Research on regenerative braking control of electric vehicles based on game theory optimization

Abstract

The energy-efficient, clean, and quiet attributes of electric vehicles offer solutions to conventional challenges related to resource scarcity and environmental pollution. Consequently, thorough research into harmonizing energy recuperation during braking, enhancing vehicle stability, and ensuring occupant comfort in electric vehicles is imperative for their effective advancement. The study introduces a regenerative braking control strategy for electric vehicles founded on game theory optimization to enhance braking performance and optimize braking energy utilization. Develop a regenerative braking control approach based on the dynamic model of an electric vehicle equipped with hub motors. Employing game theory, we establish participants, control variables, strategy sets, benefit functions, and constraints to optimize the coefficient K for regenerative braking. The efficacy and superiority of the control strategy model are validated through joint simulations using Matlab/Simulink and AVL Cruise. Research findings indicate: (1) Speed tracking error remains below 3% in both NEDC and CLTC-P simulations, underscoring the effectiveness of the dynamic model and control strategy devised in this study. (2) The energy recovery rate achieved by the game theory-based optimization strategy surpasses that of the Cruise self-contained strategy and fuzzy control strategy by 18.06% and 4.5% in the NEDC simulation, and by 13.48% and 3.85% in the CLTC-P simulation, respectively. The adhesion coefficient curves implemented on the front and rear axles, derived from the game theory optimization control strategy, closely approximate the ideal adhesion coefficient curve, leading to a substantial enhancement in the car's braking stability. The degree of jerk magnitude regulated by the game theory optimization strategy consistently falls within the ±3 m/s³ threshold, resulting in a considerable enhancement in the comfort of vehicle occupants. These outcomes underscore the efficacy of the game theory-based optimized control strategy in enhancing energy recovery, braking stability, and comfort throughout the braking process of the vehicle.

Introduction

In light of recent challenges related to traditional resource scarcity and environmental degradation, there is a rising global focus on the advancement of new energy vehicles by governments and businesses. China is dedicated to bolstering its energy independence and has established ambitious targets to peak carbon dioxide emissions by 2030 and attain carbon neutrality by 2060. ¹ Studies on regenerative braking in new energy vehicles have notably advanced the energy recovery rate during braking, leading to extended electric vehicle range. This enhancement is pivotal in improving the energy efficiency of electric vehicles and mitigating carbon emissions.

Effective regenerative braking control strategies involve determining the allocation of braking forces between the front and rear axles, as well as the division between motor-mechanical braking forces. Presently, the management of braking stability in hub-motor electric vehicles predominantly relies on adhering to the prescribed front and rear braking force distribution standards outlined by the Economic Commission of Europe (ECE). Three primary categories exist for front and rear-axle brake power distribution strategies: ideal brake force distribution, optimal regenerative braking, and parallel brake force distribution strategies. ² Ideal braking force distribution strategy results in the braking force of the front and rear axles being distributed according to the neighborhood of the ideal braking force curve (I curve), meaning that the front and rear wheels of the car have been held at the same time when braking. This approach optimally utilizes adhesion conditions, resulting in enhanced braking stability and energy recovery.^3,4 The optimal regenerative braking strategy, which is the series braking force distribution strategy, has been to prioritize the use of motor braking while ensuring braking stability. When ensuring the maximum power of the motor, the remaining braking power is supplemented by mechanical braking to maximize the energy recovery efficiency.^5,6 The parallel brake force distribution strategy, maintaining the conventional mechanical braking system, involves the direct addition of motor braking torque to the mechanical brake system. Although this method exhibits lower energy recovery efficiency, it boasts a straightforward structure and dependable safety features.^7,8 Research indicates that each strategy presents unique benefits. The ideal brake force distribution and optimal regenerative braking strategies necessitate high-precision linear control systems, rendering their implementation more complex. Conversely, the parallel brake force distribution strategy offers easier implementation, albeit with lower energy recovery rates and reduced comfort. Consequently, this paper introduces a brake power distribution strategy that leverages the I curve, ECE regulations, f-line, and r-line for improved stability and driving comfort through the appropriate front and rear axle brake power distribution, alongside maximizing energy recovery efficiency.

