1. Introduction
Air pollution refers to the presence in the air of substances that have a detrimental and poisonous impact on the environment and ecological system. The air pollutants include carbon monoxide, carbon dioxide, hydrocarbons, ammonia, sulfur dioxide, nitrous oxides, chlorofluorocarbons, and pollutant particulate matter. In urban areas, more solar energy is absorbed due to the type of land cover, which increases the environmental temperature.
In urban areas, extensive construction impedes air circulation and fresh air influx. Large buildings reduce wind speed, and the pollutant emissions from a large number of vehicles further intensify the increase in temperature. The ratio of pollution emission from the traffic can be lower than other urban sources, but it is large in the roadside vicinity. This problem is more severe in highly populated urban areas. Traffic pollution is not restricted to vehicles’ exhausts. It also depends on the type of material used in the tires, engine age, type of fuel, fuel combustion, as well as the pavement material and its conditions. These factors impact human health.
Carbon monoxide, a colorless and odorless gas, is the most hazardous gas for human health; when inhaled, it combines with hemoglobin and reduces the capacity of blood to carry oxygen. Prolonged exposure to carbon monoxide causes slow death, as it reduces the oxygen levels in the body. It causes suffocation, nasal irritation, disorientation, dizziness, and headache. Sulfur dioxide has a strong smell and is produced by impurities present in the fuel. This causes irritation in the throat, eyes, and nose. High concentrations of this gas can cause death. Sulfur dioxide contributes to the formation of sulfuric acid, which causes acidic rain, which, in turn, causes building corrosion and smog. The smog affects drivers’ visibility and contributes to traffic accidents.
The growth of the global population has led to increased air pollution. In 2019, there were 4.2 million early deaths due to air pollution worldwide, with 223,000 deaths reported in Pakistan and 196,000 in the US [
1]. Traffic emissions are a major contributor to air pollution. In 2020, traffic emissions were responsible for 45% of the pollution in the US [
2], while in 2017, the greenhouse emissions in Europe were at 72%. Passenger vehicles produced 44% of greenhouse gases in Europe, while heavy transport caused 9% [
3]. In China, road vehicles are responsible for about 80% of CO emissions and 40% of
emissions [
4]. Urban traffic emissions have increased air pollution and reduced the quality of air, which harms not only the human body but also the ecological system.
Particulate matter can have a solid or liquid form and includes engine wear, heavy metals, hydrocarbons, and sulfur and nitrogen composites. Particulate matter production is higher in diesel engines than in gasoline engines. The vehicles listed as zero pollution emission mostly use electricity charging, and electricity production has its own impact on the environment.
CO
2 emitted from fossil fuels contributes about 65% to global greenhouse gas emissions (GHGs). In 2020, CO
2 emitted from direct human involvement impacted forestry, contributing 11% of GHGs. Globally, 16% of CH
4, 6% of N
2O, and 2% of fluorinated gases (F-gases) contribute to GHGs [
2]. Harmful traffic emissions not only reduce the air quality but also affect natural biological systems. It is therefore vital to characterize traffic emissions to improve air quality and reduce emission-associated risks. Different traffic emission characterizations are being established to assess traffic pollutants.
Traffic emissions can be assessed using vehicle emission exhaust. The usual method to obtain vehicle emissions is calculating the pollutants ratio based on vehicle mobility [
5]. Pollution emission from a single vehicle is used as the pollutant ratio. It gives the pollutant quantity after 1 km of travel and considers the engine load, vehicle acceleration, and speed. The number of kilometers driven, trips made from origin to destination, and the duration of travel are considered as vehicle activity [
6]. The result for the total emissions using the emission factor and vehicle’s activity is presented as
In the literature, different methods are employed to assess traffic emission inventories and fuel consumption, characterizing the traffic situation, average speed, cycle variable, emissions, and instantaneous emission models [
7]. Speed characterization considers the different road speeds. In 1998, Ahn used a mathematical regression model and the vehicle speed and acceleration to assess the emission ratio and fuel consumption [
5]. In 1978, Evans and Herman [
8] used the average speed model to estimate the fuel consumption ratio from trips traveled over a road segment. This approach could not estimate fuel consumption on small road segments and for traffic at speeds faster than 60 km/h [
9]. They could be obtained by estimating fuel consumption and traffic pollutants instantaneously.
