Unsignalized intersection vehicle platoon formation scheduling method based on mixed traffic

Abstract

Managing unsignalized intersections is further developed in the context of automated driving with vehicle-road coordination. In this context, the virtual platoon of lining vehicles into a one-dimensional virtual queue is based on a fully automated driving environment. It cannot be used in today’s mixed traffic, and improper sequencing rules can cause significant delays. Thus, we propose a hierarchical framework to manage unsignalized intersections, where the lower-level distributed controller controls the mixed vehicles to form a vehicle platoon with CAV at the head of the platoon. The upper-level collaborative controller determines the passing order of the vehicle platoon. In deciding the passing order, we propose the Mixed Platoon Scheduling Model (MPSM) to improve the traffic safety and efficiency of unsignalized intersections in mixed traffic environments and to obtain the optimal vehicle passing order without collisions. First, MPSM transforms the conflict scheduling optimization problem into a graphical optimization problem to get a Mixed conflict-directed graph (MCDG) of nodes. Second, the spanning tree’s depth and average width are optimized by coexisting undirected graphs so that the number of vehicle platoons passing through the intersection simultaneously increases. Then, we change the order of the spanning tree nodes to reduce potential delay. The effectiveness of the intersection management framework was evaluated experimentally. The results show that MPSM possesses good delay performance and has an advantage over Adaptive Signal Control Method (Adaptive-SIM), FCFS, and DFST algorithms, which can be applied in different traffic environments.

Introduction

Connected and Automated Vehicles (CAVs) are considered critical players in the next generation of transportation systems, providing a different approach to traffic management control¹. Vehicles share information through V2I communication, which helps to coordinate vehicle movement better and improve system operational efficiency and safety. In the past decade, researchers have proposed various scheduling control strategies focusing on unsignalized intersections, where the critical issue is determining the optimal sequence of CAVs passing through the intersection².

While there are two representative strategies for vehicle conflict sequence optimization, namely rule-based and negotiation-based strategies, Meng compared the performance between the two representative strategies³. The rule-based strategy aims to enumerate all possible passing sequences to find a globally optimal solution. Most studies formulate the problem as a mixed-integer linear programming problem for scheduling vehicle passing times, and the objective is usually to minimize the delays of all vehicles⁴. Xu equivalently formulated the scheduling optimization problem as a tree search problem, where each tree node represents a partial vehicle order, equivalent to searching for the leaf node corresponding to the minimum delay value⁵. Jiang discussed the advantages of unsignalized omnidirectional lanes on this basis and similarly transformed the optimal scheduling problem into a tree search process and improved it by introducing the Monte Carlo search⁶. However, this type of enumeration strategy is susceptible to the number of vehicles, and the search time of the algorithm rises sharply with the increase in the number of vehicles, especially in the case of unsignalized intersections. Before solving the dimensional catastrophe problem, to the best of our knowledge, almost no algorithms can be used to obtain the optimal scheduling solution. Therefore, this shortcoming hinders the practice of such methods. The goal of negotiation-based strategies is to find an acceptable local optimal vehicle passing order in a relatively short time using some heuristic rules, and such strategies require self-organized collaborative movement among vehicles based on V2V. At an early stage, the Autonomous Intersection Management (AIM) strategy was proposed by Stone⁷, which divides intersections into raster spatiotemporal resources and allocates these raster resources to each vehicle in a first-come-first-served (FCFS) manner. Of course, this strategy has also been improved by most scholars later, and many variants currently exist, e.g., the reservation occupancy strategy¹. However, this approach does not lead to the ideal vehicle passing sequence in all cases, and the test by Meng et al. demonstrated that some special negotiation-based strategies do not lead to an acceptable vehicle passing sequence.

These problems are complex to solve at the algorithmic level; based on this, Xu et al. transformed the problem of conflict-free scheduling vehicle sequencing into a virtual mapping process by introducing a conflict-directed graph to represent the conflict relationships between vehicles⁸. The optimal spanning tree is obtained using the minimum breadth search algorithm based on the conflict-directed graph to represent the movement relationship of vehicles. Chen⁹ improved on Xu⁸ by representing the vehicle relationships using four conflict types, where the reachability conflict expresses the relationship between motion and position among vehicles, which pushes the conflict-directed graph to a conflict-mixed-directed graph, further optimizing the scheduling efficiency. Some scholars have paid attention to the problem of lateral scheduling of vehicles. In a directed lane, the lane function determines the conflict relationship of the vehicles. Chen¹⁰ scheduled the lateral movement of the vehicles to change the lanes they are in to get the optimal conflict relationship and then used the method of Xu⁸ to realize conflict-free scheduling. Zhou proposed a virtual projection strategy based on a flow control method, which avoids the problem of solution catastrophe due to oversized vehicles in the virtual projection strategy¹¹.

The above studies focus on the control scheduling of individual CAVs, which is difficult to break through in terms of algorithmic performance and system communication capabilities. Cai^12,13 have shown that vehicles crossing intersections in platoons can significantly improve efficiency and throughput while crossing intersections in a platoon can also save fuel¹⁴. Li proposed the COOR-PLT framework for coordinating vehicle formations to pass through intersections adaptively while considering the size of the vehicle platoon with the goals of efficiency, fairness, and energy saving and validated it effectively on SUMO¹⁵. Wang solved the problem of collaborative passing of vehicle platoons through adjacent consecutive intersections¹⁶. They proposed a joint control model by combining vehicle motion with signal switching that further improves the overall operational efficiency of vehicles. These studies usually assume that all vehicles are CAVs. However, we still have quite a long time to deal with scenarios where human-driven vehicles are involved in traffic before fully automated driving technology becomes commonplace.

