Optimization Models of Actuated Control Considering Vehicle Queuing for Sustainable Operation

Abstract

How to sustainably conduct intersection operations is a key issue of the current research. For an actuated control intersection, queued vehicles, control parameters, and phase schemes all affect the operation effect. This paper discusses queued vehicles at actuated intersections and their influence on signal timing. First, this paper establishes an improved traffic wave model and proposes a vehicle queuing model on this basis. Second, by analyzing the queuing and dispersion process of queued vehicles, a minimal green time calculation model is proposed. Then, this paper establishes a maximal green time calculation model aiming at minimizing average vehicle delay and maximizing traffic capacity under different phase schemes, and considers the influence of queued vehicles. Lastly, the models are verified separately; results show that the average error of the minimal green time model was 4.18%, and the average optimization rate of the maximal green time model was 9.27%. It is proved that the models achieved great accuracy and optimization effects, which could potentially improve intersection sustainability.

1. Introduction

A large amount of practical experience has proven that only increasing the supply level cannot essentially solve traffic congestion problems [1]. In contrast, using management and control strategies can achieve better results. As an important part of urban road systems, intersections are not only key to solving congestion, but also determine the entire system’s sustainability. Therefore, optimizing intersection signal control parameters is a hot issue in current research.

Intelligent methods such as deep learning [2,3], fuzzy control theory [4,5], and intelligent algorithms [6,7] have been proposed that highly rely on high-precision and real-time traffic data to ensure reliability. Some scholars believe that using these methods to solve congestions will become the main trend. However, the current intelligence and automation levels of most intersections cannot meet the requirements. In fact, more than 90% of the intersections in China are still not equipped with modern detectors such as video and radar detectors [8], so the methods above are not applicable at this stage.

Considering the above problem, this paper focuses on actuated signal control, an effective and widely used control strategy relying on traffic detectors to detect traffic flow and make corresponding control decisions [9]. In actuated control, the minimal green time ensures the dispersion of all queued vehicles, and the maximal green time limits the maximal time of the phase. These two basic parameters can avoid the secondary queuing of vehicles and the waste of green time, so it is greatly significant to optimize their value. Furthermore, in view of the current intelligent transportation development level in China, further exploration of actuated control optimization could be highly compatible with realistic needs.

The rest of the paper is organized as follows. Section 2 reviews the existing literature relevant to this paper. In Section 3, we establish an improved traffic wave model, and propose a vehicle queuing model on this basis. In Section 4, we construct a minimal green time calculation model by analyzing the queuing and dispersion process of queued vehicles, and construct a maximal green time calculation model considering the influence of queued vehicles. The solving algorithm for the model is given in Section 5. Verifications are conducted in Section 6, and our contributions are concluded in Section 7.

2. Literature Review

In the 1950s, Webster [10] first proposed an optimal cycle length model with minimal average vehicle delay, and suggested that the maximal cycle length is approximately 1.25–1.5 times the optimal cycle length, which provided a reference for setting the maximal green time. Pappis and Mamdani [11] first applied fuzzy logic in signal control, taking vehicle arrival and queuing information as inputs, and outputting the actuated green time. Murat and Gedizioglu [12] also established a fuzzy control model on the basis of that of Pappis [11] that considered the phase sequence influence. By obtaining the real-time vehicle delay and queue length, Cowan [13] improved actuated control, which could effectively evaluate the control effect. By studying vehicle arrival characteristics, traffic capacity, saturation flow, vehicle delay, queue length, and parking rate, Akcelik [14] proposed cycle length and green time calculation models that can be used for semiactuated and fully actuated control. Furth et al. [15] decomposed the cycle length into two parts: full utilization time and loss time, and a cycle length calculation model was constructed on this basis. Viti and Van Zuylen [16] proposed a green time calculation model based on probability theory that considered queue length. Moghimi et al. [17] analyzed different green time demand levels, and deduced the cycle length by using a time sequence forecast method. Shiri and Maleki [18] determined the maximal green time through fuzzy logic, and dynamically adjusted it by monitoring the traffic operating conditions. Jing [19] established a minimal green time optimization model considering the stochastic characteristics of traffic flow. Hao [20] proposed a framework for actuated control that took into account uncertainties in potential driver distraction issues that could reduce energy consumption and emissions. Bao [21] analyzed traffic flow according to collected data, including occupancy and vehicle queuing information, and then proposed an actuated control strategy that could reduce vehicle delay. Christian [22] designed a fully actuated control using logical controls that were able to perceive the pedestrian density on refuge islands, prioritizing pedestrians more. Wang [23] used RFID technology to record arrival flow and predict future traffic volume to obtain the cycle length of the actuated control. Wu [24] presented a new volume-occupancy-based actuated control system for a diamond interchange that used real-time traffic volume and occupancy to improve the traffic by adaptively adjusting the signal control plan.

