A Cooperative Car-Following Eco-Driving Strategy for a Plug-In Hybrid Electric Vehicle Platoon in the Connected Environment

Abstract

The development of the Connected and Autonomous Vehicle (CAV) and Hybrid Electric Vehicle (HEV) provides a new effective means for the optimization of eco-driving strategies. However, the existing research has not effectively considered the cooperative speed optimization and power allocation problem of the Connected and Autonomous Plug-in Hybrid Electric Vehicle (CAPHEV) platoon. To this end, a hierarchical eco-driving strategy is proposed, which aims to enhance driving efficiency and fuel economy while ensuring the safety and comfort of the platoon. Firstly, an improved car-following model is proposed, which considers the motion states of multiple preceding vehicles. On this basis, a platoon cooperative car-following decision-making method based on model predictive control is designed. Secondly, a distributed energy management strategy is constructed, and a bionic optimization algorithm based on the behavior of nutcrackers is introduced to solve nonlinear problems, so as to solve the energy distribution and management problems of powertrain systems. Finally, the tests are conducted under the driving cycle of the Urban Dynamometer Driving Schedule (UDDS) and the Highway Fuel Economy Test (HWFET). The results show that the proposed strategy can ensure the driving safety of the CAPHEV platoon in different scenes, and has excellent tracking accuracy and driving comfort. Compared with the rule-based strategy, the equivalent energy consumption of UDDS and HWFET is reduced by 20.7% and 5.5% in the battery’s healthy charging range, respectively.

1. Introduction

With the continuous growth of the global economy and the expansion of the population scale, the energy crisis and environmental problems have become the key challenges to restrict the sustainable development of modern society. Under this background of transportation, as the main field of global oil consumption, tackling low-carbon transformation and upgrading demand is particularly urgent. As a new energy vehicle, the Hybrid Electric Vehicle (HEV) optimizes the operating points of the power system through the collaborative work of the electric drive system and the Internal Combustion Engine (ICE), so as to improve fuel efficiency. Compared with HEVs, Plug-in Hybrid Electric Vehicles (PHEVs) show more prominent energy-saving potential due to their large-capacity batteries that can be recharged from the external grid to supplement energy. In addition, PHEVs/HEVs can obtain braking kinetic energy from deceleration through regenerative braking, which is impossible for traditional fuel vehicles to achieve [1]. Therefore, to fully utilize the energy-saving advantages of PHEVs/HEVs, high-performance eco-driving strategies are essential.

Eco-driving strategies can be divided into two parts: speed planning and energy management. With the rapid development of intelligent transportation systems, Connected and Autonomous Vehicles (CAVs) can obtain information such as the location, speed, and acceleration of other vehicles through vehicle-to-vehicle (V2V) and vehicle-to-infrastructure (V2I) communication. Based on the multi-source heterogeneous data obtained by V2V and V2I communication, the speed planning system can reduce the frequency of sudden braking and acceleration by dynamically optimizing the vehicle speed, and improve the driving comfort and fuel economy under the premise of ensuring driving safety and traffic efficiency [2]. At present, a large amount of research has been conducted on vehicle speed planning under connected conditions. Yu and Shi [3] proposed an extended Full Velocity Difference (FVD) model, which considers the influence of vehicle-to-vehicle distance fluctuations on the speed of the target vehicle. The results show that the model can improve the stability of traffic flow and fuel efficiency. Based on this, Guo et al. [4] proposed an improved FVD model considering the speed fluctuations of multiple preceding vehicles. Han et al. [5] proposed a car-following model considering the average speed of the preceding vehicle in the connected environment. The results showed that its fitting accuracy was higher than that of the FVD model. Wen et al. [6] proposed a longitudinal control algorithm based on the FVD model and model predictive control (MPC) to improve the tracking accuracy, driving safety, and fuel economy of the CAV platoon. Wang et al. [7] constructed a simulation framework based on the following model and cellular automata. However, the car-following models have limitations in speed planning, which are not only difficult to support cooperative planning at the platoon formation, but are also difficult to realize multi-objective optimization, such as energy consumption and comfort. Zhao et al. [8] proposed an eco-driving strategy based on MPC, which utilizes V2I and V2V technologies to reduce the fuel consumption of the CAV platoon in intersection scenarios. Viadero-Monasterio et al. [9] proposed a control strategy based on static output feedback for the stability problem of vehicle queues caused by quality uncertainty. They adopted the predecessor–leader-following topology and designed the controller through linear matrix inequality. Bertoni et al. [10] proposed an ecological cooperative adaptive cruise control algorithm based on nonlinear MPC to predict the short-term trajectory of the vehicle in front, thereby reducing vehicle energy consumption. Kim et al. [11] proposed a predictive cruise controller based on learning MPC, which utilizes historical travel data to enhance vehicle fuel economy. Kim et al. [12] proposed an energy-aware motion planning framework that integrates longitudinal speed and lateral lane-changing decisions for connected electric vehicles in urban transportation. Viadero-Monasterio et al. [13] addressed the problem of network disconnection in heterogeneous platoons by adopting an adaptive switching between the predecessor–leader-following and predecessor-following topologies, and combining an event-triggering mechanism to reduce control instructions. To enhance the fuel efficiency of the CAV platoon in highway scenarios, Zhai et al. [14] proposed a collaborative control strategy based on the distributed MPC framework. Zhang et al. [15] proposed a hierarchical velocity planning algorithm based on MPC, which combines the multiple targets shooting method to achieve optimal velocity distribution. Sharma et al. [16] proposed an eco-driving algorithm for heavy-duty trucks. The algorithm can predict the speed distribution of the leading vehicle and plan the speed of the vehicle to achieve low fuel consumption. Lu et al. [17] proposed a hierarchical eco-driving method. The upper layer adopted a dynamic programming model to generate the best driving reference trajectory. At the lower layer, a dual-mode speed controller based on MPC was developed to handle infeasible reference trajectories. Although the model predictive control algorithm can achieve the cooperative planning of single vehicles and platoons, it has the limitations of high computational complexity and long time consumption.

