# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021192
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## A computational modular approach to evaluate ${\mathrm{NO_{x}}}$ emissions and ozone production due to vehicular traffic

 1 Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma, Rome, 00161, Italy 2 Istituto per le Applicazioni del Calcolo "M. Picone", Consiglio Nazionale delle Ricerche, Rome, 00185, Italy 3 Department of Mathematical Sciences, Rutgers University, Camden, NJ 08102, USA

* Corresponding author: Caterina Balzotti

Received  November 2020 Revised  June 2021 Early access July 2021

Fund Project: C. B., M. B. and B. D. F. were supported by the Italian Ministry of Instruction, University and Research (MIUR) under PRIN Project 2017 No. 2017KKJP4X, SMARTOUR Project No. B84G14000580008, and by the CNR TIRS Project FOE 2020. B. P.'s work was supported by the National Science Foundation under Cyber-Physical Systems Synergy Grant No. CNS-1837481

The societal impact of traffic is a long-standing and complex problem. We focus on the estimation of ground-level ozone production due to vehicular traffic. We propose a comprehensive computational approach combining four consecutive modules: a traffic simulation module, an emission module, a module for the main chemical reactions leading to ozone production, and a module for the diffusion of gases in the atmosphere. The traffic module is based on a second-order traffic flow model, obtained by choosing a special velocity function for the Collapsed Generalized Aw-Rascle-Zhang model. A general emission module is taken from literature, and tuned on NGSIM data together with the traffic module. Last two modules are based on reaction-diffusion partial differential equations. The system of partial differential equations describing the main chemical reactions of nitrogen oxides presents a source term given by the general emission module applied to the output of the traffic module. We use the proposed approach to analyze the ozone impact of various traffic scenarios and describe the effect of traffic light timing. The numerical tests show the negative effect of vehicles restarts on emissions, and the consequent increase in pollutants in the air, suggesting to increase the length of the green phase of traffic lights.

Citation: Caterina Balzotti, Maya Briani, Barbara De Filippo, Benedetto Piccoli. A computational modular approach to evaluate ${\mathrm{NO_{x}}}$ emissions and ozone production due to vehicular traffic. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021192
##### References:
 [1] L. J. Alvarez-Vázquez, N. García-Chan, A. Martínez and M. E. Vázquez-Méndez, Numerical simulation of air pollution due to traffic flow in urban networks, J. Comput. Appl. Math., 326 (2017), 44-61.  doi: 10.1016/j.cam.2017.05.017.  Google Scholar [2] L. J. Alvarez-Vázquez, N. García-Chan, A. Martínez and M. E. Vázquez-Méndez, Optimal control of urban air pollution related to traffic flow in road networks, Math. Control Relat. F., 8 (2018), 177-193.  doi: 10.3934/mcrf.2018008.  Google Scholar [3] R. Atkinson, Atmospheric chemistry of $\mathrm{VOC}s$ and $\mathrm{NO_x}$, Atmos. Environ., 34 (2000), 2063-2101.  doi: 10.1016/S1352-2310(99)00460-4.  Google Scholar [4] R. Atkinson and W. P. L. Carter, Kinetics and mechanisms of the gas-phase reactions of ozone with organic compounds under atmospheric conditions, Chem. Rev., 84 (1984), 437-470.  