A signal design problem is studied for efficiently managing autonomous vehicles (AVs) and regular vehicles (RVs) simultaneously in transportation networks. AVs and RVs move on separate lanes and two types of vehicles share the green times at the same intersections. The signal design problem is formulated as a bilevel program. The lower-level model describes a mixed equilibrium where autonomous vehicles follow the Cournot-Nash (CN) principle and RVs follow the user equilibrium (UE) principle. In the upper-level model, signal timings are optimized at signalized intersections to allocate appropriate green times to both autonomous and RVs to minimize system travel cost. The sensitivity analysis based method is used to solve the bilevel optimization model. Various signal control strategies are evaluated through numerical examples and some insightful findings are obtained. It was found that the number of phases at intersections should be reduced for the optimal control of the AVs and RVs in the mixed networks. More importantly, incorporating AVs into the transportation network would improve the system performance due to the value of AV technologies in reducing random delays at intersections. Meanwhile, travelers prefer to choose AVs when the networks turn to be congested.
In the past decade, the automobile industries have made significant technological development by bringing computerization into driving. Such development is constantly accelerated under the circumstance where quite a few companies such as Google, Volvo, and BMW have been advocating autonomous vehicles (AVs) that navigate without direct human operations. Once AVs enter into the market, there is a mixed traffic in the transportation networks. Moreover, the traffic pattern and related management in a transportation system change accordingly when AVs are involved.
With no doubt, AVs could alleviate vehicle crashes dramatically. According to the study of NHTSA, more than 40% of fatal crashes are attributed to human fails, such as alcohol, distraction, and fatigue [
AVs can pass intersections more efficiently by shortening start-up times and headways among vehicles at signal controlled intersections [
In order to keep up with the technical development of AVs and make full use of the related merits, it is critical for the government to design tangible policies and control strategies to adapt the deployment of the technology. In this paper, we focus on designing optimal signal control strategies of AVs.
Despite the infusive technical developments, there is still a long time, perhaps several decades, to go for the AV vehicles to reach a high market share or dominate the full market. In this case, a heterogeneous traffic stream consisting of both RVs and AVs is more reasonable in the foreseeable future. In fact, it is believed that the government can identify critical locations for managing and/or separating AVs from the heterogeneous traffic steam. Meanwhile, in spirit of applying exclusive bus lanes, an exclusive AV lane would be an attractive and effective choice to implement AVs in urban traffic system. For example, Tientrakool et al. reported that the efficiency of the AVs lane is three times the common lane with mixed traffic flows [
So far, much of the research in AV filed from micro perspective focused on analyzing AVs’ on-road vehicle-following behaviors [
The research of AVs from the macro perspective of travel mode split, network design, and parking behavior analysis has attracted an increasing attention in recent years. For example, quite a few studies suggested that a shared taxi fleet composed of AVs may replace the traditional taxi pattern. More importantly, incorporating AVs into the transportation system could alleviate traffic congestion in the networks [
Some researchers studied the traffic improvement methods based on transportation networks, considering the behaviors of drivers or travelers. Network optimization problems for the mixed AV traffic flows include AV lane configuration problem [
In this paper, we investigate the signal design problem in the transportation system consisting of AVs and RVs. The signal design problem in this paper shows some new features from previous studies. Firstly, the routing choice decisions of two types of vehicle in a mixed network give rise to a mixed equilibrium, rather than the conventional UE. Specifically, AV users follow the CN principle in routing decision-making and travelers who drive RVs behave as UE players to seek their respective shortest paths. Secondly, as mentioned above, the travel (random) delay at interactions can be reduced by applying the merits of V2V and V2I techniques. Therefore, it is attractive for the government to design an optimal signal control scheme to make better use of the green times so as to enhance network performance. However, it is a challenging task to formulate the network signal design problem by taking into the mixed routing choice behaviors of AV and RV users. We attempt to propose a general mathematical programming model to help governments design and implement tangible signal control strategies to minimize the total travel cost of a transportation network with mixed traffic flows.
