A PDE-ODE model for a junction with ramp buffer

International audienceWe consider the Lighthill-Witham-Richards traffic flow model on a junction com- posed by one mainline, an onramp and an offramp, which are connected by a node. The onramp dynamics is modeled using an ordinary differential equation describing the evolution of the queue length. The definition of the solution of the Riemann problem at the junction is based on an optimization problem and the use of a right-of-way parameter. The numerical approximation is carried out using Godunov scheme, modified to take into account the effects of the onramp buffer. We present the result of some simulations and check numerically the convergence of the method

On the optimization of conservation law models at a junction with inflow and flow distribution controls

The paper proposes a general framework to analyze control problems for conservation law models on a network. Namely we consider a general class of junction distribution controls and inflow controls and we establish the compactness in $L^1$ of a class of flux-traces of solutions. We then derive the existence of solutions for two optimization problems: (I) the maximization of an integral functional depending on the flux-traces of solutions evaluated at points of the incoming and outgoing edges; (II) the minimization of the total variation of the optimal solutions of problem (I). Finally we provide an equivalent variational formulation of the min-max problem (II) and we discuss some numerical simulations for a junction with two incoming and two outgoing edges.Comment: 29 pages, 14 figure

Control Problems for Conservation Laws with Traffic Applications

Conservation and balance laws on networks have been the subject of much research interest given their wide range of applications to real-world processes, particularly traffic flow. This open access monograph is the first to investigate different types of control problems for conservation laws that arise in the modeling of vehicular traffic. Four types of control problems are discussed - boundary, decentralized, distributed, and Lagrangian control - corresponding to, respectively, entrance points and tolls, traffic signals at junctions, variable speed limits, and the use of autonomy and communication. Because conservation laws are strictly connected to Hamilton-Jacobi equations, control of the latter is also considered. An appendix reviewing the general theory of initial-boundary value problems for balance laws is included, as well as an appendix illustrating the main concepts in the theory of conservation laws on networks

Control Problems for Conservation Laws with Traffic Applications

Conservation and balance laws on networks have been the subject of much research interest given their wide range of applications to real-world processes, particularly traffic flow. This open access monograph is the first to investigate different types of control problems for conservation laws that arise in the modeling of vehicular traffic. Four types of control problems are discussed - boundary, decentralized, distributed, and Lagrangian control - corresponding to, respectively, entrance points and tolls, traffic signals at junctions, variable speed limits, and the use of autonomy and communication. Because conservation laws are strictly connected to Hamilton-Jacobi equations, control of the latter is also considered. An appendix reviewing the general theory of initial-boundary value problems for balance laws is included, as well as an appendix illustrating the main concepts in the theory of conservation laws on networks

Macroscopic traffic flow optimization on roundabouts

The aim of this paper is to propose an optimization strategy for traffic flow on roundabouts using a macroscopic approach. The roundabout is modeled as a sequence of 2 × 2 with one mainline and secondary incoming and outgoing roads. We consider two cost the total travel time and the total waiting time, which give an estimate of the time by drivers on the network section. These cost functionals are minimized with respect to the of way parameter of the incoming roads. For each cost functional, the analytical expression given for each junction

Autonomous vehicles: From vehicular control to traffic control

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Adjoint-based optimization on a network of discretized scalar conservation law PDEs with applications to coordinated ramp metering

International audienceThe adjoint method provides a computationally efficient means of calculating the gradient for applications in constrained optimization. In this article, we consider a network of scalar conservation laws with general topology, whose behavior is modified by a set of control parameters in order to minimize a given objective function. After discretizing the corresponding partial differential equation models via the Godunov scheme, we detail the computation of the gradient of the discretized system with respect to the control parameters and show that the complexity of its computation scales linearly with the number of discrete state variables for networks of small vertex degree. The method is applied to solve the problem of coordinated ramp metering on freeway networks. Numerical simulations on the I15 freeway in California demonstrate an improvement in performance and running time compared to existing methods

Traffic flow optimization on roundabouts

International audienceThe aim of this article is to propose an optimization strategy for traffic flow on roundabouts using a macroscopic approach. The roundabout is modeled as a sequence of 2 × 2 junctions with one main-lane and secondary incoming and outgoing roads. We consider two cost functionals: the total travel time and the total waiting time, which give an estimate of the time spent by drivers on the network section. These cost functionals are minimized with respect to the right of way parameter of the incoming roads. For each cost functional, the analytical expression is given for each junction. We then solve numerically the optimization problem and show some numerical results

Continuous Modeling and Optimization Approaches for Manufacturing Systems

This thesis is concerned with two macroscopic models that are based on hyper- bolic partial differential equations (PDE) with discontinuous flux functions. The first model describes the material flow of an entire production line with finite buffers. We consider different solutions of the model, present the novel DFG- method (Discontinuous Flux Godunov), and compare the results with other established numerical methods. Additionally, we investigate a restricted optimization problem with respect to partial differential equations with discontinuous flux functions and consider two different solution approaches that are based on the adjoint method and the mixed integer problem (MIP). Further, we extend the model and its optimization problem to network structures. The second model describe the material flow on conveyor belts with obstacle interactions. We introduce a novel two dimensional model with a discontinuous and a non-local flux function. We consider a finite volume method and the discon- tinuous Galerkin method for solving this model. Finally, we validate the model with real data and present a numerical study with respect to the introduced solution methods