Adaptive control for reliable cooperative intersection crossing of connected autonomous vehicles

Rapid advances in vehicle automation and communication technologies enable connected autonomous vehicles (CAVs) to cross intersections cooperatively, which could significantly improve traffic throughput and safety at intersections. Virtual platooning, designed upon car‐following behavior, is one of the promising control methods to promote cooperative intersection crossing of CAVs. Nevertheless, demand variation raises safety and stability concerns when CAVs adopt a virtual platooning control approach. Along this line, this study proposes an adaptive vehicle control method to facilitate the formation of a virtual platoon and the cooperative crossing of CAVs, factoring demand variations at an isolated intersection. This study derives the stability conditions of virtual CAV platoons depending on the time‐varying traffic demand. Based on the derived stability conditions, an optimization model is proposed to adaptively control CAVs dynamics by balancing approaching traffic mobility and safety to enhance the reliability of cooperative crossing at intersections. The simulation results show that, compared to the nonadaptive control, our proposed method can increase the intersection throughput by 18.2%. Also, time‐to‐collision results highlight the advantages of the proposed adaptive control in securing traffic safety.


| INTRODUCTION
Traffic congestion is a severe problem in modern cities that increases travel time, air pollution, fuel consumption, and economic losses. Urban congestion caused about 8.7 billion hours of travel delay and 35 billion gallons of wasted fuel for an estimated congestion cost of $190 billion in 2019. 1 Intersections represent major road bottlenecks where traffic flows from multiple directions cross at the location with a limited capacity. The delay at intersections alone contributes 5%-10% of total traffic delay. 2 Meanwhile, the intense conflicting movements of vehicles result in frequent accidents at intersections. According to the European Union community database on road accidents (CARE), more than 20% of traffic fatalities are intersection-related in the European Union. 3 In the United States, around 30% of fatal crashes occurred in intersection areas between 2015 and 2019. 4 These statistics demonstrate the need to improve traffic mobility and safety at intersections.
Traffic controls at intersections have evolved from traditional fixed-time signal control to recent adaptive signal control to manage intersection traffic efficiently. The traditional fixed-time control approaches determine signal phases and green splits based on historical data. Therefore, they cannot adapt to the nonrecurrent traffic pattern. 5 Adaptive traffic signal systems such as the Split Cycle Offset Optimization Technique (SCOOT) 6  Depending on the existence of a central operator for supervising the control process, traffic management strategies for signal-free intersections can be classified as centralized and distributed control. 10 Centralized cooperative controls can be further categorized as resource reservation approaches 11,12 or trajectory planning approaches. [13][14][15][16] The resource reservation approaches discretize the intersection space into cells along the temporal and spatial dimensions. The discretized spatiotemporal cells will then be allocated to the approaching vehicles. For example, Dresner and Stone proposed a reservation approach where the vehicle agent reserves a space-time block in the intersection. The intersection controller manages the reservations following the "first come, first serve" policy. 11 For the other centralized cooperative control strategy, trajectory planning approaches seek to simultaneously determine all vehicles' paths to ensure that all vehicles enter the signal-free intersection without any conflicts. Trajectory planning-based studies design cooperative intersection control by assigning vehicles the optimized paths. For example, Li and Wang 13 proposed a scheduling algorithm that assigns vehicles to cross an intersection. Their scheduling algorithm finds the minimal travel times for all approaching vehicles based on a spanning tree. Mirheli et al. 14  Besides using tokens, virtual vehicle platooning is an efficient alternative for distributed cooperative intersection control.