There is also a lot of current research focusing on hybrid traffic flows where CAVs coexist with Connected Human Vehicles (CHVs), which is more in line with the current traffic scenarios. Ploeg demonstrates the stability and controllability of mixed-vehicle platoons, which creates a foundation for scheduling control of vehicle platoons¹⁷. Li utilized deep Q-networks to optimize intersection signaling schedules to optimize the operational efficiency of CAVs and CHVs and verified the beneficial effects of the method under different penetration rates¹⁸. Jiang developed a hierarchical control method for mixed platoons at unsignalized intersections, where vehicle queueing control was designed at the upper level, and eco-driving strategies were designed at the lower level using optimal control theory and discussed different demand levels under different performance and metric sensitivity¹⁹. These studies are modeled and solved in a Mixed Integer Programming (MIP) framework, and they are all limited by vehicle size and high model complexity.

In summary, the shortcomings of existing studies and the main contributions of this paper are as follows:

1. Most studies only focus on a pure CAV environment, which is not in line with the mixed traffic flow scenarios that will be faced soon. This paper proposes a “1 + N” mixed platoon approach to stabilize the intersection with CAV-led CHVs.

2. In the past, rule-based and negotiation-based strategies have almost always been modeled under the MIP framework, although some studies have used heuristic rules to reduce the computational cost. We propose a graph optimization-based vehicle ordering method based on Xu⁸, where we additionally consider the reachability problem due to the vehicles’ maximum motion performance constraints and design the corresponding communication topology for the vehicles.

3. In conflict-directed graphs, the studies represented by Xu⁸ and Chen⁹ have not considered the problem of conflict duration, which is objective in the actual conflict process, and it is unreasonable to regard the evacuation time of each depth in the spanning tree as the same. We build a Mixed Conflict-Directed Graph (MCDG), assign weights to the edges according to the possible durations of different conflicts, and design the Mixed Platoon Scheduling Model (MSPM) algorithm to optimize node exchanges to achieve vehicle platoon sequence.

Problem description

Xu describes a conflict-free scheduling problem for vehicles at a regular intersection, introducing the concept of a conflict-directed graph⁸. Based on this work, we propose to develop a conflict-free scheduling method for multi-lane intersections while also considering the dynamic reachability between vehicles and the influence of CHVs on the platoons. We design a hierarchical framework to manage unsignalized intersections, where the lower-level distributed controller controls the mixed vehicles to form a vehicle platoon with CAV at the head of the platoon and controls the operation of the vehicle platoon, and the upper-level collaborative controller determines the passing order of the vehicle platoon.

Assumptions:

(1) Like most existing intersection traffic studies, lane-changing behavior is prohibited to ensure vehicle safety and improve traffic efficiency. For safety reasons, vehicles with conflicting relationships are not allowed to travel in the intersection conflict area at the same time. (2) Considering the ideal communication conditions without communication delays or packet loss, CAVs and CHVs can transmit their speeds and positions to the central cloud coordinator via wireless communication (e.g., V2I communication). (3) The vehicles can autonomously form a platoon of vehicles with the same steering as they approach the intersection. (4) The CAV is capable of fully autonomous driving and is assumed to have perfect steering performance. Therefore, we focus only on the longitudinal control of the vehicle.

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Fig. 1

Intersection collaboration and control areas.

Vehicle dynamics model

Vehicle equation of motion

As shown in Fig. 1, vehicles enter the intersection collaboration zone (shown as blue dashed lines), which has a length of . After entering the zone, Vehicles can interact with the central coordinator for information, and vehicles autonomously form a vehicle platoon in the COOP zone. All vehicles in the platoon have the same steering, and the platoon is organized into the correct lane before entering the CTRL zone. The intersection control zone (shown by the red dashed line), which is of length , is where the central coordinator controls platoon CAV vehicles to form a safe and efficient passing sequence. We index the vehicles sequentially according to the position of the platoon in which they are located and the distance of the platoon from the stop line. Note that since we are concerned more with the vehicle scheduling problem than the vehicle control problem, a simple second-order vehicle dynamics model is used rather than a more complex one, as shown in Eq. (1). Vehicle dynamics are very complex due to the many nonlinear components such as the engine, transmission, gears, wheels, and external forces such as aerodynamic drag and rolling resistance, and here we use a simple linear model¹⁷.

1

Where is the state space of vehicle,and denote the remaining distance and speed from the CAV to the parking line at time , denotes the acceleration of the vehicle, denotes the vehicle power delay, and denotes the control input. We do not consider the impact of the dynamic performance between vehicles; therefore, using this model of uniformly variable motion, we call it a continuous time linear system .

In addition, the vehicle’s physical speed and acceleration must be constrained. Assume that is the maximum velocity constraint,, are the deceleration and acceleration constraints, respectively, as shown in Eq. (2).

2

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Fig. 2

Schematic diagram of vehicle platoons.

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Fig. 3

Intersection vehicle conflict analysis.

Vehicle platooning model

As Jiang et al. concluded¹⁹, vehicle platoons at intersection scenarios are usually less than four vehicles, which is different from the characteristics of highways, and the advantage of short-range scheduling is that it can be organized into small columns of vehicles in a simple and fast way. Note that since we focus on conflict scheduling of vehicles at intersections, we ignore the process of forming vehicle platoons in COOP zones, and detailed scheduling of vehicle platoons for lane changing can be found in the work of Cai et al.^12,13.

In the study of vehicle platoon control in mixed traffic environments, the “1 + n” platoon structure is the most common²⁰. It can adapt to scenarios with various traffic compositions and CAV penetration rates, so we also adopt the 1 + n platoon mode. The CHV can only communicate with the central coordinator in one way, i.e., the central coordinator can obtain the CHV’s operation and position information in real time. The platoon head CAV can communicate with other CAVs and the central coordinator with each other. Therefore, a vehicle platoon can be regarded as an agent, and the CHVs following inside the platoon are determined by Eq. (3)²¹, which is obtained by improving the Helly following model.

