Regenerative braking is the core process of braking energy recovery in electric vehicles. It not only maximizes the use of vehicle kinetic energy while ensuring vehicle braking safety, but also plays a key role in extending the driving range and improving energy efficiency (Soltani, 2024). The principle of braking energy recovery is shown in Figure 1.

As shown in Figure 1, during the entire process, the drive motor acts as a generator unit, efficiently recovering vehicle kinetic energy and converting it into electrical energy. This improves overall vehicle efficiency and extends the driving range. During the process of braking energy recovery, the motor is required to dynamically switch modes and finely regulate power output under complex operating conditions, with its control being affected by multiple factors such as vehicle speed, battery state, and braking force intensity. Therefore, the study first designs a fuzzy controller and defines membership functions to achieve adaptive control of the regenerative braking ratio under multiple input variables. A vehicle dynamics model is constructed to characterize the effects of multiple resistance forces on vehicle motion during braking, and the critical components influencing energy recuperation in regenerative braking are also systematically modeled. The power balance expression for the vehicle is shown in Equation 1. m · a = F b r a k i n g − F r e s i s t a n c e (1)

In Equation 1, m represents the vehicle’s curb weight, a is the vehicle’s acceleration. F b r a k i n g and F r e s i s t a n c e represent the total braking force applied by the vehicle and the total resistance encountered by the vehicle, respectively. It should be noted that both the braking force and the total resistance are defined as acting in the direction opposite to the vehicle’s motion; therefore, when F b r a k i n g > F r e s i s t a n c e , the vehicle undergoes deceleration. The expression for total resistance is shown in Equation 2. F r e s i s t a n c e = F r o l l i n g + F d r a g + F g r a d e F r o l l i n g = m · g · f F d r a g = 0.5 · ρ · A · C d · v 2 F g r a d e = m · g · ⁡ sin θ (2)

In Equation 2, F r o l l i n g , F d r a g , and F g r a d e represent rolling resistance, air resistance, and grade resistance, respectively. f is the rolling resistance coefficient, ρ is the air density, A is the vehicle’s frontal area, C d is the air resistance coefficient, v is the vehicle speed, θ is the road gradient angle. The study further constructs a battery charge and discharge model, as shown in Figure 2.

From Figure 2, during regenerative braking, the motor converts the braking energy into electrical energy to charge the battery. The battery’s state of charge (SOC) is updated using the integration method, with the specific expression shown in Equation 3. S O C t = S O C 0 − 1 Q ∫ 0 t I c h τ · η c h · d τ (3)

In Equation 3, S O C t represents the battery’s SOC at time t , within a range of 0 , 1 , S O C 0 is the initial SOC. Q is the battery capacity, I c h τ is the charging current during regenerative braking, η c h is the charging efficiency. The powertrain transmits the motor’s output torque to the wheels, and its efficiency affects energy conversion. During motor braking, the approximate relationship between the motor’s output torque and the generated regenerative braking force is shown in Equation 4. T r e g = F r e g · R η m (4)

In Equation 4, T r e g and F r e g represent the motor braking torque and regenerative braking force, respectively, R is the wheel radius, η m is the efficiency of the motor and powertrain. To ensure braking safety, the distribution of braking force between the front and rear axles is crucial. Based on the vehicle’s center of gravity, the ideal braking force for the front axle is expressed in Equation 5. F b f = G · a L (5)

In Equation 5, G represents the vehicle’s gravity, a is the horizontal distance from the vehicle’s center of gravity to the front axle, L is the vehicle’s wheelbase. The braking force for the rear axle is expressed in Equation 6. F b r = G − F b f (6)

In Equation 6, F b r represents the braking force at the rear axle. To achieve proper distribution of braking force between motor braking and mechanical braking on the front axle, a fuzzy controller is designed. To ensure accurate evaluation of the objective function and effective braking force distribution, the controller design incorporates a longitudinal vehicle dynamics model. This model integrates mass-related inertial force, aerodynamic drag, rolling resistance, and grade resistance, forming a comprehensive dynamical representation that governs vehicle deceleration behavior under braking input. The input and output variables are shown in Table 1.

In Table 1, the definitions of these input and output variables provide the foundation for the fuzzy controller’s design. The controller aims to adjust the regenerative braking ratio in real-time based on the vehicle’s actual operating conditions, maximizing energy recovery and ensuring smooth braking. The variables in the controller are set and described, and to help the fuzzy controller adapt to complex conditions like vehicle speed, braking intensity, and battery SOC, a Type-2 Gaussian membership function is used to finely describe the input. The advantage of Type-2 is that it can accurately describe the precision of input variables under uncertainty and noise interference. Therefore, Type-2 is adopted to help the fuzzy controller adapt to complex conditions, and the form of the Type-2 Gaussian membership function is shown in Equation 7. μ x = exp − x − c 2 2 σ 2 (7)

In Equation 7, x represents the input variable, c is the center value of the membership function, σ is the standard deviation that determines the width of the function. Considering measurement and environmental uncertainties, friction optimization uses parameter Δ to construct upper and lower membership functions, expressed in Equation 8. μ − x = exp − x − c + Δ 2 2 σ 2 μ − x = exp − x − c − Δ 2 2 σ 2 (8)

In Equation 8, μ − x and μ − x represent the lower and upper membership functions, respectively, Δ is the uncertainty range of the membership function’s center position. The parameter settings are detailed in Table 2.

