4.1 CO2 incremental emissions analysis of arrived vehicles
In actual traffic, vehicles passing through signalized intersections can be divided into three modes: complete stop, incomplete stop and normal driving. A complete stop means that, due to the influence of signal lights and queuing vehicles, the vehicle must slow and stop to wait and then follow the acceleration to a certain speed when the vehicle in front begins to accelerate. An incomplete stop means that the front vehicle has not completely accelerated when the vehicle arrives and must decelerate to a certain speed and then follow the front vehicle to accelerate again. Normal driving means that the arriving vehicle is not affected by the signal lights and queuing vehicles and will pass through the intersection at the road section speed. Since the root cause of the increase in the CO
2 emissions of vehicles at intersections is the change in vehicle driving trajectory, only the CO
2 emissions of acceleration, deceleration and idling stage of complete stop and incomplete stop vehicles must be considered in the analysis process. Then, in the analysis process, we assume that the incomplete stop behavior is a complete stop behavior without an idling process, and the sum of CO
2 emissions generated by a vehicle at deceleration and acceleration stage is
ER. Because it is difficult to determine the deceleration amplitude of the vehicle with incomplete stops in the signal cycle, we consider the incomplete stop a complete stop behavior without an idling process. In addition, we assume that the average sum emissions of each vehicle during deceleration and acceleration stage is
ER. However, this parameter will be affected by many traffic conditions; there are great differences in the value of this parameter under different traffic conditions, making it difficult to calibrate accurately. Therefore, there are some differences between the assumptions here and the actual traffic scenarios. However, we make such assumptions only to facilitate the theoretical analysis of vehicle CO
2 emissions and qualitatively understand the relationship between vehicle CO
2 emissions and the delay and stop rate. The model established in section 4.2 of this paper has no relationship with this assumption. Thus, regardless of fuel vehicles or electric vehicles, the average incremental CO
2 emissions per vehicle in a signal cycle can be approximately estimated as Eq. (
9). It is worth noting that although the electric vehicle can achieve energy recovery in the deceleration stage, and it is affected by the energy recovery efficiency, the energy recovery in the braking stage is less than the consumption of the vehicle accelerating to the road section speed again.
$$ Afe=\frac{St\times qC\times ER+ It\times qC\times FR- St\times qC\times SR}{qC}= St\times \left( ER- SR\right)+ It\times FR $$
(9)
In Eq. (
9),
FR represents the vehicle CO
2 emissions rate in the idling stage;
SR represents the average CO
2 emissions of the vehicle passing through the distance of deceleration and acceleration processes at road section speed;
C is the signal cycle of the intersection;
q is the vehicle arrival rate of the intersection approach;
St is the stop rate of the intersection approach; and
It is the average vehicle idle time at the intersection approach. According to the analysis of Shao [
47], the control delay of vehicles at intersections is equal to the sum of the lost time in the process of acceleration and deceleration and idle time (stopped delay), and the U.S. Road Capacity Manual proposes that the average stopped delay is approximately 0.76 times the average control delay through repeated observations and measurements of actual traffic phenomena. Thus,
St and
It can be calculated as follows [
48]:
$$ St=f\left(\frac{1-\eta }{1-q/s}+\frac{N_o}{qC}\right) $$
(10)
$$ It=0.76\times d $$
(11)
$$ d=\frac{C{\left(1-\eta \right)}^2}{2\left(1-\eta x\right)}+\frac{xN_o}{q} $$
(12)
$$ {N}_o=\left\{\begin{array}{l}\frac{QT}{4}\left(\left(x\hbox{-} 1\right)+\sqrt{{\left(x\hbox{-} 1\right)}^2+\frac{12\left(x-{x}_0\right)}{QT}}\right),x>{x}_0\\ {}0,x\le {x}_0\end{array}\right. $$
(13)
$$ {x}_0=0.67+\frac{sg}{600} $$
(14)
In Eqs. (
10) to (
14),
g is the effective green light time of the intersection approach;
η is the green time ratio of the intersection approach;
s is the vehicle saturation flow rate of the intersection approach;
\( x=\raisebox{1ex}{$ qC$}\!\left/ \!\raisebox{-1ex}{$ sg$}\right. \) is the saturation level of the intersection approach;
No is the average number of vehicles held up at the intersection approach in one signal cycle;
Q is the traffic capacity of the intersection approach;
T is the time duration for which the vehicle arrival rate is equal to
q;
f is the correction factor for a complete stop; and
d is the control delay of the intersection approach.