Above all, we found the following: (a) The optimization models constructed in recent years mostly rely on the real-time data obtained by intelligent detectors, but these data are not easy to obtain in most intersections, and the accuracy and reliability of the data still need to be further studied. Therefore, these models lack certain applicability. (b) There is less research on the minimal green time, and these studies mainly focused on the arrival characteristics of vehicles, ignoring the impact of queued vehicles. (c) Studies on maximal green time mainly focused on traffic flow characteristics with no indepth discussion of phase schemes and queued vehicles.

To deal with the abovementioned issues, the following work is presented in this paper: (a) An improved traffic wave model is first established, and a vehicle queuing model is proposed on this basis. Then, a minimal green time calculation model considering queued vehicles is proposed that fully considers the impact of the queued vehicles on the control effect. (b) A maximal green time calculation model considering phase schemes and queued vehicles is established that deeply discusses the phase schemes and impact of queued vehicles. (c) The data needed in this paper can be accurately obtained at almost all intersections through coil detectors, which means that the constructed models in this paper have stronger practical value.

3. Vehicle Queuing Model Based on Improved Traffic Wave Model

3.1. Improved Traffic Wave Model

The starting wave transmission process is shown in Figure 1a. The traffic flow in front of the wave front was congested traffic flow with a speed of 0 and density of congestion density k_j. The traffic flow behind the wave front was the traveling traffic flow with the speed of critical speed v_m, density of critical density k_m, and headway of the saturated headway h_m.

The standing wave transmission process is shown in Figure 1b. The traffic flow in front of the wave front was the traveling traffic flow with speed v₁, density k₁, and headway h₁. The traffic flow behind the wave front was the congested traffic flow with speed of 0 and the density of congested density k_j.

There are three existing traffic wave calculation methods, namely, the Greenhill [25], the Greenberg traff [26] and the LWR [27,28] traffic wave models. The LWR traffic wave model is as follows:

$$u_{\mathrm{start}}=-\frac{{\nu }_m{k}_m}{k_m-k_j}$$

(1)

$$u_{\mathrm{stop}}=-\frac{{\nu }_1{k}_1}{k_j-k_1}$$

(2)

where u_start represents the starting wave (m/s), and u_stop represents the standing wave (m/s).

The relationship among traffic volume q, speed v, and density k is as follows:

$$q=k·v$$

(3)

The relationship between space headway s and headway h is as follows:

$$s=h·v$$

(4)

The relationship between space headway s and density k is as follows:

$$s=\frac{1}{k}$$

(5)

Combining Equations (1)–(5), the improved traffic wave model is shown in Equations (6) and (7).

$$u_{\mathrm{start}}=\frac{v_m}{1-v_m{h}_m{k}_j}$$

(6)

$$u_{\mathrm{stop}}=\frac{v_1}{1-v_1{h}_1{k}_j}$$

(7)

As we can see in the LWR model’s Equations (1) and (2), traffic density k₁ is needed for calculation, which is difficult to obtain directly. In the improved model, shown in Equations (6) and (7), headway h₁ was adopted instead of k₁, which could be accurately obtained in real time thanks to the widely used coil detectors in actuated control.

3.2. Vehicle Queuing and Dispersion Process

On the basis of unsaturated traffic flow, all vehicles queued in the previous cycle were assumed to be completely released. Figure 2 describes the vehicle-queuing and traffic wave transmission processes in the nth cycle.