The economy of electric vehicles in braking primarily relies on the distribution of motor regenerative braking and mechanical friction braking torque. The allocation of motor-mechanical braking torque determines the amount of recovered braking energy, directly impacting energy efficiency and driving range. Braking torque allocation control strategies can be categorized into rule-based and optimization-based approaches. Rule-based composite control strategies, such as those incorporating fuzzy control or neural networks, utilize practical experience to establish torque distribution rules based on torque demand under various braking conditions and constraints restricting regenerative braking. ⁹ A regenerative braking control strategy based on a fuzzy logic controller typically considers inputs such as braking intensity, vehicle speed, power battery state of charge (SOC), and brake pedal position. The output of the controller is commonly the ratio coefficient between regenerative braking torque and total braking torque.^4,10–13 He et al. ¹⁴ introduced a neural network (NN) controller for composite braking systems aiming to enhance energy efficiency and braking stability concurrently. Simulation results revealed a 3.04% enhancement in the final SOC with the NN controller compared to the parallel braking approach, indicating superior energy efficiency. ¹⁴ Maia et al. ¹⁵ utilized the reinforcement learning (RL) technique to refine the regenerative braking fuzzy logic model. By gathering data from 12 urban areas and focusing on regenerative factors, they successfully demonstrated the ability of the model to learn the regenerative braking factor. An optimization algorithm-based control strategy was developed to distribute braking torque, taking into account vehicle structure, motor and battery features, braking safety regulations, and other constraints. The corresponding objective function was formulated based on diverse optimization objectives related to braking modes, followed by the application of various optimization algorithms for online or offline optimal braking torque distribution solutions. ⁹ Common optimization control strategies, such as genetic algorithms, particle swarm optimization, game theory, and model prediction, were explored. Pei et al. ¹⁶ utilized a genetic algorithm to derive look-up tables of allocation coefficients under varying speeds and braking intensities. In comparison with the I curve allocation strategy, their approach demonstrated enhanced performance in terms of energy regeneration and braking stability. Zhang et al. ¹⁷ introduced a predictive regenerative braking control strategy primarily utilizing particle swarm optimization, coupled with the integration of the ant colony algorithm into the iterative process, showcasing enhanced stability and economic efficiency in the experimental results. Cheng et al. ¹⁸ proposed a game theory-based control strategy to harmonize the electric motor, internal combustion engine, and braking system to achieve a multi-objective equilibrium, leading to improved vehicle tracking, reduced fuel consumption, and enhanced driving comfort. Tang et al. ¹⁹ suggested a model predictive control strategy for coordinating a regenerative braking system with an anti-lock braking system, resulting in superior braking performance with a 22.76% boost in energy recovery efficiency compared to the conventional control strategy. Zhang et al. ²⁰ presented a novel predictive control method integrating adaptive cubic exponential prediction and dynamic programming to optimize motor braking torque and wheel cylinder pressure, yielding increased energy recovery efficiency. The above findings indicate that the current technology for controlling for distributing the electric-mechanical composite braking force in pure electric vehicles is relatively mature. Both rule-based control strategies and optimized control strategies have achieved satisfactory results. However, the design of the braking control strategy for electric vehicles with hub motors should also consider the vehicle's stability, comfort, and safety during braking, while also aiming to improve braking economy. There is insufficient research considering multiple properties during the braking process of hub-motor electric vehicles. This article addresses this gap by designing a regenerative braking control strategy for electric vehicles using game-theoretic optimization. It constructs a multi-objective optimization model incorporating the benefit function of economy, stability, and comfort. The strategy ensures vehicle stability and comfort during braking while maximizing energy recovery.

To address the aforementioned background, the current study centers on hub-motor electric vehicles, conducting a dynamic analysis to propose a control strategy that optimizes both energy efficiency and control performance during braking. The strategy prioritizes three critical elements: economy, stability, and comfort, ultimately leading to significant enhancement in the braking economy and control performance of hub-motor electric vehicles. A judicious allocation of braking force between the front and rear axles is implemented to bolster stability and driving comfort during braking. A regenerative braking control strategy leveraging game theory optimization is crafted to increase the proportion of regenerative braking torque through multi-objective game optimization. This approach guarantees braking stability and comfort, maximizes braking energy recovery, extends the operational range of electric vehicles, and contributes to accomplishing the “double carbon” objective.

Modeling hub-motor electric vehicle dynamics

Hub-motor electric vehicle system structure

The hub-motor distributed four-wheel drive system propels the wheels directly propelled by motors, eliminating numerous transmission parts like transmissions, driveshafts, differentials, and half shafts. Such a design enhances the drive system's efficiency and facilitates the achieving of a lightweight vehicle structure. ²¹

The hub-motor distributed four-wheel drive system consists of several components, including the distributed drive controller, vehicle control unit, motor control unit, hybrid battery (Battery H), battery management system, electronic control unit, electro-hydraulic brake, and four hub-motor drive units. Figure 1 illustrates the system structure.

Figure 1.

Structure of hub-motor electric vehicle composite braking system.