Through the analysis of Eq. (
9), it can be seen that, on the premise that parameters
ER,
FR and
SR have been determined, the average incremental CO
2 emission per vehicle of a signal cycle can be determined by the stop rate and idle time. At the same time, according to analysis of the work of Zhao et al. [
12] and actual traffic phenomena, we can draw the general conclusion that the changes in vehicle CO
2 emissions at an intersection approach are caused by the different stop behaviors and idle times of vehicles. The incremental emissions of a vehicle will change with changes in the stop rate and idle time. When the idle time is similar, the incremental emissions will increase with an increase in the stop rate. In the case of a similar stop rate, the incremental emissions will increase with an increase in the idle time. However, the average incremental CO
2 emissions of vehicles at signalized intersections will be affected by many factors, such as the random arrival of vehicles, speed fluctuations, incomplete stop behaviors, etc., and the accuracy of incremental CO
2 emission of vehicles cannot be guaranteed through the method of equivalent averages of parameters
ER,
FR and
SR. Therefore, it can be explained to some extent that there is a specific mapping relationship between the changes in incremental emissions and the stop rate and delay, but the estimation accuracy of the model must be further improved if establishing a simple linear relationship, as in Eq. (
9).
4.2 Modeling
According to the polynomial combination of vehicle speed and acceleration/deceleration, Ahn et al. [
11] used a statistical regression method to determine the instantaneous fuel consumption and emissions model of fuel vehicles. Yao et al. [
17] and Zhang and Yao [
18] also used the same method to study a statistical model of the instantaneous energy consumption of electric vehicles. Inspired by this idea, we analyze the different polynomial combinations of stop rate and control delay based on the data of vehicle average CO
2 incremental emissions under different traffic conditions and establish a statistical model of vehicle CO
2 incremental emissions. It should be noted that the delay in vehicles at intersections mainly includes three stages—deceleration, idling, and acceleration—for comprehensively evaluating the operating efficiencies of signalized intersections in actual traffic, and researchers have conducted a series of studies based on the control delay rather than the stopped delay [
49], and the control delay and stopped delay approximately follow a linear relationship. Therefore, we use the stop rate and control delay models shown in Eqs. (
10) and (
12), respectively, to establish incremental emissions models from the perspective of regression statistics.
The establishment of a vehicle emissions model eventually turns to practical applications. Although it is possible to build a more realistic model relying only on the relevant data obtained from a large number of observations and in-depth analysis of actual traffic phenomena, the CO
2 emissions of vehicles at signalized intersections are affected by many factors, such as signal timing and vehicle arrival rate, resulting in vehicle energy consumption and emissions that will also differ under different traffic conditions. Therefore, it is difficult to collect the actual emissions data of different vehicles at signalized intersections, and the vehicle emissions under different traffic conditions are difficult to cover comprehensively. In addition, it is easy to cause certain deviations in many data processing processes. To overcome the difficulty of data collection, Zhao et al. [
12] used the full velocity difference (FVD) car-following model to simulate vehicle trajectories at signalized intersections and combined the model with vehicle-specific power emissions model to analyze the influences of signal timing, arrival rate and road section speed on fuel vehicle emissions. This approach provides a simple and convenient way to analyze vehicle energy consumption and emissions at signalized intersections. Therefore, we use the FVD car-following model combined with the instantaneous emissions model of fuel vehicles in section 2 and the instantaneous energy consumption model of electric vehicles in section 3 to simulate the CO
2 emissions of fuel vehicles and electric vehicles under different traffic conditions, respectively. The simulation conditions in the process of data acquisition are shown in Table 5 in
Appendix.