3.2.1. Vehicle Queuing Process

As shown in Figure 2a, when the red light turned on, the driving vehicles stopped in sequence behind the stop line. The standing wave formed at this time transmitted backward from the stop line, which is shown in Equation (8).

$$\left|u_{\mathrm{stop}}^{(n)}\right|=\frac{V_a^{\left(n\right)}}{V_a^{\left(n\right)}·h_a^{\left(n\right)}·k_j-1}$$

(8)

where v_a⁽ⁿ⁾ is the average speed of the arriving vehicles (m/s), and h_a⁽ⁿ⁾ is the average headway (s).

3.2.2. Vehicle Dispersion Process

As shown in Figure 2b, when the green light turned on, the static vehicles started to leave the intersection. The starting wave formed at this time transmitted backwards from the stop line, which is as follows:

$$\left|u_{\mathrm{start}}^{(n)}\right|=\frac{{\nu }_m}{{\nu }_m·h_m·k_j-1}$$

(9)

As shown in Figure 2c, when the starting wave and the standing wave met, all queued vehicles turned to the driving state and were released at critical speed v_m.

3.3. Vehicle-Queuing Model

Assuming that the studied traffic flow started with red time in the nth cycle, and the vehicles queued in the previous cycle had been completely released, the distance between the leading vehicle and the stop line was ignored.

When t_rb⁽ⁿ⁾ ≤ t ≤ t_gb⁽ⁿ⁾

$$\left\{\begin{array}{l}{x}_0^{\left(n\right)}=0\hfill \\ x_l^{\left(n\right)}=\left(t-t_{rb}^{\left(n\right)}\right)\left|u_{\mathrm{stop}}^{\left(n\right)}\right|\hfill \\ L^{\left(n\right)}=x_l^{\left(n\right)}-x_0^{\left(n\right)}\hfill \end{array}\right.$$

(10)

where t_rb⁽ⁿ⁾ is the start time of the red light, t_gb⁽ⁿ⁾ is the start time of the green light, x₀⁽ⁿ⁾ is the position of the leading vehicle relative to the stop line (m), x_l⁽ⁿ⁾ is the position of the trailing vehicle relative to the stop line (m), and L⁽ⁿ⁾ is the queue length (m).

When t_gb⁽ⁿ⁾ ≤ t ≤ t_qd⁽ⁿ⁾

$$\left\{\begin{array}{l}{x}_0^{\left(n\right)}=\left(t-t_{gb}^{\left(n\right)}\right)\left|u_{\mathrm{start}}^{\left(n\right)}\right|\hfill \\ x_l^{\left(n\right)}=\left(t-t_{rb}^{\left(n\right)}\right)\left|u_{\mathrm{stop}}^{\left(n\right)}\right|\hfill \\ L^{\left(n\right)}=x_l^{\left(n\right)}-x_0^{\left(n\right)}\hfill \end{array}\right.$$

(11)

where t_qd⁽ⁿ⁾ is the moment when the starting wave and the standing wave meet, and is shown in Equation (12).

$$t_{qd}^{\left(n\right)}=\frac{t_{rb}^{\left(n\right)}\left|u_{\mathrm{stop}}^{\left(n\right)}\right|-t_{gb}^{\left(n\right)}\left|u_{\mathrm{start}}^{\left(n\right)}\right|}{\left|u_{\mathrm{stop}}^{\left(n\right)}\right|-\left|u_{\mathrm{start}}^{\left(n\right)}\right|}$$

(12)

The longest queue length L_max⁽ⁿ⁾ formed at time t_qd⁽ⁿ⁾ is as follows:

$$L_{\max }^{\left(n\right)}=\left(t_{gb}^{\left(n\right)}-t_{rb}^{\left(n\right)}\right)·{\left(\frac{1}{\left|u_{\mathrm{stop}}^{\left(n\right)}\right|}-\frac{1}{\left|u_{\mathrm{start}}^{\left(n\right)}\right|}\right)}^{-1}$$

(13)

After t_qd⁽ⁿ⁾, the vehicles passed through the intersection, as shown in Figure 2c. At this time, there were no queued vehicles.

