Ecological Cooperative Adaptive Control of Connected Automate Vehicles in Mixed and Power-Heterogeneous Traffic Flow

Abstract

The development of vehicle electrification and intelligent network technologies has led to a new type of mixed and power-heterogeneous traffic flow, comprised of regular vehicles (RVs) and connected and automated vehicles (CAVs), fuel vehicles (FVs) and electric vehicles (EVs). To reduce the energy consumption of mixed and power-heterogeneous traffic flow operating at a signalized intersection, the Ecological Control Unit–Cooperative Adaptive Control (ECU-CACC) is proposed in this paper. The vehicle platoon is divided into units which are named minimum ecological control units (min-ECUs). A bi-level control framework is designed to improve traffic efficiency and reduce energy consumption. The lower-level aims to plan the best ecological trajectory for every min-ECU, and the upper-level optimizes the passing strategies for efficiency through speed coordination. Scenario numerical experiments are performed to verify the effectiveness of the bi-level optimal control model and analyze the energy-saving effect of ECU-CACC under different vehicle mixing situations. The results from the experiment prove the excellent energy-saving potential of the proposed ECU-CACC, which helps the min-ECUs save about 10–20% energy consumption compared with a regular pattern.

1. Introduction

As the internet of vehicles gathered steam, the technology of cooperative adaptive cruise control (CACC), which aims to automate the longitudinal behavior of road vehi-cles by regulating the inter-vehicle distance to a desired value, is the main field of vehicle control study and commericial application. In an actual urban traffic environment, vehicles will have high fuel consumption rates and emission levels while approaching signalized intersections, because vehicles are forced to stop ahead of traffic signals when encountering red indications, thus producing shock waves within the traffic stream, in turn resulting in vehicle acceleration or deceleration maneuvers and idling events, which increases the vehicle fuel consumption and emission levels [1,2]. With the increasingly prominent problem of automobile exhaust and energy shortage, it is necessary to propose new solutions for the platoon control at an intersection through a combination of platooning technology and eco-driving technology, i.e., the technology of Eco-CACC (ecological cooperative adaptive cruise control).

In recent years, research on the applacation of Eco-CACC technology has gradually emerged. Tajeddin et al. [3] designed a Multi-Lane Adaptive Cruise Controller (MLACC) which determined the optimal velocity and lane-to-drive in real-time. Encompassing multiple objectives including safety, energy efficiency and desired velocity tracking, the cruise controller solved lane-specific optimization problems to compute an instantaneous trip cost for each lane and selected the lane that posed the lowest cost. Ahn et al. [4] used a novel integrated Eco-Cooperative Automated Control (Eco-CAC) system to route internal combustion engine vehicles (ICEVs), hybrid electric vehicles (HEVs) and battery-only electric vehicles (BEVs) in a fuel/energy-efficient manner, to select vehicle speeds based on anticipated traffic network evolution, minimize vehicle energy consumption near signalized intersections and intelligently modulate the longitudinal motion of vehicles along freeways within a cooperative platoon to minimize fuel/energy consumption. Among them, research on Eco-CACC strategy at a signalized intersection has been a hot research topic. Most research pays attention to the signalized intersection [1,2,5,6,7,8], attempting to improve the efficiency [9,10,11,12,13,14,15], string stability [7,10,11] and energy economy [1,2,5,6,7,8,9,11,12,13,14,15] of vehicle platoons. Ma et al. [5] proposed the use of Eco-CACC for both a single vehicle and a platoon of vehicles, then developed the system for signalized intersections, achieving energy-oriented adaptive velocity planning under varying traffic flow velocity and realized optimal velocity trajectory of a connected and automated vehicle (CAV) platoon at successive signalized intersections with a pass at green (PaG) algorithm property. Naus et al. [16] designed a CACC system that focuses on the feasibility of implementation, and defined a corresponding sufficient frequency-domain condition for the string stability of heterogeneous traffic promising the security and stability of vehicles, which indicated the necessity of the development of CACC technology.

The main control subject of CACC is a CAV, however, there is still a long way to go for a pure CAV traffic environment. Thus, a mixed traffic flow including CAVs and regular vehicles (RVs) will be the regular traffic condition for a long time into the future. Some researchers conducted studies focusing on a mixed platoon of CAVs and RVs [9,17,18,19,20,21]. An et al. [16] modeled the impact of a CAV’s speed reduction on mixed traffic flow, which had an accuracy of over 80%. What’s more, Zhou et al. [19] denoted that a larger CAV penetration is expected to improve a mixed traffic flow. Wang et al. [21] described a cooperative eco-driving (CED) system targeted for signalized corridors, focusing on how the penetration rate of CAVs affected the energy efficiency of the traffic network. Extensive experiments have confirmed that the behavior of RVs could be guided and controlled by controlling the speed profile of CAVs in a mixed traffic flow, which could reduce energy consumption during operation.

As for the energy attributes of vehicle control under CACC, there are many studies concentrated on a homogeneous platoon of conventional fuel vehicles (FVs) [1,7,10,15,17,18,22,23,24] or electric vehicles (EVs) [2,3,7,9,12,25]. In order to improve traffic flow and reduce fuel consumption, Wang et al. [24] formulated a space–time lattice based model to construct vehicle trajectories considering boundary conditions of kinematic limits, vehicle-following safety and lane-changing rules. Kim et al. [25] focused on a safe CACC design and analyzed the value of vehicle-to-vehicle (V2V) communication in the presence of uncertain front vehicle acceleration. In addition, to quantify the potential improvement in energy consumption and road throughput from CACC, they characterized experimentally the relationship between inter-vehicular gap, vehicle speed and reduction of energy consumption for a compact plug-in HEV. However, there are few studies conducted on power-heterogeneous traffic flow including FVs and EVs. FVs usually have high fuel consumption and emission levels due to theacceleration/deceleration events associated with stop-and-go traffic and idling along a trip [20]. The EVs directly account for zero emissions to the environment by using a renewable source of energy, but the behavior of stop-and-go still creates unnecessary energy consumption. For the different operation features between EVs and FVs, the most energy-saving speed profiles of these two kinds of vehicles would be different as well. In addition, unlike the kinetic energy recovery of EVs while braking, FVs always consume extra fuel during the trip, even idling. Hence, the optimal speed trajectories for EVs and FVs are quite different [6]. Eco-CACC strategies that focus on the power-heterogeneous traffic flow including the above two kinds of vehicles are more challenging.