doi: 10.1021/cr00063a002.  Google Scholar [5] A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938.  doi: 10.1137/S0036139997332099.  Google Scholar [6] M. Barth, F. An, T. Younglove, G. Scora, C. Levine, M. Ross and T. Wenzel, Development of a Comprehensive Modal Emissions Model: Final Report, Technical report, National Research Council, Transportation Research Board, National Cooperative Highway Research Program, NCHRP Project 25–11, 2000. Google Scholar [7] D. C. Carslaw, S. D. Beevers, J. E. Tate, E. J. Westmoreland and M. L. Williams, Recent evidence concerning higher $\mathrm {NO_x}$ emissions from passenger cars and light duty vehicles, Atmos. Environ., 45 (2011), 7053-7063.  doi: 10.1016/j.atmosenv.2011.09.063.  Google Scholar [8] D. de la Fuente, J. M. Vega, F. Viejo, I. Díaz and M. Morcillo, Mapping air pollution effects on atmospheric degradation of cultural heritage, J. Cult. Herit., 14 (2013), 138-145.  doi: 10.1016/j.culher.2012.05.002.  Google Scholar [9] European Environment Agency, Air Quality in Europe – 2019 Report, Technical Report, 2019. Google Scholar [10] S. Fan, M. Herty and B. Seibold, Comparative model accuracy of a data-fitted generalized Aw-Rascle-Zhang model, Netw. Heterog. Media, 9 (2014), 239-268.  doi: 10.3934/nhm.2014.9.239.  Google Scholar [11] S. Fan, Y. Sun, B. Piccoli, B. Seibold and D. B. Work, A collapsed generalized Aw-RascleZhang model and its model accuracy, arXiv preprint, arXiv: 1702.03624. Google Scholar [12] F. J. Fernández, L. J. Alvarez-Vázquez, N. García-Chan, A. Martínez and M. E. Vázquez-Méndez, Optimal location of green zones in metropolitan areas to control the urban heat island, J. Comput. Appl. Math., 289 (2015), 412-425.  doi: 10.1016/j.cam.2014.10.023.  Google Scholar [13] M. Garavello, K. Han and B. Piccoli, Models for Vehicular Traffic on Networks, American Institute of Mathematical Sciences, 2016.  Google Scholar [14] E. Hairer and G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential-Algbraic Problem, Second edition, Springer Series in Computational Mathematics, 1996. doi: 10.1007/978-3-642-05221-7.  Google Scholar [15] D. J. Jacob, Heterogeneous chemistry and tropospheric ozone, Atmos. Environ., 34 (2000), 2131-2159.  doi: 10.1016/S1352-2310(99)00462-8.  Google Scholar [16] M. Z. Jacobson, Fundamentals of Atmospheric Modeling, Cambridge University Press, 2005. doi: 10.1017/CBO9781139165389.  Google Scholar [17] T. Koto, IMEX Runge-Kutta schemes for reaction-diffusion equations, J. Comput. Appl. Math., 215 (2008), 182-195.  doi: 10.1016/j.cam.2007.04.003.  Google Scholar [18] J. D. Lambert, Numerical Methods for Ordinary Differential Systems, John Wiley & Sons, Ltd., Chichester, 1991.  Google Scholar [19] J.-P. Lebacque, S. Mammar and H. Haj-Salem, Generic second order traffic flow modelling, in Transportation and Traffic Theory, Elsevier, (2007), 755–776. Google Scholar [20] M. J. Lighthill and G. B. Whitham, On kinematic waves II. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. A, 229 (1955), 317-345.  doi: 10.1098/rspa.1955.0089.  Google Scholar [21] T. Luspay, B. Kulcsar, I. Varga, S. K. Zegeye, B. De Schutter and M. Verhaegen, On acceleration of traffic flow, in Proceedings of the 13th International IEEE Conference on Intelligent Transportation Systems (ITSC 2010), IEEE, (2010), 741–746. doi: 10.1109/ITSC.2010.5625204.  