottleneck have shown that discharge flow near entrances and exits are less than the capacity of the highway the theoretical value. This phenomenon is called capacity drop. Figure 3: Collected measurements (a) just downstream and (b) upstream of a bottleneck. On figure 3, we can observe measurement just downstream and upstream of a bottleneck. The discharge flow is less than the capacity of the highway. For years, LWR model has been enable to reproduce this so-called capacity drop. It leads researcher to second order macroscopic traffic flow model without more success. 8

Recently some hints was given to prove that capacity drop are caused by lanechanging. Indeed, vehicles willing to exit trend to driver slower giving a lively interest to faster vehicles to overtake them. Locally, bottlenecks appear on the weaving section (lanes on the right) reducing the global capacity of the road. It is now an inescapable fact that model must represent with a well accuracy the lanechanging phenomenon. 4.2 Lane aggregation Macroscopic measurements are made by loop detectors. Even if there is often one loop by lane, only one fundamental relationship is estimated for all the lanes. This aggregation is a tremendous source of errors. Indeed, fundamental relationship represents successive equilibrium state of traffic. Lanes can be macroscopically at the same time in different equilibrium states. In this case, aggregation of various states leads to a estimated relationship with no physical meaning. As an example, consider the diagram 4, which shows three lanes in very fluid situation and left lane hardly congested. The global equilibrium state is in congested area even if most of the lanes are in free-flow situation. Figure 4: Estimation of fundamental diagram from different lanes traffic states Thereby, we can say that each lane has its own fundamental relationship. It seems logical that maximum speed on the right lane is smaller than on the left lane. These observations affect parameters of each fundamental diagram. As it is shown on diagram 4, estimating a global relationship is difficult due to different diagram from a lane to another. This is not the case in very congested situation where behavior is almost identical on each lane. We consider that finding a global relationship is hopeless. This implies that our model is able to reproduce multipipes and to simulate lane-changing. 9

4.3 Multiclasses behavior In real life influence of trucks on the global traffic is easily observable. A mixedflux LWR model have been proposed by [1]. Two classes are involving with interactions between them. Complex phenomenon such as lower capacities on highways, deceleration of light vehicles and formation of temporary queues caused by passing are captured by the model. Another multi-pipe approach have been proposed by [7] based on the theory of moving bottleneck. A slower car behave as a bottleneck for the faster flow. In the same time, a lane-change model is added. We will us the above presented model to reproduce multiclasse traffic. 5 Newest improvements of the LWR model In this paper, we will focus on lane-changing to illustrate what the variational principle enables to do. Two latest improvement will be presented: the relaxation phenomenon in lane-changing process and the multiclasses adaptation of the LWR model. 5.1 Relaxation phenomenon The relaxation phenomenon takes place whenever a lane change occurs at a short spacing that falls outside the fundamental diagram. Such non-equilibrium spacing poses problems to car-following rules that can only handle equilibrium ones. Consider the diagram in figure 5, which shows the trajectory of the three vehicles involved in a lane change maneuver. The lane-changing vehicle (c in the figure) cuts-in between the leader (l) and the follower ( f ) vehicles in the target lane at time t 0. At this time the spacing of the follower changes from s f (t0 ) to s f (t 0 + ) as its new leader becomes c, which sees a spacing of s c (t 0 + ) in front of him. When s f (t 0 + ) and/or s c(t 0 + ) are non-equilibrium values [6] proposes a macroscopic model based on the LWR model represent those phenomenon. In [9], it is found that for a given driver there exists a value of ε that reproduces the relaxation process with uncanny accuracy and that the mean e value does not worsen the fit significantly. 10

x (l) (c) (f) s f (t 0 ) s c (t + 0 ) s f (t + 0 ) Relaxation Time t 0 t Figure 5: Relaxation phenomena Macroscopically, by identifying lane-changer at an on-ramp and observing the same vehicle 300 meters downstream, we can conclude that vehicles involved in the relaxation process are outside the FD at the time they changed lane but converge to the FD. Their paths in the (k,q) space can be observed. Furthermore formal paths can be calculated. As shows the figure 6, predicted paths are quite consistent with observed paths for same initial speed and acceleration of the leader. This result is reassuring because it indicates that the model is well defined and produces reasonable predictions. (a) A3 space, m A2 A1 time, s Figure 6: Relaxation phenomenon at a macroscopic scale Microscopically, the model matches the field measurements data very well. We choose to validate by solving the lead-vehicle problem. We compute our model using the initial position of the following car as initial data and the trajectory of the lead vehicle as the boundary conditions. Then, we find the value minimizing SSE between simulated solution and experimental data. Epsilon parameter is optimal for the considered pair of vehicle. 11

