A Logic Threshold Control Strategy to Improve the Regenerative Braking Energy Recovery of Electric Vehicles

Abstract

With increasing global attention to climate change and environmental sustainability, the sustainable development of the automotive industry has become an important issue. This study focuses on the regenerative braking issues in pure electric vehicles. Specifically, it intends to elucidate the influence of the braking force distribution of the front and rear axles on access to energy recovery efficiency. Combining the I curve of a pure electric vehicle and the boundary line of the Economic Commission of Europe (ECE) regulations, the braking force distribution relationship between the front and rear axles is formulated to satisfy braking stability. The maximum regenerative braking force of the motor is determined based on the motor torque characteristics and battery charging power, and the regenerative braking torque is optimized by combining the constraints of the braking strength, battery state of charge (SOC), and vehicle speed. Six road working conditions are built, including the New European Driving Cycle (NEDC), the World Light-Duty Vehicle Test Cycle (WLTC), Federal Test Procedure 72 (FTP-72), Federal Test Procedure 75 (FTP-75), the China Light-Duty Vehicle Test Cycle—Passenger (CLTC-P), and the New York City Cycle (NYCC). The efficiency of the regenerative braking strategy is validated by using the Simulink/MATLAB simulation. The simulation results show that the proposed dynamic logic threshold control strategy can significantly improve the energy recovery effect of electric vehicles, and the energy recovery efficiency can be improved by at least 25% compared to the situation without regenerative braking. Specifically, under the aforementioned road working conditions, the braking energy recovery efficiency levels are 27.69%, 42.18%, 49.54%, 47.60%, 49.28%, and 51.06%, respectively. Moreover, the energy recovery efficiency obtained by the current dynamic logic threshold is also compared with other published results. The regenerative braking control method proposed in this article makes the braking control of electric vehicles more precise, effectively reducing energy consumption and improving the driving range of electric vehicles.

1. Introduction

Since the emergence of fuel vehicles, the energy conversion efficiency of commonly used engines is less than 50%. In order to improve the energy conversion efficiency of fuel vehicles, many measures to reduce energy loss have been developed, such as an aerodynamic design, an exhaust gas turbocharging design, combustion process optimization, thermal energy recovery (including exhaust heat optimization, engine heat optimization), and so on [1,2]. With the increasingly serious problems of environmental pollution and energy crisis, the use of new energy vehicles to replace fuel vehicles is receiving more and more attention. The electrification and intelligent development of vehicles is an irreversible trend, and electric vehicles have become a hotspot of research around the world. In recent years, electric vehicles have developed vigorously; however, their battery range and charging speed still cannot meet people’s special needs [3,4]. In addition to the improvement of energy storage and driving methods, the importance of braking energy recovery technology as an effective way to extend the range is becoming more and more prominent [5,6].

Brake energy recovery technology aims to reduce the heat that is lost during braking; the working process will make the traveling vehicle produce a corresponding resistance to achieve the effect of braking, and the recovered mechanical energy is recovered in the form of mechanical energy storage, electromagnetic energy storage, or chemical energy storage [7,8]. Mechanical energy storage includes compressed air energy storage and flywheel energy storage [9,10]; electromagnetic energy storage includes superconductivity and supercapacitors [11,12]; and electrochemical energy storage includes lead acid batteries, nickel–hydrogen batteries, lithium-ion batteries, sodium–sulfur batteries, and redox flow batteries [13,14]. In addition to the influence of energy storage devices, the factors that affect the braking energy recovery efficiency of electric vehicles include the drive type [15,16], motor performance [17,18], driving conditions [19,20], and control strategy [21,22]. Regenerative braking systems can be categorized into three groups based on the layout of the driving motors: central motor regenerative braking systems, side-wheel motor regenerative braking systems, and in-wheel motor regenerative braking systems. The energy that can be recovered during the braking process is only the driving kinetic energy on the driving wheel, while the kinetic energy on the driven wheel can only be consumed by the heat generated by mechanical friction braking. Therefore, while ensuring braking safety, distributing power to the driving wheels as much as possible is beneficial for improving the efficiency of braking energy recovery. The power generation efficiency of the motor also has a significant impact on the recovery of braking energy. As a key component of the regenerative braking system, the better the braking capacity of the motor, the higher the proportion of a regenerative braking force and the greater the braking energy recovery effect when distributing a regenerative braking force and a mechanical braking force. The driving conditions have the most direct impact on the recovery of braking energy. If electric vehicles are driving on congested urban roads and require frequent starting, acceleration, and deceleration, there are more braking conditions, which increases the number of regenerative braking times and can increase the energy recovery effect. If an electric vehicle is driving on a highway, there will be very few braking deceleration conditions and less braking energy recovery. Improving energy recovery efficiency requires a reasonable regenerative braking control strategy. Due to the high operating speed of electric vehicles, relying solely on regenerative braking during braking makes it difficult to decelerate in a timely manner, which requires mechanical braking to provide corresponding braking force. Therefore, the ratio of regenerative braking force to mechanical braking force during the braking process is particularly important. The higher the proportion of regenerative braking force, the more conducive it is to the recovery of braking energy. The most crucial aspect of regenerative braking control strategies is to optimize the distribution of regenerative braking force and mechanical friction braking force between the front and rear wheels, while ensuring braking safety, in order to achieve maximum energy recovery and optimize the driver’s experience.

In recent years, researchers have conducted extensive research on the energy recovery efficiency of regenerative braking and made certain progress. He et al. [23] studied a torque optimization strategy aimed at minimizing energy loss in regenerative braking systems for pure electric vehicles driven by dual motors. They increased the energy recovery by 3.35% under the WLTC. He et al. [24] put forward a regenerative braking control strategy through the optimization of the dynamic distribution coefficient of the dual electric mechanism. The energy recovery under NYCC conditions increased by 1.18%. Jiang et al. [25] proposed a control strategy aimed at distributing frictional braking force between the front and rear wheels in a fixed proportion. They achieved a braking energy recovery of 17.4% under NEDC conditions. Geng et al. [26] proposed a dual-layer multi parameter control strategy for braking energy recovery, which significantly improves the energy recovery efficiency of the vehicle. Ning et al. [27] proposed an energy recovery efficiency optimization algorithm based on fuzzy Q-learning. Under the conditions of the UDDS cycle, the energy recovery efficiency was improved by approximately 9%. Zhao et al. [28] determined the control parameters through a fuzzy optimization algorithm. They reduced battery consumption by 1.22% under NEDC conditions. Wu et al. [29] designed a regenerative braking torque distribution rule dominated by dual motor energy recovery, which improved the energy recovery rate and stability of electric vehicles. Yin et al. [30] proposed a regenerative braking control strategy based on a deep Q-learning network with adaptive weight coefficients. The braking energy recovery efficiency of their proposed strategy increased by 7.4% compared to the Q-learning strategy. Chen et al. [31] proposed an online collaborative estimation method for road slope and vehicle mass based on neural networks and least squares methods. The energy recovery of regenerative braking can be improved by up to 9.62%. Ashok et al. [32] showed that the fuzzy PID control strategy performs well in the energy recovery of electric two-wheeled vehicles, with an energy recovery rate of 44%. Liu et al. [33] adopted an adaptive braking force distribution control strategy. Under the NEDC and NYCC conditions of high-adhesion road surfaces, the vehicle’s braking energy recovery rates were 52.62% and 47.45%, respectively. Sandrini et al. [34] proposed a regenerative braking control strategy that can save 29.5% to 30.3% of energy consumption in the WLTC, and 23.9–24.4% and 19.01%, respectively, in the US06 driving cycle.

In the context of the increasingly significant negative impact of traditional fuel vehicles on the environment, countries worldwide are vigorously developing electric vehicles with zero emissions and zero pollution [35,36]. However, mileage anxiety is a topic that has always accompanied new energy. Regenerative braking, as one of the important ways to save energy in electric vehicles, has also received increasing attention, and its application and popularization have become an inevitable trend for the sustainable development of the automotive industry [37,38]. Facing the dual pressure of energy and environmental protection, improving the energy utilization rate of regenerative braking is of great significance and value for energy conservation and sustainable development [39,40]. The working principle of brake energy recovery control is to maximize energy recovery on the basis of sufficient braking torque to meet the braking safety distance and braking performance of new energy vehicles. The combination of regenerative braking and hydraulic braking makes the distribution of braking power more complicated, and it must be distributed on the basis of ensuring operational safety, otherwise it will affect the control effect of regenerative braking, and is prone to unstable and uncontrollable problems. It can be seen that the amount of recovered braking energy is closely related to the braking energy recovery strategy. The logic threshold control strategy algorithm is simple, easy to implement, and has good robustness, as long as the state variable is limited to the corresponding region. This control method can basically ensure that electric vehicles operate in an efficient range. The innovation of this article lies in the use of a logic threshold control strategy to allocate the relationship between hydraulic braking and regenerative braking, coordinating the distribution ratio between them, in order to further improve energy utilization efficiency.