Longitudinal braking dynamics model

During braking, the vehicle experiences a combined effect of braking force, air resistance, and road resistance, including ramp and rolling resistance based on force analysis.

$$F_j=F_w+F_i+F_f+F_b$$ (1)

$$-\sigma m{\displaystyle \frac{dv}{dt}={\displaystyle \frac{C_{D}A{v}^2}{21.15}+mg\sin \alpha +mg{f}_r\cos \alpha +F_b}}$$ (2)

where $F_w$ is the air resistance generated by the longitudinal motion of the vehicle, $F_i$ is the ramp resistance, $F_f$ is the rolling resistance, $F_j$ is the acceleration resistance, $F_b$ is the total braking force, $C_D$ is the air resistance coefficient, A is the longitudinal windward area of the vehicle, v is the longitudinal speed of the vehicle, m is the vehicle mass, g is the gravitational acceleration, $\alpha$ is the angle of the current ramp, $f_r$ is the traveling resistance coefficient, and $\sigma$ is the vehicle rotating mass conversion factor, which is greater than 1.

Battery modeling

The power battery, serving as an energy storage device for energy recovery, which serves as an energy storage device for energy recovery, significantly impacts regenerative braking. In this study, a hybrid power battery is chosen.

$$SO{C}_t=SO{C}_0-{\displaystyle \frac{100{\int }_0^t{I}_b\left(\sigma \right)d_{\sigma }}{Q_a}}$$ (3)

$$I_b={\displaystyle \frac{U_b-\sqrt{P_m{R}_b}}{R_b}}$$ (4)

$$P_r=U_b{I}_b$$ (5)

where $SO{C}_t$ is the current battery SOC value, $SO{C}_0$ is the initial battery SOC value, $P_r$ is the battery regenerative power, $U_b$ and $I_b$ are the terminal voltage and current of the battery, respectively, $Q_a$ is the capacity of the battery, $R_b$ is the equivalent internal resistance of the battery, and $P_m$ is the motor power.

Motor model

Hub motors function based on the same principle as centralized drive motors and can employ various motor types, including permanent magnet synchronous motors, asynchronous servo motors, and switched reluctance motors, among others. Among these options, the brushless permanent magnet synchronous hub motor stands out as the favored choice due to its attributes such as high power density, efficiency, and broad speed range. Consequently, this study opts for the brushless permanent magnet synchronous hub motor with the motor power equation as follows:

$$P_m=\{\begin{array}{c}{\displaystyle \frac{T_m{n}_m}{9550{\eta }_m}motor}\\ {\displaystyle \frac{T_m{n}_m{\eta }_m}{9550}generator}\end{array}$$ (6)

where $T_m$ is the output torque of the motor, $n_m$ is the motor speed, and ${\eta }_m$ is the motor efficiency.

Analyze the regenerative braking control mechanism

The strategy for regenerative braking control plays a crucial role in influencing the rate of energy recovery during braking. It enhances the economy and stability of electric vehicles by effectively distributing the braking forces between the front and rear axles and the motor-mechanical braking force. By ensuring braking stability and safety, increasing the proportion of regenerative braking can increase the rate of energy recovery, thereby extending the vehicle's driving range. This section introduces the strategies for braking force distribution between the vehicle's front and rear axles in the vehicle's braking system, as well as the strategy for distributing the electric motor-mechanical braking force.

Front and rear axle brake force distribution

Front and rear axle braking force distribution determines the vehicle braking stability and the degree of utilization of adhesion conditions, the braking system should ensure that the braking system should ensure that the front wheels hold before the rear wheels, or that the front and rear wheels hold simultaneously, to prevent potential side slipping of the rear wheels, these two are stable braking conditions.

Define Z as the braking strength, represented by the ratio of vehicle deceleration to gravitational acceleration, given by equation (7):

$$Z=\left|{\displaystyle \frac{dv}{dt}{\displaystyle \frac{1}{g}}}\right|$$ (7)

Automobile braking process when the front and rear wheels at the same time hold, the relationship between the front and rear wheels according to the ground adhesion formula, which is represented by equation (8), the resulting curve, known as the “I curve,” represents the ideal relationship between the braking force of the front and rear wheels.

$$F_r={\displaystyle \frac{1}{2}\left[{\displaystyle \frac{mg}{h_g}\sqrt{L_b^2+{\displaystyle \frac{4h_{g}L}{mg}{F}_f}}-\left({\displaystyle \frac{mg{L}_b}{h_g}+2F_f}\right)}\right]}$$ (8)

where $F_f$ and $F_r$ for the front and rear wheel brake force, respectively, $h_g$ for the height of the car's center of gravity, L for the distance between the front and rear axles of the car, $L_a$ for the center of gravity of the car to the centerline of the front axle, $L_b$ for the center of gravity of the car to the distance from the centerline of the rear axle, and $\phi$ for the road adhesion coefficient.

