*2.1. Eco-CACC-I for BEVs*

In previous studies, an eco-driving system for gasoline-powered vehicles, named ICEV Eco-CACC-I, was developed and tested under a simulated environment and real-world field tests [18–21]. The ICEV Eco-CACC-I system computes real-time, fuel-optimized speed profiles that vehicles can follow so that they can proceed through signalized intersections while consuming minimum amounts of fuel. In the previous field test study, we implemented the developed eco-driving system into a real-world automated vehicle as an adaptive cruise control system [19]. Note that the developed Eco-CACC-I system does not directly optimize the collaboration between multiple vehicles. Here, the term "cooperative" means the vehicles equipped with the developed system can cooperate with signalized intersections under a connected-vehicle environment. During the previous network-level simulation test, the recommended speed computed by the developed system was used as a variable speed limit, which worked together with other constraints, such as the car following model and collision avoidance constraint, to control vehicle speed [21]. The same control logic of the previously developed algorithm was used in this study to develop an Eco-CACC-I controller that allows BEVs to drive smoothly through signalized intersections with minimal energy consumption and thus extend their range.

The same control environment setup for ICEV Eco-CACC-I was used here to develop the BEV Eco-CACC-I. The interested reader may read about the previous work in [19,21]. The control region was defined as a distance upstream of the signalized intersection (*dup*) to a distance downstream of the intersection (*ddown*) in which the BEV Eco-CACC-I controller optimizes the speed profiles of vehicles approaching and leaving signalized intersections. Upon approaching a signalized intersection, the vehicle may accelerate, decelerate, or cruise (maintain a constant speed) based on a number of factors, such as vehicle speed, signal timing, phase, distance to the intersection, road grade, and headway distance, etc. [2]. We assumed no leading vehicle ahead of the BEV so that we could compute the energy-optimized vehicle trajectory for the BEV without considering the impacts of other surrounding vehicles. The computed optimal speed was used as a variable speed limit, denoted by *ve*(*t*), which is one of the constraints on the BEV longitudinal motion. When a BEV travels on the roadway, there are other constraints to be considered, including the allowed speed constrained by the vehicle dynamics model, steady-state car following mode, collision avoidance constraint, and roadway speed limit. All of these constraints work together to control the vehicle speed. In this way, the proposed system can also be used in the situation that the BEV follows a leading vehicle and the vehicle speed can be computed by *v*(*t*) = *min*(*v*1(*t*), *v*2(*t*), *v*3(*t*), *v*4(*t*), *ve*(*t*)) using the following constraints:


Within the control region, the vehicle's behavior can be categorized into one of two cases: (1) the vehicle can pass through the signalized intersection without decelerating or (2) the vehicle must decelerate to pass through the intersection. Given that vehicles drive in different manners for cases 1 and 2, the BEV Eco-CACC-I strategies were developed separately for the two cases.

Case 1 does not require the vehicle to decelerate to traverse the signalized intersection. In this case, the cruise speed for the vehicle to approach the intersection during the red indication can be calculated by Equation (1) to maximize the average vehicle speed during the control region.

$$
\mu\_c = \min \left( \frac{d\_{\rm up}}{t\_r}, \mu\_f \right) \tag{1}
$$

When the vehicle enters the control region, it should adjust speed to *uc* according to the vehicle dynamics model illustrated later in Equations (5) through (7). After the traffic light turns from red to green, the vehicle accelerates from the speed *uc* to the maximum allowed speed (speed limit *uf*) by following the vehicle dynamics model until it leaves the control region.

In case 2, the vehicle's energy-optimized speed profile is illustrated in Figure 1. After entering the control region, the vehicle with the initial speed of *u*(*t*0) needs to brake at a deceleration level denoted by *a*, then cruise at a constant speed of *uc* to approach the signalized intersection. After passing the stop bar, the vehicle should increase speed to *uf* per the vehicle dynamics model, and then cruise at *uf* until the vehicle leaves the control region. In this case, the only unknown variables are the upstream deceleration rate *a* and the downstream throttle *fp*. The following optimization problem was formulated to compute the optimum vehicle speed profile associated with the least energy consumption.

**Figure 1.** Optimum vehicle speed profile in case 2.

Assuming a BEV enters the Eco-CACC-I control region at time *t*<sup>0</sup> and leaves the control region at time *t*<sup>0</sup> + *T*, the objective function entails minimizing the total energy consumption level as:

$$\min \int\_{t\_0}^{t\_0 + T} EC(u(t)) \cdot dt \tag{2}$$

where *EC* denotes the electric energy consumption at instant *t* using Equations (8) through (11). The constraints to solve the optimization problem can be built according to the relationships between vehicle speed, location, and acceleration/deceleration, as presented below:

$$u(t):\begin{cases} u(t) = u(t\_0) - at & t\_0 \le t \le t\_1\\ u(t) = u\_c & t\_1 < t \le t\_r\\ u(t+t) = u(t) + \frac{\Gamma\left(\frac{f}{f\_r}\right) - \Gamma(u(t))}{m} t & t\_r < t \le t\_2\\ u(t) = u\_f & t\_2 < t \le t\_0 + T \end{cases} \tag{3}$$

$$\begin{aligned} u(t\_0) \cdot t - \frac{1}{2}at^2 + u\_c(t\_r - t\_1) &= d\_{up} \\ u\_c = u(t\_0) - a(t\_1 - t\_0) &\\ \int\_{t\_r}^{t\_2} u(t)dt + u\_f(t\_0 + T - t\_2) &= d\_{down} \\ u(t\_2) = u\_f &\\ a\_{min} < a \le a\_{max} \\ f\_{min} \le f\_p &\le f\_{max} \\ u\_c > 0 \end{aligned} \tag{4}$$

In Equation (3), function *F* denotes the vehicle tractive force calculated by Equation (6), and function *R* represents all the resistance forces (aerodynamic, rolling, and grade resistance forces) calculated by Equation (7). Note that the maximum deceleration was limited by the comfortable threshold felt by average drivers [2]. The throttle value ranges between 0 and 1. To solve the optimization problem, dynamic programming was used to list all the candidate solutions with the associated electric energy consumption levels. This allowed calculation of optimal parameters for upstream deceleration *a* and downstream throttle *fp* by finding the candidate solution associated with the minimum energy consumption for vehicles passing the control region. To solve the proposed optimization problem in real-time, an A-star search method was selected to ensure fast and efficient computations. The A-star search method is one of the best and most popular path search methods to find the lowest cost path using a heuristic function [22]. The deceleration and throttle levels are considered as constant values in the A-star algorithm when computing the future cost. However, given that the optimal solution is recomputed every decisecond, the acceleration/deceleration level can also be updated every decisecond, thus producing a varying acceleration/deceleration maneuver. In the proposed optimization problem, first, a constant throttle level was assumed (e.g., 0.6) to find the optimal deceleration level, which corresponds to the minimal energy consumption for the entire trip from *ddown* to *dup*. In this way, the starting speed (cruise speed *uc*) and the ending speed (speed limit

*uf*) on the downstream roadway are known, so the optimal throttle level which corresponds to the minimal energy consumption for the downstream trip can be located. The details of how the A-star algorithm outperforms other pathfinding algorithms and the steps to implement the A-star algorithm can be found in [22].