In response to the above issues, this paper takes a front-wheel-drive pure electric vehicle as the research object, and proposes a regenerative braking control strategy based on dynamic logic threshold control by combining the braking stability and the influence of motor and battery characteristics. In Section 2, we detail the structure of electric vehicle braking systems and the principle of energy recovery. Six road working conditions are built, and the distribution of braking force between the front and rear axles is proposed to satisfy the braking stability. In Section 3, the logical threshold control method is used to allocate motor and hydraulic braking forces for front axle regenerative braking, ensuring effective and stable braking. We first propose a regenerative braking distribution strategy with the maximum braking torque of the front axle motor as a logical threshold. The distribution of motor regenerative braking force exhibits nonlinear control characteristics due to the constraints of system state variables. Then, we propose to optimize the motor participation in the regenerative braking process with brake strength, the battery SOC value, and vehicle speed as constraints. Then, we propose to optimize the motor’s participation in the regenerative braking process with brake strength, the battery SOC value, and vehicle speed as limitations. In Section 4, the efficiency of the regenerative braking strategy is validated by using the Simulink/MATLAB R2022b simulation software. The regenerative braking force control strategy proposed in this article is of great significance for improving the energy utilization efficiency of electric vehicles and reducing environmental impacts.

2. Model Construction for Electric Vehicles

2.1. Physical Properties of Electric Vehicles

2.1.1. Mechanical Structure of Regenerative Braking

The braking energy regeneration system of electric vehicles mainly includes two parts: the motor regenerative braking part and the hydraulic friction braking part. Although regenerative braking can recover braking energy and provide partial braking force to the wheels, the regenerative braking of the motor is limited by many conditions such as motor characteristics, battery, vehicle speed, etc., and cannot independently complete the braking requirements during emergency braking and high-intensity braking. In order to ensure the safety of a vehicle’s braking, traditional hydraulic friction braking should also be used as an auxiliary when regenerative braking is used [41]. Figure 1 shows a front-wheel-drive pure electric vehicle, whose regenerative braking system is a composite braking system composed of an electric motor system and a hydraulic system. This composite braking system mainly includes two parts: a power system and an energy storage system. The power system includes hydraulic brakes, motors, and electronic control systems. Hydraulic brakes utilize the pressure of liquid to achieve braking. During braking, regenerative braking switches the electric motor to a generator, using the vehicle’s inertia to drive the motor rotor and generate reverse torque. This process converts some of the vehicle’s kinetic or potential energy into electrical energy. The electronic control system is responsible for monitoring the braking situation of the vehicle and controlling and adjusting the motor. The energy storage system is the part of the regenerative braking system used to store electrical energy, which can store the electrical energy generated during the braking process and release it for use by the vehicle when needed. Common energy storage systems include battery packs, supercapacitors, and flywheels [42].

The regenerative braking control strategy is integrated into the electronic control system. When a vehicle brakes, the electronic control system calculates the total braking force required by the brake pedal signal collected by the sensor, and combines signals such as the vehicle speed, wheel speed, motor speed, battery SOC, battery voltage, and braking strength to identify the driver’s braking intention and analyze the vehicle’s driving status, and thus determine whether energy recovery is necessary [43]. If energy can be recovered, based on the analysis results and the corresponding regenerative braking distribution strategy, the electronic control system calculates the braking force that the hydraulic braking system and the motor braking system need to bear, respectively, and sends a control signal. If the energy cannot be recovered, the hydraulic brake system can independently complete the braking task or the anti-lock brake system (ABS) can perform anti-lock control. The execution unit (hydraulic brake and motor) receives control signals to complete the task of braking energy recovery.

2.1.2. Driving Formula for Electric Vehicles

The torque generated by the electric motor in an electric vehicle is transmitted to the driving wheel through the transmission mechanism, causing the driving wheel to generate a circumferential force on the road surface. When there is sufficient adhesion between the driving wheel and the road surface, that is, when the driving wheel does not slip on the road surface, the reaction force generated by the road surface to drive the vehicle on the road is the driving force of the vehicle, F_t. The driving resistance of a vehicle consists of four parts: rolling resistance (F_f), acceleration resistance (F_j), slope resistance (F_i), and air resistance (F_w). Due to the elastic hysteresis loss caused by the internal friction of the tire, the work that is carried out on the tire during deformation cannot be fully recovered. It is this elastic hysteresis loss in the tire that causes rolling resistance. When a vehicle accelerates, it overcomes the inertia force caused by its mass acceleration motion, which is the acceleration resistance. Acceleration resistance includes the inertia force of the translational mass and the inertia couple moment of the rotational mass. The component of the gravity of a vehicle along a slope is expressed as the resistance of the vehicle to the slope. The component of the air force acting on the vehicle when traveling in a straight line along the direction of travel is called air resistance. Air resistance is divided into two parts: pressure resistance and frictional resistance. The combined force of normal pressure acting on the exterior surface of a vehicle in the direction of travel is called pressure resistance, and frictional resistance is the combined force of tangential force generated by the viscosity of air on the body surface along the direction of travel. The driving formula of a vehicle refers to the relationship between the driving force and various driving resistances when driving on the road surface, which is expressed as follows [44]:

$$F_t=F_f+F_w+F_i+F_j.$$

(1)

After expanding the equation, we obtain the following:

$$\frac{T_{tq}{i}_g{i}_0{\eta }_T}{r}=Gf\cos \alpha +G\sin \alpha +\frac{C_{D}A{u}_{\mathrm{a}}^2}{21.15}+\delta m\frac{\mathrm{d}u}{\mathrm{d}t},$$

(2)

where T_tq is the motor output torque, i_g is the transmission ratio, i₀ is the transmission ratio of the main reducer, η_T is the mechanical efficiency of the transmission system, r is the rolling radius of the wheel, G is the vehicle weight, f is the rolling resistance coefficient, α is the road slope, C_D is the air resistance coefficient, A is the windward area, u_a is the vehicle speed, δ is the conversion coefficient of the rotational mass, m is the vehicle mass, and du/dt is the acceleration.

When a pure electric vehicle is traveling, the driving force and driving resistance are balanced, and the motor power and resistance power should also be balanced. In other words, the output power of the motor is always equal to the sum of the driving resistance power and mechanical transmission loss power at every moment during the driving process of the pure electric vehicle. The driving resistance power of a pure electric vehicle includes the rolling resistance power (P_f), air resistance power (P_w), slope resistance power (P_i), and acceleration resistance power (P_j). The power balance equation is as follows [45]:

$$P_{\mathrm{e}}=\frac{1}{{\eta }_T}\left(\frac{Gf{u}_a}{3600}+\frac{C_{D}A{u}_a^3}{76140}+\frac{Gi{u}_a}{3600}+\frac{\delta m{u}_a}{3600}\frac{\mathrm{d}u}{\mathrm{d}t}\right).$$

(3)

2.1.3. Parameters of the Drive Motor

A motor is an electrical device that transforms electrical energy into mechanical energy. When it operates in this manner, it functions as a motor. However, when it transforms mechanical energy into electrical energy, it operates as a generator. When combined with new energy vehicles, they exhibit electric motor characteristics when driving the vehicle forward or backward in a discharge state. When the vehicle’s driver releases the accelerator pedal or presses the brake pedal, it exhibits the characteristics of the generator. At present, the commonly used driving motors for new energy vehicles include two types: permanent magnet synchronous motors and alternating current asynchronous motors. Most new energy vehicles use permanent magnet synchronous motors, while only a small number of vehicles use alternating current asynchronous motors. From the working characteristics of the driving motor, we can know that the maximum torque, rated torque, rated power, peak power, rated voltage, maximum rotational speed, and rated rotational speed are the main parameters of the driving motor.

Table 1. Requirements for power performance indicators of pure electric vehicles.

Parameter	Value	Parameter	Value
Maximum speed umax/km·h−1	140	Climbing speed ui/km·h−1	35
Acceleration time (0–100 km·h−1) tm/s	13	Driving range S/km	250
Maximum climbing degree αmax/%	30	Overtaking time (80–100 km·h−1) ts/s	8

provides the design requirements for electric vehicles.

For the peak power demand of the driving system, the main considerations are the maximum speed, the maximum gradient at a given speed, and the acceleration time from a standing start.

Table 2. Parameters of the vehicle.

Parameter	Value	Parameter	Value
Unloaded weight, mu/kg	1370	Air resistance coefficient, CD	0.35
Full-loaded weight, mf/kg	1640	Position of COM, hg	0.54
Wheelbase, L/m	2.560	Rolling resistance coefficient, f	0.016
Distance from the front axle to COM, a/m	1.240	Windward area, A/m2	2.1
Distance from the rear axle to COM, b/m	1.320	Tire rolling radius, r/m	0.326
Coefficient of rotational mass, δ	1.1	Mechanical efficiency, ηT	0.95

gives the structural parameters of the electric vehicle [46].