3

.

where and are parameters determined according to the driver response-stimulus model, is the driver’s reaction time. For simplicity of processing, we consider that all drivers have the same reaction time, is the desired distance between the CHV and the vehicle in front of it. By defining the position and velocity errors of the CHV, we can reconstruct Eq. (3) as follows:

4

In Eq. (4), and denote the position error and velocity error between the CHV i and the head-of-row CAV; and are the distance and velocity of the head-of-row CAV from the parking line, respectively; is the distance between the CHV i and the head-of-row CAV. Then, according to Eq. (4), the error equation shown in Eq. (5) can be obtained.

5

Where, is an element of the adjacency matrix of the CHV. represents the communication connection status between vehicle i and vehicle . Specifically, when equals 1, it indicates that vehicle i can obtain the status information (such as position, speed, etc.) of vehicle through V2V communication or sensors, meaning there is an information transmission link from to i. Otherwise, it means vehicle i cannot acquire information about vehicle , and there is no communication connection. Defining, ,, we can recharacterize Eq. (5) as the system shown in (6). In Eq. (6), , and are expressed as follows, respectively, where.

6

According to Jia et al.‘s definition of mixed vehicle platoon stability²², the position and velocity difference between the CHV i in the vehicle platoon and the CAV at the head of the platoon must satisfy boundedness, as shown in Eq. (7).

7

When a vehicle operates with system Eq. (6) satisfying the conditions shown in Eq. (7), then the 1 + n mixed platoon is a stable and controllable vehicle platoon²⁰. The following section will describe the controlled scheduling problem for vehicle platoons conflicting at intersections.

Scheduling model for mixed vehicle platoons

Conflict analysis

Conflict analysis is one of the most fundamental areas in the study of multi-agent cooperation. In the work of Xu⁸, the authors use the virtual projection method to organize vehicles from different directions into a virtual lane and control the scheduling by designing the communication topology between them. We also use a similar technique to introduce a conflict-directed graph to describe the conflicted relationship generated by vehicle motion. The difference is that the object of our control scheduling is a Vehicle Platoon (VP), and here, we take the lead vehicle of a VP as the information receiving and processing center of the whole platoon. Therefore, we can regard a VP as an agent (Fig. 2).

The intersection’s geometry does not impact our control scheduling model because the conflict analysis is generated based on the location of the lanes where the vehicles are located. As shown in Fig. 3, vehicles from different lanes form multiple conflict points for a non-regular intersection. Although the conflict scenarios are more complex, they can be categorized into three types: crossing, converging, and diverging. A few VPs are selected to illustrate the conflicting relationships in Fig. 3. Note that the following analysis can be applied to any intersection geometry.

(1) Crossing conflict. Vehicles from different lanes will likely collide when crossing the conflict point (indicated by the red circle). For example, VP2 and VP3 have crossing conflict points.

(2) Converging conflict. Vehicles from different lanes cannot enter the same lane simultaneously, as indicated by the orange arrows. For example, VP 2 and VP 4 have converging conflict points.

(3) Diverging conflict. Assumption 1 states that lane changing and overtaking are not allowed. Therefore, vehicles in the same lane cannot pass through the intersection simultaneously, as shown by the green squares. For example, there are diverging conflict points for VP6 and VP7.

Interestingly, in addition to the above three types of conflicts describing VP motion routes, we believe a different kind of conflict is caused by the speed and acceleration constraints of VPs^9,10. For example, in Fig. 3, when VP5 is about to reach the stop line at the designed virtual platoon speed, VP7 arrives at the control area. If we schedule VP5 and VP7 to pass simultaneously, VP5 will have to wait for a long time for VP7 to be near the stop line, which can jeopardize traffic safety and efficiency. In this case, due to the limitation of VP7’s speed and acceleration, VP7 can’t catch up with VP5 before reaching the stop line, even if VP7 does not have a conflicted relationship. Therefore, we further introduce a fourth type of conflict, namely, reachability conflict. This is usually not considered in similar studies^11,23.

(4) Reachability conflicts. Even if the VPs do not have route conflicts, their acceleration and speed constraints still restrict them from passing through the intersection simultaneously. For example, as mentioned earlier, VP5 and VP7 have a reachability conflict. Suppose VP5 arrives at the control area boundary with the designed virtual platoon speed . From Eq. (1) and Eq. (2) shows if VP7 immediately accelerates with maximum acceleration until it reaches maximum velocity, it still takes time to get the stop line, as shown in Eq. (8).

8

.

Where is the difference between the VP length and the VP length, and is the distance from the stop line of different inlet lanes to the center of the intersection. When determining whether the current VP has a reachability conflict with the VP, i.e., when the current VP i reaches the CTRL boundary of the control area, the current speed and position of the VP need to satisfy Eq. (9). We expect the VP to operate at the desired speed to obtain stable traffic near the stop line.

9

In Eq. (9), the remaining distance between the VP and the VP corresponds to a closer distance from the stop line. If a new platoon of vehicles cannot catch up with some of them due to speed/acceleration constraints, reachability conflicts are used to maintain consistency in scheduling results. We define different conflict sets to characterize the conflicting relationship between VPs. For any VP, , the crossing conflict set is defined as , the converging conflict set is defined as, the diverging conflict set is defined as , the reachability conflict set is defined as. Note that a conflict set is determined when a VP reaches a control zone boundary. At this point, the VP is at the boundary of the control area while the other vehicle platoons are inside the control area. Therefore, the indexes of the other VP in the VP conflict set are smaller than itself, and the elements in the VP conflict set satisfy Eq. (10).

