Lighthill Whitham Richards Model Research Articles

Hysteresis and stop-and-go waves in traffic flows

Stop-and-go waves, also called phantom jams, are often observed in real traffic flows but can be produced neither by the classical Lighthill–Whitham–Richards (LWR) model nor by its known variants. To capture stop-and-go waves, we add hysteresis to the LWR model. For the model we propose, all possible viscous waves are found, and necessary and sufficient conditions for their existence are provided. In particular, deceleration and acceleration shocks appear; the latter were never rigorously defined before, in spite of the fact that they were observed in real traffic flows. Stop-and-go waves can be constructed by a pair of deceleration and acceleration shocks that completes a hysteresis cycle, illustrating how hysteresis loops lead to stop-and-go waves. In contrast, in the phase region where anticipation (i.e. negative hysteresis) loops exist, stop-and-go waves are not present, and speed variations decay. Riemann solutions are then found for all possible Riemann data. We explicitly show that, in the phase region where hysteresis loops exist, a sufficient deviation in speed of a few vehicles in an otherwise uniform car platoon can generate stop-and-go waves, confirming observations of real traffic experiments.

Multiclass information flow propagation control under vehicle-to-vehicle communication environments

Most existing models for information flow propagation in a vehicle-to-vehicle (V2V) communications environment are descriptive. They lack capabilities to control information flow, which may preclude their ability to meet application needs, including the need to propagate different information types simultaneously to different target locations within corresponding time delay bounds. This study proposes a queuing-based modeling approach to control the propagation of information flow of multiple classes. Two control parameters associated with a vehicle, the number of communication servers and the mean communication service rate, are leveraged to control the propagation performance of different information classes. A two-layer model is developed to characterize the information flow propagation wave (IFPW) under the designed queuing strategy. The upper layer is formulated as integro-differential equations to characterize the spatiotemporal information dissemination due to V2V communication. The lower layer characterizes the traffic flow dynamics using the Lighthill–Whitham–Richards model. The analytical solution of the asymptotic density of informed vehicles and the necessary condition for existence of the IFPW are derived for homogeneous traffic conditions. Numerical experiments provide insights on the impact of the mean communication service rate on information spread and its spatial coverage. Further, a numerical solution method is developed to solve the two-layer model, which aids in estimating the impacts of the control parameters in the queuing strategy on the IFPW speed under homogenous and heterogeneous conditions. The proposed modeling approach enables controlling the propagation of information of different information classes to meet application needs, which can assist traffic managers to design effective and efficient traffic management and control strategies under V2V communications.

Influence of coarse toll on the dynamic properties of traffic flow in a single-entry traffic corridor

In this paper, we investigate the effects of coarse toll on the user equilibrium (UE) state in a single-entry traffic corridor with early and late arrivals from analytical and numerical perspective. Unlike previous studies on traffic tolls, the flow congestion model is used to take into account the dynamic propagation of traffic flow. The LWR (Lighthill–Whitham–Richards) model and the Greenshields’ velocity–density function are used to describe the dynamic properties of traffic flow. The quasi-analytic solutions of the three possible cases in UE due to different charges are derived. Both analytical and numerical results indicate that rarefaction waves and shock waves appear in the corridor under the coarse toll. The three-dimensional evolution diagrams of traffic flow and space–time diagrams illustrate that the introduction of course toll lets flow patterns more complicated. This extends the knowledge given by the classical bottleneck model.

An existence result for a constrained two-phase transition model with metastable phase for vehicular traffic

In this paper we study a phase transition model for vehicular traffic flows. Two phases are taken into account, according to whether the traffic is light or heavy. We assume that the two phases have a non-empty intersection, the so called metastable phase. The model is given by the Lighthill–Whitham–Richards model in the free-flow phase and by the Aw–Rascle–Zhang model in the congested phase. In particular, we study the existence of solutions to Cauchy problems satisfying a local point constraint on the density flux. We prove that if the constraint F is higher than the minimal flux \(f_\mathrm{c}^-\) of the metastable phase, then constrained Cauchy problems with initial data of bounded total variation admit globally defined solutions. We also provide sufficient conditions on the initial data that guarantee the global existence of solutions also in the case \(F < f_\mathrm{c}^-\). These results are obtained by applying the wave-front tracking technique.

