Optimal control of automated left-turn platoon at contraflow left-turn lane intersections

Abstract

Purpose

This study aims to propose a centralized optimal control model for automated left-turn platoon at contraflow left-turn lane (CLL) intersections.

Design/methodology/approach

The lateral lane change control and the longitudinal acceleration in the control horizon are optimized simultaneously with the objective of maximizing traffic efficiency and smoothness. The proposed model is cast into a mixed-integer linear programming problem and then solved by the branch-and-bound technique.

Findings

The proposed model has a promising control effect under different geometric controlled conditions. Moreover, the proposed model performs robustly under various safety time headways, lengths of the CLL and green times of the main signal.

Originality/value

This study proposed a centralized optimal control model for automated left-turn platoon at CLL intersections. The lateral lane change control and the longitudinal acceleration in the control horizon are optimized simultaneously with the objective of maximizing traffic efficiency and smoothness

Citation

Yang, H., Zhao, J. and Wang, M. (2022), "Optimal control of automated left-turn platoon at contraflow left-turn lane intersections", Journal of Intelligent and Connected Vehicles, Vol. 5 No. 3, pp. 206-214. https://doi.org/10.1108/JICV-03-2022-0007

1. Introduction

Intersections are key nodes of urban road networks. To improve the efficiency of spatial and temporal resources utilization at intersections, design measures including geometric design and signal control have been proposed. Among them, the contraflow left-turn lane (CLL) intersection, also called the exit lanes for left-turn (EFL) intersection, is a special intersection design mode. The CLL design sets part of the exit lanes at the intersection as a comprehensive functional area, which can be used dynamically as the exit lane and left-turn lane within a cycle. At the upstream of the intersection, there is a median opening and a presignal to control the position and time period that left-turn vehicles are allowed to enter the comprehensive functional area.

To optimize the operational efficiency of the CLL design, Zhao et al. (2013) first proposed an optimization model, in which all the variables including geometric and signal timing for both main and presignals were integrated. Wu et al. (2016) proposed analytical delay models for estimating the delay of left-turn vehicles and optimized the position of the median opening and the signal timing of presignal. Liu et al. (2019b) proposed an improved shockwave-based method to estimate the maximum left-turn queue length. Zhao et al. (2019) proposed a saturation flow rate adjustment model for EFL control based on field data. Wu et al. (2019) proposed an actuated signal control strategy for accommodating the traffic flow fluctuation. The results show that the proposed design concept outperforms the fixed-time control strategy in increasing capacity and reducing delay for left-turning vehicles.

Several studies have been conducted to evaluate the safety of the CLL design. Zhao et al. (2015) analyzed the driver’s reaction to such design under various traffic signs and markings using a high-fidelity driving simulator. Zhao and Liu (2017) evaluated the safety of the CLL design based on empirical data. The results indicated that safety risks can be relieved by providing more guiding information.

However, as an unconventional intersection, the operation performance of CLL design depends on the adaptability of driver behaviors (Liu et al., 2021). Autonomous driving is a promising technology to deal with the heterogeneity of drivers. A large number of studies considered the optimal trajectory guidance for autonomous vehicles when they traverse crossing the intersection. Some studies focused on optimizing vehicular trajectories under the consideration of the signal phase and timing information (Jiang et al., 2017; Kamalanathsharma et al., 2015), while some others aimed to establish autonomous intersection control algorithms to enable autonomous vehicles to pass the intersection cooperatively without traffic signals (Ahmane et al., 2013; Lee and Park, 2012) or to optimize vehicle trajectories and signal control in a unified framework (Feng et al., 2018; Xu et al., 2019; Yu et al., 2018; Liu et al., 2022). Modeling the two-dimensional movement of vehicles at intersections is also been conducted in some studies (Ma et al., 2017; Yang et al., 2019; Bichiou and Rakha, 2018; Zhao et al., 2020). For the CLL design, Wu et al. (2021) proposed a “cooperative vehicle sorting” strategy that seeks to optimally sort connected and automated vehicles in a multilane platoon to reach the desired configuration.