The level of traffic emissions from the vehicle’s engine at the start depends on the engine temperature, which is low in engines that are inactive for extended periods. Cold engines produce high emissions during starting, while the warmer engines have lower emissions at startup. Evaporative emissions are due to fuel evaporation when the vehicle’s engine is hot at the end of a trip. Models for inventorying and estimating traffic emissions, such as ARTEMIS [
10], calculate evaporative emissions, as well as different emissions from hot- and cold-start engines. The ARTEMIS model requires detailed information on the vehicle composition and the distance traveled (vehicle activity) to provide more consistent data on the vehicle and diverse driver behaviors. However, the model requires substantial data validation and collection resources, which have a large financial impact.
As the number of vehicles increases due to population growth and complex urban traffic circumstances, traffic pollutant modeling becomes more important in reducing air pollution (congestion). Some organizations struggle to meet the national air quality and transportation emissions goals. More realistic traffic flow models are needed to characterize traffic emissions more correctly. As a result, data on traffic emissions and flow characteristics are maintained consistently. In this paper, a more realistic second-order traffic model based on traffic emissions is proposed.
Traffic emissions are characterized at different levels. Microscopically, the instantaneous engine revolution per minute, acceleration, and speed of individual vehicles are considered. Macroscopically, the aggregate flow, density, and speed are considered. In order to forecast traffic emissions and flow, the authors of this paper used the microscopic traffic emission parameters in a macroscopic traffic model that was easy to implement. The vehicles’ speed, mass airflow, the ratio of fuel to air, and the engine speed were obtained through onboard diagnostic units.
The traffic models are classified as macroscopic, microscopic, and mesoscopic. The macroscopic models consider the speed, the number of vehicles on a road segment (traffic density), and traffic flow. This model type provides the average changes in traffic parameters over the length of the road. The temporal and spatial traffic evolutions are also characterized. These types of models are developed using partial differential equations systems. This paper considered aggregate flow (macroscopic) models in conjunction with instantaneous emission models to characterize the average pollutants’ emissions.
The second-order macroscopic systems of aggregate traffic behavior are based on traffic conservation: acceleration and deceleration. The first macroscopic traffic model of this type was developed by Richards, Lighthill, and Whitham [
11]. The second element was based on the presumption of changes by a driver (anticipation) and speed alignment at those changes (relaxation) to attain equilibrium conditions. Payne and Whitham (PW model) were the first to propose a second-order traffic system with anticipation and relaxation terms. That model estimated that at traffic changes, vehicles had uniform interactions. Consequently, traffic density and speed progressed idealistically. Realistically, the model should have considered the presumptions of drivers. In 1996, Helbing improved the gas kinetic theory (GKT) model with the help of numerical schemes. That characterization employed velocity and density correlations and helped estimate complicated traffic changes. Their model provided an anisotropic traffic progress [
12]. In 2010, Caligaris et al. suggested numerical stability conditions to estimate traffic progress accurately with the PW model. In 2015, Bosire et al. implemented Zhang’s characterization in Kenya to correctly estimate traffic progress on a 500 m road segment, which justified the use of aggregate traffic parameters [
13]. In 2017, Mohan and Ramadurai improved the PW model by incorporating the area occupancy parameter into the model instead of traffic density. The model captured the bottleneck effect more accurately than previous models but ignored the drivers’ anticipation of the traffic conditions [
14]. In 2016, Khan and Gulliver improved the PW model by considering the traffic speed changes, which improved the model’s inaccurate traffic estimation [
15]. They also suggested another new macroscopic model that considered traffic speed management and the drivers’ physiological and psychological states and reactions. According to their assumptions, the drivers’ anticipation should be characterized based on physiology and psychology. Khan and Gulliver improved the model by incorporating the traffic transition velocities acquired during traffic alignment [
16]. Zhang [
17], Aw Rascle [
18], Daganzo [
19], and Berg et al. considered mathematical techniques to reduce the large changes flow, which was unrealistic. Khan and Gulliver’s models improved the traffic characterization based on the traffic’s physical parameters, resulting in more realistic and accurate representations of the traffic changes.