Social optimum for evening commute in a single-entry traffic corridor with no early departures

In this paper, we investigate the evening commute behaviors on the social optimum (SO) state in a single-entry traffic corridor with no early departures. Differing from the previous studies on evening commute, the dynamic properties of traffic flow are analyzed with the LWR (Lighthill–Whitham–Richards) model. The properties of optimum cumulative inflow curve with general desired departure time distribution curve are deduced, and then the analytic solutions for common desired departure time in SO are obtained. Three numerical examples are carried out to capture the characteristics of evening commuting behaviors under different values of time. The analytic and numerical results both indicate that the rarefaction wave originating from the first entry point influences the whole or part of the outflow curve. No shock wave exists through the commuting process. In addition, the cost curves show that the trip cost increases and the departure delay cost decreases with departure time, whereas the travel time cost first increases then decreases with departure time under the SO principle.

Follow-the-Leader models can be viewed as a numerical approximation to the Lighthill-Whitham-Richards model for traffic flow

We show how to view the standard Follow-the-Leader (FtL) model as a numerical method to compute numerically the solution of the Lighthill-Whitham-Richards (LWR) model for traffic flow. As a result we offer a simple proof that FtL models converge to the LWR model for traffic flow when traffic becomes dense. The proof is based on techniques used in the analysis of numerical schemes for conservation laws, and the equivalence of weak entropy solutions of conservation laws in the Lagrangian and Eulerian formulation.

A macroscopic traffic flow model that includes driver sensitivity to the number of free spaces ahead

This paper addresses the first-order extension of the Lighthill–Whitham– Richards (LWR) macroscopic traffic flow model. Although previous studies have focused on the fluid aspect of traffic flow, none have addressed the sensitivity of drivers to the number of free spaces within a certain distance ahead of the subject driver. To incorporate driver behavior, we used the number of free spaces ahead of subject drivers and their sensitivity to the number of free spaces within a certain distance ahead. The resulting model is a convection-diffusion model. By computing Einstein's diffusion equation and comparing it with the diffusion coefficient in the extended model, a theoretical relation for the driver's sensitivity was derived. A representation of the numerical results of the LWR model, the convection-diffusion model with a typical diffusion coefficient, and the extended model showed that the proposed model has a lower mean relative error value.

Boundary coupling of microscopic and first order macroscopic traffic models

We couple, at a fixed interface, the microscopic Follow the Leader model and the macroscopic Lighthill–Whitham–Richards model, both used for describing vehicular traffic. The coupling is obtained by suitable boundary conditions. We prove existence of solutions for the corresponding Cauchy problem, by using the wave-front tracking method and classical results on ordinary differential equations.

Analysis of user equilibrium for staggered shifts in a single-entry traffic corridor with no late arrivals

In this paper, we investigate the effects of staggered shifts on the user equilibrium (UE) state in a single-entry traffic corridor with no late arrivals from the analytical and numerical perspective. The LWR (Lighthill–Whitham–Richards) model and the Greenshields’ velocity–density function are used to describe the dynamic properties of traffic flow. Propositions for the properties of flow patterns in UE, and the quasi-analytic solutions for three possible situations in UE are deduced. Numerical tests are carried out to testify the analytical results, where the three-dimensional evolution diagram of traffic flow illustrates that shock and rarefaction wave exist in UE and the space-time diagram indicates that UE solutions satisfy the propagation properties of traffic flow. In addition, the cost curves show that the UE solutions satisfy the UE trip-timing condition.

Optimal urban development density along a multimodal linear travel corridor with time–distance toll scheme

ABSTRACTConsider a travel corridor with a multimodal transport system (highway and railway) that connects continuous residential locations to the city center. All commuters travel along the corridor from home to work in the morning peak hour. The spatial dynamics of the traffic congestion on both transportation systems are determined by the trip-timing condition. The flow dynamics on the highway will be considered by applying the Lighthill–Whitham–Richards model. A time–distance road pricing scheme is applied to achieve the system optimal condition. The urban population is assumed to be located continuously along the corridor. This study aims to find the optimal urban population density distribution leading to optimal transportation system performance, with a basic assumption that it follows some given distribution pattern. The problem is eventually formulated into a mathematical program with complementarity constraints and an efficient solution algorithm is applied. Finally, numerical examples are applied to test the validity of the model formulation.

