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Connected and automated vehicles (CAVs) trajectories not only provide more real-time information by vehicles to infrastructure but also can be controlled and optimized, to further save travel time and gasoline consumption. This paper proposes a two-level model for traffic signal timing and trajectories planning of multiple connected automated vehicles considering the random arrival of vehicles. The proposed method contains two levels, i.e., CAVs’ arrival time and traffic signals optimization, and multiple CAVs trajectories planning. The former optimizes CAVs’ arrival time and traffic signals in a random environment, to minimize the average vehicle’s delay. The latter designs multiple CAVs trajectories considering average gasoline consumption. The dynamic programming (DP) and the General Pseudospectral Optimal Control Software (GPOPS) are applied to solve the two-level optimization problem. Numerical simulation is conducted to compare the proposed method with a fixed-time traffic signal. Results show that the proposed method reduces both average vehicle’s delay and gasoline consumption under different traffic demand significantly. The average reduction of vehicle’s delay and gasoline consumption are 26.91% and 10.38%, respectively, for a two-phase signalized intersection. In addition, sensitivity analysis indicates that the minimum green time and free-flow speed have a noticeable effect on the average vehicle’s delay and gasoline consumption.

Traffic congestion has become a common traffic phenomenon in many cities [

As one of the effective methods to alleviate urban traffic congestion [

Many works have been conducted to search the optimal traffic signal and vehicle trajectories in the CAVs environment [

However, there are several limitations to current integrated optimization methods. First, Feng et al. [

The contribution of this paper consists of extending the optimal framework in Feng et al. [

The remainder of the paper is organized as follows. Section

Connected and automated vehicles (CAVs) have great potential in improving traffic efficiency and reducing traffic congestion and have gained a wide application in the transportation field during the last decade [

To our knowledge, the first approach focuses on vehicle trajectory planning [

The second method optimizes signal timing plans by CAVs data [

Therefore, to address this gap, the third approach simultaneously optimizes CAVs trajectories and traffic signals. Xu et al. [_{2} emission. Feng et al. [

This study proposes a two-level model for traffic signal timing and trajectories planning of multiple CAVs considering the random arrival of vehicles. The integrated optimization problem is modeled as a two-level model. Firstly, the traffic signal and arrival time for CAVs are optimized by the signal timing model to minimize the average vehicle’s delay. Secondly, considering average gasoline consumption, an optimal control method is proposed to optimize trajectories for all CAVs. Finally, the proposed method is tested in a simulation experiment, and numerical studies and sensitivity analysis are carried out based on a simple two-phase intersection.

The following necessary assumptions are made to facilitate modeling and analysis.

The interarrival time of all CAVs follows the shifted negative exponential distribution, which is verified at an isolated intersection [

All CAVs can share information (such as location, speed, and arrival time) through V2V; hence, their arrival time can be predicted more accurately [

All CAVs arrive at the boundary of the control zone and through the downstream intersection with the desired speed, which can refer to Ghiasi et al. [

All CAVs cannot change lanes in the control zone; that is, only the longitudinal movement is considered [

In this study, no left-turn and right-turn are considered; only through traffic flow it is modeled, which is shown in Figure

A simple intersection with four arms.

The vehicle trajectories planning of each arm

Notation of major symbols used in this paper.

Symbol | Description |
---|---|

Gasoline consumption | |

| The parameter of gasoline consumption rate |

| The weight of the vehicle |

| The parameter relevant to the energy efficiency of the engine |

| The parameter associated with positive acceleration |

| The vehicle’s speed |

| The vehicle’s acceleration |

| The power (kW) required to drive the vehicle |

Traffic signals | |

| Set of arms at the intersection |

| A signal timing plan |

| The effective green time for arm |

| The effective red time for arm |

| The lost time of a traffic signal cycle |

| The length of the control zone at arm |

| The desired speed at each arm |

| The vehicle arrival rate at arm |

| The saturation flow rate at arm |

| The minimum green time duration for arm |

| The maximum green time duration for arm |

Vehicle trajectory | |

| The set of CAVs arriving at the border of control zone at arm |

| The set of CAVs trajectories functions at arm |

| The location of the |

| The speed of the |

| The acceleration of the |

| The time of |

| The expected arrival time at the stop line of the |

| The optimal arrival time at the stop line of the |

| The delay of control and communication |

| The safety spacing between two consecutive CAVs |

| The minimum acceleration of CAVs |

| The maximum acceleration of CAVs |

As shown in Figure

The proposed method consists of two levels, i.e., vehicle’s arrival time and traffic signal timing, and vehicle trajectories planning. The former optimizes traffic signals and vehicles’ arrival time for CAVs to minimize the average vehicle’s delay. The latter optimizes trajectories for all CAVs considering average gasoline consumption based on the optimal traffic signal timing plan. To better understand the proposed model, the vehicle’s trajectories are optimized by giving the optimal traffic signal plan of a two-phase intersection:

The time of CAVs (

The analysis indicates that the optimal arrival time at the stop line is determined by the expected arrival times, traffic signals, and saturation flow rate. Taking the optimal arrival time of the

The first CAV of a signal cycle at each arm

If the expected arrival time of the first CAV is during the red signal period, to minimize the vehicle’s delay, the first CAV’s optimal arrival time is equal to the start time of the green signal in the next signal cycle.

If the expected arrival time of the first CAV is during the green signal duration, to minimize the vehicle’s delay, the first CAV’s optimal arrival time is equal to the expected arrival time.

