**1. Introduction**

Traffic signal control is one of the most effective methods to alleviate traffic congestion [1,2]. The existing signal control strategies can be divided into three categories, namely fixed time, actuated, and adaptive [3]. Fixed-time-based traffic control utilizes the historical traffic volume at an intersection to calculate the parameters for signal timing (e.g., cycle length, green time split, and offset) [4,5]. Fixed-time-based traffic control may not be able to adapt to demand fluctuation caused by the stochastic nature of traffic arrivals, which may degrade its benefits of signal control. In order to tackle the real-time demand fluctuation, actuated and adaptive traffic control methods have been proposed. By collecting real-time traffic arrivals from preinstalled sensors (e.g., loops, microwaves, and videos), actuated control methods can adjust their signal timing on the basis of a few simple rules, that is, green extension and minimum and maximum green duration constraints [6–9]. For adaptive control

methods, they predict traffic arrivals first by utilizing the data collected from preinstalled sensors and then optimize the predefined metrics (e.g., minimize delays, maximize throughputs) [10–12]. Whether actuated control or adaptive control is involved, their signal plans are optimized based on the data collected from preinstalled sensors. However, these sensors (e.g., loops, microwaves, and videos) are only installed at a few fixed points near the intersection and can only collect traffic data at those limited points. Indeed, intersection signal controllers are not able to grasp the accurate and complete information of traffic arrivals based on these limited sensors, which may result in the inefficiency of the traffic signal control. Furthermore, the preinstalled sensors (e.g., loops, microwaves, and videos) can only monitor vehicles as they are approaching an intersection; they are not able to influence or change traffic arrivals based on the current signal timings to achieve better performance.

With the development of wireless communication technology such as radio-frequency identification (RFID), dedicated short-range communication (DSRC), and 5G (the fifth generation of cellular network technology), a connected vehicle (CV) technique has emerged and is being employed in recent years [13–19]. By equipping vehicles with an on-board unit (OBU), the connected vehicles are able to communicate with surrounding vehicles and infrastructures (e.g., signal controllers) in real time [3,20–22], and these connected vehicles can be regarded as moving sensors. Therefore, first, the trajectory data collected from the connected vehicle such as real-time speed, acceleration, and location can be transferred to the signal controller, then signal plans can be optimized with more precision [23,24]. Second, with connected vehicles, some critical metrics such as queue length can be detected or estimated [25–28], which can be incorporated to reflect the performance and influence the optimization of signal plans. This may further improve the efficiency of traffic control.

Most existing studies merely focus on how to optimize signal plans based on the trajectory data collected from the connected vehicle [3,20,29,30]. In particular, based on the trajectory data, vehicles' arrival times (i.e., the time for vehicles to arrive at the stop line) around the intersection can be estimated with more precision, then signal plans can be optimized by taking these vehicles' arrival times as exogenous input. However, based on the two-way communication between vehicles and the signal controller in a connected vehicle environment, a speed guidance or speed advisory can be deployed. For example, Wan et al. [31] proposed a speed advisory system (SAS) for reducing idling and improving energy efficiency at a red light. Ramezani and Benekohal [32] claimed that the accuracy of data and the effectiveness of speed guidance could be enhanced based on a connected vehicle technique, and they developed an optimization program to determine advisory speeds for connected vehicles in work zones. Tajalli and Hajbabaie [33] regulated the movement of vehicles to achieve a "smoother" flow of traffic by harmonizing the speed of connected vehicles.

By using a speed guidance or speed advisory, the arrival time of connected vehicles can be dynamically optimized by adjusting their travel speed according to the current signal status and traffic conditions [34–36]. This means that not only signal plans can be adjusted to adapt to traffic arrivals but also traffic arrivals can be adjusted likewise to adapt to the signal plans in a connected vehicle environment. Specifically, Tang et al. [35] for the first time introduced driver's bounded rationality into a speed guidance model. The impacts of driver's bounded rationality on fuel consumption and emissions were studied. Tang et al. [36] further introduced a speed guidance strategy into a car-following model. Wu et al. [34] proposed a speed guidance model for a fixed-time-based signal controller. However, signal timing plans were not simultaneously optimized along with the speed guidance strategy in those studies. In particular, the existing studies focused mainly on modelling speed guidance strategies without optimizing signal timing plans at the same time, especially when many connected vehicles were coming from conflicting movements (e.g., through vehicles from the east and through vehicles from the south).

This paper fills the mentioned gap and contributes to the literature in several ways. First, this study examines the dynamic interaction between signal timing and vehicle arrival time (or travel speed). This has been rarely modelled in the literature. In particular, two speed guidance strategies associated with dynamic signal timing plans and three types of vehicle delays (i.e., control delay, queuing delay, and signal delay) are modelled. Second, by using the Standard North American NEMA (National Electrical Manufacturers Association) Dual-Ring structure, the proposed model is among the first to simultaneously optimize intersection signal timings and recommended travel speed for connected vehicles coming from conflicting movements at an intersection in a united framework. Specifically, the proposed model calculates the optimal signal timing to decrease the total delay for all approaching connected vehicles as well as optimizes the travel speed for each connected vehicle at the same time. It should be noted that both signal timing and travel speed adjustments may improve travel efficiency for one vehicle; however, it remains a great challenge to simultaneously allocate green time to each phase and calculate the optimal travel speed for all approaching connected vehicles, especially when these connected vehicles are coming from conflicting movements.

The remainder of this paper is organized as follows. Section 2 starts with a simple example of a connected vehicle to clear the intersection with either signal timing or speed adjustments. Notations and descriptions are also provided in Section 2. Section 3 models the problem, and the speed guidance model, delay model, and objective function are proposed to simultaneously optimize vehicle arrival time and signal timing plans. Numerical examples are presented in Section 4 to illustrate the model and analysis. Section 5 concludes the paper.

#### **2. Problem Description and Parameter Definition**

Taking the end of the green light as an example, as shown in Figure 1, if the vehicle proceeds without speed adjustment, it reaches the intersection at *Ta* and stops because the signal turns red at *Tr*. If the vehicle needs to pass the intersection without stopping, the common practice is to extend the duration of the green light for *Ta* − *Tr* seconds, which means the existing methods need to obtain the vehicle arrival time as inputs to optimize the signal timing and extend the green light. However, vehicles could also clear the intersection without stopping by other means. For example, guiding the driving speed from *V*<sup>0</sup> to *V*<sup>1</sup> can help the vehicle arrive at the intersection during green time without recalculating the signal timing. This means that either optimizing the signal timing or adjusting the arriving time (or speed) can make the vehicle clear the intersection without stopping. However, most existing studies focus mainly on signal timing optimization or speed guidance. In this paper, both signal timing and speed are assumed to be optimized simultaneously. Note that when these two variables are optimized for each vehicle, the complexity and optimization space of the model are greatly increased. The research problem in this paper can be specified as follows: how to obtain the optimal vehicle arrival time and the signal timing for all approaching vehicles around the intersection to ensure the maximum performance of the intersection.

**Figure 1.** Signal timing and vehicle arrival.

The main notations and their definitions and descriptions are presented in Table 1.


**Table 1.** Notations and descriptions.