10

.

In addition, we need to set the virtual leading vehicle 0 in front of the VP closest to the intersection at a constant speed. denotes the predefined desired platoon speed in virtual platoon coordination. Without virtual lead vehicle 0, the VP closest to the intersection will accelerate to, thus failing to form a virtual platoon⁸. With the spanning tree generated by this predefined scheduling method, the virtual platoon is controlled like a typical vehicle platoon. Accordingly, the virtual front vehicle 0 is added to the set of VPs closest to the intersection in each lane.

The work of Xu et al. and Chen et al. both consider homogeneous single intelligent vehicles^{8, 9–10}; they can ignore the effect of the length of the vehicle itself on the band. For platoons of vehicles, each of which has a different length, the size of the conflict area between two platoons is related to their routes, while the conflict duration is related to their lengths. For this reason, we assume that the vehicles can move according to a predefined route and generate a conflict region, as shown in Fig. 4.

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Fig. 4

conflict region.

We set the overlapping part of each conflict route as the conflict region, , and are the locations where the VP i enters the intersection, enters the conflict region and exits the conflict region, respectively. Then, for a VP, the length of the conflict area can be described as shown in Eq. (11).

11

Where, denotes the distance from the intersection’s entrance to the conflict area’s entrance, denotes the distance from the intersection’s entrance to the conflict area’s exit, and the difference between the two is the length of the conflict area of the VP.

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Fig. 5

Conflict directed graph and coexistence undirected graph.

Mixed platoon scheduling model

The Mixed Platoon Scheduling Model (MPSM) proposed in this paper is based on an extension of the virtual vehicle platoon approach, and our goal is to propose a collaborative approach to improve traffic safety and efficiency at unsignalized intersections in a mixed-traffic environment to obtain the optimal sequence of vehicle passing without collision. Let be the time step for the VP to enter the control zone and be the time for the last vehicle of the VP to exit the conflict zone; then, our objective 1 can be formulated as shown in Eq. (12), defined as the evacuation time of the intersection.

12

The metrics in objective 1 can only evaluate the system’s overall performance, so we introduce the average vehicle delay as the second metric. The running time difference between the VP in the actual state and in the free state is the delay. Objective 2 can be formulated as shown in Eq. (13).

13

Since Objective 2 considers the travel time of each VP, it indicates the individual benefit, which is a secondary optimization objective for traffic efficiency. In this study, we mainly focus on the optimal VP passing order to minimize the evacuation time, as described in objective 1 in Eq. (12).

Based on the previous section’s conflict relationship analysis, we adopt the Conflict Directed Graph (CDG) to define the conflicting relationship between VPs. A CDG can be denoted as , if there are K VPs in the control area, then the set of nodes is denoted as, and the set of edges denotes the conflicting relationship between nodes, which is the same as the definition of Xu⁸. All edges are unidirectional, i.e., pointing from the parent to the child, indicating that the parent passes first and the child then passes. However, in real-world scenarios, the passage order of crossing and converging conflicts can be exchanged, which brings more flexibility to the overall scheduling. Therefore, we categorize edges into two types, unidirectional edges, and bidirectional edges, and the sum of the edges of the two types is equal to the edges of the conflict-directed graph . Further, we can obtain the Mixed Conflict Directed Graph (MCDG) as shown in Fig. 5(a), denoted as. Black unidirectional edges represent diverging conflicts and reachability conflicts, and unidirectional edges imply that VP is not allowed to overtake VP due to Assumption 1, or VP cannot catch up with VP due to the fulfillment of Eq. (9). Red bidirectional edges represent crossing conflicts and converging conflicts, implying that the arrival order of VP and VP can be interchanged.

For , it has a virtual guide vehicle with root node 0. The MCDG describes the conflict relationships of all vehicle platoons and the conflict types. We can construct a spanning tree to obtain a feasible passing order through the MCDG, as known in Xu et al.‘s work. The depth of each node in the spanning tree is calculated based on its distance to the root node. In other words, the depth of a node represents the passing order of the VPs, i.e., VPs corresponding to nodes at the same depth should enter the intersection at the same time, which has been proved in the study of Xu⁸ as shown in Lemma 1.

Lemma 1

, if satisfies , then, where, and denotes the depth of the node .

According to Lemma 1, it is easy to associate the maximum depth of a spanning tree node with the evacuation time in objective 1, and associate the average depth of a spanning tree node with the average delay of vehicles in objective 2. Therefore, we assign nodes with conflicts to different depths. This also implies that the vehicles in the intersection are always conflict-free. Further, we can understand the depth of a spanning tree in graphical language as the sparsity time , then minimizing the sparsity time is equivalent to finding the smallest depth among the feasible spanning trees, and we can re-describe objective 1 as objective 3 as shown in Eq. (14). Let be the set of feasible spanning trees for , and be the feasible spanning tree with the most minor depth in

14

Recall that in the conflict scenario shown in Fig. 3, up to six VPs can pass through the intersection at the same time; in other words, it is possible to optimize by optimizing the number of VPs that can enter the intersection at the same time. The opposite view of the conflict relation is the coexistence relation. Here, we can introduce the concept of coexistence graph^9,10 as shown in Fig. 5(b) for the coexistence relation corresponding to the scenario shown in Fig. 3, which we define as a Coexisting Undirected Graph (CUG), denoted as. Where, . To optimize the overall spanning tree depth, we need to increase the average width of the spanning tree, i.e., the number of intersection VPs that can pass through the intersection simultaneously. Therefore, solving Eq. (14) is equivalent to finding the maximum number of VPs that can be combined in . This is in line with our general knowledge that the maximum combination of VPs represents the maximization of the efficiency of the intersection, which reduces the overall evacuation time, i.e., reduces . In graph theory, cliques are often used to describe the coexistence of nodes, as shown in Lemma 2.