Google Scholar [22] S. Manahan, Environmental Chemistry, CRC press, 2017.  doi: 10.1201/9781315160474.  Google Scholar [23] H. Omidvarborna, A. Kumar and D.-S. Kim, $\mathrm{NO_x}$ emissions from low-temperature combustion of biodiesel made of various feedstocks and blends, Fuel Process. Technol., 140 (2015), 113-118.  doi: 10.1016/j.fuproc.2015.08.031.  Google Scholar [24] L. I. Panis, S. Broekx and R. Liu, Modelling instantaneous traffic emission and the influence of traffic speed limits, Sci. Total Environ., 371 (2006), 270-285.  doi: 10.1016/j.scitotenv.2006.08.017.  Google Scholar [25] B. Piccoli, K. Han, T. L. Friesz, T. Yao and J. Tang, Second-order models and traffic data from mobile sensors, Transp. Res. Part C: Emerg. Technol., 52 (2015), 32-56.  doi: 10.1016/j.trc.2014.12.013.  Google Scholar [26] V. Ramanathan and Y. feng, Air pollution, greenhouse gases and climate change: {G}lobal and regional perspectives, Atmos. Environ., 43 (2009), 37-50.  doi: 10.1016/j.atmosenv.2008.09.063.  Google Scholar [27] P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.  Google Scholar [28] M. Rößler, T. Koch, C. Janzer and M. Olzmann, Mechanisms of the NO$_2$ formation in diesel engines, MTZ Worldw., 78 (2017), 70-75.  doi: 10.1007/s38313-017-0057-2.  Google Scholar [29] S. Samaranayake, S. Glaser, D. Holstius, J. Monteil, K. Tracton, E. Seto and A. Bayen, RealTime estimation of pollution emissions and dispersion from highway traffic, Comput.-Aided Civ. Inf., 29 (2014), 546-558.  doi: 10.1111/mice.12078.  Google Scholar [30] J. H. Seinfeld and S. N. Pandis, Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, John Wiley & Sons, 2016. doi: 10.1063/1.882420.  Google Scholar [31] R. Smit, L. Ntziachristos and P. Boulter, Validation of road vehicle and traffic emission models – A review and meta-analysis, Atmos. Environ., 44 (2010), 2943-2953.  doi: 10.1016/j.atmosenv.2010.05.022.  Google Scholar [32] F. Song, J. Y. Shin, R. Jusino-Atresino and Y. Gao, Relationships among the springtime ground–level $\mathrm{NO_x}$, $\mathrm{O}_3$ and $\mathrm{NO_3}$ in the vicinity of highways in the US East Coast, Atmos. Pollut. Res., 2 (2011), 374-383.  doi: 10.5094/APR.2011.042.  Google Scholar [33] B. Sportisse, Fundamentals in Air Pollution: From Processes to Modelling, Springer-Verlag, 2010. Google Scholar [34] J. M. Stockie, The mathematics of atmospheric dispersion modeling, SIAM Rev., 53 (2011), 349-372.  doi: 10.1137/10080991X.  Google Scholar [35] J. Tidblad, K. Kreislová, M. Faller, D. de la Fuente, T. Yates, A. Verney-Carron, T. Grøntoft, A. Gordon and U. Hans, ICP materials trends in corrosion, soiling and air pollution (1987–2014), Materials, 10 (2017). doi: 10.3390/ma10080969.  Google Scholar [36] Transportation Research Board, Critical Issues in Transportation 2019, Technical report, The National Academies of Sciences, Engineering, Medicine, 2019. Google Scholar [37] TRB Executive Committee, Special Report 307: Policy Options for Reducing Energy and Greenhouse Gas Emissions from U.S. Transportation, Technical Report, Transportation Research Board of the National Academies, 2011. Google Scholar [38] US Department of Transportation and Federal Highway Administration, Next generation simulation (NGSIM), http://ops.fhwa.dot.gov/trafficanalysistools/ngsim.htm. Google Scholar [39] T. Wang, L. Xue, P. Brimblecombe, Y. F. Lam, L. Li and L. Zhang, Ozone pollution in China: A review of concentrations, meteorological influences, chemical precursors, and effects, Sci. Total Environ., 575 (2017), 1582-1596.  