Abstract: To improve the traffic capacity of urban arterial roads and alleviate traffic congestion, this paper proposes to first assess and predict the traffic flow at opposite intersections on arterial roads in the current and future periods to determine its distribution pattern, and then select specific signal coordination control schemes accordingly. For example, when the opposite traffic flow is symmetrically distributed, a graphical method for bidirectional green wave coordination control under the traditional symmetrical release mode can be selected. When the opposite traffic flow is asymmetrically distributed, this paper presents a new method using numerical methods to obtain green wave coordination control under the asymmetrical release mode. The green wave coordination effect of this control method is achieved by establishing an ideal intersection that is infinitely close to the actual intersection, determining the optimal signal phase timing, common signal period, phase difference, and green wave bandwidth of the intersection group, and proving the correctness and accuracy of the control method through numerical examples.
0 Introduction
With the continuous improvement of my country's social and economic level, people's quality of life has also greatly improved. More and more people gather in cities, and rural areas are gradually transforming into urban areas. This means that my country's social productivity and science and technology have also reached a certain level. However, at the same time, the rapid development has also brought about the phenomenon of "big city disease", such as traffic congestion, resource shortage, and environmental pollution. These problems seriously restrict the development of cities. Among them, the common method for solving the traffic congestion problem is green wave coordination control. It coordinates the signal settings of multiple adjacent intersections on the main road to increase the probability of vehicles encountering green lights when passing through each intersection, reduce parking time and parking rate, and thus form a continuous traffic flow like "waves" on the main road, so as to pass through multiple intersections continuously and uninterruptedly under the green light, thereby alleviating the huge pressure of traffic congestion [1].
Graphical method, numerical method and model method[2] are already familiar to researchers and scholars as green wave coordination control methods, and many people have done a lot of work on improving the algorithms of these three coordination control methods. Among them, the coordination control scheme determined by numerical method has better comprehensive performance and stronger controllability. However, these common green wave coordination control methods are not omnipotent. For the case of symmetrical distribution of traffic flow on the trunk line in a group of intersections with two-way straight traffic, the above-mentioned methods are very effective, but they are useless for traffic trunk lines with asymmetrical traffic flow. Therefore, this paper proposes a green wave coordination control method based on numerical method for the case of asymmetrical traffic flow through the trunk line, and analyzes and verifies its accuracy and effectiveness through numerical examples.
1. Arterial Road Traffic Flow Prediction Model
Suppose there is a north-south urban arterial road with n intersections. Straight traffic from south to north is the direction of vehicles going up, and straight traffic from north to south is the direction of vehicles going down. Let the traffic flow in the direction of going up be and the traffic flow in the direction of going down be . Asymmetric traffic flow refers to the traffic flow values in both directions in the same phase being significantly different, with one direction being significantly more than the other. That is, the traffic flow difference a and saturation b in both directions satisfy: , and , which is the sum of the saturation flow of all corresponding straight lanes. Then the traffic flow in the opposite direction is considered to be asymmetric [3]. Since the traffic flow changes throughout the day, there may be a large difference in traffic flow at different times. In view of this situation, the use of time-sharing or adaptive arterial traffic signal coordination control is generally considered to meet the coordination control effect [4]. Therefore, it is very necessary to predict the traffic flow before choosing which signal timing scheme to use. Here, a traffic flow prediction model is introduced. It uses an iterative method to make macroscopic predictions. The basic process is as follows:
(1) Taking a main road (representing the i-th intersection on the main road, and the road segment between the i-th intersection and the (i+1)-th intersection) as an example, its simplified route diagram is shown in Figure 1. The figure shows the number of vehicles entering the intersection from right turns, straight ahead, and left turns. The formula for calculating the number of vehicles entering the intersection is as follows:
Figure 1. Basic route map of the trunk line
At step k: represents the traffic flow originating from the intersection in the approach lane; represents the traffic flow about to turn into the approach lane; represents the number of vehicles reaching the end of each queue; represents the traffic capacity of the line containing the approach lane (in terms of the number of vehicles); represents the proportion of vehicles leaving the line and entering each approach lane; represents the saturation flow rate on the line; (k) represents the number of vehicles turning into the intersection when the line traffic is not yet saturated; then represents the number of vehicles turning into the intersection when the line is saturated; is a binary parameter that takes only 0 or 1. If the value is 0, the signal light for turning into the approach lane at the intersection is red; if the value is 1, the signal light for turning into the approach lane at the intersection is green.
