Journal of Advanced Transportation

Volume 2017, Article ID 5649823, 13 pages

https://doi.org/10.1155/2017/5649823

## Optimal Signal Design for Mixed Equilibrium Networks with Autonomous and Regular Vehicles

Key Laboratory of Road and Traffic Engineering of the Ministry of Education, Tongji University, Shanghai 200092, China

Correspondence should be addressed to Nan Jiang; nc.ude.ijgnot@ujtnangnaij9891

Received 30 April 2017; Accepted 13 June 2017; Published 16 August 2017

Academic Editor: Xiaobo Qu

Copyright © 2017 Nan Jiang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1]. In this regard, the AVs have potential to reduce quite a number of fatal accidents in that the related human failings can be overcome by AVs essentially. In addition, to improve driving safety, AVs are introduced in the transportation system to mitigate traffic congestion. Baking and acceleration actions of leading vehicles can be well detected and anticipated by the advanced automated driving technology. Such technology could reduce the traffic-destabilizing shockwave propagation and consequently enhance the capacity of road links. Traffic speed could be increased by 8% to 13% in freeway, depending on the communication method and traffic smoothing approaches [2]. In fact, other techniques such as vehicle-to-vehicle (V2V) and vehicle-to-infrastructure (V2I) communication can be applied to alleviate traffic congestion as well as automated driving technology.

AVs can pass intersections more efficiently by shortening start-up times and headways among vehicles at signal controlled intersections [3]. Moreover, by utilizing the V2V and V2I communication, random delays at signal controlled intersections can be decreased substantially. Therefore, AVs can more effectively use green times at signals and eventually better use intersection capacities. In the presence of V2V communication, all involved AVs could be programmed to choose their own routes to maximize the group interest by following the principle of Cournot-Nash (CN) equilibrium. In other words, AVs are under the control of a central manager and behave cooperatively in travel routing decisions. In contrast, RVs behave fully competitively and follow user equilibrium to find the path with the shortest travel cost. Overall, the AVs have the potential and inherent advantage in reducing traffic congestion in the network.

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 [4]. Therefore, the use of AV lanes can save travel time substantially, which can in turn enhance the market share of AVs.

So far, much of the research in AV filed from micro perspective focused on analyzing AVs’ on-road vehicle-following behaviors [5–10]. Different from the comparatively simple on-road moving condition, the driving actions of AVs at intersections turn to be more complicated because several conflict points resulted from crossing, moving in different directions. To characterize such problem, Dresner and Stone proposed an intersection management algorithm for AVs by utilizing a cell-based intersection reservation system [3].

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 [11]. Fagnant and Kockelman developed an agent-based model to investigate the impact of AVs on urban transport system, and they found that one AV would be able to replace eleven RVs but would only lead to 10% more travel cost [12].

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 [13] and AV zone design problem [14]. For example, Chen et al. [13] proposed a time-dependent model to optimally deploy AV lanes on a general network with mixed AVs and RVs. Zhou et al. found that cooperative autonomous vehicles can substantially improve the efficiency of freeway merging [15].

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 [16].

The rest of this paper is organized as follows. Section 2 defines and formulates the signal design problem and proposes the solution algorithm. Numerical examples are provided in Section 3. The last section concludes the paper.

#### 2. Signal Design Problem with Mixed Traffic Flows

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.

##### 2.1. The Lower-Level Problem

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 (1) represent link travel time with respect to the link flow for RVs and AVs, respectively. Parameters , , and represent the sensitivities of link flow to the travel time. Due to the technical advantages of AVs, we assume and .

The intersection delay times of RVs and AVs of each phase at the intersection can be expressed as

Delay functions (2) consist of the average delay and the random delay; and represent the weight of the random delay in the total delay. Due to the technical improvement of AVs, we believe that AVs can effectively reduce the random delay (), thus reducing the total delay at the intersections.

Assume that the path set of the RV is and that of the AVs is . We can establish a user equilibrium model as follows:

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 (17), the following necessary complementary conditions can be derived:

Functions (18) and (19) satisfy the UE principle of RVs. Functions (18) and (19) satisfy the CN principle of AVs. Functions (31)–(33) satisfy the logit-based mode choice. It is obvious that the solution of the minimization problem (3)–(11) satisfies the logit-based mode choice, namely, the mixed equilibrium condition of RVs and AVs.

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 and are both strictly convex functions. Furthermore, the link performance functions are strictly increasing and convex. Thus, objective function (3) is strictly convex. So the lower-level model has a unique solution. Therefore, the existence and uniqueness of the solution of the lower-level model are guaranteed.

##### 2.2. The Upper-Level Model

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: and are the matrixes to represent the relationship among different phases at the intersection , so that function (35) implies that the phases at the intersection should satisfy certain linear constraints.

##### 2.3. Solution Algorithm

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 [17]. The procedure of the algorithm is outlined below.

*Step 0*. Determine an initial set of green time ratio and let .

*Step 1*. By using the given , solve the lower-level mixed equilibrium problem which yields the initial vector of traffic volume , intersection delay , and link travel time .

*Step 2*. Calculate the derivatives, using the sensitive analysis, and derive , , and .

*Step 3*. Formulate local linear approximations of upper-level objective function (36) by using derivative information and solve the linear programming to obtain an auxiliary solution.

*Step 4*. Update the toll vector .

*Step 5*. Terminate the algorithm when , where is the convergence criterion; otherwise and go to Step 1.

For detailed equations of calculating the derivatives, please refer to [18, 19].

#### 3. Numerical Examples

The road network with 14 links and 6 nodes in the numerical examples is provided in Figure 1. The road network has 12 OD pairs: 1-3, 1-5, 1-6, 3-1, 5-1, 6-1, 5-6, 6-5, 3-5, 5-3, 3-6, and 6-3. The solid lines in the figure represent the lanes for RVs. The dash lines indicate the lanes for AVs. Relevant parameters of the road network are shown in Table 1. Nodes 2 and 4 represent signal controlled intersections. For the sake of simplicity, we assume that, at intersections 2 and 4, for RVs the capacity proportion of left turn lane, through lane, and the right turn lane is 1 : 2 : 1. At intersection 2, for AVs, the capacity proportion of left turn lane, through lane, and right turn lane is 1 : 2 : 1. At intersection 4, the AVs lane has only through lane. In this study, intersection 4 takes one control strategy, while there are four different signal control strategies for intersection 2 as candidates. In our numerical examples, we examine the optimal green time ratios and some sensitive analyses under different signal control strategies. In the numerical examples, we have , and .