Lighthill Whitham Richards Model Research Articles

Numerical relaxation limit and outgoing edges in a central scheme for networked conservation laws

AbstractA recently introduced scheme for networked conservation laws is analyzed in various experiments. The scheme makes use of a novel relaxation approach that governs the coupling conditions of the network and does not require a solution of the Riemann problem at the nodes. We numerically compare the dynamics of the solution obtained by the scheme to solutions obtained using a classical coupling condition. In particular, we investigate the case of two outgoing edges in the Lighthill–Whitham–Richards model of traffic flow and in the Buckley–Leverett model of two phase flow. Moreover, we numerically study the asymptotic preserving property of the scheme by comparing it to its preliminary form before the relaxation limit in a 1‐to‐1 network.

Higher-Order Traffic Flow Model Extended to Road Networks

In this work, a conserved higher-order (CHO) macroscopic traffic flow model is extended to road networks. We first introduce the Riemann problem at a junction for the CHO model and then provide a general framework to solve it, considering the compatibility between the CHO and Lighthill-Whitham-Richards (LWR) models. Through the presented framework, Riemann solvers for the LWR model can be extended to the CHO model. Specifically, a kind of Riemann solver for the LWR model is extended to derive the Riemann solver for the CHO model, in which the total actual flow at the junction is maximized. The Riemann solvers for three typical junctions (one-in-two-out junction, two-in-one-out junction, and two-in-two-out junction) are discussed. The first-order finite-volume method is adopted to solve the extended model. Numerical examples are given to validate the extended model and the numerical algorithm.

A dynamically bi‐orthogonal solution method for a stochastic Lighthill‐Whitham‐Richards traffic flow model

AbstractMacroscopic traffic flow modeling is essential for describing and forecasting the characteristics of traffic flow. However, the classic Lighthill–Whitham–Richards (LWR) model only provides equilibrium values for steady‐state conditions and fails to capture common stochastic variabilities, which are a necessary component of accurate modeling of real‐time traffic management and control. In this paper, a stochastic LWR (SLWR) model that randomizes free‐flow speed is developed to account for the stochasticity incurred by the heterogeneity of drivers, while holding individual drivers’ behavior constant. The SLWR model follows a conservation law of stochastic traffic density and flow and is formulated as a time‐dependent stochastic partial differential equation. The model is solved using a dynamically bi‐orthogonal (DyBO) method based on a spatial basis and stochastic basis. Various scenarios are simulated and compared with the Monte Carlo (MC) method, and the results show that the SLWR model can effectively describe dynamic traffic evolutions and reproduce some commonly observed traffic phenomena. Furthermore, the DyBO method shows significant computational advantages over the MC method.

Front-tracking transition system model for traffic state reconstruction, model learning, and control with application to stop-and-go wave dissipation

Connected and Autonomous Vehicles is a technology that will be disruptive for all layers of traffic control. The Lagrangian, in-the-flow nature of their operation offers untapped new potentials for sensing and actuation, but also presents new fundamental challenges. In order to use these vehicles for traffic state reconstruction and control, we need suitable traffic models, which should be computationally efficient and able to represent complex traffic phenomena. To this end, we propose the Front-tracking Transition System Model, a cell-free modelling approach that can incorporate Lagrangian measurements, and has a structure that yields itself to on-line model learning and control. The model is formulated as a transition system, and based on the front-tracking method for finding entropy solutions to the Lighthill–Whitham–Richards model. We characterize the solution of this model and show that it corresponds to the solution of the underlying PDE traffic model. Algorithms for traffic state reconstruction and model learning are proposed, exploiting the model structure. The model is then used to design a prediction-based control law for stop-and-go wave dissipation using randomly arriving Connected and Autonomous Vehicles. The proposed control framework is able to estimate the traffic state and model, adapt to changes in the traffic dynamics, and achieve a reduction in vehicles’ Total Time Spent.

Macroscopic flow characterization at T-junctions

At T-junctions, a minor road meets a main road and merging vehicles from the minor road create traffic bottlenecks as significant interactions between vehicles occur. These interactions reduce road capacity resulting in congestion. The Lighthill-Whitham-Richards (LWR) model is widely employed to characterize traffic flow. However, this model only considers small changes in flow which can be inadequate for T-junctions. Thus, a new model is proposed which is based on driver response during transitions at these junctions. The proposed and LWR models are evaluated using the first order upwind scheme (FOUS). The Courant-Friedrich-Lewy (CFL) condition is employed to ensure model stability. Results are presented which show that the proposed model better characterizes traffic flow at T-junctions.