In this study, we focus on the trajectory planning of automated platoons for left-turn maneuvers under the given geometric design and signal control. In literature, trajectory planning and control at intersections can be categorized into three groups:

1. trajectory control at signal-free intersections,

2. cooperative trajectory planning under the consideration of exogenous signal control; and

3. joint optimization of the trajectory and signal control.

Signal-free intersections organize autonomous vehicles to traverse intersections without collision (Ahmane et al., 2013; Lee and Park, 2012; Zohdy and Rakha, 2016; Yu et al., 2019; Liu and Fan, 2021). Control algorithms arrange the sequence of vehicles passing the intersection to ensure operational safety without the necessity of having vehicle trajectory information before arriving. The optimality from the platoon’s point of view may not be guaranteed. The cooperative CAV trajectory planning algorithms optimize vehicle accelerations at an isolated intersection or along a corridor, assuming that signal timings are known to the optimization models as exogenous inputs. As to the trajectory planning systems at isolated intersections, the objective functions simply consider comfort and/or fuel consumption (Jiang et al., 2017; Zhao et al., 2018; Li and Ban, 2019; Typaldos et al., 2020). Providing a fixed signal cycle length, the red phases at isolated intersections are normally represented as constraining the terminal conditions of vehicle position, speed and acceleration using terminal costs and/or equality constraints. These terminal conditions are normally estimated as the position of stop-line, the maximal speed and zero acceleration, respectively. On the other hand, the trajectory planning systems along a corridor are usually designed for an individual vehicle (Asadi and Vahidi, 2011; Kamal et al., 2013; He et al., 2015; Wan et al., 2016; HomChaudhuri et al., 2017), while a few control approaches consider the overall vehicle platooning (Liu et al., 2020; Liu et al., 2019a).

Although plenty of studies on trajectory planning of autonomous vehicles at intersections have been reported, they are not suitable for the control of left-turn vehicles at contraflow left-turn intersections because of the particular challenges brought by the intersection design, such as the lateral lane change control in limited space (central divider opening area) and limited time (presignal green light period). This study proposed a centralized left-turn vehicle trajectory control model for CLL intersections based on optimal control. The contributions are:

• both the longitudinal acceleration of left-turn vehicles and their lateral lane changes are optimized; and

• the proposed model is cast into a mixed-integer linear programming problem that can be solved efficiently.

We demonstrate the plausibility of the trajectory control approach and the efficiency of the approach through numerical experiments and sensitivity analysis.

The remainder of the paper is structured as follows: Section 2 describes the optimal control model in detail. The proposed model is linearized in Section 3, and then a global optimal solution can be obtained. A case study is used to analyze the plausibility of the planned platoon trajectories in Section 4. The effectiveness of the model is tested by sensitivity analysis in Section 5. Conclusions are presented in Section 6.

2. Optimal control model

As illustrated in Figure 1, we will adopt centralized control to optimize the trajectories of the left-turn platoon at the CLL intersections. The exogenous input variables to the centralized controller include the intersection geometry design and signal timing. The decision variables include the lateral lane change decision and the longitudinal acceleration.

2.1 Assumptions

In view of the characteristics of intelligent connected vehicles, the following assumptions are made:

A1.

At the intersection, all vehicles are fully autonomous driving vehicles, and the state information of vehicles in the control region, e.g., from x_s upstream of the intersection to the exits, can be collected by the vehicle onboard sensors and exchanged in real-time to the centralized controller.

A2.

The centralized controller also has knowledge regarding intersection geometric parameters and signal phase and timing information.

A3.

The centralized controller plan vehicle trajectories in a future horizon to minimize some cost function. The planned trajectories will be sent to the vehicles and tracked by the vehicle actuators.

2.2 Vehicle motion dynamics

Let X_i denote the state of vehicle i. The state contains the longitudinal position (m), located lane and longitudinal velocity (m/s), as shown in equation (1):

(1) Xi(t)=[xi(t), li(t),vi(t) ]′

where x_i(t) is the longitudinal position of vehicle i, in m; l_i(t) is the lane number that vehicle i is located, 0-on the regular left-turn lane, 1-on the CLL; v_i(t) is the longitudinal velocity of vehicle i, in m/s.

