Energy-Saving Optimization for Electric Vehicles in Car-Following Scenarios Based on Model Predictive Control

Abstract

In this paper, an economy-oriented car-following control (EOCFC) strategy is proposed for electric vehicles in car-following scenarios. Specifically, a controller based on model predictive control (MPC) is developed to optimize the host vehicle’s speed for better energy economy while ensuring good car-following performance and ride comfort. The vehicle’s energy consumption is accurately quantified in the form of demand power, which is incorporated in the cost function for energy optimization. The proposed EOCFC strategy is evaluated using three standard test cycles, i.e., New European Driving Cycle (NEDC), Urban Dynamometer Driving Schedule (UDDS) and Worldwide Harmonized Light Vehicles Test Cycle (WLTC), in comparison with a typical multi-objective adaptive cruise control strategy. The evaluation results demonstrate that the proposed EOCFC improves the energy economy of the host vehicle by 0.53%, 3.33% and 1.51%, under the NEDC, UDDS and WLTC test cycles respectively.

1. Introduction

1.1. Background

In recent years, the deterioration of the environment through pollution and energy shortage have imposed more stringent requirements on the energy consumption of road vehicles [1]. Compared with single-motor-drive electric vehicles (EVs), front-and-rear-independent-drive electric vehicles (FRIDEVs) can further exploit the energy-saving potential of EVs by means of appropriate torque distribution between the front and rear motors [2].

In addition to economical hardware configurations such as FRIDEVs, software such as eco-driving strategies also plays a crucial role in energy consumption reduction. It has been shown in the literature that by using eco-driving strategies [3,4,5], the energy consumption of vehicles can be effectively reduced. In recent years, advanced driving assistance systems (ADASs) have entered a new era of fast development and commercialization. Adaptive cruise control (ACC), as a typical type of ADAS, is an ideal carrier for eco-driving strategies to maximize their energy-saving potentials [6]. ACC assists the driver in achieving longitudinal control by adaptively adjusting the throttle or brake to maintain a certain cruise speed or to ensure an appropriate inter-vehicle distance, based on the state information of the preceding vehicle obtained by onboard sensors. Compared to human drivers who control longitudinal motions in an intuitive pattern, the longitudinal automation provided by ACC enables EVs to better execute eco-driving strategies [7], and as a result, the economy of EVs can be improved to a greater extent by designing an economy-oriented ACC system.

1.2. Literature Review

Economy-oriented ACC is originated from economy-oriented cruise control which improves vehicle economy by planning the economical cruise speed for the host vehicle [8]. Driven by the fast development of ACC systems, in recent years attention has been increasingly paid to energy-saving control in car-following scenarios [9]. Two main categories of car-following control strategies have been proposed in the literature to improve energy economy: rule-based strategies [10,11,12,13,14] and optimization-based strategies [15,16,17,18,19,20,21].

The rule-based strategies are mainly to achieve energy savings based on driving experience. Li et al. [10,11] proposed a PnG control strategy to minimize fuel consumption in car-following scenarios, which led to the reduction of fuel consumption by 20% compared to the LQ-based controller. Ioannou and Stefanovic [12] pointed out that the fuel economy of the host vehicle can be improved by smoothing its acceleration. Zhang and Ioannou [13] designed a proportional-integral (PI) controller for trucks to reduce fuel consumption by avoiding unnecessary acceleration and braking. Wu et al. [14] proposed a fuel economy optimization system in which the Lagrange multipliers method was used to calculate the desired acceleration or deceleration for fuel consumption optimization. However, the above rule-based strategies highly rely on human expertise and intuition, and as a result, the energy-saving potential of vehicles has not been fully exploited.

For optimization-based strategies, the control action is determined by optimizing a certain cost function which normally consists of one or more performance indicators. Li et al. [15] first proposed to utilize model predictive control (MPC) to achieve multi-objective optimization for better energy economy in car-following scenarios. Specifically, in their approach [15,16], the host vehicle’s acceleration was used to characterize its energy consumption, and a cost function was designed by taking into account car-following error and fuel consumption. Additionally, the requirements for safety, car-following performance and longitudinal ride comfort were also incorporated in this optimization problem as constraints. Luo et al. [17] also employed acceleration as an indicator of energy consumption. However, in their approach, acceleration was not directly incorporated into the cost function; instead, it was controlled to track a pre-designed smooth curve. Schmied et al. [18] designed an MPC-based car-following controller, in which vehicle speed and acceleration were employed to measure static fuel consumption. Jia et al. [19,20] developed an economy-oriented ACC system based on MPC for EVs. This approach took into account the effect of surrounding traffic information on vehicle speed and planned optimal speed profile for the host vehicle, aiming to achieve car-following control with low energy consumption. Madhusudhanan et al. [21] proposed a control scheme suitable for signalized intersections. In this method, to reduce energy consumption of the host vehicle, its speed was regulated according to future traffic signals and states of the preceding vehicle. In the above studies, energy economy was optimized by means of acceleration smoothing. However, acceleration is only an indirect measure (or indicator) of energy consumption, and it cannot accurately quantify energy consumption of the host vehicle. As a result, control strategies based on acceleration regulation may not lead to optimal economy performance.