Energy management aims to optimize the powertrain system without altering the vehicle’s motion state, so as to enhance fuel efficiency and reduce energy consumption. Energy management strategies can be divided into two methods: optimization-based methods and rule-based methods. The rule-based methods are simple to design and easy to implement, but the formulation of rules usually depends on expert experience or prior knowledge, and the adaptability to driving conditions is poor. The optimization-based methods can be further classified into global optimization and instantaneous optimization. Common global optimization methods include dynamic programming (DP) [18,19,20], and Pontryagin’s Minimum Principle (PMP) [21]. Although the methods based on DP and PMP can obtain the global optimal solution, their performance depends on complete prior driving information. However, the time-varying characteristics of actual driving cycles and the uncertainty of information make it difficult to apply such methods in practical engineering. The equivalent consumption minimization strategy (ECMS) is an instantaneous optimization method to solve the problem of optimal control in energy management [22,23,24]. The basic idea is to convert electrical energy consumption into equivalent fuel consumption, thereby solving for the optimal power distribution ratio. Compared with global optimization methods, ECMS avoids the prior knowledge requirements for the entire driving cycle by solving the problem of the optimal instantaneous equivalent energy consumption at each time step. Model predictive control is another commonly used instantaneous optimization method. This method predicts the equivalent energy consumption in a finite time step in the future based on the PHEV power system model, thereby solving the optimal power distribution between the motor and the ICE [25,26,27]. Compared with the single-step instantaneous optimization strategy of the ECMS, the MPC algorithm has stronger robustness by rolling to solve the optimal control sequence in the predicted horizon.

So far, much research has been conducted on the energy management strategies of PHEVs. Dehkordi et al. [28] constructed a distance-based DP algorithm and proposed an eco-driving strategy based on this. The results show that this strategy has significant energy-saving performance under the condition of limited security constraints. Wang and Lin [25] proposed a two-layer eco-driving strategy for the Connected and Autonomous Hybrid Electric Vehicles in a mixed driving scenario to reduce fuel consumption. Hu et al. [29] proposed a distributed control algorithm based on V2V technology. The algorithm combines switch feedback control and eco-MPC methods to optimize the fuel consumption of the platoon. Zhu et al. [30] addressed the eco-driving problem of PHEVs by constructing it as a partially observable Markov decision process and applying the proximal policy to optimize the deep reinforcement learning algorithm for the solution. The results show that this controller reduces fuel consumption by more than 17% compared to the baseline controller for human driving. Viadero-Monasterio et al. [31] addressed the issue of heterogeneous electric vehicle platoons affected by measurement noise, external disturbances, and actuator failures. They adopted a predecessor–leader-following topology, combined with Kalman filters, to estimate and compensate for faults and disturbances. Nie and Farzaneh [32] proposed a real-time dynamic predictive cruise control system to minimize the energy consumption of electric vehicles under comprehensive traffic conditions. Chada et al. [33] proposed a predictive eco-driving assistance system, which uses information from on-board sensors and geographic information of the driving route to calculate the optimal driving speed. Zhou et al. [19] proposed an energy management strategy based on MPC, which integrates navigation data to reduce fuel consumption. Deshpande et al. [18] embedded the equivalent consumption minimization strategy into the DP algorithm to optimize the speed planning and energy management of PHEVs. In reference [34], Bian et al. proposed a fuel economy optimization method for distributed platoons. Du et al. [35] designed a bi-level eco-driving strategy for electric trucks. The upper level calculates the optimal solution of the velocity planning problem by adopting the alternating direction multiplier method. The lower level achieves optimal energy consumption by calculating the power distribution of the motor. Shieh et al. [36] considered the pulse and coasting control methods of HEVs and proposed an MPC algorithm based on this for alternating control, thereby saving fuel. Wingelaar et al. [37] proposed a branch and boundary MPC algorithm to achieve eco-driving. Zhu et al. [38] constructed the eco-driving problem of CAPHEVs as a three-state rolling time-domain optimal control problem, including vehicle speed, SoC, and driving time, and designed a DP parallel architecture based on CUDA on an NVIDIA GPU to reduce the solution time. Viadero-Monasterio et al. [39] constructed power demand models for different vehicle types at signal intersections, designed algorithms for the length of the control area and the optimal arrival time, and took into account the acceleration preferences of drivers at the same time. In addition, due to the strong nonlinear characteristics, multi-variable coupling relationships, and complex constraint conditions of the nonlinear model of the PHEV power system, its optimization solution process faces a significant challenge that traditional methods find difficult to converge efficiently. Intelligent optimization algorithms based on bionics have demonstrated remarkable computational efficiency and strong global optimal search capabilities in such nonlinear optimization problems by simulating biological behaviors, such as fish foraging and bird migration [40,41].

However, existing studies have not systematically solved the cooperative optimization problem of eco-driving for the Connected and Autonomous Plug-in Hybrid Electric Vehicle (CAPHEV) platoon. The analysis of the existing literature reveals the following limitations. Firstly, although references [19,20,22,24,25,26,27,36] involve the energy management problem of the PHEV, this research is limited to single-vehicle car-following scenarios, and they lack in-depth research on the cooperative eco-driving strategy of the platoon. Secondly, the research in references [6,8,15,16,17,28,29,34] is still confined to the speed planning of traditional vehicles and fails to fully consider the unique energy management strategies and coupling characteristics of the power system of PHEVs. Note that existing research generally focuses on taking driving safety and fuel economy as the parallel optimization goals of PHEV eco-driving strategies. However, driving safety is directly related to the driver’s life safety, and its importance is particularly prominent. Therefore, driving safety should be the primary consideration rather than an optimization objective level that is parallel to fuel economy.

To further explore the potential of the eco-driving strategy of the CAPHEV platoon, a cooperative car-following eco-driving strategy is proposed. At the upper level, the proposed strategy realizes cooperative decision-making on vehicle driving behavior based on the current motion state of the platoon and other environmental factors. The lower-level controller aims to optimize the torque distribution of the motor and ICE of each vehicle to enhance fuel efficiency and reduce energy consumption. Figure 1 shows the hierarchical control scheme of the proposed strategy, which consists of the speed planning layer, the energy management layer, and the power execution layer. As shown in Figure 1, the target platoon collects real-time road environment information through V2V and V2I technologies and on-board sensors. In upper-level decision-making, the leading vehicle formulates motion decisions based on road information using the improved car-following model. The following vehicles first calculate the initial value of acceleration using the improved car-following model, and then optimize them through the MPC algorithm to generate a safe, comfortable, and efficient motion decision. This step-by-step strategy not only retains the advantages of MPC in multi-objective cooperative optimization of platoons but also has the characteristics of the low computational complexity of car-following models, thus achieving a balance between optimization performance and computational efficiency. The lower-level strategy will comprehensively consider nonlinear constraints, such as the state of charge of the battery and fuel consumption efficiency, and adopt the energy management strategy (EMS) based on nonlinear time-varying MPC to achieve the optimal distribution of engine torque and motor torque. Finally, the result is sent to the PHEV powertrain system, thereby achieving online eco-driving control.