(1)

When the vehicle is fully loaded and running on a flat road surface, the peak power demand of the motor corresponding to its maximum speed (u_max) is calculated as follows:

$$P_a=\frac{u_{max}}{3600{\eta }_T}(mgf+\frac{C_{D}A{u}_{max}^2}{21.15}),$$

(4)

(2)

The power required to meet the maximum climbing slope (α_max) requirement at a certain speed (u_i) is as follows:

$$P_b=\frac{u_i}{3600{\eta }_T}(mgf\cos {\alpha }_{max}+mg\sin {\alpha }_{max}+\frac{C_{D}A{u}_i{}^2}{21.15}),$$

(5)

(3)

The acceleration time (t_m) at u_e = 0~u_a = 100 km/h can reflect the acceleration performance in the commonly used speed range of the vehicle. The peak power required for a pure electric vehicle from 0 to 100 km/h can be obtained:

$$P_c=\frac{1}{3600{\eta }_T}(\frac{2}{3}mgf{u}_a+\frac{4}{5}\frac{C_{D}A{u}_a^3}{21.15}+\frac{1}{2t_m}\delta \mathrm{m}\frac{u_a^2+u_e^2}{3.6}),$$

(6)

Taking into account the power requirements of pure electric vehicles under various conditions such as maximum speed, maximum slope climbing, and acceleration, the peak power of the driving motor is as follows:

$$P_{max}=\max (P_a,P_b,P_c).$$

(7)

The maximum power of a motor refers to the maximum power that the motor can withstand in a short period of time, while the rated power refers to the power that the motor can withstand during long-term continuous operation. The rated power is determined based on the maximum power of the motor, and the rated power of a pure electric vehicle drive motor is as follows:

$$P_e=P_{max}/\lambda ,$$

(8)

where the motor load factor (λ) is 2.5. The small load factor of the motor can lead to the motor being unable to withstand overload operations for a short period of time, resulting in faults such as short circuits, overheating, and even damage to the motor. A large load factor may cause the motor to operate for a long time under overload, reducing the reliability and lifespan of the motor [47].

The maximum speed (n_max) of the drive motor depends on the maximum vehicle speed (u_max) and is influenced by the minimum transmission ratio (i_min) and wheel radius (r), as shown below:

$$n_{max}=\frac{u_{max}{i}_{min}}{0.377r}.$$

(9)

Under the condition of meeting the maximum speed, a margin of 200–300 r/min should be reserved for the maximum speed of the matched drive motor. The selection of the rated speed (n_e) and maximum speed (n_max) of the drive motor should comply with the torque speed characteristics of the drive motor:

$$n_e=\frac{n_{max}}{\xi },$$

(10)

where ξ is the constant power zone coefficient for the motor, and ξ = 3 [48].

The motor’s rated torque can be calculated using its rated power and rated speed, and the calculation formula is as follows:

$$T_e=\frac{9550P_e}{n_e}.$$

(11)

The peak torque of the drive motor is as follows:

$$T_{max}=\xi T_e.$$

(12)

The transmission ratio has a significant impact on the performance of pure electric vehicles. Under different driving conditions, an appropriate transmission ratio can effectively meet the performance requirements of electric vehicles, allowing the motor to operate within a high efficiency range, while reducing the load on the motor and power supply, and reducing power loss. Electric vehicles are driven in the minimum transmission ratio gear (i_min) at maximum speed, and in the maximum transmission ratio gear (i_max) at maximum slope climbing [49].

(1)

Determine the upper limit of the transmission ratio based on the maximum motor speed and maximum design speed:

$$i_{min}\le 0.377\frac{n_{max}r}{u_{max}}.$$

(13)

(2)

Determine the lower limit of the transmission ratio based on the constraint conditions of the short-term maximum climbing slope:

$$i_{max}\ge n_{max}r\left(mg(f\cos {\alpha }_{max}+\sin {\alpha }_{max})+\frac{C_{d}A{u}_i^2}{21.15}\right)/9550P_{me}{\eta }_T.$$

(14)

Due to the excellent speed regulation characteristics of the motor, the gearbox matched with the motor generally does not exceed three gears. If a single gear can also meet the power requirements, a single gear has more cost advantages compared to two gears. Currently, mostly single-gear reduction schemes are used, mainly because the characteristics of the motor are different from those of internal combustion engines. The driving motor generally has the characteristics of a low-speed constant torque and a high-speed constant power, which can generate a large amount of torque at very low speeds, unlike internal combustion vehicles that require deceleration to increase torque. Therefore, this article chooses a fixed gear ratio reducer (i_g) and a main reducer (i₀). This article sets the maximum speed of the driving motor as 7200 r/min. The rated speed of the drive motor is set at 6000 r/min. The transmission ratio of the fixed gear ratio reducer is set to i_g = 1.4, and the transmission ratio of the main reducer is set to i₀ = 4.176.

Table 3. Parameters of motor.

Parameter	Value	Parameter	Value
Rated power, Pe/kW	30	Peek speed, nmax/r·min−1	7200
Peak power, Pmax/kW	75	Rated torque, Te/N·m	120
Rated speed, ne/r·min−1	2400	Peek torque, Tmax/N·m	300

presents the technical parameters of the motor.

2.1.4. SOC Calculation of Power Battery

The driving range is an important indicator for evaluating pure electric vehicles. The larger the battery pack capacity equipped with pure electric vehicles, the longer the driving range. However, a large-capacity battery pack can increase the cost of the vehicle. So it is necessary to reasonably match the battery capacity of the vehicle. The reverse calculation of battery capacity is carried out using the constant speed method (u_c = 60 km/h) and the target value of range design. The energy (E_bat) and capacity (Q_cap) of the power battery are as follows [50]:

$$E_{bat}\ge \frac{mgf+C_{d}A{u}_a{}^2/21.15}{3600{\eta }_{soc}\cdot {\eta }_T\cdot {\eta }_{cut}}\cdot S,$$

(15)

$$Q_{cap}=1000\frac{E_{bat}}{U_e},$$

(16)

where the power battery discharge depth (η_soc) is 0.8, the discharge current efficiency (η_cut) is 0.9, and U_e is the power battery rated voltage.

Table 4. Characteristics of power batteries.

Parameter	Value	Parameter	Value
Rated voltage, Ue/V	336	Maximum charging power, PBmax/kW	110
Battery capacity, Qcap/Ah	120	Specific energy, E/Wh/kg	120

provides the performance parameters of the battery.

The equivalent circuit model uses an ideal voltage source to simulate the potential difference between the positive and negative electrode materials of the battery, representing the electromotive force characteristics of the battery, while internal resistance is used to simulate the resistance generated during electrochemical reactions and ion movement. Because of their different formation mechanisms, the internal resistance of batteries can be categorized into Ohmic internal resistance and polarization internal resistance. Ohmic internal resistance is composed of electrode material, electrolyte, membrane internal resistance, and contact internal resistance, while polarization internal resistance is generated when polarization occurs during the electrochemical reaction process. Equivalent circuit models are divided into the Thevenin model, the Partnership for a New Generation of Vehicles (PNGV) model, the dual polarization model, and the Rint model [51]. The Rint model is the simplest equivalent circuit model, consisting of a nonlinear voltage source and a resistor connected in series. The voltage source and resistor constantly change with the battery state, making parameter identification easy but not accurate, and cannot reflect the dynamic characteristics of the battery during polarization and polarization elimination processes. The formula for the Rint model is as follows:

$$U_{oc}=U_e-I_{m}R,$$

(17)

where I_m is the working current, R is the internal resistance of the power battery, and R = 0.015 Ω.

To ensure the normal operation of electric vehicles, the driving power output of the driving motor must be greater than the sum of the power that overcomes various resistance and energy losses. The driving current of electric vehicles constantly changes according to driving needs. Based on power balance, the output current of the battery has the following relationship:

$$I_m=\frac{{\eta }_{soc}\cdot {\eta }_T\cdot {\eta }_{cut}{T}_{tq}n}{9.55U_{ec}}.$$

(18)

where T_tq is the driving torque of the motor.

The regenerative braking of electric vehicles is achieved using the principle of motor/generator reversibility of the motor. When electric vehicles need to slow down or coast down, the control circuit of the driving motor can be used to generate electricity and operate the motor, converting the energy generated during deceleration and braking into current for charging the battery, thereby achieving regenerative utilization. The current (I_m) generated during motor braking is expressed as follows:

$$I_m=\frac{-{\eta }_{soc}\cdot {\eta }_T\cdot {\eta }_{cut}{T}_{tm}n}{9.55U_{ec}},$$

(19)

where T_tm is the motor braking torque, n is the motor rotational speed, and U_ec is the power battery terminal voltage.

The SOC (state of charge) of a battery refers to the state of the remaining battery charge. Knowing the value of the SOC also determines the “fuel consumption” of the vehicle. Moreover, the voltage, current, and energy management are directly related to the SOC. Since the SOC cannot be directly obtained, we can only estimate it indirectly. The estimation methods can be roughly divided into three types, including the ampere-hour integration method, the open-circuit voltage method, and the Kalman filtering method. The ampere-hour integration method calculates the amount of charge change during a certain period of time by integrating the current. Subtracting the amount of charge change from the initial amount of charge can estimate the current SOC. The calculation formula is as follows [52]:

$$SOC=SO{C}_{init}-\frac{1}{3600Q_{\mathit{cap}}}{\displaystyle \int {\eta }_b{I}_{m}dt},$$

(20)

in which SOC_init denotes the battery’s initial SOC, and η_b = 0.95 is the motor power generation efficiency.