In this study, it is assumed that the initial state of a vehicle X_i0 is given, which is described as equation (2):

(2) Xi0=[xi0, li0,vi0]′

where X_i0 is the state of vehicle i at the initial time; x_i0 is the longitudinal position of vehicle i at the initial time, in m; l_i0 is the lane number in which vehicle i is located at the initial time; v_i0 is the longitudinal velocity at the initial time, in m/s.

The system dynamics can be expressed by equation (3). The control of lateral lane change and longitudinal acceleration constitute the control U_i(t), as shown in equation (4):

(3) Xi(t+1)=[xi(t+1)li(t+1)vi(t+1)]=[xi(t)+vi(t)Δt+12ai(t)Δt2li(t)+ci(t)vi(t)+ai(t)Δt]

where Δt is the interval of a control step, in s; c_i(t) is the control of lateral lane change of vehicle i, 0 -no lane change, 1-lane change; a_i(t) is the longitudinal acceleration of vehicle i, in m/s²:

(4) Ui(t)=[ci(t), ai(t) ]′

where U_i(t) is the control vector of vehicle i at time t.

2.3 Cost function

The vehicle platoon plans its trajectory in a fixed horizon T by controlling its longitudinal acceleration and lane change strategies. Considering the operation efficiency and smoothness of the vehicle platoon trajectories passing through the intersection, the cost of the optimal control model includes three parts: the number of vehicles not crossing the intersection, the position during operation and the absolute acceleration during operation, as shown in equation (5). The control goal is to minimize the cost, as shown in equation (6). As the main goal of the control is to maximize the operational efficiency of the left-turn traffic flow at the intersection, the three costs are associated with the weights set as α₁ ≫ α₂ ≫ α₃:

(5) C(X,U)=α1∑i=1n(1−zi(T))−α2∑i=1N∑t=0Txi(t)+α3∑i=1N∑t=0T|ai(t)|

where C(X,U) is the cost of vehicles in the control system; N is the total number of vehicles in the control system, in veh; T is the control horizon, in s; z_i(t) is a binary variable indicating whether vehicle i passes the intersection at discrete time t, 1 indicates passed, 0 indicates unpassed; α₁, α₂ and α₃ are the relative weights of the three parts of the cost:

(6) C∗=min⁡UC(X,U)=min⁡Uα1∑i=1n(1−zi(T))−α2∑i=1N∑t=0Txi(t)+α3∑i=1N∑t=0T|ai(t)|

where C* is the cost of vehicles in the control system when the optimal control is applied.

Whether the vehicle has passed the intersection at any given time can be formulated by an auxiliary variable z_i(t), as shown in equation (7). When vehicle i passes the intersection at time t (x_i (t) ≥ x_s), z_i (t) can only be 1. When vehicle i did not pass the interaction at time t (x_i (t) ≤ x_s), z_i (t) can only be 0:

(7) M(zi(t)−1)≤xi(t)−xs≤Mzi(t), ∀i∈[1,N];t∈[0,T]

where x_s is the position of the stop-line, in m; M is a large positive constant number.

2.4 Constraints

The state and the control variable values of vehicles should be restricted according to the traffic rule and the characteristics of the vehicle.

• Constraints of control variable a_i

The acceleration of vehicles at each time step should be restricted within a reasonable minimum and maximum acceleration limitation, as shown in equation (8):

(8) amin≤ai(t)≤amax, ∀i∈[1,N];t∈[0,T]

where a_min and a_max are the minimum and maximum acceleration limitations, respectively, in m/s².

• Constraints of control variable c_i

The control variable of lateral lane change of a vehicle can be 0 and 1, which indicates the vehicle keeping the original approach left-turn lane and changing to the CLL, respectively. The lane change is assumed to be accomplished instantaneously. The lane change positions of vehicles are restricted at the median opening. Equations (9) and (10) restrict the start and end permitted lane change position, respectively. When vehicle i is in the range of the median opening [x_o1 ≤ x_i (t) ≤ x_o2], the right-side values of equations (9) and (10) are positive. Then, c_i(t) can be 0 (no lane change) or 1 (lane change). When vehicle i is not in the range of the median opening [x_i (t) < x_o1 or x_i(t) > x_o2], one of the right-side values of equations (9) and (10) is negative. Then, c_i(t) can only be 0 (no lane change):

(9) M(ci(t)−1)≤xi(t)−xo1, ∀i∈[1,N];t∈[gsp,gep]

(10) M(ci(t)−1)≤xo2−xi(t), ∀i∈[1,N];t∈[gsp,gep]

where x_o1 and x_o2 are the start and end positions of the median opening, respectively, in m; gsp is the start of green at the presignal, in s; and gspis the end of green at the presignal, in s.