doi: 10.1016/j.scitotenv.2016.10.081.  Google Scholar [40] R. P. Wayne, Chemistry of Atmospheres, Clarendon Press, Oxford, 1991.   Google Scholar [41] S. K. Zegeye, B. De Schutter, J. Hellendoorn, E. A. Breunesse and A. Hegyi, Integrated macroscopic traffic flow, emission, and fuel consumption model for control purposes, Transp. Res. Part C: Emerg. Technol., 31 (2013), 158-171.  doi: 10.1016/j.trc.2013.01.002.  Google Scholar [42] H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transp. Res. B, 36 (2002), 275-290.  doi: 10.1016/S0191-2615(00)00050-3.  Google Scholar [43] K. Zhang and S. Batterman, Air pollution and health risks due to vehicle traffic, Sci. Total Environ., 450-451 (2013), 307-316.  doi: 10.1016/j.scitotenv.2013.01.074.  Google Scholar

show all references

##### References:
 [1] L. J. Alvarez-Vázquez, N. García-Chan, A. Martínez and M. E. Vázquez-Méndez, Numerical simulation of air pollution due to traffic flow in urban networks, J. Comput. Appl. Math., 326 (2017), 44-61.  doi: 10.1016/j.cam.2017.05.017.  Google Scholar [2] L. J. Alvarez-Vázquez, N. García-Chan, A. Martínez and M. E. Vázquez-Méndez, Optimal control of urban air pollution related to traffic flow in road networks, Math. Control Relat. F., 8 (2018), 177-193.  doi: 10.3934/mcrf.2018008.  Google Scholar [3] R. Atkinson, Atmospheric chemistry of $\mathrm{VOC}s$ and $\mathrm{NO_x}$, Atmos. Environ., 34 (2000), 2063-2101.  doi: 10.1016/S1352-2310(99)00460-4.  Google Scholar [4] R. Atkinson and W. P. L. Carter, Kinetics and mechanisms of the gas-phase reactions of ozone with organic compounds under atmospheric conditions, Chem. Rev., 84 (1984), 437-470.  doi: 10.1021/cr00063a002.  Google Scholar [5] A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938.  doi: 10.1137/S0036139997332099.  Google Scholar [6] M. Barth, F. An, T. Younglove, G. Scora, C. Levine, M. Ross and T. Wenzel, Development of a Comprehensive Modal Emissions Model: Final Report, Technical report, National Research Council, Transportation Research Board, National Cooperative Highway Research Program, NCHRP Project 25–11, 2000. Google Scholar [7] D. C. Carslaw, S. D. Beevers, J. E. Tate, E. J. Westmoreland and M. L. Williams, Recent evidence concerning higher $\mathrm {NO_x}$ emissions from passenger cars and light duty vehicles, Atmos. Environ., 45 (2011), 7053-7063.  doi: 10.1016/j.atmosenv.2011.09.063.  Google Scholar [8] D. de la Fuente, J. M. Vega, F. Viejo, I. Díaz and M. Morcillo, Mapping air pollution effects on atmospheric degradation of cultural heritage, J. Cult. Herit., 14 (2013), 138-145.  doi: 10.1016/j.culher.2012.05.002.  Google Scholar [9] European Environment Agency, Air Quality in Europe – 2019 Report, Technical Report, 2019. Google Scholar [10] S. Fan, M. Herty and B. Seibold, Comparative model accuracy of a data-fitted generalized Aw-Rascle-Zhang model, Netw. Heterog. Media, 9 (2014), 239-268.  doi: 10.3934/nhm.2014.9.239.  