So, for the foreseeable future, a mixed (CAVs and RVs) power-heterogeneous (FVs and EVs) traffic flow will be a new traffic operation form. For this new form, there are three problems that need to be solved on how to conduct CACC technology (1) Considering the vehicle difference of energy consumption, dynamic operation, etc., how to rapidly determine the optimal ecological operation state (i.e., the velocity trajectory of a vehicle platoon) of a mixed power-heterogeneous vehicle set has not been effectively solved. (2) How to guide the trajectory of RVs through CAVs to obtain the optimal ecological operation state of a whole platoon is challenging. (3) Under a signalized intersection scenario, how to guide the heterogeneous platoon through the signalized intersection in a reasonable way is a difficult problem.

To reduce the energy consumption of mixed and power-heterogeneous vehicle platoon operating at a signalized intersection, as well as to seek a more energy-saving and efficient platoon control strategy, a novel CACC control method based on an ecological control unit (ECU-CACC) is proposed in this paper. Firstly, according to the mixed operation characteristics of CAVs and RVs, a minimum ecological unit (min-ECU) is first proposed, which splits the whole vehicle platoon into basic control units for the ECU-CACC model. Next, oriented platoon operation and an intersection passing process, a bi-level optimal control strategy of ECU-CACC, is constructed. In the lower-level control, the speed profile of a CAV is dynamically optimized to minimize the energy consumption of the corresponding min-ECU, and the optimization results of each min-ECU are fed back to the upper-level. In the upper-level control, the speed profiles of related min-ECUs are re-optimized to maximize the efficiency within the intersection signal cycle. Through the joint optimization between the lower-level and the upper-level, the energy-saving and efficient guidance of a mixed and power-heterogeneous vehicle platoon is achieved. Finally, the effectiveness of the ECU-CACC is verified and relative influence factors of the performance of the ECU-CACC are analyzed in detail by extensive numerical simulation experiments.

This paper is structured as follows: Section 2 describes the min-ECU’s definition and construction. The bi-level optimal control model of the ECU-CACC, including the best ecological speed profile optimization and the creation of a passing strategy for related min-ECUs, is conducted in Section 3. While the validation of and analysis with the proposed control method are presented in Section 4. Subsequently, the paper is concluded in Section 5.

2. Minimum Ecological Control Unit Construction

The mixed (CAVs and RVs) and power-heterogeneous (FVs and EVs) vehicle platoon is treated as the research and control objective of this paper. Referring to existing research [16,17,18,19,20,21], this paper controls the speed profile of a CAV to guide the following RVs’ driving, thus realizing the overall ecological operation of the vehicle platoon. For the convenience of analysis, the min-ECU containing a CAV and several following RVs is defined as shown in Figure 1. At the same time, it is assumed that the vehicle always follows the vehicle in front without changing lanes or overtaking, and the power type of vehicles can be determined by a detection device along the roadside.

Different from a traditional CACC which only considers the individual status of a CAV, the overall status of a min-ECU will be used as the optimization target in this paper, so it is easier to realize the status control of the overall vehicle platoon.

As is shown in Figure 1, the design of the min-ECU divides thr vehicle platoon into different units, which facilitates the independent optimization of the speed profile and passing scheme of each min-ECU. This distributed structure will reduce the difficulty of CAV speed trajectory optimization and improve the calculation efficiency. In addition, the min-ECUs could have various combining forms, which can the fulfil the vehicle platoon control requirements under different penetrance of RV and CAV. Moreover, by recombining and splitting multiple min-ECUs, the platoon’s cooperative control could be realized with regards to different traffic scenarios (such as intersection scenarios) to achieve the control of the whole traffic flow condition.

3. ECU-CACC Method

To alleviate intersection congestion and energy waste during vehicle operation, this paper proposes a bi-level control model based on min-ECUs, which aims to improve the efficiency of a vehicle platoon and reduce energy consumption. Specifically, considering the preceding vehicle’s behavior, velocity hardness and vehicle dynamic performance, the model firstly minimizes the energy consumption of a min-ECU by optimizing the speed profile of the head CAV, and then properly improves the whole vehicle platoon’s efficiency by adjusting the speed profile of partial min-ECUs which would pass the intersection in the next green period. In addition, in order to simplify the model, this paper only studies the control of vehicles in a single lane ignoring the behavior of a changing lane and overtaking, i.e., the vehicle couldn’t overtake the front vehicle.

In this Section, the framework of the bi-level optimal control model is first proposed with its control process. After this, the lower-level model which aims to find the best ecological speed profile of a min-ECU is presented. Subsequently, the upper-level with the passing strategies of some min-ECUs is described. Finally, the methods for solving the above two models are provided.

3.1. The Control Process of the Bi-Level Optimal Control Model

In the optimization process of the ECU-CACC, the goal of the lower-level is ecological speed profile planning, while the upper-level is devoted to maximizing the number of vehicles passing the intersection in the green period. As is shown in Figure 2, to achieve the purpose of minimum energy consumption, the lower-level model calculates the energy consumption of a min-ECU utilizing the instantaneous electric and fuel model with a unified energy consumption coefficient ${\phi }_{u\_ev}$ and ${\phi }_{u\_fv}$. Then, considering the vehicle dynamic characteristics and the proceeding vehicle’s behavior, the ecological velocity is calculated. Subsequently, the speed profile is planned smoothly from the head vehicle’s initial velocity to a target velocity by the harness. In addition, the lower-level model updates the optimal speed trajectory of the CAV at every $T$ interval. In the upper-level model, on the basis of the trajectory of the CAV, i.e., $v_i(t)$, from the lower-level, and considering passing points as well as the constraint of a vehicle’s motion trajectory, the number of vehicles passing the intersection would be obtained by judging the possibility of the unit negotiating the intersection during the current green period. To maximize the efficiency, a new minimum velocity constrain, $v_{\min \_j}^{*}$, is calculated and conveyed to the ecological velocity planning part for the min-ECUs at the beginning of the next red period. Therefore, by the above calculations, the min-ECUs between two intersections would operate ecologically and efficiently.