The lane change should be forbidden during the red time of the presignal, as shown in equation (11):

(11) ci(t)=0, ∀i∈[1,N];t∈[0,gsp]∪[gep,T]

Moreover, as the allowed lane change position is limited, we restrict that the number of lane-changes a vehicle executes to not more than one during the control horizon T, which is shown in equation (12):

(12) ∑t=0Tci(t)≤1, ∀i∈[1,N]

• Constraints of state variable v_i

The velocity of vehicles at each time step should be restricted between stop (reversing is prohibited) and the road speed limit, as shown in equation (13):

(13) 0≤vi(t)≤vmax, ∀i∈[1,N];t∈[0,T]

where v_max is the road speed limit, in m/s.

• Constraints of state variable l_i

Two traffic lanes are considered in the control system. One is the regular left-turn lane, which is numbered 0. The other is CLL, which is numbered 1. To avoid the conflict on the CLL, the left-turn vehicles cannot be on the CLL from the end of green of the left-turn phase at the main signal to the start of green at the presignal when vehicles have not passed the stop-line, as shown in equation (14). In equation (14), the signal timing parameters, gem, gsp, and C, are external input parameters. The range of t in this constraint can be determined before the optimization process:

(14) li(t)≤zi(t), ∀i∈[1,N];t∈[gem+(q−1)C,gsp+qC];q∈Z

where gem is the end of green of the left-turn phase at the main signal, in s; C is the cycle length of the intersection, in s.

• Constraints of state variable x_i

The distance between any two vehicles on the same traffic lane should satisfy the safety distance requirement, as shown in equation (15). When vehicles i and j on the same traffic lane, the right-side value of equation (15) is 0. Then, the constraint is active. Otherwise, the right-side value of equation (15) is –M, and the constraint remains inactive:

(15) xi(t)−xj(t)−Dj(t)≥−M|li(t)−lj(t)|, ∀i,j∈[1,N];i<j;t∈[0,T]

where D_j (t) is the safety distance requirement of the following vehicle j at time step t, in m, which can be determined by equation (16):

(16) Dj(t)=dv+ds+vj(t)h, ∀j∈[1,N];t∈[0,T]

where d_v is the vehicle length, in m; d_s is the required safety distance for stopped vehicles, in m; and h is the safety time headway, in s.

Moreover, to avoid the red-light violation, vehicles cannot pass the stop-line during the red time, as shown in equation (17):

(17) (xi(t1)−xs)(xi(t2)−xs)≥0, ∀i∈[1,N];t1=gem+(r−1)C;t2=gbm+rC;r∈Z

where gbm is the start of green of the left-turn phase at the main signal, in s.

3. Solution method

The optimal control problem of the automated left-turn platoon is established as a mixed-integer nonlinear programming problem with the objective function of equation (6) and constraints of equations (1)–(4) and (7)–(17). Equations (6), (15) and (17) are nonlinear, which prevents the application of efficient solvers. Therefore, we apply the following steps to linearize them. Note that the linearization steps in our problem do not change the optimal solution.

• Linearization of the objective function equation (6)

As the objective is to minimize the costs, we can linearize equation (6) by replacing |a_i (t)| by an auxiliary variable A_i (t) and adding two constraints, as shown in equations (18)–(20):

(18) C(X,U)=α1∑i=1n(1−zi(T))+α2∑i=1N∑t=0TAi(t)−α3∑i=1N∑t=0Txi(t)

where A_i (t) is an auxiliary variable, in m/s²:

(19) ai(t)≤Ai(t), ∀i∈[1,N];t∈[0,T]

(20) −ai(t)≤Ai(t), ∀i∈[1,N];t∈[0,T]

• Linearization of the constraint equation (15):

