Lax–Hopf Based Incorporation of Internal Boundary Conditions Into Hamilton–Jacobi Equation. Part I: Theory

Claudel, Christian; Bayen, Alexandre M.

doi:10.1109/tac.2010.2041976

Cited by 134 publications

(146 citation statements)

References 48 publications

(139 reference statements)

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“…In many traffic flow applications [4,24] a piecewise linear flow function (called triangular) is used. However, our focus is the consideration of a nonlinear flow function that will lead to a more challenging nonlinear mixed-integer problem.…”

Section: B55mentioning

confidence: 99%

Partial Outer Convexification for Traffic Light Optimization in Road Networks

Göttlich¹,

Potschka²,

Ziegler³

2017

SIAM J. Sci. Comput.

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Abstract. We consider the problem of computing optimal traffic light programs for urban road intersections using traffic flow conservation laws on networks. Based on a partial outer convexification approach, which has been successfully applied in the area of mixed-integer optimal control for systems of ordinary or differential algebraic equations, we develop a computationally tractable two-stage solution heuristic. The two-stage approach consists of the solution of a (smoothed) nonlinear programming problem with dynamic constraints and a reconstruction mixed-integer linear program without dynamic constraints. The two-stage approach is founded on a discrete approximation lemma for partial outer convexification, whose grid-independence properties for (smoothed) conservation laws are investigated. We use the two-stage approach to compute traffic light programs for two scenarios on different discretizations and demonstrate that the solution candidates cannot be improved in a reasonable amount of time by global state-of-the-art mixed-integer nonlinear programming solvers. The two-stage solution candidates are not only better than results obtained by global optimization of piecewise linearized traffic flow models but also can be computed at a faster rate.Key words. partial outer convexification, traffic networks, discretized conservation laws, optimization, mixed-integer programming AMS subject classifications. 35L65, 49J20, 90C11, 90C35 DOI. 10.1137/15M10481971. Introduction. With their seminal papers [22,25] in the 1950s on timedependent models for traffic flow with continuous densities instead of individual vehicles, Lighthill and Whitham and Richards established a new area of mathematical research in traffic modeling. The area is still highly active, e.g., for the case of traffic intersections (see, e.g., [5,8]), which constitute a building block of larger road networks. Our goal in this paper is to devise efficient heuristics to approximately solve the problem of optimal traffic light settings for road networks. We base our work on the model and scenarios presented in [12] and refer the reader to the discussion therein for further references regarding the wide variety of different approaches for traffic light control or related problems such as spillover dissipation [11,23].The resulting mathematical optimization problems, which we elaborate in full detail in section 2, can be described as nonlinear mixed-integer optimal control problems constrained by scalar hyperbolic conservation laws on networks using boundary control. These problems are extremely challenging for several reasons: First, the problems are infinite dimensional in nature and must be discretized appropriately. Special care has to be exercised for the discretization of the scalar conservation laws in order to guarantee well-posedness and, on the discrete level, consistency and sta-

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Section: B55mentioning

confidence: 99%

Partial Outer Convexification for Traffic Light Optimization in Road Networks

Göttlich¹,

Potschka²,

Ziegler³

2017

SIAM J. Sci. Comput.

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“…The analysis of the signalized junction involves a semi-analytical solution representation of the Hamilton-Jacobi equation (3.10), namely the generalized Lax-Hopf formula (Aubin et al, 2008;Claudel and Bayen, 2010). For the compactness of our presentation we only show relevant results below, and refer the reader to for more detailed discussion and proof.…”