Homogenization of second order discrete model with local perturbation and application to traffic flow

The goal of this paper is to derive a traffic flow macroscopic model from a second order microscopic model with a local perturbation. At the microscopic scale, we consider a Bando model of the type following the leader, i.e the acceleration of each vehicle depends on the distance of the vehicle in front of it. We consider also a local perturbation like an accident at the roadside that slows down the vehicles. After rescaling, we prove that the cumulative distribution functions of the vehicles converges towards the solution of a macroscopic homogenized Hamilton-Jacobi equation with a flux limiting condition at junction which can be seen as a LWR (Lighthill-Whitham-Richards) model.

Data Assimilation Using a Mesoscopic Lighthill–Whitham–Richards Model and Loop Detector Data: Methodology and Large-Scale Network Application

Traffic managers and operators need decision support systems able to provide online traffic flow monitoring and short-term traffic predictions on large-scale networks. Data assimilation (DA) techniques are used to combine observed data and a traffic model. This paper proposes a comprehensive data assimilation framework, based on a mesoscopic Lighthill–Whitham–Richards model, that has short computational times, is well suited for network discontinuities, provides individual vehicle tracking, and can easily be coupled with any dynamic traffic assignment model. The framework also relies on state variables that require adjustments of the DA framework. The requirements proposed by the paper are concerned with ( a) the model numerical scheme, ( b) the traffic state transformation operators, and ( c) updating of the model. The proposed DA framework is first applied to a simplified network. It validates the ability of the proposed framework to update and propagate traffic states accordingly. The proposed framework is then applied to a real large-scale network. The results demonstrate its ability to monitor and forecast traffic conditions with online capabilities.

New Extended Discrete First-Order Model to Reproduce Propagation of Jam Waves

This paper proposes an extension of the discrete Lighthill–Whitham–Richards model of the cell transmission model type to reproduce capacity drop and the propagation of jam waves. Recent studies have tried to incorporate the capacity drop into discrete first-order traffic flow models for traffic optimization purposes. It was found that the inflow to a discharging cell predicted by these models might have been overestimated, and this overestimation influenced the propagation of a jam wave. An empirical analysis was carried out to confirm this assumption. It was found that the extent of the flow reduction depended on the state difference between the targeting cell and its upstream cell. On the basis of these findings, a new mathematical model formulation is given. Simulations with both a hypothetical freeway stretch and a real-life freeway stretch are performed to test the behavior of the proposed model. The previously mentioned models are also simulated for comparison. The simulation results indicate that the proposed model is better able to reproduce jam waves. In addition, the proposed model can be used in a linear model predictive control framework and formulated as a linear optimization problem, which may be beneficial for a real-life, real-time application.

On the local fractional LWR model in fractal traffic flows in the entropy condition

Entropy plays an important role in the simulation of traffic flow distribution. This paper studies the entropy condition for the Lighthill–Whitham–Richards model of the fractal traffic flows described by local fractional calculus. We also discuss the solutions of non‐differentiability with graphs by using the local fractional variational iteration method. Copyright © 2015 John Wiley &amp; Sons, Ltd.

Bayesian analysis of traffic flow on interstate I-55: The LWR model

Transportation departments take actions to manage traffic flow and reduce travel times based on estimated current and projected traffic conditions. Travel time estimates and forecasts require information on traffic density which are combined with a model to project traffic flow such as the Lighthill-Whitham-Richards (LWR) model. We develop a particle filtering and learning algorithm to estimate the current traffic density state and the LWR parameters. These inputs are related to the so-called fundamental diagram, which describes the relationship between traffic flow and density. We build on existing methodology by allowing real-time updating of the posterior uncertainty for the critical density and capacity parameters. Our methodology is applied to traffic flow data from interstate highway I-55 in Chicago. We provide a real-time data analysis of how to learn the drop in capacity as a result of a major traffic accident. Our algorithm allows us to accurately assess the uncertainty of the current traffic state at shock waves, where the uncertainty is a mixture distribution. We show that Bayesian learning can correct the estimation bias that is present in the model with fixed parameters.

Homogenization of second order discrete model and application to traffic flow

The goal of this paper is to derive traffic flow macroscopic models from microscopic models. At the microscopic scales, we consider a Bando model, of the type following the leader, i.e., the acceleration of each vehicle depends on the distance to the vehicle in front of it. We take into account the possibility that each driver can have different characteristics such as sensibility to other drivers or optimal velocities. After rescaling, we prove that the solution of this system of ODEs converges to the solution of a macroscopic homogenized Hamilton-Jacobi equation which can be seen as a LWR (Lighthill-Whitham-Richards) model.

