In order to analyze the effect of U-turn vehicle on traffic performance, the work develops a game theory-based description of drivers’ interactions in U-turn scene, considering the decision-making uncertainty. The hybrid strategy of the game is obtained. The relevant parameters of model are calibrated by collected video data in Changchun, China. A two-way four-lane cellular automaton model with the game model imbedded is constructed for identifying the effect of U-turn vehicle on traffic performance. The influencing factors are identified with their correlation analyzed by numerical simulation of different traffic conditions. According to the simulation results, U-turn traffic has a significant influence on traffic delay in the lane of same direction, compared with opposite direction. The severity of conflict between vehicles is classified and the causes are identified by analyzing the arrival rate of the U-turn vehicle and the conflicting straight vehicle and the relationship with one another. In addition, the threshold of traffic flow causing traffic conflict and traffic delay are proposed. The results show that the proposed models reconstructed the traits of traffic flow and conflict phenomenon in the presence of U-turn vehicles on road section.
The road safety has become a high-priority issue to traffic engineers and traffic authorities for decades. Conflicts between different participants, including cars, motors, buses, bicycles, and pedestrians [
U-turn facilities are used as open areas for two-way traffic flow on the road, often set at the entrance of the intersection or the middle of the road section. U-turn vehicles have significant impact on the traffic performance. Median openings (including U-turns) are considered as the crash-prone locations of Thai highways [
In theory, straight vehicle should get priority over U-turn vehicle all the time. The gap acceptance mechanism of U-turn vehicle is the process of selecting suitable vehicle gap after arriving at a mid-block median opening. In this process, the U-turn vehicle may be successful or unsuccessful, depending on the available gaps appearance of straight vehicle. In reality, the U-turning vehicles often do not wait for an acceptable gap in the on-coming straight-traffic. They gradually move onto the conflicting lane to show their intention to go; particularly when the U-turn traffic is in a long queue or has waited for a long time, the drivers tend to be more aggressive. At the same time, the conflicting straight vehicle will be reluctant to yield by increasing speed, changing lanes, or flashing their headlights. The interaction between U-turn vehicle and straight vehicle can be described as two persons’ noncooperative game behavior. Game theory has the advantage of take into account the impact of stochasticity and decision uncertainty, providing a powerful tool for analyzing the interaction behavior in transport system. Game theory has been applied widely in different disciplines, such as lane changing, car-following, and driverless vehicle control [
In addition, there have been extensive studies done on the simulation of the vehicle-vehicle interaction. The most popular simulation model is rule-based models [
This paper makes two contributions. The first is quantifying the dynamic process of decision-making varied with traffic character over each lane at U-turn road section. The second contribution is evaluating the impact of the relationship between the arriving rate of U-turn vehicles on one side and the arriving rate of straight vehicles on opposite side lane on the lane-based delay and vehicle-vehicle conflict as well as exploring the threshold of different levels of traffic delay and conflict severity, which would be helpful for transportation agencies to meet driver’s time-cost and safety or comfort needs when they choose the opening for U-turn vehicles in road section.
The work is organized as follows. In Section
A typical U-turn scenario is shown in Figure
Firstly, we formulate the game, taking into account the players, strategy, and payoff.
In this game theory model, there are two players,
Table
Game payoff matrix.
U-turn vehicle | |||
---|---|---|---|
Pass | Wait | ||
Straight vehicle | Pass | | |
Wait | | |
Game payoff matrix quantified.
U-turn vehicle | |||
---|---|---|---|
Pass | Wait | ||
Straight vehicle | Pass | | |
| | ||
Wait | | | |
| |
As shown in Table
U-turn vehicle (U-turn, wait) probability is as follows:
Straight vehicle (travel, wait) probability is as follows:
From the perspective of social optimal efficiency, U-turn vehicles and straight vehicles should be polite in the U-turn area; that is,
Tanimoto et al. [
According to (
The relationship between the variables and the Nash Equilibrium solution. (a) The headway is equal to 1. (b) The headway is equal to 2.75.
The social dilemma structure can be divided into Prisoner’s dilemma and Chicken dilemma. Prisoner’s dilemma indicates that Nash Equilibrium is close to one of the pure strategic situations. At this time, the probability that a certain strategy appears is close to 1. Chicken dilemma has the characteristics of polymorphic Nash Equilibrium, not close to any pure strategic situation; it means that no strategy has a probability close to 1. Through the analysis of Figure
We conducted an empirical survey at the U-turn point (at the east side of the intersection of Guanghua Street and Weishan Road in Changchun City, China), as shown in Figure
Scene of the U-turn point in Changchun, China.