For the constraint equation (15), as |l_i (t)–l_j (t)| consists of two binary variables, we can linearize it by replacing it by an auxiliary binary variable δ_ij (t) and adding four constraints, as shown in equations (21)–(25):

(21) xi(t)−xj(t)−Dj(t)≥−Mδij(t), ∀i,j∈[1,N];i<j;t∈[0,T]

where an auxiliary binary variable:

(22) δij(t)≥li(t)−lj(t), ∀i,j∈[1,N];i<j;t∈[0,T]

(23) δij(t)≥lj(t)−li(t), ∀i,j∈[1,N];i<j;t∈[0,T]

(24) δij(t)≤li(t)+lj(t), ∀i,j∈[1,N];i<j;t∈[0,T]

(25) 1−δij(t)≥li(t)lj(t), ∀i,j∈[1,N];i<j;t∈[0,T]

In equation (25), the item l_i (t) l_j (t) should further be linearized by replacing it by an auxiliary binary variable y_ij (t) and adding three constraints, as shown in equations (26)–(29):

(26) 1−δij(t)≥γij(t), ∀i,j∈[1,N];i<j;t∈[0,T]

where y_ij (t) an auxiliary binary variable:

(27) li(t)+lj(t)−γij(t)≤1, ∀i,j∈[1,N];i<j;t∈[0,T]

(28) γij(t)≤li(t), ∀i,j∈[1,N];i<j;t∈[0,T]

(29) γij(t)≤lj(t), ∀i,j∈[1,N];i<j;t∈[0,T]

• Linearization of the constraint equation (17)

The multiplication of x_i (t₁) and x_i (t₂) makes equation (17) nonlinear. We can linearize it by using the auxiliary variable z_i (t) from equation (7). The states of each vehicle passing the stop-line or not at the beginning and end of the red time should be the same. Therefore, equation (17) can be equivalent to equation (30). As the signal timing parameters, gem, gbm and C, are external input parameters, the range of t in this constraint can be determined before the optimization process:

(30) zi(t2)−zi(t1)=0, ∀i∈[1,N];t1=gem+(r−1)C;t2=gbm+rC;r∈Z

Then, the problem is cast into a mixed-integer linear programming problem with objective function of equation (18) and constraints of equations (1)–(4), (7)–(14), (16), (19)–(24) and (26)–(30). It can be solved by the branch-and-bound technique in commercial numerical computation software, such as MATLAB.

4. Case Study

The proposed trajectory control method optimizes the lateral lane change and the longitudinal acceleration at each time step. A case study is used to analyze the plausibility of the planned platoon trajectories.

4.1 Input parameters

A CLL approach with one regular left turn lane and one CLL is used for the case study. The left-turn platoon consists of eight vehicles with an initial position of 20–160 m away from the stop line, and each vehicle is located 20 m apart. The remaining parameters are shown in Table 1.

4.2 Trajectory planning results

The optimized trajectories are shown in Figure 2. The expected behavior based on the objective function is:

• vehicles pass the stop line as much as possible;

• each vehicle travels to the furthest distance; and

• no strong fluctuations exist in the acceleration rate.

As can be seen from Figure 2(a), all eight vehicles pass the stop line during the green light of the main signal, and travel at the maximum speed after passing the stop line. The first four vehicles use the regular left-turn lane, while the remaining four vehicles use the CLL. The usage of the regular and contraflow left-turn lanes is balanced, which ensures the operational efficiency of the approach. We find that there is fluctuation in acceleration/speed profiles of Vehicles 5–8 at around 10–15 s. It is because the lane changing process is not considered in the study. The lane change is assumed to be accomplished instantaneously. Therefore, the acceleration/speed may fluctuate before and after the lane change, e.g., Vehicle 6 follows Vehicle 5 before the lane change. When Vehicle 5 makes a lane change, Vehicle 6 follows Vehicle 4 and accelerates because Vehicle 4 is far away. However, after Vehicle 6 makes a lane change, it should follow Vehicle 5 again. Vehicle 6 has to decelerate to meet the safety time headway constraint. This results in fluctuation in the acceleration/speed. All of the results are reasonable and meet the constraint requirements of the controller formulation.