2.1.5. Analysis of Energy Conversion

The energy recovered and stored in the battery (E_r) can also be calculated as follows:

$$E_r={\displaystyle {\int }_{braking}{U}_{oc}}{I}_{m}dt.$$

(21)

The output electrical energy of the battery pack is ultimately converted into mechanical energy to drive the vehicle, and the total energy required to drive the vehicle is as follows:

$$E_{total}={\displaystyle {\int }_{driving}{U}_{oc}}{I}_{m}dt.$$

(22)

When an electric vehicle without regenerative braking brakes, the change in the state of charge (ΔSOC_n) between the final SOC_end and the initial SOC_int is directly related to the total energy (E_total). For an electric vehicle with regenerative braking, the difference (ΔSOC_y) contains the energy recovered and stored in the battery, E_r. Thus, the regenerative braking recovery efficiency (ε) can also be calculated as follows [53]:

$$\epsilon =\frac{E_r}{E_{\mathit{total}}}=\frac{\Delta SO{C}_n-\Delta SO{C}_y}{\Delta SO{C}_n}.$$

(23)

2.2. Braking Force Distribution of Front and Rear Axles

2.2.1. I Curve

In the design of automobile braking systems, if braking under different road conditions can ensure that the front and rear brakes hold at the same time, then the relationship curve between the front and rear brakes’ braking force, F_μ1 and F_μ2, is known as the ideal distribution of the front and rear brakes’ braking force, usually referred to as the I curve [54]. It is easy to see from the interaction force between the ground and the wheels during braking that, due to the braking force exerted by the ground on the wheels, a torque is generated that causes the vehicle to flip forward. Therefore, the normal reaction force on the ground must be redistributed, meaning that the axle load is transferred from the rear to the front. This transfer amount is related to the position of the vehicle’s center of mass, wheelbase, and braking strength. The ground normal reaction force of the front and rear axles can be obtained from the torque balance:

$$\{\begin{array}{l}{F}_{Z1}L=Gb+m\frac{\mathrm{d}u}{\mathrm{d}t}{h}_{\mathrm{g}}\\ F_{Z2}L=Ga-m\frac{\mathrm{d}u}{\mathrm{d}t}{h}_{\mathrm{g}}\end{array},$$

(24)

where L is the height of the center of mass, and a and b are the distances from the center of mass to the front and rear axles, respectively. In other words, for a given vehicle, the amount of axle load transfer is related to the braking strength, z = du/dt. The greater the braking strength, the greater the ground normal reaction force on the front wheels and the smaller the one on the rear wheels. If both the front and rear wheels are held, the braking strength (z) is equal to the coefficient of adhesion (φ), giving the following relation:

$$\{\begin{array}{l}{F}_{Z1}=\frac{G}{L}\left(b+\phi h_{\mathrm{g}}\right)\\ F_{Z2}=\frac{G}{L}\left(a-\phi h_{\mathrm{g}}\right)\end{array}.$$

(25)

On any φ road surface, in order for both the front and rear wheels to lock at the same time, the combined braking force of the front and rear brakes must be equal to the ground adhesion, and F_μ1 and F_μ2, respectively, must be equal to their respective adhesion:

$$\{\begin{array}{l}{F}_{\mu 1}+F_{\mu 2}=\phi G\\ F_{\mu 1}=\phi F_{Z1}\\ F_{\mu 2}=\phi F_{Z2}\end{array}.$$

(26)

Substituting the equations for the ground normal reaction forces, F_z1 and F_z2, of the front and rear wheels into the above equation, we obtain the following:

$$\{\begin{array}{l}{F}_{\mu 1}+F_{\mu 2}=\phi G\hfill \\ \frac{F_{\mu 1}}{F_{\mu 2}}=\frac{b+\phi h_{\mathrm{g}}}{a-\phi h_{\mathrm{g}}}\hfill \end{array},$$

(27)

The first term in the above equation is the equal braking force line group, and the second term is the braking force distribution line group. The line connecting the intersection points of the corresponding φ values of the above two line groups is the ideal distribution curve of the braking force of the front and rear brakes, which is referred to as the I curve. By eliminating φ, the front and rear terms of the above equation are reduced to one equation:

$$F_{\mu 2}=\frac{1}{2}\left[\frac{G}{h_{\mathrm{g}}}\sqrt{b^2+\frac{4h_{\mathrm{g}}L}{G}{F}_{\mu 1}}-\left(\frac{G}{h_{\mathrm{g}}}+2F_{\mu 1}\right)\right].$$

(28)

2.2.2. f Curve and r Curve

The f-line set examines the correlation between the front and rear wheel ground braking forces (F_Xb1 and F_Xb2) when the front wheels are locked and the rear wheels are not locked. When the front wheels are locked, the following relationship is established [55]:

$$F_{X\mathrm{b}1}=\phi F_{Z1}=\phi \left(\frac{Gb}{L}+\frac{F_{X\mathrm{b}1}+F_{X\mathrm{b}2}}{L}{h}_{\mathrm{g}}\right).$$

(29)

Reorganization can take the following forms:

$$F_{X\mathrm{b}2}=\frac{L-\phi h_{\mathrm{g}}}{\phi h_{\mathrm{g}}}{F}_{X\mathrm{b}1}-\frac{Gb}{h_{\mathrm{g}}}.$$

(30)

The f-line set for different values of φ has an intercept with the vertical-coordinate F_Xb2 of −Gb/h_g, and all lines pass through the point (0, −Gb/h_g).

The r-line set refers to the relationship curve between the ground braking force of the front and rear wheels when the front wheels are not locked and the rear wheels are locked. When the rear wheels are locked, the following relationship is established [56]:

$$F_{X\mathrm{b}2}=\phi F_{Z2}=\phi \left(\frac{Ga}{L}-\frac{F_{X\mathrm{b}1}+F_{X\mathrm{b}2}}{L}{h}_{\mathrm{g}}\right).$$

(31)

Reorganization can take the following forms:

$$F_{X\mathrm{b}2}=\frac{-\phi h_{\mathrm{g}}}{L+\phi h_{\mathrm{g}}}{F}_{X\mathrm{b}1}+\frac{\phi Ga}{L+\phi h_{\mathrm{g}}}.$$

(32)

For different values of φ, the intercept of the r-line set with the transverse-coordinate F_Xb1 is Ga/h_g, and all lines cross the point (Ga/h_g, 0).

The analysis of the braking process of a vehicle on various road surfaces shows that (1) the intersection point of the f-line set and the r-line set, that is, the front and rear wheels locking simultaneously, and the line connecting the intersection point of the f-line set and the r-line set is the I curve; (2) the area between the f-line set above the I curve and the r-line set below the I curve indicates that both the front and rear wheels are locked, the ground braking force will no longer increase, and thus the above areas are meaningless.

2.2.3. ECE Regulations

The Economic Commission for Europe (ECE) has developed a series of braking regulations to ensure the safety and reliability of automotive braking systems. The ECE regulations stipulate that for road surface conditions of φ = 0.2 to 0.8, the braking strength must meet the following conditions:

$$\mathrm{z}\ge 0.1+0.85\left(\varphi -0.2\right).$$

(33)

The braking forces of the front and rear wheels of a vehicle under braking should satisfy the following:

$$\left\{\begin{array}{l}{F}_{\mu 1}=\frac{z+0.07}{0.85}\frac{G}{L}(b+z{h}_g)\hfill \\ F_{\mu 2}=Gz-F_{\mu 1}\hfill \end{array}\right..$$

(34)

By eliminating the parameter z, we obtain the following:

$${\left(F_{\mu 1}+F_{\mu 2}\right)}^2+G\left(0.077+\frac{b}{h_g}\right)\left(F_{\mu 1}+F_{\mu 2}\right)-\frac{0.85GL}{h_g}{F}_{\mu 1}+\frac{0.07G^{2}b}{h_g}=0.$$

(35)

According to the above equation, the relationship between the front and rear brake forces is also known as the M curve [57].

2.2.4. Allocation for Braking Forces of Front and Rear Axles

The distribution of braking force between the front and rear axles of a vehicle’s braking system is crucial for the utilization of braking stability and adhesion conditions of the vehicle. Different braking force distribution conditions will produce different braking effects. For vehicles with a fixed ratio of front and rear brake force, only when braking on a road surface with a synchronous adhesion coefficient can the front and rear wheels lock up simultaneously. When braking on a road surface with an adhesion coefficient greater than the synchronous adhesion coefficient, the rear wheels will first lock up and then the front wheels will lock up. When braking on a road surface with an adhesion coefficient less than the synchronous adhesion coefficient, the front wheels will first lock up and then the rear wheels will lock up. This method not only fails to take into account the changes in loading mass during vehicle operation, but also fails to consider the variability of road adhesion conditions during vehicle operation, making it difficult to meet the requirements of the braking force distribution ratio. Therefore, by combining the I curve with the braking force distribution curve formulated by the ECE regulations, we propose a multi-stage distribution of front and rear wheel braking forces based on different braking strengths.

As shown in Figure 2, according to the braking intensity, the working process of vehicle braking can be divided into four stages: low-intensity braking, medium-intensity braking, high-intensity braking, and emergency braking. In Figure 2, point O is the coordinate origin; point A is the transverse coordinate point when z = 0.1; the AB line is the tangent line of the M curve from point A; point B is the intersection of the AB line and z = 0.505; point C is the intersection of the I curve and z = 0.665; point D is the intersection of braking intensity z = 0.9; and point E is the intersection of the AB line and the M curve. The OA section is low-intensity braking, the AB section is medium-intensity braking, the BC section is high-intensity braking, an the CD section is emergency braking. In order to ensure the braking efficiency and braking safety, the CD section only allows the participation of hydraulic braking.