Physics-Informed Deep Learning for Traffic State Estimation: Illustrations With LWR and CTM Models

We present a physics-informed deep learning (PIDL) approach to tackle the challenge of data sparsity and sensor noise in traffic state estimation (TSE). PIDL strengthens a deep learning (DL) neural network with the knowledge of traffic flow theory to accurately estimate traffic conditions. The ‘physics’—a priori information of the system—acts as a regularization agent during training. We illustrate the implementation of the proposed approach with two commonly used models representing traffic physics: Lighthill-Whitham-Richards (LWR) model and the cell transmission model (CTM). The LWR implementation is illustrated with Greenshields’ and inverse-lambda fundamental diagrams; whereas, CTM model implementation works with any fundamental diagram of choice. Two case studies validate the approach by reconstructing the velocity-field. Case study-I uses synthetic data generated to resemble the trajectory of connected and autonomous vehicles as captured by roadside units. Case study-II employs NGSIM data mimicking scant probe vehicle observations. We observe that the proposed PIDL approach is particularly better in state estimation with a lower amount of training data, illustrating the capability of PIDL in making precise and timely TSE even with sparse input. E.g., With 10% CAV penetration rate and a 15% added-noise, relative error for PIDL was at 22.9% compared to 30.8% for DL.

A statistical mechanics approach to macroscopic limits of car-following traffic dynamics

We study the derivation of macroscopic traffic models from car-following vehicle dynamics by means of hydrodynamic limits of an Enskog-type kinetic description. We consider the superposition of Follow-the-Leader (FTL) interactions and relaxation towards a traffic-dependent Optimal Velocity (OV) and we show that the resulting macroscopic models depend on the relative frequency between these two microscopic processes. If FTL interactions dominate then one gets an inhomogeneous Aw–Rascle–Zhang model, whose (pseudo) pressure and stability of the uniform flow are precisely defined by some features of the microscopic FTL and OV dynamics. Conversely, if the rate of OV relaxation is comparable to that of FTL interactions then one gets a Lighthill–Whitham–Richards model ruled only by the OV function. We further confirm these findings by means of numerical simulations of the particle system and the macroscopic models. Unlike other formally analogous results, our approach builds the macroscopic models as physical limits of particle dynamics rather than assessing the convergence of microscopic to macroscopic solutions under suitable numerical discretisations.

Lighthill-Whitham-Richards Model for Traffic Flow Mixed with Cooperative Adaptive Cruise Control Vehicles

In the future, road traffic will incorporate a random mix of manual vehicles and cooperative adaptive cruise control (CACC) vehicles, where a CACC vehicle will degrade to an adaptive cruise control (ACC) vehicle when vehicle-to-vehicle communications are not available. This paper proposes a generalized framework of the Lighthill-Whitham-Richards (LWR) model for such mixed vehicular flow under different CACC penetration rates. In this approach, the kinematic wave speed propagating through the mixed platoon was theoretically proven to be the slope of mixed fundamental diagram. In addition, the random degradation from CACC to ACC was captured in mathematical expectation for traffic scenarios where the CACC only monitors one vehicle ahead. Three concrete car-following models, the intelligent driver model (IDM) and CACC/ACC models validated by Partners for Advanced Transit and Highways (PATH) program, were selected as examples to investigate the propagation of small perturbations and shock waves. Numerical simulations were also performed based on the selected car-following models. Moreover, the derived mixed LWR model was applied to solve some traffic flow problems. It indicates that the proposed LWR model is able to describe the propagation properties of both small perturbations and shock waves. The mixed LWR model can also be used to solve some practical problems, such as the queue caused by a traffic accident and the impact of a moving bottleneck. More importantly, the proposed generalized framework admits other CACC/ACC/regular car-following models, including those developed from further experiments.