Google Scholar [11] S. Fan, Y. Sun, B. Piccoli, B. Seibold and D. B. Work, A collapsed generalized Aw-RascleZhang model and its model accuracy, arXiv preprint, arXiv: 1702.03624. Google Scholar [12] F. J. Fernández, L. J. Alvarez-Vázquez, N. García-Chan, A. Martínez and M. E. Vázquez-Méndez, Optimal location of green zones in metropolitan areas to control the urban heat island, J. Comput. Appl. Math., 289 (2015), 412-425.  doi: 10.1016/j.cam.2014.10.023.  Google Scholar [13] M. Garavello, K. Han and B. Piccoli, Models for Vehicular Traffic on Networks, American Institute of Mathematical Sciences, 2016.  Google Scholar [14] E. Hairer and G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential-Algbraic Problem, Second edition, Springer Series in Computational Mathematics, 1996. doi: 10.1007/978-3-642-05221-7.  Google Scholar [15] D. J. Jacob, Heterogeneous chemistry and tropospheric ozone, Atmos. Environ., 34 (2000), 2131-2159.  doi: 10.1016/S1352-2310(99)00462-8.  Google Scholar [16] M. Z. Jacobson, Fundamentals of Atmospheric Modeling, Cambridge University Press, 2005. doi: 10.1017/CBO9781139165389.  Google Scholar [17] T. Koto, IMEX Runge-Kutta schemes for reaction-diffusion equations, J. Comput. Appl. Math., 215 (2008), 182-195.  doi: 10.1016/j.cam.2007.04.003.  Google Scholar [18] J. D. Lambert, Numerical Methods for Ordinary Differential Systems, John Wiley & Sons, Ltd., Chichester, 1991.  Google Scholar [19] J.-P. Lebacque, S. Mammar and H. Haj-Salem, Generic second order traffic flow modelling, in Transportation and Traffic Theory, Elsevier, (2007), 755–776. Google Scholar [20] M. J. Lighthill and G. B. Whitham, On kinematic waves II. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. A, 229 (1955), 317-345.  doi: 10.1098/rspa.1955.0089.  Google Scholar [21] T. Luspay, B. Kulcsar, I. Varga, S. K. Zegeye, B. De Schutter and M. Verhaegen, On acceleration of traffic flow, in Proceedings of the 13th International IEEE Conference on Intelligent Transportation Systems (ITSC 2010), IEEE, (2010), 741–746. doi: 10.1109/ITSC.2010.5625204.  Google Scholar [22] S. Manahan, Environmental Chemistry, CRC press, 2017.  doi: 10.1201/9781315160474.  Google Scholar [23] H. Omidvarborna, A. Kumar and D.-S. Kim, $\mathrm{NO_x}$ emissions from low-temperature combustion of biodiesel made of various feedstocks and blends, Fuel Process. Technol., 140 (2015), 113-118.  doi: 10.1016/j.fuproc.2015.08.031.  Google Scholar [24] L. I. Panis, S. Broekx and R. Liu, Modelling instantaneous traffic emission and the influence of traffic speed limits, Sci. Total Environ., 371 (2006), 270-285.  doi: 10.1016/j.scitotenv.2006.08.017.  Google Scholar [25] B. Piccoli, K. Han, T. L. Friesz, T. Yao and J. Tang, Second-order models and traffic data from mobile sensors, Transp. Res. Part C: Emerg. Technol., 52 (2015), 32-56.  doi: 10.1016/j.trc.2014.12.013.  Google Scholar [26] V. Ramanathan and Y. feng, Air pollution, greenhouse gases and climate change: {G}lobal and regional perspectives, Atmos. Environ., 43 (2009), 37-50.  doi: 10.1016/j.atmosenv.2008.09.063.  Google Scholar [27] P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.  Google Scholar [28] M. Rößler, T. Koch, C. Janzer and M. Olzmann, Mechanisms of the NO$_2$ formation in diesel engines, MTZ Worldw., 78 (2017), 70-75.  doi: 10.1007/s38313-017-0057-2.  Google Scholar [29] S. Samaranayake, S. Glaser, D. Holstius, J. Monteil, K. Tracton, E. Seto and A. Bayen, RealTime estimation of pollution emissions and dispersion from highway traffic, Comput.