4. Suggested Traffic Model Based on Traffic Emissions
Zhang’s traffic characterization is
which assumes that the flow is smooth with no exits and entrances to a road when deceleration and acceleration do not occur. In the case of transitions, Equation (15) can calculate traffic changes. A driver’s presumption considers the density changes. That is, speed changes corresponds to changes in density. The speed changes occur during relaxation time τ and adapt to equilibrium speed
. This model ignores the impact of emissions on speed and traffic density. However, traffic emission modeling and mitigation are equally important to improve the environmental quality of the road surroundings. Equilibrium speeds are characterized to confirm a traffic situation [
25]. For ease of implementation, Greenshield’s equilibrium speed Equation (11) was employed in the proposed model.
For effective traffic emissions management, traffic characterization must consider emission fluctuations. The pollutant emissions correspondingly vary with speed adjustments, and more fluctuation is experienced at transitions. To model the traffic more realistically,
emissions were regressed from the acquired density and speed. From Equations (12) and (13),
emissions fluctuated as the speed and density varied. Speed changes in traffic are density-dependent. A large fluctuation in speed occurs as a large variation in density occurs. Similarly, changes in velocity are small for small variations in density. A driver presumes speed changes while observing the changes in density. Consequently, considering the changes in emission with speed and density from (12) and (13), respectively, can result in a driver’s presumption to effectively manage emissions. That is, considering the equilibrium speed
(12) results in
Changes in equilibrium velocity distribution with emissions from
result in
Changes in vehicle emission with density from Equation (13) result in
The speed changes with density having an impact on vehicle emissions are obtained with Equations (16) and (18) as
This means that equilibrium speed changes are small at a faster speed and large at a lower speed. In other words, the tendency toward changes in equilibrium speed is more prominent in congested traffic than in free flow. Furthermore, the changes in equilibrium speed in a higher traffic density are large, and they are small in a lower density. Equation (19) suggests that emission variations are more significant in congested, high-density traffic, as emissions are larger in slower traffic. To include the emissions’ impact in Zhang’s model, from Equation (19) is substituted with (15).
The distance
between vehicles is small in congested traffic and larger in free flow. In other words, as the distance between the vehicles reduces, density increases, resulting in higher emissions. That is,
Drivers try to keep a safe distance in addition to the distance required to maneuver a vehicle to prevent accidents. The distance required to maneuver a vehicle depends on speed and relaxation time At higher speeds, the control distance covered is larger than at the slower speed.
Further,
takes the form
From Equation (21), Equation (20) becomes
Then, from Equations (19) and (22), the coefficient
of
(driver anticipation) in Equation (15) becomes
Combining Equation (23) with (15), the proposed model based on
emissions results in
Equations (24) and (25) are the proposed traffic characterizations that consider traffic emissions. Equation (24) is the same as Equation (14) [
25]. The speed and density estimated with the suggested model can forecast
emissions using the regression models developed in
Section 3.
5. ROE Discretization of Models
A numerical decomposition of traffic models is required to divide the continuous functions into small discrete parts to obtain algorithms for the numerical simulations. Many schemes are used to decompose traffic models, including finite-volume methods, finite-difference methods, and higher-resolution schemes. The ROE discretization scheme [
26,
27,
28] was used to implement the proposed and Zhang’s models in MATLAB to provide more accurate results for traffic behavior. This high-resolution scheme solved the partial differential systems with large changes more accurately. For a more accurate traffic behavior with large changes, an entropy fix was employed [
16].