400 380 360 (a) ε = 0.2 c l SSE = 0.504 c l (l) (c) 480 460 ε c/l = 0.6 SSE c l = 1,206 (l) (c) Position [m] 340 320 300 280 260 240 220 Position [m] 440 420 400 380 200 0 5 10 15 20 Time [s] 360 0 5 10 15 20 Time [s] Figure 7: LVP solutions for two typical cases Ninety percent of non-equilibrium lane-changes exhibit a RMSE smaller than 4 meter. That is to say experimental trajectories are correctly estimated by the relaxation model we propose. Furthermore, we sum up in table 1 results of simulation with an optimal value and a mean value, and we compared model with relaxation process or not. It suggests us that the assumption of a mean value for epsilon only slightly degrades the fit. Indeed, the difference in RMSE is only one meter. Without relaxation the RMSE is 4 meter which requires us to use a relaxation process in our global model. ε ε optimal ε mean No relaxation RMSE (m) 1,5 2,5 4 Table 1: Results of simulation with an optimale value of ε, a mean value of ε and no relaxation The main result of these paper is that even non-equilibrium situation of lanechanging can be modeled by a first order macroscopic traffic flow model. Moreover, one single parameter seems to capture most of the complexity of the relaxation phenomenon. 12

5.2 Multiclasses LWR rules The latest development of the LWR model is its multiclass rule. As we saw in 3.3, the α parameter is particular to each vehicle n and linked to the FD. Assuming that time-step t and volume-step n stay constant during the simulation and common to all the vehicles, wave speed w and maximal density k m could be distributed. Formally every probability distribution could be used. But we are focusing on normal distribution, Poisson and bimodal processes. Physical meaning of the bimodal distribution is a flow composed only by heavy trucks and light vehicles. Both of the parameters have the same process centered on a mean value. Durée du cycle : 90 Durée du vert : 90 Nbr voies amont : 2 Nbr voies aval : 1 200 0 Espace (m) 200 400 600 800 Vitesse (m/s) Kx (veh/m) 0 100 200 300 400 500 600 12 10 8 6 0 100 200 300 400 500 600 0.2 0.1 0 0 100 200 300 400 500 600 Temps (s) Figure 8: Multiclasses simulation 13

Figure 8 picts indivual trajectories for a normal distribution. A lane reduction leads to the creation of a congestion going backwards. Various speed can be observe in fluid situation where slopes of trajectories are quite different from one to another. Individual maximum densities appear on the picture in congestion. Some vehicles have a bigger headways with their leader. Durée du cycle : 90 Durée du vert : 90 Nbr voies amont : 2 Nbr voies aval : 1 200 100 0 100 200 Espace (m) 300 400 500 600 700 800 900 0 50 100 150 200 250 300 350 400 Temps (s) Figure 9: Simulation with 20% of trucks Figure 9 illustrate a two-classes simulation with 20% rate of heavy trucks. Lane-changing process is not implemented for the moment. The first slow vehicle smooths all the difference in speed, but difference in jam density can be seen in congested state. 6 Discussion The main lacks of the first order macroscopic traffic flow model LWR have been fulfilled in the last years. A microscopic and exact numerical resolution has been identified. Thereof, authors proposed lane-changing processes for the lane-changing decision but also for capturing complex phenomenon such as relaxation, fully consistent with the macroscopic dynamic and approach. With dropping exactness in numerical resolution but keeping a bounded error, multiclasses flow can now be simulated by the LWR model. Finally, it is worth noting that implementing distribution in parameters leads to a replication problem. How to be sure that we made sufficient number of runs to well-estimated our measure of effectiveness? This has been investigated by the authors [2]. 14

References [1] S. CHANUT. Modélisation macroscopique de l écoulement d un trafic routier mixte poids lourds et véhicules légers. PhD thesis, INSA, Lyon, 2005. [2] N. CHIABAUT and Ch. BUISSON. Towards a determination of the minimal number of replications for stochastic traffic models. In Traffic and Granular Flow 2007, Paris, France, 2007. submitted. [3] C. F. DAGANZO. The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory. Transportation Research Part B, 28:269 287, 1994. [4] C. F. DAGANZO. A variational formulation of kinematic waves: basic theory and complex boundary conditions. Transportation Reseatch Part B, 39:187 196, 2005. [5] C. F. DAGANZO. In traffic flow, cellular automata = kinematic waves. Transportation Research Part B, 40:396 403, 2006. [6] J. A. LAVAL and L. LECLERCQ. A microscopic theory of lane-changing. Submitted for publication, 2006. [7] J.A. LAVAL. Hybrid Models of Traffic Flow: Impacts of Bounded Vehicle Accelerations. PhD thesis, University of California, Berkeley, 2004. [8] J-P. LEBACQUE. The godunov scheme and what it means for first order traffic flow models. In Jean-Baptiste Lesort, editor, 13th International Symposium on Transportation and Traffic Theory, pages 647 678, Lyon, France, 1996. Pergamon. [9] L. LECLERCQ, N. CHIABAUT, J. A. LAVAL, and Ch. BUISSON. Relaxation phenomenon after changing lanes: an experimental validation with the ngsim dataset. Transportation Research Record, to be published. [10] L. LECLERCQ, J. LAVAL, and E. CHEVALLIER. The lagrangien coordinates and what it means for first order traffic flow models. In 17th International Symposium on Transportation and Traffic Theory, London, U.K., 2007. accepted for presentation. [11] M. J. LIGHTHILL and G. B. WHITHAM. On kinematic waves ii. a theory of traffic flow on long crowded roads. Proceedings of the Royal Society A, 229:317 345, 1955. [12] G. F. NEWELL. A simplified theory of kinematic waves in highway traffic, part i: general theory. Transportation Research B, 28(4):281 287, 1993. [13] P. I. RICHARDS. Shockwaves on the highway. Operations research, 4:42 51, 1956. 15