-Aided Civ. Inf., 29 (2014), 546-558.  doi: 10.1111/mice.12078.  Google Scholar [30] J. H. Seinfeld and S. N. Pandis, Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, John Wiley & Sons, 2016. doi: 10.1063/1.882420.  Google Scholar [31] R. Smit, L. Ntziachristos and P. Boulter, Validation of road vehicle and traffic emission models – A review and meta-analysis, Atmos. Environ., 44 (2010), 2943-2953.  doi: 10.1016/j.atmosenv.2010.05.022.  Google Scholar [32] F. Song, J. Y. Shin, R. Jusino-Atresino and Y. Gao, Relationships among the springtime ground–level $\mathrm{NO_x}$, $\mathrm{O}_3$ and $\mathrm{NO_3}$ in the vicinity of highways in the US East Coast, Atmos. Pollut. Res., 2 (2011), 374-383.  doi: 10.5094/APR.2011.042.  Google Scholar [33] B. Sportisse, Fundamentals in Air Pollution: From Processes to Modelling, Springer-Verlag, 2010. Google Scholar [34] J. M. Stockie, The mathematics of atmospheric dispersion modeling, SIAM Rev., 53 (2011), 349-372.  doi: 10.1137/10080991X.  Google Scholar [35] J. Tidblad, K. Kreislová, M. Faller, D. de la Fuente, T. Yates, A. Verney-Carron, T. Grøntoft, A. Gordon and U. Hans, ICP materials trends in corrosion, soiling and air pollution (1987–2014), Materials, 10 (2017). doi: 10.3390/ma10080969.  Google Scholar [36] Transportation Research Board, Critical Issues in Transportation 2019, Technical report, The National Academies of Sciences, Engineering, Medicine, 2019. Google Scholar [37] TRB Executive Committee, Special Report 307: Policy Options for Reducing Energy and Greenhouse Gas Emissions from U.S. Transportation, Technical Report, Transportation Research Board of the National Academies, 2011. Google Scholar [38] US Department of Transportation and Federal Highway Administration, Next generation simulation (NGSIM), http://ops.fhwa.dot.gov/trafficanalysistools/ngsim.htm. Google Scholar [39] T. Wang, L. Xue, P. Brimblecombe, Y. F. Lam, L. Li and L. Zhang, Ozone pollution in China: A review of concentrations, meteorological influences, chemical precursors, and effects, Sci. Total Environ., 575 (2017), 1582-1596.  doi: 10.1016/j.scitotenv.2016.10.081.  Google Scholar [40] R. P. Wayne, Chemistry of Atmospheres, Clarendon Press, Oxford, 1991.   Google Scholar [41] S. K. Zegeye, B. De Schutter, J. Hellendoorn, E. A. Breunesse and A. Hegyi, Integrated macroscopic traffic flow, emission, and fuel consumption model for control purposes, Transp. Res. Part C: Emerg. Technol., 31 (2013), 158-171.  doi: 10.1016/j.trc.2013.01.002.  Google Scholar [42] H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transp. Res. B, 36 (2002), 275-290.  doi: 10.1016/S0191-2615(00)00050-3.  Google Scholar [43] K. Zhang and S. Batterman, Air pollution and health risks due to vehicle traffic, Sci. Total Environ., 450-451 (2013), 307-316.  doi: 10.1016/j.scitotenv.2013.01.074.  Google Scholar
A schematic representation of the four computational modules
Top: Flow-density relationship (left) and velocity-density relationship (right) from the NGSIM dataset. Bottom: Family of flux functions (7) (left) and family of velocity functions (8) (right) for the calibrated parameters
Comparison between ground-truth emission rate and modeled emission rate computed using discrete acceleration (15) on density and speed via kernel density estimation (left). Comparison of emission rate computed with the discrete (15) and analytical (9) acceleration (right). Both the results refer to 500 meters of road and 13 minutes of simulation (data from 4:01 pm - 4:14 pm of NGSIM dataset)