The conserved traffic model in vector form is
where U represents data variables,
represents the function vector, and R represents the source vector. Subscripts
x and
t show the spatial and temporal gradients of the data variables, respectively.
For the conserved form of the suggested model, Equation (24) is multiplied by
, which gives
and multiplying Equation (25) by
gives
Combining Equations (27) and (28) gives
Rearranging Equation (29) gives
The subscripts
and
denote the temporal and spatial gradients, respectively. This is the conserved form of the proposed model from Equation (25). The proposed model from Equations (24) and (30) in the vector form of Equation (26) is
The vector form of the traffic models in Equation (26) in quasi-linear form is
where
is the Jacobian matrix, which denotes
and helps approximate eigenvectors and eigenvalues. The eigenvalues then approximate the second-order partial differential traffic systems, which analyze the traffic behavior evolution.
From Equation (31),
for the proposed model is
To find the eigenvalues and eigenvectors from Equation (31), we use
where
I is the 2 × 2 identity matrix, and
represents eigenvalues. From Equation (34), Equation (35) gives the quadratic relation of eigenvalues as
From Equation (35), the eigenvalues are given in Equation (36)
which gives the velocity changes. That is, a change in traffic behavior is based on traffic emissions. To optimize the emissions, the coefficients of density and velocity can be controlled. To find the eigenvectors, we use
and
x =
, which gives eigenvector
as
To obtain the average velocity, ∆f is
From Equation (39),
where from Equation (31),
and f (
)
from Equation (41) is
Comparing Equation (40) with (43), the quadratic relation of velocity is
Thus, the solution of Equation (44) is
, where the subscript
denotes the road segments. Furthermore,
and
; thus, Equation (45) takes the form
Equation (46) denotes the average velocity of the proposed model at
and
. The average density of the proposed model is the geometric mean of the density [
27].
Zhang’s model has the same average density and velocity as the proposed model. The computing array for the performance analysis of the traffic model was created by segmenting the solution domain spatially and temporally. The time step was equal to
, and a road segment had a width of
, which was the distance between adjacent points. Using
and
, the temporal and spatial traffic behavior could be estimated over the entire road segment with the ROE decomposition scheme [
29].
6. Simulation Results
The performance of the proposed and Zhang’s models is presented in this section for periods of 75 s. In these simulations, the temporal and spatial traffic evolution on a 300 m circular road was considered to assess the worst traffic scenario [
16]. The circular road was chosen as the traffic dynamics within the circle are easy to observe. In addition, the worst traffic scenario can be observed on a circular rather than a straight road. The time period was 0.5 s, while the distance length was 10 m to guarantee Courant–Friedrichs–Levy conditions [
28]. The relaxation time was 1 s [
16]. The safe distance headway was 15 m. The traffic for both the proposed and Zhang’s models achieved Greenshield’s equilibrium velocity (speed, Equation (11)) distribution during alignment [
25]. The traffic density on the road was normalized between zero and one. The maximum velocity and density were 9 m/s and 1, respectively. The initial density condition was
A drastic change in the initial density was considered to assess the traffic behavior of Zhang’s and the proposed models. The simulation parameters for both models are summarized in
Table 5.
