Integrated Optimization of Traffic Signals and Vehicle Trajectories at Intersection With the Consideration of Safety During Signal Change

This paper develops an integrated optimization of traffic signals and vehicle trajectories. The signal is optimized to improve the intersection efficiency and the calculation of intergreen interval (IGI) serves as constraints to guarantee vehicle safety during signal changing. Then the vehicle trajectory in the approach lane and inside intersection is optimized to increase fuel efficiency. The proposed method is evaluated by microscopic simulation, comparing with the actuated signal control (ASC) method and an ad hoc cooperation method between traffic signals and vehicles. Results indicate the proposed control algorithm is effective to prevent conflicts during signal changing periods. Operation Efficiency and fuel efficiency are improved. The benefit is 24.9% on vehicle delay, and is 5.5% on fuel efficiency. The proposed system can potentially be used in real-time.


I. INTRODUCTION
Accidents at intersections have been a serious problem. Researches show 22% of total crashes happen at intersections in the United States [1] and around 30% of urban traffic accidents took place at or near signalized intersections in China [2]. Signalized intersections play a significant role in the urban roadway network [3], [4]. The present study avoids vehicle collisions at signalized intersections through precise signal control, most notably by optimizing intergreen interval (IGI) [5]. IGI is generally defined as the interval between the end of the green time of the last traffic stream and the beginning of the green time of the next [6]. Inadequate IGI leads to potential conflicts for the vehicles from the next phase may drive into the conflicting point before the vehicles from the previous phase leaves. Overlong IGI is a waste of time [7]. Therefore, the design of IGI should be optimized for efficiency and ensuring safety.
The associate editor coordinating the review of this manuscript and approving it for publication was Rashid Mehmood .
In a conventional traffic environment with human-driven vehicles, IGI equals the sum of the yellow signal and all-red signal (red clearance interval). There are two categories of design methods for IGI based on different critical conditions. The first category is based on the maximum safety condition, which is widely applied in America [8] and Japan [7]. In the first category, vehicles from the next phase are allowed to pass the stop line only after the last vehicle from the last phase passes through the whole intersection [9]. The second category is based on the maximum efficiency condition, which is applied in German [10]. In this category, the first vehicle of the next phase can arrive at the conflict area after the last vehicle of the last phase passes the conflicting area. In both two categories, IGI design depends on vehicle speed inside the intersection. Researches often calibrate vehicle speed using aggregated historical data. This simplification results in safety risks.
Thanks to the technologies of the connected and autonomous vehicle (CAV) which collect detailed vehicle information, a new source of data for traffic control can be obtained for signal control [11]. This data helps the design of IGI. But current researches of signal timing under CAV technology ignore the design of IGI. They simply choose the suggested value in conventional traffic environment such as 3s [12], [13], 4s [14], [15] or 6s [16]. This simplification will lead to both safety and efficiency problems in practical application. If the last vehicle passes the intersection with low speed, the constant IGI is sometimes insufficient for its leaving. This may lead to lateral conflict between the last vehicle and vehicles from the next conflicting phase. In contrast, if vehicles can quickly pass the intersection than average, or there is not any traffic in one phase at all, the constant IGI is a waste of time.
To summarize, previous studies have not specifically focus on safety during signal change. IGI was designed without considering vehicle states and intersection characteristics in these autonomous intersection management methods. However, the design of IGI is crucial for it affects both safety and efficiency. Thus, the objective of this research is to establish an integrated optimization of traffic signals and vehicle trajectories at an intersection which ensures the safety of vehicle operation during signal changing. Besides, this method must be fast enough for potential real-time implementation.
The remainder of the paper is organized as follows: Section 2 presents the problem and the key notations. Section 3 describes the appliance of IGI calculation in the integrated optimization of traffic signals and vehicle trajectories. Section 4 conducts a simulation evaluation. Finally, conclusions are delivered in Section 5.