The two major contributions of this study are as follows:

(1) A cooperative car-following decision-making method integrating the improved Intelligent Driver Model (IDM) and MPC is designed. This method effectively improves the car-following performance of the platoon and fuel economy while ensuring driving safety.

(2) A distributed energy management strategy is proposed. This strategy transforms the power distribution problem into a mathematical model by establishing an equivalent fuel consumption model and optimizing it using the MPC algorithm to achieve real-time optimization and adjustment of power distribution. For the nonlinear problems in the optimization solution process, an intelligent optimization algorithm based on the behavior of nutcrackers is introduced, which effectively improves the computational efficiency and optimization accuracy.

The rest of this article is organized as follows: In Section 2, the CAPHEV system model and the hierarchical eco-driving strategy are introduced. In Section 3, the evaluation results of the proposed strategy under different driving cycles are provided. In Section 4, the conclusion is given.

2. Methodology

In this section, the hypotheses about the CAPHEV platoon are first made. Secondly, to construct a hierarchical eco-driving strategy, the PHEV power system is modeled. Finally, the proposed cooperative car-following decision-making method and distributed energy management strategy are introduced in detail.

2.1. Hypothesis

To facilitate the development of the eco-driving strategy, the following hypotheses are made in this study.

Hypothesis 1.

In this study, all vehicles are equipped with a cooperative perception network composed of V2V and V2I communication modules, on-board sensors, and Road Side Units (RSUs), which can obtain the motion state parameters (such as position, velocity, and acceleration) of surrounding vehicles.

Hypothesis 2.

To establish an analyzable vehicle dynamics model, it is assumed that all vehicles within the traffic flow have isomorphic dynamic characteristics, and the road adhesion coefficient remains constant.

Hypothesis 3.

This study is limited to the longitudinal motion of vehicles on urban roads, and the yaw dynamics and lane-changing behavior are ignored.

2.2. Vehicle Powertrain Modeling

In this paper, a single-axle parallel PHEV is selected as the research object, which is equipped with a 63 kW/1.9 L engine, a 49 kW motor, an Automated Mechanical Transmission (AMT), and a battery pack [23,25,27]. The specific powertrain structure and key parameters are shown in Figure 2 and

Table 1. Parameter configuration of the PHEV powertrain.

Component	Parameter	Value
Engine	Maximum power (kw)	63
	Maximum torque (N·m)	145
Motor	Maximum power (kw)	49
	Maximum torque (N·m)	250
Transmission	Final transmission ratio	4.55
	AMT transmission ratio	[3.79 2.17 1.41 1.00 0.86]
Battery pack	Maximum battery capacity (Ah)	13
	Nominal voltage (V)	288

.

(1) Vehicle Longitudinal Kinematic Model

According to reference [42], considering the rolling resistance, slope resistance, air resistance, and acceleration resistance of the vehicle during running, the longitudinal kinematic model of the vehicle is constructed:

$$F_d\left(t\right)=mg\mu cos\theta +mgsin\theta +{\displaystyle \frac{1}{2}}\rho A{C}_D{v}^2\left(t\right)+ma\left(t\right),$$

(1)

where $F_d\left(t\right)$ represents the driving force, $\mu$ and $C_D$ are the rolling resistance coefficient and aerodynamic resistance coefficient, respectively, m is the vehicle mass, A is the windshield area of the vehicle, $\rho$ is the air density, g is the gravitational acceleration, $\theta$ is the road slope, and $v\left(t\right)$, $a\left(t\right)$ are the instantaneous velocity and acceleration of the target vehicle, respectively.

The relationship between the required driving force and the output torque of the motor and engine is expressed as follows:

$$F_d·r_w=T_{req}·i_0{i}_g{\eta }_t^{sign\left(T_w\right)}+T_b{T}_{req}=T_{em}+T_{eng},$$

(2)

where $r_w$ represents the radius of the wheel, $i_g$ and $i_0$ are the gear transmission ratios of the gearbox and the final drive, respectively, and ${\eta }_t$ is the conversion efficiency. $T_{eng}$ and $T_{em}$ represent the engine torque and the motor torque, respectively. $T_{\mathrm{req}}$ is the torque required by the vehicle, and $T_b$ is the braking torque acting on the wheels, $sign\left(T_w\right)=\left\{\begin{array}{c}1,T_w\ge 0\\ -1,T_w<0\end{array}\right.$. According to the structural characteristics of the parallel PHEV, the relationship between wheel velocity $n_w$, motor velocity $n_{em}$, and engine velocity $n_{eng}$ is described as follows [22]:

$$n_{em}=n_{eng}=n_w·i_0{i}_g.$$

(3)

(2) Engine Model

The engine provides the main power for the PHEV. During the vehicle’s driving, the engine propels the vehicle by converting fuel into mechanical energy. Referring to references [19,25,26], the fuel consumption rate of the engine can be expressed as a function of the engine’s instantaneous torque $T_{eng}$ and rotational velocity $n_{eng}$, as shown below:

$$Q_{BSFC}=f_b\left(T_{eng},n_{eng},\right)$$

(4)

where $Q_{BSFC}$ represents the brake fuel consumption rate; the unit is g/kWh.

From this, the instantaneous fuel consumption of the vehicle can be derived, such that

$${\dot{m}}_{fc}={\displaystyle \frac{T_{eng}·n_{eng}·Q_{BSFC}}{3.6\times {10}^6}},$$

(5)

where the unit of instantaneous fuel consumption ${\dot{m}}_{fc}$ is g/s. Note that the engine cannot operate at a speed lower than idle.

(3) Battery Model

Referring to the literature [23,43,44,45], an equivalent circuit model is adopted to model the battery pack, which includes open-circuit voltage and equivalent internal resistance.

$$P_{batt}=V_{oc}{I}_b-I_b^2{R}_b,$$

(6)

where $P_{batt}$, $V_{oc}$, $I_b$, and $R_b$ represent the battery power, open-circuit voltage, current, and equivalent resistance, respectively. The relationship between open-circuit voltage, equivalent resistance, and the state of charge (SoC) is shown in Figure 3.