Section: Approximation Efficacy Of the Continuum Signal Modelmentioning

confidence: 99%

A bi-level model of dynamic traffic signal control with continuum approximation

Han

Sun

Liu

et al. 2015

Transportation Research Part C: Emerging Technologies

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This paper proposes a bi-level model for traffic network signal control, which is formulated as a dynamic Stackelberg game and solved as a mathematical program with equilibrium constraints (MPEC). The lower-level problem is a dynamic user equilibrium (DUE) with embedded dynamic network loading (DNL) sub-problem based on the LWR model (Lighthill and Whitham, 1955;Richards, 1956). The upper-level decision variables are (time-varying) signal green splits with the objective of minimizing network-wide travel cost. Unlike most existing literature which mainly use an on-and-off (binary) representation of the signal controls, we employ a continuum signal model recently proposed and analyzed in , which aims at describing and predicting the aggregate behavior that exists at signalized intersections without relying on distinct signal phases. Advantages of this continuum signal model include fewer integer variables, less restrictive constraints on the time steps, and higher decision resolution. It simplifies the modeling representation of large-scale urban traffic networks with the benefit of improved computational efficiency in simulation or optimization. We present, for the LWR-based DNL model that explicitly captures vehicle spillback, an in-depth study on the implementation of the continuum signal model, as its approximation accuracy depends on a number of factors and may deteriorate greatly under certain conditions. The proposed MPEC is solved on two test networks with three metaheuristic methods. Parallel computing is employed to significantly accelerate the solution procedure.

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“…Assuming a previous update exists, the validity of the next update can be determined based on the computed speed and the temporal/spatial gap from previously filtered GPS reading called PrevLocationFiltered. We consider the current location invalid if it is updated long after the previous update (line [10][11][12][13][14], if it has not traveled a minimum distance (line 16-18, e.g., stopped at the traffic signal), or if has a speed glitch (line [19][20][21][22][23][24][25][26][27][28][29][30].…”

Section: B Virtual Trip Line Measurementsmentioning

confidence: 99%

“…While our work [10], [11], [42] explicitly derives techniques to reconstruct traffic from VTL type data, such a reconstruction becomes harder for arterials. Also, while macroscopic flow models such as the ones used in [10], [11], [42] exist and can be used for secondary networks [17], [39], their parameters are in general unknown or inaccessible and only documented for few cities, making their use impossible without going to the field and measuring them. In addition, even if they were known, the complexity of the underlying flows makes it challenging to perform estimation of the full macroscopic state of the system at low penetration rates.…”

Section: Challenges In Arterial Roads Traffic Estimationmentioning

confidence: 99%

Enhancing Privacy and Accuracy in Probe Vehicle-Based Traffic Monitoring via Virtual Trip Lines

Hoh

Iwuchukwu

Jacobson

et al. 2012

IEEE Trans. on Mobile Comput.

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Abstract-Traffic monitoring using probe vehicles with GPS receivers promises significant improvements in cost, coverage, and accuracy over dedicated infrastructure systems. Current approaches, however, raise privacy concerns because they require participants to reveal their positions to an external traffic monitoring server. To address this challenge, we describe a system based on virtual trip lines and an associated cloaking technique, followed by another system design in which we relax the privacy requirements to maximize the accuracy of real-time traffic estimation.We introduce virtual trip lines which are geographic markers that indicate where vehicles should provide speed updates. These markers are placed to avoid specific privacy sensitive locations. They also allow aggregating and cloaking several location updates based on trip line identifiers, without knowing the actual geographic locations of these trip lines. Thus, they facilitate the design of a distributed architecture, in which no single entity has a complete knowledge of probe identities and fine-grained location information. We have implemented the system with GPS smartphone clients and conducted a controlled experiment with 100 phone-equipped drivers circling a highway segment, which was later extended into a year-long public deployment.

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Lax–Hopf Based Incorporation of Internal Boundary Conditions Into Hamilton–Jacobi Equation. Part I: Theory

Cited by 134 publications

References 48 publications

Partial Outer Convexification for Traffic Light Optimization in Road Networks

Partial Outer Convexification for Traffic Light Optimization in Road Networks

A bi-level model of dynamic traffic signal control with continuum approximation

Enhancing Privacy and Accuracy in Probe Vehicle-Based Traffic Monitoring via Virtual Trip Lines

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