# Design of Optimized Energy Management Strategy for All-Wheel-Drive Electric Vehicles
## 2 Design of All-Wheel Driving Model
### Power System Modeling
\tag{23}\]  
So the efficiency objective function can be acquired by the following equation:  
\[\min J_{\rm d}=w\left\{\frac{\beta T_{\rm f}}{\eta_{\rm f1}(T_{\rm f},w)}+ \frac{(1-\beta)T_{\rm r}}{\eta_{\rm r1}(T_{\rm r},w)}+\frac{\beta T_{\rm f}}{ \eta_{\rm f2}(\frac{T_{\rm f}}{2},w)}+\frac{(1-\beta)T_{\rm r}}{\eta_{\rm f2}( \frac{T_{\rm r}}{2},w)}\right\} \tag{24}\]  
In the whole driving cycle, the control vector is defined as torque split ratio \(u_{\beta}({\rm k})\), and \(u_{\rm c}({\rm k})\), indicating which motor is working. Further, in the whole drive, the time interval is selected as 1s, so the discrete-time state equation is given by (25). \[X(k+1)=f(X(k),U(k))+X(k) \tag{25}\]  
where X(k) is the system state vector, \(U({\rm k})\) is the system control vector. In the offline optimization process of all-wheel-drive (AWD), energy management is carried out in different driving conditions. For example, the Manhattan bus driving cycle, West Virginia Suburban driving cycle, and typical cities in China driving cycles. With the help of the vehicle longitudinal dynamic theory, the driven torque can be calculated. Therefore, the system state vector can be confirmed as battery \(SOC\) and motors' state. \[X(k)=[SOC(k),u_{\rm s}(k)]^{\rm T} \tag{26}\]  
\[u_{\rm s}(k)=\left\{ \begin{array}{l}1,\ {\rm participation}\\ 0,\ {\rm withdrawal}\end{array} \right. \tag{27}\]  
For four in-wheel motor independence drives, the input torque consists of rear motors and front motors torque, and the power split ratio which is defined as torque split ratio. Therefore, the control vector could be simplified by  
\[U(k)=[u_{\beta}(k),u_{\rm c}(k)]^{\rm T} \tag{28}\]  
So, the optimal energy management strategy is described as finding optimal control vector \(U({\rm k})\) to minimize cost function \(J\)d and sustain relatively higher \(SOC\) in terminal driving. To insurance security and drivability in the driving cycle, the penalty function of torque increases sharply, and motor temperatures are considered in the cost function. \[S(k)=m|T(k+1)-T(k)|-n|Q(k+1)-Q(k)| \tag{29}\]  
The m is \(n\) plenty factors in the optimal process, and \(Q({\rm k})\) is the temperature predictor function which is measured in the experiment of different working conditions and recorded. When the temperature of the motor exceeds the threshold, the front driving mode will be replaced by the rear driving mode, corresponding to this condition, when the vehicle working in all driving modes the power of the motor will be limited. Based on [PERSON]'s optimization thought, all processes of optimization could be decomposed into a serious minimum optimization problem. The sub-problems are each step of optimal decision, where all processes of optimization control vector \(U^{*}(k)\) consist of the stage of optimal control vector u(k). \[U(k)=[u_{\beta}(k),u_{\rm c}(k)]^{\rm T} \tag{30}\]  
In all optimization, procedures are described as Equations (31) and (32)Step N - 1:  
\[J^{*}_{N-1}(X(N-1))=\min L(X(N-1),U(N-1))\] (31) Step k(0 < k < N - 1):  
\[J^{*}_{k}(X(k))=\min\big{[}L(X(k),U(k)+J^{*}_{k-1}(X(k-1))\big{]} \tag{32}\]  
where the \(J^{*}_{k}(X(k))\) is the optimal function of indicator, the optimal control strategy is solved by the above formula in the reserve method. In each moment k, there is always a control vector \(U^{*}(k)\) to optimal indicator function. All processes of dynamic programming are shown in Figure 5. The optimal control vector \(U^{*}(k)\) including different driving cycles is determined through objective function and record with the help of state-space updating.