A total of 38 vehicles preparing for U-turn in road section are selected as samples in this research, and there are 198 games shown in Figure
Observed game results in different scenes.
We collected statistics for probability of U-turn vehicles’ action under two kinds of headway intervals. The results of surveys are shown in Table
Headway statistics in different scenes.
Headway interval | The number of turning vehicles | The number of waiting vehicles | Total by row | U-turn vehicle (U-turn, wait) probability |
---|---|---|---|---|
(0-2] | 24 | 150 | 174 | (0.14, 0.86) |
| ||||
(2-3.5] | 14 | 10 | 24 | (0.58, 0.42) |
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Total by column | 38 | 160 | 198 | (0.19, 0.81) |
We calibrated the model parameters of
A two-way four-lane cellular automata traffic flow model with a U-turn facility in middle of road is established (see Figure
Scene of vehicle U-turn cellular automata model.
① Lane A is in the same direction of Lane B, and Lanes C and D are opposite. ② One cell represents 3.75 m. ③ The road length is set as 200 cells, so the road length equals 750 m.
① The simulation time step is set to 1s. ② It is assumed that all the vehicles have the same length of 2 cells (7.5m) and the maximum speed of 5 cells (
U-turn vehicles specific behavior process is as follows: The vehicle completes the U-turn through (B0, B1) to (B1, C1), (B1, C1) to (C1, D1), and (C1, D1) to (D1, D0).
According to the division of the road area, there are two types of lane changing rules: The forced lane changing rule of the U-turn vehicle in S2 of Lane A and the general lane changing rule of the other vehicles on the road. Using the classic lane changing rule proposed by Chowdhury [
We use the classic NaSch model as the vehicle update rule. In the NaSch model, time, space, and speed are discretized, which is the minimum model that can reproduce the basic characteristics of road traffic flow. Although the NaSch model [
It is also stipulated that the U-turn vehicle on Lane B will no longer change the lane in the area S2 but eventually moves to the U-turn position B1. U-turn vehicle in Lane A, without meeting lane change condition, will wait at Position A1 until it can safely change to B and eventually to B1. Combining the above two lane changing rules constitutes the lane changing rule of the model.
The opening boundary conditions are used to control the access of the vehicle, according to the monitor of locations of the head vehicle and the last vehicle. When the head vehicle position is greater than the lane length, it is already out of the road.
Otherwise, when the last vehicle position from the road entrance is greater than
At the same time, the rule of vehicle into the road is modified: The vehicles into Lanes A and B with probability
As stated in Section
In order to quantify the relationship between the utilities of delay and threat perception, the results are normalized. According to the analysis of the UTG model, the U-turn rule is divided into two states:
U-turn vehicle (pass, wait) probability in Lane C is as follows:
Straight vehicle (pass, wait) probability in Lane C is as follows:
U-turn vehicle (pass, wait) probability in Lane D is as follows:
Straight vehicle (pass, wait) probability in Lane D is as follows:
Here,
Parameter setting:
By changing the probability of vehicle entering Lanes A and B (marking the probability as
The most frequently used conflict indexes are the time of collision (TTC) and the postinvasion time (PET). In the work, the TTC index is selected to evaluate the severity of the U-turn conflict. TTC means that if the current speed, direction, and trajectory are maintained, two road users expect to have a collision. Smaller TTC leads to the more serious conflict.
In order to quantitatively study the severity of the conflict of the U-turn, the conflict was divided into discrete severity levels according to the different thresholds of the conflict index. At the same time, as the conflicts will happen in Lanes C and D, the severity of the conflict is calculated based on the lane.
Figure
TTC in lane. (a) Lane C. (b) Lane D.
Severe: The average time for the straight vehicle to arrive at the conflict point is between 1.5 and 2 time steps. The U-turn vehicle takes up two cells, at least 2 time steps to leave the area without stopping. It means that the U-turn vehicle and the straight vehicle will inevitably change their behavior strategy because of conflict; if not, it will inevitably produce traffic accidents. The severity of the conflict is defined as “severe” in this condition.
