Lighthill-Whitham-Richards Research Articles

Real-time queue length estimation for congested signalized intersections

How to estimate queue length in real-time at signalized intersection is a long-standing problem. The problem gets even more difficult when signal links are congested. The traditional input–output approach for queue length estimation can only handle queues that are shorter than the distance between vehicle detector and intersection stop line, because cumulative vehicle count for arrival traffic is not available once the detector is occupied by the queue. In this paper, instead of counting arrival traffic flow in the current signal cycle, we solve the problem of measuring intersection queue length by exploiting the queue discharge process in the immediate past cycle. Using high-resolution “event-based” traffic signal data, and applying Lighthill–Whitham–Richards (LWR) shockwave theory, we are able to identify traffic state changes that distinguish queue discharge flow from upstream arrival traffic. Therefore, our approach can estimate time-dependent queue length even when the signal links are congested with long queues. Variations of the queue length estimation model are also presented when “event-based” data is not available. Our models are evaluated by comparing the estimated maximum queue length with the ground truth data observed from the field. Evaluation results demonstrate that the proposed models can estimate long queues with satisfactory accuracy. Limitations of the proposed model are also discussed in the paper.

Kinetic description of bottleneck effects in traffic flow

This paper deals with the effects of traffic bottlenecks using an extended Lighthill-Whitham-Richards (LWR) model. The solution structure is analytically indicated by the study of the Riemann problem characterized by a discontinuous flux. This leads to a typical solution describing a queue upstream of the bottleneck and its width and height, and informs the design of a δ-mapping algorithm. More significantly, it is found that the kinetic model is able to reproduce stop-and-go waves for a triangular fundamental diagram. Some simulation examples, which are in agreement with the analytical solutions, are given to support these conclusions.

Difference schemes, entropy solutions, and speedup impulse for an inhomogeneous kinematic traffic flow model

The classical Lighthill-Whitham-Richards (LWR) kinematic traffic model is extended to a unidirectional road on which the maximum density $a(x)$ represents road inhomogeneities, such as variable numbers of lanes, and is allowed to vary discontinuously. The car density $\phi = \phi(x,t)$ is then determined by the following initial value problem for a scalar conservation law with a spatially discontinuous flux: $\phi_t + (\phi v(\phi/{a(x)})_x = 0, \quad \phi(x,0)=\phi_0(x),\quad x \in \mathbb{R},\quad t\in (0,T),$ (*) where $v(z)$ is the velocity function. We adapt to (*) a new notion of entropy solutions (Burger, Karlsen, and Towers [Submitted, 2007]), which involves a Kružkov-type entropy inequality based on a specific flux connection $(A,B)$, and which we interpret in terms of traffic flow. This concept is consistent with both the driver's ride impulse and the desire of drivers to speed up. We prove that entropy solutions of type $(A,B)$ are unique. This solution concept also leads to simple, transparent, and unified convergence proofs for numerical schemes. Indeed, we adjust to (*) new variants of the Engquist-Osher (EO) scheme (Burger, Karlsen, and Towers [Submitted, 2007]), and of the Hilliges-Weidlich (HW) scheme analyzed by the authors [ J. Engrg. Math. , to appear]. It is proven that the EO and HW schemes and a related Godunov scheme converge to the unique entropy solution of type $(A,B)$ of (*). For the Godunov version, this is the first rigorous convergence and well-posedness result, since no unnecessarily restrictive regularity assumptions are imposed on the solution. Numerical experiments for first-order schemes and formally second-order MUSCL/Runge-Kutta versions are presented.

Explicit construction of entropy solutions for the Lighthill–Whitham–Richards traffic flow model with a piecewise quadratic flow–density relationship

In this paper we explicitly construct the entropy solutions for the Lighthill–Whitham–Richards (LWR) traffic flow model with a flow–density relationship which is piecewise quadratic, continuous, concave, but not differentiable at the junction points where two quadratic polynomials meet, and with piecewise linear initial condition and piecewise constant boundary conditions. The proposed model is a generalization of the well-known piecewise linear flow–density relationship in the LWR model. As observed traffic flow data can be well fitted with such continuous piecewise quadratic functions, the explicitly constructed solutions provide a fast and accurate solution tool which may be used for predicting traffic or as a diagnosing tool to test the performance of numerical schemes. We implement these explicit entropy solutions for two representative traffic flow cases and also compare them with numerical solutions obtained by a high order weighted essentially non-oscillatory (WENO) scheme.

