Autonomous-Vehicle Intersection Control Method Based on an Interlocking Block

Abstract

Non-signalized intersections have only ever been suitable for low traffic flow; however, with the development of autonomous driving technology and new control methods, the operation efficiency of this kind of intersection may be improved. In view of the shortcomings of existing non-signalized intersection control methods in multilane situations and inspired by railway trains, an interlocking-block intersection control model is proposed. In this study, vehicles between parallel lanes are combined into a few combos, and the combo shape can be determined according to a pairing model and the interlocking angle range, and the gaps between the front and rear vehicles are simulated as blocks in a railway system, which are added into the intersection control model as virtual blocked cars (VBCs) for optimization. In setting the optimization objectives, the connotation and realization of fairness are discussed. Experimental results show that compared with signalized intersections, roundabouts, and non-signalized intersections without control, the interlocking-block intersection control model greatly reduces vehicle delay. Compared with an existing model, the calculation speed in a multilane situation has been greatly improved, while the vehicle delay is similar.

1. Introduction

On urban roads, intersections are the main cause of congestion. In addition, vehicle safety accidents occur frequently at intersections. A study showed that the average number of accidents per kilometer at intersections is 1.37 times that of other sections [1]. In particular, non-signalized intersections are not suitable for large flows and are more dangerous than signalized intersections. Discovering how to increase vehicle capacity and reduce accident hazards at non-signalized intersections is becoming increasingly important, and autonomous driving provides a possibility. In recent years, autonomous driving technology has made great progress and is expected to provide safer and more efficient means of transportation [2,3]; Vehicle-to-Vehicle (V2V) and Vehicle-to-Infrastructure (V2I) communications can realize cooperative safety (e.g., real-time situational awareness and vulnerable pedestrian protection) [4]. With autonomous driving coming to people’s view, non-signalized intersections are, necessarily, a very likely trend in the future. Therefore, intersection control under autonomous driving is a problem that must be faced [5,6,7].

Recently, many researchers have performed additional research on intersection vehicle traffic control methods in intelligent transportation systems considering an environment of vehicle road cooperation or autonomous driving, including more intelligent signal control [8,9,10,11,12,13,14,15] as well as non-signal centralized and distributed control [16,17,18,19,20,21,22,23,24,25]. However, most of these studies pay little attention to the number of lanes and do not discuss the impact of the number of lanes on intersection control and countermeasures. Clearly, with the increase in the number of lanes, the number of conflict areas will increase faster, and the computational complexity will remarkably increase. Previous studies have not proposed a solution to this problem.

In addition, no matter the method, the object of calculation is almost every vehicle or longitudinal platoon, while the horizontal relationship of vehicles has not been taken into account, especially in centralized coordination algorithms, which take the individual vehicle as the minimum control unit and excessively pursue efficiency yet ignore the impact of passenger feelings and communication quality on safety.

Interlocking is a technology of high importance in railway systems. It is used to describe the restrictive relationship between the signal, turnout, and route, which is often between two pieces of equipment [26]. An interlocking system is a part of railway signal systems and is used to ensure the operational safety of equipment. Another key technology in railway systems is the moving block. Moving-block signaling is an intelligent operation interval control system based on communication and computer technology. By dynamically setting the distance between trains, the operation density and carrying capacity of trains can be improved [27].

Clearly, in road traffic, the vehicle control of autonomous vehicles is similar to that of rail trains, especially at intersections. The premise for a vehicle passing safely through an intersection is that other vehicles will not conflict with it during each crossing, similar to 1-to-n interlocking. Meanwhile, for multilane intersections, a special interlocking relationship can be built between vehicles in different lanes to maintain a certain relative position relationship between vehicles and facilitate the passage of vehicles from other directions. The interval in a moving block can correspond to the gap between vehicles. If each vehicle in an intersection can complete the crossing through the block created by other vehicles, the intersection is conflict-free. In other words, we can achieve a similar conflict-free interlocking effect as long as we establish a 1-to-1 corresponding relationship between blocks and passing vehicles.

Based on the ideas above, this paper realizes the control of autonomous vehicles at intersections by constructing the interlocking relationship between vehicles in different lanes and the corresponding relationship between blocks and real vehicles.

The main contributions of this paper can be summarized as follows: (i) The concept of vehicle interlocking is proposed, which forms a horizontal interlocking for vehicles with similar spatiotemporal characteristics between parallel lanes, transforms the multilane problem into a single-lane problem, and provides a solid foundation for creating vehicle crossing space. This process can be completed in a distributed manner, greatly reducing the number of variables and constraints that need to be input to the centralized control, so the difficulty of intersection centralized control is reduced. A matching method for interlocking vehicles and the adjustment range of the combined form are also given in detail. (ii) According to the concepts of vehicle interlocking and railway moving blocks, a virtual blocked car is proposed as a direct research object so that the problem of vehicles avoiding each other in an intersection can be transformed into a finite 1-to-1 encounter problem to realize the interlocking between vehicles and reduce the complexity of the model. Through the calculation of the intersection interlocking block model, the interval to be maintained is obtained, which ensures that all vehicles can pass through the intersection safely and efficiently. (iii) The right of way is embedded into the control model. In this paper, the priority of vehicles is no longer calibrated as a series of simple weights but based on an interlocking block, that is, the number of vehicles passing through the gap in front of each vehicle, which provides a certain theoretical basis for the fairness of the algorithm.