3.2. The Best Ecological Speed Profile Planning of Min-ECU

In this part, with the purpose of minimizing energy consumption, the lower-level model is formulated, with consideration of a vehicle’s dynamic characteristics, the impact of the preceding vehicle and the smooth transition of speed.

3.2.1. Energy Consumption Model

The min-ECU this paper studied is composed of FVs and EVs, thus the vehicle’s instantaneous energy consumption requires inclusion of the fuel consumption of FVs and electric consumption of EVs.

The model from [21] is used to estimate the electricity of EVs, which could adapt to different operating conditions considering the meaning of unit mileage. It can be expressed by:

$$f_{e\_i}(v_i(t),a_i(t))=\left\{\begin{array}{ll}\frac{1}{\eta }(\frac{Mgu{v}_i(t)}{3600}+\frac{C_{d}A{v}_i^3(t)}{76140}+\frac{\delta M{v}_i(t)a_i(t)}{3600})& a_i(t)\ge 0\\ 0& a_i(t)<0\end{array}\right.,$$

(1)

where $v_i(t)$, $a_i(t)$, $\eta$, $M$ and $A$ represent the velocity, acceleration, powertrain’s efficiency, quality and windward area of a vehicle, respectively. $\mu$, $C_d$ and $\delta$ are the coefficient of rolling resistance, air resistance and rotating mass. $g$ denotes the acceleration of gravity.

This paper utilizes the Australian Road Research Board (ARRB) instantaneous energy consumption model [26], which is widely used in the field of Eco-CACC, to calculate the fuel consumption of a FV:

$$f_{\mathrm{f}\_\mathrm{i}}(v_i(t),a_i(t))=\left\{\begin{array}{ll}a+{\beta }_1{R}_T{v}_i(t)+\frac{{\beta }_{2}M{a}_i^2(t)v_i(t)}{1000}& a_i(t)>0,R_T>0\\ a+{\beta }_1{R}_T{v}_i\left(\mathrm{t}\right)& a_i(t)\le 0,R_T>0\\ a& R_T\le 0\end{array}\right.$$

(2)

and

$$R_T=b_1+b_2{v}_i(t)+b_3{v}_i^2(t)+\frac{Ma}{1000}+\frac{MgG}{100000},$$

(3)

where $R_T$ is the torque of a vehicle, $G$ is the grade of road, $\alpha$, ${\beta }_1$, ${\beta }_2$, $b_1$, $b_2$ and $b_3$ are the fit coefficients of the model.

To calculate and reduce the whole energy consumption of the min-ECU, it is necessary to unify the power consumption of EVs and the fuel consumption of FVs. Thus, the unified energy consumption coefficient is defined for an EV and FV, respectively.

$${\phi }_{u\_ev}=p_{e\_ev}\ast {\lambda }_{w\_ev}$$

(4)

and

$${\phi }_{u\_fv}=p_{e\_fv}\ast {\lambda }_{w\_fv},$$

(5)

where $p_{e\_ev}$ and $p_{e\_fv}$ are economic parameters which refer to the real-time price of electricity and fuel, respectively, and ${\lambda }_{w\_ev}$ and ${\lambda }_{w\_fv}$ are the weight coefficients. The calculation of a min-ECU’s instantaneous energy consumption could be expressed as:

$$J_{e\_whole\_ins}=\min {\displaystyle \sum _{i=1}^m{\epsilon }_i}{\phi }_{u\_fv}{f}_{\mathrm{f}\_i}(a_i(t),v_i(t))+{\displaystyle \sum _{i=m+1}^n(1-{\epsilon }_i){\phi }_{u\_ev}{f}_{e\_i}(a_i(t),v_i(t))}.$$

(6)

3.2.2. The Model of Ecological Speed Profile Planning

In order to achieve the objective of minimizing energy consumption when a min-ECU is driving between two intersections, with the consideration of the vehicle’s dynamic characteristic, the preceding vehicle’s behavior and the smooth transition of velocity, the ecological velocity profile optimization model is formulated as:

$$Z=\min J_{e\_whole},$$

(7)

$$J_{e\_whole}={\displaystyle \sum _{i=1}^m{\epsilon }_i}{\phi }_{u\_fv}{\displaystyle {\int }_{t_0}^{t_0+T}{f}_{f\_i}(a_i(t),v_i(t))dt}+{\displaystyle \sum _{i=m+1}^n(1-{\epsilon }_i){\phi }_{u\_ev}{\displaystyle {\int }_{t_0}^{t_0+T}{f}_{e\_i}(a_i(t),v_i(t))}}dt,$$

(8)

and

$${\epsilon }_i=\left\{\begin{array}{ll}1& \mathrm{vehicle} i is FV\\ 0& \mathrm{vehicle} i is EV\end{array}\right.,$$

(9)

where $J_{e\_whole}$ is the whole energy consumption of the min-ECU; ${\epsilon }_i$ is a constant variable whose value is 0 or 1, $t_0$ is the initial time; $T$ represents the time interval of optimization; $n$ is the number of vehicles in the min-ECU; $m$ denotes the code of FVs in the min-ECU. In the lower-level, the objective of minimum energy consumption is realized by optimizing $v_i(t)$ and $a_i(t)$.

The best ecological speed profile is constrained by the vehicle’s own dynamic characteristics, the behavior of the preceding vehicle and the smooth transition of speed. The above factors will be described and defined in the following content.

(1)

The constraint of vehicle dynamic performance

Considering the dynamic characteristic of a vehicle, the constraints about velocity and acceleration are given by:

$$a_{\min }\le a_i(t)\le a_{\max }$$

(10)

and

$$v_{\min }\le v_i(t)\le v_{\max },$$

(11)

where $v_{\min }$, $v_{\max }$, $a_{\min }$ and $a_{\max }$ are minimum velocity, maximum velocity, minimum acceleration and maximum acceleration, respectively.