The other CAVs of a signal cycle at each arm

If the estimated arrival time is shorter than the expected arrival time, the optimal arrival time is equal to the expected arrival time at the stop line.

If the estimated arrival time is not shorter than the expected arrival time, the optimal arrival time is equal to the estimated arrival time.

Therefore, the average vehicle’s delay for this intersection is formulated as

The average gasoline consumption function for arm

Therefore, the average gasoline consumption for this intersection is formulated as

where

In this study, a dynamic programming (DP) algorithm and the GPOPS are adopted to solve the traffic signal timing problem and multiple vehicle trajectories planning problem, respectively.

Many DP-based traffic signal timing methods have been developed [

When the state variable

After determining

Step 1: Set initial stage

Step 2: For

Record

Step 3: If (

Else if (

Else

The first recursion starts with stage 1 and the cumulative value function as 0. For each stage, the DP searches the optimal solution

After optimal value function is determined, the optimal decision

Finding

If (

As an optimal control problem, the vehicle trajectories planning can be handled numerically by GPOPS [

In summary, the two-level optimization algorithm is as follows. (i.e., Algorithm

Initialize:

(1)Set the total simulation time as

(2)Simulate the arrival times of CAVs at arm

Iterate:

(3)

(4) Get the arrival times (

(5) Calculate

(6) Optimize the traffic signal timing plan by DP algorithm.

(7)

(8) Obtain signal time plan

(9)

(10) Obtain

(11) Calculate

(12) Optimize the

(13) Save the vehicle trajectories

(14) Calculate

(15)

(16) Calculate

(17)

(18) Save vehicle trajectories, signal timing plan, gasoline consumption, and average delay at the current time planning horizon

(19)

(20)

Output:

Output vehicle trajectories, signal timing plan, gasoline consumption, and average delay at the total time planning horizon

The simulation duration of every scenario with a different traffic volume is 900 seconds. Every scenario is repeated five times with different random seeds. Besides, vehicle arrival conforms to the Poisson distribution [

In signal optimization, a four-arm and two phases of a cycle are selected. The time planning horizon is

In the vehicle trajectories planning, the delay of control and communication ^{2} and ^{2}, respectively.

The two-level integrated optimization model, denoted as “IO”, is compared with Signal-fixed. Three volume levels, namely, 600, 800, and 1200 vph, are created in this study [

The average vehicle’s delay and gasoline consumption in different scenarios.

Delay (s/veh) | Gasoline consumption (mL/veh) | ||||||
---|---|---|---|---|---|---|---|

IO | Signal-fixed | Decrease | IO | Signal-fixed | Decrease | ||

1200 | 1200 | 5.8598 | 8.0169 | −26.91% | 25.0633 | 27.9663 | −10.38% |

1200 | 800 | 5.4404 | 6.4440 | −15.57% | 24.3265 | 25.6891 | −5.30% |

800 | 800 | 5.0441 | 6.6516 | −24.17% | 23.2960 | 25.4607 | −8.50% |

800 | 600 | 4.9859 | 6.2673 | −20.45% | 23.2806 | 25.0732 | −7.15% |

Average | 5.3326 | 6.8450 | −21.77% | 23.9916 | 26.0473 | −7.83% |

Trajectories of CAVs in arm 1 and 2 as an example. (a) 1200/1200 vph. (b) 1200/800 vph. (c) 800/800 vph. (d) 800/600 vph.

As shown in Table

Figure

In this study, the minimum green time (

Minimum green time is to ensure the safety of drivers and pedestrians. A minimum green time that is too long may result in increased delay; one that is too short may violate pedestrian needs. Therefore, different geometric shapes of intersections can set different minimum green time. To avoid the influence of other parameters, scenario one (1200/1200 vph) is selected as a sensitivity analysis of the minimum green time. In the sensitivity analysis,

Sensitivity analysis on minimum green time.

As shown in Figure

The free-flow speeds influence CAVs arrival time, which is an essential parameter for traffic signal optimization and trajectories planning of this study. Scenario no.1 (1200/1200 vph) is selected as a sensitivity analysis of the free-flow speeds. In the sensitivity analysis,

Sensitivity analysis on free-flow speed.

The sensitivity analysis (Figure

This study developed a two-level model for traffic signal timing and trajectories planning of multiple connected automated vehicles considering the random arrival of vehicles. Based on the numerical experiments, the following conclusions can be drawn:

Compared with the Signal-fixed, the reduced average vehicle’s delays with four scenarios are 26.91%, 15.57%, 24.17%, and 21.77%, and the reduced gasoline consumption with four scenarios are 10.38%, 5.30%, 8.50%, and 7.15%.

The proposed two-level model could reduce both vehicle’s delay and gasoline consumption by 26.91% and 10.38%, compared with Signal-fixed control in these studied scenarios, respectively.

Sensitivity analysis suggests that the minimum green time and free speed have a significant impact on the two-level model’s performance.

A shorter minimum green time results in a significantly less average vehicle’s delay and gasoline consumption. The optimal free-flow speed is 13 m/s in the study scenario.

In the current work, this work applied the proposed model to a single intersection, similar to vehicle merging behavior [

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

This work are supported by the National Natural Science Foundation of China (no. 52002339), the Science and Technology Program of Sichuan Province (2021YJ0535), and the Fundamental Research Funds for the Central Universities (2682021CX058).