Entropy solutions for a two-phase transition model for vehicular traffic with metastable phase and time depending point constraint on the density flow

We consider a macroscopic two-phase transition model for vehicular traffic flow subject to a point constraint on the density flux. The two phases correspond to light and heavy traffic and their dynamics are described respectively by the Lighthill–Whitham–Richards model and the Aw–Rascle–Zhang model. Their intersection, the so-called metastable phase, is assumed to be non empty. The discrete in time point constraint mechanism, inducing flux limitation at bottlenecks, is explored within this two-phase model. We introduce a new definition of admissible solutions for the Cauchy problem, for which we prove existence and we provide a characterization. In particular, these admissible solutions attain the maximal flux allowed by the constraint whenever it is enforced, which guarantees compatibility of the constructed solutions with the modeling assumption imposed at the level of the Riemann solver. These results rely on the wave-front tracking method and on adaptation of the specific entropies and renormalization properties introduced in Andreainov, Donadello, Rosini, M3AS (2016) while dealing with the Aw–Rascle–Zhang model with point constraint.

Variable Speed Limit and Ramp Metering Control of Highway Networks Using Lax-Hopf Method: A Mixed Integer Linear Programming Approach

This paper presents a novel optimization formulation to solve the problem of variable speed limit control on road networks modeled by the Lighthill-Whitham-Richards (LWR) partial differential equation. It also presents some mathematical rules that allow for a reduction in the size and computational time of the optimization problem. Using the analytical solutions to the LWR model, an optimization problem is formulated for the variable speed limit and ramp metering control of traffic on highway networks using the Lax-Hopf algorithm. The resulting problem, which is non-linear in the decision variables, is transformed into a Mixed Integer Linear Program. An example is presented to show the effectiveness of the approach, including its application to a real-world highway network with multiple ramp connections. The possibility of linear relaxation of integer variables in the problem is also considered. Lastly, the method is compared to a classical Link Transmission Model formulation of the variable speed limit control problem.

A Control-Theoretic Approach for Scalable and Robust Traffic Density Estimation Using Convex Optimization

Monitoring and control of traffic networks represent alternative, inexpensive strategies to minimize traffic congestion. As the number of traffic sensors is naturally constrained by budgetary requirements, real-time estimation of traffic flow in road segments that are not equipped with sensors is of significant importance - thereby providing situational awareness and guiding real-time feedback control strategies. To that end, firstly we build a generalized traffic flow model for stretched highways with arbitrary number of ramp flows based on the Lighthill Whitham Richards (LWR) flow model. Secondly, we characterize the function set corresponding to the nonlinearities present in the LWR model, and use this characterization to design real-time and robust state estimators (SE) for stretched highway segments. Specifically, we show that the nonlinearities from the derived models are locally Lipschitz continuous by providing the analytical Lipschitz constants. Thirdly, the analytical derivation is then incorporated through a robust SE method given a limited number of traffic sensors, under the impact of process and measurement disturbances and unknown inputs. The estimator is based on deriving a convex semidefinite optimization problem. Finally, numerical tests are given showcasing the applicability, scalability, and robustness of the proposed estimator for large systems under high magnitude disturbances, parametric uncertainty, and unknown inputs.

A multiclass Lighthill-Whitham-Richards traffic model with a discontinuous velocity function

The well-known Lighthill-Whitham-Richards (LWR) kinematic model of traffic flow models the evolution of the local density of cars by a nonlinear scalar conservation law. The transition between free and congested flow regimes can be described by a flux or velocity function that has a discontinuity at a determined density. A numerical scheme to handle the resulting LWR model with discontinuous velocity was proposed in [J.D. Towers, A splitting algorithm for LWR traffic models with flux discontinuities in the unknown, J. Comput. Phys., 421 (2020), article 109722]. A similar scheme is constructed by decomposing the discontinuous velocity function into a Lipschitz continuous function plus a Heaviside function and designing a corresponding splitting scheme. The part of the scheme related to the discontinuous flux is handled by a semi-implicit step that does, however, not involve the solution of systems of linear or nonlinear equations. It is proved that the whole scheme converges to a weak solution in the scalar case. The scheme can in a straightforward manner be extended to the multiclass LWR (MCLWR) model, which is defined by a hyperbolic system of \begin{document}$ N $\end{document} conservation laws for \begin{document}$ N $\end{document} driver classes that are distinguished by their preferential velocities. It is shown that the multiclass scheme satisfies an invariant region principle, that is, all densities are nonnegative and their sum does not exceed a maximum value. In the scalar and multiclass cases no flux regularization or Riemann solver is involved, and the CFL condition is not more restrictive than for an explicit scheme for the continuous part of the flux. Numerical tests for the scalar and multiclass cases are presented.

