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Rear-end accidents are the most common accident type at signalized intersections because of the different driving tendencies in the dilemma zone (DZ), where drivers are faced with indecisiveness of making “stop or go” decisions at yellow onset. In various researches, the number of vehicles in the DZ has been used as a safety indicator—the more the vehicles in the DZ, the higher the probability of rear-end accidents. However, the DZ-associated rear-end accident potential varies depending on drivers’ driving tendencies and the situations (position and speed) at the yellow onset. This study’s primary objective is to explore how the driving tendency impacts the DZ distribution and the probability of rear-end accidents. To achieve this, three types of driving tendencies were classified using K-means clustering analysis based on driving variables. Further, the boundary of the DZ is determined by logistic regression model of drivers’ stop/go decision. Then, we proposed the conditional probability model of rear-end accidents and developed a Monte Carlo simulation framework to calculate the model. The results indicate that the rear-end accident probability is dependent on the driving tendency even at the same position with the same speed in the DZ. The aggressive type has the highest risk probability followed by conservative and then the normal types. The quantitative results of the study can provide the basis for rear-end accident assessments.

Rear-end accidents are common at signalized intersections in China. Besides driver errors, indecisiveness in the dilemma zone (DZ) at signalized intersections is a leading cause of such accidents [

Although some negative effects of the GSCDs installation on intersection safety have been observed by some researchers [

An issue is created when intersections do not have GSCDs (NGSCDs), as drivers do not know when the green light changes to the yellow light. When the lights change, if a driver is close to the stop line, they have to stop abruptly, which leads to a higher risk of rear-end accidents. Finally, the regulation and enforcement were forced to revise again after being implemented for a week that drivers should stop if they can stop safely when the signal turns to yellow; however, those who cross the stop line on the yellow light will not be punished [

The “solid yellow” regulation was only implemented for a week; however, the discussion it triggered is not over. The big issue of NGSCDs during the regulation’s implementation has made the public and urban traffic managers aware of the advantages of GSCDs, causing GSCDs to be increasingly used in many cities of China, such as Beijing, Shanghai, Guangzhou, Nanjing, Shijiazhuang, and Harbin [

We believe that some of the negative impact of GSCDs on the DZ may partially come from the effect of characteristics of drivers. Since the last decade, a growing number of Chinese people have obtained their own cars and licenses, which has directly resulted in an increase in the number of novice drivers in the population [

DZ can be divided into two categories: type I and type II [

As shown in Figure _{c}>_{0}, the area between_{c} and_{0} is the dilemma zone (_{dz}=_{c}-_{0}); however, when_{c}_{0}, the zone is termed the option zone. In the option zone, drivers can either pass the intersection or slow down and stop at the stop line while the yellow signal is on or they can go. If the driver hesitates, it will lead to the emergence of the DZ. In fact, the boundaries of type I DZ are not fixed, and it will change depending on the driver’s speed on the yellow light’s onset [

Formation of the dilemma zone and option zone (source literature [

If the concept of type I DZ is from the view of stopping and yellow light passing distances, then the concept of type II DZ is from the drivers’ choice of stopping, which Zegeer [

Estimating accident risk probabilities of the DZ is a key issue in the DZ protection systems. The traditional approach is to take the number of vehicles in the DZ as a risk measurement indicator of the DZ at yellow onset. This approach is based on the assumption that the number of vehicles in the DZ is equivalent to the corresponding accident risk probabilities of the DZ, regardless of where the vehicle is located. However, the accident risk probabilities of the DZ will change based on factors such as the location and speeds of vehicles, the drivers’ behavior, and the power and braking performance of the vehicles. Some scholars have been exploring this field from the perspective of transition interval phase [

Past studies made significant contribution to understand the impact of countdown timers, driver behavior, the phase transition interval (usually referred to as the yellow signal), and the boundaries of the DZ on the accident risk probabilities of the DZ. However, the following issues remain to be addressed:

(1) Previous studies only investigated the impact of GSCDs or NGSCDs concerning the stop/go decisions of drivers but neglected the different driving maneuvers (e.g., aggressive acceleration, abrupt stop, and headway) of different driving tendencies (aggressive, normal, and conservative) under the condition of GSCDs. As we know, with an increasing number of Chinese people obtaining their own cars and licenses, the diversity of drivers is also increasing. Differences in driving tendency add meaning to “heterogeneous traffic”, which increases the complexity of driving behavior in the DZ and results in serious rear-end accidents.

