Analytical and grid-free solutions to the Lighthill–Whitham–Richards traffic flow model

Mazaré, Pierre-Emmanuel; Dehwah, Ahmad H.; Claudel, Christian; Bayen, Alexandre M.

doi:10.1016/j.trb.2011.07.004

Cited by 107 publications

(74 citation statements)

References 28 publications

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“…which is the same as the description of the dynamics of each section i ∈ N r of sub-network r ∈ [R] given in (8). Thus, if y r (0) = x r (0) = x r 0 for all r ∈ [R] then x r (t) = y r (t) and (i) holds.…”

Section: K−1] Then the Closed-loop System (3) Under The Decentmentioning

confidence: 96%

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A linear programming approach to routing control in networks of constrained nonlinear positive systems with concave flow rates

2016

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We consider control design for positive compartmental systems in which each compartment's outflow rate is described by a concave function of the amount of material in the compartment. We address the problem of determining the routing of material between compartments to satisfy time-varying state constraints while ensuring that material reaches its intended destination over a finite time horizon. We give sufficient conditions for the existence of a time-varying state-dependent routing strategy which ensures that the closed-loop system satisfies basic network properties of positivity, conservation and interconnection while ensuring that capacity constraints are satisfied, when possible, or adjusted if a solution cannot be found. These conditions are formulated as a linear programming problem. Instances of this linear programming problem can be solved iteratively to generate a solution to the finite horizon routing problem. Results are given for the application of this control design method to an example problem.

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Section: K−1] Then the Closed-loop System (3) Under The Decentmentioning

confidence: 96%

“…Examples motivating the use of this type of outflow model come from road traffic [8] and air traffic management research. The authors of [7] point out that, although the outflow of a section of airspace will increase as the density of traffic increases, there is an upper bound on the outflow rate.…”

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confidence: 99%

A linear programming approach to routing control in networks of constrained nonlinear positive systems with concave flow rates

2016

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“…The solutions to (2.2) need to satisfy additional equality constraints associated with initial, upstream, and downstream boundary conditions, denoted as c ini (x), c up (t) and c down (t), respectively. Work in [29] presents the Cauchy problem in the form of…”

Section: B Cauchy Problem On Single Highway Linkmentioning

confidence: 99%

“…More details can be found in [29]. However, equalities in Cauchy problem (2.3) cannot be incorporated into the traffic flow dynamic model due to the continuous time t and unknown variable x.…”

Section: B-j/f Solutions and Model Constraintsmentioning

confidence: 99%

Hybrid Optimal Control for Time-Efficient Highway Traffic Management

Liu

Dai

et al. 2018

2018 Annual American Control Conference (ACC)

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This article examines the hybrid traffic control problem to minimize total travel time (TTT) of a highway network through traffic management infrastructures, including dynamic speed limit signs, ramp metering, and information board.We first build the traffic flow model based on the Moskowitz function for each highway link to predict traffic status within a control horizon. The traffic density is predicted based on the flow dynamic model and corrected periodically by measured traffic flow data. The minimum TTT traffic control problem is then formulated as a mixed-integer quadratic programming problem with quadratic constraints. Numerical simulation of a real world highway network is provided to demonstrate significant reduction of TTT and alleviation of traffic congestion compared to results obtained from ALINEA and PI-ALINEA methods. Abstract-This article examines the hybrid traffic control problem to minimize total travel time (TTT) of a highway network through traffic management infrastructures, including dynamic speed limit signs, ramp metering, and information board.We first build the traffic flow model based on the Moskowitz function for each highway link to predict traffic status within a control horizon. The traffic density is predicted based on the flow dynamic model and corrected periodically by measured traffic flow data. The minimum TTT traffic control problem is then formulated as a mixed-integer quadratic programming problem with quadratic constraints. Numerical simulation of a real world highway network is provided to demonstrate significant reduction of TTT and alleviation of traffic congestion compared to results obtained from ALINEA and PI-ALINEA methods.

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“…Among others, it allows to improve modeling flexibility e.g. taking into account moving bottlenecks [14,15] and it provides accurate algorithms for numerical treatment and data assimilation [7,8,20,45]. Up to our best knowledge, existing numerical procedures relying on variational formulation of the LWR model use the seminal paper of Claudel and Bayen [6] which proposes a semiexplicit form of the solution to Hamilton-Jacobi equations using control techniques based on viability theory [2,3].…”

Section: Motivation Of Lax-hopf Algorithmsmentioning

confidence: 99%

A variational formulation for higher order macroscopic traffic flow models: Numerical investigation

Costeseque

Lebacque²

2014

Transportation Research Part B: Methodological

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This paper deals with numerical methods providing semi-analytic solutions to a wide class of macroscopic traffic flow models for piecewise affine initial and boundary conditions. In a very recent paper, a variational principle has been proved for models of the Generic Second Order Modeling (GSOM) family, yielding an adequate framework for effective numerical methods. Any model of the GSOM family can be recast into its Lagrangian form as a Hamilton-Jacobi equation (HJ) for which the solution is interpreted as the position of vehicles. This solution can be computed thanks to Lax-Hopf like formulas and a generalization of the inf-morphism property. The efficiency of this computational method is illustrated through a numerical example and finally a discussion about future developments is provided.Keywords: Traffic flow, Hamilton-Jacobi equation, Lax-Hopf algorithm, Lagrangian. Introduction General backgroundMacroscopic traffic flow modeling. In order to get a realistic estimation of the real-time traffic states on networks, traffic operators and managers need macroscopic traffic flow models. These models must be simple, robust, allowing to get solutions at a low computational cost. The main macroscopic models are based on conservation laws or hyperbolic systems (see [30,19] for a review). The seminal LWR model (for Lighthill-Whitham and Richards) was proposed in [42,51] as a single conservation law with unknown the vehicles density. This model based on a first order Partial Differential Equation (PDE) is very simple and robust but it fails to recapture some empirical features of traffic. In particular, it does not allow to take into account non-equilibrium traffic states mainly in congested situation. More sophisticated models referred to as higher order models were developed to encompass kinematic constraints of real vehicles or also the wide variety of driver behaviors, even at the macroscopic level. In this paper we deal with models of the Generic Second Order Modeling (GSOM) family. Even if these models are more complicated to deal with, they permit to reproduce traffic instabilities (such as the so-called stop-and-go waves, the hysteresis phenomenon or capacity drop) which move at the traffic speed and differ from kinematic waves [53] Traffic flow monitoring. Before the wide propagation of internet handsets, traffic monitoring has mainly been built on dedicated infrastructure which imply quite important installation and maintenance costs. Traffic flow monitoring and management has been deeply modified with the development of new technologies in mobile sensing aiming to provide a quite important quantity of floating car data. Traffic flow models are needed to be well suited such that managers could use both Eulerian and Lagrangian data for improving traffic state estimation. The term Eulerian refers to "classical" fixed equipment giving records of occupancy or flow of vehicles on a freeway section. This kind of measurements come from e.g. fixed inductive loop detectors, Radio Frequency Identification (R...

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Analytical and grid-free solutions to the Lighthill–Whitham–Richards traffic flow model

Cited by 107 publications

References 28 publications

A linear programming approach to routing control in networks of constrained nonlinear positive systems with concave flow rates

A linear programming approach to routing control in networks of constrained nonlinear positive systems with concave flow rates

Hybrid Optimal Control for Time-Efficient Highway Traffic Management

A variational formulation for higher order macroscopic traffic flow models: Numerical investigation

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