A dynamic variety model to describe the traffic flow of target lane affected by lane change on urban road

In order to discover the variation of the traffic flow caused by lane change, the article analyzes the minimum safety distance and the lane change time between the lane change vehicle and the vehicles of the target lane based on the theory of vehicle lane change model and Bureau of Public Roads model. The influence time of lane change is determined by choosing two adjacent lanes as the research object according to the minimum safety distance of three kinds of combination. A model to describe the affected speed on lane 2 was established when the vehicle changed lane from 3 to 2 based on the measured data. The research results show that the speed of the target lane affected by the lane change is closely related to the volume of the adjacent two lanes and the acceleration of lane change vehicle. Then, while the more small acceleration of the lane change vehicle was, it would be more big variation traffic speed of the target lane. Finally, when the acceleration of the lane change is between 0.1 and 2.0 m/s2, the influence is more serious than others.


Introduction
Lane change is the common driving behavior of traffic flow, which involves the traffic characteristics of the driver's feeling, perception, judgment, and operation. When the traffic density is low, it is beneficial for driver to obtain the desired speed and improve the efficiency of traffic. When traffic density is heavy, the opportunity to change lane is small. If the driver is forced to change lane, it will affect the speed of current and target lane, which is easy to result in traffic conflict and form potential traffic safety hazard. 1 Usually, we focus on the models of lane change and aim at different conditions such as lane change behavior decisions, safe distance, and lane change time by the method of simulation and experimental data analysis. [2][3][4] And the model to describe the impact of the lane change on traffic flow is rare. Similar research mainly focuses on the interference effects. For example, Jing et al. 5 analyzed the relationship between the number of vehicles and traffic density. LY Wei 6 analyzed the behavior of vehicles affected by the change lane of bus with the method of cellular automata model. S Oh 7 investigated a variety state of traffic flow and found that it will affect the efficiency of the traffic flow and reduce traffic capacity of the downstream. Based on the measured data, HZ Xu 8 established the influence speed model of traffic flow owing to lane change. W Yu 9 analyzed the relationship between lane change and traffic congestion by using the traffic wave model. ML Wang 10 established a lane change model based on fuzzy control theory and quantitatively analyzed the influence of different lane change times on the traffic flow through the simulation comparison. Some research shows that different accelerations, safe distances, and traffic condition of two adjacent lanes will lead to different effects on the target lane during the different condition of lane change. 11 , 12 Less studies can be found toward the analysis of lane change time, especially lack of the micro-changes characteristics of each lane under the influence of multifactor coupling while the existing studies mainly focus on the macro-change characteristics of traffic flow.
The article established a calculation model of lane change time through the theoretical analysis of the characteristics of safe distance. In addition, combining with the instance, we established a model to describe the dynamic variety traffic speed of a single lane influenced by lane change. It has been verified by simulation and reality that the model has some practical significance to research the characteristics of traffic safety and variation of the delay on motorized road in the future.

The model of the dynamic variety traffic speed affected by lane change
In order to study the variety performance of the vehicles affected by lane change, this article tries to use the theoretical method to find some related orderliness through analyzing the behavior of lane change. Because we mainly focus on the influence of target lane caused by lane change, this article neglects the distance of the vehicle back and forward, which plans to change lane.
Some rules were made that the order number of the lane which is more close to the center of the road would be named 1, the next one should be 2, and so on. Figure 1 shows the initial position of the vehicles on lanes 2 and 3. In order to analyze the trajectory of vehicle, we establish a coordinate system and define left edge as X-axis and upward as Y-axis. As shown in Figure 1