(2) After a certain period of time, vehicles exiting from the upstream intersection reach the end of the queue at the intersection. This delay time is recorded as:
At step k: represents the traffic flow capacity of the line (statistically based on the number of vehicles); represents the sum of the number of vehicles queuing in each lane of the line; represents the total number of lanes on the line; represents the average length of the vehicles; represents the free speed of the vehicles, i.e., unconstrained by any external conditions.
(3) Based on the prediction model, we can predict the number of vehicles that will reach the end of the queue at step k. The formula is as follows: , which represents the number of vehicles that leave the intersection entrance lane and enter the line; derived from the formula.
(4) The number of vehicles that reach the end of the convoy and enter different lanes on the route is expressed by the following formula:
(5) The expression for the queue length of vehicles entering the intersection at step k+1 is denoted as: , and the expression for the total queue length is:
(6) Taking into account the entry and exit of traffic flow on the line during the period [k, k+1], the storage capacity on the line is recalculated as follows:
(7) The traffic flow on the line at step k+1 is derived as follows:
Therefore, as long as the road traffic flow state at step k is known, the road traffic situation n steps later can be predicted using the above iterative prediction method.
If the difference in traffic flow between the two directions, *a*, and the saturation level, *b*, satisfy the condition, then it is an asymmetric case; otherwise, it is a symmetric case. Then, the appropriate signal coordination control method is selected based on the traffic flow distribution. If the opposing traffic flow is symmetrical, a graphical method for bidirectional green wave coordination control under a symmetrical release mode can be selected; if the opposing traffic flow is asymmetric, a numerical method for green wave coordination control under an asymmetric release mode can be considered. The following will focus on the numerical method for green wave coordination control under asymmetric traffic flow.
2 Numerical Green Wave Coordination Control under Flow Asymmetry
The guiding principle of the numerical solution algorithm for bidirectional green wave coordinated control under asymmetric traffic flow is that the ideal green wave bandwidth should exist in any arterial direction of travel and be large enough to enable the arterial control system to achieve efficient green wave coordinated control [6]. It is mainly achieved by analyzing the characteristics of the intersection phase design method, using the time distance analysis method in arterial traffic coordinated control, finding an ideal intersection location, minimizing the interval between it and the actual intersection, and determining the optimal signal phase timing, common signal period and phase difference of the intersection group [7]. The specific calculation steps are as follows:
(1) Based on the spatial rigid geometry of the arterial road intersection group and the actual traffic demand, determine the space and setting method of its signal phase optimization. The signal phase setting methods available include the common symmetrical release method at the entrance, the separate release method at the entrance, and the overlapping release method [8][9].
In comparison, symmetrical and individual approach release methods are quite influential in existing intersection signal phase design approaches. If the hard conditions of a trunk intersection are symmetrical, then choosing symmetrical approach release for coordinated control can greatly improve intersection capacity and reduce traffic delays. Similarly, individual approach release has its own unique advantages in balancing traffic flow saturation. Asymmetrical phase overlap release involves properly arranging release times for different traffic volumes in traditional four-phase release. In other words, it allows busier intersections to "borrow" some time from less busy intersections, thus significantly reducing vehicle dwell time and naturally improving intersection tidal flow phenomena and overall traffic efficiency.
(2) The green ratio is reasonably allocated to the intersection group under each signal phase setting method.
(3) Determine the range of values for the common signal cycle of the arterial road intersection group and the optimization space[10].
(4) Using the time-distance diagram [11], the ideal intersection spacing under different signal periods and different signal phase sequences is derived.
(5) Determine the optimal signal timing design scheme for the arterial road intersection, and determine the optimal common signal cycle for the arterial road and the optimal signal phase setting method for each intersection.
(6) Determine the bandwidth of the bidirectional green wave.