The SoC is an important parameter for measuring the remaining power of a battery. It is modeled as follows:

$$\stackrel{•}{SoC}=-{\displaystyle \frac{I_b}{Q_b}},$$

(7)

where $Q_b$ represents the maximum capacity of the battery pack. Combined with Equations (6) and (7), the instantaneous SoC change rate can be expressed as follows:

$$\stackrel{•}{SoC}=-{\displaystyle \frac{V_{oc}-\sqrt{V_{oc}^2-4R_b{P}_{em}}}{2R_b{Q}_b}}.$$

(8)

(4) Electric Motor (EM) Model

The motor provides auxiliary power for the PHEV to improve fuel economy. On the one hand, it can be used as an electric motor to obtain electricity from the battery pack and convert it into mechanical energy. On the other hand, it can also be used as a generator to absorb mechanical energy from redundant power from the engine or regenerative braking power to charge the battery. Referring to the literature [18,22,24], the motor efficiency can be expressed as a function of the instantaneous motor torque $T_{em}$ and rotational velocity $n_{em}$, as shown below:

$${\eta }_{em}=f_{em}\left(T_{em},n_{em}\right).$$

(9)

According to the working state of the motor, the motor power $P_{em}$ can be expressed as follows:

$$P_{em}=\left\{\begin{array}{c}{T}_{em}·{\displaystyle \frac{n_{em}}{{\eta }_{em}}} , T_{em}\ge 0\\ T_{em}·n_{em}·{\eta }_{em} , T_{em}<0\end{array},\right.$$

(10)

where $T_{em}\ge 0$ indicates that the EM is working as an electric motor, and $T_{em}<0$ indicates that the EM is working as a generator. Additionally, the energy required by the motor comes entirely from the battery pack, so $P_{em}=P_{batt}$.

2.3. Hierarchical Eco-Driving Strategy

2.3.1. Cooperative Car-Following Decision-Making for CAPHEV Platoon

(1) Improved Car-following Model

The IDM is a microscopic traffic flow simulation model for car-following behavior analysis, which is widely used in the development of adaptive cruise control and a cooperative autonomous driving system [46]. The model consists of the expected acceleration term in the free state and the deceleration adjustment term based on the safety distance constraint. The mathematical expression of the IDM is as follows:

$$a_n\left(t\right)=a_{\mathrm{m}}\left[1-{\left({\displaystyle \frac{v_n\left(t\right)}{v_{\mathrm{m}}}}\right)}^{\delta }-{\left({\displaystyle \frac{s\ast }{\Delta x_n\left(t\right)}}\right)}^2\right],$$

(11)

$$s\ast =s_0+v_n\left(t\right)·T_d-frac{v}_n\left(t\right)·\Delta v_n\left(t\right)2\sqrt{a_m{b}_n},$$

(12)

where $a_n\left(t\right)$ and $v_n\left(t\right)$ represent the acceleration and velocity of the nth vehicle at time t, respectively. $\Delta x_n\left(t\right)=x_{n-1}\left(t\right)-x_n\left(t\right)$ and $\Delta v_n\left(t\right)=v_{n-1}\left(t\right)-v_n\left(t\right)$ represent the distance and velocity difference between the $n-1$th and the nth vehicles, respectively. $a_m$, $v_m$, $s_0$, $T_d$, and $b_n$ represent the maximum acceleration, maximum velocity, minimum safe distance, expected headway, and maximum deceleration, respectively.

However, the IDM only relies on the information of a preceding vehicle to decide the motion state of the vehicle and ignores the influence of the dynamic changes in the road conditions in front on the driving behavior of the vehicle [47,48]. This leads to frequent sudden acceleration and deceleration of vehicles in unstable road traffic environments, which seriously affects the safety, comfort, and economy during the driving process. To this end, an improved IDM is constructed, which considers the motion state parameters of multiple preceding vehicles, such as position, velocity, and acceleration. The mathematical expression for the improved IDM is as follows:

$$a_n\left(t\right)=a_{\mathrm{m}}\left[1-{\left({\displaystyle \frac{v_n\left(t\right)}{v_{\mathrm{m}}}}\right)}^{\delta }-{\left({\displaystyle \frac{s\ast }{\Delta x_n\left(t\right)}}\right)}^2\right]+\gamma ·\sum _{i=1}^q{\xi }_i\Delta v_{n+1-i}\left(t\right)+\kappa ·\sum _{i=1}^q{\xi }_i\Delta a_{n+1-i}\left(t\right),$$

(13)

$${\xi }_i=\left\{\begin{array}{c}{\displaystyle \frac{q-1}{q^i}},i\ne q\\ {\displaystyle \frac{1}{q^{i-1}}},i=q\end{array}(i=1,2,\cdots ,q),\right.$$

(14)

where q represents the number of vehicles in front of the target vehicle within the platoon, $\gamma$ and $\kappa$ are the constant gain coefficients, and ${\xi }_i$ is the weighted coefficient of different positions of the leading vehicles. The parameter configurations of the improved IDM are shown in

Table 2. Ear stone model parameters.

Parameter	Value	Parameter	Value
$a_m$ (m/s2)	2	$v_m$ (m/s)	30
$s_0$ (m/s)	3	$T_d$ (s)	1.5
$b_n$ (m/s2)	3	$\gamma$	0.2
$\kappa$	0.1

.

(2) Cooperative Decision-making Method

To further enhance the car-following performance of the platoon, i.e., ensure driving safety and minimize the headway, relative velocity, and frequency of acceleration and deceleration, a cooperative decision-making method of vehicle platoon is proposed, which integrates the car-following model and model predictive control. First, a vehicle kinematic model, including acceleration, velocity, and position state variables, is constructed as follows:

$$\left\{\begin{array}{c}x(k+1)=x\left(k\right)+v\left(k\right)·T\\ v\left(\mathrm{k}+1\right)=v\left(k\right)+a\left(k\right)·T\\ a(k+1)=a\left(k\right)+\Delta a\left(k\right)·T\end{array}.\right.$$

(15)