In detail, the possible trajectory can be divided into three categories: stop without lane change; stop and lane change; and lane change without stopping.

• Category 1: stop without lane change

According to the overall planning of the platoon, Vehicles 1–4 adopt the strategy of stopping at the conventional left-turn lane without changing lanes and waiting for release, shown as the solid lines in Figure 2(a). When Vehicle 1 arrives the intersection, it reaches the red light and stops for release. Because of the red light of the presignal, Vehicles 1–4 cannot take access to the integrated function zones. At this time, Vehicles 2–4 adopt the minimum acceleration change, so they queue in the conventional lane behind Vehicle 1, waiting for the release of the main signal.

• Category 2: lane change with stop

Vehicle 5 chooses to enter the opening of the central separation zone into the CLL to wait for release, shown as the dotted lines in Figure 2(a). The presignal is green when Vehicle 5 passes the median opening with a slight deceleration, then it can preferentially enter the CLL and wait for release, as shown in Figure 2(b). Through the stop and lane change strategy, Vehicle 5 can quickly pass the intersection. More importantly, considering the overall planning of the whole group of vehicles through the intersection, it is a reasonable choice because the number of left-turn vehicles on the regular and contraflow left-turn lanes is balanced, which can maximize the efficiency of the whole platoon passed at the intersection.

• Category 3: lane change without stop

Vehicles 6–8 adopt the control strategy of lane change without stopping, shown as the dashed trajectories in Figure 2(a). They reach the control area when the presignal changes to green. Moreover, the queue length of the conventional lane is longer than that of the CLL. Thus, these vehicles choose to use the left-turn lane in the mixed-usage area. As the signal light turns to green and the queue length at the mixed-usage area is short, they decrease the speed according to the safety headway and pass the intersection without stopping.

In summary, the control strategies in the test case are reasonable in the numerical simulation, and the whole platoon can make reasonable decisions. Compared with the platoon vehicles before control, the release order of some vehicles is earlier, and the model control strategy significantly improves the traffic efficiency of this case. Therefore, the trajectory control model of the platoon on the CLL is reasonable, and it improves the traffic efficiency of the whole intersection.

5. Sensitivity analysis

This section explores the benefits of autonomous platoons under different geometric and signal control conditions through sensitivity analysis. The impact of three key parameters on the operation of the intersection is analyzed in this section, including the effects of safe headway, green time of the main signal and the length of the CLL on the number of vehicles passed through the intersection. To evaluate the effectiveness of the proposed trajectory control method, a platoon of 20 vehicles is simulated and the other parameters are maintained the same as in the case above Table 1. We also implement the full velocity difference (FVD) model to represent human driving behavior as the baseline group for comparison. The number of passing vehicles during the green time of the main signal is selected as the evaluation index.

5.1 Effect of safe headway

The distance required between two vehicles in this section is determined by the safety headway, the length of the vehicles and velocity. The safety headway is selected as a parameter to explore the impact of autonomous platoons with different control capabilities. The better the acceleration and braking capabilities of autonomous vehicles, the smaller the safe headway distance and the higher the chance of crossing the intersection. In our analysis, the safety headway varies from 0.5 s to 3 s, and the initial distance is 40 m to meet the initial safety distance requirement constraint position. As shown in Figure 3, the number of vehicles passing the intersection is negatively correlated with safety headway. When the safety headway is less than 2 s, 12 vehicles out of the 20-vehicle platoon can pass the intersection. When the safety headway is 2.5 s, the number of passing vehicles is 10, smaller than that of the FVD model. In conclusion, the model works well on autonomous vehicles when the safety headway is less than 2 s. When the safety headway is set in this range of fewer than 2 s, the number of vehicles through the intersection remains constant. The results show that the model is less sensitive to the safety headway, and the number of vehicles passing the intersection never changes within the normal value range.

