1. Equilibrium Manifolds in 2D Fluid Traffic Models
- Author
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Liudmila Tumash, Maria Laura Delle Monache, Carlos Canudas de Wit, Systèmes Commandés en Réseau (NECS-POST), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-GIPSA Pôle Automatique et Diagnostic (GIPSA-PAD), Grenoble Images Parole Signal Automatique (GIPSA-lab), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Grenoble Images Parole Signal Automatique (GIPSA-lab), Université Grenoble Alpes (UGA), ERC, European Project: 694209,Advanced Grant (AdG), ERC-2015-AdG ,Scale-FreeBack(2016), Networked Controlled Systems (NECS), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Département Automatique (GIPSA-DA), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Stendhal - Grenoble 3-Université Joseph Fourier - Grenoble 1 (UJF)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Stendhal - Grenoble 3-Université Joseph Fourier - Grenoble 1 (UJF)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Grenoble Images Parole Signal Automatique (GIPSA-lab), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Stendhal - Grenoble 3-Université Joseph Fourier - Grenoble 1 (UJF)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Stendhal - Grenoble 3-Université Joseph Fourier - Grenoble 1 (UJF)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), Département Automatique (GIPSA-DA), Université Stendhal - Grenoble 3-Université Joseph Fourier - Grenoble 1 (UJF)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP)-Centre National de la Recherche Scientifique (CNRS)-Université Stendhal - Grenoble 3-Université Joseph Fourier - Grenoble 1 (UJF)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP)-Centre National de la Recherche Scientifique (CNRS)-Grenoble Images Parole Signal Automatique (GIPSA-lab), Université Stendhal - Grenoble 3-Université Joseph Fourier - Grenoble 1 (UJF)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP)-Centre National de la Recherche Scientifique (CNRS)-Université Stendhal - Grenoble 3-Université Joseph Fourier - Grenoble 1 (UJF)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Grenoble - Rhône-Alpes, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)
- Subjects
050210 logistics & transportation ,Partial differential equation ,Steady states ,Field (physics) ,Plane (geometry) ,05 social sciences ,Mathematical analysis ,Coordinate system ,Flux ,Traffic flow ,Network topology ,Differential geometric methods ,01 natural sciences ,Partial differential equations ,010101 applied mathematics ,Control and Systems Engineering ,11. Sustainability ,0502 economics and business ,Urban systems ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,0101 mathematics ,Constant (mathematics) ,ComputingMilieux_MISCELLANEOUS ,Mathematics ,Road traffic - Abstract
International audience; The main goal of this paper is to analytically find a steady-state in a large-scale urban traffic network with known and constant demand and supply on its boundaries. Traffic dynamics are given by a continuous two-dimensional macroscopic model, where state corresponds to the vehicle density evolving in a 2D plane. Thereby, the flux magnitude is given by the space-dependent fundamental diagram and the flux direction depends on the underlying network topology. In order to find a steady-state, we use the coordinate transformation such that the 2D equation can be rewritten as a parametrized set of 1D equations. This technique allows us to obtain the curves along which the traffic flow evolves, which are essentially the integral curves of the flux field constructed from the network geometry. The results are validated by comparing the obtained steady-state with the one estimated by using a microsimulator.
- Published
- 2020