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Fig. 6

Spanning tree structure corresponding to different solution methods.

Lemma 2

In , is a clique for any, and, all satisfy, , where denotes the existence of an edge between node and node , and denotes its absence.

As in Fig. 5(b), there is a clique, which means every two distinct nodes are adjacent, but it is not the largest clique. Here, we refer to the number of nodes in the clique as the number of cliques, denoted as . Note that in the overall intersection scenario of Fig. 3, if there are sufficient VPs in the coexisting lanes, then there exists . For , a cluster cover is the partition of all nodes into k disjoint subsets, such that ,, the subgraph derived from is a cluster, it satisfies the condition of a complete graph, and the minimum number of cluster-covered subsets of a graph is denoted as ⁹. At this point, we can summarize that the number of clusters of a CUG denotes the maximum number of VPs that can pass through the intersection at the same time, while the Minimum Clique Covering (MCC) number denotes the solution corresponding to the smallest possible evacuation time passing order. Further, assuming that the evacuation times of the nodes at each level of the feasible spanning tree are theoretically closer, we solve for the spanning tree corresponding to the minimum of . As shown in Table 1, we solve the scenarios shown in Fig. 3, and obtain solutions that are all , but they have different average number of clusters .

Table 1. Feasible MCC solution sets ().

Solution	V1	V2	V3	V4
1	1,4,5,6	7,8,9	2	3
2	1,4,5,6	2,7	8,9	3
3	4,5,6,8	1,2	7,9	3
4	4,5,6,8	1,3	2,7	9

Observing the feasible solutions given in Table 1, it is easy to associate the average number of cliques in the spanning tree with the average vehicle delay defined in objective 2, which responds to the delay level of each vehicle platoon. Then, we can similarly translate objective 2 into graphical language and reformulate it into objective 4, as shown in Eq. (15).

15

where denotes the depth of the clique, and denotes the optimal clique coverage of the CUG . Note that objective 4 and objective 2 are not exactly equivalent. Objective 2 denotes the average delay per vehicle, and objective 4 denotes the average delay per vehicle platoon, which we have converted to the graphical language shown in objective 4 to simplify the computational process, and therefore, see the work of Chen¹⁰.

In addition, we can see that Solution3, and Solution4 in Table 1 are the same and , but their evacuation times and average delays are not the same during actual operation. This is likely to be related to the specific conflicts between VPs, recalling that the conflict types of VPs, crossing conflicts and converging conflicts can be exchanged in order, and the conflict duration and evacuation time of each pair of conflicting VPs are different. Therefore, we can still further optimize Objective 3 and Objective 4 from the graphical point of view. Taking the scenario shown in Fig. 3 as an example, Xu et al. are using Depth-First Spanning Tree (DFST) for vehicle passing order solution⁸, as shown in Fig. 6(a); Chen used the MCC method to generate the spanning tree⁹, as shown in Fig. 6(b). The spanning tree of MCC is much smaller, but there is no specification of spanning tree edges in Chen et al.‘s work⁹, which have a specific physical meaning. As shown in Fig. 6(c), we add swappable edges between cliques , and the red edges indicate conflict pairs of swappable order. We use the conflict duration of a conflict as the weight of an edge , and minimize the sum of the conflict durations of all nodes connected by bidirectional edges by swapping node orders.

In the spanning tree , we define the set of nodes with bidirectional edge connections as, and the set of nodes with only unidirectional edge connections as, then the weight of a conflict can be formulated as shown in Eq. (16).

16

Where, is the length of the vehicle platoon , is the difference in distance of the current position of the vehicle platoon from the conflict region , is the speed of the vehicle platoon , and is the minimum distance of the security guarantee. Our goal is to minimize the sum of the weights of all by changing the order of the nodes, to satisfy Eq. (17), which is taken as objective 5.

17

Where, , is the parent node and is the child node. We can also find that in the spanning tree , not all nodes have conflicts among the clusters . As shown in Fig. 6(d), there are three scenarios: no conflict , non-exchangeable conflict , and exchangeable conflict . For edges with actual physical conflicts, they all obey the weight assignment in (16). Then, non-exchangeable conflicts also have weights determined by the duration of the conflict, but the weight of is determined by the order of the clique . In other words, when the order of the clique is determined, the weight of is also specified. Similarly, we can minimize the sum of the weights of all , which can be expressed as shown in Eq. (18).

18

Where, , is the parent node and is the child node. Obviously when optimizing , is also changing dynamically, and the evacuation time of a conflicting pair depends on the weight size of the edge with conflict during the actual vehicle movement. Thus, in a spanning tree , when the sum of the weights of all edges with conflicts is minimized, the potential delay of the WCDG is minimized at that time. After considering the weights of the one-way edges, we can reformulate objective 5 as shown in Eq. (19).

19

Where, is the minimum potential delay of . At this point, we have transformed a cross-over conflict scheduling optimization problem into a graph optimization problem. Among the above six objectives, only three objectives are required to be deployed inside the algorithm, , , and , and there exists an optimization order for these three objectives. is to obtain a spanning tree with minimum depth , is to reduce the average depth of the nodes without increasing the total depth , and is to reduce the potential delay of the conflicting nodes based on and .

Vehicle scheduling control process

In the previous section, we described a specific approach for vehicle scheduling control, which has a detailed flow, as shown in Fig. 7, and can be divided into three phases: the grouping, the optimization, and the implementation phases. The figure shows the behaviors of the vehicles corresponding to the different phases, where the grouping phase is classified as the collaboration area, the optimization and implementation phases are classified in the control areas as in Fig. 1.

