A computational modular approach to evaluate $ {\mathrm{NO_{x}}} $ emissions and ozone production due to vehicular traffic

Caterina Balzotti; Maya Briani; Barbara De Filippo; Benedetto Piccoli

doi:10.3934/dcdsb.2021192

Article Contents

2022, Volume 27, Issue 6: 3455-3486. 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
^* Corresponding author: Caterina Balzotti

Received: November 2020

Revised: June 2021

Published: June 2022

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 35L65, 62P12; Secondary: 90B20.

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Figure 1. A schematic representation of the four computational modules

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Figure 2. 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

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Figure 3. 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)

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Figure 4. 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 $

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Figure 5. Numerical grid and adaptive time steps (black crosses) required by the solver

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Figure 6. Flowchart of the complete procedure

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Figure 7. Traffic dynamic 1: Variation of density (a), speed (b), analytical acceleration (c) and $ {\mathrm{NO_{x}}} $ emissions (d) in space and time

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Figure 8. 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)

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Figure 9. Traffic dynamic 2: Variation of density (a), speed (b), analytical acceleration (c) and $ {\mathrm{NO_{x}}} $ emissions (d) in space and time

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Figure 10. 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)

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Figure 11. 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)

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Figure 12. 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 $

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Figure 13. 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

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Figure 14. Vertical diffusion of ozone concentration ($ {\mathrm{g}}{/}{\mathrm{km}}^{3} $) in $ \Omega $ at different times with (bottom) and without (top) traffic lights

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Figure 15. Diffusion of ozone concentration ($ {\mathrm{g}}{/}{\mathrm{km}}^{3} $) in time at $ 1\, {\mathrm{m}} $ height with (right) and without (left) traffic lights

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Figure 16. Horizontal diffusion of ozone concentration ($ {\mathrm{g}}{/}{\mathrm{km}}^{3} $) in $ \Omega $ at different times with (bottom) and without (top) traffic lights

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Table 1. 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 $

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Table 2. $ {\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

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Table 3. 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

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Table 4. 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} $

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Table 5. 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

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