4. Optimization Models of Basic Parameters

4.1. Minimal Green Time Optimization Model

In the actuated control, the minimal green time is defined as the time during which all queued vehicles can pass through the intersection [29]. As shown in Figure 3, the time for queued vehicles of length L to pass through includes two parts: (a) the time required for the starting wave to pass from the leading vehicle to the trailing vehicle, which was recorded as T₁; and (b) the time required for vehicles gradually leave the intersection, which was recorded as T₂.

Minimal green time G_p is as follows:

$$\begin{array}{l}{G}_p=T_1+T_2\\ \left\{\begin{array}{l}{T}_1=L·{\left|u_{\mathrm{start}}\right|}^{-1}\hfill \\ T_2=L·v_m^{-1}\hfill \end{array}\right.\end{array}$$

(14)

4.2. Maximal Green Time Optimization Model

This study establishes an optimization model for the maximal green time considering the queued vehicles, and gives the objective function and constraints under different phase schemes. The cycle length was calculated by the formula of Webster [10], and the objective function simultaneously pursued the smallest average vehicle delay and largest traffic capacity. In order to unify the dimensions and facilitate the solution, this study expresses the minimal average vehicle delay as the ratio of the optimized vehicle delay to the current vehicle delay, and the maximal capacity is expressed as the reciprocal of the ratio of the optimized capacity to the current capacity. The objective function is as follows:

$$\min Z={\mu }_1\left(\frac{{\displaystyle \sum d_i·q_i}}{{\displaystyle \sum q_i}}·D_0^{-1}\right)+{\mu }_2\left(CA{P}_0·{\left({\displaystyle \sum \frac{s_i·g_{ei}}{c}}\right)}^{-1}\right)$$

(15)

where d_i is the average vehicle delay in flow direction i (s), q_i is the traffic volume in flow direction i (pcu), D₀ is the traffic capacity under current signal timing plan (pcu/h), s_i is the saturated flow in flow direction i (pcu/h), g_ei is the effective green time in flow direction i (s), c is the cycle length (s), $\mu$₁ is the weight coefficient of the average vehicle delay, and $\mu$₂ is the weight coefficient of the capacity. The traffic engineer can decide the weight coefficient according to the actual operating conditions of the intersection. In this study, $\mu$₁ = 2/3 and $\mu$₂ = 1/3 were chosen.

Average vehicle delay d_i was calculated with the Webster model [30], which is as follows:

$$d_i=\frac{c·{\left(1-g_{ei}/c\right)}^2}{2\left(1-y_i\right)}$$

(16)

where y_i is the flow ratio in flow direction i, which equals to saturated flow dividing q_i [10].

In the constraints, this study considers the length of queued vehicles, which is shown in Equation (17).

$$H\ge L_{\max }$$

(17)

where H is the distance between the upstream and downstream intersections (m), and L_max is the longest queue length calculated in Formula (13). Incorporating Equation (13) into Equation (17) is shown as Equation (18).

$$g_i\ge c-H·\left(u_{\mathrm{stop}}{}^{-1}-u_{\mathrm{start}}{}^{-1}\right)$$

(18)

The basic constraints for signal timing are as follows:

$$\left\{\begin{array}{l}{g}_i=g_{ei}+A-l_i\hfill \\ g_i^{\left(\min \right)}\le g_i\le g_i^{\left(\max \right)}\hfill \end{array}\right.$$

(19)

where g_i is the green time in flow direction i (s), A is the yellow time (s), l_i is the starting loss time in flowing direction i (s), g_i^(min) is the minimal green time in flow direction i (s), and g_i^(max) is the maximal green time in flow direction i (s).

When considering the influence of phase schemes, this study adopted the NEMA phase structure [31]. Since its structure could be flexibly adjusted, such as Phases 1 and 5 shown in

Table 1. Phase schemes.

Phase Schemes	Schematic Diagrams
F1
F2
F3
F4

, it has better adaptability to different traffic situations.

The objective function and constraints under different phase schemes are shown in

Table 2. Objective function and constraints.