Then, the problem of vehicle platoon cooperative control is transformed into a constrained multi-objective optimization problem. The acceleration value calculated by the improved car-following model is taken as the feedforward input, and the optimal acceleration within each control horizon is solved online through the MPC algorithm. The problem of vehicle platoon cooperative driving optimization can be expressed as follows:

$$\begin{array}{c}minJ\left(k\right)={\displaystyle \sum _{t=k}^{k+Np-1}}{\displaystyle \sum _{i=0}^n}[{(\Delta x_i-s_0)}^2+\Delta {v_i}^2],k=0,1,2,\dots \hfill \\ statevariable:{\xi }_k={\left[x_1,v_1,a_1,\dots ,x_n,v_n,a_n\right]}^T\hfill \\ controlvariable:u_k={[\Delta a_2,\dots ,\Delta a_n]}^T\hfill \\ subjectto\left\{\begin{array}{c} 0\le v\le v_{max}\\ a_{min} \le a\le a_{max}\\ \Delta a_{min}\le \Delta a\le \Delta a_{max},\end{array}\right.\hfill \end{array}$$

(16)

where $\Delta x_i$ and $\Delta v_i$ represent the distance and the velocity difference between the ith vehicle and the preceding vehicle, respectively. $x_i$, $v_i$, and $a_i$ represent the position, velocity, and acceleration of the ith vehicle.

2.3.2. Distributed Energy Management Strategy

(1) Optimization Objectives

Due to the lack of a direct quantitative correlation between the fuel consumption and SoC, and excessive consumption of battery energy storage, this will lead to an accelerated decline in battery health, thereby generating additional battery life cycle costs. The ECMS design method proposed in reference [23] has been widely verified and applied in the existing research on energy management strategies for Hybrid Electric Vehicles [19,22,25], which achieves an optimal balance between fuel economy and battery life maintenance. Therefore, the ECMS is adopted to set the SoC range within 0.5–0.75 to achieve a balance between battery life and energy consumption. The equivalent factor $s_e$ is defined as

$$s_e\left(SoC\right)=\left\{\begin{array}{c}0.01 \hspace{1em}\hspace{1em}SoC>0.75\\ \begin{array}{c}-6\times SoC+5 0.75\ge SoC>0.5\\ \begin{array}{c} 10 0.5\ge SoC>0.2\\ 1000 SoC\le 0.2\end{array}\end{array}\end{array}.\right.$$

(17)

The instantaneous equivalent fuel consumption of EM can be expressed as

$${\dot{m}}_{ec}=s_e\left(SoC\right)·{\displaystyle \frac{{\tilde{Q}}_{BSFC}·P_{em}}{3.6\times {10}^6·{\overline{\eta }}_{em}}},$$

(18)

where ${\tilde{Q}}_{BSFC}$ represents the equivalent fuel consumption rate, with its value set at 240 g/kWh. ${\overline{\eta }}_{em}$ represents the average efficiency of the motor, with its value set at 88%.

The cost function $J_{eco}$ of the energy management strategy consists of the instantaneous fuel consumption ${\dot{m}}_{fc}$ of the engine and the instantaneous equivalent energy consumption ${\dot{m}}_{ec}$ of the motor, which is expressed as follows:

$$J_{eco}={\dot{m}}_{ec}+{\dot{m}}_{fc},$$

(19)

where the instantaneous fuel cost ${\dot{m}}_{fc}$ can be referred to Equation (5).

(2) Model Predictive Control

Model predictive control is a control strategy based on a rolling horizon optimization framework. Due to its explicit constraint handling ability for multi-variable coupled systems, strong robustness in dynamic environments, and the collaborative mechanism of online optimization and feedback correction, it is widely applied in the field of real-time optimization control for complex dynamic systems [49]. Therefore, an energy management algorithm is based on ECMS and nonlinear time-varying MPC to solve the power distribution problem of PHEVs. Specifically, in each discrete time step, based on the system state variables (such as speed, SoC, and expected acceleration) obtained in real time, the controller predicts the future vehicle motion state and battery state in a finite horizon, according to the constructed nonlinear kinetic model and battery model. Then, a multi-objective optimization problem in a finite prediction horizon is constructed, which generates the optimal control sequence in the prediction horizon by solving the comprehensive cost function, including engine fuel consumption and motor equivalent energy consumption. Finally, the first control variable of the sequence is applied in real time to the powertrain system. The process is repeated with each time step until the system reaches its final state. The energy management optimization problem at the time step can be described as follows:

$$\begin{array}{c}min{J}_{opt}\left(k\right)={\displaystyle \sum _{t=k}^{k+N_p-1}}{J}_{eco}\left({\xi }_t,u_t\right)·T_s,k=0,1,2,\dots \hfill \\ statevariable:{\xi }_k={\left[T_{req},SoC\right]}^T\hfill \\ controlvariable:u_k=T_{eng}\hfill \\ subjectto\left\{\begin{array}{c}{T}_{eng}^{min}\le T_{eng}\le T_{eng}^{max}\\ T_{em}^{min}\le T_{em}\le T_{em}^{max}\\ So{C}_{min}\le SoC\le So{C}_{max}\end{array}\right.\hfill \end{array},$$

(20)

where $T_s$ is the sampling time, and $T_{eng}^{min}$ and $T_{eng}^{max}$ are the minimum and maximum values of the engine torque, respectively. $T_{em}^{min}$ and $T_{em}^{max}$ are the minimum and maximum values of the EM torque, respectively. $So{C}_{min}$ and $So{C}_{max}$ are the minimum and maximum values of the SoC, respectively. To enhance computational efficiency, the control horizon $N_c$ is set to 1, which means there is only one group of elements in the control sequence.

(3) Nutcracker Optimization Algorithm

The objective function of energy management optimization problems is nonlinear and non-explicit, which leads to the inability to obtain its analytical expression. Therefore, the nonlinear MPC problem is usually solved by the numerical method. However, traditional numerical methods are prone to getting stuck in local optima and require a large amount of computation. The nutcracker optimization algorithm (NOA) achieves a balance between global search and local development by simulating the cooperative behavior of nutcrackers in the processes of foraging, storing, and retrieving food [41]. In the energy management of the PHEVs, multi-objective nonlinear optimization problems are effectively addressed through biomimetic mechanisms, which avoid falling into the limitations of local optima or high computational complexity. Before the introduction of the NOA algorithm, it is necessary to give some basic definitions. The position of the ith nutcracker at the time t is represented as $X_i^t$. The cost function can be represented by the quality of food at a certain position, i.e., $Y=f\left(X_i^t\right)$, where Y is the optimization objective. Assume that the information is shared within the nutcracker group, which facilitates the nutcrackers in finding high-quality food. According to the behavior of nutcrackers, the NOA algorithm can be expressed as follows:

Initialization. The position of nutcrackers is initialized as a uniform distribution within the feasible region defined by Constraint (20).