To fully explore the energy-saving potential of EVs, an economy-oriented car-following control (EOCFC) scheme is proposed in this paper. For the host vehicle, three types of control objectives (i.e., car-following performance, ride comfort and energy economy) are taken into consideration in the controller design, and the demand power of the host vehicle is employed as an indicator to reflect its energy consumption. Then, a cost function which incorporates the above three objectives is constructed and solved. The effectiveness of the proposed control scheme is evaluated and verified under various typical driving cycles.

1.3. Original Contributions

The main contributions of this paper include:

• A nonlinear multi-objective model predictive control framework is developed for a FRIDEV under car-following scenarios, in which safety, car-following performance, ride comfort and energy economy are optimized simultaneously;
• The demand power of the host vehicle is used as an indicator to accurately reflect the energy consumption and incorporated in the cost function to achieve enhanced energy economy.

1.4. Outline of the Paper

The rest of the paper is organized as follows. System modeling (vehicle longitudinal dynamics and electric drive system) is introduced in Section 2. Details of the proposed EOCFC strategy are elaborated in Section 3. The verification results under different driving cycles are discussed in Section 4. Conclusions and future work are given in Section 5.

2. System Modeling

2.1. Vehicle Longitudinal Dynamics

To tackle this multi-objective optimization problem, a state equation for the ACC system must be first established. In this study, the constant time headway (CTH) was employed as the desired inter-vehicle distance; so, the longitudinal kinematics of the preceding vehicle and the host vehicle, as shown in Figure 1, are expressed by:

$$\left\{\begin{array}{l}\Delta d=d-d_{\mathrm{des}}\\ \Delta v=v_{\mathrm{p}}-v_{\mathrm{h}}\\ d_{\mathrm{des}}=th\cdot v_{\mathrm{h}}+d_0\end{array}\right.,$$

(1)

where $\mathsf{\Delta }d$ is the spacing error; $d$ denotes the actual inter-vehicle spacing; $d_{\mathrm{des}}$ represents the desired inter-vehicle spacing; $th$ is the time headway; $v_{\mathrm{h}}$ represents the speed of the host vehicle; $d_0$ is the minimum allowable distance between the preceding vehicle and the host vehicle when both vehicles come to a complete stop; $\Delta v$ is the speed error (i.e., the relative speed); and $v_{\mathrm{p}}$ denotes the speed of the preceding vehicle.

Taking the derivative of both sides of the above equations leads to:

$$\left\{\begin{array}{l}\Delta \dot{d}=\Delta v-th\times a_{\mathrm{h}}\\ \Delta \dot{v}=a_{\mathrm{p}}-a_{\mathrm{h}}\end{array}\right.,$$

(2)

where $a_{\mathrm{p}}$ denotes the acceleration of the preceding vehicle, and $a_{\mathrm{h}}$ represents the acceleration of the host vehicle. Considering the response characteristics of the vehicle driving and braking systems, the acceleration of the host vehicle $a_{\mathrm{h}}$ can be modelled as a first-order system, as follows:

$$a_{\mathrm{h}}=\frac{K_{\mathrm{s}}}{T_{0}s+1}{a}_{\mathrm{des}},$$

(3)

where $K_{\mathrm{s}}$ denotes the DC gain of the system; $T_0$ represents the time constant of the system; and $a_{\mathrm{des}}$ stands for the desired acceleration.

Based on Equations (1)–(3), the following governing equations for the longitudinal inter-vehicle dynamics can be established:

$$\left\{\begin{array}{l}\dot{x}=A_{0}x+B_{0}u+G_{0}w\\ y=C_{0}x\end{array}\right.,$$

(4)

with

$$x={[\Delta d, \Delta v, a_{\mathrm{h}}]}^T, u=a_{\mathrm{des}}, w=a_{\mathrm{p}}$$

$$A_0=\left[\begin{array}{ccc}0& 1& -th\\ 0& 0& -1\\ 0& 0& -1/T_0\end{array}\right], B_0=\left[\begin{array}{c}0\\ 0\\ K_{\mathrm{s}}/T_0\end{array}\right], G_0=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right], C_0=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$$

By means of zero-order hold, the above governing Equation (4) can be discretized as follows:

$$\left\{\begin{array}{l}x(k+1)=Ax(k)+Bu(k)+Gw(k)\\ y(k)=Cx(k)\end{array}\right.,$$

(5)

with

$$A={\displaystyle \sum _{k=0}^{\infty }\frac{{A_0}^k{T_{\mathrm{s}}}^k}{k!}}, B={\displaystyle \sum _{k=0}^{\infty }\frac{{A_0}^{k-1}{T_{\mathrm{s}}}^k}{k!}}{B}_0, G={\displaystyle \sum _{k=0}^{\infty }\frac{{A_0}^{k-1}{T_{\mathrm{s}}}^k}{k!}}{G}_0, C=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$$

where $k$ denotes the time index, and $T_{\mathrm{s}}$ represents the sampling time.