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Fig. 7

Vehicle scheduling control flowchart.

The adaptation of MPSM to general intersections

The MPSM is also applicable to general unsignalized intersections, such as other four-way intersections, three-way intersections, and roundabouts. As shown in Fig. 8, similarly, assuming vehicles follow our guidelines, we can identify conflict areas and conflict types based on the guidelines in each direction of the intersection. Figure 8(a) shows a three-way intersection with a shared lane for through and left-turning vehicles. Figure 8(b) depicts a roundabout, where different line types or colors are used to distinguish guide lines for easy observation. The characteristic of a roundabout is that vehicles from different directions merge and diverge within the intersection, which does not affect our model. Since merging conflicts exist during convergence, our model ensures that conflicting vehicles do not enter the intersection simultaneously. Therefore, within a roundabout, when vehicles from different directions have merging conflicts followed by diverging conflicts, we only need to consider the first conflict (merging conflict). Similarly, we use a conflict-directed graph to describe conflicts, as shown in Fig. 9, to obtain the conflict-directed graph and coexistence undirected graph, and further solve for preliminary feasible solutions (OBJ3 and OBJ4) and calculate conflict durations (OBJ6) to find solutions with the minimum conflict duration. The solution process is the same as that for the intersection scenario in Fig. 3.

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Fig. 8

Simulation scene.

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Fig. 9

Conflict directed graph and coexistence undirected graph of Fig. 8.

Therefore, the MPSM can be applied to general unsignalized intersection scenarios, regardless of whether the intersection is irregular, has multi-direction shared lanes, roundabouts, or other configurations. The approach involves constructing a conflict-directed graph based on the intersection scenario, followed by generating a coexistence undirected graph. Through the coexistence undirected graph, the method seeks to determine the minimum number of cliques that can pass through the intersection simultaneously and the minimum average node depth, further aiming to find the minimum conflict duration, reduce potential delays for conflicting nodes, and obtain the optimal intersection passage sequence.

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Fig. 10

Simulation scene.

Simulation testing and analysis

Simulation description

The tests are co-simulated by SUMO 1.19.0, and Python 3.8, dependency libraries include Traci, matplotlib, time, math, numpy, yaml, random, itertools, copy, cvxpy, etc. All our test simulations are run on a 13th-Gen Intel(R) Core (TM) i9-13900HX 2.20 GHz processor. The control and collaboration zones of the vehicles are shown in Fig. 10, where is the optimized control zone, where all the vehicles execute according to the scheduling instructions, and is the collaboration zone, where the vehicles change lanes and group.

The vehicle inputs are randomized to make the simulation as close to the actual intersection, i.e., the vehicle arrivals obey the Poisson distribution. In the simulation, the generation of vehicles is completed within 30 s. In the vehicle input, 24 streams in 8 lanes are set up. CAV and CHV streams are set up in all four inlets and three directions to flexibly regulate the combination of streams. We conducted simulation tests on the effects of the algorithm with vehicle numbers of 40, 80, 120, 160, and 200, which approximately correspond to single-lane traffic flows of 585pcu/h, 1170pcu/h, 1800pcu/h, 2385pcu/h, and 2880pcu/h, respectively. The parameters are set as shown in Table 2.

In this paper, we tested Adaptive Signal Control Method (Adaptive-SIM), FCFS, DFST, MCC-ORI, and MSPD, also named MCC-WO. The Adaptive-SIM is a signal control method, while the other four are non-signal control methods. MCC-ORI is the algorithm corresponding to the WCDG without exchanging the vehicle order, and MCC-WO is the algorithm of the WCDG for exchanging the vehicle order. The performance of the five algorithms is tested under different vehicle demands, control zones, and lengths of the collaboration zones.

Table 2. Key Parameters.

Parameter	Symbol	Value
Distance error		0.1
Velocity error		0.01
Desired velocity		15 m/s
Desired distance		30 m
Maximum acceleration		5 m/s2
Minimum acceleration		– 6 m/s2
Maximum velocity		20 m/s
Minimum velocity		0 m/s
Safe time		0.2s
The Poisson parameter for a single traffic flow		0.05,0.11,0.17,0.22,0.28

Performance testing under different vehicle requirements

Our inputs satisfy the same mathematical expectation of arriving vehicles in each lane, so we only need to control the total number of vehicles at the overall intersection. Our tests are performed conditional on the length of the control zone, the CAV penetration rate, and, the length of the formation change zone 300 m.

As shown in Fig. 11, the delay performance corresponds to the five algorithms under different vehicle inputs. It should be noted that our total delays include scheduling delays and grouping changing delays, i.e., scheduling delays are the delays incurred by the vehicles from the beginning of the controlled process until they pass through the conflict point, and grouping changing delays are the delays incurred by the vehicles in the regions . The figure shows the delay performance corresponding to the five different algorithms. The adaptive signal control method (Adaptive-SIM) has the maximum delay compared with the other four non-signal control algorithms. The delay difference of FCFS is more considerable compared to the other three algorithms because the conflict logic of FCFS is different from the other three algorithms. FCFS does not allow more than two vehicles to enter the conflict zone. At the same time, the conflict zone of FCFS is the whole intersection, which results in the vehicle only waiting for the vehicle inside the intersection to drive out before it can enter the intersection, which makes the delay of the FCFS algorithm high. The other three algorithms have the same conflict logic, but the scheduling logic is not used, so there is little difference in delay performance. Regarding different vehicle number inputs, MCC-WO performs excellently, which shows that our proposed method has some advantages in delay performance. It can also be seen by comparing the difference between Fig. 11(a) and (b) that the delay incurred by vehicles in the grouping change zone (the collaboration zone) also increases with the number of vehicles.