2. Literature Review

In terms of vehicle cooperative control at non-signalized intersections, many basic studies have been performed. The control methods used at intersections can be divided into two categories: distributed control and centralized control. The methods used in these studies have shown better control effects. For distributed control, Raravi et al. proposed an intelligent vehicle-merging algorithm in a cooperative vehicle infrastructure environment. The optimal vehicle merging maneuver is obtained by solving an optimization problem based on vehicle kinematics to minimize the maximum travel time of each vehicle from two conflicting paths to the intersection [16]. Andreas et al. solved the problem of connection through urban intersections and optimal control of CAVs without any clear traffic signals to minimize energy consumption while meeting the requirements of throughput maximization [17]. Compared with distributed control, optimal solutions are more likely under centralized control, so many scholars pay attention to this field. Dresner and Stone developed automatic intersection management (AIM), which uses an intersection reservation system based on discrete units. Vehicles send spatiotemporal reservations to request the controller to arrange their path. Vehicles with licenses can leave, while other vehicles slow down and stop to reserve again [18]. The subsequent research results show that AIM greatly improves the traffic conditions of isolated intersections [19,20]. Rakha et al. developed a cooperative vehicle intersection control (CVIC). The intersection controller adjusts vehicle trajectories and solves a nonlinear constrained optimization problem [21]. This method is said to greatly reduce the total stopped time, but the calculation result is not necessarily the minimum intersection delay. Similarly, Lee et al.’s goal was to minimize the total length of possible overlapping trajectories [22], but it stipulates that to ensure collision avoidance, only one conflicting vehicle is allowed to enter the intersection, so it is not applicable to large multilane intersections.

For the conflict distribution of traffic flow at a two-way single-lane intersection, Ghaffarian et al. [23] designed an intersection coordination control system without traffic lights. In the research, the vehicle flow conflict resolution was modelled as an integer programming problem. The simulation results show that if the weight is appropriate, the effect of this system is better than traditional single-point signal control. A new study allocates vehicle reservations through AIM [24] and proposes mixed-integer linear programming to optimize vehicle reservation under a selection target. The results show that the optimal reservation allocation is feasible and significantly reduces the vehicle delay. To prove the effectiveness of the algorithm, a reservation-based intersection control algorithm was implemented for hurricane evacuation in a connected and autonomous vehicle environment [25].

Another method used in autonomous intersection management is auction-based scheduling [28,29], in which the right of way is determined by bidding. Based on the intersection of the auction, only the request of the auction winner is accepted, and all other requests are rejected. Each agent should participate in the auction repeatedly until one is won. However, the problem is that vehicles with high time values may be trapped behind vehicles with low time values, which is not reasonable in terms of time value [30].

Interlocking is a kind of relationship. The implementation mode in railway systems experiences three stages: mechanical interlocking, relay interlocking, and computer interlocking [26]. Currently, computer interlocking is being increasingly used in rail transit. These advanced interlocking technologies have improved network information security and interface standardization [31,32].

In terms of moving blocks, Lee and others in Korea took the lead in making a more detailed study on moving blocks. They simulated two typical highspeed train sets in Korea, tgv-k and khst-20, of which the former adopts a moving block. They simulated a section of a highspeed rail line using the static data of a simplified line. The simulation results show that with the addition of a moving block, the line capacity increases by approximately 60% [33]. Dick et al. studied the effect of a moving-block signal control system on different train heterogeneities and numbers of tracks [34]. The study calculated the train delay in North American railways. The experimental results show that the moving-block system always shows a lower average train delay than a three-phase or four-phase block signal system. The conclusion is that a moving block can increase the capacity of a single-track corridor by several trains per day, which can effectively replace the construction of an additional second main track infrastructure in the short term.

Compared to methods represented by CPIC, we no longer directly consider each single vehicle as the control object and calculation input of the main model but propose interlocking. Interlocking can be seen as a horizontal formation, rather than generating traditional longitudinal vehicle platoons. Through interlocking, the calculation cost can be significantly reduced under multilane conditions, and the proposed virtual blocked car also reduces the model complexity, which allows the model to handle control problems with more vehicles more efficiently in a single calculation, while also making it more convenient to control the passage of vehicles in each gap.

3. Interlocking Vehicles

3.1. Multilane Intersection Vehicle Traffic Problem

There are two models in many existing studies for the analysis of intersection conflict. One is the point conflict mode, that is, ignoring the geometric parameters, only calculating whether the two vehicles reach a point at the same time and considering that the scope of the conflict is only one point. The other is the regional conflict mode; for example, Dresner and other researchers used a grid method or similarly extended a certain safety boundary around a vehicle. The way to avoid conflict is to ensure the two boundaries do not intersect.

