A Riemann solver for a system of hyperbolic conservation laws at a general road junction

Jin, Wen-Long

doi:10.1016/j.trb.2016.12.007

Cited by 18 publications

(13 citation statements)

References 51 publications

(67 reference statements)

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“…Substituting the flows computed above into the update scheme of the traffic density on each cell, it follows that the explicit forms of A k , B ρ k , B q k , and B φ k in (19) are…”

Section: 2mentioning

confidence: 99%

Error bounds for Kalman filters on traffic networks

Sun¹,

Work²

2018

Networks &Amp; Heterogeneous Media

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This work analyzes the estimation performance of the Kalman filter (KF) on transportation networks with junctions. To facilitate the analysis, a hybrid linear model describing traffic dynamics on a network is derived. The model, referred to as the switching mode model for junctions, combines the discretized Lighthill-Whitham-Richards partial differential equation with a junction model. The system is shown to be unobservable under nearly all of the regimes of the model, motivating attention to the estimation error bounds in these modes. The evolution of the estimation error is investigated via exploring the interactions between the update scheme of the KF and the intrinsic physical properties embedded in the traffic model (e.g., conservation of vehicles and the flow-density relationship). It is shown that the state estimates of all the cells in the traffic network are ultimately bounded inside a physically meaningful interval, which cannot be achieved by an open-loop observer.

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“…Substituting the flows computed above into the update scheme of the traffic density on each cell, it follows that the explicit forms of A k , B ρ k , B q k , and B φ k in (19) are…”

Section: 2mentioning

confidence: 99%

Error bounds for Kalman filters on traffic networks

Sun¹,

Work²

2018

Networks &Amp; Heterogeneous Media

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“…This subsection briefly describes the experiment setup and the next subsection constructs an optimal on-ramp metering controller using the convex program (19) and the additional constraints (23).…”

Section: Simulation Configurationmentioning

confidence: 99%

“…It is well known that conservation of vehicles across the junction, i.e., s∈Sv q s (t, b s ) = r∈Rv q r (t, a r ), is insufficient to uniquely define the flows at the junction. To address this issue, a variety of junction models [10,15,18,19,20,22,23,24,27] have been proposed to define a unique internal boundary flow solution using additional rules governing the distribution or priority of the flows. Compared to the merge junction, for which relatively few models have been proposed, the diverge junctions have led to a number of modeling efforts.…”

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confidence: 99%

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A Convex Formulation of Traffic Dynamics on Transportation Networks

Claudel

Piccoli³

et al. 2017

SIAM J. Appl. Math.

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This article proposes a numerical scheme for computing the evolution of vehicular traffic on a road network over a finite time horizon. The traffic dynamics on each link is modeled by the Hamilton-Jacobi (HJ) partial differential equation (PDE), which is an equivalent form of the Lighthill-Whitham-Richards PDE. The main contribution of this article is the construction of a single convex optimization program which computes the traffic flow at a junction over a finite time horizon and decouples the PDEs on connecting links. Compared to discretization schemes which require the computation of all traffic states on a time-space grid, the proposed convex optimization approach computes the boundary flows at the junction using only the initial condition on links and the boundary conditions of the network. The computed boundary flows at the junction specify the boundary condition for the HJ PDE on connecting links, which then can be separately solved using an existing semi-explicit scheme for single link HJ PDE. As demonstrated in a numerical example of ramp metering control, the proposed convex optimization approach also provides a natural framework for optimal traffic control applications.

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“…In (Jin, 2012b), it was shown that such junction flux functions can be used as entropy conditions to pick out unique and physical solutions of the multi-commodity LWR model (1c) at network junctions. In this study, we use the invariant junction model developed in (Jin, 2012a), which substantially simplifies analyses by eliminating interior states from solutions. Here the upstream demands and turning proportions are defined at (L − a , t) (a ∈ I j ), downstream supplies at (0 + , t) (b ∈ O j ), and fluxes at the junction point (L a for upstream link a and 0 for downstream link b).…”

Section: A Network Kinematic Wave Modelmentioning

confidence: 99%

On the stability of stationary states in general road networks

Jin

2017

Transportation Research Part B: Methodological

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In (Jin, 2015), with a discrete map in critical demand levels, it was proved that there exist stationary states for the kinematic wave model of general road networks with constant origin demands, route choice proportions, and destination supplies. In this study we further examine the stability property of stationary states with the same map, and the results will help us to understand the long-term trend of a network traffic system. We first review a network kinematic wave model and properties of stationary states on a link, define the criticality of junctions in stationary states, and discuss information propagation in stationary states on links and junctions. We then present the map and examine information propagation in the map. We apply the map to analytically study the stability of stationary states on ring roads and divergemerge networks with circular information propagation and compare them with results obtained from the Poincaré map (Jin, 2013). We further study the stability property of general stationary states in a grid network. We find that the stability of fixed points of the map is the same as that of stationary states in a network, and the new approach is more general than the Poincaré map approach. We conclude the study with future directions and implications.

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A Riemann solver for a system of hyperbolic conservation laws at a general road junction

Cited by 18 publications

References 51 publications

Error bounds for Kalman filters on traffic networks

Error bounds for Kalman filters on traffic networks

A Convex Formulation of Traffic Dynamics on Transportation Networks

On the stability of stationary states in general road networks

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