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Fig. 11

Comparison of the delay performance of each algorithm under different vehicle demands.

The traffic capacity of an intersection refers to the maximum number of vehicles that the intersection can stably pass through per unit time²⁴. As shown in Fig. 11, under different vehicle numbers, the total delay of MCC-WO is lower than that of other strategies. In low-traffic scenarios (40 and 80 vehicles), the delay of MCC-WO is relatively low, which indicates that it can enable vehicles to pass through more efficiently, so the initial traffic efficiency is higher. In medium-traffic and high-traffic scenarios (120, 160, and 200 vehicles), as the number of vehicles increases, the delay of all strategies will increase (traffic pressure becomes greater), but the delay of MCC-WO increases more gently. For example, when there are 200 vehicles, the delays of Adaptive-SIM and FCFS soar sharply, while that of MCC-WO remains at a relatively low level. And the lower the delay, the shorter the time for vehicles to pass through the intersection, and the more vehicles can pass through per unit time, so the higher the traffic capacity. This means that MCC-WO can handle more vehicles per unit time, thereby improving the traffic capacity of the intersection.

Comparison of algorithmic solution performance

Through Fig. 11, we can see that MCC-ORI and MCC-WO have advantages in delay performance. Compared with DFST, these advantages are because we build a spanning tree with a smaller search depth, which makes the overall evacuation time of vehicles shorter, i.e., scheduling time. For FCFS, the algorithm does not need to deal with a complex search process, and the vehicle passing order is the arrival order. In DFST, a depth-first spanning tree is built to find a feasible spanning tree, and its algorithmic complexity is mainly reflected in the number of nodes in the spanning tree. In MCC-ORI and MCC-WO, the algorithms need to find the spanning tree with the minimum depth, which means the algorithms need to deal with the exponential permutation problem that varies with the number of vehicles. Therefore, we also improved the method, reducing the algorithm’s solution time by introducing the Simulated Annealing Approach (SAA) to perform a random search. As shown in Fig. 12, the CPU time of each non-signal algorithm varies with the number of processed vehicles. In the figure, we set the maximum value of CPU time to 100; when it exceeds 100, the maximum value is used. We observe that the algorithms are in an acceptable time range when the number of processed vehicles is less than 100. In contrast, MCC-WO has the longest computation time, which is because MCC-WO needs to perform two searches: the first one is the search of the optimal spanning tree, and the second one is the search of the optimal vehicle sequences, especially the second one is the search of the optimal vehicle sequences when the depth of the spanning tree is greater than 15, the algorithm is already unable to obtain results within an acceptable time frame. Therefore, although the algorithm MCC-WO has better delay performance, it is suitable for the case of a small number of vehicles.

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Fig. 12

CPU time of each algorithm for different numbers of vehicles.

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Fig. 13

Iterative process of MCC-ORI/WO spanning tree searching.

Figure 13 shows the iterative change graphs of the search process of algorithm MCC-ORI and algorithm MCC-WO in the minimum depth spanning tree, where the vertical coordinate cost is that shown in Eq. (15). It is shown that the algorithm improved by the SAA can effectively improve the search performance of the MCC. Combined with the CPU time curve of the MCC-ORI in Fig. 12, the algorithm improved by the SAA can overcome the increase in the number of vehicles brought about by the dimensionality disaster. However, MCC-WO consumes too much time in the search for the optimal order of vehicles, which is an area that needs to be improved in the future of this study. If only heuristic algorithms are used for improvement, it may lead to solution performance results that are inferior to those of MCC-ORI. This requires that the improved algorithm should possess both delay performance and solution performance simultaneously.

Algorithm sensitivity analysis

In Fig. 11, we analyze the delay performance of five algorithms under different vehicle demands. MCC-WO has good advantages, but its solution time is too long, especially after the number of vehicles reaches a specific size. Therefore, we analyze the pattern of delay changes of different algorithms by changing the number of vehicles as input. We also discuss the performance sensitivity of different algorithms under different vehicle demands. As shown in Fig. 14, the horizontal coordinate is the amount of change in the number of vehicles, and the vertical coordinate is the increase in delay. The five algorithms present the general trend of delay increase with the number of vehicles. Still, each algorithm has a different procedure, among which the increasing trend of Adaptive-SIM is the largest, and the growing trend of MCC-ORI and MCC-WO is the same. Analyzing the gradient, Adaptive-SIM shows an almost linear increase. FCFS and DFST algorithms show a trend of increasing fast and then slowly, which aligns with the “logarithmic” curve. At the same time, MCC-ORI and MCC-WO show a trend of growing slowly and then fast, which aligns with the “exponential” curve. This indicates that FCFS and DFST increase with the number of vehicles. This suggests that the two algorithms have significant marginal benefits in terms of delay as the number of vehicles increases, and the MCC-ORI and MCC-WO are not suitable for scenarios with large numbers of vehicles.

Figure 15 shows the average vehicle scheduling delays under different vehicle demands. In general, it still exhibits roughly the same trend as the total delays, with the maximum average delay per vehicle of the algorithm Adaptive-SIM and the minimum average delay per vehicle of the algorithm MCC-WO. In contrast, the change in the average vehicle scheduling delays with increased vehicles does not show significant changes. Compared with Fig. 14, the average vehicle scheduling delay shown in Fig. 15 does not have a considerable sensitivity to change, which may be due to the motion characteristics of different vehicles.

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Fig. 14

Total delay Change with different numbers of vehicles.

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Fig. 15

Average vehicle scheduling delays.