In summary, most research objects, whether the former or the latter, only consider two vehicles. They believe that the conflict in an intersection can be understood as the relationship between two vehicles, which is resolvable. However, if only the conflict between two vehicles is considered in isolation, the efficiency seems to be slightly low because every two vehicles require a calculation judgment. This disadvantage is more common in distributed control. It is well-known that the main roads in cities are often multilane, which is more common in large cities. Therefore, when vehicles from a certain lane cross an intersection, there will be four or more potential conflict points. If we only consider the relationship between two vehicles from the beginning of modelling, we need to perform at least four calculations for each vehicle that may conflict with other vehicles. Although the operation may be simple, it consumes more resources.

• Demand and historical data. The outputs are the subarea division results and the set-point. The fast-update layer is responsible for sampling the traffic state of each link at time-step k and predicting the state of the next step, both of which will be used to build the final optimization function. Based on the output of other layers, an optimization function will be constructed and further solved at the control layer to obtain the optimal control law of the next step.

From another point of view, the problem caused by multiple lanes is not only the increase in conflict points; the gap that a vehicle needs to cross is created by many other vehicles. If we can create more and better gaps, it can be expected that the crossing efficiency of vehicles will be improved; however, for multilane entry lanes, due to the irregular arrival distribution of vehicles in each lane, they show irregular dispersion. The insertion gap they provide to nonparallel traffic flows is no longer the gap between the front and rear vehicles in a single lane but the intersection of multilane gaps, which will lead to different lengths and random positions of gaps, and many gaps are divided into small and unmeaningful gaps because of the intersection operation. Clearly, this is not conducive to vehicles crossing each other, and previous intersection control research has not focused on this. In brief, the methods proposed by existing studies do not focus on and solve the impact of multilane entrance on the alternate crossing of vehicles at an intersection but evaluate the space–time position relationship of two vehicles and do not consider how to increase the alternating space of vehicles from the principle and root. To solve this problem, this paper proposes some definitions to analyze how to address the adverse effects of multiple lanes.

3.2. Equivalent Gap and Effective Gap

First, we define the concept of an equivalent gap. The equivalent gap is specifically aimed at multiple lanes. The equivalent gap refers to the gap of the whole entrance lane obtained by calculating the intersection of the gaps between vehicles in each lane for the same entrance. Second, we define the concept of an effective gap. As the name suggests, if a gap can allow at least one vehicle in other directions to pass safely, the gap is effective. Therefore, based on the above idea, we actually want to find an effective equivalent gap to let a vehicle pass, and the first step is to create a larger equivalent gap.

However, in a multilane situation, creating effective clearance is more complex than for a single lane, especially when the traffic flow is large. Figure 1 shows the arrival of vehicles on two kinds of entrance lanes. Figure 1a shows a single lane, and Figure 1b shows multiple lanes, both of which contain the same lane 2 with vehicle distribution. The upper side of the two figures shows the gap distribution in the section area. The green rectangular segment represents the equivalent gap, while yellow indicates that the space is occupied. The lower part of the figure shows the position–time diagram of vehicles passing through the intersection at a uniform speed, and the width of the slash represents the time lag of the vehicle length. Compared with Figure 1a, the equivalent gap in Figure 1b is greatly compressed and segmented, which is caused by the irregular distribution of multilane vehicles. Therefore, if we want to increase the size and number of gaps, the distribution of multilane vehicles must be standardized.

3.3. Vehicle Interlocking

It can be seen from the analysis of Figure 1 that for the multilane problem, when vehicles randomly enter entry lanes, their equivalent gap is greatly compressed due to the disordered transverse and longitudinal positions, and this problem is simple for a single lane, as shown in Figure 1. Therefore, we introduce the interlock concept.

An interlock in electrical control refers to the mutual restriction relationship between two control lines. For example, when one starts, the other must be closed. Interlocking in railways refers to the establishment of a mutual restriction relationship between the annunciator, turnout, and other equipment to ensure the safety of train operation and shunting operation in railway stations.

Inspired by this, we select some vehicles from multiple lanes to establish such synchronous restriction relationships and control their relative position to enhance the regularity of vehicle distribution in the road section and weaken the reduction effect of equivalent gaps caused by multiple lanes.

3.3.1. Benchmark Methods and Metrics

Similar to an interlock in electrical engineering and railways, the main problem lies in the interlocking object and mechanism. First, it is necessary to determine which vehicles need to be interlocked together. In some studies on adaptive cruise and fleet control, researchers only consider the longitudinal spacing between vehicles. When applied to intersection control, they mainly enter the form of longitudinal fleets using default vehicles. However, as mentioned above, for the traffic problem of multilane intersections (hereinafter referred to as MLIP), the transverse position relationship of parallel-lane vehicles is very important. Therefore, the object of interlocking in this paper is not longitudinal vehicles but vehicles between parallel lanes. The relative position of the transverse vehicles is fixed through interlocking. Here, the vehicles in the same interlocking are regarded as a whole, which is called a combo. Clearly, after interlocking is realized, multiple interlocks can also show a certain longitudinal shape; this content will be discussed in Section 3.3.3.

