# Design of Optimized Energy Management Strategy for All-Wheel-Drive Electric Vehicles
## 2 Design of All-Wheel Driving Model
### Power System Modeling
The configuration of the rear-left and right motor is the same as the front. The EV's control unit (VCU) is connected with F-MCU and R-MCU by CAN bus. \[P_{\text{m}}=\frac{T_{\text{m}}\cdot w_{\text{m}}}{\eta_{\text{m}}} \tag{7}\]3 Configuration of PHEV test platform distribution. \[P_{\text{m}}=\frac{T_{\text{m}}\cdot w_{\text{m}}}{\eta_{\text{m}}} \tag{7}\]4 Battery’s open-circuit voltage corresponding to \(SOC\). There are three driving forms in driving mode, front driving mode, rear driving mode, and coupling operation of front and rear drive mode, respectively. Thus,  
\[\text{Pdem}=\left\{ \begin{array}{ll}P_{\text{f1}}+s_{\text{f}}(t)P_{\text{f2}}& P_{\text{r1}}=P_{\text{r2}}=0\\ P_{\text{r1}}+s_{\text{r}}(t)P_{\text{f2}}&P_{\text{f1}}=P_{\text{f2}}=0\\ P_{\text{r1}}+P_{\text{r2}}+P_{\text{f1}}+P_{\text{f2}}\end{array} \right. \tag{13}\]  
\[s(t)=\left\{ \begin{array}{ll}1,\text{ motor on}\\ 0,\text{ motor off}\end{array} \right. \tag{14}\]  
When the VCU receives the signal of the brake pedal of the four in-wheel motors working together in breaking mode, the braking power is given by Equation (15). According to the demand level of the brake pedal, the electric braking will work firstly, and once the demand level of the breaking pedal exceeds 30% of all ranges, the mechanical braking and electric braking will work together to maintain braking force. Because electric braking can recover electric energy, and the motor can transfer mechanical energy to electrical energy, then electric breaking can improve the utilization of energy. \[P_{\text{gen}}=\eta_{\text{g}}(T_{\text{gen}}(t),\omega_{\text{m}}(t))\cdot T_{ \text{gen}}(t)\cdot\omega_{\text{m}}(t),T_{\text{gen}}(t)<0 \tag{15}\]  
\[m\cdot r\frac{dv}{dt}=T_{\text{gen}}(t)+T_{\text{mec}}(t) \tag{16}\]  
\[f_{\text{fb}}(t)=\frac{L_{\text{r}}}{L}+\frac{a(t)}{g}\frac{H}{L} \tag{17}\]  
\[f_{\text{rb}}(t)=(1-f_{\text{fb}}(t))\cdot c_{\text{g}} \tag{18}\]  
where the \(f_{\text{fb}}(t)\) and \(f_{\text{rb}}(t)\) are the front and rear distribution rate of braking force, respectively. \(L_{\text{r}}\) is the distance between the rear axle and center of mass, \(L\) is the distance between the rear and front wheels. \(H\) is the height of the center of the mass from the ground. \(a(t)\) is the deceleration in breaking. \(c_{\text{g}}\) is distribution ratio between breaking torque of front and rear axle. Where the \(P_{\text{gen}}\) is power generation, \(T_{\text{gen}}\) (\(t\)) is the braking torque produced by the motor, \(T_{\text{mec}}\) (\(t\)) is mechanical braking torque. In the condition of normal operation and without considering other electrical devices, the torque distribution coefficient \(\beta\) is defined as the ratio of front axle torque to driving torque, such as Equation (19). \[\beta=\frac{T_{\text{f}}}{T_{\text{f}}+T_{\text{r}}} \tag{19}\]  
In real working conditions and modeling, we make the following consumptions for simplicity [8]. 1. The two front and rear in-wheel motors have no relative slip in synchronous rotation. 2. The two front and rear in-wheel motors' configuration is the same, but the output torque depends on applications. Based on those assumptions and considering longitudinal dynamics in this paper, when the rear or front motor work singularly, the assumptions can be expressed as follows. \[\left\{ \begin{array}{ll}T_{\text{f1}}=T_{\text{f}}\\ T_{\text{r1}}=T_{\text{r}}\end{array} \right. \tag{20}\]  
\[\left\{ \begin{array}{ll}T_{\text{f1}}=\beta T_{\text{req}}\\ T_{\text{r1}}=\ (1-\beta)\ T_{\text{req}}\end{array} \right. \tag{21}\]In the case of front and rear motor working together, the equation is expressed as:  
\[\left\{ \begin{array}{l}T_{\rm f1}=T_{\rm f2}=\frac{Tf}{T}\\ T_{\rm r1}=T_{\rm r2}=\frac{Tf}{2}\end{array} \right. \tag{22}\]  
\[\left\{ \begin{array}{l}T_{\rm f1}+T_{\rm f2}=\beta T_{\rm req}\\ T_{\rm r1}+T_{\rm r2}=\ (1-\beta)\ T_{\rm req}\end{array} \right.