(1)

When z ≤ 0.1, i.e., the OA section in Figure 2, the braking strength is not high. Therefore, under this working condition, the front wheels provide all the required braking force for the vehicle, and the distribution rule is as follows:

$$\{\begin{array}{l}{F}_1=Gz\\ F_2=Gz-F_1\end{array},$$

(36)

where F₁ and F₂ are the braking forces of the front and rear axles, respectively.

(2)

When the braking strength is 0.1 < z ≤ 0.505, i.e., in section AB in Figure 2, due to the ECE regulation, the front and rear braking power distribution must be located above the M curve; therefore, in this case, the front and rear wheels provide the braking power at the same time:

$$\{\begin{array}{l}{F}_1=\frac{Gz+0.02268G}{1.2268}\\ F_2=Gz-F_1\end{array}.$$

(37)

(3)

When the braking strength is 0.505 < z ≤ 0.665, i.e., the BC section in Figure 2, as the braking strength increases, the weight of the rear wheel braking force is appropriately increased to take into account the braking stability of the vehicle, and the distribution rule is as follows:

$$\{\begin{array}{l}{F}_1=0.95\phi h_g\frac{Gz+Gb/h_g}{L}\\ F_2=Gz-F_1\end{array}.$$

(38)

(4)

Emergency braking occurs when the braking strength is z > 0.665. Under this condition, the braking force of the front and rear wheels is distributed according to the I curve, i.e., the CD segment in Figure 2, and the distribution rule is as follows:

$$\{\begin{array}{l}{F}_1=G\phi \frac{b+z{h}_g}{L}\\ F_2=G\phi \frac{a-z{h}_g}{L}\end{array}.$$

(39)

2.3. Six Driving Cycle Conditions

The driving conditions of a vehicle are the basis for the development of testing methods and limit standards for vehicle energy consumption, emissions, and other aspects. These driving cycles are mainly used to evaluate the pollutants emitted by vehicles under different driving conditions to determine whether they comply with relevant international, regional, or national emission standards. The characteristics, testing conditions, and testing standards of each driving cycle are different, but they are all designed to improve the accuracy and authenticity of vehicle emissions.

As Figure 3a shows, the NEDC consists of four urban cycles and one suburban cycle [58]. The entire NEDC test distance is 11.022 km, and the test time is 19 min and 20 s. From 0 to 780 s, it simulates urban conditions. During the test, it includes accelerating, maintaining speed, decelerating, stopping, and repeating this four times. The parking and constant speed time account for about 30%. In the test, the highest speed is 50 km/h, and the average speed is 18.77 km/h. Starting from the 780th second, the second driving condition, the suburban driving condition, is tested. The cycle time is 400 s, and the average speed is 62.6 km. The constant speed driving condition accounts for 52.25%.

As described in Figure 3b, the duration of the WLTC test is 1800 s, and the test is divided into four zones: low-speed zone, medium-speed zone, high-speed zone, and super high-speed zone [59]. The corresponding durations are 589 s, 433 s, 455 s, and 323 s, respectively. The corresponding maximum speeds are 56.5 km/h, 76.6 km/h, 97.4 km/h, and 131.3 km/h, respectively. The WLTC sets different operations such as parking, braking, and rapid acceleration, and also incorporates factors such as rolling resistance, gear position, and vehicle weight into the test. The maximum speed of the WLTC test reached 113.3 km per hour, and the actual road driving test included 30% urban roads, 33% rural roads, and 33% highway driving cycles.

As shown in Figure 3c, FTP-72 consists of cold transition conditions (0–505 s) and steady-state conditions (506–1370 s) [60]. In 1975, on the basis of FTP-72, a 600 s hot soak vehicle and a hot transition condition (repeated cold transition condition) were added to form FTP-75. The full time span of the process was about 2474 s.

As shown in Figure 3d, FTP-75 and FTP-72 are two variants of the EPA Urban Dynamometer Driving Schedule (UDDS) in the United States [61]. The FTP-75 cycle is the third stage where FTP-72 has an added 505 s. The new stage is the same as the first stage of FTP-72, but it belongs to the hot start condition. The third stage starts 10 min after the engine stops. Therefore, the entire FTP-75 cycle consists of the following sections: cold start transient phase: 0–505 s; stable stage: 506–1372 s; hot soak (minimum 540 s, maximum 660 s); and hot start transient: 0–505 s.

As presented in Figure 3e, the cumulative driving distance of the CLTC-P cycle is 14.48 km, including three driving modes, low speed, medium speed, and high speed, which are matched with the urban driving mode, suburban driving mode, and high-speed driving mode used by Chinese drivers on a daily basis [62]. The operating time is 1800 s, and the low-speed operating time is 674 s, with a time ratio of 37.4%. The medium-speed working condition lasts for 693 s, with a time ratio of 38.5%. The high-speed working condition lasts for 433 s, with a time ratio of 24.1%. During this period, the average speed is 28.96 km/h, and the maximum speed under the high-speed working condition is 114 km/h. The idle (stop) time accounts for 22.11%.

The NYCC, representing urban driving conditions with multiple stop-and-go motions, is presented in Figure 3f [63].

2.4. Front and Rear Axle Braking Forces under Six Cycle Conditions

Due to the power distribution method of front-wheel-drive electric vehicles, when the car brakes, it will cause an axle load transfer, and the front wheels will add some additional weight. The stronger the braking, the greater the transfer of axle load, and the greater the load borne by the front wheels. In this case, the braking force of the front wheels must increase accordingly. Figure 4 shows the braking force distribution of the front and rear axles under six cycle conditions. It can be seen that, under the six cycle conditions mentioned above, the braking strength almost does not exceed 0.15, that is, z ≤ 0.15. To ensure maximum regenerative braking force, almost all braking torque is distributed to the front axle, while the braking force on the rear axle is almost zero. Moreover, the front axle braking under the NEDC driving cycle is significantly smaller than the other five driving cycles.

3. Control Strategy for Front Axle Braking Force

3.1. Definition of Basic Patterns Describing Allocation Strategy

The distribution strategy of motor braking force and hydraulic braking force at the front wheel is as follows: (1) under slight braking conditions, only the motor provides regenerative braking force; (2) under moderate and high braking conditions, the motor and hydraulic brake jointly participate in braking, and the regenerative braking force of the motor is given by the distribution coefficient; (3) under emergency braking conditions, the regenerative braking function is turned off and the brake provides all braking force.

As shown in Figure 5, the logic threshold control algorithm for regenerative braking is as follows: (1) the maximum braking force of the front wheel motor will be used as the threshold value when the logic threshold control algorithm determines that the total braking force of the front axle exceeds the motor’s maximum braking force, indicating that the front wheel motor has reached the maximum regenerative braking degree and the remaining braking force is provided by hydraulic braking; (2) when the logic threshold control algorithm determines that the total braking force is less than the motor maximum braking force, the hydraulic braking of the front axle does not provide braking torque, and all braking force is provided by the motor braking force for regenerative braking.

3.2. Maximum Braking Torque Model of Electric Motor

The torque output characteristics of the motor determine the regenerative braking performance of electric vehicles. The regenerative braking torque of the motor is constrained by the generated power and speed, and when the braking intensity is too high, the motor cannot meet the braking requirements. When the speed is below the rated speed, the motor maintains a constant torque operating characteristic, while above the rated speed, the motor operates at a constant power. At the same time, the electrical energy generated by regenerative braking is stored in the battery pack, and there is an upper limit to the maximum charging current that the battery pack can accept. In addition, when the motor speed is lower than the minimum speed, the significance of energy recovery is not significant. To avoid excessive use of the motor, regenerative braking can be omitted. Therefore, the maximum regenerative braking torque (T_emax) can be described as the relationship shown in the following equation [64]:

$$T_e{}_{max}=\left\{\begin{array}{l}\min \left[\frac{9550P_{max}}{n}\hspace{1em}\frac{9550P_B{}_{max}}{{\eta }_{cut}n}\right], n>n_e\\ \min \left[T_{max}\hspace{1em}\hspace{1em} \frac{9550P_B{}_{max}}{{\eta }_{cut}{n}_e}\right], n_{min}\le n\le n_e\\ 0, n\le n_{min}\end{array}\right.,$$

(40)

where P_max is the peak power of the motor, P_Bmax denotes the maximum charging power of the battery, η_cut is the battery charging efficiency, T_max indicates the peak torque of the motor, n_min refers to the minimum speed that the motor can achieve, and n_min = 356 r·min⁻¹.

3.3. Distribution Coefficient

In the whole braking process, to ensure the braking stability of electric vehicles, and to recover the braking energy to extend the electric vehicle’s range as much as possible, there are many factors that affect the distribution of regenerative braking force, among which the effects of braking intensity, vehicle speed, and battery SOC on the motor braking force exhibit uncertainty and nonlinear characteristics. Therefore, these three factors are used as limiting quantities to impose secondary restrictions on the distribution of regenerative braking force in the motor [65].