Analysis of the safety distance characteristics between O and A
Trajectory of the vehicles O and A was shown in Figure  2. In order to elaborate the relationship between A and B, the vehicle has finished the process of lane change. Based on the coordinate system, this article assumes that the horizontal and vertical coordinates of the lane change vehicle are x o (t) and y o (t), respectively, at one moment.
The horizontal and vertical coordinates of A are x A (t) and y A (t), respectively. Assuming that the moment while O 1 reaches the boundary between lanes 2 and 3 as t adj + t c and t adj is the moment to start speeding up laterally, and t c is the duration time after acceleration. At the moment of t adj + t c , O 1 crosses the horizontal line LA right of the vehicle A and assumes that they meet with the line LA at C. So t adj + t c is the time when the lane change vehicle moves to C. Then, some equations can be drawn as follows where u(t) is the angle between the vehicle O and the horizontal direction at the moment t adj + t c , which can be figure out as follows and w 2 = y A (t adj + t c ) and w 1 = y 0 (t adj + t c ). Substituting them into formula 1, we can get the equations as follows If t ø t adj + t c , when the lane change vehicle crosses the line LA, it easily leads to side collision with A. When the vehicle finishes the lane change, it easily leads to tailgating with A, so if T ø t ø t adj + t c , it is dangerous for lane change vehicles. In order to avoid collision, they need to be satisfied with equations as follows Assuming the horizontal distance between the left front point of O 1 and the right rear point of A 3 as S(t), then we can calculate S(t) as follows where S AO (0) is the initial longitudinal distance between the headstock of O and the tailstock of A at the moment ready to change lane. Other symbols are synonymous. Therefore, if the collision of the lane change vehicle should be avoided, it must meet with two requirements, which is S(t).0 and S AO (0), and should meet the minimum safe distance requirement. 13 The formula is as follows and S A (t) À L A À w O : sin½u(t) À S o (t) + S AO (0).0.

Analysis of the safety distance characteristics between O and B
Similarly, in order to avoid friction or collision, the vehicle O must keep the minimum safe distance with B after changing lane; the position of each vehicle is shown in Figure 3.
Assuming the horizontal distance between the left rear point of O 2 and the right front point of B 2 as S(t), then where S BO (0) is the initial longitudinal distance between the headstock of O and the tailstock of B at the moment just ready to change lane. Other symbols are synonymous. Therefore, if the collision of the lane change vehicle should be avoided, it must meet with two requirements, which is S(t).0 and S BO (0), and should meet the minimum safe distance requirement. The formula is as follows

Analysis of influence time caused by the lane change vehicle
Related studies have shown that 14 each driver has desirable speed during certain traffic density, which is related to vehicle's mechanical performance, driver's  characteristics, speed limit rule, and so on. The vehicle will intend to change lane while the influence of slow one ahead leads to lower speed than expected within a certain range. Usually, the vehicle changes lane from lower speed lane to higher speed one, so this article mainly aims to this type and carries on some research. From the analysis of formula (8), we can know that if we want to compute the safe distance between the lane change vehicle and target vehicle, we need to measure the speed and acceleration of them. But the actual measurement is difficult. In order to find the general rule of lane change, the article assumes that lane change vehicle is driving at a constant acceleration and other vehicles keep constant motion while changing lane.
The time for changing lane between O and A. The changing time is defined that there is a safe distance between O and A after changing. Figure 2 shows the trajectory of the vehicles O and A. We define t 0 as the beginning of changing lane, t 1 as the end time, t a as the time for changing, so t a = t 1 À t 0 . After changing O's speed in accordance with those in target line, that is, v o (t) = v A (t), the time for accelerating is and A's driving distance is

So the longitudinal distance between O and A after changing is
If we avoid O not to collide with A, it must satisfy the condition S(t).0, then Figure out in equation (12), then Because the usual value of u is about 3°; 5°and w o is 1.5 m, 0:1\w o sin u(t)\0:2. Generally, , and when formula (13) is satisfied The time for changing lane between O and B. The changing time is defined that there is a safe distance between O and B after changing. Figure 3 shows the trajectory of the vehicles O and B. We defined t 0 as the beginning of changing lane, t 1 as the end time, t a as the time for changing, so t a = t 1 À t 0 . After changing O's speed in accordance with those in target line, that is, v o (t) = v B (t), the time for accelerating is During this time (t a ), O's driving distance is and B's driving distance is

