March  2018, 8(1): 177-193. doi: 10.3934/mcrf.2018008

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

Received  April 2017 Revised  September 2017 Published  January 2018

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.

Citation: Lino J. Alvarez-Vázquez, Néstor García-Chan, Aurea Martínez, Miguel E. Vázquez-Méndez. Optimal control of urban air pollution related to traffic flow in road networks. Mathematical Control and Related Fields, 2018, 8 (1) : 177-193. doi: 10.3934/mcrf.2018008
References:
[1]

L. J. Alvarez-VázquezN. García-ChanA. Martínez and M. E. Vázquez-Méndez, An application of interactive multi-criteria optimization to air pollution control, Optimization, 64 (2015), 1367-1380. 

[2]

L. J. Alvarez-VázquezN. García-ChanA. Martínez and M. E. Vázquez-Méndez, Stackelberg strategies for wastewater management, J. Comput. Appl. Math., 280 (2015), 217-230. 

[3]

L. J. Alvarez-VázquezN. García-ChanA. Martínez and M. E. Vázquez-Méndez, Numerical simulation of air pollution related to traffic flow in urban networks, J. Comput. Appl. Math., 326 (2017), 44-61. 

[4]

S. CanicB. PiccoliJ. Qiu and T. Ren, Runge-Kutta discontinuos Galerkin method for traffic flow model on networks, J. Sci. Comput., 63 (2015), 233-255. 

[5]

E. Casas, Pontryagin's principle for state constrained boundary control problems of semilinear parabolic equations, SIAM J. Control Optim., 35 (1997), 1297-1327. 

[6]

E. CasasC. Clason and K. Kunisch, Parabolic control problems in measure spaces with sparse solutions, SIAM J. Control Optim., 51 (2013), 28-63. 

[7]

E. CasasR. Herzog and G. Wachsmuth, Analysis of spatio-temporally sparse optimal control problems of semilinear parabolic equations, ESAIM Control Optim. Calc. Var., 23 (2017), 263-295. 

[8]

G. M. CocliteM. Garavello and M. Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886. 

[9]

M. Garavello and B. Piccoli, Conservation laws on complex networks, Ann. I. H. Poincare, 26 (2009), 1925-1951. 

[10]

N. García-ChanL. J. Alvarez-VázquezA. Martínez and M. E. Vázquez-Méndez, On optimal location and management of a new industrial plant: Numerical simulation and control, J. Franklin Institute, 351 (2014), 1356-1371. 

[11]

N. García-Chan, L. J. Alvarez-Vázquez, A. Martínez and M. E. Vázquez-Méndez, Numerical simulation for evaluating the effect of traffic restrictions on urban air pollution, in: Progress in Industrial Mathematics at ECMI 2016, (eds. P. Quintela et al.), Springer, in press.

[12]

P. GoatinS. Goettlich and O. Kolb, Speed limit and ramp meter control for traffic flow networks, Eng. Optimiz., 48 (2016), 1121-1144. 

[13]

S. GoettlichM. Herty and U. Ziegler, Modeling and optimizing traffic light settings in road networks, Comput. Oper. Res., 55 (2015), 36-51. 

[14]

S. P. Han, A globally convergent method for nonlinear optimization, J. Optim. Theory Appl., 22 (1977), 297-309. 

[15]

H. Holden and H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads, SIAM J. Math. Anal., 26 (1995), 999-1017. 

[16]

A. O. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, 1968.

[17]

J. LelieveldJ. S. EvansM. FnaisD. Giannadaki and A. Pozzer, The contribution of outdoor air pollution sources to premature mortality on a global scale, Nature, 525 (2015), 367-371. 

[18]

M. J. Lighthill and G. B. Whitham, On kinetic waves Ⅱ. Theory of traffic flows on long crowed roads, Proc. Roy. Soc. London Ser. A., 229 (1995), 317-345. 

[19]

G. I. Marchuk, Mathematical Models in Environmental Problems, Elsevier, New York, 1986.

[20]

A. MartínezM.E. Vázquez-MéndezR. Muñoz and L. J. Alvarez-Vázquez, A local regularity result for Neumann parabolic problems with nonsmooth data, Indagat. Math., 28 (2017), 494-515. 

[21]

J. A. Nelder and R. Mead, A simplex method for function minimization, Computer J., 7 (1965), 308-313. 

[22]

P. A. Nguyen and J. P. Raymond, Control localized on thin structures for the linearized Boussinesq system, J. Optim. Theory Appl., 141 (2009), 147-165. 

[23]

D. Parra-Guevara and Y. N. Skiba, Elements of the mathematical modeling in the control of pollutants emissions, Ecol. Model., 167 (2003), 263-275. 

[24]

P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51. 