When the braking intensity is z ≤ 0.1, the driver’s braking demand is not significant and the motor braking ratio can be maximized. When the braking intensity is 0.1 < z ≤ 0.505, the braking demand is moderate. In order to measure braking safety, the motor should be allocated as much braking force as possible. When the braking intensity is 0.505 < z ≤ 0.665, the proportion of the motor participating in braking should gradually decrease as the braking intensity increases. The car needs emergency braking when z > 0.665. At this point, the main focus should be on ensuring braking safety, so energy recovery will no longer be carried out, and the hydraulic friction braking system will complete the braking. The limiting distribution coefficient of regenerative braking force determined by the braking strength, k_z, is as follows:

$$k_z=\{\begin{array}{l}1, z\le 0.1,\\ -\frac{500}{405}z+\frac{455}{405}, 0.1<z\le 0.505,\\ 3.125(0.665-z), 0.505<z\le 0.665,\\ 0, z>0.665,\end{array}.$$

(41)

When a vehicle brakes, the equivalent speed of the vehicle will directly affect the maximum regenerative braking torque that the motor can provide. When the speed is below 8 km/h, the motor speed is very low, the inertia energy of the car is low, and the back electromotive force generated on the motor is also too low, resulting in less recoverable energy or the direct failure of the regenerative braking function. Therefore, the power of the electric mechanism should gradually exit the braking process. When the motor speed is too high, this can easily lead to excessive charging power, leading to battery damage. So, we introduce the speed impact factor, k_u; three speed thresholds, 8 km/h, 10 km/h, and 120 km/h, are set to limit the energy input into the battery during the energy recovery process. The impact of vehicle speed on motor braking force can be expressed as follows:

$$k_u=\{\begin{array}{l}0, u\le 8,\\ -0.5(8-u), 8<u\le 10,\\ 1, 15<u\le 120,\\ -(u-200)/80, 120<u\le 200,\\ 0, u>200,\end{array}.$$

(42)

The SOC value of a battery has a significant impact on the charging power of the battery. In order to protect the battery system, during the braking energy recovery process, when the SOC value of the battery exceeds the upper limit by 90%, energy recovery will no longer be carried out. Therefore, the battery SOC influence factor (k_SOC) is introduced. We set two threshold points for SOC, 0.8 and 0.9, to limit the energy input of the battery during the energy recovery process. The impact of the battery SOC value on motor braking force can be expressed as follows:

$$k_{SOC}=\{\begin{array}{l}0, SOC>0.95,\\ 20(0.95-SOC), 0.95<SOC<0.9,\\ 1, SOC\le 0.9,\end{array}.$$

(43)

Therefore, the total distribution coefficient can be expressed as follows:

$$k=k_z\cdot k_u\cdot k_{SOC}.$$

(44)

In the above equation, the influence factors of vehicle speed and the SOC are introduced to correct the actual output regenerative braking force [66].

The regenerative braking system consists of regenerative braking and hydraulic braking. It is necessary to allocate the regenerative braking force and hydraulic braking force reasonably in order to achieve the stability of the braking system and recover more braking energy. At the front axle, the hydraulic braking system and motor work together to provide braking force for the car, which means that F₁ = F_hyd + F_reg, where F_hyd is the hydraulic braking force and F_reg is the regenerative braking force of the motor. When the required braking torque during the braking process is greater than the maximum braking torque that the motor itself can provide, hydraulic braking and regenerative braking need to work simultaneously. At this point, to ensure maximum energy recovery, hydraulic braking only provides the difference between the actual required braking record and the maximum braking torque that the motor can provide. The distribution relationship between the hydraulic braking force and regenerative braking force is as follows:

$$\{\begin{array}{l}{F}_{reg}=T_{emax}{i}_g{i}_0/r,F_{hdy}=F_1-F_{reg}, k{F}_{1}r/\left(i_g{i}_0\right)\ge T_{emax}\\ F_{reg}=F_{1}k, F_{hdy}=F_1-F_{reg}, k{F}_{1}r/\left(i_g{i}_0\right)<T_{emax}\end{array}.$$

(45)

The recovery of braking energy is jointly completed by hydraulic braking and regenerative braking. The motor will generate its maximum braking torque, and the remaining braking force will be compensated for by the mechanical braking system. For any given braking intensity, motor braking is preferred to achieve as much energy recovery as possible. In cases where the braking strength is less than 0.1, the braking force is completely dependent on regenerative braking, while in cases where the braking strength is greater than 0.1, the hydraulic brakes will be involved in the braking process. As can be seen in Figure 6, the regenerative braking force is generally greater than the hydraulic braking force, which is due to the fact that the braking intensity is generally less than 0.1 in the six driving cycles mentioned above. Meanwhile, in some individual time periods, the hydraulic braking force is greater than the regenerative braking force, which is due to the fact that the braking intensity becomes greater and hydraulic braking must be added to complete the braking demand. Under urban conditions, the vehicle speed is relatively low, the braking intensity is not high, and the required motor braking torque is about 500 N. In suburban conditions with higher vehicle speeds, the required motor braking force is also greater. Electric vehicles have a higher braking force and can recover more energy, with a braking force of over 1500 N. Under the WLTC working conditions, there are more braking times than in other working conditions, and the participation of the motor is also relatively large. In the NEDC condition, there is significantly less braking and significantly less motor involvement.

3.4. Discussion on Control Stability

The logic threshold control strategy can be divided into two types based on the variable and immutable control parameters: the static logic threshold and dynamic logic threshold [67]. That is, the working area of the motor is limited by a set of static or dynamic parameters. The logic threshold control strategy is simple to implement and has a wide range of practical applications. However, static control strategy also has shortcomings, as it does not take into account the dynamic changes in operating conditions, making it difficult to ensure that the vehicle achieves optimal economic performance. Dynamic logic thresholds are beneficial for energy management strategies to adapt to different driving conditions, further improving vehicle economy. In this article, the logic threshold control strategy limits the working area of the motor based on its maximum braking torque to enhance the energy recovery efficiency.

If a system is disturbed and deviates from its original equilibrium state, and after the disturbance disappears, it can recover to its original state with a certain degree of accuracy after a sufficient period of time, then the system is said to be stable, otherwise it is said to be unstable. The primary condition for any automatic control system is that the system can operate normally and stably. An unstable system cannot function properly. The Lyapunov stability theory is a theory established by Russian mathematician and mechanist A.M. Lyapunov in 1892 to analyze system stability [68]. Stability can be determined using Lyapunov functions. We select a Lyapunov function as follows:

$$V=\left({(-SOC)}^2+{({\displaystyle \int I_m}dt)}^2\right)/2.$$

(46)

Next, the derivative of this Lyapunov function is obtained analytically:

$$\begin{array}{c}V\prime =-SOCSOC\prime +{\displaystyle \int I_m}dt{I}_m=SOC{I}_m+{\displaystyle \int I_m}dt{I}_m\\ =SOC{I}_m+(SO{C}_{int}-SOC)I_m=SO{C}_{int}{I}_m\le I_m,\end{array}$$

(47)

in which I_m is always less than 0 during the braking process. It can be seen that, during the braking process, it is easy to obtain V′ < 0, thus indicating that the system is asymptotically stable [69].

4. Result Analysis

4.1. Construction of Simulink Simulation

The numerical simulations are solved in the Matlab/Simulation software, the version of which is Matlab R2022b. The Simulink model for the regenerative braking of electric vehicles adopts a modular modeling method, which can be applied to various operating scenarios by building various sub modules. It provides a reliable simulation and analysis tool for vehicle design, research and development, and testing work. As shown in Figure 7, the Simulink model includes a cycle driving condition input module, a front and rear axle brake force distribution module, a maximum braking torque module of the motor, a regenerative braking distribution coefficient module, a hydraulic and motor brake force distribution module, and a battery SOC calculation module, which can comprehensively analyze and simulate the motion state and dynamic parameters of a vehicle.

The cycle driving condition input module includes speed and acceleration inputs for the six cycle conditions, used to simulate the motion state and dynamic parameter changes of vehicles under different working conditions. The front and rear axle brake force distribution modules determine whether the electric vehicle is in a driving or braking state based on acceleration, and can be used to calculate the motor input torque and motor braking torque. If the electric vehicle is in a braking state, the front and rear axle braking force shall be distributed according to the distribution rules of the front and rear axle braking force in Section 2.2.4. The maximum braking torque module of the motor determines the maximum braking torque that the motor can reach at the current speed. The regenerative braking distribution coefficient module determines the distribution coefficient based on the SOC, vehicle speed, and braking intensity. The hydraulic and motor braking force distribution module is used to distribute the hydraulic braking force and motor braking force at the front axle. The battery SOC calculation module can be used to calculate real-time changes in the SOC.

4.2. Variations in Distribution Coefficient (k) and Current (I_m)

Due to the different energy recovery effects under different driving cycles, the same initial simulation conditions are set for the six driving cycles to conduct simulation research on the regenerative braking strategy based on dynamic logic thresholds proposed in this paper. Figure 8 shows the variation curve of the distribution coefficient under six driving conditions. For the six driving cycle conditions, their average distribution coefficients are 0.2223, 0.3746, 0.3290, 0.2506, 0.3222, and 0.2104, respectively. Therefore, during the WLTC, the proportion of motors participating in regenerative braking is relatively high, while that of the NYCC is relatively small.