Foraging. Nutcrackers will fly from their current position to other positions to search for food, as shown in Equation (21):

$$X_i^{t+1}=\left\{\begin{array}{c}{X}_i^t,if{\tau }_1<{\tau }_2\\ \left\{\begin{array}{c}{X}_m^t+\lambda (X_A^t-X_B^t)+\mu (r^{2}u-l),ift\le T/2\\ X_C^t+\varpi (X_A^t-X_B^t)+\mu (r^{2}u-l)(r_1<\delta ),\mathrm{otherwise}\end{array},\mathrm{otherwise},\right.\end{array}\right.$$

(21)

where A, B, and C are three different nutcracker individuals randomly selected from the group, and $X_m^t$ represents the average position of the nutcracker group at time t. ${\tau }_1$, ${\tau }_2$, r, and $r_1$ are random numbers between 0 and 1. $\lambda$ is the random number generated by the Levy flight function, T is the maximum number of iterations, and u and l represent the upper and lower limitation of the feasible region, respectively. $\delta$ represents the probability of a global search for nutcrackers. $\varpi$ is a random number generated based on the uniform distribution (${\tau }_3$), normal distribution (${\tau }_4$), and Levy flight (${\tau }_5$), as shown in Equation (22):

$$\varpi =\left\{\begin{array}{c}{\tau }_3,if{r}_1<r_2\\ \begin{array}{c}{\tau }_4,if{r}_2<r_3\\ {\tau }_5,if{r}_1<r_3\end{array}\end{array} ,\right.$$

(22)

where $r_2$ and $r_3$ are consistent with $r_1$ and are random numbers between 0 and 1.

Storage. If high-quality food is successfully found, the nutcrackers will transport it to the storage area, as shown in Equation (23):

$$X_i^{t+1}=\left\{\begin{array}{c}{X}_i^t+\mu (X_{opt}^t-X_i^t)\left|\lambda \right|+r_1(X_A^t-X_B^t),if{\tau }_1<{\tau }_2\\ X_{opt}^t+\mu (X_A^t-X_B^t),if{\tau }_1<{\tau }_3\\ X_{opt}^t·p,\mathrm{otherwise}\end{array},\right.$$

(23)

where $X_{opt}^t$ represents the best position of the individual at time t, and p is a random factor that linearly decreases from 1 to 0, according to the number of iterations.

Cache search. When winter comes, the nutcrackers shift their behavior from foraging and storing to seeking the storage areas. The nutcrackers will locate the storage areas based on the reference points in their memory. The reference point can be expressed as

$$\left\{\begin{array}{c} R_{i,1}^t=X_i^t+\alpha cos\left(\theta \right)(X_A^t-X_B^t)\\ R_{i,2}^t=X_i^t+\alpha cos\left(\theta \right)((u-l){\tau }_3+l)·\varsigma \end{array},\right.$$

(24)

where $R_{i,k}^t$ represents the position of the kth reference point of the ith nutcracker at time t ($k=1,2$). $\alpha$ is a factor that prevents premature convergence to a possible solution, and its value is $\left\{\begin{array}{c}{(1-t/T)}^{2t/T},if{r}_1>r_2\\ {(t/T)}^{2/t},otherwise\end{array}\right.$. $\varsigma$ is a factor that selects between global search and local search, and its value is $\left\{\begin{array}{c} 1,ifr<0.2\\ 0,otherwise\end{array}\right.$.

Based on the reference point information, the nutcrackers search for the storage areas according to Equation (25). Note that the nutcrackers will randomly select a reference point to locate the storage spot. If food is not found, the nutcrackers will flee in all directions to search, thus avoiding getting stuck in a local optimum.

$$X_i^{t+1}=\left\{\begin{array}{c}\left\{\begin{array}{c} X_i^t,if{\iota }_1<{\iota }_2\\ X_i^t+r_1(X_{opt}^t-X_i^t)+r_2(R_{i,1}^t-X_C^t),otherwise\end{array}\right.,if{\iota }_3<{\iota }_4\\ \left\{\begin{array}{c} X_i^t,if{\iota }_5<{\iota }_6\\ X_i^t+r_1(X_{opt}^t-X_i^t)+r_2(R_{i,2}^t-X_C^t),otherwise\end{array},otherwise\right.\end{array},\right.$$

(25)

where ${\iota }_k(k=1,2,\dots ,6)$ is a random number between 0 and 1.

Retrieval. The nutcrackers select the optimal position based on different reference points and the current position. The specific equation is as follows:

$$X_i^{t+1}=\left\{\begin{array}{c} X_i^t,iff\left(X_i^t\right)<f\left(R_{i,1}^t\right)andf\left(X_i^t\right)<f\left(R_{i,2}^t\right)\\ R_{i,1}^t,iff\left(R_{i,1}^t\right)<f\left(R_{i,2}^t\right)andf\left(R_{i,1}^t\right)<f\left(X_i^t\right)\\ R_{i,2}^t,iff\left(R_{i,2}^t\right)<f\left(R_{i,1}^t\right)andf\left(R_{i,2}^t\right)<f\left(X_i^t\right)\end{array}.\right.$$

(26)

To solve the complex nonlinear optimization problem in the energy management of PHEVs, an intelligent algorithm based on the behavior of nutcrackers is introduced. The algorithm extensively seeks the optimal feasible solution in the initial stage and can effectively escape from the local optimal solution. Meanwhile, compared with other optimization algorithms, such as the ant colony algorithm, particle swarm algorithm, and genetic algorithm, the NOA algorithm has significantly improved in computational efficiency and the ability to find the global optimum. This means it is suitable for the application of eco-driving in the CAPHEV platoon.

3. Results and Discussion

In this section, the simulation platform and the configuration of experimental parameters are introduced. Then, the simulation comparison results of the proposed method and the algorithms based on the IDM and rules under different driving cycles are presented, and the control accuracy and energy-saving effects are also compared and analyzed.

3.1. Simulation Experiment Parameter Configuration

MATLAB/Simulink and Carsim are used to co-simulate the proposed method. The configuration of the computer used for simulation is as follows: an Intel Core i5-13400 CPU, an NVIDIA RTX 3060 GPU, and 32 GB of RAM. The parameters used in the simulation experiment are shown in

Table 3. Simulation parameter settings.