For an entrance road with the number of lanes n, the range of the number of vehicles CP of combo P is [1, n]. Considering the purpose of increasing the equivalent gap, the more vehicles an interlocking body contains, the better, thus reducing the number of joint ventures. Therefore, the ideal situation is that the number of vehicles in all the combos is n, but if considering an actual road traffic situation, there are two problems: 1. the number of vehicles to be interlocked in each lane is not necessarily equal, and 2. the distribution of vehicles is not necessarily uniform. When the number of vehicles in each lane is not equal, it is obviously impossible to realize that the number of vehicles in each joint venture is n while the distribution of vehicles is uneven, especially if the difference in vehicle arrivals between parallel lanes is large, and the distance between some vehicles is too far. If the vehicles are set as a joint venture without economic benefits, on the one hand, it is difficult to adjust the vehicles; on the other hand, if there is a large equivalent gap adjustment, it will not be significant.

Therefore, we need to determine the maximum longitudinal distance between interlocking vehicles, which can directly use the number of lanes at the corresponding intersection and the minimum effective gap under vehicle conditions (referred to as the effective gap). After determining the maximum longitudinal distance between interlocking vehicles, the data can be cleaned. If the longitudinal distance between a vehicle on the entrance road and any vehicle in the other parallel lanes is greater than this value, the vehicle will not be interlocked and matched, and it is removed as the departure point. The longitudinal distance is the forward distance of the vehicle driving direction, and the opposite is the negative distance.

As shown in Figure 2: (1) When the distance between two vehicles is greater than the effective gap, no interlocking relationship is established between them. Therefore, the combos AB and BE in Figure 2 do not exist. (2) When the distance is less than the effective gap, interlocking can be established theoretically, but it is easy to find that for vehicle D, combo DC or combo DE can be selected. If combo DC is selected, vehicle B cannot be interlocked. Therefore, we need to establish certain pairing criteria to solve the problem of mutual competition in vehicle pairing.

3.3.2. Selection of Interlocked Vehicles

• Objects. The core of the interlocking problem is to determine which vehicles need to be connected together. As shown in Figure 3, the traditional method [35] is to pair objects based on certain consensus principles or independent principles on both sides. This type of matching is only based on preference selection, such as cost, age, and other information, without strict restrictions on the order. To solve this problem, we propose an interlocking pairing.
• Model. Considering that the computer representation of some multidimensional data constraints is complex, we can reduce the dimension of multilane pairing into a small number of two-lane pairing problems. The basis for this is that the maximum longitudinal distance between interlocking vehicles is certain, so the interlocking classification obtained is meaningful.

Hence, a concise model can be established for the solution. Given a two-lane scenario, $v1$ and $v2$ correspond to the vehicles in lanes $L1$ and $L2$. Variable x is a binary variable that indicates whether the two vehicles from the two lanes are paired successfully.

Since we want to interlock vehicles closer to each other to reduce the difficulty of adjustment, the objective function is:

$$\min {\displaystyle {\displaystyle \sum _{v1=1}^{L1}}{\displaystyle \sum _{v2=1}^{L2}{x}_{v1,v2}}}\ast d_{v1,v2}$$

(1)

where $d_{v1,v2}$ is the distance parallel to the lane direction between two car heads, and the minimum value is 0.

Constraint 1: Each vehicle in lane1 must only be allocated in one combo and cannot be arranged in two interlocks at the same time.

$${\displaystyle \sum _{v1=1}^{L1}{x}_{v1,v2}=1}$$

(2)

Constraint 2: Each vehicle in lane2 must only be allocated in one combo, as well. For the case of equal quantity, match directly in order. However, the number of vehicles to be paired in the two lanes may not be equal, which would make the equality constraint meaningless. Therefore, assuming the number of vehicles ${\mathrm{N}}_{L1}$ is less than ${\mathrm{N}}_{L2}$, the following constraints can be written as:

$${\displaystyle \sum _{v2=1}^{L2}{x}_{v1,v2}\le 1}$$

(3)

Constraint 3: The maximum interlocking distance is less than or equal to the effective gap length.

$$d_{v1,v2}\le d_{valid}$$

(4)

Constraint 4: $x$ is a 0−1 variable, indicating whether the two vehicles are paired.

$$x_{v1,v2}=\left\{\begin{array}{l}0,v_1\mathrm{and}{v}_2\mathrm{are}\mathrm{not}\mathrm{paired}\hfill \\ 1,v_1\mathrm{and}{v}_2\mathrm{are}\mathrm{paired}\hfill \end{array}\right.$$

(5)