The current of the motor during a single driving cycle is shown in Figure 9. Figure 9 shows that, during the acceleration process of the vehicle, the motor provides driving torque, the current at the motor end is the input, and the motor is in the driving state; during the deceleration process of the vehicle, the motor provides braking torque, the current at the motor end is the output to charge the battery, and the motor is in the braking state. When the vehicle acceleration is positive, the motor input current is positive; when the vehicle acceleration is negative, and the motor output current is negative.

For the six driving cycle conditions, their maximum discharge currents are 84.72 A, 106.97 A, 78.47 A, 89.09 A, 94.16 A, and 62.95 A, respectively, and their maximum charging currents are −45.57 A, −63.5574 A, −50.65 A, −51.86 A, −85.21 A, and −39.48 A, respectively. In the positive current region, the maximum current of the WLTC is the highest, followed by the CLTC-P, FTP-75, the NEDC, FTP-72, and the NYCC. In the negative current region, the maximum current of the CLTC-P is the highest, followed by the WLTC, FTP-75, FTP-72, the NEDC, and the NYCC. From the perspective of the maximum discharge current, the driving resistance of vehicles under the WLTC is relatively high, while from the perspective of the maximum charging current, the braking process of vehicles under the CLTC-P driving cycle is more intense. For the six driving cycle conditions, their average discharge currents are 12.48 A, 15.65 A, 10.03 A, 8.02 A, 9.17 A, and 4.93 A, respectively, and their average charging currents are −3.45 A, −6.60 A, −4.97 A, −3.81 A, −4.51 A, and −2.51 A, respectively. In the positive current region, the average current of the WLTC is the highest, followed by the NEDC, FTP-72, the CLTC-P, FTP-75, and the NYCC. In the negative current region, the average current of the WLTC is the highest, followed by FTP-72, the CLTC-P, FTP-75, the NEDC, and the NYCC. From the perspective of the average discharge current, vehicles under the WLTC consume electricity the fastest, while from the perspective of the average charging current, vehicles under the WLTC also have better regenerative braking energy recovery.

4.3. SOC Variations

The state of charge of the battery reflects the energy state of a pure electric vehicle. The initial value of the SOC simulation is set to SOC_int = 0.9. As the cycle progresses, the SOC of the power battery gradually decreases. As shown in Figure 10, the SOC decrease of electric vehicles with regenerative braking is not a monotonic decrease, but a pulsating decrease. The SOC of electric vehicles without regenerative braking shows a monotonic downward trend. This is because, during regenerative braking conditions, the battery is charged and the SOC shows an increase. When the electric vehicle is driven again, the SOC will decrease again. At the initial stage, the SOC values of the battery are almost in a state of overlap, and as the vehicle travels/brakes, the SOC curves of electric vehicles with and without regenerative braking diverge.

From

Table 5. SOC comparison and energy recovery efficiency.

Driving Cycle	No Regenerative Braking			Regenerative Braking			ε/%
	SOCint/%	SOCendn/%	ΔSOCn/%	SOCint/%	SOCendy/%	ΔSOCy/%
NEDC	90	86.75	3.25	90	87.65	2.35	27.69
WHTC	90	83.80	6.20	90	86.41	3.59	42.18
FTP-72	90	86.97	3.03	90	88.47	1.53	49.54
FTP-75	90	85.63	4.37	90	87.71	2.29	47.60
CLTC-P	90	86.37	3.63	90	88.15	1.85	49.28
NYCC	90	89.35	0.65	90	89.68	0.32	51.06

, it can be seen that, under the NEDC, the final value of the battery SOC of the electric vehicle with regenerative braking is SOC_endy = 87.65%, while without regenerative braking, the final value of the battery SOC of the electric vehicle is SOC_endn = 86.75%. The SOC values of electric vehicles with and without regenerative braking decrease by ΔSOC_y = 2.35% and ΔSOC_n = 3.25%, respectively, thus achieving an energy recovery efficiency of ε = 27.69%. Under the WLTC, FTP-72, FTP-75, CLTC-P, and NYCC conditions, the values of SOC_endy are 86.41%, 88.47%, 87.71%, 88.15%, and 89.68%, respectively, while the values of SOC_endn are 83.80%, 86.97%, 85.63%, 86.37%, and 89.35%, respectively. Therefore, the energy recovery efficiency is ε = 42.18%, 49.54%, 47.60%, 49.28%, and 51.06%, respectively. The driving cycle with the highest energy recovery efficiency is the NYCC, while the driving cycle with the lowest energy recovery efficiency is the NEDC. Under the NEDC working conditions, due to the short cycle time and less braking, there is less energy recovery. Under the NYCC condition, the difference in SOC values is greater due to the greater amount of energy recovered.

Figure 11 shows variations in the driving distance and energy recovery efficiency. From Figure 11, it can be seen that the curve changes for the energy recovery efficiency under the NEDC have fluctuating characteristics. The curve of energy recovery efficiency under the WLTC has a large and flat value in the early stage, but shows a downward trend in the later stage. The curve of energy recovery efficiency under FTP-72 shows significant fluctuations in the early stage, while in the later stage, the value of energy recovery efficiency shows significant and flat changes. The changes under FTP-75 are similar to those under FTP-72, but the energy recovery efficiency curve under FTP-72 fluctuates in the later stage and shows a downward trend. The curve of energy recovery efficiency under the CLTC-P shows a significant and flat change in the value of energy recovery efficiency throughout the entire process. The energy recovery efficiency under the NYCC is similar to that of the CLTC-P, but the curve variation for energy recovery efficiency is more flat. For the six driving cycle conditions, their driving distances are s = 11.02 km, 23.26 km, 12.02 km, 17.75 km, 14.50 km, and 1.89 km, respectively. Their average driving speeds are v_ave = s/t_total = 33.62 km/h, 46.52 km/h, 31.53 km/h, 25.79 km/h, 29.01 km/h, and 11.37 km/h, respectively, where t_total is the total driving time.

4.4. Energy Recovery Performance of Regenerative Braking

When an electric vehicle brakes, the electric motor can serve as a generator for energy recovery. Figure 12 shows the trends of the total energy consumption (E_tol) and total energy recovery (E_reg) of electric vehicles under the six driving conditions. The change curves of E_tol and E_reg show an upward trend, and E_tol is significantly larger than E_reg.

Figure 12a shows the energy variation over time under the NEDC conditions. The total energy consumption under the entire cycle is E_tol = 1306.71 kJ, and regenerative braking recovers E_reg = 361.84 kJ of the braking energy. Under the WLTC, FTP-72, FTP-75, CLTC-P, and NYCC driving cycles, the total energy consumption is E_tol = 2498.25 kJ, 1220.93 kJ, 1761.66 kJ, 1463.80 kJ, and 261.59 kJ, respectively, while the total energy recovery is E_reg = 1053.87 kJ, 604.84 kJ, 838.54 kJ, 721.29 kJ, and 133.57 kJ, respectively. It can be seen that the total energy consumption under the WLTC is the highest, while the total energy consumption under the NYCC is the lowest. The total energy recovery under the WLTC is the highest, while the total energy recovery under the NYCC is the lowest.

Two parameters are introduced to measure the energy consumption and energy recovery characteristics of electric vehicles under different driving cycle conditions. The energy consumption per kilometer, defined as E_c = E_tol/s, refers to the ratio of total energy consumption to total driving distance throughout the entire driving cycle. Energy recovery per kilometer, defined as E_r = E_reg/s, refers to the ratio of total energy recovery to total distance traveled throughout the entire driving cycle. From

Table 6. Comparison of energy.

Parameter	NEDC	WLTC	FTP-72	FTP-75	CLTC-P	NYCC
Energy consumption per kilometer, Ec/kJ/km	118.57	107.40	101.57	99.24	100.95	138.40
Energy recovery per kilometer, Er/kJ/km	32.83	45.30	50.31	47.24	49.74	70.67

, it can be seen that the energy consumption per kilometer (E_c) under the NYCC is the highest, while the energy consumption per kilometer (E_c) under the FTP-75 driving cycle is the lowest. The analysis suggests that under the FTP-75 driving cycle, vehicles have a relatively high driving speed, causing a larger rolling resistance and wind resistance. The energy recovery per kilometer (E_r) under the NYCC is the highest, while the energy recovery per kilometer (E_r) under the WLTC is the lowest. This is because under the NYCC, the vehicle has a lower speed and is often in a braking state. The frequent and slight braking behavior is beneficial for regenerative energy recovery.

5. Discussion and Comparison

5.1. Comparison of Energy Recovery Performance

In order to increase readers’ understanding and learning of regenerative braking, we conduct a similarity and difference comparison with studies published on energy recovery efficiency. References [70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94] list the energy recovery efficiency achieved via different control strategies, as shown in

Table 7. Comparison and discussion of energy recovery efficiency.