(2)

The constraint of the preceding vehicle behavior

In addition to velocity and acceleration, the preceding vehicle’s behavior including queuing and normal operation is also considered. In this paper, the behavior of the preceding vehicle at an intersection is regarded as a constraint to the following min-ECU. Referring to the method of [27], a static path constraint (i.e., the constraint of a red period) is given when the preceding vehicle is queuing. As is shown in Figure 3, a CAV and its following RVs could not negotiate an intersection until the queue releases when the preceding vehicle is queuing, thus the behavior of a CAV is constrained by the process of the downstream vehicle’s queuing. Essentially, the static path constraint represents the impact of the red period in the signalized intersection on the CAV, which is defined as:

$$x_{p\_i}(t)\le s_{p\_i}(t)$$

(12)

and

$$\stackrel{•}{x_{p\_i}(t)}=v_i(t),$$

(13)

where $x_{p\_i}(t)$ represents the position of vehicle $i$, $s_{p\_i}(t)$ is the static path constraint and its determination refers to [27].

In Figure 3, $s_0$ denotes the headway when vehicles are queuing, $d_{line\_i-1}$ represents the distance between the position of the preceding vehicle and the parking line, which can be obtained easily under an intelligent connected environment. A fictitious vehicle following the preceding one is defined, which is $d_{line\_i-1}+s_0$ far away from the parking line. It is apparent that a CAV trajectory (as the blue line in Figure 3)would not operate under a safe condition unless its trajectory does not exceed the fictitious one’s, i.e., $s_{p\_i}(t)$.

Considering the disturbance from the object in front of the CAV, a dynamic path constraint is necessary. Among these objects, most of them are RVs or CAVs. Therefore, the Gipps model [27], a typical model of vehicle safety following, is introduced to mirror this situation by reflecting the following behavior of a CAV.

$$v_i(t)\le \min (v_g^1,v_g^2),$$

(14)

$$v_g^1=v_i(t)+2.5a_{\max }\tau (1-\frac{v_i(t)}{v_{\max }})\sqrt{0.025+\frac{v_i(t)}{v_{\max }}}$$

(15)

and

$$v_g^2=a_{\min }\tau +\sqrt{a_{\min }^2{\tau }^2-a_{\min }[2[x_{p\_i-1}(t)-s_0-x_{p\_i}(t)]-v_i(t)\tau -\frac{v_{i-1}{(t)}^2}{a_{\min }}]},$$

(16)

where $v_g^1$ and $v_g^2$ are the intermediate quantities of the model and $\tau$ is the reaction time of a driver.

(3)

The constraint of velocity harshness

During driving, the comfort of drivers and passengers, as well as the stability of the min-ECUs are defined by the acceleration’s smooth transformation. Using a trigonometric function [28] as a hard constraint, which facilitates the guidance of following RVs and avoids acceleration or deceleration and reduces the computations, finally the whole unit could achieve ecological operation. Figure 4 shows the speed profile of CAV.

The final exporting velocity equation is defined by:

$$v_i(t)=\left\{\begin{array}{ll}{v}_{opt}\pm \Delta v_{dis}& t_0<t\le t_f\\ v_a& t_f<t\le t_0+T\end{array}\right.,$$

(17)

where $v_{opt}$ is the trigonometric function this paper used, and it can be expressed by Equation (18). $\Delta v_{dis}$ denotes the perturbation variable and $T$ is the time interval of the lower-level model.

$$v_{opt\_i}(t)=\left\{\begin{array}{ll}\overline{v_{a\_i}}-v_{d\_i}\cos (\omega t)& 0\le t\le \frac{\pi }{2\omega }\\ \overline{v_{a\_i}}-v_{d\_i}\frac{\omega }{\phi }\cos \phi (t-\frac{\pi }{2\omega }+\frac{\pi }{2\phi })& \frac{\pi }{2\omega }\le t<\frac{\pi }{2\omega }+\frac{\pi }{2\phi }\\ \overline{v_{a\_i}}+v_{d\_i}\frac{\omega }{\phi }& \frac{\pi }{2\omega }+\frac{\pi }{2\phi }\le t<t_f\end{array}\right.,$$

(18)

where $v_{opt\_i}(t)$ is the velocity of a vehicle after optimization by the trigonometric function; $\overline{v_{a\_i}}$ is the mean target velocity and $\overline{v_{a\_i}}=\frac{v_i(t_0)+v_{a\_i}}{2}$; $v_a$ is the result of Equation (7) and is named the target velocity; $v_{d\_i}$ is the difference between the vehicle’s average target velocity $\overline{v_{a\_i}}$ and its initial velocity $v_i(t_0)$, i.e., $v_{d\_i}=v_{a\_i}-v_i(t_0)$; $\omega$ is the rate of change of acceleration/deceleration when the velocity of a vehicle is between $v_i(t_0)$; $\overline{v_{a\_i}}$, $\phi$ is the rate of change of acceleration/deceleration when the speed is below $\overline{v_a}$ and $\phi =\max \left\{\phi \right\}$. $J_v$ is the rate of change of acceleration, which satisfies:

$$\left|J_v\right|=\left|v_{d\_i}\omega \phi \right|\le 10.$$

(19)

The acceleration $a_i$ is constrained by $\left|a_{\max }\right|\le 2.5$. $d_{\mathrm{int}\_i}$ represents the target distance of vehicle:

$$\begin{array}{l}{d}_{\mathrm{int}\_i}={\displaystyle {\int }_0^{\frac{\pi }{2\omega }}(v_{a\_i}-v_{d\_i}\cos (\omega t))dt+(v_{a\_i}+v_{d\_i}\frac{\omega }{\phi })(\frac{d_{\mathrm{int}}}{v_{a\_i}}-\frac{\pi }{2\phi }-\frac{\pi }{2\omega })}\\ +{\displaystyle {\int }_{\frac{\pi }{2\omega }}^{\frac{\pi }{2\phi }+\frac{\pi }{2\omega }}(v_{a\_i}-v_{d\_i}\frac{\omega }{\phi }\cos \phi (t-\frac{\pi }{2\omega }+\frac{\pi }{2\phi }))dt}\end{array}$$

(20)

where $t_f$ is the ratio of target distance $d_{\mathrm{int}\_i}$ to average target velocity $\overline{v_{a\_i}}$.