Macroscopic Traffic Flow Characterization for Stimuli Based on Driver Reaction

The design and management of infrastructure is a significant challenge for traffic engineers and planners. Accurate traffic characterization is necessary for effective infrastructure utilization. Thus, models are required that can characterize a variety of conditions and can be employed for homogeneous, heterogeneous, equilibrium and non-equilibrium traffic. The Lighthill-Whitham-Richards (LWR) model is widely used because of its simplicity. This model characterizes traffic behavior with small changes over a long idealized road and so is inadequate for typical traffic conditions. The extended LWR model considers driver types based on velocity to characterize traffic behavior in non lane discipline traffic but it ignores the stimuli for changes in velocity. In this paper, an improved model is presented which is based on driver reaction to forward traffic stimuli. This reaction occurs over the forward distance headway during which traffic aligns to the current conditions. The performance of the proposed, LWR and extended LWR models is evaluated using the first order upwind scheme (FOUS). The numerical stability of this scheme is guaranteed by employing the Courant, Friedrich and Lewy (CFL) condition. Results are presented which show that the proposed model can characterize both small and large changes in traffic more realistically. Doi: 10.28991/cej-2021-03091632 Full Text: PDF

Studying perturbations and wave propagations by lane closures on traffic characteristics based on a dynamic approach

Lane closures cause perturbations and wave propagations on traffic characteristics of freeways and consequently affect traffic flow. To conduct a dynamic analysis regarding the effects of lane closures on traffic characteristics, this study aimed to examine the effects of lane closures as multi-static bottlenecks on the evolution and wave propagation of uniform traffic flow in a four-lane divided freeway based on the field study and macroscopic traffic flow model. One-lane closure is a work zone and two-lane closure occurs when the second lane near the one-lane closure is blocked due to an incident caused by a lane change. In addition, the classical Lighthill–Whitham–Richards (LWR) model was combined with the non-linear Greenshield’s speed–density relation to expand the models for characterizing the behavior of shockwaves under lane closures. Thus, based on numerical procedures regarding the Lax–Friedrichs scheme, the results indicated that wave propagations by two-lane closure have significant effects on the perturbations of traffic flow density which are 1.5 times as much as the wave propagations in one-lane closure. Moreover, one-lane closure reduced the capacity by 21% although the two-lane closure caused a 45% decrease in the capacity. A comparison between expanded LWRρ and LWRu models with the LWR model demonstrated that expanded models under two-lane closure and one-lane closure improved LWR model for the congested condition, and between uncongested and congested traffic condition 15.61% and 39%, respectively. Therefore, expanded models revealed a better performance as a dynamic approach to analyze the behavior of wave propagations in jam traffic conditions compared to LWR model under lane closures.

Properties of the LWR model with time delay

In this article, we investigate theoretical and numerical properties of the first-order Lighthill-Whitham-Richards (LWR) traffic flow model with time delay. Since standard results from the literature are not directly applicable to the delayed model, we mainly focus on the numerical analysis of the proposed finite difference discretization. The simulation results also show that the delay model is able to capture Stop & Go waves.

A Fast Lax–Hopf Algorithm to Solve the Lighthill–Whitham–Richards Traffic Flow Model on Networks

Efficient and exact algorithms are important for performing fast and accurate traffic network simulations with macroscopic traffic models. In this paper, we extend the semianalytical Lax–Hopf algorithm in order to compute link inflows and outflows with the Lighthill–Whitham–Richards (LWR) model. Our proposed fast Lax–Hopf algorithm has a very low computational complexity. We demonstrate that some of the original algorithm’s operations (associated with the initial conditions) can be discarded, leading to a faster computation of boundary demands/supplies in network simulation problems for general concave fundamental diagrams (FDs). Moreover, the computational cost can be further reduced for triangular FDs and specific space–time discretizations. The resulting formulation has a performance comparable to the link transmission model and because it solves the original LWR model for a wide range of FD shapes, with any initial configuration, it is suitable to solve a broad range of traffic operations problems. As part of the analysis, we compare the performance of the proposed scheme with that of other well-known computational methods.