(2) Although GSCDs have been installed at many urban intersections in China, with the significantly diverse driving behavior and China’s fast-growing roadway system, more field studies are needed to enrich the dataset for developing regulations.

This paper is aimed at developing a model to quantify the risk probabilities of rear-end accidents under different driving tendencies during the phase transition interval. More specifically, the research objective includes the following tasks:

We chose two urban signalized intersections as the experimental observation points, in Nanjing, China. The intersections were chosen because they are representative and have the same traffic flow patterns (97 percent of passenger cars and 3 percent of buses), the same signal phase (follow the same sequence of “steady green-flashing green (FG; 3s)-yellow (Y; 3s)-red”, and each intersection is not set all-red time, and the yellow light is the clearance time), and the same speed limit value (60km/h). For each intersection, one camera was set up on top of a roadside building to cover a studied approach road. The first studied approach is an eastbound approach along ZhongShan East Avenue (≈120 m); the second studied approach is an eastbound approach along HeXi Avenue (≈280 m). The video for each intersection approach was recorded from 8:00 a.m. to noon and 1:00 p.m. to 5:00 p.m. For each studied approach, over 150 hours of recorded video data were collected. The video recording process is shown in Figures

Data collection and processing.

Camera’s position at the first intersection

Camera’s position at the second intersection

The extraction trajectory data with Tracker

Transform coordinates with the imaging principle

The extraction method of trajectory data involved two steps. Firstly, the Tracker Video Analysis and Modeling Tool were used to extract the vehicles’ coordinates in the video. Secondly, the imaging principle was used to match the coordinates in the video with the location in the real world.

Tracker [

In this study, we include only two-vehicle accidents or the collision of the first two vehicles in an accident involving more than two vehicles. Approximately 796 couples’ trajectory data was extracted. Then, the driving parameters were derived, such as the speed, the acceleration/deceleration, and the headways.

In previous studies, driving tendency classification is completed according to the comparison of drivers’ actual stop/go decisions with the expected stop/go decision [_{0} (the theoretical calculating value of the maximum distance a vehicle can pass the intersection at the highest driving efficiency before the red signal is shown). With this definition, one can exclude aggressive drivers who stop at the stop line using aggressive acceleration or abrupt braking during their driving. In addition, from Liu [

Therefore, in this paper, the K-means clustering analysis method was adopted to classify the type of driving tendency. The standardized score of variables (speed, entry times (for the last-to-go vehicles), headway (for car-following vehicles, headway time is less than 7s [

Statistical results of driving variable parameters of different driving tendencies.

Driving tendencies | Variables | N | Min | Max | Mean | STD |
---|---|---|---|---|---|---|

Aggressive | Distance from stop line (m) | 220 | 24.34 | 69.79 | 41.1618 | 7.47832 |