The proposed model’s density behavior on a 300 m road at 5 s, 15 s, 37.5 s, and 50 s is shown in
Figure 12 and indicates that the density was smoothed over time. The density range at 5 s was from 0.07 to 0.62. At 0 m, it was 0.0948, and at 40 m, the density reduced to 0.075. Then, it increased to 0.097 at 70 m. At 110 m, the density marginally increased to 0.117, and again to 0.619 at 140 m, then marginally reduced to 0.583 at 190 m. At 240 m, it reduced to 0.107 and remained at 0.099 till 300 m. At 15 s, the range of traffic density was 0.1 to 0.64. At 0 m, it was 0.120, and at 140 m, it was reduced to 0.100. It increased to 0.64 at 160 m, and at 190 m, it marginally reduced to 0.57. The traffic density smoothly aligned to 0.140 at 300 m. At 37.5 s, the range of traffic density was from 0.120 to 0.420. At 0 m, it was 0.300, which smoothly aligned to 0.120 at 200 m. The density increased to 0.410 at 225 m and marginally increased again to 0.420 at 235 m. The density decreased to 0.310 at 300 m. At 50 s, the range of traffic density was from 0.13 to 0.36. At 0 m, it was 0.35 and smoothly aligned to 0.13 at 250 m. The density decreased to 0.37 at 275 m and was uniform up to 300 m. The density of the proposed model was in the employed limits of zero and one.
The corresponding velocity behavior of the proposed model on a circular 300 m road at 5 s, 15 s, 37.5 s, and 50 s are shown in
Figure 13. At 5 s, the velocity range was from 8.3 m/s to 3.6 m/s. At 0 m, it was 8.14 m/s. It was reduced from 8.32 m/s at 40 m to 8.13 m/s at 70 m. At 110 m, the velocity marginally decreased to 7.94 m/s and was further reduced to 3.49 m/s at 140 m. The velocity marginally increased to 3.72 m/s at 190 m and rose further to 7.98 m/s at 240 m. It remained uniform at 8.10 m/s up to 300 m. At 15 s, the velocity range was from 3.47 m/s to 8.28 m/s. At 0 m, it was 7.90 m/s, then marginally increased to 8.28 m/s at 140 m. It was reduced to 3.47 m/s at 160 m. At 190 m, the velocity increased to 3.8 m/s, then smoothly increased to 7.8 m/s at 300 m. At 37.5 s, the traffic velocity range was from 5.30 m/s to 8.00 m/s. At 0 m, it was 6.30 m/s and marginally increased to 8.00 m/s at 190 m. It decreased to 5.30 m/s at 240 m and then increased smoothly to 6.10 m/s at 300 m. At 50 s, the range of traffic velocity was from 5.7 m/s to 7.80 m/s. At 0 m, the velocity was 5.80 m/s, marginally increased to 7.80 m/s at 240 m, decreased to 5.80 m/s at 290 m, and was finally reduced to 5.70 m/s at 300 m. The proposed model velocity was in the range of 0 to 9 m/s.
The traffic behavior of Zhang’s density model on a circular 300 m road at 5 s, 15 s, 37.5 s, and 50 s is shown in
Figure 14. At 5 s, the range of traffic density was from 0.05 to 0.45. The density increased from 0.093 at 0 m to 0.43 at 170 m, then smoothly decreased to 0.154 from 170 m up to 260 m. The density was more oscillatory and abruptly changed over short distances between 240 m and 300 m, as shown in
Figure 13. At 15 s, the range of density was from 0.16 to 0.32. The density reduced from 0.21 at 0 m to 0.16 at 60 m. It oscillated over a short distance of 10 m and increased between 60 m and 160 m. It rose to 0.32 at 200 m and fell to 0.24 at 300 m. At 37.5 s, the range of density was from 0.24 at 0 m to 0.23 at 140 m. At 50 s, the range of density was from 0.24 at 0 m to 0.25 at 70 m. The oscillatory behavior was observed with Zhang’s model at 5 s and 15 s, as shown in
Figure 14.