Parameter	Value	Parameter	Value
Vehicle Mass m (kg)	1500	Air Resistance Coefficient $C_D$	0.44
Frontal Area A (m2)	2.66	Air Density $\rho$ (kg/m3)	1.206
Road Slope $\theta$ (°)	0	Road Resistance Coefficient $\mu$	0.0125
Tire Radius (m)	0.343	Sampling Time $T_s$ (s)	0.1
Transmission Efficiency ${\eta }_t$ (%)	90	Control Horizon $N_c$	1
Initial $SoC$	0.75	Prediction Horizon $N_p$	10
$v_e$(m/s)	90	$v_{max}$ (m/s)	30
$a_{max}$ (m/s2)	2	$\Delta a_{max}$ (m/s3)	1.5
$a_{min}$ (m/s2)	−3	$\Delta a_{min}$ (m/s3)	−1.5

.

To verify the effectiveness of the proposed eco-driving strategy in real driving scenarios, two common driving cycles are set, that is, the Urban Dynamometer Driving Schedule (UDDS) [21,27,43,50,51] and the Highway Fuel Economy Test (HWFET) [27,43,51]. The parameters are shown in

Table 4. Parameters of the driving cycles.

Driving Cycle	Time (s)	Average Velocity (m/s)	Maximum Velocity (m/s)	Mileage (km)
UDDS	1370	8.8	25.3	12.0
HWFET	765	21.5	26.8	16.5

.

3.2. Rule-Based Eco-Driving Strategy

To evaluate the performance of the proposed strategy, the IDM and the rule-based Electric Assisted Control Strategy (EACS) are adopted as the benchmark control strategy. The EACS is one of the most commonly used rule-based strategies in parallel PHEVs, where the engine serves as the main energy source, and the motor only operates when the required torque is too high or the engine’s operating efficiency is too low. Therefore, this strategy improves overall fuel economy and is widely used as the benchmark eco-driving control strategy [22,25,43,50]. Since the EACS strategy is not the core of this research, it will be briefly introduced. The EACS strategy determines the torque distribution between the engine and the motor based on factors such as the current SoC, speed, and the maximum torque of the engine. Figure 4 shows the workflow of the EACS.

Table 5. Parameter settings of the EACS.

Parameter	Description	Value
$So{C}_H$	The highest SoC allowed	0.7
$So{C}_L$	The lowest SoC allowed	0.5
$v_L$	The threshold of velocity	4 m/s
$t_{min}$	The threshold factor for the ICE to charge	0.5
$t_{ch}$	The charging factor	50
$t_{off}$	The threshold factor for the ICE to shut off	0.2
$t_{dis}$	The discharging factor	0.2

defines the parameters involved in the EACS.

3.3. Simulation Results

3.3.1. Analysis of Car-Following Performance

(1) UDDS

To evaluate the car-following performance of the CAPHEV platoon under the UDDS, vehicle No. 0 is set as the leading vehicle and follows the UDDS velocity trajectory. And the following five vehicles are taken as the research objects, i.e., No. 1, 2, 3, 4, and 5. Figure 5 shows the velocity, space headway, and their error curves during the car-following process under the UDDS. Note the driving cycle of UDDS data derived from real urban road collection; its frequent start–stop complex dynamic incentives put forward higher requirements for the robustness of the model.

The simulation results in Figure 5a,b show that the proposed method demonstrates excellent tracking accuracy for the driving behavior of the preceding vehicle, while the IDM has significant deficiencies in this aspect. Specifically, when the preceding vehicle is in the acceleration–deceleration transition stage, the tracking error of the IDM shows a deteriorating trend as the platoon size expands, which seriously affects the safety, comfort, and economy of the platoon. Note that the proposed method still shows excellent robust tracking performance in the process of speed mutation and is not affected by the size of the platoon. It can be seen from Figure 5c,d that the fluctuation amplitude of the vehicle-to-vehicle space headway is significantly suppressed by the proposed method. And a compact and uniform vehicle-to-vehicle distance is maintained for the platoon under the premise of ensuring driving safety. However, obvious deficiencies in the control of the space headway are revealed in the IDM. The IDM cannot achieve stable adjustment of the distance, which leads to frequent acceleration and deceleration operations of the vehicle during the following process, thereby causing inefficient utilization of road space resources and ultimately having a negative impact on traffic flow efficiency and driving safety. Figure 5e–h show the longitudinal velocity error curves of each vehicle and the error curves of the distance between vehicles under the UDDS driving cycle. The results show that the velocity errors of the proposed method are all less than 3 m/s, and the space headway errors are all less than 5 m. Both indices are significantly better than the IDM model, which fully verifies the excellent longitudinal dynamic control accuracy of the proposed method under complex urban scenarios.

Figure 6 shows the acceleration and the jerk of the proposed method under the UDDS driving cycle. It can be seen from Figure 6 that the peak acceleration of the proposed method is maintained within 2 m/s², and the peak rate of acceleration change is less than 1.5 m/s³. Severe acceleration fluctuations are avoided by this control input, which ensures that the vehicle’s longitudinal dynamic parameters are maintained within a reasonable threshold range for driving safety and comfort.

(2) HWFET

Similarly, the performance of the proposed method is evaluated in the HWFET driving cycle. Figure 7 shows the simulation results of different methods under the HWFET driving cycle.

It can be seen from Figure 7a–d that, compared with the IDM, the velocity of the proposed method under the HWFET driving cycle is closer to the velocity trajectory of the target vehicle, and the change in the space headway between vehicles is smoother. Figure 7e–h show the longitudinal velocity error curves of each vehicle and the error curves of the distance between vehicles under the HWFET driving cycle. The results show that in the stable speed stage, the velocity error of the proposed method is less than 1 m/s, and the headway error is all less than 2 m. Therefore, the established method is demonstrated to achieve superior velocity tracking accuracy and enhanced platoon stability compared to the IDM model, with this advantage consistently observed across both urban road and highway driving conditions. Therefore, it can be further verified that, compared with the IDM, the proposed method has higher speed tracking accuracy and car-following stability in both urban road and highway driving cycles.