Speed (m/s) | 220 | 10.11 | 18.48 | 13.3690 | 1.79014 | |

Acceleration/Deceleration (m/s^{2}) | 220 | -4.16 | 1.40 | -1.0895 | 1.66172 | |

Entry time^{1} (s) | 131 | 2.64 | 5.22 | 3.5204 | .57218 | |

Headway (s) | 213 | .67 | 2.99 | 1.7348 | .58343 | |

Stop time^{2} (s) | 89 | 4.10 | 6.01 | 4.8156 | .43181 | |

Reaction time (s) | 220 | .32 | 1.31 | .8276 | .15027 | |

| ||||||

Normal | Distance from stop line (m) | 352 | 19.97 | 67.49 | 36.3597 | 6.62826 |

Speed (m/s) | 352 | 5.27 | 18.53 | 12.6959 | 1.63167 | |

Acceleration/Deceleration (m/s^{2}) | 352 | -2.64 | 4.24 | -.5775 | 2.06079 | |

Entry time (s) | 150 | 1.63 | 3.19 | 2.3965 | .33242 | |

Headway (s) | 296 | .65 | 3.73 | 1.9953 | .61555 | |

Stop time (s) | 202 | 4.27 | 8.01 | 5.8085 | .69168 | |

Reaction time (s) | 352 | .35 | 1.95 | 1.0744 | .25885 | |

| ||||||

Conservative | Distance from stop line (m) | 224 | 15.89 | 76.42 | 38.9934 | 11.93947 |

Speed (m/s) | 224 | 3.98 | 17.13 | 11.7563 | 1.76093 | |

Acceleration/Deceleration (m/s^{2}) | 224 | -2.37 | 2.45 | -.6588 | 1.02378 | |

Entry time (s) | 116 | 1.62 | 3.28 | 2.4757 | .29486 | |

Headway (s) | 170 | .98 | 3.99 | 2.0137 | .59860 | |

Stop time (s) | 108 | 5.59 | 12.10 | 8.5019 | 1.43955 | |

Reaction time (s) | 224 | .44 | 2.46 | 1.2434 | .41100 |

Note: (1) Entry time refers to the time from flashing green onset to the time when the vehicle has just entered the intersection. (2) Stop time refers to the time from flashing green onset to the time when the vehicle stops at the stop line.

Vehicular speed refers to point speed with flashing green onset. The descriptive statistics of different driving tendencies are shown in Table

Acceleration refers to the average acceleration of the last-to-go vehicle from the flashing green onset to entering the intersection. It can be calculated by the following equation: _{c} denotes the distance between the last-to-go vehicle to the stop line;

Deceleration refers to the average deceleration of the first-to-stop vehicle. It can be calculated by the following equation:

The distribution probability density is expressed as

where

The calibration of the parameters.

Driving tendencies | Acceleration | Deceleration | Headway | ||||
---|---|---|---|---|---|---|---|

Normal distribution | Normal distribution | Weibull (3p) distribution | |||||

| | | | | | | |

Aggressive | 0.565 | 0.221 | 0.276 | -3.019 | 2.193 | 1.357 | 0.534 |

| |||||||

Normal | 0.926 | 1.698 | 0.222 | -2.267 | 2.584 | 1.640 | 0.542 |

| |||||||

Conservative | 0.946 | 0.052 | 0.291 | -1.423 | 2.031 | 1.312 | 0.850 |

where

To understand drivers’ decision during the phase transition interval and to determine the boundaries of type II DZ, a binary logistic regression was presented. Before the model coefficients were calibrated, the Pearson correlation coefficient formula was used to test the correlation of the variables to reduce the repetitive effect of the variables on the model. We found that the correlation coefficients between the speeds and headways are greater than 0.6, which belongs to strong correlation. Therefore, the headways variable is deleted. The model can be expressed as follows:

where ^{2});

The final logistic regression models of drivers’ stop/go decision can be presented by

As shown in (

According to (

Based on type II DZ, we know the upper boundary of the DZ is located where the vehicles have 90 percent probability of stopping and the lower boundary is at 10 percent. In addition,

① Under the condition of aggressive driving tendency

② Under the condition of normal driving tendency

③ Under the condition of conservative driving tendency

As discussed, when the phase transition interval starts, drivers in DZ need to decide to “go” or “stop” at first and then take the appropriate action (such as selecting a perceived safe acceleration/deceleration rate). A rear-end accident is likely to occur when a leading vehicle makes a stop decision and decelerates and the target vehicle fails to stop in the available stopping distance (scene 1:_{1} denotes the probability) or continues driving (scene 1:_{2} denotes the probability).