Vehicle braking process when the front wheels hold, the front and rear wheels’ ground adhesion relationship is shown in equation (9), the line drawn from the equation is the f line.

$$F_r={\displaystyle \frac{(L-\phi h_g)F_f}{\phi h_g}-{\displaystyle \frac{mg{L}_b}{h_g}}}$$ (9)

Vehicle braking process when the rear wheels hold and the front wheels roll, the relationship between the ground adhesion of the front and rear wheels is shown in equation (10), and the line plotted from this equation is the r line.

$$F_r={\displaystyle \frac{(mg{L}_a-F_f{h}_g)\phi }{L+\phi h_g}}$$ (10)

To ensure optimal braking stability and efficiency, Regulations set by the United Nations Economic Commission for Europe govern the distribution of braking force between the front and rear axle brakes of automobiles. These regulations are represented by equations (11) and (12), and the curves resulting from these equations represent the ECE regulations.

$$F_f={\displaystyle \frac{Z+0.07}{0.85}{\displaystyle \frac{G}{L}(L_b+Z{h}_g)}}$$ (11)

$$F_r=GZ-F_f$$ (12)

This study is centered on the hub-motor electric vehicle and analyzes the relationship curves for distributing braking force between the front and rear axles. These curves comprise five lines: I, ECE, f (φ = 0.3), f (φ = 0.7), and r (φ = 0.7). The intersection points of these curves, namely O, A, B, and C, represent the key points of braking force distribution. Furthermore, D corresponds to a point on the I curve. In terms of braking stability, the curves for front and rear axle braking force distribution should remain within the bounds defined by the I curve, ECE curve, and x-axis, as shown in Figure 2.

Figure 2.

Front and rear axle brake force distribution.

By utilizing equations (8) to (12), the coordinates of the characteristic point intersections can be calculated, determining the corresponding values of braking strength, denoted as Z. The braking strength Z can be categorized into four intervals, as illustrated in Table 1.When 0 < Z ≤ 0.18, the smaller the motor torque under a certain speed the lower the motor efficiency, and in the case of very small torque motor efficiency with the torque decreases rapidly, in scenarios with low-strength braking, the front axle continuously provides the required braking force, to enhance the efficiency of wheel motor power generation, 0.18 < Z ≤ 0.3, the braking force in accordance with the brake When the braking power is distributed according to the front wheel holding curve at braking intensity Z_AB, as braking intensity increases, all four hub motors contribute to the braking process to maximize energy recovery, when 0.3 < Z ≤ 0.7, on the basis of ensuring the braking stability of the vehicle and the driving comfort, the distribution curves of braking power for the front and rear axles are adjusted to align as closely as possible with the I curve, aims to maximize the efficiency of energy recovery, Z > 0.7 emergency braking is considered, in order to ensure the safety of braking and the battery life, the braking power distribution curves are as close as possible to the I-curve. To ensure the safety of braking and the service life of the battery, the motor is not involved in braking, and the braking power is provided by mechanical braking. To summarize, the front and rear axle braking force distribution curves are the OABCD curves shown in Figure 2.

Table 1.

Front and rear axle brake force distribution.

Braking strength	Braking status	Brake force distribution curve
0 < Z ≤ 0.18	Mild braking	OA
0.18 < Z ≤ 0.3	Medium braking	AB
0.3 < Z ≤ 0.7	Medium to high braking	BC
Z > 0.7	Emergency braking	CD

Motor-mechanical braking force distribution

The rate of regenerative braking energy recovery is determined by the distribution of motor-mechanical braking force. During braking, prioritizing vehicle braking safety and maximizing the energy recovery rate is crucial. To achieve this, increasing the regenerative braking force of the front and rear axle motors is essential to enhance energy recovery during braking and improve the vehicle's driving range.

$$F_e=K{F}_t$$ (13)

where $F_e$ is the total regenerative braking force of the four hub motors, $F_t$ is the total demand braking force, and K is the regenerative braking scaling factor.

The following will optimize the value of the ratio coefficient K of the front and rear axle motor regenerative braking force to the mechanical braking force based on the game theory optimal control strategy. The ratio of motor-mechanical braking force allocation for both the front and rear wheels of the vehicle is K. Once the allocation of braking force for the front and rear axles is completed, the optimized value K will be used in the computation module, which calculates the motor-mechanical braking force and mechanical braking force for both the front and rear wheels.

Regenerative braking game model

Game theory optimization model

Non-cooperative game

The paper focuses on an energy-saving control strategy by framing it as an optimization issue concerning torque allocation ratios. To achieve a balance between higher energy efficiency and improved control performance during the vehicle braking process, a non-cooperative game control strategy is proposed. Game theory is commonly used in cooperative games and non-cooperative games. In non-cooperative games, participants are mutually constrained during the game process, with each seeking to maximize their revenue. In a Nash equilibrium, any participant's alteration would lead to diminished revenue. Determining the Nash equilibrium in a non-cooperative game disregards the participants’ weights. This methodology minimizes human subjectivity, resulting in increased objectivity and robustness.

Game modeling

A finite non-cooperative game consists of three components: $G=\{N,{(S_i)}_{i\in N},{(f_i)}_{i\in N}\}$ , where

1. $N=\{1,2,\dots ,n\}$ denotes that there are n participants here.