3.3.2. Analysis of Energy Management Performance

In this section, the energy-saving performance of the proposed eco-driving strategy is evaluated under UDDS and HWFET driving cycles. The benchmark strategy consists of the IDM and the rule-based energy management strategy. To comprehensively analyze the energy consumption performance of PHEVs, two evaluation metrics are adopted in this study, i.e., the equivalent consumption of fuel and electricity combined and the fuel consumption. In addition, the SoC of the battery is also the most important evaluation metric for PHEVs. Therefore, the fuel consumption, equivalent energy consumption, and SoC are taken as evaluation metrics. The usage of the engine and motor of the PHEV under the control of the proposed method in the UDDS and HWFET driving cycles is shown in Figure 8.

The results show that under the UDDS driving cycle, the proposed method preferentially calls the motor for low-speed drive, makes full use of the high torque response characteristics of the motor to meet the frequent acceleration and deceleration requirements, and avoids the ICE running in the inefficient interval. Under the HWFET driving cycle, the proposed method automatically switches to the ICE-based power distribution mode and realizes the energy consumption optimization during high-speed cruise by optimizing the engine operating point to the efficient working area. To evaluate the energy-saving performance of the proposed method, the change curves of fuel consumption, equivalent energy consumption, and SoC, and the distribution of engine efficiency under two driving cycles are analyzed. Figure 9a,b show the equivalent energy consumption, fuel consumption, and SoC change curves under the UDDS and HWFET driving cycles, respectively.

Algorithm 1 represents the IDM + rule-based strategy.

Algorithm 2 represents the cooperative car-following method + rule-based strategy.

Algorithm 3 represents the proposed eco-driving strategy.

As can be seen from Figure 9, under the same driving task, the target vehicle controlled by the proposed method shows lower fuel consumption and equivalent energy consumption, and its core lies in the precise regulation of the SoC of the battery. Specifically, the relatively high SoC of the vehicle is maintained throughout the driving process, which allows the deep involvement of the motor in complex operating conditions (such as the inefficiency interval of the ICE) to retain a sufficient energy margin, thus achieving the reduction in energy consumption. At the same time, the stable SoC change trend avoids the deep charging and discharging of the power battery, which effectively inhibits the power battery attenuation and prolongs the service life.

Table 6. Simulation results of different algorithms under different driving cycles.

Algorithm	Driving Cycle	Fuel Consumption (g)	Equivalent Energy Consumption (g)	SoC (%)	Average Running Time (ms)
Algorithm 1	UDDS	378.5	379.7	72.1	-
Algorithm 2		310.8 (−17.9%)	333.0 (−12.3%)	65.1 (−9.7%)	385
Algorithm 3		300.1 (−20.7%)	301.1 (−20.7%)	72.3 (+0.3%)	422
Algorithm 1	HWFET	635.1	638.4	72.7	-
Algorithm 2		624.5 (−1.7%)	624.6 (−2.2%)	74.5 (+2.5%)	398
Algorithm 3		605.7 (−4.6%)	603.0 (−5.5%)	77.5 (+6.6%)	430

presents the simulation results of different algorithms under different working conditions in terms of the fuel consumption, equivalent energy consumption, SoC, and average running time. It can be seen that the proposed algorithm shows energy-saving advantages under different driving conditions. Under the UDDS driving cycle with frequent start–stop urban roads, the fuel consumption and equivalent energy consumption are significantly reduced by 20% with the efficient power allocation strategy led by the motor. Under the HWFET driving cycle of the highway steady-state scenario, the energy consumption is still reduced by 4.5–5.5% through the optimization of the ICE economic operating point, and the high level of the SoC is maintained in the whole process, which reflects the significant economic advantages of the energy management strategy from complex driving cycles to stable scenarios. In addition, the average running time of the proposed algorithm is less than 500 ms, which satisfies the real-time requirements of platoon control. Thus, the proposed algorithm avoids control lag and reduces string stability caused by excessive time consumption.

Figure 10a,b show the distribution of operating points of the ICE under UDDS and HWFET driving cycles, respectively. The blue, red, and yellow points indicate the ICE operating points of Algorithms 1, 2, and 3, respectively. It can be seen that most of the ICE operating points using the proposed method are located in the efficient region (i.e., $Q_{BSFC}<240$ g/kWh).

4. Conclusions

Aimed at the difficulties of cooperative control, complex multi-objective optimization, and high computational load faced by the CAPHEV platoon, a hierarchical cooperative car-following eco-driving strategy is proposed in this study. In upper-level decision-making, this strategy integrates the car-following model considering information from multiple preceding vehicles with the nonlinear time-varying MPC algorithm. Under the premise of ensuring driving safety, it improves the platoon tracking accuracy, reduces energy consumption, and achieves a balance between car-following performance and energy consumption. In lower-level decision-making, the power distribution is optimized by this strategy through the integration of MPC and the NOA, without the expected motion state of the CAPHEV being affected. The results show that, compared with the IDM + rule-based strategy, the proposed method reduces both fuel consumption and equivalent energy consumption by 20.7% under the UDDS driving cycle. And the method reduces the fuel consumption and equivalent energy consumption by 4.6% and 5.5% under the HWFET driving cycle, respectively. Meanwhile, the proposed method can control the speed error and the distance between the vehicles within 3 m/s and 5 m, respectively, and maintain a high state of charge during driving, which verifies its applicability in urban and high-speed traffic scenarios.

However, there are still some limitations in this study. Firstly, the road slope parameter is not taken into account. Secondly, the current control framework only focuses on longitudinal motion and does not involve the coordinated optimization of lateral and longitudinal motion. In actual scenarios, the slope of the road can alter the driving force requirements of vehicles. For instance, going uphill increases the power consumption of the motor and affects the balance of battery discharge. Going downhill easily wastes the regenerative braking opportunities of gravitational potential energy, which deviates from the energy consumption optimization goals of ecological driving. Lane-changing behavior can cause instantaneous fluctuations in energy consumption, which disrupts the stability of energy distribution. The increased tire resistance when turning at small corners will also push up actual energy consumption and reduce the eco-driving adaptability of the strategy. In follow-up research, we plan to incorporate these two factors into the model. By integrating high-precision map slope data and collecting lateral motion energy consumption features, we aim to construct an eco-driving strategy that integrates multiple scenarios to enhance its actual eco-driving performance in complex road conditions. The hierarchical control architecture proposed in this study offers a new approach that takes into account both economy and car-following performance for the cooperative eco-driving of a platoon, which enriches the theoretical system of cooperative control for the CAPHEV platoon, and provides technical support for the large-scale platoon operation of new energy vehicles.