Phase Schemes	Objective Function and Constraints
F1	Delay	$D=\left({\displaystyle \sum _{i=1}^8{d}_i·q_i}\right)·{\left({\displaystyle \sum _{i=1}^8{q}_i}\right)}^{-1}$
	Capacity	$CAP={\displaystyle \sum _{i=1}^8\frac{s_i·g_{ei}}{c}}$
	Constraints	$\left\{\begin{array}{l}{g}_1+g_2-g_5-g_6=0\hfill \\ g_3+g_4-g_7-g_8=0\hfill \\ g_1+g_2+g_3+g_4+4I=c\hfill \end{array}\right.$
F2	Delay	$D=\left({\displaystyle \sum _{i=1}^2{d}_i·q_i}+{\displaystyle \sum _{i=5}^6{d}_i·q_i}+d^{\prime }·q^{\prime }\right)·{\left({\displaystyle \sum _{i=1}^2{q}_i}+{\displaystyle \sum _{i=5}^6{q}_i}+q^{\prime }\right)}^{-1}$
	Capacity	$CAP={\displaystyle \sum _{i=1}^2\frac{s_i·g_{ei}}{c}}+{\displaystyle \sum _{i=5}^6\frac{s_i·g_{ei}}{c}}+\frac{s^{\prime }·g_e{}^{\prime }}{c}$
	Constraints	$\left\{\begin{array}{l}{g}_1+g_2-g_5-g_6=0\hfill \\ g_1+g_2+g^{\prime }+3I=c\hfill \end{array}\right.$
F3	Delay	$D=\left({\displaystyle \sum _{i=3}^4{d}_i·q_i}+{\displaystyle \sum _{i=7}^8{d}_i·q_i}+d^{\prime }·q^{\prime }\right)·{\left({\displaystyle \sum _{i=3}^4{q}_i}+{\displaystyle \sum _{i=7}^8{q}_i}+q^{\prime }\right)}^{-1}$
	Capacity	$CAP={\displaystyle \sum _{i=3}^4\frac{s_i·g_{ei}}{c}}+{\displaystyle \sum _{i=7}^8\frac{s_i·g_{ei}}{c}}+\frac{s^{\prime }·g_e{}^{\prime }}{c}$
	Constraints	$\left\{\begin{array}{l}{g}_3+g_4-g_7-g_8=0\hfill \\ g_3+g_4+g^{\prime }+3I=c\hfill \end{array}\right.$
F4	Delay	$D=\left(d^{\prime }·q^{\prime }+d^{″}·q^{″}\right)·{\left(q^{\prime }+q^{″}\right)}^{-1}$
	Capacity	$CAP=\frac{s^{\prime }·g_e{}^{\prime }}{c}+\frac{s^{″}·g_e{}^{″}}{c}$
	Constraints	$g^{\prime }+g^{″}+2I=c$

Note: s′, s″, d′, d″, g_e′, g_e″, g′, g″, q′, q″ represent the saturated flow, average vehicle delay, effective green time, shown green time, and traffic volume in a higher traffic volume direction, respectively, when the phase schemes are F₂, F₃, F₄.

.

5. Solving Algorithm for the Optimization Model

The maximal green time calculation model is a nonlinear integer optimization model. For this model, there are many variables, and each variable must meet certain constraints. It is very difficult to apply traditional optimization methods to solve the problem, and heuristic algorithms are generally used. The more widely used methods are the genetic, simulated annealing, and particle swarm algorithms. Among them, the genetic algorithm has the advantages of good robustness and global search ability. Therefore, this study uses the genetic algorithm to solve the maximal green time calculation model [32].

5.1. Variable Coding

Since green time generally does not exceed two digits, the chromosome length of 16 consisting of 8 loci was chosen.

5.2. Fitness Function

When the optimization problem is a minimization problem, the fitness function takes the opposite value of the objective function. Therefore, this study takes −Z as the fitness function.

5.3. Genetic Manipulation

The manipulation procedures include selection, crossover, and mutation. The specific steps are as follows:

Step 1: Initialization. Set population number (100), chromosome length (16), number of iterations (800), crossover probability (0.8), mutation probability (0.001), and roulette selection operator.

Step 2: use real encoding to assign values to the parameters, and randomly generate 100 individuals that meet the constraints as the initial group.