In the network, AVs and RVs in the road links are managed to, respectively, use exclusive lanes (common lane and AV lane). In this case, no interference between two different types of vehicles exists in the links so that the advantage of AV technology would be fully utilized. At the same intersections, AVs and RVs share the green time. The merit of lower random delay in AV lane will attract more travelers to use AVs. The signal design problem is a Stackelberg game between the government (network management authority) and the travelers. The government is the leader of designing signal control schemes. And the travelers who use RVs and AVs behave as followers. Therefore, we can formulate the signal design problem as a bilevel programming model, where in the upper level the government designs optimal signal control schemes to manage the movement of AVs and RVs, and in the lower level a CN-UE mixed equilibrium is given to characterize the AV users’ and RV users’ routing behaviors. Note that the link traffic model used in the paper is a BPR static function, and some recently calibrated link model can be utilized in the future study [
The rest of this paper is organized as follows. Section
In this section, we firstly describe the signal design problem where both AVs and RVs share a transportation network. Then, a bilevel programming model is developed, to characterize the leader-follower behavior of the manager and the travelers. The upper-level model optimizes the total travel time in the network by determining optimal green time ratios of all phases at the signal control intersections. The lower-level problem is the CN-UE mixed equilibrium problem that determines the mixed traffic pattern. To facilitate the model formulations, the following assumptions are made. For each link, the AVs and RVs use their exclusive lanes, which means the traffic streams of AVs and RVs would not interfere with each other. The travel mode split for each commuter is determined by the discrete choice model (logit model). The capacity of AV lane is much larger than of the capacity of RV lane. The performance functions of AV and RV links are strictly increasing, as well as convex functions with respect to link flows. At the intersections, AVs and RVs share the same green times. The total delay at the intersections consists of average delay and random delay, and the AVs technologies can reduce the random delay. Thus, for AVs, the weight of random delay in the total delay is smaller than that for RVs. AVs follow the CN principle to choose the routes while the RVs follow the UE principle.
The lower level of the model is a logit-based traffic equilibrium problem. RVs follow user equilibrium (UE) for traffic assignment, while AVs follow Cournot-Nash equilibrium (CN) for traffic assignment. Logit model is used to split the travel demands of the two modes.
We assume that travel time consists of road travel time and intersection delay time. The travel time of RVs and AVs can be represented by the following BPR functions, respectively:
Functions (
The intersection delay times of RVs and AVs of each phase at the intersection can be expressed as
Delay functions (
Assume that the path set of the RV is
For a mathematical programming problem, any local minimum solution satisfies the first-order conditions. If the first-order conditions of the model satisfy the path and mode choice principles, then the mixed equilibrium holds. We construct the following Lagrangian function:
The first-order conditions are
By function (
Functions (
And the local optimal solution of a convex program is also the global optimal solution. If the objective function is a strictly convex function, there is only one optimal solution of the model. Obviously
The upper-level model introduces the optimal signal control scheme for AVs and RVs, with the aim of minimizing system travel time:
The decisional variables are the green time ratios of all phases at the signal controlled intersections. Meanwhile, different phases have to meet certain requirements at the same signal controlled intersections; namely, the following constraint needs to be added to the upper model:
This problem belongs to the second-best network design problem; the sensitivity analysis based method is used to solve this model. Thus, we must evaluate the changes in equilibrium link flows caused by the changes in the green time ratios. It is difficult to evaluate the changes in equilibrium link flows directly because of the implicit, nonlinear function form of equilibrium link flows. The linear function can be used to approximate the nonlinear function of equilibrium link flows. Relative algorithm was proposed by Yang and Yagar [
For detailed equations of calculating the derivatives, please refer to [
The road network with 14 links and 6 nodes in the numerical examples is provided in Figure
Relevant parameters of the road network.
