Trajectory planning for connected and automated vehicles at isolated signalized intersections under mixed traffic environment
Introduction
Recent advances of connected and autonomous vehicle (CAV) technologies are regarded as one of the most promising solutions to improve traffic safety and efficiency, which have become a major topic of concern for policymakers and researchers. The connected vehicle technology enables real-time communication between vehicles (V2V) and between vehicles and infrastructures (V2I); and the automated vehicle technology enables precise control of vehicle trajectories. The combination of connected vehicle and automated vehicle technologies further enables the trajectory planning for CAVs and offers new approaches to traffic operations.
By designing CAV trajectories, a number of existing studies aim to improve traffic operational performance such as safety, efficiency, and environmental friendliness at particular facilities, such as to reduce environmental impacts along highway segments (Lu et al., 2019) and to coordinate vehicles going through the highway merging area (Hu and Sun, 2019). At urban intersections, vehicles arriving during red lights may stop at stop bars and then accelerate when traffic lights turn green. This process raises the travel time and fuel consumption of vehicles as well as reduces intersection capacity. By appropriate trajectory planning, CAVs can slow down in advance to avoid stops and queues at a stop bar (Feng et al., 2018, He et al., 2015). Therefore, CAV trajectory planning at intersections is widely investigated for both optimizing vehicle driving behaviors and improving traffic flow performances.
For the energy saving and emission reduction of vehicles at intersections, CAV trajectory planning is adopted in the development of eco-drive systems. Eco-driving systems usually provide ecological speed profiles to a vehicle based on the predicted behaviors of its preceding vehicles in a look forward horizon and traffic signal timings (Xia et al., 2013, He et al., 2015, Yang et al., 2017). The current states of the preceding vehicles can be received through vehicle-to-x (V2X) communication (Hu et al., 2016), or detected by the on-board sensors of CAVs (Kamal et al., 2015) and loop detectors (Jiang et al., 2017). And their future trajectories are predicted using longitudinal driving behavior models, such as Gipps car-following model (Kamal et al., 2015) and the intelligent driver model (Jiang et al., 2017). The optimal speed profiles of the target vehicle are usually generated in an optimal control framework to reduce fuel consumption and improve mobility and comfort, under the constraints of traffic rules (He et al., 2015, Hu et al., 2016, Jiang et al., 2017).
In addition, CAV trajectory planning is introduced in the research area of CAV-based traffic control at signalized and “signal-free” intersections to mitigate congestion, lessen the risk of crashes, and reduce fuel consumption and emissions under both fully and partially CAV environment. In the fully CAV environment, the conflicts between vehicles with incompatible movements at intersections can be avoided by controlling CAV trajectories without explicit traffic lights (Dresner and Stone, 2008, Mirheli et al., 2019). A planned trajectory strategy could lead a CAV to slow down in advance to avoid stops and queues at stop bars for the elimination of start-up lost time and the improvement of driving experience at signalized intersections (He et al., 2015, Mirheli et al., 2018, Feng et al., 2018, Yu et al., 2018, Zhang and Cassandras, 2019, Yu et al., 2019, Kamal et al., 2020). However, the fully CAV environment cannot be realized in the near future. It is widely expected that the mixed traffic with human-driven vehicles (HVs) and autonomous vehicles (AVs) will exist in the next 20–30 years (Zheng et al., 2020, Guo et al., 2021). Compared with the fully CAV environment, the trajectory planning in the partially CAV environment needs to consider the driving behaviors of HVs, whose trajectories cannot be precisely controlled directly. Most of related studies focus on the optimization of the longitudinal speed profiles of CAVs based on predicted future trajectories of HVs (Yang et al., 2016; Pourmehrab et al., 2019; Zhao et al., 2018, Ghiasi et al., 2019, Yao and Li, 2020, Niroumand et al., 2020). Yang et al. (2016) used kinematic wave theory to predict the queue length of HVs at a signalized intersection and then proposed a bi-level model to optimize both signal timings and CAV longitudinal trajectories. Pourmehrab et al. (2019) proposed an Intelligent Intersection Control System (IICS) for mixed traffic flows at signalized intersections. HV trajectories were first predicted by the Gipps car-following model and CAV trajectories were then optimized for minimum delay. Guo et al. (2019) proposed an efficient dynamic programming with shooting heuristic (DP-SH) algorithm for the integrated optimization of CAV trajectories and signal timings. HV trajectories were predicted based on the entry information from vehicle detectors upstream of the intersection at the beginning of the trajectory control section. Zhao et al. (2018) proposed a model predictive control (MPC) method to minimize the fuel consumption for platoons of mixed CAVs and HVs passing a signalized intersection. Niroumand et al. (2020) developed a mixed-integer non-linear program to optimize signal timings and vehicle-group trajectories at an intersection under the mixed traffic environment. A “white” phase was introduced for CHVs to pass the intersection. The movements of CHVs were estimated by a linear car-following model based on the estimated/planned future trajectories of lead vehicles. Yao and Li (2020) proposed a decentralized control model for CAV trajectory planning at a signalized intersection with a single-lane road to minimize the travel time, fuel consumption, and safety risks of each CAV. The results showed that the decentralized model overperformed the benchmark centralized control model in terms of computational efficiency without significant loss of the system optimality. Above studies have also validated that CAV trajectory planning can not only improve the driving experience of CAVs but also influence their following vehicles and optimize the overall traffic operational performances.