Additionally, the longitudinal force equilibrium equation for the host vehicle can be characterized as follows:

$$\begin{array}{cc}{T}_{\mathrm{d}}& =\left(mgf\cos \alpha +mg\sin \alpha +\frac{C_{\mathrm{D}}A}{21.15}{v}^2+\delta m\frac{\mathrm{d}v}{\mathrm{d}t}\right)\cdot r\hfill \\ & =\left(T_{\mathrm{m}\_\mathrm{f}}\times {\eta }_{\mathrm{f}}+T_{\mathrm{m}\_\mathrm{r}}\times {\eta }_{\mathrm{r}}\right)i_0\hfill \end{array}$$

(6)

where $T_{\mathrm{d}}$ denotes the demand torque of vehicle; $T_{\mathrm{m}\_\mathrm{f}}$ and $T_{\mathrm{m}\_\mathrm{r}}$ represent the front and rear motor output torques; ${\eta }_{\mathrm{f}}$ and ${\eta }_{\mathrm{r}}$ are the front and rear reducer efficiencies; $i_0$ denotes the speed ratio of the front and rear reducers; $r$ represents the tire radius; $m$ is the vehicle curb mass; $g$ is the gravitational acceleration; $f$ denotes the rolling resistance coefficient; $\alpha$ stands for the road slope; $C_{\mathrm{D}}$ is the drag coefficient; $A$ represents the frontal area; $v$ is the vehicle longitudinal speed; and $\delta$ is the rotational mass coefficient.

2.2. Electric Drive System

As mentioned above, a FRIDEV is considered in this study. The layout of the electric drive system is shown in Figure 2. This powertrain is composed of two permanent-magnet synchronous motors (PMSMs), each of which is connected to two driving wheels via a reducer and a differential. These two motors are not mechanically connected to each other, so they can be independently controlled by their motor control units (MCUs). The battery provides power to the two PMSMs when driving and receives energy during regenerative braking, supervised by an onboard battery management system (BMS). The vehicle control unit serves as a control center for the entire powertrain, which receives signals from and sends control commands to the BMS and MCUs.

The rotational speeds of the front and rear motors are governed by the following equation:

$$n_{\mathrm{m}\_\mathrm{f}}=n_{\mathrm{m}\_\mathrm{r}}=\frac{60v{i}_0}{2\pi r}v{i}_0,$$

(7)

where $n_{\mathrm{m}\_\mathrm{f}}$ and $n_{\mathrm{m}\_\mathrm{r}}$ denote the rotational speeds of the front and rear motors on the host vehicle.

The total input power $P_{\mathrm{m}\_\mathrm{in}}$ of the two motors can be expressed as follows:

$$P_{\mathrm{m}\_\mathrm{in}}=\frac{T_{\mathrm{m}\_\mathrm{f}}{n}_{\mathrm{m}\_\mathrm{f}}+T_{\mathrm{m}\_\mathrm{r}}{n}_{\mathrm{m}\_\mathrm{r}}}{9550},$$

(8)

Two motor efficiency maps, as shown in Figure 3, were used in this study to characterize the efficiencies of these two PMSMs under different operating conditions (i.e., different combinations of speeds and torques). Note that the maximum torque, maximum power and efficiency characteristics of these two PMSMs are different, which necessitates torque distribution between these two motors for economy optimization purposes.

As for the battery system, a straightforward Rint battery model [22] was employed in this study, and the schematic of this model is shown in Figure 4. In this model, the battery is represented by an equivalent circuit containing a power source and a resistor.

Based on this Rint battery model, the battery output power can be written as:

$$P_{\mathrm{b}}=UI,$$

(9)

where $P_{\mathrm{b}}$ denotes the battery output power; U represents the terminal voltage; and I is the charge or discharge current. The terminal voltage can be further expressed as:

$$U=E_{\mathrm{oc}}-IR,$$

(10)

where $E_{\mathrm{oc}}$ represents the electromotive force, and R stands for the internal resistance.

