In recent years, electric vehicles have gained significant attention due to their rapid response and environmental benefits, with electric SUVs becoming a focal point in the automotive industry. However, challenges such as limited charging infrastructure and driving range persist. Dual-motor drive systems offer advantages in power transmission efficiency and regenerative braking compared to single-motor systems, making torque distribution optimization crucial for enhancing performance and economy. This study addresses the torque distribution problem in dual-motor driven electric SUVs by developing a mathematical model for drive system efficiency and proposing a hybrid optimization algorithm that combines simulated annealing with a particle swarm optimization algorithm featuring a contraction factor (SA-CFPSO). The algorithm dynamically allocates torque between the front and rear axles to maximize overall system efficiency. Simulations under CLTC-P and WLTC driving cycles demonstrate that this strategy effectively adjusts motor outputs, meets power demands, and improves driving range by 4.2% and 5.1%, respectively, thereby enhancing the cost-effectiveness of electric SUVs.
The architecture of a typical electric SUV includes key components such as front and rear motors, a power battery, reducers, differentials, and wheels. For instance, consider an electric SUV with a curb mass of 2570 kg and a gross vehicle mass of 3160 kg. The vehicle’s parameters, including wheel radius, rolling resistance coefficient, frontal area, and drag coefficient, are essential for modeling and optimization. The drive system must satisfy power requirements based on maximum speed, gradeability, and acceleration performance. The total power needed for the electric SUV is derived from these criteria, ensuring that the combined peak and rated power of the motors meet the vehicle’s demands. The front and rear motors are selected with specific rated and peak torques and speeds to operate efficiently across various driving conditions.
To formulate the efficiency model, the output power of the drive system is expressed as $$P_{\text{out}} = T_d \cdot n_d$$, where \(T_d\) is the total demanded torque and \(n_d\) is the rotational speed. The overall drive system efficiency \(\eta_d\) is given by:
$$\eta_d = \frac{P_{\text{out}}}{P_{\text{in}}} = \frac{T_d n_d}{\alpha T_d n_d / \eta_1 + (1 – \alpha) T_d n_d / \eta_2} = \frac{1}{\alpha / \eta_1 + (1 – \alpha) / \eta_2}$$
Here, \(\alpha\) represents the torque distribution ratio for the front motor, while \(\eta_1\) and \(\eta_2\) denote the efficiencies of the front and rear motors, respectively. The optimization objective is to maximize \(\eta_d\) subject to constraints such as \(0 \leq \alpha \leq 1\), torque limits of each motor, and speed boundaries. This model forms the basis for applying the SA-CFPSO algorithm to determine the optimal torque split for the electric SUV.
The SA-CFPSO algorithm integrates the global search capabilities of simulated annealing with the convergence properties of a particle swarm optimization that includes a contraction factor. In this approach, each particle’s position \(x_i\) corresponds to the front motor torque ratio \(\alpha\), and its velocity \(v_i\) represents the change in \(\alpha\). The velocity and position update equations are:
$$v_i(y+1) = \gamma \left\{ v_i(y) + c_1 r_1 [p_i(y) – x_i(y)] + c_2 r_2 [\bar{p}_g(y) – x_i(y)] \right\}$$
$$x_i(y+1) = x_i(y) + v_i(y+1)$$
where \(c_1\) and \(c_2\) are learning factors, \(r_1\) and \(r_2\) are random numbers between 0 and 1, and \(\gamma\) is the contraction factor calculated as:
$$\gamma = \frac{2}{|2 – C – \sqrt{C^2 – 4C}|}, \quad C = c_1 + c_2, \quad C > 4$$
To prevent premature convergence, simulated annealing introduces a jump probability based on temperature \(T\), allowing the algorithm to escape local optima. The fitness function \(F\) is defined as the system efficiency \(\eta_d\), guiding the search toward optimal torque distribution for the electric SUV.
Key parameters for the electric SUV and the optimization algorithm are summarized in the following tables. Table 1 outlines the basic vehicle parameters, while Table 2 details the motor specifications. Table 3 provides the SA-CFPSO algorithm parameters used in the simulations.
| Parameter | Value |
|---|---|
| Curb Mass (kg) | 2570 |
| Gross Vehicle Mass (kg) | 3160 |
| Wheel Radius (m) | 0.381 |
| Rolling Resistance Coefficient | 0.010 |
| Frontal Area (m²) | 2.5 |
| Drag Coefficient | 0.29 |
| Rotational Mass Conversion Factor | 1.5 |
| Maximum Speed (km/h) | 150 |
| Maximum Gradeability (%) | 30 |
| Acceleration Time (0-100 km/h, s) | 10 |
| Parameter | Front Motor | Rear Motor |
|---|---|---|
| Rated Torque (N·m) | 105 | 159 |
| Peak Torque (N·m) | 230 | 340 |
| Rated Speed (r/min) | 4983 | 2528 |
| Peak Speed (r/min) | 16000 | 6000 |
| Rated Power (kW) | 55 | 42 |
| Peak Power (kW) | 120 | 92 |
| Parameter | Value |
|---|---|
| Population Size | 40 |
| Maximum Iterations | 100 |
| Particle Dimension | 1 |
| Position Range | 0 to 1 |
| Learning Factors \(c_1\), \(c_2\) | 2.1 |
| Annealing Rate | 0.5 |
The simulation process involves modeling the electric SUV in software environments like Cruise and Simulink to validate the proposed strategy. Under the CLTC-P and WLTC driving cycles, the SA-CFPSO algorithm dynamically adjusts the torque distribution, ensuring that both motors operate within their high-efficiency zones. For example, during acceleration or high-speed scenarios, the algorithm allocates higher torque to both motors, leveraging their combined capacity while maintaining efficiency. The state of charge (SOC) of the battery is monitored to assess energy consumption, with results indicating slower SOC depletion under the SA-CFPSO strategy compared to rule-based methods.
Energy consumption and driving range are critical metrics for evaluating the performance of electric SUVs. The following table compares the results under different control strategies for the CLTC-P and WLTC cycles.
| Parameter | Cycle | Rule-Based Strategy | SA-CFPSO Strategy | Improvement |
|---|---|---|---|---|
| Energy Consumption (kWh/100km) | CLTC-P | 19.39 | 18.69 | -0.70 |
| Energy Consumption (kWh/100km) | WLTC | 19.61 | 18.67 | -0.94 |
| Driving Range (km, SOC: 95% to 5%) | CLTC-P | 455.63 | 475.18 | 19.55 |
| Driving Range (km, SOC: 95% to 5%) | WLTC | 448.78 | 471.96 | 23.18 |
The SA-CFPSO strategy demonstrates superior performance by reducing energy consumption and extending the driving range of the electric SUV. This improvement is attributed to the algorithm’s ability to optimize torque distribution in real-time, adapting to varying driving conditions and ensuring that the motors operate at their most efficient points. The integration of simulated annealing helps avoid local optima, while the contraction factor in PSO enhances convergence speed and stability.
In conclusion, the proposed hybrid particle swarm optimization algorithm effectively addresses the torque distribution challenge in dual-motor driven electric SUVs. By maximizing the overall drive system efficiency, this strategy significantly enhances the vehicle’s economy and practicality. Future work could explore adaptive parameter tuning and real-world implementation to further validate the benefits for electric SUVs in diverse operating environments.