Slight: The average time for the straight vehicle to arrive at the conflict point is between 2 and 3 time steps. In this condition, if the U-turn vehicle does not stop to U-turn directly, the straight vehicle do not have to change the behavior. If the U-turn vehicle needs to wait for a peripheral cause (for example, the U-turn vehicle in Lane C is waiting for the pass of the straight vehicle in Lane D), the straight vehicle needs to change its behavior to avoid the collision. The severity of the conflict is defined as “slight” in this condition.
Potential: The average time for the straight vehicle to arrive at the conflict point is more than 3 time steps, meaning that this time can meet the certain demand of the U-turn, and the straight vehicles do not need to change their behavior. However, there will be a conflict if the U-turn vehicle keeps waiting in the conflict point. The severity of the conflict is defined as “potential” in this condition.
Using SPSS to analyze the influencing factors of traffic conflicts, Table
Correlation matrix for conflict.
| | | ||
---|---|---|---|---|
TC of lane C | Correlation coefficient | -0.045 | -0.548 | 0.241 |
Significance | 0.722 | | 0.055 | |
| ||||
TTC of lane D | Correlation coefficient | -0.097 | -0.415 | -0.012 |
Significance | 0.447 | | 0.924 |
We analyze the impact of
Impact of factors on TTC in lane. (a) Lane C. (b) Lane D.
The TTC value and
The reason for the above phenomenon is related to the traffic flow of Lane C affected by
The U-turn of the vehicle at the U-type facility will result in the delay of the same/opposite way vehicles. The former is caused by the lane changing behavior in interweaving zone of S2 area, and the latter is caused by the conflict with U-turn vehicle.
In the work, based on the theory of traffic wave, the influence of U-turn on delay is analyzed by comparing the change of traffic flow in U-turn area. Since the steering area has a significant delay on the traffic flow of the road, only the upper steering area of each lane is considered in the statistics (see Figure
Flow in lane. (a) Lane A, (b) Lane B, (c) Lane C, and (d) Lane D.
Using SPSS to analyze the factors influencing the traffic volume, Table
Correlation matrix of influencing factors for delay.
| | | ||
---|---|---|---|---|
Lane A | Correlation | 0.171 | -0.127 | -0.828 |
Significance | 0.130 | 0.262 | | |
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Lane B | Correlation | 0.160 | -0.122 | -0.829 |
Significance | 0.157 | 0.281 | | |
| ||||
Lane C | Correlation | -0.039 | 0.554 | -0.470 |
Significance | 0.731 | | | |
| ||||
Lane D | Correlation | -0.059 | 0.572 | -0.448 |
Significance | 0.606 | | |
We analyze the impact of
Impact of factors on volume in lane. (a) Lane A. (b) Lane C.
By comparing the traffic volume with
When
In the case of
The traffic volume of C forms a crowded flow in the vicinity of
When the U-turn vehicle in A and B is queued, it will obviously block the straight travel of these lanes, and this delay will spread in the form of waves upstream, resulting in continuous decrease of traffic flow. For C and D, the U-turn vehicles can U-turn in U-turn gap; although affecting the driving of the straight vehicles, they will not cause the stairs of the flow, so traffic volume will not be reduced to a minimum, but in the state of crowded flow.
We can get the following conclusions by analyzing conflict and delay.
Thus,
Figure
Space-time diagram in lane when
In order to analyze the traffic conflict caused by vehicle U-turning and its influence on traffic flow of road section, we establish the UTG model to describe drivers’ interactions between U-turn vehicle and straight vehicle and run simulation under different traffic condition based on cellular automata. Our works are mainly summarized in three aspects:
This paper makes a useful attempt to construct the U-turn model by considering the driver’s decision-making uncertainty and analyze the effect of U-turn behavior on traffic through the construction of cellular automata simulation. This work provides some information for traffic managers to choose the opening for U-turn vehicles in road section. Due to the difficulty in exploring drivers psychological aspect from video data, the parameters in model can be verified by other approaches (e.g., questionnaires or driving simulator). Further research can also be made on the identification of the impact of U-turn traffic on the different lanes including the same direction lanes which will be influenced by the lane change behaviors of U-turn vehicles. Moreover, some parameters such as the size of the opening and the form of the waiting area which are important to the U-turn behavior of vehicles will been considered.
The data used to support the findings of this study are included within the article.
The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
This work was supported by the National Key Research and Development Program of China [Grant no. 2016YFB0101601] and Jilin Transportation and Transportation Science and Technology Project