[25]

Y. N. Skiba and D. Parra-Guevara, Control of emission rates, Atmosfera, 26 (2013), 379-400. 

[26]

M. Treiber and A. Kesting, Traffic Flow Dynamics: Data, Models and Simulation, Springer-Verlag, Berlin, 2013.

show all references

References:
[1]

L. J. Alvarez-VázquezN. García-ChanA. Martínez and M. E. Vázquez-Méndez, An application of interactive multi-criteria optimization to air pollution control, Optimization, 64 (2015), 1367-1380. 

[2]

L. J. Alvarez-VázquezN. García-ChanA. Martínez and M. E. Vázquez-Méndez, Stackelberg strategies for wastewater management, J. Comput. Appl. Math., 280 (2015), 217-230. 

[3]

L. J. Alvarez-VázquezN. García-ChanA. Martínez and M. E. Vázquez-Méndez, Numerical simulation of air pollution related to traffic flow in urban networks, J. Comput. Appl. Math., 326 (2017), 44-61. 

[4]

S. CanicB. PiccoliJ. Qiu and T. Ren, Runge-Kutta discontinuos Galerkin method for traffic flow model on networks, J. Sci. Comput., 63 (2015), 233-255. 

[5]

E. Casas, Pontryagin's principle for state constrained boundary control problems of semilinear parabolic equations, SIAM J. Control Optim., 35 (1997), 1297-1327. 

[6]

E. CasasC. Clason and K. Kunisch, Parabolic control problems in measure spaces with sparse solutions, SIAM J. Control Optim., 51 (2013), 28-63. 

[7]

E. CasasR. Herzog and G. Wachsmuth, Analysis of spatio-temporally sparse optimal control problems of semilinear parabolic equations, ESAIM Control Optim. Calc. Var., 23 (2017), 263-295. 

[8]

G. M. CocliteM. Garavello and M. Piccoli, Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886. 

[9]

M. Garavello and B. Piccoli, Conservation laws on complex networks, Ann. I. H. Poincare, 26 (2009), 1925-1951. 

[10]

N. García-ChanL. J. Alvarez-VázquezA. Martínez and M. E. Vázquez-Méndez, On optimal location and management of a new industrial plant: Numerical simulation and control, J. Franklin Institute, 351 (2014), 1356-1371. 

[11]

N. García-Chan, L. J. Alvarez-Vázquez, A. Martínez and M. E. Vázquez-Méndez, Numerical simulation for evaluating the effect of traffic restrictions on urban air pollution, in: Progress in Industrial Mathematics at ECMI 2016, (eds. P. Quintela et al.), Springer, in press.

[12]

P. GoatinS. Goettlich and O. Kolb, Speed limit and ramp meter control for traffic flow networks, Eng. Optimiz., 48 (2016), 1121-1144. 

[13]

S. GoettlichM. Herty and U. Ziegler, Modeling and optimizing traffic light settings in road networks, Comput. Oper. Res., 55 (2015), 36-51. 

[14]

S. P. Han, A globally convergent method for nonlinear optimization, J. Optim. Theory Appl., 22 (1977), 297-309. 

[15]

H. Holden and H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads, SIAM J. Math. Anal., 26 (1995), 999-1017. 

[16]

A. O. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, 1968.

[17]

J. LelieveldJ. S. EvansM. FnaisD. Giannadaki and A. Pozzer, The contribution of outdoor air pollution sources to premature mortality on a global scale, Nature, 525 (2015), 367-371. 

[18]

M. J. Lighthill and G. B. Whitham, On kinetic waves Ⅱ. Theory of traffic flows on long crowed roads, Proc. Roy. Soc. London Ser. A., 229 (1995), 317-345. 

[19]

G. I. Marchuk, Mathematical Models in Environmental Problems, Elsevier, New York, 1986.

[20]

A. MartínezM.E. Vázquez-MéndezR. Muñoz and L. J. Alvarez-Vázquez, A local regularity result for Neumann parabolic problems with nonsmooth data, Indagat. Math., 28 (2017), 494-515. 

[21]

J. A. Nelder and R. Mead, A simplex method for function minimization, Computer J., 7 (1965), 308-313. 

[22]

P. A. Nguyen and J. P. Raymond, Control localized on thin structures for the linearized Boussinesq system, J. Optim. Theory Appl., 141 (2009), 147-165. 

[23]

D. Parra-Guevara and Y. N. Skiba, Elements of the mathematical modeling in the control of pollutants emissions, Ecol. Model., 167 (2003), 263-275. 

[24]

P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51. 

[25]

Y. N. Skiba and D. Parra-Guevara, Control of emission rates, Atmosfera, 26 (2013), 379-400. 

[26]

M. Treiber and A. Kesting, Traffic Flow Dynamics: Data, Models and Simulation, Springer-Verlag, Berlin, 2013.

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