So the longitudinal distance between O and B after changing is
If we avoid O not to collide with B, it must satisfy the condition S(t).0, that is Figure out in equation (18), then Generally, , and when for- The time influenced by the lane change. From the analysis above, we know that in order to avoid collision between O and A, the time for changing is required to satisfy the formula as follows: , where t A a is the changing time that can ensure minimum safe distance between O and A after changing. That is to say, it should be as short as possible and not go beyond the threshold, so that O can avoid rear-end collision with A. For O and B, it requires t B a . ( , where t B a is the changing time that can ensure minimum safe distance between O and B after changing. It should be as long as possible and greater than the limit so that they can avoid collision.
When the vehicle changes from lane 3 to lane 2, if B leaves far enough from O, we can neglect the influence between them and just need to meet the longest changing time of A. If A is far enough from O, we also can neglect the influence between them, and only need to meet the shortest change time of B. If the distance between A and B, and the lane change vehicle O, is moderate, then the influence of them should be considered.
Based on the analysis of the actual data, we find that the vast majority of vehicles consider the front vehicle on the target lane as the object priority. So, this article first studies the situation, which meets the requirement of the vehicle A, where If we want to meet the requirement of B at the same time, it needs t B a . ( We assume that the vehicles' transport condition on the target lane is consistent, that is, v B = v A , and consider different minimum safe distances and suggest vehicles are changed with the same acceleration a o . That is, O satisfies the shortest changing time t a 2 (t A a \ t B a ), where there is no influence on A or B.
At this situation, the vehicle O can meet the safe change requirement with A, but it will impact B. The influential time is where 0\t a \ If the vehicle O meets the safe changing requirement with B, it will fail to meet with A. It means O will collide with A; thus, we do not consider this state.
3. S AO 0 ð Þ\S BO 0 ð Þ The limit of t A a is less than t B a . In order to meet the requirement for changing lane, there are various types. So, we analyze with different time caused by lane change.
It can meet the requirement of A and B for changing lane at the same time. Vehicle O needs to satisfy the shortest changing time t a 2 (t A a \ t B a ) = [, but this situation is impossible because the intersection is an empty set.
where 0\t a \ In the formula, Relevant research 15 indicates that the minimum safe distance between the lane change vehicle and the vehicle A on the target lane is Analyzing formula (24), we can know that the value of S AO (0) is ½0, L A + w o sin u(t), and the minimum safe distance between O and B is Analyzing formula (25), we can know that the value of S BO (0) is ½L B + w o sin u(t), + '. Since this article assumes that the vehicles' transport condition in the target lane is consistent, the vehicle O is the same type as the vehicle on target lane, so it means v B = v A and L A = L O . Then, S AO (0)\L BO (0); obviously, the condition is impossible. Thus, if the vehicle O has clearance with B, it will have an impact on B. So, the influential time is where Related studies have shown that, 16 vehicle's speed after affected is closely related to the social vehicles' initial velocity, impact time and the distance between the location of the parking vehicles began to slow and entrance.

Application of the model
And it can be described as v = À 2:031 + 0:842v J À 0:04T y + 0:101 s. Through the data analysis, we found that the average speed of society vehicles is closely related to traffic and its lane change rule can use BPR (Bureau of Public Roads) function model to describe, 17 when the road conditions and traffic conditions are generally identical. If it substitutes other data for v J , T y , and s in the model, it can also reflect the lane change rule.
Because this article assumes that the vehicle changes from low speed way to high speed way, and that there is no reduction behavior after finishing changing, the S value is 0. When the vehicle changes from lane 3 to lane 2, the model that lane change vehicle impact vehicles on target lane 2 can be described as v = À2:031 + 0:842v 2 À 0:04    change from lane 3 to lane 2. Figure 6 compared velocity on lane 2 before and after. It can be seen that lane change caused vehicles slow down; meanwhile, if the      volume of lane 2 increases, then amplitude of speed change will be reduced. It is accordance with the actual situation and proves that model is accurate.

Conclusion
The speed of the target lane affected by the lane change is closely related to the volume of the adjacent two lanes, and the acceleration of lane change vehicle and the influential time can be calculated as follows: À t a . When the volume of the lane was from 500 to 600 pcu/ h while the target lane was from 0 to 500 pcu/h or when the volume of the lane was from 100 to 300 pcu/h while the target lane was from 1000 to 1500 pcu/h, the speed of the target was deeply affected by the lane change.
While the more small acceleration of the lane change vehicle was, it would be more big variation traffic speed of the target lane. Finally, when the acceleration of the lane change is between 0.1 and 2.0 m/s 2 , the influence is more serious than others.

Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.