A Convex Formulation of Traffic Dynamics on Transportation Networks

Abstract

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.

Error bounds for Kalman filters on traffic networks

“…Junction solver on traffic networks. This subsection introduces the junction solver proposed in [24]. As shown in Figure 2a, the junction solver computes flux f(ρ l k , ρ i k ) and f(ρ l k , ρ j k ) between the connecting cells l, i and j forming a diverge junction.…”

“…The diverge junction solver is posed as a convex program with a carefully constructed objective function to accommodate the throughput maximization (R.2) and the flow distribution (R.3). The mathematical formula of the diverge junction solver is given below (see [24,Section 3.2] for more details).…”

“…Solution point: Solution point: models have been proposed to resolve this issue [14,20,22]. In the same spirit of these models, the diverge junction solver applied in this article is developed to produce similar traffic condition dependent solutions without introducing additional complexity on the traffic dynamics [24]. As a related note, the results proved in this article can be extended to FIFO models with minor changes to the proof.…”

“…For conciseness of the presentation, we provide next the explicit formulas of A k , B ρ k , B q k , and B φ k when the fluxes are computed according to (24), and the construction of the system dynamics when the fluxes are given by (25) can be done in a similar fashion. The matrices A k , B ρ k , B q k , and B φ k read This property is due to the CFL condition [33] in the discretization scheme (8).…”

“…The SMM-J is an extension of the SMM, in the sense that the SMM-J describes the traffic density evolution on a road section with a junction, while the SMM only considers one-dimensional (i.e., without junctions) road sections. The SMM-J combines a switched linear system representation of the CTM with the junction model developed in [24], however, other junction models can also be incorporated in a similar fashion to derive linear traffic models on junctions. We also provide the observability result on each mode of the SMM-J, and show that nearly all modes are unobservable due to the irreversibility of the conservation laws in the presence of shocks and junctions.…”