Combining Equations (9) and (10) leads to:

$$I=\frac{E_{\mathrm{oc}}-\sqrt{{E_{\mathrm{oc}}}^2-4\times R\times P_{\mathrm{b}}}}{2\times R},$$

(11)

In this study, the battery state of charge (SOC) was calculated using the ampere-hour integral method, as follows:

$$SOC={SOC}_0-\frac{1}{C}{\displaystyle {\int }_{t_1}^{t_2}{\eta }_{\mathrm{b}}I dt},$$

(12)

where $SO{C}_0$ denotes the battery initial SOC; C represents the battery rated capacity; $t_1$ and $t_2$ are the starting and ending time for charging (or discharging); and ${\eta }_{\mathrm{b}}$ stands for the battery charging (or discharging) efficiency. The battery efficiency map used in this study is shown in Figure 5.

3. Economy-Oriented Car-Following Control Strategy

In the car-following mode, the host vehicle follows the preceding vehicle and maintains a safe inter-vehicle spacing. It is a challenging task to optimize the economy performance in this mode. On the one hand, the reduction in energy consumption can result in deterioration of car-following performance (i.e., increased spacing error). On the other hand, if the host vehicle strictly follows a desired inter-vehicle spacing, then its energy consumption is deterministic, and economy optimization becomes impossible. Hence, an appropriate trade-off between economy performance and car-following performance should be made. In this paper, the car-following requirement is properly relaxed without jeopardizing vehicle safety, and inter-vehicle spacing is maintained within a safe range. By this means, economy optimization in the car-following mode becomes achievable, as different inter-vehicle spacing can lead to different energy consumption. Indeed, the economy optimization task can be formulated as a multi-objective optimization problem, as shown in Figure 6.

3.1. Control Objectives

This section introduces the performance requirements that the ACC system needs to satisfy. The performance requirements should be properly determined, so that (1) the car-following performance, ride comfort and energy consumption are accurately quantified, and (2) the cost function appropriately reflects the interactions between the above performances. In this section, the above three types of performances are individually analyzed, followed by the design of a cost function which incorporates these performances.

A.

Car-following performance

The car-following performance of an ACC system is normally evaluated by means of two indicators—spacing error and speed error (i.e., relative speed). Using these indicators, in this study, three important requirements were taken into consideration in quantifying the car-following performance. Firstly, the desired inter-vehicle spacing is strictly restrained within the allowable range (i.e., the green area in Figure 7). Secondly, both spacing error and speed error are maintained within a small range. Lastly, strict safety constraint is imposed on the actual inter-vehicle spacing.

For the first requirement, as shown in Figure 7, the upper and lower boundaries of the desired inter-vehicle spacing $d_{\mathrm{des}}$ can be written as:

$$d_{max}=t{h}_{\max }\times v_{\mathrm{h}}+d_{0\_\max },$$

(13)

$$d_{min}=t{h}_{\min }\times v_{\mathrm{h}}+d_{0\_\min },$$

(14)

where $d_{\max }$ and $d_{\min }$ are the maximum and minimum values of $d_{\mathrm{des}}$; $t{h}_{\max }$ and $t{h}_{\min }$ denote the maximum and minimum values of $th$; and $d_{0\_\max }$ and $d_{0\_\min }$ represent the maximum and minimum values of $d_0$. In this study, $t{h}_{\max }$ and $t{h}_{\min }$ were chosen as 2.5 s and 1.2 s, respectively, while $d_{0\_\max }$ and $d_{0\_\min }$ are chosen as 3 m and 6 m, respectively. Note, that the desired inter-vehicle spacing $d_{\mathrm{des}}$ must fall between these two boundaries, i.e., it must fall within the green area shown in Figure 7.

For the second requirement, the spacing error $\Delta d$ and the speed error $\Delta v$ are constrained according to the following inequalities:

$$\Delta d_{\min }\le \Delta d\le \Delta d_{\max },$$

(15)

$$\Delta v_{\min }\le \Delta v\le \Delta v_{\max },$$

(16)

where $\Delta d_{\min }=d_{\min }-d_{\mathrm{des}}$; $\Delta d_{\max }=d_{\max }-d_{\mathrm{des}}$; and $\Delta v_{\max }$ and $\Delta v_{\min }$ are the upper and lower boundaries of $\Delta v$.

For the third requirement, to guarantee safety during ACC operation, the following constraint is imposed on the actual inter-vehicle spacing [23]:

$$d\ge \max \left\{TTC\cdot \Delta v,d_{\mathrm{s}}\right\},$$

(17)

where $TTC$ denotes the time to collision, and $d_{\mathrm{s}}$ represents a fixed safe distance.

As mentioned above, car-following performance can be evaluated by means of spacing error and speed error (i.e., relative speed). Given that the above requirements (i.e., Equations (15)–(17)) are met, the car-following performance can be quantified as follows:

$$J_1={\sigma }_{\Delta d}\Delta d^2+{\sigma }_{\Delta v}\Delta v^2,$$

(18)

where $J_1$ is the performance index quantifying the car-following performance, and coefficients ${\sigma }_{\Delta d}$ and ${\sigma }_{\Delta v}$ are the weighting factors for these two terms.