_{1} can be estimated using

_{i} taken by the leading vehicle when it decides to stop; and

In (

where_{f} is the target vehicle’s speed;_{h} denotes the headway between the leading vehicle and the target vehicle;_{f} denotes the distance between the target vehicle and the stop line; and_{lower} is the lower boundary of DZ._{h} follows the Weibull (3P) distribution (see Table

① Under the condition of aggressive driving tendency

② Under the condition of normal driving tendency

③ Under the condition of conservative driving tendency

_{0} and then the leading vehicle brakes and moves a distance of_{1} in t seconds. Meanwhile, the following vehicle driver perceives, reacts, and moves a distance of_{2}. The final distance between the two vehicles becomes h. If h>0, it is safe; otherwise, a rear-end accident occurs.

The process of car-following.

Relative position of two adjacent vehicles before and after braking

Change process diagram of the deceleration rate of the target vehicle

When the leading vehicle brakes, the performance of the driver in the following vehicle consists of three successive components as shown in Figure _{r}), set as 0.7s for the aggressive type, 0.9s for the normal type, and 1.1s for the conservative type, respectively. When a driver takes an action, it also takes time for the vehicle to reach its maximum deceleration and finally stop. This includes the brake system response (_{b}), which is set as 0.05s; the time needed to reach the maximum deceleration (_{u}), which is set as 0.2s; and the braking time until the following vehicle stops (_{c}).

In an interval_{1,} which is expressed as

The distance moved by the following vehicle_{2} is expressed as

Then, the final distance

The maximum deceleration of_{1} is determined by the product of the wheel-road adhesive coefficient (^{2}). Thus,

Considering the worst situation, the critical deceleration of the leading vehicle is

The required minimum deceleration (_{r}) of the leading vehicle to stop safely before the red signal starts as

where

Assume _{c}; otherwise

Given the normal distribution of deceleration discussed previously, the probability of the target vehicle fails to avoid a collision under different driving tendency conditions as follows:

① Under the condition of aggressive driving tendency

② Under the condition of normal driving tendency

③ Under the condition of conservative driving tendency

Based on the previous work, the overall model for scene 1 can be written as follows:

where

The overall risk probability of a rear-end collision is shown in

Calculating the

^{2}, -7.84m/s^{2}, respectively.

Estimation results of rear-end collision probability under different driving tendencies are shown in Figure

Simulation results of rear-end collision probability.

Probability of rear-end collisions at scene 1

Probability of rear-end collisions at scene 2

Total probability of rear-end collisions

It can be inferred from Figure

The total collision rear-end probability of different driving tendency is shown in Figure

The DZ problem is a leading cause for rear-end accidents at signalized intersections. In this paper, based on field observations and over 300 hours of recorded video data collection at two intersections and data analysis, we found that the collision rear-end probability is different even if a vehicle is at the same position with the same speed because the driver’s tendency is a key factor that cannot be ignored. To discover how the driving tendency impacts the DZ distribution and then impacts the probability of rear-end accidents, we proposed a conditional probability model of rear-end accidents and developed a Monte Carlo simulation framework to calculate the model. The conclusion that the aggressive type is the highest collision rear-end probability followed by the conservative and normal is drawn. However, to draw general conclusions, observations and data collection are needed for the different types of intersections with various geometric settings, traffic compositions, and signal timings.

The trajectory data used to support the findings of this study have not been made available because the National Natural Science Foundation of China (No. 51478110) has not completed them.

The authors declare that they have no conflicts of interest.

This work was supported by the National Natural Science Foundation of China (Nos. 51478110, 51508122) and Science and Technology Program of Jiangsu Province (No. BY2016076-05).