Continuity of the path delay operator for dynamic network loading with spillback

This paper establishes the continuity of the path delay operators for dynamic network loading (DNL) problems based on the Lighthill–Whitham–Richards model, which explicitly capture vehicle spillback. The DNL describes and predicts the spatial-temporal evolution of traffic flow and congestion on a network that is consistent with established route and departure time choices of travelers. The LWR-based DNL model is first formulated as a system of partial differential algebraic equations. We then investigate the continuous dependence of merge and diverge junction models with respect to their initial/boundary conditions, which leads to the continuity of the path delay operator through the wave-front tracking methodology and the generalized tangent vector technique. As part of our analysis leading up to the main continuity result, we also provide an estimation of the minimum network supply without resort to any numerical computation. In particular, it is shown that gridlock can never occur in a finite time horizon in the DNL model.

A robust optimization approach for dynamic traffic signal control with emission considerations

We consider an analytical signal control problem on a signalized network whose traffic flow dynamic is described by the Lighthill–Whitham–Richards (LWR) model (Lighthill and Whitham, 1955; Richards, 1956). This problem explicitly addresses traffic-derived emissions as constraints or objectives. We seek to tackle this problem using a mixed integer mathematical programming approach. Such class of problems, which we call LWR-Emission (LWR-E), has been analyzed before to certain extent. Since mixed integer programs are practically efficient to solve in many cases (Bertsimas et al., 2011b), the mere fact of having integer variables is not the most significant challenge to solving LWR-E problems; rather, it is the presence of the potentially nonlinear and nonconvex emission-related constraints/objectives that render the program computationally expensive.To address this computational challenge, we proposed a novel reformulation of the LWR-E problem as a mixed integer linear program (MILP). This approach relies on the existence of a statistically valid macroscopic relationship between the aggregate emission rate and the vehicle occupancy on the same link. This relationship is approximated with certain functional forms and the associated uncertainties are handled explicitly using robust optimization (RO) techniques. The RO allows emissions-related constraints and/or objectives to be reformulated as linear forms under mild conditions. To further reduce the computational cost, we employ a link-based LWR model to describe traffic dynamics with the benefit of fewer (integer) variables and less potential traffic holding. The proposed MILP explicitly captures vehicle spillback, avoids traffic holding, and simultaneously minimizes travel delay and addresses emission-related concerns.

On the Braess Paradox with Nonlinear Dynamics and Control Theory

We show the existence of the Braess paradox for a traffic network with nonlinear dynamics described by the Lighthill–Whitham–Richards model for traffic flow. Furthermore, we show how one can employ control theory to avoid the paradox. The paper offers a general framework applicable to time-independent, uncongested flow on networks. These ideas are illustrated through examples.

Multiscale Traffic Flow Model Based on the Mesoscopic Lighthill–Whitham and Richards Models

This article reviews the multiscale modeling concept as an adaptive strategy in traffic flow modeling. The central objective for the multi-scale model is to describe and to predict traffic phenomena (e.g., individual accelerations, queuing) that occur at different scales by switching between modeling paradigms. The emphasis in this paper is placed on resolving inconsistencies that arise in the process of integrating models, which are addressed with the mesoscopic Lighthill–Whitham and Richards models—together, widely known as the Lighthill–Whitham–Richards (LWR) model—as the coarse-scale model in the framework. The mesoscopic model is fully consistent at the macroscopic scale with the classic LWR model but does facilitate tracking of individual vehicles. Because of this characteristic, this model can be combined with microscopic models, because no disaggregation or aggregation process is needed. It is thus implied, for example, that travel information (e.g., destination) can be carried over this model without loss of information. Also, vehicle heterogeneity can be incorporated in the mesoscopic model and, as a result, is preserved in the process of switching between models. A brief overview of the mesoscopic LWR model and its solution by the variational theory is presented. General boundary conditions, including internal boundaries (e.g., moving bottlenecks) and interfaces between models, are analyzed and evaluated. The solution of some cases, in which the stationary and moving bottlenecks interact, is provided. The paper concludes with a case study that demonstrates the integration of microscopic and mesoscopic models in a multiscale model and investigates the influence of various boundary conditions on traffic features.