# Research on Multi-Mode Braking Energy Recovery Control Strategy for Battery Electric Vehicles
## 2 Brake Energy Recovery Control Strategy
### Single-Pedal Brake Energy Recovery Control Strategy  
#### 2.2.1 Power Calculations for Electric Machines  
According to the external characteristic curve of the motor, the relationship between motor braking torque \(T_{e\_motor}\) and motor speed \(n\) can be expressed as follows [22]:  
\[T_{e\_motor}=T_{max}(1-\frac{n}{n_{max}}) \tag{1}\]  
where \(T_{max}\) is the maximum output torque of the motor, \(n_{max}\) is the maximum speed of the motor and \(n\) is the current speed of the motor.  
The torque output from the motor is transferred to the wheels after passing through the gearbox and transmission system, and the wheel torque can be expressed as follows:  
\[T_{e\_wheel}=T_{e\_motor}\cdot i\cdot\eta_{T} \tag{2}\]  
where \(T_{e\_wheel}\) is the wheel braking torque, \(i\) is the reduction gear ratio and \(\eta_{T}\) is the transmission efficiency.  
Figure 1: Mode switching logic schematic.  
Then, the wheel braking force \(F_{e\_wheel}\) is as follows:  
\[F_{e\_wheel}=\frac{T_{e\_motor}\cdot i\cdot\eta_{T}}{r} \tag{3}\]  
where \(i\) is the reduction gear ratio, \(\eta_{T}\) is the transmission efficiency, and \(r\) is the effective radius of drive wheels.  
Substituting the motor braking torque into the above equation, the braking force of the motor \(F_{e\_motor}\) can be obtained as follows:  
\[F_{e\_motor}=\frac{T_{max}(1-\frac{n}{n_{max}})\cdot i\cdot\eta_{T}}{r} \tag{4}\]  
Regardless of the mode, the energy recovered from braking is ultimately converted to electrical energy and stored in the vehicle's HV battery. During this process, the battery voltage and charging current will change with the continuous transfer of energy, affecting the energy recovery rate; so, the battery charging power also limits the motor's maximum power.  
The formula for the maximum charging power of the battery is as follows:  
\[\left\{\begin{array}{l}P_{bat\_cc}=I_{max}(V_{bat}+I_{max}R)\\ P_{bat\_cv}=U_{max}(U_{max}-V_{bat})/R\end{array}\right. \tag{5}\]  
where \(P_{bat\_cc}\) and \(P_{bat\_cv}\) are the maximum charging power of the HV battery at constant current and constant voltage charging, respectively; \(I_{max}\) is the maximum charging current, \(U_{max}\) is the maximum charging voltage, \(V_{bat}\) is the battery voltage; and \(R\) is the battery internal resistance.  
The equation for the relationship between battery charging power and motor braking torque is as follows [23]:  
\[T_{e\_bat}=\frac{9550P_{bat\_cc}}{n} \tag{6}\]  
where \(T_{e\_bat}\) is the motor braking torque.  
The power equation for the battery charging power limit of the motor mechanism can be obtained by association as follows:  
\[F_{e\_bat}=\left\{\begin{array}{l}\frac{9550P_{bat\_cc}i\eta_{T}}{nr}\\ \frac{9550P_{bat\_cc}i\eta_{T}}{nr}\end{array}\right. \tag{7}\]  
where \(n\) is the motor speed; \(i\) is the primary reducer ratio \(\eta_{T}\) is the transmission efficiency, and \(r\) is the wheel radius.  
In summary, the maximum braking force of the motor is as follows:  
\[F_{e\_\max}=\left\{\begin{array}{l}\min(\frac{T_{max}i\eta_{T}}{r},\frac{9550 P_{bat\_cc}i\eta_{T}}{nr},\frac{9550P_{bat\_cc}i\eta_{T}}{nr}),n<n_{e}\\ \min(\frac{9550P_{bat\_cc}i\eta_{T}}{nr},\frac{9550P_{bat\_ci}i\eta_{T}}{nr}, \frac{9550P_{bat\_ci}i\eta_{T}}{nr}),n>n_{e}\end{array}\right. \tag{8}\]