Step 3: calculate the objective function and fitness value of each individual.

Step 4: perform selection operations on all individuals and select a new population.

Step 5: perform crossover operations on random paired individuals according to crossover probability.

Step 6: perform mutation operations on individuals according to mutation probability.

Step 7: if number of iterations g = 800, the individual with the best fitness is taken as the optimal solution output, and the algorithm can be terminated.

6. Verification

6.1. Verification of Minimal Green Time Calculation Model

In order to verify the calculation accuracy of the minimal green time calculation model, a VISSIM simulation experiment was designed as shown in Figure 4.

Taking a signal lane as the research object, two data collection detectors were set at the stop line and distance L to collect parameters. The traffic volume input was set to be 800 pcu/h, the speed distribution was 40 and 50 km/h, and 25 s was set as the vehicle release time every 30 s. The control group was the minimal green time calculation model provided in the 2010 Highway Capacity Manual (HCM), which is as follows:

$$G_p=3+h·Integer\left(\frac{L}{s}\right)$$

(20)

Taking the queue length L = 55 m as an example, the statistics of 30-period simulations are shown in

Table 3. Simulation results of 55m queue length.

Index	Simulation Time (s)	HCM Model (s)	Relative Error	Optimization Model (s)	Relative Error
1	22.40	22.80	1.79%	21.60	3.57%
2	20.80	22.80	9.62%	21.60	3.85%
3	23.80	25.40	6.72%	25.20	5.88%
4	19.79	22.80	15.21%	21.60	9.15%
5	20.00	22.80	14.00%	21.60	8.00%
6	26.34	29.40	11.62%	28.60	8.58%
7	21.20	23.00	8.49%	22.00	3.77%
8	22.65	24.60	8.61%	23.40	3.31%
9	21.30	22.80	7.04%	21.60	1.41%
10	21.56	22.80	5.75%	21.60	0.19%
11	21.30	22.80	7.04%	21.60	1.41%
12	22.62	24.60	8.75%	23.40	3.45%
13	21.70	22.80	5.07%	21.60	0.46%
14	24.38	22.80	6.48%	21.60	11.40%
15	21.79	22.80	4.64%	21.60	0.87%
16	21.90	23.40	6.85%	23.80	8.68%
17	20.52	22.80	11.11%	21.60	5.26%
18	22.40	24.60	9.82%	23.40	4.46%
19	18.36	21.00	14.38%	19.80	7.84%
20	19.51	21.00	7.64%	19.80	1.49%
21	21.00	22.80	8.57%	21.60	2.86%
22	26.10	28.30	8.43%	27.60	5.75%
23	20.33	21.00	3.30%	19.80	2.61%
24	16.90	19.20	13.61%	18.00	6.51%
25	24.13	27.20	12.72%	26.40	9.41%
26	24.89	27.20	9.28%	26.40	6.07%
27	21.12	22.80	7.95%	21.60	2.27%
28	18.57	20.60	10.93%	19.80	6.62%
29	20.80	21.00	0.96%	19.80	4.81%
30	26.58	28.20	6.09%	28.00	5.34%

.

In order to intuitively reflect the difference between the simulated and calculated values, the result was drawn as a scatter diagram, as shown in Figure 5.

The figure shows that the minimal green time calculation model proposed in this paper was closer to the simulated value, while the traditional HCM calculation model had larger calculation errors.

A 30-period simulation was conducted for queue lengths L of 25, 35, 45, 55, and 65 m, and the simulation results are shown in

Table 4. Simulation results under different queue lengths.

Queue Length (m)	Simulation Time (s)	HCM Model		Optimization Model
		Time (s)	Relative Error	Time (s)	Relative Error
25	9.92	11.56	18.13%	10.06	3.90%
35	12.01	13.20	10.41%	12.13	3.39%
45	18.41	19.73	7.29%	19.13	3.83%
55	21.82	23.54	8.42%	22.53	4.84%
65	24.54	26.17	7.94%	25.22	4.91%
Average relative error		10.44%		4.18%

.