In summary, the pairing model can be written as:

$$\begin{array}{l}\min {\displaystyle \sum _{v1=1}^{L1}{\displaystyle \sum _{v2=1}^{L2}{x}_{v1,v2}}}\ast d_{v1,v2}\hfill \\ s.t.\left\{\begin{array}{l}{\displaystyle \sum _{v1=1}^{L1}{x}_{v1,v2}=1}\hfill \\ {\displaystyle \sum _{v2=1}^{L2}{x}_{v1,v2}\le 1}\hfill \\ d_{v1,v2}\le d_{valid}\hfill \\ x_{v1,v2}\in \left\{0,1\right\}\hfill \end{array}\right.\hfill \end{array}$$

(6)

Assuming that the number of vehicles ${\mathrm{N}}_{L1}$ is less than ${\mathrm{N}}_{L2}$, if ${\mathrm{N}}_{L1}$ = ${\mathrm{N}}_{L2}$, they are paired according to the sequence of vehicles and the effectiveness is verified according to Formula (4).

However, for such pairing problems, the same solution may occur. As shown in Figure 4, A and B are two-lane vehicle queues, and EF and GH are vehicles in the two lanes. According to the different longitudinal position relationships of the vehicles in the two lanes, they can be divided into three categories.

The first type is contained, that is, EF is within the coverage of GH.

The third type is localized, and the coverage of EF and GH does not intersect.

In the three types, the vertical lines are made from E and F to the straight line where GH is located; the vertical feet are E’ and H’; and the total longitudinal distance required for pairing is calculated and analyzed below. Two cases are calculated in each type, case 1: if E and G are paired and F is paired with H; case 2: if E and H are paired, F and G are paired.

In the first type, the total distance of case 1 is E’G + F’H; the total distance of case 2 is E’H + F’G = GH + E’G. The former is less than the latter. In the second type, the total distance of case 1 is E’G + F’H; the total distance of case 2 is E’H + F’G = 2E’H + E’G + F’H. The former is less than the latter.

In the third type, the total distance of case 1 is E’G + F’H; the total distance of case 2 is E’H + F’G = E’G + F’H. The former is equal to the latter.

In short, in any case, the distance between the two vehicles is less than or equal to the cross-pairing. Therefore, when the cross situation occurs, the problem can be solved by exchanging cross vehicles so that the pairing scheme can be easily realized with lower difficulty than applying constraints. The process can be expressed as

$$\begin{array}{l}If{x}_{i,j}=1,x_{p,q}=1,(i-j\left)\right(p-q)<0\hfill \\ Then{x}_{i,j}=0,x_{p,q}=0;x_{i,q}=1,x_{p,j}=1\hfill \end{array}$$

3.3.3. Interlocking Angle

After determining the interlocking relationship between vehicles, it is necessary to determine the shape of the vehicle interlocking. If the shape is not limited, the closed figure covered by the combination can be convex or concave, that is, the shape of the combination can be arbitrary, and some are meaningless for traffic efficiency at the intersection. An overly broad adjustment range can affect the efficiency of vehicles passing through the intersection, so the shape needs to be specified.

In particular, Figure 5 shows that the vehicles in the two intersection directions pass through the intersection alternately. At this point, vehicles from the north and south have priority. Figure 5 shows the relative position state in which the three vehicles traveling from west to east can enter at the earliest, which is called the optimal shape (OS). If the three vehicles forming the combo are regarded as particles, their connecting line is not a line perpendicular to the driving direction but a slash. In the two cases, the angle between the combo and the horizontal line is different. The angle of the slash is related to the driving direction, which can be positive or negative. We call the angle of these two directions the interlocking angle.

As shown in Figure 6, based on a certain lane and driving direction, the interlocking angle between two lanes can be positive or negative, and the angle here corresponds to the angle of the OS. Through more specific research, a meaningful interlocking angle range must exist. When the absolute value of the angle is less than that of the OS angle, it means a waste of space and time. Because the entry time of the combo is delayed, it can be specified that the range of the interlocking angle is the yellow triangle in Figure 6.

Under certain road speed limit conditions, if a vehicle passes through an intersection at a fast speed and the vehicle length is more than 4 m, the shape of the combo cannot be significantly changed in the process of passing through the intersection, so we can keep the shape of the combo unchanged in the process of passing through the intersection. Furthermore, as shown in Figure 5, a combo may encounter two vertical vehicle crossings in the process of passing through the intersection. For a combo shape, either the minimum positive interlocking angle or the minimum negative interlocking angle in Figure 6 may not reduce the delay at the same time, that is, the minimum interlocking angle shown in Figure 6 is only one-way optimal; therefore, according to the different arrival times of vehicles from the north and south directions, a flexible angle setting may achieve more efficient traffic, which is the meaning of the interlocking angle in the triangular area in Figure 6.