Link |
|
|
|
|
---|---|---|---|---|
1 | 18 | 2000 | 18 | 2000 |
2 | 18 | 2000 | 18 | 2000 |
3 | 26 | 2000 | 26 | 2000 |
4 | 26 | 2000 | 26 | 2000 |
5 | 21 | 2000 | 21 | 2000 |
6 | 21 | 2000 | 21 | 2000 |
7 | 14 | 2000 | 14 | 2000 |
8 | 14 | 2000 | 14 | 2000 |
9 | 35 | 2000 | — | — |
10 | 35 | 2000 | — | — |
11 | 40 | 2000 | — | — |
12 | 40 | 2000 | — | — |
13 | 25 | 2000 | 25 | 2000 |
14 | 25 | 2000 | 25 | 2000 |
Topology of the transportation network.
Based on the experience and theories of traffic engineering, we make the following assumptions for the traffic organization and signal control at the intersections as follows: Conflict points between RVs and AVs are not allowed in the same phase, because it is difficult to establish the communication between RVs and AVs. So these conflict points should be eliminated by signal control strategies for the sake of safety. The conflicts among AVs are allowed in one phase, but the number should be limited in one signal phase, because the V2V technologies could help the AVs to avoid crash if there is enough space. But if too many conflict points exist in the same phase, the space is not insufficient for AVs to prevent crashes. So we assume, in the numerical examples, that there are at most four conflict points in the same phase.
For intersection 4, we propose only one control strategy. Illustration of green time ratios of different movements is shown in Figure
Illustration of green time ratios in intersection 4.
A five-phase control strategy is used at intersection 4; in the strategy, 4 phases are utilized by RVs; one phase is utilized by AVs. The illustration is shown in Figure
Five-phase signal control strategy in intersection 4.
For intersection 2, illustration of green time ratios of different directions is shown in Figure
Illustration of green time ratios in intersection 2.
And then we propose four signal control strategies for intersection 2.
Illustration of control strategy 1 in intersection 2.
And the relationship among the green time ratios in the strategy can be represented by the following linear functions:
Illustration of control strategy 2 in intersection 2.
And the relationship among the green time ratios in the strategy can be represented by the following linear functions:
Illustration of control strategy 3 in intersection 2.
Illustration of control strategy 4 in intersection 2.
The relationship among the green time ratios in the strategy can be represented by the following linear functions:
The first example is simulated to examine the performance of signal control strategies. In the example, there are 3000 travelers for each OD pair. For AVs, we set the weight of the random delay in the total delay
Optimal green time ratios under different strategies.
Strategy 1 | Strategy 2 | Strategy 3 | Strategy 4 | ||
---|---|---|---|---|---|
Optimal green time ratios of intersection 4 | Phase 1 | 0.288 | 0.265 | 0.276 | 0.225 |
Phase 2 | 0.212 | 0.158 | 0.208 | 0.156 | |
Phase 3 | 0.225 | 0.164 | 0.254 | 0.278 | |
Phase 4 | 0.202 | 0.285 | 0.162 | 0.228 | |
Phase 5 | 0.073 | 0.128 | 0.1 | 0.113 | |
|
|||||
Optimal green time ratios of intersection 2 | Phase 1 | 0.252 | 0.229 | 0.201 | 0.189 |
Phase 2 | 0.164 | 0.182 | 0.223 | 0.301 | |
Phase 3 | 0.284 | 0.256 | 0.189 | 0.156 | |
Phase 4 | 0.192 | 0.165 | 0.172 | 0.123 | |
Phase 5 | 0.108 | 0.168 | 0.119 | 0.165 | |
Phase 6 | — | — | 0.096 | 0.066 | |
|
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Total system cost | 2422422 | 2508682 | 2897253 | 2995322 |
The second example is proposed to examine the sensitivity of the parameter
Relation between
The third example is proposed to exam the sensitivity of the parameter
Relation between
The fourth example is proposed to examine the effect of travel demand on the total travel time in the four strategies. In the case of
Total travel time and travel ratio of AVs under different travel demand.