Since we used the improved traffic wave model to calculate the queue length, the table shows that the minimal green time calculation model proposed in this paper (4.18% average relative error) was more accurate than the traditional HCM calculation model (10.44% average relative error) under different queue lengths, which verified the effectiveness and accuracy of the model.

6.2. Verification of Maximal Green Time Calculation Model

It was assumed that each flow direction of the intersection was released at the maximal green time, and this study did not consider the influence of the maximal green time on the overall control effect. Information of a certain intersection is shown in

Table 5. Intersection information.

Traffic Flow Direction		Traffic Volume (pcu/h)	Saturated Flow (pcu/h)	Ratio
SB	TH	912	2750	0.33
	LT	356	1550	0.23
NB	TH	832	2750	0.30
	LT	144	1550	0.09
WB	TH	288	2450	0.08
	LT	428	2150	0.20
EB	TH	248	2750	0.09
	LT	216	1550	0.14

Note: SB, southbound direction; NB, northbound direction; WB, westbound direction; EB, eastbound direction; TH, straight-through vehicle; and LT, left-turn vehicle.

.

We assumed that yellow time A = 3 s, all red period r = 1 s, and starting loss time l = 2 s. Two sets of experiments were designed as follows: (a) calculation results of traditional Webster method (traditional model); (b) calculation results of the maximal green time calculation model proposed in this paper (optimization model). The objective function and constraints were input into MATLAB, and the maximal green time of each flow direction was obtained as shown in

Table 6. Solution results.

Phase Schemes	Index	WB LT (s)	B TH (s)	NB LT (s)	SB TH (s)	EB LT (s)	WB TH (s)	SB LT (s)	NB TH (s)
F1	①	32	15	37	54	32	15	37	54
	②	18	22	20	53	17	23	20	53
F2	①	24	10	39	39	24	10	39	39
	②	25	15	36	36	14	26	36	36
F3	①	16	16	18	24	16	16	18	24
	②	13	13	12	36	13	13	22	26
F4	①	12	12	20	20	12	12	20	20
	②	10	10	22	22	10	10	22	22

.

Table 7. Optimization results of average delay and capacity.

Phase Schemes	Index	Average Vehicle Delay (s)	Traffic Capacity (pcu/h)	Optimization Ratio
F1	①	59.62	4279	/
	②	42.69	4472	12.79%
F2	①	24.75	3185	/
	②	24.15	3742	10.75%
F3	①	26.07	3328	/
	②	22.57	3741	11.45%
F4	①	7.84	7733	/
	②	7.57	7668	2.10%
Average optimization ratio				9.27%

shows the optimization results of average vehicle delay and capacity under different phase schemes.

We can see from the table that, under phase schemes F₁, F₂, and F₃, the optimization ratios all exceeded 10%. Under phase scheme F₄, since there were only two green time parameters that could be adjusted, it also had an optimization ratio of 2.10%. Thus, the maximal green time calculation model established in this paper could effectively reduce the average vehicle delay and improve traffic capacity (9.27% average optimization ratio), which verified the effectiveness of the model.

7. Conclusions

Compared with previous studies, this paper focused on the influence of queued vehicles, and two basic control parameters, namely, the minimal and maximal green times, were optimized to achieve a better control effect. Through this study, more reasonable control parameters could be obtained to avoid green loss, reduce vehicle delay, and improve traffic operation sustainability.

The main contributions of this paper are as follows: (a) According to the relationship between basic traffic flow parameters, an improved traffic wave model was established. Then, by analyzing the vehicle-queuing process, a vehicle-queuing model was proposed. (b) By analyzing the queuing and dispersion process of queued vehicles, a minimal green time calculation model was established. Through simulation, the average error between calculation and simulation values was 4.18%, verifying the model’s accuracy. (c) The maximal green time calculation model with the objective function of minimizing the average vehicle delay and maximizing traffic capacity under different phase schemes was established. The influence of queued vehicles was considered in the constraints, and the a genetic algorithm solution process was given. Through calculation, the average optimization rate was 9.27%, verifying its effeteness.

In summary, this paper proposed two basic parameter optimization models of actuated control, and verifications successfully proved their validity of improving green time utilization, reducing vehicle delay, and enlarging traffic capacity, which is essential for improving traffic sustainability.