4. Intersection Conflict Resolution Based on a Moving Block

4.1. Moving Block and Virtual Blocked Car

In railways, an important concept is a moving block. A moving block refers to a communication-based block mode in which a train automatically sets the running speed according to the distance and route conditions between the subsequent train and the previous train. In rail transit, the distance between each train is not fixed, and the actual operation position and state of a train affect the minimum operation interval. Therefore, the block partition and its length need to be constantly adjusted to adapt to different operation states, which is the core idea of a moving block.

Compared with railways, vehicles in road traffic also have similar characteristics. When driving manually, a driver will estimate the speed between their own side and the front to judge the distance to be maintained. This distance is based on the driver’s rational and perceptual knowledge. In rail transit, the concept of blocking first appeared because of the possible beyond-visual-range problem. With the trial operation of autonomous driving in many countries and cities, when vehicle operation no longer depends on or mainly depends on driver visual perception and cognitive judgment, a similar blocking mechanism can be applied to vehicles operating on roads to ensure a higher degree of transportation safety, especially for vehicles operating in areas with a high possibility of conflict, such as intersections.

Given the above, as shown in Figure 7, this paper puts forward a moving blocking mechanism for intersections; that is, according to the traffic conditions of the intersection, the subsequent vehicles in the same lane need to maintain a certain blocking interval with the preceding vehicles so that all conflicts can be resolved in the blocking interval. Furthermore, the same is true for the interlocked combo; that is, the block section between the front and rear combos needs to ensure the safe passage of crossing vehicles. To achieve this, we need to take the blocking interval as a research object. For convenience, we can make the transformation shown in Figure 8 The vehicles in a road section are taken as gaps, and the real gaps between vehicles are converted to virtual blocked cars (VBCs) with variable lengths.

4.2. Interlocking Block Conflict Resolution Model

4.2.1. Notation

For a multilane intersection with four entrances, vehicles in each direction only go straight through the intersection, and the vehicles complete interlocking on the road section to form a combo to enter the intersection. Before entering the intersection, they reach the speed limit $tarv$, and the vehicles maintain this speed through the intersection. Among the four entrances, N S and S N are a group, W E and E W are a group and one group is converted to the form of a virtual blocked car (Section 4.1). The VBCs are marked as $\check{𝚤}$, and the RCs 410 that do not change are $\widetilde{𝚤}$.

4.2.2. Longitudinal Position Relationship of Vehicles

Regardless of which entrance is concerned, it is a necessary condition that the rear vehicle cannot collide with the front vehicle.

$${\tau }_{\tilde{i}}^j\le {\tau }_{\tilde{i}}^{j+1}-\frac{l_{\tilde{i}}^j}{tar{v}_{\tilde{i}}^j}$$

(7)

where ${\tau }_{\tilde{i}}^j$ is the time when real car (RC) $j$ with distance $l_{\tilde{i}}^j$ from entrance $\tilde{i}$ arrives at the intersection stop line. In particular, according to the definition in Section 4.1, the distance between two adjacent VBCs is a fixed value: the length of the RC between them.

4.2.3. Encounter between RCs and VBCs

If an RC and VBC meet at an intersection and the total time of meeting is enough for the RC to cross the conflict area, it means that conflict is avoided. Therefore, it is necessary to construct a variable to describe the encounter between the two types of vehicles. The first question is which two cars meet, and we establish a variable ALLOC to describe it.

$$ALLO{C}_{\left(\widehat{i},j\right)\left(\tilde{i},j^{\prime }\right)}=\left\{\begin{array}{l}0,\mathrm{the}\mathrm{two}\mathrm{cars}\mathrm{didn}’\mathrm{t}\mathrm{meet}\hfill \\ 1,{\mathrm{car}}_{\widehat{\mathrm{i}}}^{\mathrm{j}}\mathrm{met}{\mathrm{car}}_{\mathrm{i}˜}^{\mathrm{j}\prime }\hfill \end{array}\right.$$

(8)

Each VBC and RC have this variable to describe whether they met. The total number of ALLOC variables is constant and can be expressed as: ${\displaystyle \sum {\displaystyle \sum ca{r}_{\widehat{i}}^j}}\cdot {\displaystyle \sum {\displaystyle \sum ca{r}_{\tilde{i}}^{j^{\prime }}}}$.

It should be noted that a VBC can meet multiple RCs. Theoretically, the number can reach the maximum number of RCs, which means that multiple vehicles have passed through a blocking gap continuously. In contrast, an RC can only meet a limited number of VBCs.

$${\displaystyle \sum _j^{c_{\widehat{i}}}ALLO{C}_{\left(\widehat{I},j\right),\left(\tilde{i},j^{\prime }\right)}}=1$$

(9)

This formula ensures that for vehicles from a certain direction, one RC meets only one VBC in the intersection range. When a VBC and RC meet, they are described as follows:

$${\tau }_{\widehat{i}}^j+\Delta {\tau }_{\widehat{i}}^j-{\tau }_{\tilde{i}}^j\le 0$$

(10)

$$\left({\tau }_{\widehat{i}}^j+\Delta {\tau }_{\widehat{i}}^j+{\delta }_{\widehat{i}}^j\right)-({\tau }_{\tilde{i}}^j+len+\Delta {\tau }_{\tilde{i}}^j)\ge 0$$

(11)

where $\Delta \tau$ is the time required to pass through the conflict zone, $len$ is the length of RC, and ${\delta }_{\widehat{i}}^j$ is the length of VBC.

