Dynamic user equilibrium based on a hydrodynamic model

doi:10.1016/j.trb.2012.10.001

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Dynamic user equilibrium based on a hydrodynamic model

2013-01-01

PRE

Terry L. Friesz *

Ke Han

P.A. da Mota Silveira Neto

Amir H. Meimand

Tao Yao

DOI：10.1016/j.trb.2012.10.001

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Abstract

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In this paper we present a continuous-time network loading procedure based on the Lighthill-Whitham-Richards model proposed by Lighthill and Whitham (1955) and Richards (1956). A system of differential algebraic equations (DAEs) is proposed for describing traffic flow propagation, travel delay and route choices. We employ a novel numerical apparatus to reformulate the scalar conservation law as a flow-based partial differential equation (PDE), which is then solved semi-analytically with the Lax-Hopf formula. This approach allows for an efficient computational scheme for large-scale networks. We embed this network loading procedure into the dynamic user equilibrium (DUE) model proposed by Friesz et al. (1993). The DUE model is solved as a differential variational inequality (DVI) using a fixed-point algorithm. Several numerical examples of DUE on networks of varying sizes are presented, including the Sioux Falls network with a significant number of paths and origin-destination pairs (OD). The DUE model presented in this article can be formulated as a variational inequality (VI) as reported in Friesz et al. (1993). We will present the Kuhn-Tucker (KT) conditions for that VI, which is a linear system for any given feasible solution, and use them to check whether a DUE solution has been attained. In order to solve for the KT multiplier we present a decomposition of the linear system that allows efficient computation of the dual variables. The numerical solutions of DUE obtained from fixed-point iterations will be tested against the KT conditions and validated as legitimate solutions. (C) 2012 Elsevier Ltd. All rights reserved.

Keywords:

LWR model

Dynamic user equilibrium

Dynamic network loading

Lax-Hopf formula

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