2. $(S_i{)}_{i\in N}=\prod _{i=1}^n{S}_i$ is the action set (strategy set) of the participants, where $S_i=\{S_1^i,S_2^i,\dots ,S_{k_i}^i\},i=1,2,\dots ,n$ is the strategy set of the $i$ th participants, which indicates that the $i$ th participant has $ki$ th strategies to choose from.

3. $f=(f_1,f_2,\dots ,f_i)$ , where $f_i:S\to \mathit{R}$ is the benefit function of the $i$ th participant.

In this article, the two participants are energy efficiency $\eta$ and control performance $C$ , two benefit functions $Y_e,Y_c$ are set to quantify energy efficiency $\eta$ and control performance C, respectively.

$$G=\{(\eta ,C),(T_m,T_b),(Y_e,Y_c)\}$$ (14)

where $\eta ,C$ are Participant 1 and Participant 2, respectively, $T_m,T_b$ are optimization variables, and $Y_e,Y_c$ are benefit functions.

The steps to find the Nash equilibrium solution for a non-cooperative game are

1. According to the determined participants, optimization variables, and benefit function, build the game theory optimization model $G=\{(\eta ,C),(T_m,T_b),(Y_e,Y_c)\}$ .

2. Based on the designed participants $\eta ,C$ and optimization variables $T_m,T_b$ , perform sensitivity analysis on the variables, enumerate the optimization variables, and classify them into strategy sets $S_{T_m}$ and $S_{T_b}$ .

3. Initialize and randomly generate feasible strategy sets, optimize the benefit function $Y_e,Y_c$ of each participant in the space of the respective strategy sets, get the corresponding optimal optimization variable values among the participants, and complete an iteration.

4. Calculate the number of norms between the combinations of strategies before and after, and judge whether the convergence condition $\parallel s^{(1)}-s^{(0)}\parallel \le \epsilon ={10}^{-6}$ is satisfied, if so, the game process ends, and the Nash equilibrium solution $\mathbb{N}=\{T_m^{Nash},T_b^{Nash}\}$ is obtained, otherwise, $s^{(1)}$ replace $s^{(0)}$ , and return to step 3.

Benefit function modeling

The objective is to achieve enhanced energy efficiency and superior control performance, which includes economy, stability, and comfort. It is quantified by two benefit functions: ${\mathit{Y}}_{\mathit{e}}$ and ${\mathit{Y}}_c$ . Energy efficiency, that is, economy, is quantified by the benefit function ${\mathit{Y}}_{\mathit{e}}$ of the energy recovery rate

$$T_m={\displaystyle \frac{9550P_m}{n_m}}$$ (15)

$${\mathit{Y}}_{\mathit{e}}={\displaystyle \frac{E_r}{E_z}={\displaystyle \frac{1000\int T_m{n}_m/9550dt}{{\sum }_{v_i>v_j}{\displaystyle \frac{1}{2}m\left({\left({\displaystyle \frac{v_i}{3.6}}\right)}^2-{\left({\displaystyle \frac{v_j}{3.6}}\right)}^2\right)-\int fmgvdt-\int {\displaystyle \frac{C_{D}A}{21.15}{(3.6v)}^{2}vdt}}}}}$$ (16)

where ${\mathit{Y}}_{\mathit{e}}$ is the braking energy recovery rate $E_r$ indicates the energy recovered from braking, $E_z$ is the total braking energy consumption (the amount of kinetic energy change minus the braking energy consumed by rolling resistance and wind resistance), $v_i$ is the start of braking, $v_j$ is the termination speed of braking, and f is the rolling resistance coefficient.

Control performance, including stability, comfort, and safety, is quantified by the use of the coefficient of adhesion benefit function $Y_1$ for the front and rear axles. The formula indicates that as the utilization attachment coefficients ${\phi }_f$ and ${\phi }_r$ approach the braking strength Z, the ground attachment conditions are fully utilized, leading to a more reasonable distribution of braking force for the vehicle and increased car stability.

$${\phi }_f={\displaystyle \frac{(F_{ef}+F_{mf})L}{mg(L_b+Z{h}_g)}}$$ (17)

$${\phi }_r={\displaystyle \frac{(F_{er}+F_{mr})L}{mg(L_a-Z{h}_g)}}$$ (18)

$$a={\displaystyle \frac{dv}{dt}}$$ (19)

$$F_b=m*gZ=F_{ef}+F_{mf}+F_{er}+F_{mr}=(F_{mf}+F_{mr})/(1-K)$$ (20)

$$T_m=(F_{ef}+F_{er})r_w$$ (21)

$$T_b=(F_{mf}+F_{mr})r_w$$ (22)

$$K={\displaystyle \frac{T_m}{T_m+T_b}}$$ (23)

$${\phi }_i={\displaystyle \frac{F_{Xbi}}{F_{Zi}}}$$ (24)