A. NOTATIONS
Notations used in the paper are summarized in Table. 1.

B. CONTROL STRUCTURE
In this paper, a typical signalized intersection is shown in Fig. 1. All vehicles are CAVs. Control zone refers to the area where signal timings and vehicle trajectories get optimized. It is assumed that vehicles and the Intersection Manager (IM) can communicate with each other inside the control zone. Each vehicle gets trajectory guidance from IM and follows it.
Signal control is adopted for intersection control in this paper. Signal control guarantees non-motorized vehicles and pedestrians crossing the intersection and facilitates the VOLUME 8, 2020  application of this method in the mixed-traffic environment in the future.
As shown in Fig. 2, the proposed control structure is divided into two modules. Module 1 focuses on efficiency improvement. It completes the signal optimization. Module 2 optimizes the specific trajectory of each vehicle to reduce fuel consumption. The optimal control system is activated when a vehicle drives into the control zone.
Module 1: Signal optimization: Based on the current state of vehicles, this module aims at letting all vehicles pass the intersection as early as possible. Safety during signal change is guaranteed by detailing IGI design. This module aims at optimizing the efficiency of the intersection.
Module 2: Trajectory optimization: After the signal plan is determined, IM optimizes the trajectory of each vehicle based on the required arrival time and the trajectory plan of the preceding vehicle. The trajectory plan received by each vehicle guides them to safely cross the intersection during the green light. This module aims at reducing fuel consumption by optimized trajectories.

III. METHODOLOGY
In this section, the formulation of Module 1 is entailed in Section 3.1, and Module 2 is entailed in Section 3.2. The planning horizon procedure is presented in Section 3.3.

A. SIGNAL OPTIMIZATION
This study first optimizes the signal timing plan. The signal plan determines the time of each vehicle to cross the intersection and is therefore related to the efficiency. The input to this section is the current vehicle status, including position and speed. This section generates an optimal signal plan and the arrival time of vehicles in t . The arrival time information is then sent to module 2.

1) MODEL OBJECTIVE
Previous studies of signal optimization usually adopt a phasebased signal structure. The movement of vehicles from several approach lane are combined into phases, such as the order-fixed four-phase structure [12], [17]- [19], orderunfixed four-phase structure [15], [20], and the standard North American NEMA dual-ring, eight-phase control system [21]- [23]. These signal structures limit the flexibility of phase sequence optimization. This paper adopts a flexible signal structure [24]. A phase is defined as a green timing unit of one approach lane. If vehicles from lane k 1 have no conflict with the vehicle from lane k 2 , the phase of lane k 1 can associate with the phase of lane k 2 . Therefore, a wider variety of phase associations are possible [14], [25].
The model objective is minimizing total vehicle delay. The definition of vehicle delay D w is shown in Fig.3. The signal plan S = i k , G i k , ∀k ∈ K; i = 1, . . . , N and the arrival states of vehicles A w = t w a , v w a , ∀w ) are formulated as the solution to the following optimal control problem P1.

2) SIGNAL SETTINGS
This section summarizes the basic settings for the signal optimization model. For each approach lane k, (3) shows each green time meets the constraint of maximum/ minimum green time.
For different approach lanes k 1 and k 2 , (4) and (5) show a certain interval lying between the green light of conflicting approach lanes, and there is no limit to the green light of nonconflicting approach lanes.
where M is a sufficiently big number. γ k 1 ,k 2 = 1, if the movements of the approach lane k 1 and k 2 are incompatible; Meanwhile, the optimization results cannot conflict with the previously determined signal plan.

3) PREDICTION OF VEHICLE STATE IN THE INTERSECTION
This section first calculates the earliest time of each vehicle w to arrive at the stop line t w e . Its calculation has two cases, depending on whether the vehicle can accelerate to the speed limit.
Case1. the vehicle speed can reach V max before entering the intersection.
Case 2. the vehicle speed can never reach V max before entering the intersection.
Then, take the preceding vehicle w into consideration. Each CAV needs to keep a stationary time headway with its preceding vehicle w .
where h c is the minimum time headway of a CAV to its preceding vehicle. Finally, take the signal into consideration. Each vehicle only passes the stop lane during the green light. (16) and (17)

4) GREEN INTERVAL CALCULATION
IGI aims at ensuring the safety of conflicting vehicles from adjacent green periods. The calculation method in this paper set the critical condition based on the maximum efficiency condition. For example, vehicle w 1 from approach lane k 1 get right of way before vehicle w 2 from approach lane k 2 in Fig. 4. Thus, to avoid a crash at the conflict point, their depart time should meet: where t x is the time required for a vehicle to cross another vehicle safely. When the trajectories of two vehicles intersect. The calculation of IGI needs to meet the most adverse scenario. In extreme cases, t w 1 a is the end of the last green period, and t w 2 a is the start time of the next green period.  After vehicles pass the stop line, they speed up to the speed limit of intersection and then pass the intersection as quickly as possible. Therefore, t k 1 and t k 2 are calculated as: where d k 1 ,k 2 is the distance between the stop line of approach lane k 1 and the conflict point of approach lane k 1 and k 2 along the longitudinal vehicle trajectory. The vehicle trajectory inside the intersection is determined using the method in [26].