B.

Ride comfort

Usually, acceleration and its time derivative (i.e., jerk) are employed to evaluate ride comfort of an ACC system [24]. To ensure ride comfort during ACC operation, the magnitudes of both acceleration and jerk should be properly constrained.

In this paper, the desired acceleration, desired jerk and actual acceleration of the host vehicle are employed to quantify ride comfort during ACC operation. Namely, the following performance index for ride comfort is defined:

$$J_2={\sigma }_{a1}{a}_{\mathrm{des}}^2+{\sigma }_{a2}{\dot{a}}_{\mathrm{des}}^2+{\sigma }_{a3}{a}_{\mathrm{h}}^2,$$

(19)

where $a_{\mathrm{des}}$, ${\dot{a}}_{\mathrm{des}}$, and $a_{\mathrm{h}}$ denote the desired acceleration, desired jerk and actual acceleration of the host vehicle, respectively, and ${\sigma }_{a1}$, ${\sigma }_{a2}$, and ${\sigma }_{a3}$ are the weighting factors for these three terms.

Additionally, the above-mentioned desired acceleration, desired jerk and actual acceleration are constrained as follows:

$$\left\{\begin{array}{l}{a}_{\min }\le a_{\mathrm{des}}\le a_{\max }\\ j_{\min }\le {\dot{a}}_{\mathrm{des}}\le j_{\max }\\ a_{\min }\le a_{\mathrm{h}} \le a_{\max }\end{array}\right.,$$

(20)

where $a_{\max }$ and $a_{\min }$ are the upper and lower boundaries for acceleration, while $j_{\max }$ and $j_{\min }$ denote the upper and lower boundaries for jerk.

C.

Energy consumption

In the existing literature, energy consumption during ACC operation is mainly quantified in three different ways: (a) energy consumption is represented by the 2-norm of acceleration and the 2-norm of jerk [15], (b) energy consumption is quantified using the 2-norm of acceleration error [17], and (c) energy consumption is approximated as a convex function of acceleration and speed [20].

Note that in the above approaches the energy consumption is not directly quantified as the product of power and time; instead, it is indirectly represented as a function of energy-related quantities, such as acceleration and speed. The underlying reason is that by doing so, the computation difficulties caused by nonlinearities in the subsequent receding-horizon optimization stage can be avoided. However, the above representations of energy consumption are only reasonable approximations, and errors are inevitably present in these forms of approximate representations. Considering this limitation, in this paper, a more direct and accurate representation of energy consumption is employed. Specifically, by definition, the energy consumed during ACC operation is the integral of power over time, as follows:

$$J_3={\sigma }_{\mathrm{p}}{\displaystyle \int P\mathrm{d}t=}{\sigma }_{\mathrm{p}}{\displaystyle \int \frac{Tn}{9550}\mathrm{d}t},$$

(21)

where $J_3$ denotes the economy performance index; ${\sigma }_{\mathrm{p}}$ represents the weighting factor for this performance; $P$ is the output power of the powertrain; $T$ is the output torque of the powertrain; and $n$ is the rotational speed of motor. Note that $T$ and $n$ can be further expressed as follows:

$$\frac{T{i}_0}{r}=mgf\cos \alpha +mg\sin \alpha +\frac{C_{\mathrm{D}}A}{21.15}{v}_{\mathrm{h}}^2+\delta m{a}_{\mathrm{des}},$$

(22)

$$n=\frac{v_{\mathrm{h}}{i}_0}{0.377r},$$

(23)

It should be emphasized that Equation (21)—the integral of power over time—is the most precise representation of energy consumption during ACC operation. By this means, the errors existing in the current forms of energy consumption representations are eliminated, which in turn improves the economy performance of ACC systems. This is also the major innovation and contribution of this paper.

3.2. Overall Cost Function

Combining the above three performance indices for car-following performance, ride comfort and energy consumption, the overall performance index (i.e., cost function) for the ACC car-following mode is obtained:

$$\begin{array}{ll}{J}_0& =J_1+J_2+J_3\hfill \\ & ={\sigma }_{\Delta d}\Delta d^2+{\sigma }_{\Delta v}\Delta v^2+{\sigma }_{a1}{a}_{\mathrm{des}}^2+{\sigma }_{a2}{\dot{a}}_{\mathrm{des}}^2+{\sigma }_{a3}{a}_{\mathrm{h}}^2+{\sigma }_{\mathrm{p}}{\displaystyle \int P\mathrm{d}t}\hfill \end{array}$$

(24)

Equation (24) can be further formatted as follows:

$$J_0=y^T{\sigma }_{y}y+{\sigma }_{a1}{u}^2+{\sigma }_{a2}{\dot{u}}^2+{\sigma }_{\mathrm{p}}{\displaystyle \int Pdt},$$