Efficient implementation of the shock‐fitting algorithm for the Lighthill–Whitham–Richards traffic flow model

AbstractThis paper firstly presents the existence and uniqueness properties of the intersection time between two neighboring shocks or between a shock and a characteristic for the analytical shock‐fitting algorithm that was proposed to solve the Lighthill–Whitham–Richards (LWR) traffic flow model with a linear speed–density relationship in accordance with the monotonicity properties of density variations along a shock, which have greatly improved the robustness of the analytical shock‐fitting algorithm. Then we discuss the efficient evaluation of the measure of effectiveness (MOE) of the analytical shock‐fitting algorithm. We develop explicit expressions to calculate the MOE–which is the total travel time that is incurred by travelers, within the space‐time region that is encompassed by the shocks and/or characteristic lines. A numerical example is used to illustrate the effectiveness and efficiency of the proposed method compared with the numerical solutions that are obtained by a fifth‐order weighted essentially non‐oscillatory scheme. Copyright © 2007 John Wiley & Sons, Ltd.

A continuous whole-link travel time model with occupancy constraint

The continuous dynamic network loading problem (CDNLP) aims to compute link travel times and path travel times on a congested network, given time-dependent path flow rates for a given time period. A crucial element of CDNLP is a model of the link performance. Two main modeling frameworks have been used in link loading models: The so-called whole-link travel time (WTT) models and the kinematic wave model of Lighthill–Whitham–Richards (LWR) for traffic flow. In this paper, we reformulate a well-known whole-link model in which the link travel time, for traffic entering a time t, is a function of the number of vehicles on link. This formulation does not require the satisfying of the FIFO (first in, first out) condition. An extension of the basic WTT model is proposed in order to take explicitly into account the maximum number of vehicles that the link can accommodate (occupancy constraint). A solution scheme for the proposed WTT model is derived. Several numerical examples are given to illustrate that the FIFO condition is not respected for the WTT model and to compare the travel time predictions effected by LWR and WTT models.

Adjoint-based control of a new eulerian network model of air traffic flow

An Eulerian network model for air traffic flow in the National Airspace System is developed and used to design flow control schemes which could be used by Air Traffic Controllers to optimize traffic flow. The model relies on a modified version of the Lighthill-Whitham-Richards (LWR) partial differential equation (PDE), which contains a velocity control term inside the divergence operator. This PDE can be related to aircraft count, which is a key metric in air traffic control. An analytical solution to the LWR PDE is constructed for a benchmark problem, to assess the gridsize required to compute a numerical solution at a prescribed accuracy. The Jameson-Schmidt-Turkel (JST) scheme is selected among other numerical schemes to perform simulations, and evidence of numerical convergence is assessed against this analytical solution. Linear numerical schemes are discarded because of their poor performance. The model is validated against actual air traffic data (ETMS data), by showing that the Eulerian description enables good aircraft count predictions, provided a good choice of numerical parameters is made. This model is then embedded as the key constraint in an optimization problem, that of maximizing the throughput at a destination airport while maintaining aircraft density below a legal threshold in a set of sectors of the airspace. The optimization problem is solved by constructing the adjoint problem of the linearized network control problem, which provides an explicit formula for the gradient. Constraints are enforced using a logarithmic barrier. Simulations of actual air traffic data and control scenarios involving several airports between Chicago and the U.S. East Coast demonstrate the feasibility of the method.

Description and weno numerical approximation to nonlinear waves of a multi-class traffic flow LWR model

A strict proof of the hyperbolicity of the multi-class LWR (Lighthill-Whitham-Richards) traffic flow model, as well as the descriptions on those nonlinear waves characterized in the traffic flow problems were given. They were mainly about the monotonicity of densities across shocks and in rarefactions. As the system had no characteristic decomposition explicitly, a high resolution and higher order accuracy WENO (weighted essentially non-oscillatory) scheme was introduced to the numerical simulation, which coincides with the analytical description.

Convergence of a Discretised Travel-Time Model

In network models for dynamic traffic assignment (DTA), the travel time on a link is often treated as a function of the number of vehicles on the link. Instead of applying this model to the whole link, we divide the link into segments, apply the model (suitably adjusted) sequentially to these segments, and investigate how the solution is affected by various levels of discretisation (as the discretisation is refined, the solution converges to the solution of the Lighthill-Whitham-Richards (LWR) model). We also restrict the link (and segment) travel-time function to ensure that it satisfies a first-in-first-out (FIFO) property and explore how this affects and restricts the form of the flow-density functions used in the LWR model. We numerically illustrate the solution of the discretised model for various travel-time functions and patterns of inflows, for both homogeneous and inhomogeneous links. Subject to the above restriction on the flow-density function, the numerical results suggest that dividing “long” links into even a few segments can make the model solution closely approximate the LWR solution, while retaining tractability in the network model. We also observe, for example, that the whole-link (undescretised) travel-time model has a “flattening” effect on the profiles of flows and travel times (this effect can be reduced to any desired extent by using discretisation); and that the travel time and outflow for an inhomogeneous link can be approximated very closely by treating the link as homogeneous, with capacity parameters set equal to the average capacity from the inhomogeneous link.