B. TRAJECTORY OPTIMIZATION
In this section, a trajectory optimization model is formulated to reduce fuel consumption and improve smoothness.
The trajectory optimization model is responsive. This module is activated when a new vehicle enters the control zone or the signal plan updates. When a new vehicle w enters the control zone, the model first predicts the time when vehicle w passes the stop line t w a based on the current signal plan, similar to the steps in Section 3.1.3. Then the trajectory plan with the lowest fuel consumption is optimized for vehicle w. When the signal plan is updated, the trajectory plans are also updated according to the new signal plan and the arrival state through the stop line of the vehicles optimized in P1.
Previous studies optimized the trajectory of vehicles at the approach lanes whereas neglecting the trajectory inside the intersection [14], [27], [28]. Nonetheless, the trajectory inside the intersection is not only directly related to whether the vehicle will collide inside the intersection, but also affects the intersection efficiency [29]. Thus, the trajectory of CAVs in the approach lane and inside the intersection is optimized in this section.
The trajectory optimization is formulated as the solution to the following optimization problem P2. The optimized trajectory is designed based on CAV's current status α w t = x w t−1 , v w t−1 , a w t−1 , ∀t ∈ t w 0 , t w i and future arrival state.
P2 : min The objective function (22) aims to minimize the fuel consumption of each CAV. It is a simplified representation of fuel consumption commonly applied in optimal control theory [14], [30]. (23) and (24)

C. ITERATIVE SOLVING PROCESS
The flow chart in Fig. 6 summarizes the overall procedure of the proposed integrated optimization.
As shown in Fig. 7, the signal optimization model starts every other period w . In each optimization, the planning horizon length is op . The green lights started before t op + op are reserved. All vehicles can pass the intersection within the planning horizon.  The trajectory optimization model starts once a vehicle drives into the control zone. Therefore, each vehicle can get a future trajectory plan and keep following it.

IV. SIMULATION EVALUATION A. EXPERIMENTAL DESIGN
The intersection safety, efficiency, and fuel efficiency are evaluated to verify the improvement of the proposed method. The proposed integrated optimization (Proposed) is compared with two benchmark intersection control methodologies: 1) Actuated signal control (ASC). All the control structure and optimization parameters are the same as the proposed system, except for the IGI. Two widely adopted values of IGI are chosen. Benchmark 1: IGI = 3s. Benchmark 2 IGI = 4s.

2) Cooperation between traffic signals and vehicles (CTV).
A control method proposed in [31]. When comparing with CTV, the experimental design keeps consistent with the settings in [31] such as vehicle input and speed limit.
To evaluate the proposed intersection management, a typical intersection with four approach lanes shown in Fig. 1 is applied. Only left-turning and through vehicles are considered, for right-turning vehicles are not controlled by the signal. The length of the control zone is set as 300 meters. The length is select to be consistent with the reliable communication range of Dedicated Short-Range Communications (DSRC). Vehicles are generated according to Poisson distribution. Speed when vehicles enter the control zone v w 0 follows a normal distribution. v w 0 ∼ N (15, 0.7). The constraint of turning velocity is set based on the intersection design manual [32], [33]. The proposed control system is written in Python. The experiment adopts Gurobi 9.0 [34] on a computer with Intel R CoreTMi5 -1.80GHz to solve the two models P1 and P2. The settings for all parameters adopted are presented in Table. 2.

B. SIMULATION RESULTS AND DISCUSSIONS
The simulation results confirm the proposed approach is superior to the ASC method in safety, efficiency, and fuel efficiency. Besides, the proposed approach has larger throughput and fuel efficiency than the CTV method.
The proposed system can potentially be used in realtime. The computational time of P1 is 0.46 s, given a 15 s optimization interval and 0.1 s time step. Fig. 8 shows the computational time of P1 can be further reduced with shorter optimization time horizon. The computational time of P2 is 0.0020 s, given a 0.1 s time step. Fig. 9 shows the computational time of P2 can be further reduced with greater time step size.