5.2 Effect of green time of the main signal

The green time of the main signal varies from 10 s to 50 s, with 10 s intervals in our sensitivity analysis. As shown in Figure 4, the number of vehicles passing the intersection increases with the increase of the main signal green time when the green time of the main signal is more than 30 s. After that, all 20 vehicles in the platoon can pass the intersection for both the proposed model and the FVD model. The FVD model has the worse controlled effectiveness when the main signal is under 30 s. In general, the main signal green time is shorter than 20 s, and within this reasonable green time duration, the traffic efficiency can be optimized by the proposed model. Moreover, the proposed model is less sensitive to the green time length of the main signal than the FVD model. When the main signal green time is 10 s, only five cars cannot pass the intersection and need to queue up twice with the proposed model, while ten cars have to wait for the next cycle with the FVD model.

5.3 Effect of length of the contraflow left-turn lane

The length of the CLL varies from 10 m to 90 m with 20 m intervals in the sensitivity analysis. As shown in Figure 5, with our proposed model, when the length of the CLL is more than 50 m, all 20 vehicles of the platoon can pass. When the length of the CLL is 30 m, 19 vehicles can pass the intersection. When the length of the CLL is only 10 m, 16 vehicles can pass the intersection. For the FVD model, the number of vehicles passing the intersection increased more rapidly than that of the proposed method when the length of the CLL is less than 30 m. Therefore, it can be concluded that the proposed model is not very sensitive to the length of the CLL, and the FVD model is more sensitive than the proposed model. The results of the study indicate that the proposed model can plan the trajectory reasonably and changing the length of the CLL has little effect on the number of vehicles passed.

More novel intersection designs will be proposed under the automated driving environment. Sun et al. (2018) put forward an innovative intersection operation scheme named MCross, which is able to serve two-direction traffic from one road in one signal phase and maximizes the intersection capacity by utilizing all lanes from the road at any time. It remains an interesting topic to compare them under various conditions. However, in this study, we focus on the trajectory planning of automated platoons for left-turn maneuvers under CLL design. The signal control with CLL can accommodate vulnerable road users and mixed CAV-HV flows, while many signal-free concepts and the MCross concept fall short when considering vulnerable road users and mixed flows.

6. Conclusions

The optimal platoon trajectory control model established in this paper gives priority to the driving trajectory strategy of the whole platoon of vehicles at the CLL intersection. The model is verified by simulation and the FVD model are carried out for comparison. The following conclusions can be drawn:

• The vehicles of the platoon follow the planned trajectory, which not only improves the use of available lanes, but also improves the utilization rate of functional areas and the traffic efficiency of intersections. Thus, it is beneficial to plan the trajectory for the vehicle platoon.

• In the case study we set, the operation strategy of the platoon vehicle mainly includes three classes: stop without lane change, lane change with stop and lane change without stop. Considering the actual situation of the vehicle operation, if the main signal is still green, the vehicle can leave the intersection directly, and the vehicle can adopt the no lane change-no stop strategy.

• The proposed model performs robustly under various safety time headways, lengths of the CLL and green times of the main signal. It indicates that the proposed model has extensive applicability.

This paper adopts a centralized optimal control strategy to ensure that the adopted control strategy is globally optimal. Global information is available for each vehicle. However, the initial position of the vehicle's departure is fixed in this model. In future studies, we can extend the model to under a model predictive control framework (Wang et al., 2014a; Wang et al., 2014b) to optimize the dynamic traffic flow.

Figures

Figure 1

Layout of CLL intersections

Figure 2

Trajectory planning results

Figure 3

Effect of safe headway

Figure 4

Effect of green time of the main signal

Figure 5

Effect of length of the contraflow left-turn lane

Table 1

Input parameters of the case study

Parameters	Values	Parameters	Values
Start of green of the left-turn phase at the main signal, gbm	20 s	Maximum acceleration, amax	5 m/s2
End of green of the left-turn phase at the main signal, gem	35 s	Minimum acceleration, amin	−5 m/s2
Start of green at the presignal, gsp	10 s	Vehicle length, dv	5 m
End of green at the presignal, gep	30 s	Required safety distance, ds	1 m
Cycle length, C	100 s	Initial velocity, vi0	10 m/s
Position of the stop-line, xs	300 m	Safety headway, h	1 s
Start position of the median opening, xo1	230 m	Control horizon, T	40 s
End position of the median opening, xo2	240 m	Interval of a control step, Δt	1 s
Road speed limit, vmax	60/3.6 m/s	Large positive constant number, M	10,000