In a word, the ecological velocity trajectory optimizing model proposed in this paper is shown as the following:

$$Z=\min J_{e\_whole}.$$

(21)

Subject to:

$$J_{e\_whole}={\displaystyle \sum _{i=1}^m{\epsilon }_i}{\phi }_{u\_fv}{\displaystyle {\int }_{t_0}^{t_0+T}{f}_{\mathrm{f}\_i}(a_i(t),v_i(t))}dt+{\displaystyle \sum _{i=m+1}^n(1-{\epsilon }_i){\phi }_{u\_ev}{\displaystyle {\int }_{t_0}^{t_0+T}{f}_{e\_i}(a_i(t),v_i(t))}}dt,$$

(22)

$$J_{e\_whole\_ins}=\min {\displaystyle \sum _{i=1}^m{\epsilon }_i}{\phi }_{u\_fv}{f}_{\mathrm{f}\_i}(a_i(t),v_i(t))+{\displaystyle \sum _{i=m+1}^n(1-{\epsilon }_i){\phi }_{u\_ev}{f}_{e\_i}(a_i(t),v_i(t))}$$

(23)

and

$${\epsilon }_i=\left\{\begin{array}{ll}1& \mathrm{vehicke} i is FV\\ 0& \mathrm{vehicke} i is EV\end{array}\right..$$

(24)

The constraint of vehicle dynamic performance:

$$a_{\min }\le a_i(t)\le a_{\max }$$

(25)

and

$$v_{\min }\le v_i(t)\le v_{\max }.$$

(26)

The constraint of the preceding vehicle’s behavior:

$$\left\{\begin{array}{ll}\mathrm{if} v_{i-1}(t)=0& \mathrm{then} x_{p\_i}(t)\le s_{p\_i}(t)\\ \mathrm{if} v_{i-1}(t)\ne 0& \mathrm{then} v_i(t)\le \min (v_g^1,v_g^2)\end{array}\right..$$

(27)

The constraint of velocity harshness:

$$v_i(t)=\left\{\begin{array}{ll}{v}_{opt}\pm \Delta v_{dis}& t_0<t\le t_f\\ v_a& t_f<t\le t_0+T\end{array}\right..$$

(28)

3.2.3. The Car Following Model of RVs

In the min-ECU, RVs would follow the CAVs according to the following behavior. The Intelligent Driver Model (IDM) [29] is used in this paper, which could reflect the following characteristic of RVs in a mixed traffic condition, and its equation can be expressed by:

$$a_i=a_{\max }[1-{(\frac{v_i}{v_f})}^4-{(\frac{s^{*}(v_i,\Delta v_i)}{\Delta x_i-L_i})}^2]$$

(29)

and

$$s^{*}(v_i,\Delta v_i)=s_0+v_i{h}_s+\frac{v_i\Delta v_i}{2\sqrt{a_{\max }{a}_{com}}},$$

(30)

where $v_f$ is the velocity of free traffic flow; $\Delta v_i$ and $\Delta x_i$ are the difference in velocity and position between the vehicle $i$ and the vehicle $i-1$, i.e., $\Delta v_i=v_i-v_{i-1}$, $\Delta x_i=x_{p\_i}-x_{p\_i-1}$; $s^{*}$ is the target space distance between the vehicle $i$ and the vehicle $i-1$; $h_s$ is the safe time headway between two vehicles; $a_{com}$ is the comfortable retarded velocity.

3.3. The Passing Strategy Optimization of Min-ECU

When the min-ECUs approach the signalized intersection, according to the speed profiles from the lower-level model, some may reach the parking line at the beginning of the red period. Because of this, these min-ECUs would have to wait for a whole red period, which causes extra energy consumption.

Considering the above situation, a cooperative control is constructed in this section, improving the number of passing min-ECUs and operation efficiency. In this part, the situation that there is a green period is redundant, and there are min-ECUs that arrive at the parking line at the beginning of the red period is the target traffic condition this paper optimized. As is shown in the following equations, the optimization objective in the upper-level is to maximize the number of vehicles which pass the intersection within the green period.

$$N^{*}=\max \left({\displaystyle \sum _{j=1}^{N_v}{n}_j}+\rho {\displaystyle \sum _{j=N_v}^{N_v+N_m}{n}_j}\right),$$

(31)

where $\rho$ is a constant variable whose value is 0 or 1.

The maximum number of vehicles passing the intersection $N^{*}$ is formed by two parts considering the $k$ min-ECUs in the current cycle. One is the number of vehicles passing the intersection in the green period under the ecological speed profile from the lower-level, which could be obtained by $N_v$. The other one, i.e., $N_m$, is the number of min-ECUs reaching the passing line at the beginning of the red period, whose speed profile will be re-optimized to improve the whole efficiency of the intersection. The first part could be expressed by:

$$N_v={\displaystyle \sum _{j=1}^k-\mathrm{sgn}\{\mathrm{sgn}[(t_{d\_j}-t_{green\_l})(t_{d\_j}-t_{green\_r})]-1\}}.$$

(32)

Equation (32) should satisfy the flowing constrains:

$${\displaystyle {\int }_t^{t_{d\_j}}{v}_j(t)} dt=d_{line\_j},$$

(33)

$$v_{\min }\le v_j(t)\le v_{\max },$$

(34)

$$t_{red,r}\le t_{d\_j}\le t_{green\_r} \mathrm{or} t_{green,l}\le t_{d\_j}\le t_{red\_l},$$

(35)

$$v_j(t)=\stackrel{•}{x_{p\_j}(t)}$$

(36)

and

$$x_{p\_j}(t)<x_{p\_j-1}(t)+s_d,$$

(37)

where $t_{d\_j}$ is the time when min-ECU $j$ reaches the passing line; $t_{green,l}$ and $t_{green,l}$ denote the beginning and end time of the green light; $t_{red,l}$ and $t_{red,r}$ denote the beginning time and end time of the red light; $n_j$ is the number of vehicles passing the intersection in the min-ECU; $s_d$ denotes the safe distance between two min-ECUs; $d_{line\_j}$ is the distance between the min-ECU $j$ and the passing line. $N_v$ means the number of min-ECU passing the intersection, whose passing point is between $t_{green,l}$ and $t_{green,r}$ and $\mathrm{sgn}$ is the jump function.