$${\mathit{Y}}_1=\sqrt{{({\phi }_f-Z)}^2-{({\phi }_r-Z)}^2}$$ (25)

where $F_{Xbi}$ is the ground braking force generated by the $i$ th axle of the car corresponding to the braking strength Z , $F_{Zi}$ is the normal reaction force of the ground on the $i$ th axle when the braking strength is $Z$ , ${\phi }_i$ is the utilized coefficient of adhesion of the $i$ th axle corresponding to the braking strength Z. $F_{\mathrm{ef}}$ front axle motorized force, $F_{\mathrm{er}}$ rear axle motorized force, $F_{\mathrm{mf}}$ front axle mechanical braking force, $F_{\mathrm{mr}}$ rear axle mechanical braking force, $T_b$ total mechanical braking torque, a vehicle braking deceleration, and $r_w$ dynamic rolling radius.

Occupant comfort is mainly evaluated by the degree of jerk of the vehicle, the lower the degree of jerk, the higher the ride comfort, and the benefit function ${\mathit{Y}}_2$ is shown in equation (26):

$${\mathit{Y}}_2={\displaystyle \frac{da}{dt}={\displaystyle \frac{d^{2}v}{d{t}^2}}}$$ (26)

$${\mathit{Y}}_{\mathit{c}}={\mathit{Y}}_1+{\mathit{Y}}_2$$ (27)

Benefit functions ${\mathit{Y}}_e$ and ${\mathit{Y}}_{\mathit{c}}$ are determined by the control variables of each and interdependence, while participants $\eta$ and C are considered self-interested, aiming to minimize their benefits to enhance braking energy efficiency and improve control performance, individually.

Constraints

This study aims to balance increased energy efficiency and enhanced control performance during braking in a hub-motor electric vehicle. Within the constraints of vehicle speed, SOC, braking strength, and motor torque, the distribution of wheel braking force prioritizes the engagement of the hub-motor braking for enhanced stability, comfort, and the maximization of braking energy recovery.

• (1) When the battery's SOC surpasses 90%, the motor is not employed for braking to prevent overcharge. In such cases, braking exclusively relies on the mechanical friction brake to extend the battery's lifespan. With the SOC ranging between 80% and 90%, the use of motor braking is progressively diminished.

  $$k_1=\{\begin{array}{c}1SOC\le 0.8\\ 9-10SOC0.8<SOC\le 0.9\\ 0SOC>0.9\end{array}$$ (28)
• (2) When the vehicle speed drops below 5 km/h, no energy recovery occurs during braking, and only the mechanical friction brake functions in the braking operation. In the speed range of 5–10 km/h, the motor functions at a low speed, leading to instability and minimal braking current generation. As a result, energy recovery efficiency is notably low, necessitating a reduction in electric power proportion.

  $$k_2=\{\begin{array}{c}0v\le 5\mathrm{km}/\mathrm{h}\\ 0.2v-15\mathrm{km}/\mathrm{h}<v<10\mathrm{km}/\mathrm{h}\\ 1v\ge 10\mathrm{km}/\mathrm{h}\end{array}$$ (29)
• (3) When the braking strength Z surpasses 0.7, the vehicle enters an emergency braking scenario. For optimal braking stability and safety, the motor is disengaged from braking, leaving the mechanical friction brake to execute the braking task independently.

  $$k_3=\{\begin{array}{c}1Z\le 0.7\\ 0Z>0.7\end{array}$$ (30)
• (4) The braking torque for each hub motor should not surpass its maximum torque limit at the current speed.

  $$k_4=\{\begin{array}{c}1T_i\le T_{\mathrm{nmax}\mathrm{\_}i}\\ {\displaystyle \frac{T_{\mathrm{nmax}\mathrm{\_}i}}{T_i}{T}_i>T_{\mathrm{nmax}\mathrm{\_}i}}\end{array}(i=\mathrm{flb},\mathrm{frb},\mathrm{rlb},\mathrm{rrb})$$ (31)

$$K=K_b{k}_1{k}_2{k}_3{k}_4$$ (32)

where $T_i$ is the braking torque of the left-front, right-front, left-rear, and right-rear motors, $T_{\mathrm{nmax}\mathrm{\_}i}$ is the maximum torque of the left-front, right-front, left-rear, and right-rear motors at this rotational speed point, and $K_b$ is the regenerative braking proportionality coefficient before constraints.

Game theory optimization model construction

The construction of the game theory optimization model is conducted using Matlab/Simulink software, while Figure 3 showcases the flowchart illustrating the regenerative braking control strategy derived from game theory optimization.

Figure 3.

Flowchart of regenerative braking control strategy based on game theory optimization.