Comparison of modeled (black-dotted), modeled with correction factors $r_{j}$ (red-circles) and ground-truth (blue-solid) emission rates along 500 meters of road during 13 minutes of simulation for the three time periods of the NSGIM dataset. The top row is computed for $r_{1} = 1.42$, the central row for $r_{2} = 1.35$ and the bottom row for $r_{3} = 1.15$
Numerical grid and adaptive time steps (black crosses) required by the solver
Flowchart of the complete procedure
Traffic dynamic 1: Variation of density (a), speed (b), analytical acceleration (c) and ${\mathrm{NO_{x}}}$ emissions (d) in space and time
Traffic dynamic 1: ${\mathrm{NO_{x}}}$ emission rate (${\mathrm{g}}{/}{\mathrm{h}}$) as a function of speed and acceleration (left); variation in time of the total emission rate (${\mathrm{g}}{/}{\mathrm{h}}$) along the entire road (right)
Traffic dynamic 2: Variation of density (a), speed (b), analytical acceleration (c) and ${\mathrm{NO_{x}}}$ emissions (d) in space and time
Traffic dynamic 2: ${\mathrm{NO_{x}}}$ emission rate (${\mathrm{g}}{/}{\mathrm{h}}$) as a function of speed and acceleration (left); variation in time of the total emission rate (${\mathrm{g}}{/}{\mathrm{h}}$) along the entire road (right)
Traffic dynamic 2.1: Variation in time of the total ${\mathrm{NO_{x}}}$ emission rate (${\mathrm{g}}{/}{\mathrm{h}}$) along the entire road with $r = 3/2$ and varying the traffic light duration $t_c$ in minutes: $t_c = 7.5$ with $t_r = 3$ (left); $t_c = 5$ with $t_r = 2$ (center); $t_c = 2.5$ with $t_r = 1$ (right)
Traffic dynamic 2.2: Variation in time of the total emission rate (${\mathrm{g}}{/}{\mathrm{h}}$) along the entire road by varying the ratio $r$
Variation in time of the total concentration (${\mathrm{g}}/{\mathrm{k}}{\mathrm{m}}^3$) of ${\mathrm{O_{3}}}$ (left) and ${\mathrm{O_{2}}}$ (right), in the case of dynamics with (red-circles) and without (blue-solid) traffic light
Vertical diffusion of ozone concentration (${\mathrm{g}}{/}{\mathrm{km}}^{3}$) in $\Omega$ at different times with (bottom) and without (top) traffic lights
Diffusion of ozone concentration (${\mathrm{g}}{/}{\mathrm{km}}^{3}$) in time at $1\, {\mathrm{m}}$ height with (right) and without (left) traffic lights
Horizontal diffusion of ozone concentration (${\mathrm{g}}{/}{\mathrm{km}}^{3}$) in $\Omega$ at different times with (bottom) and without (top) traffic lights
Parameters for CGARZ model (1) calibrated on NGSIM dataset
 ${V^{\mathrm{max}}}$ $\rho_f$ ${\rho^{\mathrm{max}}}$ $\rho_{c}$ ${w_{L}}$ ${w_{R}}$ ${65}\, {{\mathrm{k}}{\mathrm{m}}{/}{\mathrm{h}}}$ ${110}\, {\mathrm{veh}{/}{\mathrm{k}}{\mathrm{m}}}$ ${800}\, {\mathrm{veh}{/}{\mathrm{k}}{\mathrm{m}}}$ ${\rho^{\mathrm{max}}}/2$ $5687$ $13000$
 ${V^{\mathrm{max}}}$ $\rho_f$ ${\rho^{\mathrm{max}}}$ $\rho_{c}$ ${w_{L}}$ ${w_{R}}$ ${65}\, {{\mathrm{k}}{\mathrm{m}}{/}{\mathrm{h}}}$ ${110}\, {\mathrm{veh}{/}{\mathrm{k}}{\mathrm{m}}}$ ${800}\, {\mathrm{veh}{/}{\mathrm{k}}{\mathrm{m}}}$ ${\rho^{\mathrm{max}}}/2$ $5687$ $13000$
${\mathrm{NO_{x}}}$ parameters in emission rate formula (16) for an internal combustion engine car, where ${\mathrm{g}}$ denotes gram, ${\mathrm{m}}$ meter and ${\mathrm{s}}$ second