(25)

with

$$y={[\Delta d, \Delta v, a_{\mathrm{h}}]}^T, {\sigma }_y=\left[\begin{array}{ccc}{\sigma }_{\Delta d}& 0& 0\\ 0& {\sigma }_{\Delta v}& 0\\ 0& 0& {\sigma }_{a3}\end{array}\right]$$

Additionally, based on Equations (15)–(17) and (20), the corresponding constraints for the above cost function are rewritten as:

$$\left[\begin{array}{c}\Delta d_{\min }\\ \Delta v_{\min }\\ a_{\min }\end{array}\right]\le y\le \left[\begin{array}{c}\Delta d_{\max }\\ \Delta v_{\max }\\ a_{\max }\end{array}\right],$$

(26)

$$\left[\begin{array}{cc}1& -TTC-th\\ 1& -th\end{array}\right]\left[\begin{array}{l}\Delta d\\ \Delta v\end{array}\right]\ge \left[\begin{array}{l}-th\cdot v_{\mathrm{p}}-d_0\\ -th\cdot v_{\mathrm{p}}-d_0+d_{\mathrm{s}}\end{array}\right],$$

(27)

$$\left\{\begin{array}{l}{a}_{\min }\le u\le a_{\max }\\ j_{\min }\le \dot{u}\le j_{\max }\end{array}\right.,$$

(28)

3.3. Model Predictive Optimization Problem

To reduce energy consumption while satisfying car-following performance and ride comfort, in this work, a multi-objective optimization problem was established for ACC systems under the framework of model predictive control (MPC). The process of constructing this optimization problem is explained below.

Based on Equation (5), the predicted system state can be written as:

$$\left\{\begin{array}{l}X(k+N_{\mathrm{P}}|k)=\tilde{A}x(k)+\tilde{B}U(k+N_{\mathrm{C}})+\tilde{G}W(k+N_{\mathrm{P}})\\ Y(k+N_{\mathrm{P}}|k)=\tilde{D}x(k)+\tilde{E}U(k+N_{\mathrm{C}})+\tilde{F}W(k+N_{\mathrm{P}})\end{array}\right.,$$

(29)

where $N_{\mathrm{P}}$ denotes the prediction horizon; $N_{\mathrm{C}}$ represents the control horizon; and matrices $X$, $Y$, $U$, $W$, $\tilde{A}$, $\tilde{B}$, $\tilde{G}$, $\tilde{D}$, $\tilde{E}$, $\tilde{F}$ are given by:

$$X(k+N_{\mathrm{P}}|k)=\left[\begin{array}{c}x(k+1|k)\\ x(k+2|k)\\ ⋮\\ x(k+N_{\mathrm{P}}|k)\end{array}\right], Y(k+N_{\mathrm{P}}|k)=\left[\begin{array}{c}y(k+1|k)\\ y(k+2|k)\\ ⋮\\ y(k+N_{\mathrm{P}}|k)\end{array}\right]$$

$$U(k+N_{\mathrm{C}})=\left[\begin{array}{c}u(k)\\ u(k+1)\\ ⋮\\ u(k+N_{\mathrm{C}}-1)\end{array}\right], W(k+N_{\mathrm{P}})=\left[\begin{array}{c}w(k)\\ w(k+1)\\ ⋮\\ w(k+N_{\mathrm{P}}-1)\end{array}\right]$$

$$\tilde{A}=\left[\begin{array}{c}A\\ A^2\\ ⋮\\ A^{N_{\mathrm{P}}}\end{array}\right], \tilde{B}=\left[\begin{array}{cccc}B& 0& \cdots & 0\\ AB& B& \cdots & ⋮\\ ⋮& ⋮& \ddots & 0\\ A^{N_{\mathrm{P}}-1}B& A^{N_{\mathrm{P}}-2}B& \cdots & A^{N_{\mathrm{P}}-N_{\mathrm{C}}}B\end{array}\right]$$

$$\tilde{G}=\left[\begin{array}{cccc}G& 0& \cdots & 0\\ AG& G& \cdots & ⋮\\ ⋮& ⋮& \ddots & 0\\ A^{N_{\mathrm{P}}-1}G& A^{N_{\mathrm{P}}-2}G& \cdots & A^{N_{\mathrm{P}}-N_{\mathrm{C}}}G\end{array}\right]$$

$$\tilde{D}=\left[\begin{array}{c}CA\\ C{A}^2\\ ⋮\\ C{A}^{N_{\mathrm{P}}}\end{array}\right], \tilde{E}=\left[\begin{array}{cccc}CB& 0& \cdots & 0\\ CAB& CB& \cdots & ⋮\\ ⋮& ⋮& \ddots & 0\\ C{A}^{N_{\mathrm{P}}-1}B& C{A}^{N_{\mathrm{P}}-2}B& \cdots & C{A}^{N_{\mathrm{P}}-N_{\mathrm{C}}}B\end{array}\right]$$

$$\tilde{F}=\left[\begin{array}{cccc}CG& 0& \cdots & 0\\ CAG& CG& \cdots & ⋮\\ ⋮& ⋮& \ddots & 0\\ C{A}^{N_{\mathrm{P}}-1}G& C{A}^{N_{\mathrm{P}}-2}G& \cdots & C{A}^{N_{\mathrm{P}}-N_{\mathrm{C}}}G\end{array}\right]$$