Travel demand | 500 | 1000 | 1500 | 2000 | 2500 | 3000 | 3500 | 4000 | |
---|---|---|---|---|---|---|---|---|---|
Strategy 1 | Travel time | 242242 | 581380.8 | 944743.8 | 1356555 | 1816815 | 2422422 | 3391388 | 5038634 |
Travel ratio of AVs | 0.58 | 0.62 | 0.66 | 0.7 | 0.74 | 0.82 | 0.8 | 0.78 | |
Strategy 2 | Travel time | 250868 | 602083.2 | 978385.2 | 1404861 | 1881510 | 2508682 | 3512152 | 5218054 |
Travel ratio of AVs | 0.58 | 0.62 | 0.65 | 0.71 | 0.74 | 0.82 | 0.82 | 0.76 | |
Strategy 3 | Travel time | 289725 | 695340 | 1129928 | 1622460 | 2172938 | 2897253 | 4056150 | 6026280 |
Travel ratio of AVs | 0.47 | 0.51 | 0.56 | 0.6 | 0.63 | 0.75 | 0.83 | 0.78 | |
Strategy 4 | Travel time | 299532 | 718876.8 | 1168175 | 1677379 | 2246490 | 2995322 | 4193448 | 6230266 |
Travel ratio of AVs | 0.48 | 0.52 | 0.56 | 0.6 | 0.63 | 0.75 | 0.83 | 0.78 |
In order to keep up with the technical development of AVs and make full use advantages of AVs technology, it is critical for government to adopt suitable policies and control strategies to adapt the deployment of the technology. In the paper, we investigate the optimal signal control of AVs in the road network. In the network, AVs and RVs in the road link use exclusive lanes, respectively. So they do not interfere with each other in the links. But at intersections AVs and RVs share the green times. Since information and communication technologies are beneficial to reducing the random delay of AVs, it is important to design a reasonable signal system to improve the efficiency of AV movements.
A bilevel model is proposed to describe the problem of signal design for mixed networks with both AVs and RVs. The lower-level model is described as a mixed equilibrium of both types of vehicles. The AV vehicles follow the CN principle for route choice, and the RVs follow the UE principle. In the upper-level model, system managers use optimal signal control schemes to allocate appropriate green times to AVs and RVs to optimize the performance of the network. The sensitivity analysis based method is used to solve the model. Numerical examples are presented based on a road network with 14 links and 6 nodes. Four numerical examples are used to demonstrate the optimal green time ratios under different signal control strategies. Some observations have been obtained from the examples. First, for the optimal signal control of the mixed network, the number of phases at the intersections should be reduced, to improve the efficiency. Second, the system cost can be reduced due to the AVs technologies of reducing the random intersection delay, and the technology accumulation could accelerate the performance improvement. Third, system cost can be further reduced due to the AVs technologies in reducing the road link delay, but the technology accumulation could not accelerate the improvement. Finally, travelers would tend to choose the AVs as the total travel demand increases until it exceeds a threshold.
The set of intersections in the road network
The set of signal control intersections in the road network
Signal controlled intersection,
The set of road links where the signal control intersection
The total time period of signal control intersection
A set of upstream links at all intersections
A set of phases of RVs at the intersection downstream of the road link
A set of phases of AVs at the intersection downstream of the road link
The number of phases of RVs at the intersection downstream of the road link
The number of phases of AVs at the intersection downstream of the road link
A set of green time ratios of each phase of RVs at the intersection downstream of the road link
A set of green time ratios of each phase of AVs at the intersection downstream of the road link
A set of green time ratios of each phase of intersection
A set of RVs flows based on phases at the road link
A set of AVs flows based on phases at the road link
if the path
The travel time of RVs at the link
The travel time of AVs at the link
The free flow travel time of RVs at the link
The free flow travel time of AVs at the link
The traffic capacity of RVs at the link
The traffic capacity of AVs at the link
The traffic volume of RVs of the phase
The traffic capacity of RVs of the phase
The traffic volume of AVs of the phase
The traffic capacity of AVs of the phase
The average of RVs of the phase
The average of AVs of the phase
The average passengers of a RV
The average passengers of an AV vehicle.
The author declares that there are no conflicts of interest regarding the publication of this paper.
This study has been substantially supported by two grants from the National Natural Science Foundation Council of China (71531011, 71601142), a grant from Shanghai Shuguang Program (Project no. 13SG23), and a grant from Shanghai Pujiang Program (16PJC090).