For the other part of Equation (32), if the redundant green period could allow the min-ECUs at the beginning of red period to pass the intersection, these min-ECUs’ speed profile would be re-optimized, thus improving the whole intersection’s efficiency.

$$\rho =\left\{\begin{array}{ll}1& \mathrm{if} C\ge 1\\ 0& \mathrm{if} C<1\end{array}\right.,$$

(38)

$$N_m=\min (C,N_v^{\prime })$$

(39)

and

$$N_v^{\prime }={\displaystyle \sum _{j=N_v+1}^k-\mathrm{sgn}\left\{\mathrm{sgn}\left[(t_{d\_j}-t_{green\_l})(t_{d\_j}-2t_{green\_r}+t_{d\_N_v})\right]-1\right\}}.$$

(40)

Meanwhile, Equation (40) should satisfy the following constrains:

$${\displaystyle {\int }_t^{t_{d\_j}}{v}_j(t)} dt=d_{line\_j},$$

(41)

$$v_{\min \_j}^{*}\le v_j(t)\le v_{\max },$$

(42)

$$v_j(t)=\stackrel{•}{x_{p\_j}(t)},$$

(43)

$$x_{p\_j}(t)<x_{p\_j-1}(t)+s_d$$

(44)

and

$$t_{green\_r}\le t_{d\_j}\le 2t_{green\_r}-t_{d\_N_v},$$

(45)

where $t_{m\_j}$ is the safe time headway between min-ECU $j$ and $j-1$, which could easily be obtained under an intelligent connected environment. $C$ means the maximum number of min-ECUs, which would reach the parking line at the beginning of the red period and could pass the intersection at the end of the green period. In this paper, we only take the min-ECUs that reach the passing line within the red period at the same time as the redundant green period, and $N_v^{\prime }$ is the number of min-ECUs in that time interval. According to the above description there will be two conditions, and to realize the objective of the upper-level, the minimum velocity constraints of these min-ECUs which could pass through the intersection by accelerating, will be recalculated as:

$$v_{\min \_j}^{*}=\left\{\begin{array}{ll}\frac{d_{line\_j}}{t_{green\_r}-\mathrm{sgn}(N_v+N_v^{\prime }-j){\displaystyle \sum _{g=j}^{N_v+N_v^{\prime }-1}{t}_{m\_g+1}}-t} (j=N_v+N_v^{\prime },N_v+N_v^{\prime }-1,\dots ,N_v+1)& \mathrm{if} C>N_v^{\prime }\\ \frac{d_{line\_j}}{t_{d\_N_v}+{\displaystyle \sum _{g=1}^j{t}_{m\_g}}-t} (j=N_v+1,N_v+2,\dots ,C)& \mathrm{if} C\le N_v^{\prime }\end{array}\right.,$$

(46)

where $v_{\min \_j}^{*}$ is the new minimum velocity constraint of a min-ECU $j$, which is obviously the optimization variable of the upper-level. With the above outcome, the upper-level would convey the following constraint to the lower-level for a new speed profile:

$$\frac{{\displaystyle {\int }_{t_0}^{t_0+T^{\prime }}{v}_j(t)dt}}{T^{\prime }}\le v_{\min \_j}^{*},$$

(47)

where $T^{\prime }$ is the optimization time interval of the upper-level and $t_0$ is the initial time. The final control effectiveness is present in Figure 5.

3.4. Solution Method of Bi-Level Optimal Control Model

For a rapid calculation, Dynamic Programming (DP) [3] is utilized in this paper for solving the lower-level. The advantage of DP is that it can break the complicated problem into multiple sub-problems and calculate the optimal decision of each stage, i.e., the optimal sub-decisions are still optimal. According to the Bellman equation, the basic optimization principle is that, whatever the initial state and decision are, the remaining decisions are still optimal to the state resulting from the first decision. The calculation process is shown on the left of Figure 6, where the state variable $s_{\gamma }$, the decision variable $x_{d\_\gamma }$ and the stage index are the ecological velocity of stage $\gamma -1$, the ecological velocity of stage $\gamma$ and the whole energy consumption of the min-ECU, i.e., $J_d$, respectively. $J_d$ could be expressed by:

$$J_d={\displaystyle \sum _{i=1}^m{\epsilon }_i}{\phi }_{u\_fv}{\displaystyle {\int }_{t_{\gamma }}^{t_{\gamma }+T_d}{f}_{f\_i}(a_i(t),v_i(t))dt}+{\displaystyle \sum _{i=m+1}^n(1-{\epsilon }_i){\phi }_{u\_ev}{\displaystyle {\int }_{t_{\gamma }}^{t_{\gamma }+T_d}{f}_{e\_i}(a_i(t),v_i(t))}}dt.$$

(48)

As an efficient and global searching algorithm, the Genetic Algorithm (GA) [30] can automatically acquire and accumulate knowledge about the search space during the search process, and adaptively control the search process to get the best solution. Therefore, in this paper, GA is used to solve the upper-level model. The process of the upper-level is shown on the right of Figure 6. During the calculation, the values of the gene in a chromosome are $t_{d\_j}$ in the current cycle.

As is shown in Figure 6, for the lower-level model, the time interval $T$ is divided into $N$ stages with the same time length. Taking the whole energy consumption of a min-ECU as the stage index, finally the best ecological speed profile is calculated and conveyed to the upper-level. The upper-level utilizes the passing points of the min-ECUs to judge the possibility of a min-ECU traversing an intersection, then creates a new minimum velocity constraint for these min-ECUs whose passing points are during the red period to make them pass the intersection by accelerating to some extent. The new constraint will be conveyed to the lower-level to re-optimize the ecological speed profile.