Extremum Seeking for Traffic Congestion Control With a Downstream Bottleneck

Abstract This paper develops boundary control for freeway traffic with a downstream bottleneck. Traffic on a freeway segment with capacity drop at outlet of the segment is a common phenomenon that leads to traffic bottleneck problem. The capacity drop can be caused by lane-drop, hills, tunnel, bridge, or curvature on the road. If incoming traffic flow remains unchanged, traffic congestion forms upstream of the bottleneck since the upstream traffic demand exceeds its capacity. Therefore, it is important to regulate the incoming traffic flow of the segment to avoid overloading the bottleneck area. Traffic densities on the freeway segment are described with the Lighthill–Whitham–Richards (LWR) macroscopic partial differential equation (PDE) model. The incoming flow at the inlet of the freeway segment is controlled so that the optimal density that maximizes the outgoing flow is reached and the traffic congestion upstream of the bottleneck is mitigated. The density and traffic flow relation at the bottleneck area, usually described with fundamental diagram, is considered to be unknown. We tackle this problem using extremum seeking (ES) control with delay compensation for the LWR PDE. ES control, a nonmodel-based approach for real-time optimization, is adopted to find the optimal density for the unknown fundamental diagram. A predictor feedback control design is proposed to compensate the delay effect of traffic dynamics in the freeway segment. In the end, simulation results are obtained to validate a desired performance of the controller on the nonlinear LWR model with an unknown fundamental diagram.

Stable Day-to-Day Dynamics for Departure Time Choice

All existing day-to-day dynamics of departure time choice at a single bottleneck are unstable, and this has led to doubt over the existence of a stable user equilibrium in the real world. However, empirical observations and our personal driving experience suggest stable stationary congestion patterns during a peak period. In this paper, we attempt to reconcile the discrepancy by presenting a stable day-to-day dynamical system for drivers’ departure time choice at a single bottleneck. In our model, the decision variable in the execution stage is still drivers’ departure times on the next day, but in the planning stage before the execution stage, drivers determine their departure times in order to arrive at the destination at better times with lower scheduling costs. We first define within-day traffic dynamics with the point queue model, costs, the departure time user equilibrium (DTUE), and the arrival time user equilibrium (ATUE). We then identify three behavioral principles in the planning stage: (i) drivers choose their departure and arrival times in a backward fashion (backward choice principle); (ii) after choosing the arrival times, they update their departure times to balance the total costs (cost-balancing principle); (iii) they choose their arrival times to reduce their scheduling costs or gain their scheduling payoffs (scheduling cost–reducing or scheduling payoff–gaining principle). In this sense, drivers’ departure and arrival time choices are driven by their scheduling payoff choice. With a single tube or imaginary road model, we convert the nonlocal day-to-day arrival time shifting problem to a local scheduling payoff shifting problem. After introducing a new variable for the imaginary density, we apply the Lighthill–Whitham–Richards (LWR) model to describe the day-to-day dynamics of scheduling payoff choice and present splitting and cost-balancing schemes to determine arrival and departure flow rates accordingly. We define the scheduling payoff user equilibrium (SPUE) as the stationary state of the LWR model, formulate a new optimization problem for the SPUE, and prove the global stability of the SPUE and, therefore, ATUE and DTUE by using Lyapunov’s second method in which the objective function in the optimization formulation is the potential function. We also develop the corresponding discrete models for numerical solutions and use one numerical example to demonstrate the effectiveness and stability of the new day-to-day dynamical model. Different from existing ones, the new adjustment mechanism leads to stable day-to-day departure time choice dynamics by guaranteeing that drivers have better choices of departure/arrival times with larger scheduling payoffs on the next day, and such better choices are not over-chosen because of the constraint imposed by the single tube’s cross-section area, which is equal to the jam density in the LWR model. This study is the first step for understanding stable day-to-day dynamics for departure time choice, and many follow-up studies are possible and warranted.

Shape-Based Nonlinear Model Reduction for 1D Conservation Laws

We present a novel method for model reduction of one-dimensional conservation law to the dynamics of the parameters describing the approximate shape of the solution. Depending on the parametrization, each parameter has a well-defined physical meaning. The obtained ODE system can be used for the estimation and control purposes. The model reduction is performed by minimizing the divergence of flows between the original and reduced systems, and we show that this is equivalent to the minimization of the Wasserstein distance derivative. The method is then tested on the heat equation and on the LWR (Lighthill-Whitham-Richards) model for vehicle traffic.

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.