However, the trajectory planning methods in these studies usually assume no lane changing and only optimize longitudinal trajectories with the consideration of other vehicles in the same lane. The lack of considering lateral trajectories (i.e., lane-changing behaviors) makes these studies inapplicable in the real world because mandatory lane changing is inevitable at urban intersections. Although studies in the research area on automatic vehicle control have investigated two-dimensional trajectory planning problems, they usually focus on the design of precise geometry properties of trajectories of individual vehicles in a short horizon (e.g., 10 s) based on local traffic environment (González et al., 2015, Bevly et al., 2016, Feng et al., 2019). The trajectory planning for multiple vehicles considering global traffic information in a long horizon is missing. It is worth mentioning that the developed Mixed Integer Linear Programming (MILP) model in Yu et al. (2018) optimized both longitudinal and lateral vehicle trajectories at isolated intersections. All vehicles were assumed to enter the control zone in dedicated lanes. That is, only optional lane changing was considered. In addition, the approach is confined to the fully CAV environment. The integrated optimization of both longitudinal and lateral vehicle trajectory strategies under the partially CAV environment remains to be investigated.
Several challenges emerge in the trajectory planning for CAVs at intersections when considering lane-changing behaviors. Firstly, rather than just consider the car-following relationship with the vehicles in the same lane, more factors like interaction with vehicles in other lanes should also be well concerned. A notable recent study investigated CAV trajectory smoothing in mixed traffic considering cut-in lane changings of CHVs in adjacent lanes (Yao and Li, 2021). However, no lane-changing maneuvers of CAVs were assumed and only longitudinal speed profiles were optimized. The interaction between the lane-changing maneuvers of CAVs and CHVs was not considered. Secondly, the lane-changing strategy should cooperate with the longitudinal speed profile, because a CAV’s lane-changing maneuvers will affect the solution space of the longitudinal trajectory and the car-following strategy affects the lateral trajectory in return. Thirdly, computational burden may render real-time implementation difficult, especially with high traffic demand. This challenge is highlighted in several pioneering studies on centralized optimization frameworks due to the complex nature of multi-trajectory planning problems (Li and Li, 2019). One approach is to decompose the problem into several sub-problems with much less complexity (Tajalli and Hajbabaie, 2021). Another approach is using discrete time in model formulation (Miyatake et al., 2011, Li and Li, 2019, Tajalli and Hajbabaie, 2021) for the application of numerical solution algorithms (e.g., dynamic programming). In addition, a vehicle trajectory is divided into several segments as an approximation to reduce the dimension of solution space (Zhou et al., 2017, Ma et al., 2017, Feng et al., 2018). And heuristic algorithms are also investigated (Guo et al., 2019). Different from centralized optimization, decentralized schemes allow vehicles to negotiate with each other and plan their driving strategies themselves (Malikopoulos et al., 2018, Liu et al., 2018, Mirheli et al., 2019, Yao and Li, 2020), which shows advantages on computation efficiency and is more suitable for real-time applications.
