With the increasing prominence of energy and environmental issues, distributed drive electric cars have become a focal point of research globally due to their energy-saving and eco-friendly advantages. In China, the electric car industry has seen rapid growth, driven by government policies and technological advancements. China EV manufacturers are actively developing advanced control strategies to enhance vehicle performance and efficiency. This study focuses on a composite braking control strategy for distributed drive electric cars that simultaneously considers yaw stability and braking energy recovery. By leveraging the independent control of four-wheel motor braking torque and friction braking torque, this approach aims to improve both safety and economy in electric car operations.
The proposed strategy adopts a hierarchical control structure. The upper-level controller consists of a longitudinal speed tracking controller based on proportion-integration-differentiation (PID) control and a yaw moment controller utilizing model predictive control (MPC). The lower-level controller optimizes tire utilization through quadratic programming to allocate braking torques, while considering factors such as braking intensity, battery state of charge (SOC), and vehicle speed to determine the motor braking force ratio coefficient. This ensures that the electric car maximizes energy recovery during braking while maintaining stability. The integration of these elements addresses key challenges in China EV development, such as range anxiety and dynamic performance.
In the upper-level controller, the longitudinal speed tracking controller employs PID control to compute the braking intensity based on the difference between the target and actual vehicle speeds. The braking intensity \( z \) is given by:
$$ z = K_p (v_d – v) + K_i \int (v_d – v) \, dt + K_d \frac{d(v_d – v)}{dt} $$
where \( K_p \), \( K_i \), and \( K_d \) are the proportional, integral, and derivative coefficients, respectively; \( v_d \) is the target vehicle speed; and \( v \) is the current actual speed. The longitudinal braking force \( F_x \) is then calculated as:
$$ F_x = z m g $$
where \( m \) is the vehicle mass and \( g \) is the gravitational acceleration. This component ensures that the electric car follows the desired speed profile during braking, which is crucial for energy efficiency in urban driving scenarios common in China EV applications.
The yaw moment controller uses MPC to ensure that the actual yaw rate and sideslip angle track their ideal values derived from a two-degree-of-freedom vehicle model. The ideal yaw rate \( \gamma_d \) and sideslip angle \( \beta_d \) under steady-state conditions are:
$$ \beta_d = \frac{ \left( \frac{b}{v_x} – \frac{a m v_x}{2 k_r (a + b)} \right) \delta }{1 + K v_x^2} $$
$$ \gamma_d = \frac{v_x \delta}{(a + b)(1 + K v_x^2)} $$
where \( a \) and \( b \) are the distances from the center of mass to the front and rear axles, \( v_x \) is the longitudinal velocity, \( k_f \) and \( k_r \) are the cornering stiffnesses of the front and rear axles, \( \delta \) is the front wheel angle, and \( K \) is the stability factor defined as:
$$ K = \frac{m}{2 (a + b)^2} \left( \frac{b}{k_f} – \frac{a}{k_r} \right) $$
The state-space representation of the system with an additional yaw moment \( M_z \) is:
$$ \dot{x} = A x + B_1 u + B_2 \psi $$
where \( x = [\beta, \gamma]^T \), \( u = M_z \), and \( \psi \) represents disturbances such as the front wheel angle. The matrices \( A \), \( B_1 \), and \( B_2 \) are derived from the vehicle dynamics. After discretization using the Euler method, the incremental state equation is used in the MPC formulation to compute the required yaw moment, ensuring the electric car maintains stability during cornering braking events.
The lower-level controller focuses on torque distribution to the four wheels. The optimization objective is to minimize tire utilization, which represents the stability margin of the tires. The cost function \( J \) is defined as:
$$ J = \sum_{i=\text{fl}, \text{fr}, \text{rl}, \text{rr}} \frac{F_{xi}^2}{(\mu F_{zi})^2} $$
where \( F_{xi} \) is the longitudinal force on wheel \( i \), \( \mu \) is the road friction coefficient, and \( F_{zi} \) is the vertical load. This optimization is subject to equality constraints that ensure the total longitudinal force and yaw moment match the upper-level controller outputs:
$$ \sum F_{xi} = F_x $$
$$ \sum \left( F_{xi} \cdot d_i \right) = M_z $$
where \( d_i \) represents the moment arms based on the wheel positions. Inequality constraints account for tire adhesion limits and motor capabilities:
$$ F_{xi} \leq \mu F_{zi} $$
$$ T_{\text{motor}_i} \leq T_{\text{max}} $$
where \( T_{\text{max}} \) is the peak motor torque. The quadratic programming problem is solved using the active set method for efficient computation.