3. Case Analysis
Assume a north-south urban traffic artery with three intersections. The intersections are asymmetrically distributed. The distance between adjacent intersections is 540m from south to north and 550m from north to south. The green wave speed between intersections is 11m/s, and the green wave speed between intersections is 10m/s. The phase design of the east and west entrances is that each entrance is given separate access. The sum of the green wave ratios of the three intersections is 1. The green wave time allocation for each entrance direction within the signal cycle range is shown in Table 1.
Table 1. Signal Timing at Each Intersection
(1) It should be noted that there are no special requirements for the signal phase setting method of the north and south entrances of the three intersection trunk lines. The three design methods of overlapping release phase, symmetrical release phase, and separate release phase can be selected arbitrarily.
(2) The green ratio is a key parameter and plays a crucial role in signal timing design. Setting the green ratio reasonably can greatly alleviate traffic flow density, reduce dwell time, and improve the traffic efficiency of road sections. Here, according to the green ratio fixed principle mentioned above, the green ratio is allocated for overlapping release, symmetrical release, and separate release design methods, as shown in Table 2.
Table 2. Green light ratio distribution in the north-south direction at each intersection
(3) Determine the range of common signal cycle values for the arterial road intersection group, and then determine its optimization space based on the obtained range for further optimization. For the three intersections on the north-south arterial road in the example, Table 1 shows that the signal cycle value spaces for each intersection are [90s, 120s], [85s, 110s] and [80s, 115s], respectively, while the allowable range of common signal cycle values for the intersection group is [max{}, min{}], which can be denoted as []. Therefore, it can be concluded that the range of common signal cycle values for the arterial road is [90s, 110s][12].
(4) Since the calculation process is quite complicated, only one example is given here. Assume that the intersections adopt the method of separate entry and exit, and calculate the ideal intersection distance between them. For ease of explanation, let C be the common signal period, be the green time of the intersections in the four entry directions of east, south, west and north, be the green light time consumed to switch from one signal phase to another, be the green wave speed, and be the ideal intersection distance between the intersections. For different phase sequence combinations between the two intersections, the corresponding ideal intersection distance can be calculated by using the time-distance diagram shown in Figure 2 [13], and the results are shown in Table 3.
Figure 2 Time-distance diagram under different signal phase sequence settings
Table 3 shows the ideal intersection spacing when the phase sequence of the intersection is set to North-South and East-West.
(5) Select the intersection as a reference intersection and calculate the offset green ratio of the other two intersections under different common signal cycles. The calculation is based on minimizing the sum of the maximum offset green ratios. The results are shown in Table 4. It can be seen from the table that when the optimal common signal cycle is 100s, the optimal signal phase combination of the intersection is north-south east-west, north-south overlap, and south-northwest-east. The calculation results of the six signal phase design methods of the intersection are summarized in a table, as shown in Table 5.
Table 4 shows the green offset ratio of each intersection under different public signal cycles.
Table 5 Optimal Phase Combinations and Common Signal Period
(6) The calculation results of intersection, offset green ratio, actual azimuth position, green ratio above and below the center line and green wave bandwidth in the south-to-north and north-to-south driving directions are shown in Tables 6 and 7, respectively.
Table 6 Calculation of Green Wave Width for Southbound Traffic (Southbound to Northbound)
Table 7 Calculation of Green Wave Width for Northbound Travel
As can be seen from the table, the green wave widths obtained in both the south-to-north and north-to-south driving directions are ideal green wave widths, achieving the ideal effect of green wave coordinated control.
4. Conclusion
In summary, the proposed arterial road traffic flow prediction model is simple in expression and easy to solve. By predicting the traffic flow distribution, a reasonable signal coordination control scheme can be accurately selected, improving the efficiency of arterial road coordination control. Addressing the asymmetric distribution of traffic flow on arterial roads, this paper proposes a numerical green wave coordination control method under an asymmetric release mode. This method considers the actual traffic conditions on the arterial road, such as the distance between intersections, the number of lanes, and the traffic flow distribution, to comprehensively optimize the signal phase sequence at arterial road intersections, achieving the expected effect of green wave coordination control. Furthermore, analysis shows that this method is also applicable to situations where the distance between opposing intersections is unequal.