In Table 2, based on variables under different conditions, the study constructs 27 rules in the form of “if … then …”. The braking intensity is classified into three levels: Z ≤ 0.3 is light braking, 0.3 < Z < 0.6 is moderate braking, Z ≥ 0.6 is emergency braking. For Rule i , the activation degree of the Type-2 membership degree interval is expressed in Equation 9. α i = min μ ∼ S O C i S O C , μ ∼ Z i Z , μ ∼ v i v (9)

In Equation 9, μ ∼ represents the uncertain Type-2 membership degree with upper and lower bounds, α i is the activation degree range of the i -th rule, and the final value can be obtained through the Karnik-Mendel algorithm. The defuzzification process uses the centroid method to determine the final output, as shown in Equation 10. K = ∑ i α i · K i ∑ i α i (10)

In Equation 10, K i represents the center value of the output membership function corresponding to Rule i , and K is the final control output.

After selecting the input and output variables and designing the Type-2 fuzzy membership functions, the study further constructs a specific fuzzy rule base. However, relying solely on manually set membership function parameters and rule weights often cannot adapt to the complex and changing real-world conditions. PSO can search for the optimal parameter combination globally, thus improving the overall performance of the fuzzy control system. Therefore, to enhance the generalization ability and output accuracy of the fuzzy controller, the study introduces PSO for offline optimization of the controller’s parameters. The fuzzy controller output is used for front axle braking force distribution. First, the total front axle braking force F b f is determined based on the vehicle’s operating conditions. Then, the front axle braking force is divided into motor braking and mechanical braking portions, with the motor braking force expression shown in Equation 11. F r e g = K · F b f (11)

In Equation 11, F r e g represents the regenerative braking force generated by the motor. The mechanical braking force is expressed in Equation 12. F m e c h = 1 − K · F b f (12)

In Equation 12, F m e c h represents the braking force generated by the hydraulic mechanical brake. Furthermore, to obtain the optimal fuzzy controller parameters, the study uses PSO for offline search. PSO is tightly coupled with the Simulink simulation environment, where each particle update is evaluated through simulation-based fitness assessment, enabling a real-time feedback loop for parameter adjustment and control performance optimization. The fuzzy controller with PSO is shown in Figure 3.

In Figure 3, after each particle swarm update during the optimization process, a simulation verification is performed using the Simulink model. Through this process, Simulink and the particle swarm optimization algorithm are tightly integrated, ensuring that each optimization iteration adjusts the control strategy based on simulation data, ultimately achieving optimal system control. Each particle represents a parameter vector, which is encoded as X = c 1 , σ 1 , Δ 1 , . . . , c N , σ N , Δ N , w 1 , . . . , w R . c , σ , and c represent the center, width, and uncertainty range of each membership function, respectively. GG represents the weights corresponding to the rules. Considering both energy recovery and braking safety, the objective function is designed as shown in Equation 13. J = w 1 · E r e c , i d e a l − E r e c + w 2 · D i d e a l − D E r e c = ∫ F r e g t · v t · d t D = ∫ 0 T v t · d t (13)

In Equation 13, Δ represents the theoretically maximum recoverable energy, E r e c is the actual recoverable energy, D i d e a l and D are the ideal and actual braking distances, respectively. Each particle is encoded as a parameter vector that includes the center values c , widths σ , and uncertainty ranges of the membership functions Δ , as well as the weights of the fuzzy rules w ; these parameters collectively determine the performance of the fuzzy controller. For each particle, the velocity and position update formulas are shown in Equation 14. v i t + 1 = w · v i t + c 1 · r 1 · p b e s t , i − x i t + c 2 · r 2 · g b e s t − x i t x i t + 1 = x i t + v i t + 1 (14)

In Equation 14, v i t represents the velocity of particle i at time t , x i t is the parameter vector of particle i , c 1 and c 2 are learning factors, r 1 and r 2 are random numbers in the range of 0–1. p b e s t , i is the historical optimal position of particle i , g b e s t is the global optimal position. Combining fuzzy inference with PSO, the final expression for the regenerative braking ratio coefficient is shown in Equation 15. K = f f u z z y S O C , Z , v = ∑ i α i · K f i n a l , i ∑ i α i (15)

In Equation 15, K f i n a l , i represents the output value corresponding to the i -th rule, determined by PSO. To quantitatively evaluate the regenerative braking strategy, energy recovery efficiency is defined as shown in Equation 16. μ r e c = E r e c E k i n × 100 % E k i n = 1 2 · m · v 0 2 (16)

In Equation 16, E k i n represents the vehicle’s initial kinetic energy, with v 0 being the initial vehicle speed. Leveraging the PSO process, the controller’s final output function ensures an optimal distribution of front axle braking force, with the interaction between motor braking and mechanical braking governed by the braking distribution formula. The diagram illustrating the regenerative braking fuzzy control strategy is presented in Figure 4.

