Optimal control of urban air pollution related to traffic flow in road networks

Lino J. Alvarez-Vázquez; Néstor García-Chan; Aurea Martínez; Miguel E. Vázquez-Méndez

doi:10.3934/mcrf.2018008

Article Contents

2018, Volume 8, Issue 1: 177-193. Doi: 10.3934/mcrf.2018008

This issue Previous Article Frequency-sparse optimal quantum control Next Article Error analysis for global minima of semilinear optimal control problems

Optimal control of urban air pollution related to traffic flow in road networks

1.
Depto. Matemática Aplicada Ⅱ, Universidade de Vigo, E.I. Telecomunicación, 36310 Vigo, Spain
2.
Depto. Física, Universidad de Guadalajara, C.U. Ciencias Exactas e Ingenierías, 44420 Guadalajara, Mexico
3.
Depto. Matemática Aplicada, Universidade de Santiago de Compostela, E. Politécnica Superior, 27002 Lugo, Spain

* Corresponding author
* Corresponding author

Received: March 31, 2017

Revised: August 31, 2017

Published: January 2018

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Abstract

Air pollution is one of the most important environmental problems nowadays. In large metropolitan areas, the main source of pollution is vehicular traffic. Consequently, the search for traffic measures that help to improve pollution levels has become a hot topic today. In this article, combining a 1D model to simulate the traffic flow over a road network with a 2D model for pollutant dispersion, we present a tool to search for traffic operations that are optimal in terms of pollution. The utility of this tool is illustrated by formulating the problem of the expansion of a road network as a problem of optimal control of partial differential equations. We propose a complete algorithm to solve the problem, and present some numerical results obtained in a realistic situation posed in the Guadalajara Metropolitan Area (GMA), Mexico.

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Mathematics Subject Classification: Primary: 49M25, 35Q93; Secondary: 90C56.

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Figure 1. Domain $ \Omega $ considered for the GMA (Mexico). The existing road network is represented with solid line (red), and with dotted line (blue) the expansion that is intended to build.

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Figure 2. Triangular Fundamental Diagram (TFD): function $f(\rho)$ (static relation) considered in the numerical experiment.

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Figure 3. Boundary condition for the LWR model on $A_1$ and $A_2$ (functions $\rho^{in}_1(t) = \rho^{in}_2(t)$), corresponding to a weekday with typical peak and valley hours.

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Figure 4. Field of wind velocities employed in the test, and air pollution isolines corresponding to the original network (a), and to the expanded network (b), after $ T = 24 $ hours.

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Figure 5. Mean flux of cars (a), and their mean velocity (b), averaged on the whole road network, along a time interval of $ T = 24 $ hours, for the original road network (solid line) and for the expanded one (dashed line).

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Figure 6. Mean car emmisions on the road network (a), and mean CO concentration on the whole domain $\Omega$ (b), along a time interval of $ T = 24 $ hours, for the original road network (solid line) and for the expanded one (dashed line).

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