4.2.4. Optimization Objectives

Here, we describe the optimization objectives in detail. In previous studies, the time and delay through an intersection have often been taken as optimization objectives. In this study, we only consider the time from the initial time to the time allowed to enter the intersection and sum the values of all vehicles. The smaller the total time is, the smaller the delay of vehicles.

In addition, previous studies did not pay attention to fairness. In a traditional fixed-timing signalized intersection, due to the differences in vehicle arrival at each entrance lane, there will sometimes be a waste of green-light time and a long queue at individual entrance lanes. For unsignalized intersections, although vehicles can cross each other more frequently, it is still a problem of which number of vehicles go first. First come first served (FCFS) has proven that its efficiency is not optimal, but it is actually more persuasive in fairness. Therefore, we hope to find a way to make the model achieve a more convincing degree of fairness without relying on FCFS. In the following, we propose three ways to set optimization objectives, all of which are expected to improve the fairness of traffic at unsignalized intersections:

• Average time cost. Since the number of vehicles at each entrance may be different, simply summing all vehicles does not reflect the average fairness of different entrance lanes because entrances with more vehicles are easily assigned a higher priority, but this may lead to greater sacrifices for other vehicles with higher initial order. Therefore, if the vehicle passage time at each entrance is averaged according to the number of vehicles, the effect of vehicle alternating passage can be achieved while more consideration is given to the initial order.
• Road grade. Road grade is obviously also an important fixed attribute. The main road has greater vehicle relief capacity. When the main road crosses a branch road, the natural main road has a certain priority. Therefore, taking the traffic time of vehicles on the main road as the optimization goal also reflects fairness.
• Alternation of rules. The number of vehicles input into each model must be limited because the vehicle information obtained is limited, so the next round of updating must be carried out after each calculation. Therefore, if the optimization objective is adjusted once for each update, any two adjacent calculations reflect different priority criteria, which reduces the bias of the overall model to a certain extent.

  $$R_{\mathrm{n}+1}=Select\left(R_{\mathrm{n}}\right)$$

  (12)

  where $R_{\mathrm{n}}$ is the optimization objective used in round n, and $Select$ represents the replacement of another optimization objective.

5. Experimental Results

5.1. Comparison of Model Parameters

Theoretically, the CPIC model is expected to achieve the optimal traffic order of all vehicles under certain set conditions, but it may not be optimal in terms of computational complexity. First, we compared the fairest case. Under the condition of a single lane, we compared our interlocking-block conflict resolution (IBCR) model and the CPIC model because there is no interlocking under a single lane, and the number of vehicles included in the calculation is not reduced due to this process. We calculate the number of basic variables and constraints that need to be introduced into the two mathematical models under different numbers of vehicles on the four entrance lanes, in which the number of variables does not include the artificial variables of the subsequent simplex method. As seen in Figure 9, when the number of vehicles is small, the variables and constraints of the IBCR model are slightly greater, and the magnitude of the difference between the two models does not exceed 10. As the number of calculated vehicles increases, the number of variables and constraints of the CPIC model quickly exceeds that of the IBCR model, and the gap becomes increasingly larger. When the number of vehicles reaches more than 30, the number of constraints of the CPIC model exceeds that of the IBCR model by approximately 500, and the difference in the number of variables reaches 200, which is an irreversible trend. This means that with the increase in the number of calculated vehicles, the IBCR model has greater advantages in computing compression, even if our application scenario is a single lane.

After discussing the situation of a single lane, multiple lanes are obviously the focus of this paper. According to the idea of control variables, the conditions we set each time are that the entrance lanes contain one, two, three, and four vehicles. These vehicles are generated according to the same Poisson distribution parameters. The constraint quantities of the two models under the conditions of a single lane, two lanes, three lanes, and four lanes are calculated, and the relationship between the IBCR and CPIC models is described in the form of a ratio. The reason why the number of variables is not used here is that according to Figure 9, the change trends of the two dependent variables are very similar. It can be seen in Figure 10 that when the number of lanes is fixed, the number of vehicles increases, and the ratio increases faster with the increase in the number of lanes.

The number of vehicles corresponding to the maximum value is 64, and the number of lanes is four. At this time, the ratio is close to 18. This result shows that the IBCR model can greatly reduce the calculation of the model, and the greater the number of vehicles, the more obvious the effect. This is because an increase in the number of lanes means an increase in potential conflict areas, and an increase in the number of vehicles means an increase in the number of conflicts. A series of models, such as the CPIC model, do not directly face and solve this problem.