A weighted essentially non-oscillatory numerical scheme for a multi-class Lighthill–Whitham–Richards traffic flow model

In this paper, we present a high-order weighted essentially non-oscillatory (WENO) scheme for solving a multi-class extension of the Lighthill–Whitham–Richards (LWR) model. We first review the multi-class LWR model and present some of its analytical properties. We then present the WENO schemes, which were originally designed for computational fluid dynamics problems and for solving hyperbolic conservation laws in general, and demonstrate how to apply these to the present model. We found through numerical experiments that the WENO method is vastly more efficient than the low-order Lax–Friedrichs scheme, yet both methods converge to the same solution of the physical model. It is especially interesting to observe the small staircases in the solution which are completely missed out, because of the numerical viscosity, if a lower-order method is used without a sufficiently refined mesh. To demonstrate the applicability of this new, efficient numerical tool, we study the multi-class model under different parameter regimes and traffic stream models. We consider also the convergence of the multi-class LWR model when the number of classes goes to infinity. We show that the solution converges to a smooth profile without staircases when the number of classes increases.

Comparing whole-link travel time models

In a model commonly used in dynamic traffic assignment the link travel time for a vehicle entering a link at time t is taken as a function of the number of vehicles on the link at time t. In an alternative recently introduced model, the travel time for a vehicle entering a link at time t is taken as a function of an estimate of the flow in the immediate neighbourhood of the vehicle, averaged over the time the vehicle is traversing the link. Here we compare the solutions obtained from these two models when applied to various inflow profiles. We also divide the link into segments, apply each model sequentially to the segments and again compare the results. As the number of segments is increased, the discretisation refined to the continuous limit, the solutions from the two models converge to the same solution, which is the solution of the Lighthill, Whitham, Richards (LWR) model for traffic flow. We illustrate the results for different travel time functions and patterns of inflows to the link. In the numerical examples the solutions from the second of the two models are closer to the limit solutions. We also show that the models converge even when the link segments are not homogeneous, and introduce a correction scheme in the second model to compensate for an approximation error, hence improving the approximation to the LWR model.

A non-equilibrium traffic model devoid of gas-like behavior

The validity of existing higher-order continuum models has been seriously challenged in recent literature. A primary objection to these models is that they exhibit gas-like behavior, that is, they predict that vehicles from behind can influence the behavior of vehicles in front, as elucidated by Daganzo [Transp. Res. B 29 (1995) 277]. Considerable efforts devoted to remove this deficiency (del Castillo et al. [In: C.-F. Daganzo (Ed.), Transportation and Traffic Theory, Elsevier Science, Amsterdam, 1993, p. 387], Zhang [Transp. Res. B 32(7) (1998) 485; Transp. Res. B 33(6) (1999) 387; Transp. Res. B 34 (2000) 583]) have met with only partial success. In this paper we present a non-equilibrium traffic model shown to be devoid of the gas-like behavior that plagues other higher-order models. The system of partial differential equations that describes this model is hyperbolic and has two characteristic fields: one is genuinely nonlinear and the other is linearly degenerate. The first field gives rise to shock and rarefaction waves that are similar to those of the Lighthill–Whitham–Richards (LWR) model, while the second field produces contact discontinuities. All these waves travel no faster than traffic; thus the trajectory of a vehicle cannot be influenced by what happens behind it. The model is also shown to exhibit correct queue-end behavior and is able to explain some of the observed traffic phenomena that challenge old models.

Effects of Relaxation and Anticipation on Riemann Solutions of Payne-Whitham Model

The solutions of Riemann problems of a particular higher-order model—the Payne-Whitham (PW) model—are studied using Roe’s flux splitting scheme as presented by Leo and Pretty. Despite numerous works on higher-order models, little is known about Riemann solutions of these models and how relaxation and anticipation affect these solutions. Riemann solutions of the PW model are computed and compared with those of the Lighthill-Whitham-Richards (LWR) model having the same initial (density) data. It was found that faster relaxation forces the PW model to behave much like the LWR model, that strong anticipation has a stabilizing effect on traffic, and that shock waves travel at different speeds in the PW model than they do in the LWR model. These findings provide a basic checklist for experimental validation of PW-like higher models.