Performance analysis of the effect with different grouping and control zone lengths

In the above test and analysis, we are all under the, condition, in this section we change the, length respectively, keep other conditions unchanged, and analyze the delay performance of different algorithms under, . As shown in Fig. 16, we marked the specific delay values while using color shades to represent the delay size, where the maximum value (500s, 600s) represents that the model has no solving result. In 16(a), the four algorithms show better delay performance when j is more significant. Still, at 500 m, the algorithms DFST, MCC-ORI, and MCC-WO have no results for solving. This is because this time is too short, and the algorithms cannot plan the results in such a short period. It is known that the optimal boundary of k is between 400 and 500 m, and further, we can easily deduce that this boundary value decreases as the number of vehicles increases.

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Fig. 16

The effect with different grouping and control zone lengths on delay performance.

In Fig. 16(b), the four algorithms have better delay performance the smaller the. Similarly, the algorithms DFST, MCC-ORI, and MCC-WO solve for no result at . Obviously, the algorithms cannot plan the trajectory of the control area in the time corresponding to the vehicle movement of 100 m. By comparing the two figures, we can find that the corresponding delays do not differ much at the same raster. Therefore, the scheduling delay of vehicles is related to the difference . However, the determination of and is affected by the number of scheduled vehicles. It can be concluded that the boundary of the combination of the optimal and , the number of scheduled vehicles, CAV penetration, and the grouping length are all closely related.

The effects of performance at different permeabilities

As shown in Fig. 17, we measured it at a vehicle size of 100, where ,, and tested the dispatching delay and operational delay at penetration rates of 0.3, 0.5, and 0.7, respectively. The study reveals that the trends of scheduling delay and total delay are comparable because the grouping and switching delays vary by the penetration rate and have the same trend as the scheduling delay. Comparing the delay metrics under the four non-signal control methods, the analysis shows that FCFS has the worst performance and is most affected by the penetration rate.

As shown in Fig. 17(a) for scheduling delays under different penetration rates, the results indicate that FCFS has the worst sensitivity under CAV penetration changes. The other three methods (DFST, MCC-SAA, and MCC-WO) perform well in terms of sensitivity, which generally shows a gradual improvement in the delay performance with the increase in penetration rate. Figure 17(b) indicates that MCC performs best, thanks to the multi-vehicle conflict model. Just as the fundamental reason for the poor sensitivity of FCFS is that it is itself a two-vehicle conflict model, which is too conservative and is a way of sacrificing efficiency to improve safety, but is not applicable in real efficient traffic scenarios, in contrast, the MCC-WO method proposed in this paper can ensure safety while maximizing intersection efficiency.

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Fig. 17

Performance variation at different CAV penetration rates.

Conclusion

In this paper, we introduce directed graphs to describe the vehicle conflict relationship at unsignalized intersections, search for the corresponding spanning tree by SAA, and use the depth of the spanning tree as an indicator of vehicle delay. Due to the different conflict durations of vehicle platoons with different lengths, they are used as the weights of connected edges of the spanning tree’s neighboring nodes, and the spanning tree’s delay calculation method is established. The scheduling method corresponding to the best delay is obtained by exchanging the order of the nodes. Through simulation tests, we can draw the following conclusions:

1. The improved MCC-WO has an advantage over FCFS, DFST, and MCC-ORI algorithms. Still, the computation time of MPSM (MCC-WO) is generally long, and they need to be applied in combination with heuristic algorithms to improve their convergence speed.

2. Under different vehicle demands, the FCFS and DFST algorithms show better marginal benefits as the vehicle size increases. At the same time, MCC-ORI and MPSM (MCC-WO) are challenging to deal with in the case of larger vehicle sizes. The MPSM algorithm needs to search for the optimal order, making it difficult for the algorithm to get the results in a shorter period.

3. The optimal lengths of the grouping and control areas must match the number of input vehicles. Different vehicle requirements correspond to different control boundaries, and the overall scheduling delay of vehicles is related to the lengths of the grouping area and the control area. Therefore, the optimal length of the grouping and control area should be set dynamically according to the vehicle scale.

4. Compared with the mainstream FCFS, MCC-ORI and MPSM (MCC-WO) are robust. Meanwhile, MCC-WO has advantages when the number of vehicles does not exceed 100, and can be applied to different intersection scenarios.

This study analyzed the performance of MPSM through simulation, but its limitations should be noted.

1. This paper does not investigate the stability and recovery capability of mixed queues. In the future, artificial disturbance events (such as sudden stops and merging conflicts) can be further added to the model to compare the disintegration probability and recovery speed of queues under different proportions of CAVs.

2. The research did not consider communication delays and packet loss. Communication delay or packet loss will have a certain impact on the real-time acquisition of vehicle information, which may suffer from signal interference, device heterogeneity, and other factors, and the differences in communication capabilities between CHVs and CAVs are not incorporated into the model, which may lead to deviations between simulation results and actual scenarios. It is a point that needs to be further studied in the future.

3. Future research could also explore the relationship between different queue length formation times and intersection guidance zone length with CAV penetration rate, improve the algorithm’s solution performance, and investigate applications in real-world scenarios.

Acknowledgements

The authors would like to thank the editors and the anonymous referees for their valuable time and efforts in handling and reviewing this paper.

Author contributions

Conceptualization, B.R. and Y.R.; methodology, J.J. and Y.R.; software, J.J.; validation, Y.R. and B.R.; formal analysis, J.J., and Y.R.; resources, J.J. and Y.R.; writing—original draft preparation, J.J.; writing—review and editing, J.J. and Y.R.; supervision, Y.R., and B.R.; project administration, Y.R.; funding acquisition, Y.R. All authors have read and agreed to the published version of the manuscript.

Funding

This research was supported by the National Natural Science Foundation of China (Grant No.41971342).

Data availability

The dataset used in this research is partially available upon request by emailing the corresponding author.

Declarations

Competing interests

The authors declare no competing interests.

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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