Approaching CAVs from different links is mapped into one virtual platoon to ensure the safety of the cooperative crossing. The virtual platoon strategy is implemented for T-intersection and on-ramp merging conditions. 19,20 Medina et al. 21 applied the virtual platoonbased cooperation control to a four-leg intersection to reduce the risk of accidents and increase intersection throughput. Xu et al. 22 proposed a virtual platoon-based approach and proved its linear string stability.
Existing studies on cooperative intersection crossing at signal-free intersections focus mainly on designing a control strategy given a fixed demand. However, traffic demand varies from time to time, leading to additional delays or safety issues if the control is not designed properly. For example, during peak hours, the increasing demand can surge over the intersection capacity, resulting in oversaturated flows and unpredictable travel times. The overflows can propagate to adjacent intersections, causing urban gridlock. Moreover, the demand variation introduces additional uncertainty to the intensive vehicular interactions at the intersection and affects the system-level travel reliability and mobility of cooperative crossing control. In this context, adaptive control strategies can adjust the control parameters to adapt to demand variation. So far, no studies have focused on designing adaptive control for cooperative intersection crossing to accommodate demand variation.
To fill this gap, this study proposes an adaptive cooperative intersection crossing control to adapt to time-dependent traffic and enhance control reliability considering traffic demand variation. The trade-off between platoon string stability and safety to determine the WEI AND HE | 279 optimal cooperative crossing control for CAVs is investigated. Traffic information acquisition systems can collect traffic data from connected-vehicle systems and various traffic sensors, including loop detectors, ultrasonic devices, and image recognition devices. 23 By using the data from upstream traffic, including flow rate, traffic speed, density, and travel time, the traffic demand at the subject intersection can be predicted in real time. In this study, an adaptive virtual platoon control is introduced by adjusting the control parameters adaptively according to the predicted traffic demand of the subject intersection.
The proposed adaptive control aims to achieve three objectives.
The first is to ensure the virtual platoon stability to mitigate perturbation effects resulting from demand variation. Second, adaptive control seeks to maintain the traffic throughput at a satisfactory level. Third, the proposed control is intended to ensure a smooth control transition by minimizing the changes of the control parameters. To achieve the three objectives, this study formulates the adaptive virtual platooning as a discrete-time optimization problem assuming that the demand does not deviate from the predicted value in the next time interval. The platoon parameters will be adjusted dynamically based on the predicted demand variation using the proposed optimization to ensure cooperative behavior among CAVs while enhancing CAVs' travel reliability.  When these CAVs approach the intersection, they instantly broadcast their states to other vehicles through V2V communications, including position and speed.
As a distributed cooperative intersection control, virtual vehicle platooning has been proposed to improve intersection mobility and safety. 21,22 It has broad advantages in reducing computational and communication burden to enhance system security and reliability.
This study adopts the virtual platoon control strategy to manage CAV traffic at an isolated signal-free intersection. In a virtual platoon control, approaching CAVs on different links are projected into one virtual platoon according to their distance to the center of the intersection to ensure the safety of the cooperative crossing. 22 The projection of the virtual vehicles is shown in Figure 1. The projected virtual vehicle platoon is shown in the dashed line rectangle. The area within the circle is called the cooperation zone. In the cooperation zone, vehicles will form a self-organized car-following maneuver to cross the intersection. Denote R as the radius of the intersection cooperation zone and Q as the center of the intersection. To solve these issues, this paper proposes an adaptive virtual platoon control by adjusting the underlying car-following dynamics.