We set the road section speed at 10 m/s, in line with the actual speed of most urban roads. For fuel vehicles and electric vehicles, the simulations are conducted separately. It is worth noting that under these conditions, the vehicle arrival rate has 51 situations, the signal cycle has 35 situations, and the green signal ratio situations for each traffic condition composed of vehicle arrival rate and signal cycle are different and can be determined by the upper and lower bounds and the step size. Finally, the number of traffic situations, composed of different vehicle arrival rate, signal cycle and green signal ratio, is 17,010. In addition, in Zhao et al. [
12], to simplify the analysis process, under the condition that the vehicle arrival rate is determined, it is assumed that the time intervals of all vehicles arriving at the intersection are the same, but this assumption condition is too strong, making it insufficient for describing actual traffic phenomena. To overcome this shortcoming, this paper assumes that the time headway of a vehicle arriving at an intersection approach obeys a shifted negative exponential distribution when simulating the vehicle trajectory under different traffic conditions, considering the randomness of vehicle arrival at the intersection approach to render the model more applicable to actual traffic analysis. At the same time, the signal cycle average vehicle emissions within 1 h of each traffic situation are considered the simulation value of the corresponding traffic situation. Then, based on a large number of simulation data in different traffic situations, which almost achieve comprehensive cover of different traffic situations, we uses SPSS statistical software to analyze the different polynomials of the stop rate and control delay and determine the structure of the statistical model of average CO
2 incremental emissions of fuel vehicles and electric vehicles as follows:
$$ Afe=\sum \limits_{i=0}^3\sum \limits_{j=0}^3{l}_{i,j}{St}^i{d}^j $$
(15)
Table 6 in
Appendix shows the regression coefficients of the statistical model. To simplify the model and avoid the influence of multicollinearity between explanatory variables in the model as much as possible, when determining the corresponding model, a polynomial combination that cannot explain the changes in the model significantly is not considered. Through SPSS analysis, the adjusted R
2 values of the CO
2 incremental emissions statistical models of fuel vehicles and electric vehicles are both 0.932, and the regression equation and explanatory variables pass the significance test at the 95% confidence level. To evaluate the accuracy of the model more intuitively, based on the simulation results of incremental emissions under different traffic situations, the model-calculated values of Eq. (
15) are compared with the simulation values. The statistics of the comparison results show that the mean absolute percentage errors of the statistical models of CO
2 incremental emissions of fuel vehicles and electric vehicles are 10.62% and 8.56%, respectively, and the standard deviations of the absolute percentage errors are 0.10 and 0.07, respectively. It can be concluded that the statistical models of CO
2 incremental emissions are reasonable through the linear regression of different polynomials of the stop rate and control delay, and the estimation accuracy of the statistical model can be guaranteed.
However, in this paper, the model is established on the premise that the road section speed is 10 m/s. According to the simulation analysis in the work of Zhao et al. [
12], when the vehicle has different road section speeds, the driving trajectory of the vehicle under the intersection approach will change, rendering the CO
2 emissions generated in the processes of vehicle deceleration and acceleration different. Therefore, there are some deficiencies in calculating the incremental emissions using the above model under different road section speeds. In actual traffic, the speed of different urban road intersections will be different, and the speed limit range is mostly between 30 km/h and 50 km/h. Thus, to render the statistical model proposed in this paper more general, it is assumed that the range of vehicle road section speeds is from 8 m/s to 14 m/s. The incremental emissions results of fuel vehicles and electric vehicles under different traffic situations, comprising different road section speeds, signal cycles, green signal ratios and vehicle arrival rates, are obtained by the above simulation method and conditions. Based on Eq. (
15), under the premise of considering the road section speed, SPSS statistical software is used to test different polynomial combinations of the stop rate, control delay and road section speed. The statistical model structure of the average CO
2 incremental emissions in one signal cycle is reestablished, as shown in Eq. (
16), and the regression coefficient is shown in Table 7 in
Appendix.
$$ Afe=\sum \limits_{i=0}^1\sum \limits_{j=0}^3\sum \limits_{k=0}^3{l}_{i,j,k}{vol}^i{St}^j{d}^k $$
(16)
Through SPSS analysis, the adjusted R
2 values of the CO
2 incremental emissions statistical models considering the road section speed of fuel vehicles and electric vehicles are 0.916 and 0.899, respectively, and the regression equation and explanatory variables pass the significance test at the 95% confidence level. Through analysis of the vehicle incremental emissions results, it is found that, when the vehicle road section speed increases, although it can reduce the travel time of the vehicle on the road, the duration times of deceleration and acceleration at the intersection will increase, resulting in an increase in incremental emissions at the intersection. To describe the influence of the road section speed on the CO
2 incremental emissions of vehicles at the intersection approach, the statistical model shown in Eq. (
16) considers the road section speed. It can be seen from a comparison with Eq. (
15) that although the complexity of the model structure shown in Eq. (
16) is increased under the premise of considering the road section speed, the adjusted R
2 values of the models of fuel vehicles and electric vehicles are reduced correspondingly but still close to 1. This outcome shows that the statistical model considering road section speed is relatively reasonable, making it more general and broadening the application scope of the model. It is worth noting that the model in this paper aims to study the average vehicle CO
2 incremental emissions under the whole intersection approach rather than a single vehicle. Therefore, for an intersection approach, the road section speed of vehicles can be determined by the average speed of all of the vehicles before the change in vehicle speed. When the vehicle speed of different intersection approaches is different, leading to changes in vehicle CO
2 emissions, the
vol of the model is a description of this situation.