To maximize braking energy recovery in the electric car, the motor braking force ratio coefficient is determined based on braking intensity \( z \), battery SOC, and vehicle speed. The coefficient \( k \) is computed as the product of three factors:
$$ k = k_1 \cdot k_2 \cdot k_3 $$
where \( k_1 \) depends on braking intensity and motor capability:
$$ k_1 = \begin{cases}
1 & \text{if } z \leq 0.7 \text{ and } T_{\text{max}} \geq T_{\text{need}} \\
\frac{T_{\text{max}}}{T_{\text{need}}} & \text{if } z \leq 0.7 \text{ and } T_{\text{max}} < T_{\text{need}} \\
0 & \text{if } z > 0.7
\end{cases} $$
\( k_2 \) is based on battery SOC to protect battery health:
$$ k_2 = \begin{cases}
1 & \text{if } Q_{\text{SOC}} < 0.8 \\
0 & \text{if } Q_{\text{SOC}} \geq 0.8
\end{cases} $$
and \( k_3 \) accounts for vehicle speed to optimize energy recovery efficiency:
$$ k_3 = \begin{cases}
0 & \text{if } v \leq 5 \text{ km/h} \\
0.2v – 1 & \text{if } 5 < v < 10 \text{ km/h} \\
1 & \text{if } v \geq 10 \text{ km/h}
\end{cases} $$
The motor braking torque for each wheel is then:
$$ T_{\text{motor}_i} = k \cdot F_{xi} \cdot R_w $$
where \( R_w \) is the wheel radius. The remaining braking torque is supplied by friction brakes. This approach enhances the energy recovery potential of China EV systems while adhering to practical constraints.
Simulation studies were conducted using MATLAB/Simulink and CarSim to validate the proposed strategy. The electric car parameters are summarized in the following tables:
| Parameter | Value |
|---|---|
| Mass \( m \) (kg) | 1230 |
| Wheel radius \( R_w \) (m) | 0.31 |
| Distance to front axle \( a \) (m) | 1.04 |
| Distance to rear axle \( b \) (m) | 1.56 |
| Rolling resistance coefficient \( f \) | 0.018 |
| Yaw moment of inertia \( I_z \) (kg·m²) | 1343.1 |
| Parameter | Value |
|---|---|
| Rated power (kW) | 7 |
| Peak power (kW) | 15 |
| Rated torque (Nm) | 200 |
| Peak torque (Nm) | 350 |
| Rated speed (r/min) | 450 |
| Peak speed (r/min) | 1250 |
| Parameter | Value |
|---|---|
| Brake pad friction coefficient \( \mu_B \) | 0.25 |
| Brake effectiveness factor \( c_B \) | 1 |
| Brake efficiency \( \eta_B \) | 0.99 |
Two scenarios were simulated: constant braking intensity and variable braking intensity. In the constant braking case, the electric car entered a circular path with a radius of 100 m on a road with a friction coefficient of 0.8. The initial speed was 50 km/h, and braking began at 1 s with an intensity of 0.2 until stop. The battery initial SOC was 0.6. The proposed tire utilization-based optimization was compared against a proportional allocation strategy. The results showed that the optimization strategy achieved better yaw rate tracking, with a maximum deviation of 0.31 deg/s from the ideal value, compared to 0.93 deg/s for the proportional method. Additionally, the SOC increase was 0.103 for the optimization strategy versus 0.095 for the proportional method, indicating superior energy recovery for the electric car.
| Control Strategy | SOC Increase | Max Yaw Rate Deviation (deg/s) |
|---|---|---|
| Proportional Allocation | 0.095 | 0.93 |
| Tire Utilization Optimization | 0.103 | 0.31 |
In the variable braking intensity scenario, the electric car started at 72 km/h on the same path, with braking intensity starting at 0.2 and increasing to 0.7 at 2 s. The optimization strategy maintained stability with a maximum yaw rate deviation of 1.88 deg/s, while the proportional method failed to track the ideal yaw rate effectively. The SOC increase was 0.09 for the optimization strategy compared to 0.05 for the proportional method, further demonstrating the advantages of the proposed approach for China EV applications in dynamic conditions.
| Control Strategy | SOC Increase | Max Yaw Rate Deviation (deg/s) |
|---|---|---|
| Proportional Allocation | 0.05 | 1.88 |
| Tire Utilization Optimization | 0.09 | 1.88 |
In conclusion, this study presents a composite braking control strategy for distributed drive electric cars that effectively balances yaw stability and braking energy recovery. The hierarchical control structure, combining PID and MPC in the upper layer and quadratic programming-based optimization in the lower layer, ensures robust performance. By incorporating factors like braking intensity, SOC, and speed into the motor braking force allocation, the strategy enhances energy recovery without compromising stability. These findings highlight the potential for improving the efficiency and safety of electric cars, particularly in the context of China EV advancements. Future work could involve real-world testing to further validate the strategy under diverse driving conditions.