| Adaptive virtual platoon control
Traffic demand at an intersection is stochastic and could vary rapidly.
The demand exceeding the capacity of intersections during peak hours causes queues and traffic congestion. Moreover, traffic congestion will increase travel time, air pollution, fuel consumption, and economic losses. Additionally, the demand variation will introduce uncertainty to the intersection, which can lead to perturbations to the upcoming traffic. Such perturbations will result in speed oscillations of vehicles that affect the traffic control reliability, increase collision risk, and raise substantial safety concerns.
Therefore, it is crucial to consider the demand fluctuation to foster reliable cooperative intersection crossing to reduce traffic congestion and mitigate the effect of traffic perturbations to enhance traffic reliability and safety.

| Optimization model
The adaptive control is built upon the traffic demand of the cooperation zone during an upcoming time interval, which can be predicted using available traffic acquisition technology and traffic prediction algorithms. [24][25][26] Based on the predicted traffic demand, this study proposes to optimize the car-following behavior of the cooperative CAVs when they form a virtual platoon responding to the varying traffic density.
If the predicted demand of approaching traffic changes over a specific level in the next time interval as shown in "Predicted traffic demand" in Figure 2, the control system will trigger the implementation of the optimization model to optimize the parameters of the carfollowing control to enhance traffic mobility and reliability, illustrated by the "Parameter optimization" box in Figure 2. The general form of the optimization model will be shown later in Equation (3) and the formulation of the optimization model will be explicitly presented in Equation (12)   The proposed adaptive control for cooperative intersection crossing seeks to achieve three objectives. The first objective is to ensure the virtual platoon's string stability. The second objective of the proposed adaptive control aims to guarantee that the traffic throughput is maintained at a satisfactory level while reducing the congestion effect due to demand variation. The third objective seeks to mitigate the changes in the control parameters to ensure a smooth control mechanism.
String stability considered in the first objective investigates the behavior of the entire platoon in response to external perturbations. 27 If the speed of one CAV is perturbed, its following vehicles in the virtual platoon have to react to the changes due to the distributed car-following control mechanism for the cooperative intersection crossing. The string stability of the virtual platoon ensures that the speed perturbation is not amplified toward the upstream traffic. 28 The (2) then the platoon is at a string stable state.
In Equation (2), d is the space gap between adjacent vehicles at an equilibrium state. Following the derivation in existing research, 30 the string stability condition is obtained for the virtual platoon. The derived platoon string stability condition can be applied to various car-following models. This paper uses function h(·) to represent the string stability condition. In particular, h < 0 stands for the case that the platoon is under the string stable condition and h ≥ 0 stands for the unstable condition. The detailed formation of the string stability condition will be presented later.
The second objective of the proposed adaptive control aims to guarantee that the traffic throughput is maintained at a satisfactory level. This paper uses function TH (·) to denote the estimated intersection throughput given a set of car-following parameters.
Notation D t denotes the predicted traffic demand at time interval t that is the input of the control system. To maintain the intersection throughput at a satisfactory level, the condition TH D − > 0 t denotes the surplus between intersection throughput and traffic demand.
The third objective seeks to mitigate the perturbation on the control parameters to ensure a smooth control mechanism. 31 Adjustment of the CAV car-following parameters will affect the platooning vehicle's dynamics. Abrupt adjustments introduce perturbations to CAVs dynamics that raise safety concerns. Therefore, a smaller change in car-following parameter adjustment implies a smoother control. Our model denotes the car-following parameters as a vector θ. Without loss of generality, this study applies the Euclidean distance to measure the extent of adjustment of platoon control parameters. Other functions, such as L 1 -norm 32 or vehicle's jerk, 33,34 could be integrated into the measurement of control smoothness.
This paper proposes an optimization model to achieve the three control objectives. If the predicted demand of approaching vehicles changes beyond a threshold in the next time interval, the proposed optimization will optimize the parameters in CAVs' car-following control to help them form a virtual vehicle platoon adapting to the new traffic demand while enhancing the platoon reliability. The optimization problem is formulated as follows: where the CAV car-following control parameters are in θ. The coefficients α, β, and γ in the objective function denote the weights assigned to the three control objectives. Note that α > 0, β < 0, γ > 0.  37 Second, it provides collision-free behavior and smooth traffic flow. 38 Third, it is well accepted to model connected autonomous vehicles' longitudinal dynamics. 36 In the IDM, a vehicle's acceleration is formulated as the following differential equation: With the IDM model adopted, the three components in the objective function in Equation (3) can be specified. For the first objective to guarantee the platoon string stability, the stability condition h is obtained for the IDM model as follows 28,30,39 : where v e and s e denote the vehicle speed and the net distance between vehicles at equilibrium state, respectively.
The stability transition surface can be obtained from the neutral stability criterion in Equation (5). Figure 3 shows where λ is the scaling parameter.
F I G U R E 3 Stability transition surface WEI AND HE | 283 As described previously, two vehicles in a virtual platoon with nonconflict movements can cross the intersection simultaneously.
Therefore, the intersection throughput is scaled up by λ λ (1 ≤ ≤ 2) times of the platoon traffic flow at equilibrium. Here, λ = 1 represents the extreme scenario that there is only one vehicle in each schedule set, indicating that only one vehicle crosses the intersection at each time, while λ = 2 represents the other extreme scenario with maximum throughput, in which two vehicles in every schedule set form a vehicle pair to cross the intersection simultaneously without conflicts. The λ can be estimated when the schedule sets are determined. Note that the schedule sets depend on the predicted vehicle arrivals that can be given by available sensing technology.
The following analysis focuses on estimating traffic flow at equilibrium, given CAV car-following parameters. Based on Equation (4) Based on the above analysis, the formulation of the optimization problem with the specified IDM model is as follows: where θ v T a = [ , , ] e denotes the car-following control parameters of CAVs. In this paper, these parameter values are from the study 40 that provided calibrated IDM parameters to represent the common behaviors in the real world. In the model, the last three constraints represent the bounds for the car-following control parameters. This study applies the grid search method to solve the optimization problem.

| NUMERICAL STUDY
We conducted a simulation study to verify the proposed adaptive  We further analyze the optimized parameters. This study further analyzes the time to collision as a measurement to investigate the safety issue of the proposed adaptive control strategy using the safety indicator TET (time exposed time-tocollision [TTC] indicator). 42 The TET is calculated for adaptive and nonadaptive control scenarios in four quarters using three TTC thresholds: 2, 3, and 4 s. The results are summarized in