 Vehicle mode $f_{1}$ $f_{2}$ $f_{3}$ $f_{4}$ $f_{5}$ $f_{6}$ $\left[{\mathrm{g}}/{\mathrm{s}}\right]$ $\left[{\mathrm{g}}/{\mathrm{m}}\right]$ $\left[{\mathrm{g}}\, {\mathrm{s}}/{\mathrm{m}}^{2}\right]$ $\left[{\mathrm{g}}\, {\mathrm{s}}/{\mathrm{m}}\right]$ $\left[{\mathrm{g}}\, {\mathrm{s}}^{3}/{\mathrm{m}}^{2}\right]$ $\left[{\mathrm{g}} \, {\mathrm{s}}^{2}/{\mathrm{m}}^{2}\right]$ If $a_i (t) \geq -0.5\, {\mathrm{m}}{/}{\mathrm{s}}^2$ 6.19e-04 8e-05 -4.03e-06 -4.13e-04 3.80e-04 1.77e-04 If $a_i (t)<-0.5\, {\mathrm{m}}{/}{\mathrm{s}}^2$ 2.17e-04 0 0 0 0 0
 Vehicle mode $f_{1}$ $f_{2}$ $f_{3}$ $f_{4}$ $f_{5}$ $f_{6}$ $\left[{\mathrm{g}}/{\mathrm{s}}\right]$ $\left[{\mathrm{g}}/{\mathrm{m}}\right]$ $\left[{\mathrm{g}}\, {\mathrm{s}}/{\mathrm{m}}^{2}\right]$ $\left[{\mathrm{g}}\, {\mathrm{s}}/{\mathrm{m}}\right]$ $\left[{\mathrm{g}}\, {\mathrm{s}}^{3}/{\mathrm{m}}^{2}\right]$ $\left[{\mathrm{g}} \, {\mathrm{s}}^{2}/{\mathrm{m}}^{2}\right]$ If $a_i (t) \geq -0.5\, {\mathrm{m}}{/}{\mathrm{s}}^2$ 6.19e-04 8e-05 -4.03e-06 -4.13e-04 3.80e-04 1.77e-04 If $a_i (t)<-0.5\, {\mathrm{m}}{/}{\mathrm{s}}^2$ 2.17e-04 0 0 0 0 0
Errors given by (21) for the three slots of the NGSIM dataset and different correction factor $r_{1} = 1.42$, $r_{2} = 1.35$ and $r_{3} = 1.15$
 Period $\mathrm{Error}(r_{1})$ $\mathrm{Error}(r_{2})$ $\mathrm{Error}(r_{3})$ 4:01 pm - 4:14 pm 0.1604 0.1666 0.2204 5:01 pm - 5:14 pm 0.0819 0.0842 0.1625 5:16 pm - 5:29 pm 0.2304 0.1773 0.0586
 Period $\mathrm{Error}(r_{1})$ $\mathrm{Error}(r_{2})$ $\mathrm{Error}(r_{3})$ 4:01 pm - 4:14 pm 0.1604 0.1666 0.2204 5:01 pm - 5:14 pm 0.0819 0.0842 0.1625 5:16 pm - 5:29 pm 0.2304 0.1773 0.0586
Parameters $k_{1}$, $k_{2}$, and $k_{3}$ of system (26), where ${\mathrm{c}}{\mathrm{m}}$ denotes centimeter, ${\mathrm{s}}$ second and $\mathrm{molecule}$ the number of molecules
 Parameter Value $k_{1}$ ${0.02}\, {\, {\mathrm{s}}^{-1}}$ $k_{2}$ ${6.09\times 10^{-34}}\, {{\mathrm{c}}{\mathrm{m}}^6}\, \mathrm{ molecule}^{-2}\, {\mathrm{s}}^{-1}$ $k_{3}$ ${1.81\times10^{-14}}\, {{\mathrm{c}}{\mathrm{m}}^3}\, \mathrm{molecule}^{-1}\, {\mathrm{s}}^{-1}$
 Parameter Value $k_{1}$ ${0.02}\, {\, {\mathrm{s}}^{-1}}$ $k_{2}$ ${6.09\times 10^{-34}}\, {{\mathrm{c}}{\mathrm{m}}^6}\, \mathrm{ molecule}^{-2}\, {\mathrm{s}}^{-1}$ $k_{3}$ ${1.81\times10^{-14}}\, {{\mathrm{c}}{\mathrm{m}}^3}\, \mathrm{molecule}^{-1}\, {\mathrm{s}}^{-1}$
Variation of the total amount of ${\mathrm{O_{3}}}$, ${\mathrm{NO}}$, ${\mathrm{NO_{2}}}$ and ${\mathrm{O}}$ concentration (${\mathrm{g}}/{\mathrm{k}}{\mathrm{m}}^3$) computed with three different traffic light duration (Traffic dynamic 2.1) with respect the total amount of concentrations without traffic light (Traffic dynamic 1)
 $t_c=t_r+t_g$ ${(3+4.5)}\, {\mathrm{min}}$ ${(2+3)}\, {\mathrm{min}}$ ${(1+1.5)}\, {\mathrm{min}}$ ${\mathrm{O_{3}}}$ 2.95e+07 3.54e+07 3.91e+07 ${\mathrm{NO}}$ 1.09e+09 1.28e+09 1.43e+09 ${\mathrm{NO_{2}}}$ 1.55e+08 1.81e+08 2.02e+08 ${\mathrm{O}}$ 7.00e+01 8.21e+01 9.13e+01
 $t_c=t_r+t_g$ ${(3+4.5)}\, {\mathrm{min}}$ ${(2+3)}\, {\mathrm{min}}$ ${(1+1.5)}\, {\mathrm{min}}$ ${\mathrm{O_{3}}}$ 2.95e+07 3.54e+07 3.91e+07 ${\mathrm{NO}}$ 1.09e+09 1.28e+09 1.43e+09 ${\mathrm{NO_{2}}}$ 1.55e+08 1.81e+08 2.02e+08 ${\mathrm{O}}$ 7.00e+01 8.21e+01 9.13e+01
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