Simulation verification

To assess the efficacy of the proposed regenerative braking control scheme, we develop the regenerative braking control strategy model and the comprehensive vehicle model using Matlab/Simulink with AVL Cruise software. This method allows us to confirm both the effectiveness of the simulation model and the superiority of the control strategy through concurrent simulation. The regenerative braking strategy based on fuzzy control is compared with the optimized control strategy based on the game theory proposed in this article, which has the inputs of vehicle speed (v), SOC, and braking intensity (Z), the output of which is the value of regenerative braking proportionality coefficient (K).^22,23

Vehicle parameters

This study focuses on a specific model of a hub-motor electric vehicle as the subject of investigation, with Table 2 detailing the parameters of the entire vehicle.

Table 2.

Vehicle parameters.

Parameter	Values
Wheelbase (m)	2.45
Gross mass (kg)	1240
Height of center of gravity (m)	0.54
Front/rear wheelbase (m)	1.47
Windward area (m2)	1.65
Air resistance coefficient	0.345
Wheel rolling radius (m)	0.301
Rolling resistance coefficient	0.0072
Horizontal distance from center of mass to front axle (m)	1.05
Horizontal distance from center of gravity to rear axle (m)	1.40

Model validation of regenerative braking allocation strategy based on game theory optimization

Validity

The simulation test conditions chosen include the New European Driving Cycle (NEDC) and the China Light-duty Vehicle Test Cycle-Passenger Car (CLTC-P). Figures 4 and 5 display the simulated speed tracking for these conditions. The speed tracking error remains below 3% under both simulation scenarios, confirming the accuracy of the simulation model for constructing or testing the braking control strategy.

Figure 4.

Vehicle speed tracking under the New European Driving Cycle (NEDC).

Figure 5.

Vehicle speed tracking under the China Light-duty Vehicle Test Cycle-Passenger Car (CLTC-P) operating conditions.

Stability

The braking process can assess the braking stability of the front and rear axles by analyzing the relationship between the coefficient of adhesion and braking strength. $\phi =z$ for the ideal coefficient of the adhesion curve. Equations (24) and (25) show that a closer match between the coefficient of adhesion and braking strength leads to more effective utilization of ground adhesion conditions, resulting in a balanced distribution of braking force and improved vehicle stability. In the research model of this article, based on ECE R13 braking regulations, it is established that for braking strength Z = 0.15–0.8, the rear axle utilization attachment coefficient curve should not be located above the front axle attachment coefficient curve, while the rear axle utilization attachment coefficient curve should be located in the straight line $Z=0.9$ or below. When the utilization attachment coefficient φ=0.2–0.8, $Z\ge 0.1+0.7(\phi -0.2)$ . Figure 6 illustrates the relationship curve between the utilization attachment coefficient and braking strength following the optimization simulation based on the aforementioned constraints.

Figure 6.

Utilization of adhesion coefficient curve before and after optimization.

Analysis of Figure 6 indicates that the front and rear axle utilization attachment coefficient curves adhere to ECE R13 braking regulations. Furthermore, the optimized front and rear axle utilization attachment coefficient curves are closer to the ideal attachment coefficient curves, which suggest that the optimization of the control strategy based on game theory is conducive to the improvement of the braking stability of the car.

Comfort

Adjustment of input braking force during braking enhances braking comfort. ²⁴ By optimizing the distribution of braking force, the degree of jerk can be reduced, leading to improved braking comfort.

Studies suggest that a comfortable degree of jerk for occupants should not exceed ±3 m/s³. ²⁵ Figures 7 and 8 display the degree of jerk curves for the fuzzy control strategy under NEDC and CLTC-P conditions. Notably, the degree of jerk of the game theory optimization-based control is noticeably lower than that of the fuzzy control strategy. The shock degree of the control based on the optimization strategy of the game theory is significantly smaller than that of the fuzzy control strategy under the NEDC condition and CLTC-P condition. Under the NEDC condition and CLTC-P condition, the degree of jerk controlled by the fuzzy control strategy is controlled within ±5 and ±4.3 m/s³ most of the time, respectively. The degree of jerk controlled by the game theory-based optimization strategy is controlled within the range of ±3 m/s³ most of the time, which significantly improves the comfort of the vehicle occupants. The game theory optimization control strategy proves effective in improving braking comfort.

Figure 7.

The curve of impact degree under the New European Driving Cycle (NEDC) condition.

Figure 8.

Shock level curve under the China Light-duty Vehicle Test Cycle-Passenger Car (CLTC-P) condition.

Economy

The simulation, conducted under NEDC and CLTC-P conditions, starts with a battery SOC initial value of 95% and runs until the SOC reaches 50%. Figures 9 and 10 present the comparison of regenerative braking proportionality coefficients (K) between the fuzzy control strategy and the strategy proposed in this study for both conditions. Additionally, Tables 3 and 4 display comparative simulation results for the economic analysis of Cruise's self-contained strategy, the fuzzy control strategy, and the strategy introduced in this article.