Studies	Vehicle Type	Control Algorithm	Cycle	Recovery Efficiency
Yin et al. [70]	electric vehicles	fuzzy control	NEDC WHTC FTP-72 FTP-75 CLTC-P NYCC	15.01% 23.20% 30.51% 29.59% 29.16% 40.13%
Jiang et al. [71]	electric vehicles	optimization distribution algorithm	UDDS NEDC	53.1% 55.3%
Xu et al. [72]	LF620 prototype EV	fuzzy control	UDDS	25.7%
Zhou et al. [73]	electric vehicles	fuzzy control	Self-defined driving cycle	34.11%
Rizzo et al. [74]	hybrid vehicles	optimal braking modulation	Self-defined driving cycle	up to 40%
Zhang et al. [75]	electric vehicles	fuzzy logic	UDDS	37.4%
Yang et al. [76]	electrified vehicles	logic threshold control and phase plane theory	Self-defined driving cycle	23.08–38.54%
Zhang et al. [77]	electrified minivan	modified control strategy	ECE drive cycle	47%
Kusuma et al. [78]	electric buses	serial regenerative braking strategy	Blok M—Kota route and Kota—Blok M route	15%
Liu et al. [79]	hybrid electric vehicles	pseudo-spectral method	Self-defined driving cycle	36.07%
Salari et al. [80]	electric vehicles	two-stage nonlinear predictive control	NEDC	24%
Wager et al. [81]	electric vehicle	regenerative braking control proposed by authors	NEDC FTP-75	11–22%
Long et al. [82]	electric vehicles	sliding mode robust control	Self-defined cycle	17%
Xu et al. [83]	electric vehicles	hierarchical control strategy	C-WTVC	12.75%
Mondal et al. [84]	electric vehicles	regenerative braking control proposed by authors	NEDC	18.38%
Huang et al. [85]	five-ton pure electric drive loader	collaborative control strategy	Cycle operation of electric loader	42.50%
Xiao et al. [86]	electric vehicles	multi-input fuzzy control logic	NEDC LA92 JP1015	21.1% 42% 22.7%
Sun et al. [87]	electric vehicles	adaptive cruise control algorithm	Self-defined driving cycle	43.65%
Xu et al. [88]	hybrid electric vehicles	predictive control algorithm	Self-defined driving cycle	16.78%
Mariani et al. [89]	hybridized FIAT grande punto	MPC strategy	Self-defined driving cycle	18%
Ziadia et al. [90]	electric vehicles	adaptive braking strategy	Self-defined driving cycle	39.6%
Yu et al. [91]	electric vehicles	neural network and particle swarm algorithm	Self-defined driving cycle	19.1%
Xu et al. [92]	electric vehicles	exhaustive search method	NEDC UDDS	7.01% 6.69%
Enang et al. [93]	hybrid electric vehicles	heuristic control strategy	Japan 10–15 driving cycle	19.07%
Sun et al. [94]	electric vehicles	fuzzy control	Japan urban cycle conditions	40.61%

. It should be noted that some of these studies do not use prescribed driving cycles, but instead use self-defined driving cycles. We did not use their self-defined driving cycles to compare the energy recovery efficiency. According to our results, the energy recovery efficiency achieved in different driving cycles is variable. For example, in the NEDC, the energy recovery efficiency is 27.69%, while in the NYCC, it reaches 50%.

From Table 7, it can be concluded that the energy recovery efficiency levels below 20% include Kusuma et al. [78], Wager et al. [81], Long et al. [82], Xu et al. [83], Mondal et al. [84], Xu et al. [88], Mariani et al. [89], Yu et al. [91], Xu et al. [92], and Enang et al. [93]. The energy recovery efficiency levels ranging from 20% to 40% cover Yin et al. [70], Xu et al. [72], Zhou et al. [73], Rizzo et al. [74], Zhang et al. [75], Yang et al. [76], Liu et al. [79], Salari et al. [80], Huang et al. [85], Xiao et al. [86], and Ziadia et al. [90]. Energy recovery efficiency levels exceeding 40% are reported in the work of Jiang et al. [71], Rizzo et al. [74], Zhang et al. [77], Huang et al. [85], Sun et al. [87], and Sun et al. [94]. Jiang et al. [71] reported that their energy recovery efficiency reached 55%. In the NEDC, Jiang et al. [71] achieved an energy recovery efficiency of 55.3%, followed by Salari et al. [80], Wager et al. [81], Xiao et al. [86], Mondal et al. [84], Yin et al. [70], and Xu et al. [92]. Although our energy recovery efficiency is not as high as that of Jiang et al. [71], it is better than others. In other driving cycles, our energy recovery efficiency exceeds 40%, indicating the superiority of the dynamic logic threshold strategy.

The different control strategies in Table 7 exhibit different energy efficiency and performance levels under different driving cycle conditions. These data have a certain reference value for studying the control strategies of electric and hybrid vehicles, as well as promoting the sustainable development of automobiles. The dynamic logic threshold strategy demonstrates its superiority, with an energy recovery efficiency exceeding 40%. The dynamic logic threshold strategy explores how to optimize the control strategy of regenerative braking based on factors such as the vehicle speed, battery SOC, and braking intensity to achieve the best energy recovery. By comparing the relevant studies available in the literature, the dynamic logic threshold strategy has a significantly improved system response speed and control accuracy, especially in complex and nonlinear systems where its application effect is more significant. It can be seen that the dynamic logic threshold strategy has significant advantages in stability, robustness, performance indicators, and specific application scenarios, which is of great significance for improving the control quality of the system.

5.2. Discussion on Energy-Saving Management

The efficient utilization of energy is crucial for electric vehicles to achieve energy conservation and environmental protection. The importance of energy recovery technology as an effective way to extend the range of electric vehicles is becoming increasingly prominent. To achieve the maximum utilization of electricity, comprehensive energy conservation, strengthened energy recovery, the precise control of electricity consumption, and improvements in energy utilization rates can be considered. Specifically, it includes the following major aspects:

(1)

Composite energy storage [95,96,97]: The design and control strategy of energy storage systems play a crucial role in the power performance and economy of electric vehicles. Lithium batteries are the main energy source for traditional pure electric vehicles. However, excessive discharge currents and long charging times seriously affect the lifespan and basic characteristics of lithium batteries. Supercapacitors have a high power density and fast charging and discharging rate, which can be used as auxiliary energy to withstand high-current charging and discharging. The performance advantages of two energy storage components can be utilized to compensate for the shortcomings of a single power source [98,99].

(2)

Optimization of control algorithms [100,101]: The most commonly used control algorithms are PID control, sliding film variable structure control, adaptive control, predictive control, fuzzy control, and neural network control. After determining the control framework, the design problem of the controller also involves the parameter optimization of the controller. Many methods such as particle swarm optimization, genetic algorithms, ant colony algorithms, machine learning, and artificial intelligence can optimize system performance, thereby further improving energy recovery efficiency [102,103].

(3)

Intelligent transportation [104,105]: Intelligent connected vehicles can achieve multiple advantages such as autonomous driving, vehicle safety, traffic management, and energy conservation and emission reduction. Intelligent connected vehicles achieve information exchange between vehicles through network communication and data sharing, improving vehicle driving safety and economy. Thus, the decision-making behavior of intelligent connected vehicles is based on environmental perception and information output from navigation subsystems, including which lane to choose, whether to change lanes, whether to follow, whether to detour, and whether to stop [106,107].

6. Conclusions

Regenerative braking technology for electric vehicles is not only able to effectively reduce vehicle overheating and brake aging problems, but also has significant implications for energy conservation, emissions reduction, improving energy utilization efficiency, and sustainable development. In order to recover as much kinetic energy as possible while meeting the braking performance requirements, it is necessary to coordinate and control the hydraulic braking and regenerative braking subsystems. This leads to two basic problems to be solved: how to distribute the total braking force on the front and rear axles to achieve a stable braking condition, and how to distribute the required braking force at the front wheels between regenerative braking and hydraulic braking to recover as much kinetic energy as possible from the vehicle. In order to ensure the directional stability of the vehicle during braking and to have sufficient braking efficiency, the relationship between the front and rear wheel braking torques is rationally assigned based on the constraints of the ECE regulations, the I curve, and the f-curve based on the braking intensity. The maximum braking torque that can be generated by the electric motor is taken as the logical threshold value to obtain the maximum energy recovery gain. Meanwhile, the ratio of regenerative braking force to front wheel braking force is optimized by taking the braking intensity, battery SOC, and vehicle speed as limiting constraints. The effectiveness of the regenerative braking strategy based on the proposed dynamic logic threshold control is verified under six cycle conditions. The main achievements are as follows:

(1)

The dynamic logic threshold strategy constructed in this article can effectively improve energy recovery efficiency during the regenerative braking process. Compared with cars without regenerative braking, the energy recovery efficiency under the six cycle conditions is 27.69%, 42.18%, 49.54%, 47.60%, 49.28%, and 51.06%, respectively.

(2)

The energy consumption efficiency per kilometer and recovery efficiency per kilometer for the six cycle conditions are 118.57 kJ/km/32.83 kJ/km, 107.40 kJ/km/45.30 kJ/km, 101.57 kJ/km/50.31 kJ/km, 99.24 kJ/km/47.24 kJ/km, 100.95 kJ/km/49.74 kJ/km, and 138.40 kJ/km/70.67 kJ/km, respectively.

(3)

The use of dynamic logic threshold control to optimize the allocation of regenerative braking is beneficial for improving the energy recovery performance of electric vehicles. Compared with the relevant published literature, it indicates that the current control strategy achieves high energy recovery efficiency.