4. Experiments and Analysis

4.1. Scene Setting

Considering an urban road with two successive intersections, this paper mainly takes through traffic flow into account, for convenience, setting right and left traffic flow aside temporarily. The length of a road for data recording is 800 m with a fixed signal timed intersections, which are set at positions of 300 m and 700 m, respectively. The cycle time of the fixed timing is 50 s, with a 30 s red period and a 20 s period. In addition, the value of $v_{\max }$ is 70 km/h in this section. The design of this experiment referred to [27] and the value of the factors in the electric consumption model and fuel consumption model referred to [21,26]. In this part, the model is solved on the platform of MATLAB with the Sheffield GA Toolbox and the programming of DP.

Table 1. The value of parameters in the experiment.

Parameter	Value	Parameter	Value	Parameter	Value
$\eta$	0.84	$b_1$	0.04	$v_f ({\mathrm{m}/\mathrm{s}}^2)$	16
$M (\mathrm{kg})$	1600	$b_2$	0.5 × 10−3	$h_s (\mathrm{s})$	1.5
$g ({\mathrm{m}/\mathrm{s}}^2)$	9.8	$b_3$	10.8 × 10−6	$a_{\max } ({\mathrm{m}/\mathrm{s}}^2)$	1.2
$\mu$	0.015	$G$	0	$a_{com} ({\mathrm{m}/\mathrm{s}}^2)$	2
$C_d$	0.7	${\beta }_1$	8.8	$s_0 (\mathrm{m})$	2
$A ({\mathrm{m}}^2)$	3	${\beta }_2$	10.2	$L (\mathrm{m})$	5
$\delta$	1.25	$\alpha (\mathrm{mL}/\min )$	39.2

shows the value of parameters in the experiment.

The control model ECU-CACC proposed in this paper is compared with a regular pattern in the experiment. In a regular pattern, a CAV only optimizes its own driving process without any control towards RVs, and executes the same operation and following behavior as the RVs. When the verifications of the lower-level and the upper-level is conducted, for the convenience of recording energy consumption, each vehicle is numbered on the test road, and the energy reduction rate of a single vehicle is taken as the performance evaluation index. With regard to the test of the whole ecological performance, the release of vehicles will follow a Poisson distribution under a certain quantity and mixed proportion (CAV and RV, FV and EV). The ECU-CACC pattern and regular pattern will be tested respectively with a 30-min test range in the test road. All the vehicles’ overall energy consumption is recorded, and the whole reduction of energy consumption is regarded as the performance evaluation index. In order to lessen the random influence from the release of vehicles following a Poisson distribution, every condition is calculated 10 times and then the corresponding evaluation result is calculated.

4.2. Verification of the Lower-Level Model

To verify the lower-level model, as the objective of this experiment, a platoon formed by a CAV followed by seven RVs is released in the test road. In addition, there are four FVs and four EVs in this platoon. The operation of vehicles under a regular pattern and ECU-CACC pattern are presented in Figure 7.

Figure 7a is the profile of a CAV conducted by the regular strategy under the same scenario. The blue trajectory represents the same CAV, and the only difference is whether the min-ECU is implemented within the ECU-CACC or not. As we can see from Figure 7a, vehicles stop because of the red indications, creating stop-and-go waves, then pass through the intersection when the green period starts, which is the common situation in city traffic. It is noted that apparent acceleration and deceleration processes occurred during the green and red periods respectively, which are the main reasons for high energy consumption and emission levels at urban signalized intersections. However, under the ECU-CACC strategy proposed in this paper, the lower-level model does not create any stop-and-go waves, as is shown in Figure 7b. A CAV followed by RVs under the IDM model operates smoothly in light of the best velocity trajectory, thus avoiding stopping and starting which are meaningless.

In order to study the smooth transition of speed further, velocity profiles of the first three vehicles in a min-ECU, conducted under two different strategies, are shown in Figure 8.

It can be seen in Figure 8a that the velocity of vehicles varies from $v_{\min }$ to $v_{\max }$ with a large amplitude of variation. At the same time, the RVs’ behaviors indicate an apparent hysteresis. This may be caused by neglecting the smooth transition of speed because the CAV only considers its own operation. On the contrary, as is shown in Figure 8b, under the lower-level model, the amplitude of the vehicles’ speed varies smoothly to a small extent. In addition, the hysteresis of the following RVs is improved noticeably compared with Figure 8a. By an comparison of the two control patterns as shown in Figure 8c, the proposed ECU-CACC helps to save energy consumption by 11.8% compared with the regular pattern. Because of avoiding the behavior of stop-and-go, FVs reduce their energy consumption rate by 14.1%, while the EVs’ energy consumption reduction rate is about 9.8%, which is almost not affected by stop-and-go traffic condition. The result verifies that the lower-level of ECU-CACC could decrease the operation energy consumption of vehicles effectively.

Considering the impact of different proportions between FV and EV within a min-ECU, time–space diagrams under three kinds of proportions, including 1:1, pure FV and pure EV, are shown in Figure 9. On the one hand, as is shown in Figure 9b, when EVs reach the intersection under the best ecological speed profile, there will be a deceleration, and subsequently the velocity will be increased to the best ecological velocity until the EVs pass the signalized intersection. The operational strategy of EVs depends on the kinetic energy recovery while baking. Thus, to some extent, the behavior of braking aims to ensure the ecological property. On the other hand, in Figure 9c, which relates to a pure FV min-ECU, fuel consumption is closely related to acceleration and deceleration, thus the vehicles traverse the successive intersections with a constant velocity taking advantage of the green wave band to decrease energy consumption.

It can be seen apparently in Figure 9a that the profile of mixed traffic is a combination between the pure EV and the pure FV, which is mainly impacted by the comprehensive effect of proportion and weight. No matter what the proportion is, the final velocity profile is the best ecological one of the min-ECU. The above results prove that an ECU-CACC control scheme could provide different power-heterogeneous min-ECUs with a suitable guidance strategy.