In Figure 4, first, based on the total vehicle braking demand, the braking force distribution plan for the front and rear axles is calculated according to ECE regulations. Then, the input variables are processed by the PSO-optimized Type-2 fuzzy controller. Type-2 fuzzy logic is used, and the membership function is optimized using the particle swarm algorithm to output the regenerative braking ratio K . This ratio K is then used to distribute the front axle braking force, determining the ratio of motor braking to mechanical braking for the front axle. Finally, the front axle regenerative braking controller and hydraulic brake controller output the regenerative braking force and mechanical braking force for the front axle, respectively. In the second distribution stage, the rear axle’s braking controller adjusts according to the total braking demand, achieving overall vehicle braking control.

In order to enhance the generality of the proposed method, the fuzzy controller design and PSO-based optimization are developed in a modular and scalable framework, which can be adapted to other vehicle configurations or control targets. The selected parameters—membership functions, uncertainty bounds, and rule weights—are not specific to the studied vehicle model but represent universal fuzzy logic components that can be generalized across similar nonlinear control problems. Furthermore, the PSO optimization process is integrated with the Simulink simulation platform in a closed-loop structure, making it applicable to a broad range of model-based control design tasks beyond regenerative braking.

To verify the feasibility of the proposed strategy, a series of experimental setups were performed. The simulation platform used was MATLAB/Simulink, and the computer hardware configuration included an Intel Core i9-13900K processor, 64 GB DDR5 memory, NVIDIA GeForce RTX 4090 graphics card, a 1 TB NVMe SSD, and a 2 TB HDD for data storage. The system ran Windows 11, with an efficient cooling system and stable power supply to meet the high-performance requirements of MATLAB/Simulink co-simulation, complex model computation, and large-scale data processing. A full vehicle motion model was built in Simulink, along with a co-simulation system including a Type-2 fuzzy controller and PSO module. The specific parameters for the working conditions are shown in Table 3. It is worth noting that co-simulation refers to the integrated modeling of multiple subsystems—such as the vehicle dynamics model, energy management module, and braking control module—within the Simulink environment, enabling real-time coupling and interaction between the fuzzy control system and the PSO algorithm, thereby providing a more realistic simulation of system responses under actual operating conditions.

As seen in Table 3, the New York City Cycle (NYCC) and the New European Driving Cycle (NEDC) were selected as typical driving cycles. The energy recovery efficiency, braking performance, and control strategy indicators of the proposed strategy were validated. The PSO-Fuzzy control strategy was tested for braking times and distances under braking intensities of 0.2, 0.5, and 0.7, with results shown in Figure 5.

In Figure 5a, it can be seen that the higher the braking intensity, the faster the vehicle speed decreases and the shorter the braking time. For example, with a braking intensity of 0.7, the PSO-Fuzzy control strategy was able to reduce the speed from 40 km/h to 0 within 1.6s, responding quickly. In contrast, braking with intensities of 0.5 and 0.2 took 3.7s and 7.1s, respectively, to reach the same speed, showing a significant time disadvantage. The PSO-Fuzzy control strategy achieved quick deceleration at high speeds by dynamically adjusting the braking intensity, reducing braking time while maintaining a smooth braking process, making it especially suitable for urban emergency deceleration or mid-to-high speed scenarios. In Figure 5b, the strategy’s advantages were further validated from the perspective of braking distance. Under the same initial speed, the PSO-Fuzzy control strategy with an intensity of 0.7 allowed the vehicle to decelerate within 4.3 m, while the braking distances for intensities of 0.5 and 0.2 were 7.8 m and 35.1 m, respectively. Compared to fixed intensity braking, this strategy offered better braking accuracy and energy efficiency. These results demonstrate that the PSO-Fuzzy control strategy has the ability to dynamically adjust braking intensity according to actual conditions, ensuring safety, response speed, and energy recovery. Next, the SOC values and recovered energy were tested for braking intensities of 0.2 and 0.5, comparing the traditional braking strategy and the PSO-Fuzzy control strategy. The results are shown in Figure 6.