The corresponding velocity of Zhang’s model on a circular 300 m road at 5 s, 15 s, 37.5 s, and 50 s is shown in
Figure 15. At 5 s, the velocity range was from 12.6 m/s to −1.5 m/s, oscillating from 8.32 m/s at 0 m to 3.89 m/s at 100 m. The velocity decreased from 4.05 m/s at 110 m to 10.61 m/s at 270 m. At 280 m, the velocity was negative, −1.50 m/s, and at 290 m, it was 12.60 m/s. At 15 s, the velocity ranged between 8.10 m/s at 0 m and 5.5 m/s at 120 m. It oscillated over a short distance of 10 m from 60 m to 160 m and smoothly aligned from 6.0 m/s at 160 m to 7.8 m/s at 300 m. At 37.5 s and 50 s, the velocity was nearly at 6.8 m/s. The velocity behavior of Zhang’s model was unusual as a velocity cannot be negative, and it did not stay within the prescribed range. Furthermore, changes over a shorter distance of 10 m did not accurately reflect traffic behavior.
Figure 16 shows the proposed model density behavior on a 300 m circular road for 75 s. The density varied from 0.07 to 0.62 within the minimum of zero and maximum of one range.
Figure 17 shows the Zhang model’s density compared with the proposed model’s density behavior during 75 s. The Zhang model’s density behavior was more oscillatory prior to the 50th time step than in the proposed model.
Figure 18 shows the proposed model velocity behavior over a 300 m circular road for 75 s. The velocity ranged from 3.6 m/s to 8.3 m/s. The variation in velocity shown in
Figure 17 corresponds to the density behavior in
Figure 16. The velocity was within the prescribed minimum of 0 m/s and a maximum of 9 m/s.
Figure 19 compares the velocity behavior of the proposed and Zhang’s models. The velocity ranged from −1.5 m/s to 12.6 m/s, which was an oscillatory and unusual behavior. The traffic velocity over the distance cannot be negative with time. The velocity of Zhang’s model prior to the 50th time step did not stay within the employed range. For example, at the 50th distance step and 10th time step, Zhang’s model gave a velocity of 8.1 m/s, while the proposed model gave a velocity of 8.3 m/s. This suggests that both models provided comparable velocity predictions, with the proposed model slightly exceeding the Zhang model’s value. At the 200th distance step and 10th time step, Zhang’s model gave a velocity of 6.4 m/s, as shown in
Figure 17, while the proposed model’s velocity was 3.7 m/s, as shown in
Figure 18. This indicates that both models provided similar velocity predictions but with the proposed model’s velocity values being lower than those of Zhang’s model. However, at the 250th distance step and 10th time step, Zhang’s model gave a velocity of 9.1 m/s, exceeding the maximum expected velocity value of 9 m/s, as shown in
Figure 17. Thus, in some cases, Zhang’s model provided higher velocity predictions than the expected maximum. The density and velocity for Zhang’s and the proposed models at the 10th time step (5 s) for a 300 m road are summarized in
Table 6.
With Zhang’s model, the changes in velocity were significant, while the proposed model showed minor changes in velocity, as shown in
Table 7, suggesting that the Zhang model’s velocity behavior was more oscillatory and unrealistic. Thus, Zhang’s model inadequately characterized traffic behavior. Furthermore, the proposed model took into account traffic emissions and could, therefore, make traffic emission predictions, while Zhang’s model could not. In the proposed model, as the velocity increased, the emissions decreased, as expected. For example, at the 100th distance step and 10th time step, the emission value, based on velocity in the proposed model, was 8.7 kg CO
2/km. Based on the correct depiction of the corresponding density behavior shown in
Figure 17, the proposed model’s lower velocity resulted in more accurately calculated lower emissions. Furthermore, the Zhang model’s velocity behavior in
Figure 19 did not correspond to the density behavior shown in
Figure 17, as a high velocity beyond the maximum was predicted for a lower density. The emission values based on velocity for the proposed model at the 10th time step (5 s) for a 300 m road are summarized in
Table 7.
The results in
Table 7 suggested that the proposed model could mitigate the velocity, traffic density, and vehicle emissions in dynamic traffic management.