Figure 9.

Proportionality coefficient K under the New European Driving Cycle (NEDC) condition.

Figure 10.

Proportionality coefficient K under the China Light-duty Vehicle Test Cycle-Passenger Car (CLTC-P) operating condition.

Table 3.

Comparison of simulation results under the New European Driving Cycle (NEDC) operating conditions.

Parameter name	Cruise self-contained strategy	Fuzzy control strategy	Control strategy of this article
Energy consumed by the whole vehicle (kJ)	67846.0	77026.2	80851.4
Recovered energy (kJ)	6863.1	18156.7	22771.3
Effective energy recovery rate (%)	10.12	23.66	28.16

Table 4.

Comparison of simulation results under the China Light-duty Vehicle Test Cycle-Passenger Car (CLTC-P) operating conditions.

Parameter name	Cruise self-contained strategy	Fuzzy control strategy	Control strategy of this article
Energy consumed by the whole vehicle (kJ)	71351.8	78282.9	82047.3
Recovered energy (kJ)	10453.0	19010.9	23080.4
Effective energy recovery rate (%)	14.65	24.28	28.13

The comparison graph illustrates that, in the majority of instances, the proportionality coefficient K of the control strategy proposed in this article exceeds that of the fuzzy control strategy significantly. This suggests that the game-theoretic optimized control strategy enhances motor braking proportion, increases the recuperation of braking energy, and boosts the economy of vehicle braking. The control strategy presented in this paper demonstrates an effective energy recovery rate that surpasses that of both the fuzzy control strategy and Cruise's strategy by 4.5% and 18.04%, respectively, under the NEDC condition. Moreover, under the CLTC-P condition, the control strategy in this article outperforms the fuzzy control strategy with Cruise by 3.85% and 13.48%, affirming the superior economic efficiency of the designed control strategy.

In conclusion, the game theory-based optimized control strategy enhances energy recovery, braking stability, and driving comfort during the vehicle's braking process, leading to commendable performance. It achieves superior braking control in pure electric vehicles’ braking procedures and enhances energy utilization efficiency.

Conclusion

This article introduces a regenerative braking control strategy for electric vehicles, considering both energy efficiency and control performance, based on game theory optimization. A benefits function model assessing the economy, stability, and comfort of braking is developed for hub-motor electric vehicles. The regenerative braking control strategy and game-theoretic optimization model are implemented using Matlab/Simulink software. Subsequently, joint simulation is conducted with the overall vehicle model created in AVL Cruise software. The effectiveness and superiority of the proposed model and control strategy are validated through simulation verification under NEDC and CLTC-P conditions. The specific results are outlined below:

1. Model validity validation: The model's validity is confirmed under NEDC and CLTC-P conditions by comparing the target speed with the simulated speed. The speed tracking error remains below 3%, demonstrating alignment with practical control requirements for designing and developing braking control strategies.

2. Verify the optimization ability of the control strategy.

   • Enhancing vehicle braking stability: Comparing the relationship curves between the coefficient of adhesion utilization and braking strength before and after optimization under ECE R13 constraints reveals compliance with regulations. The optimized front and rear axle coefficient of adhesion curves align more closely with the ideal curves, signifying the effectiveness of the game theory-based optimization control strategy in enhancing braking stability.
   • Enhancing vehicle braking comfort: By comparing the degree of jerk curves under fuzzy and game optimization control strategies in NEDC and CLTC-P scenarios, the level of jerk governed by the game theory-based optimization strategy remained within ±3 m/s² for the majority of instances. The findings establish that the game-theoretic optimization control strategy enhances occupant comfort, consequently ameliorating the braking comfort of the vehicle.
   • Enhancing vehicle braking economy: When assessing K-value curves under the fuzzy and game optimization control strategies in NEDC and CLTC-P conditions, the K-value observed with the game optimization control strategy notably surpasses that of the fuzzy control strategy consistently. The outcomes suggest that the game-theoretic optimal control strategy facilitates increased recuperation of braking energy, thereby boosting the vehicle's braking economy.

While the optimal control strategy grounded in game theory significantly boosts the braking economy and control performance of electric vehicles equipped with hub motors, its current utility is limited to conventional braking scenarios. Future research avenues could explore identifying the braking state of vehicles intending to brake while coasting and releasing the accelerator pedal. Additionally, investigations could delve into the extent of regenerative braking engagement during emergency braking situations to enhance electric vehicle economy and foster the advancement of electric vehicle technology.

Author biographies

Chunyu Li, master, research interests are vehicle regenerative braking and vehicle chassis control.

Lu Zhang, PhD, associate professor, research direction is advanced control of vehicle system dynamics and energy.

Shiqiang Lian, master, research interests are vehicle environment sensing and vehicle intelligence control.

Menglong Liu, master, research interests include the design and intelligent control of vehicle line control dynamics systems.