"A note on the Entropy Solutions of the Hydrodynamic Model of Traffic Flow" revisited

International audienceA central question in [Velan and Florian(2002)] is the influence of a non differentiable fundamental diagram on the solutions of the LWR model. This question is crucial because experimental observations put credit on piecewise linear fundamental diagrams (PLFD) [Leclercq(2005), Chiabaut et al.(2009)] and especially on triangular ones. In [Velan and Florian(2002)] it is claimed that, with the latter diagrams, the solution of the LWR model is unique but non-entropic. This note aims to invalidate this result. Considering a triangular fundamental diagram, we will demonstrate (i) that the weak solutions of the LWR are not unique and (ii) that the solution which is usually retained (and claimed to be unique) is in fact the unique weak entropy solution in the sense of Kruskov [Kruˇzkov(1970)]. This means that PLFD cannot be disproved on an alleged mathematical basis, since contrary to what is claimed in [Velan and Florian(2002)], the entropy criterion is indeed respected by its solutions

Heterogeneous perimeter flow distributions and MFD-based traffic simulation

This paper investigates how network and traffic heterogeneities influence the accuracy of a simulation based on the Macroscopic Fundamental Diagram (MFD). To this end, the MFD modeling of a simple grid network is compared to the outputs of a mesoscopic kinematic wave model simulating traffic in the same network. Heterogeneous distributions of demand and supply at the boundaries are set to the local entries and exits of the mesoscopic model to generate heterogeneous network loadings. These boundary conditions challenge the MFD simulation, as significant discrepancies are observed between both modeling approaches in steady state. While the accurate calibration of the MFD and the average trip length can reduce the discrepancies for heterogeneous demand settings, no simple solution exists for heterogeneous supply settings, because they may drive very different internal congestion patterns in the network. We propose a correction method to adjust the MFD model outputs in such a case

Dynamic macroscopic simulation of on-street parking search: a trip based approach

This paper extends a trip-based aggregate dynamic traffic model to account for on-street parking search. The trip-based approach for a road network defined as a reservoir characterizes the internal traffic states by a macroscopic fundamental diagram (MFD) in speed while individualizing all vehicle travel distances. This paper first investigates distances to park for on-street parking based on real data in Lyon (France) and stochastic numerical experiments. An updated formulation compared to the existing literature is proposed for the relation between such distances and the parking occupancy. This new formulation is then incorporated into an event-based numerical scheme that solves the trip-based MFD model

A MILP Framework to Solve the Sustainable System Optimum with Link MFD Functions

Given the increasing consciousness toward the environmental footprint of mobility, accommodating environmental objectives in existing transport planning strategies is imperative for research and practice. In this paper, we use the link macroscopic fundamental diagram (MFD) model to develop optimal routing strategies that minimize total system emissions (TSE) in multiple origin-destination (OD) networks. Piecewise linear (PWL) functions are used to approximate MFD for individual links, and to define link-level emissions. Dynamic network constraints, non-vehicle holding constraints, and convex formulations of the PWL functions are considered. Thus, the system-optimum dynamic traffic assignment (SO-DTA) problem with environmental objectives is formulated as a mixed integer linear program (MILP). Finally, on a synthetic network, numerical examples demonstrate the performance of the proposed framework

Macroscopic urban dynamics: Analytical and numerical comparisons of existing models

In this paper we compare a single reservoir model and a trip-based model under piecewise linear MFD and a piecewise constant demand. These assumptions allow to establish the exact solution of the accumulation-based model, and continuous approximations of the trip-based model at any order using Taylor series

Cross-comparison of convergence algorithms to solve trip-based dynamic traffic assignment problems

Solving a dynamic traffic assignment problem in a transportation network is a computational challenge. This study first reviews the different algorithms in the literature used to numerically calculate the User Equilibrium (UE) related to dynamic network loading. Most of them are based on iterative methods to solve a fixed-point problem. Two elements must be computed: the path set and the optimal path flow distribution between all origin-destination pairs. In a generic framework these two steps are referred to as the outer and the inner loops, respectively. The goal of this study is to assess the computational performance of the inner loop methods that calculate the path flow distribution for different network settings (mainly network size and demand levels). Several improvements are also proposed to speed up convergence: four new swapping algorithms and two new methods for the step size initialization used in each descent iteration. All these extensions significantly reduce the number of iterations to obtain a good convergence rate and drastically speed up the overall simulations. The results show that the performance of different components of the solution algorithm is sensitive to the network size and saturation. Finally, the best algorithms and settings are identified for all network sizes with particular attention being given to the largest scale