5.2. Comparison of Computation Time and Vehicle Delay

The computation time is an important factor to judge the advantages and disadvantages of a model because it is related to the timeliness of the information of the intersection centralized control center. Although there is more tolerance for the computational complexity and time of a model with the development of computer and communication technology and the further scope of information interaction, shorter is better. In addition, under the same conditions, intersection vehicle delay is also an important index to evaluate the two models.

Figure 11 compare the delay and calculation time of the two models under the conditions of two lanes and three lanes, the same flow, and the same distribution. The number of vehicles refers to the number of vehicles input into a single lane. Under each condition, the average value is calculated using a Monte Carlo method. Due to the random generation of input vehicles, the calculation time and delay of the model are not very stable but also show some characteristics. As shown in Figure 11a, the overall difference between the vehicle delay results calculated by the two models is small. Corresponding to Figure 11b, the calculation time of the CPIC method increases significantly after the number of vehicles in a single lane exceeds four. Setting different heuristic algorithms to find the initial point also does not solve this problem. In Figure 11c,d, after the number of vehicles input is greater than four, the calculation time and delay of the CPIC method are significantly increased, and it is no longer possible to find and converge them to smaller values.

5.3. Right of Way and Vehicle Priority

In most cases, the right of way is relative rather than absolute. Subjectively, it is difficult to define the right of way. For example, how long does a vehicle on a branch road give way to a vehicle on the main road? In previous studies, this priority was directly solved by setting weights. However, the right of way and vehicle priority are not the same concept, and we hope to take these two issues into account in the IBCR model. Therefore, we describe this relative right of way by limiting the block capacity and determine the vehicle priority through the optimization of the model. The block capacity (BC) constraint is:

$${\displaystyle \sum _{j^{\prime }=1}^{c_{\tilde{i}}}ALLO{C}_{\left(\widehat{i},j\right),\left(\tilde{i},j^{\prime }\right)}}\le B{C}_{\left(\widehat{i},j\right)}$$

(13)

This constraint limits the maximum number of vehicles passing through each block in the intersection so that there is no phenomenon of too many vehicles passing through one block continuously in the process of vehicle traffic, which can be understood as a restriction on the relative right of way between vehicles.

The calculation results of some models are recorded in

Table 1. The relationship between BC and delay.

BC	Total Delay of a Single Computation Output/s					Average Delay/s
2	11.57	11.93	17.25	12.13	29.57	16.5
3	4.9	0.55	4.78	4.95	2.01	3.4
2	27.03	27.23	72.33	83.3	56.43	53.3
3	8.74	11.1	14.92	21.14	15.46	14.3

, in which the ratio of the number of vehicles in the first two lines of main roads and branches is 1.2, and in the latter two lines, it is 1.6. It can be seen that in terms of average delay, the delay increases significantly when it is set to two compared with three. Especially when the gap between the number of vehicles on the main road and the number of vehicles on a branch road increases, the average delay increase also increases. Each time different random data are input, the calculated results fluctuate greatly. This shows to a certain extent that when the number of vehicles input into the model is unbalanced at the entrance, especially when the main road crosses a branch road, the capacity should not be set too small to obtain high overall traffic efficiency.

6. Conclusions

An autonomous-vehicle intersection control method based on an interlocking block is proposed in this study to reduce the computational complexity of intersection centralized control under multilane conditions. By analyzing the characteristics of vehicle conflicts at intersections with multilane entrances, the idea of an interlocking block is introduced into the intersection vehicle control method. The interlocking of vehicles transforms the multilane vehicle control problem into a single-lane vehicle control problem by interlocking the vehicles of adjacent lanes, while the constructed virtual blocked cars transform the control problem into a few matching problems. Under the condition of multiple lanes, compared with the CPIC model, the numbers of variables and constraints are significantly reduced, and the calculated vehicle traffic efficiency is close, but the moving-block model can accommodate more vehicles in one round of calculation.

Furthermore, in previous studies, the representation of vehicle priority is usually a series of weights; nevertheless, this method fails easily when the weight 580 difference is small. In addition, drivers generally want to have a certain priority, and the weights are hard to assign. In particular, some privileged vehicles can completely block other vehicles through higher authority, without calculation in an ordinary centralized control model. Therefore, this paper describes the right of way by setting the number of vehicles allowed to pass using virtual blocked cars.

Through the above analysis, we find that there are still some details worth studying [36]. In the future, a vehicle may have the right to negotiate the control strategy with the center, which may generate the need for a more concise model and efficient algorithm. A more reasonable BC value-setting method could also be a meaningful study; this value can be obtained from the relative positions of vehicles in the space–time dimension and can also be set through historical and real-time traffic conditions. Negotiations with the center involve cooperative games, such as the need for a voting mechanism to determine the priority of a group in a certain period of time in a multi-agent situation, especially considering the need for the rapid evacuation of vehicles in some special events; when individuals disagree with the allocation scheme, a quick decision-making mechanism should be established to allow each vehicle to apply for a limited number of adjustments to their right of way in order to balance personalization and fairness.