As shown in Figure 6a, there was little difference between the two strategies within the first second of braking. However, as time progressed, the PSO-Fuzzy control strategy showed a superior SOC improvement rate. When the braking intensity was 0.2, the strategy increased the SOC from 60.000% to 60.0016% within 2.4s, whereas the traditional strategy only increased it to 60.0012%. In Figure 6b, over time, the PSO-Fuzzy control strategy at z = 0.2 performed significantly better in energy recovery, accumulating 32 KJ of recovered energy within 8s, while the traditional strategy at the same intensity only recovered 27 KJ. Under the condition of z = 0.5, the PSO-Fuzzy control strategy also outperformed the traditional control. Overall, the results indicate that the PSO-Fuzzy control strategy achieved higher energy recovery efficiency and SOC improvement across different braking intensities, validating its comprehensive advantages in dynamic regenerative braking control. The different braking strategies were then introduced into the NYCC and NEDC driving cycles for comparison, as shown in Figure 7.

In Figure 7a, under the NYCC driving cycle, the SOC value for the traditional strategy continued to decrease, eventually dropping to 2.1%. In contrast, the proposed PSO-Fuzzy control strategy effectively suppressed SOC decline throughout the entire cycle, maintaining a smoother SOC curve, and at the end of the 12-h operation period, the SOC remained at 57.8%. This demonstrates that the strategy has higher energy recovery capacity and control stability in urban driving conditions with frequent stops, low speeds, and high-intensity braking. In Figure 7b, the NEDC cycle, which includes medium and high-speed driving with different acceleration and deceleration stages, showed a greater variation in SOC than the NYCC. Under the traditional strategy, the SOC dropped rapidly to 11.2% within 4 h, while the fuzzy control strategy was more stable, although the decline was still relatively fast. In contrast, the PSO-Fuzzy control strategy significantly delayed SOC decay, maintaining 47.6% throughout the entire simulation period, demonstrating its robustness and optimized control performance under complex, multi-stage conditions. In summary, although the control strategy exhibits good stability and convergence trends in simulation, future research will incorporate theoretical stability analysis of the closed-loop system to further enhance its robustness and convergence reliability under complex and dynamic operating conditions.

Additionally, by adopting a co-simulation architecture and separating the control algorithm from vehicle dynamics modeling, the proposed control strategy demonstrates potential applicability to different electric vehicle platforms and energy recovery systems. This modular structure ensures that the methodology can be extended to various intelligent vehicle control applications that involve nonlinear multi-objective optimization and uncertainty handling.

Since the NEDC driving cycle combines various driving states, including urban and suburban driving, it realistically reflects vehicle braking response and energy recovery performance under complex conditions. Therefore, the braking performance of the control strategy under the NEDC cycle was analyzed separately in the simulation to evaluate the stability and robustness of the control strategy in low-speed steady-state and frequent acceleration-deceleration scenarios. The braking speed and braking intensity over a short period of time in the NEDC cycle were verified, with results shown in Figure 8.

In Figure 8a, the vehicle speed and braking speed under the PSO-Fuzzy control strategy were compared over time. In the multi-stage acceleration-deceleration interval from 0.4s to 0.8s, the two curves were relatively close, indicating that the strategy could quickly sense vehicle speed fluctuations and output precise braking intensity in real-time, bringing the vehicle to a complete stop at 1.2s, with a smooth deceleration process and no noticeable shocks. In Figure 8b, the PSO-Fuzzy control strategy made frequent adjustments between 0.1s and 0.6s, with more significant adjustments between 0.2s and 0.4s during the high-frequency braking phase. This shows that the strategy could adaptively adjust braking intensity according to vehicle speed, road conditions, and desired deceleration, avoiding excessive braking that would lead to energy waste while ensuring smooth and safe deceleration. Overall, the PSO-Fuzzy control strategy demonstrated precise and smooth braking process control through adaptive speed and braking intensity adjustments in the NEDC cycle. Finally, the research compared the driving range and K value optimization before and after the strategy optimization, as shown in Figure 9.

In Figure 9a, the driving range performance under the optimized strategy in the NEDC cycle was compared. Under the same energy consumption conditions, the vehicle under the optimized strategy was able to travel for a longer time and with more stable speed control. The final driving range increased from 342 km before optimization to 396 km, a 15.8% increase. In Figure 9b, the traditional strategy showed relatively stable but lower K values, mostly concentrated between 0.08 and 0.47, making it difficult to quickly adjust energy recovery intensity for different conditions. In contrast, the optimized PSO-Fuzzy control strategy showed frequent fluctuations, with a dynamic range covering from 0.08 to 0.63. In typical segments such as from 100s to 300s, where frequent acceleration and deceleration alternated, the strategy demonstrated significant adaptive adjustment ability. This strategy increased K values under conditions with high energy recovery potential to enhance energy recovery and appropriately reduced K values during stages when the SOC was higher or stability was more critical, effectively ensuring the balance between braking smoothness and energy management.