The original cost function in continuous time domain is given by Equation (25). To construct the optimization problem, this cost function is first discretized and then extended to the entire prediction horizon, as follows:

$$\begin{array}{ll}J(y,u,\Delta u,P)& ={\displaystyle \sum _{i=0}^{N_{\mathrm{P}}-1}{‖y(k+i+1|k)‖}_{{\sigma }_y}^2}+{\displaystyle \sum _{i=0}^{N_{\mathrm{P}}-1}{‖u(k+i|k)‖}_{{\sigma }_u}^2}\\ & +{\displaystyle \sum _{i=0}^{N_{\mathrm{P}}-1}{‖\Delta u(k+i|k)‖}_{{\sigma }_{\Delta u}}^2}\\ & +{\displaystyle \sum _{i=0}^{N_{\mathrm{P}}-1}\{{\sigma }_{\mathrm{p}}P(k+i+1|k)T_{\mathrm{s}}\}}\end{array}$$

(30)

where $J(y,u,\Delta u,P)$ represents the cost function for the entire prediction horizon; ${‖·‖}_{\sigma }^2$ denotes the weighted 2-norm; and ${\sigma }_y$, ${\sigma }_u$, ${\sigma }_{\Delta u}$ and ${\sigma }_{\mathrm{p}}$ are the weighting factors.

Based on Equations (26)–(28), the constraints for the above cost function can be rewritten in the following form:

$$\left[\begin{array}{c}\Delta d_{\min }\\ \Delta v_{\min }\\ a_{\min }\end{array}\right]\le y(k+i+1|k)\le \left[\begin{array}{c}\Delta d_{\max }\\ \Delta v_{\max }\\ a_{\max }\end{array}\right], i=0:N_{\mathrm{P}}-1,$$

(31)

$$\left[\begin{array}{ccc}1& -TTC-th& 0\\ 1& -th& 0\\ 0& 0& 0\end{array}\right]\cdot y(k+i+1|k)\ge \left[\begin{array}{c}-th\\ -th\\ 0\end{array}\right]v_{\mathrm{p}}(k+i+1|k)+\left[\begin{array}{l}-d_0\\ -d_0+d_{\mathrm{s}}\\ 0\end{array}\right], i=0:N_{\mathrm{P}}-1,$$

(32)

$$\left\{\begin{array}{l}{a}_{\min }\le u(k+i|k)\le a_{\max }\\ j_{\min }\cdot T_{\mathrm{s}}\le \Delta u(k+i|k)\le j_{\max }\cdot T_{\mathrm{s}}\end{array}\right., i=0:N_{\mathrm{P}}-1,$$

(33)

Based on the above derivations for system state, cost function and constraints, the multi-objective optimization problem for economy-oriented ACC systems can be established as follows:

$$\begin{array}{ll}\underset{u(k+i|k),i=0:N_{\mathrm{P}}-1}{\min }J(y,u,\Delta u,P)& ={\displaystyle \sum _{i=0}^{N_{\mathrm{P}}-1}{‖y(k+i+1|k)‖}_{{\sigma }_y}^2}+{\displaystyle \sum _{i=0}^{N_{\mathrm{P}}-1}{‖u(k+i|k)‖}_{{\sigma }_u}^2}\hfill \\ & +{\displaystyle \sum _{i=0}^{N_{\mathrm{P}}-1}{‖\Delta u(k+i|k)‖}_{{\sigma }_{\Delta u}}^2}\hfill \\ & +{\displaystyle \sum _{i=0}^{N_{\mathrm{P}}-1}\{{\sigma }_{\mathrm{p}}P(k+i+1|k)T_{\mathrm{s}}\}}\hfill \end{array}$$

(34)

Subj. to:

• The discrete system state Equation (5);
• The constraints (31)–(33).

Note that Equation (34) is a nonlinear optimization problem, as the energy consumption index is a nonlinear function of the host vehicle’s acceleration and speed. To find a solution to this problem, in this study, an S-Function was devised in MATLAB by means of exhaustive search. Note that the exhaustive search method has also been employed in relevant work in the literature, such as [5,25], to tackle similar optimization problems. Interested readers are referred to [5,25] for more details.