Realizing the research gaps, this study presents an approach to the optimization of CAV trajectories in a decentralized way at isolated signalized intersections under the mixed traffic environment, which consists of connected and human-driven vehicles (CHVs) and CAVs. All vehicles are assumed to be connected in this study as governments tend to vigorously promote the V2X technologies for their benefits on safety, mobility, and environmental friendliness. For example, National Highway Traffic Safety Administration (NHTSA) is going to require vehicle-to-vehicle (V2V) capability for all new vehicles (U.S. Department of Transportation, 2017), and the connectivity adoption rate is expected to reach 100% in 2023 according to the corresponding analysis report (U.S. Department of Transportation, 2016). A bi-level optimization model is formulated based on discrete time to optimize longitudinal and lateral trajectories of a single CAV given signal timings and predicted/planned trajectories of other CHVs and CAVs. The upper-level model optimizes the lateral trajectory, (i.e., lane choices). The lower-level model optimizes the longitudinal trajectory (i.e., acceleration profiles) based on the lane-changing strategies from the upper-level model. The objective is to minimize vehicle delay, fuel consumption, and lane-changing costs. A Parallel Monte-Carlo Tree Search (PMCTS) algorithm is applied to solve the bi-level optimization model. CAV trajectories are planned one by one according to their distance to the stop bar in a decentralized way. A rolling horizon implementation procedure is designed for the application of the proposed model to time-varying traffic conditions.
The remainder of this paper is organized as follows. Section 2 describes the addressed problem. Section 3 formulates the bi-level model of CAV trajectory optimization under mixed traffic conditions. Section 4 designs the solution algorithms for the bi-level model and the implementation procedure with time-varying traffic conditions. Section 5 conducts numerical studies and sensitivity analysis. Finally, Section 6 delivers the conclusions and future research directions.
Section snippets
Problem description
Fig. 1 shows the details of one approach arm of a typical signalized intersection with four arms. Each approach lane is dedicated to a specific vehicle movement. There is a no-changing zone close to the stop bar, where lane changing is not allowed. Vehicles have to finish lane changing before the no-changing zone, which is the current practice in the real world. CAVs and CHVs coexist in the approach lanes and both follow the signals at the intersection. Vehicles are connected within the control
Model framework
Fig. 2 shows the bi-level optimization framework of the trajectory planning for CAV . At the start time step of the planning horizon, CAV collects the information on the signal timings, the initial trajectory points and movement directions of the vehicles within the control zone (), and the planned trajectories of its preceding CAVs ().
At the initialization stage, the trajectories of the preceding CHVs () is firstly predicted with the aid of the second-order car-following
Trajectory prediction algorithm
The future trajectories of the other vehicles determine the solution space of CAV ’s trajectory planning. Therefore, in the initialization process in Fig. 2, we need to predict the future trajectories of the vehicles in during the planning horizon, and generate the initial feasible trajectories for CAV and the vehicles in . The trajectory prediction is based on current traffic conditions and vehicle driving behaviors determined by the car-following model in Eissfeldt (2004) and the
Experimental data
A micro-simulation of the typical intersection in Fig. 1 is applied to explore the benefits of the proposed trajectory optimization model. Since CAV trajectory planning in one arm is independent of that in another arm, one arm of the intersection is taken for the experiments, which has four approach lanes including a left-turning lane, two through lanes, and a right-turning lane. The length of the control zone is m. The length of the no-changing zone is m. The speed limits in the
Conclusions and recommendations
This study proposes a trajectory planning model for CAVs in a decentralized way under the mixed traffic environment which consists of CHVs and CAVs. CAV trajectories are planned one by one according to their longitudinal locations in the approach lanes. The trajectory planning of a single CAV is decomposed into a lateral lane-changing strategy and a longitudinal acceleration profile. A bi-level optimization model is then built based on signal timing plans and planned/predicted trajectories of
CRediT authorship contribution statement
Chengyuan Ma: Conceptualization, Methodology, Data curation, Writing - original draft, Writing - review & editing. Chunhui Yu: Conceptualization, Methodology, Writing - review & editing. Xiaoguang Yang: Conceptualization, Supervision.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This paper is supported by National Key R&D Program of China (No. 2018YFB1600600), the National Natural Science Foundation of China (No. 61903276), and Shanghai Sailing Program (No. 19YF1451600).
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