1) SAFETY
In safety measurement based on CAV trajectories, the post encroachment time (PET) is examined using the surrogate safety assessment model (SSAM) software [35]. PET denotes the time between the departure of the encroaching vehicle from the conflict point and the arrival of the vehicle with the VOLUME 8, 2020  right-of-way at the conflict point. A lower PET indicates a higher probability of a collision [36]. The safety threshold of PET set in this paper is 1.5s considering vehicle speed inside the intersection [37]. A PET less than the threshold is identified as one critical conflict. Fig. 10 shows the PET of the proposed method meets the safety requirement. Under the control of ASC (IGI = 3 s), sometimes PET between conflict vehicles is less than the safety threshold. Collision may happen in these situations. Under the control of ASC (IGI = 4 s), the PET keeps higher than the threshold. This means high safety but also leads to a waste of time. Table. 3 shows the optimized IGI in the proposed approach varies from 0.84 to 3.14. The variation is caused by different phase sequences. For example, the maximum IGI in the proposed evaluation is 3.14. It occurs when the green light of approach lane k 1 = 2 is right after the green light of approach lane k 2 = 5. It takes a long time for the leftturn vehicle from approach lane 5 to pass the conflict point   between approach lane 5 and 2. Thus, the first vehicle from approach lane 2 needs to wait for a longer time, which means a larger IGI.

2) EFFICIENCY
The result shows the proposed approach has higher intersection efficiency than the benchmark. Fig. 11 and Fig. 12 show the proposed approach increase the capacity of the intersection and it can decrease the average vehicle delay to 24.9%   on average when compared with ASC. Fig. 13 shows the proposed approach results in a 10.1% larger throughput than CTV at the proposed situation in [31]. This can be explained as follows. First, the proposed approach has a larger solution space in signal timing results from the flexible signal structure. Besides, the efficiency is inversely proportional to IGI since IGI is a waste of effective green time. The average IGI of the proposed method is 1.79 s for the proposed optimization method prefers the phase sequence with smaller IGI freely. Presented in Fig. 14, the small IGI appears more frequently in the optimized signal timing than large IGI. A small IGI can advance the vehicle's arrival time, thereby reducing delays.   Fig. 15 shows the proposed approach increases fuel efficiency by 5.5% averagely. On the one hand, trajectory optimization allows each vehicle to pass through the intersection without braking and idling, which causes larger fuel consumption. On the other hand, the proposed method improves efficiency means a higher average vehicle speed. Therefore, fuel efficiency is increased.

V. CONCLUSION
This paper develops a signal-trajectory optimization method for the signalized intersection under the autonomous and connected vehicles. In the signal optimization module, the signal timing problem with high flexibility in signal change is first formulated. The passing state of each CAV and their behavior inside the intersection is predicted. A dynamic IGI calculation is added to signal optimization to guarantee vehicle safety during signal change. In the trajectory optimization module, the trajectory of the vehicle at the approach lane and inside the intersection is optimized. This system makes use of the connected vehicle technology and uses present information as optimization input, which includes vehicle speed, location, road status, and dynamic speed limit.
The evaluation result of the proposed approach is compared with the performance of two benchmarks: the actuated signal control (ASC) and a cooperation method between the VOLUME 8, 2020 traffic signal and vehicles (CTV) in ad hoc research. Results showed: 1. The proposed method meets the safety requirements by the dynamic design of IGI. The IGI varies from 0.84 to 3.14 to ensure the safety of conflicting vehicles during different signal changes. 2. The proposed method has a higher efficiency than ASC and CTV. The proposed approach decreases the average vehicle delay to 24.9% comparing with ASC and the proposed approach results in a 10.1% larger throughput than CTV. 3. The proposed approach increases fuel efficiency by 5.5% averagely XIAOGUANG YANG received the B.S., M.S., and Ph.D. degrees in traffic engineering from Tongji University, China. He is currently a Professor with the College of Transportation Engineering, Tongji University. His research interests include intelligent traffic systems and urban transport systems. VOLUME 8, 2020