4. Simulation Results

To verify the effectiveness of the proposed EOCFC strategy in car-following scenarios, a complete simulation environment, including a preceding vehicle and a host vehicle, was established using CarSim and MATLAB/Simulink in this study. In the simulation studies, the preceding vehicle is controlled to track the speed profile of NEDC, UDDS and WLTC test cycles, and the host vehicle follows the preceding vehicle using either the proposed EOCFC strategy or its competing method—multi-objective ACC (MO-ACC) [15]. It should be mentioned that for both EOCFC and MO-ACC, the optimal torque distribution strategy proposed in [26] is used to determine the torques of the front and rear motors. Since torque distribution is not a focus of this paper, it is not further discussed in the following text. The simulation parameters used are listed in

Table 1. Simulation parameters.

Parameters	Unit	Value
m	kg	2270
A	m2	3.0
CD	-	0.3
f	-	0.008
r	m	0.393
i0	-	10.885
α	deg	0
th	s	1.5
thmin	s	1.2
thmax	s	2.5
d0	m	5
d0_min	m	3
d0_max	m	6
Δvmin	m/s	−3.5
Δvmax	m/s	4
TTC	s	−2.5
ds	m	3
amin	m/s2	−2.8
amax	m/s2	1.2
jmin	m/s3	−6
jmax	m/s3	6

.

4.1. Car-Following and Ride Comfort Performance

Figure 8 and Figure 9 demonstrate the host vehicle speed profile and the inter-vehicle spacing resulting from the MO-ACC strategy and the EOCFC strategy, under the typical WLTC test cycle. In terms of vehicle speed, these competing strategies perform similarly, and both closely track the speed profile of the WLTC test cycle. However, these two algorithms provide very different spacing patterns: the MO-ACC strictly follows the desired inter-vehicle spacing, while the EOCFC yields an inter-vehicle spacing located within the upper and lower spacing boundaries, as explained in Section 3.1. This difference indicates that with the proposed EOCFC strategy onboard, the requirement on car-following performance is relaxed without jeopardizing vehicle safety (namely, the actual inter-vehicle spacing is maintained within a safe range). By this means, optimization of ACC economy performance in the car-following scenario can be made possible.

The acceleration results of the host vehicle using these two competing strategies are plotted in Figure 10. It is seen that under the WLTC test cycle, the acceleration amplitude resulting from EOCFC is smaller than the amplitude produced by MO-ACC, indicating that the proposed EOCFC provides superior ride comfort. The underlying reason for this superiority is that the demand power of the host vehicle is incorporated in the cost function, which reduces the occurrence of high-power operation of the host vehicle.

4.2. Energy Economy

The energy consumption resulting from these two algorithms under three standard test cycles are given in

Table 2. Energy consumption under various test cycles (initial SOC = 0.8).

Control Scheme	Energy Consumption (kWh)
	NEDC	UDDS	WLTC
MO-ACC	1.3687	1.3762	3.5537
EOCFC	1.3614	1.3304	3.5000
Improvement	0.53%	3.33%	1.51%

. It is seen that EOCFC outperforms MO-ACC for all three test cycles; specifically, EOCFC reduces energy consumption by 0.53%, 3.33% and 1.51% under NEDC, UDDS and WLTC test cycles, respectively.

It should be noted that the above percentages may seem like small numbers, but considering the large number of existing electric vehicles in the world and especially their quickly-growing sales, a small percentage of energy economy improvement can bring about magnificent economic benefits in total. Therefore, the 0.53%~3.33% energy saving ratio reported in this paper is not negligible but indeed significant. Moreover, as mentioned in Section 3, this study is dealing with a multi-objective optimization problem in which three types of performance (i.e., car-following performance, ride comfort and energy consumption) should be simultaneously taken into consideration. In other words, when optimizing energy economy, we need to make sure that the other two performances (car-following performance and ride comfort) do not seriously deteriorate. Hence, the final result we present in this article is an optimal outcome that achieves a good balance between these three performances.

5. Conclusions and Future Work

ACC has been widely considered as a promising approach to reduction in energy consumption of modern vehicles. In this paper, an EOCFC strategy is proposed to improve energy economy of a FRIDEV in car-following scenarios. The energy consumption of this vehicle is accurately quantified by means of its demand power, which is incorporated in the cost function for energy optimization. Without jeopardizing car-following performance and ride comfort, the energy economy in car-following scenarios is improved. The proposed EOCFC strategy is evaluated under the standard NEDC, UDDS and WLTC test cycles, in comparison with a benchmark method, MO-ACC. The results show that EOCFC decreases energy consumption by 1.3687 kWh, 1.3762 kWh and 3.5537 kWh under NEDC, UDDS and WLTC driving cycles, achieving economy improvements of 0.53%, 3.33% and 1.51%, respectively.

In our future studies, car-following control strategies for vehicle platoons will be investigated, and effects of road slope and traffic information will be taken into consideration in economy optimization of vehicle platoons.