4.3. Verification of the Upper-Level Model

In order to verify the effectiveness of the upper-level model, three min-ECUs are chosen as the control objectives and their operation behaviors in the test environment are analyzed. As is shown in Figure 10, the settings of the three min-ECUs obey to the following rules: the first min-ECU is composed of three vehicles and encounters the red light when reaching the signalized intersection; the second min-ECU consists of two vehicles that will directly pass the intersection as it reaches the intersection; the third min-ECU, which is formed of two vehicles, would reach the first intersection at the beginning of the next red period if it maintains a constant velocity. In addition, to avoid that the vehicle’s energy consumption property would influence the effectiveness of the test, all the vehicles’ energy type in this part is set as EV.

As is shown in the Section 1 of Figure 10, by optimization of the upper-level model, the third min-ECU will get through the intersection during the previous green period with a reasonable acceleration strategy, while the other two min-ECUs will pass the intersection according to their best ecological operation scheme. After traversing the intersection, all the min-ECUs keep a constant velocity, i.e., the best ecological velocity as is shown in Section 2. At the same time, it can be noted obviously in Section 3 that when the three min-ECUs encounter the next signalized intersection, all the min-ECUs will decelerate appropriately to pass the intersection to form a big vehicle platoon. This operation scheme verifies the effectiveness of the upper-level model in the proposed ECU-CACC, which could improve the traffic efficiency of a single intersection.

4.4. Analysis of the Whole Ecological Effectiveness

The ecological effectiveness of the ECU-CACC is influenced by different transportation demands and mixed proportions. In order to analyze the ecological effectiveness of ECU-CACC objectively under diverse traffic flow scenarios, the penetrance of a CAV is fixed at 30%, and the mixed proportion between FVs and EVs is set as 50%, which is relatively reasonable. The vehicle releasing rate of a single lane increases from 400 pcu/h to 1200 pcu/h gradually, in other words, from an unblocked condition to a saturation condition. The experiment result is presented in Figure 11: the whole ecological effectiveness increases first before declining.

As the release of vehicles follows a Poisson distribution, in the low flow condition, some RVs’ operation almost wouldn’t be affected by CAVs, because the sparse traffic flow allows a long distance between vehicles. Under this condition, partial vehicles couldn’t form min-ECUs and their operation could not be optimized by the proposed model. So during the beginning of the vehicles’ release, the energy reduction performance of the ECU-CACC is inconspicuous for the vehicles operating independently. With an increasing number of vehicles, the operation of one vehicle would be limited by the vehicles around it, i.e., the interaction behaviors among vehicles is more and more outstanding, so the influence of a CAV’s behaviors on RVs becomes gradually obvious. In this condition, the model proposed in this paper would play a role in the optimization of velocity and passing strategy step by step, and the effectiveness of the model will reach its optimum when the traffic flow is increased to a certain extent. As is shown in the Figure 11, a single lane volume with 800 pcu/h obtains the highest reduction of energy consumption. But in the wake of further increasing of the number of vehicles, the behavior of vehicle queuing is more and more obvious, which would limit the vehicles’ operation speed further, and the control effectiveness of the ECU-CACC will then be discounted heavily. Thus, we conclude that the control effectiveness of the ECU-CACC is limited by the condition of traffic flow volume. But, at a holistic level, the ECU-CACC proposed in this paper could still decrease energy consumption under different transportation demands.

In addition, the influence from the mixed proportion of CAVs and RVs, and EVs and FVs is analyzed through fixing the traffic flow with 800 pcu/h, and the result is shown in Figure 12. Under the same proportion of EVs and FVs, the higher CAV penetrance results in a higher energy consumption reduction, because the more CAVs, the more controllable vehicles in the research region, the more prominent an impact on RVs under some traffic flow conditions, thus the better ecological control effectiveness of the ECU-CACC. At the same time, with the same CAV proportion, the higher the FV penetrance, the better the ecological control effectiveness, because compared with EVs, the energy consumption of FV is affected remarkably by accelerating, decelerating and idling events, which are the common events at a signalized intersection. Under this situation, the proposed ECU-CACC could provide a better control effectiveness by optimizing the best ecological speed profile of min-ECUs to avoid unnecessary fuel consumption. In addition, as is shown in Figure 12, there is still an energy consumption reduction rate ranging from 10% to 20% under a pure EV environment, although the driving of EVs is less affected by the signalized intersection.

In conclusion, the ECU-CACC proposed in this paper could satisfy the ecological control demand of mixed and power-heterogeneous traffic flow, and its effectiveness is verified adequately.

5. Conclusions

This paper develops the method of an ECU-CACC to improve the efficiency of an urban signalized intersection and reduce energy consumption, and is formed by a bi-level model. According to the control characteristics of a mixed power-heterogeneous traffic flow, the definition of a min-ECU is proposed. In the lower-level model, considering the ecological operation of a single min-ECU, every min-ECU achieves an ecological speed profile for optimal energy economy during their driving. With the purpose of increasing the number of vehicles passing an intersection, the upper-level adjusts the minimum velocity of some min-ECUs which would reach the intersection at the beginning of a red period under the optimization of the lower-level model. In the solution process, a dynamic programming algorithm is used to calculate the best ecological speed profile for every min-ECU, and the maximal number of vehicles passing the intersection is obtained through a genetic algorithm. Scenario numerical experiments are performed to verify the effectiveness of the bi-level optimal control model and analyze the energy saving effect of the ECU-CACC under different vehicle mixing situations. The results from the experiment prove the excellent energy-saving potential of the proposed ECU-CACC, which helps the min-ECU to reduce energy consumption by 11.8% compared with the regular pattern. The analysis of relative influence factors indicates that a higher CAV penetrance and FV penetrance bring better control performance to the ECU-CACC. However, the optimal performance of the method proposed in this paper would be discounted under unblocked and over-saturated conditions, which could be seen in the part of experiment. In addition, a free communication also plays an important role in the proposed ECU-CACC, as well as the penetrance of CAV. In the following-up research, the lane-changing behaviors of vehicles can be considered to refine the reconstruction and splitting process of the min-ECUs to improve the practicability of the ECU-CACC algorithm. The cooperative control of the ECU-CACC and a single scheme can be studied to further improve the energy-saving effect of vehicle operation at signalized intersections.