## 2 Materials and Methods
### Energy Management Strategy
#### 2.4.1 Power Distribution Control  
When the DICPS works in single-motor driving mode (M1S, M2S), the output power of the motor is directly determined by the required speed and torque of the driving wheels, without power distribution. When the DICPS is in dual-motor coupling driving mode, the output power of both motors can be valued within a certain range. It is necessary to determine the power flow between two motors. As a pure electric tractor, the output power of the battery, composed of consumed power and charging power, can be expressed as  
\[P_{bat}=\left(T_{m1}\omega_{m1}\eta_{m1}-^{sign(\omega_{m1}T_{m1})}+T_{m2} \omega_{m2}\eta_{m2}-^{sign(\omega_{m2}T_{m2})}\right)/9550 \tag{19}\]  
According to Equations (4), (5), and (19), the internal power consumption of the battery can be obtained, and the optimal control problem can be formulated as  
\[\left\{ \begin{array}{l}\min_{u}(J_{c}=\int_{l_{0}}^{t_{f}}c(x(t),u(t))dt)\\ c(x(t),u(t))=P_{e}\end{array} \right. \tag{20}\]  
The goal was to find the control set \(u\) that would minimize the cost \(J_{c}\), defined as the integral of a cost function over a time interval. \(c(x(t),\)\(u(t))\) denotes the objective function, which is the internal energy of the battery. Four variables can be found in the objective function, i.e., \(T_{m1},\)\(\omega_{m1},\)\(T_{m2},\) and \(\omega_{m2}\). However, these four variables are not independent. After the tractor speed and acceleration are known, the torques of the two motors can be expressed as  
\[\left\{ \begin{array}{l}T_{m1}=\frac{I_{f}\dot{\omega_{f}}+T_{L}}{i_{m}[\mu(I _{c}+1)}+I_{m1}\dot{\omega}_{m1}\\ T_{m2}=\frac{k(I_{f}\dot{\omega_{f}}+T_{L})}{i_{m}[\mu(I_{c}+1)}+I_{m2}\dot{ \omega}_{m2}\end{array} \right. \tag{21}\]  
where \(I_{f}\) denotes the equivalent inertia of tractor body including transmission components and \(\dot{\omega}_{f}\) is the angular acceleration of the final shaft. As to the speed, there was only one independent design variable, which was the EM_S speed in this study. Once it was determined, the EM_R speed could be calculated by  
\[\left\{ \begin{array}{l}\omega_{m2}=i_{2}\frac{\omega_{c}(1+k)-\omega_{m1}/i _{1}}{k}\\ \omega_{c}=\frac{\frac{\gamma_{e}\gamma_{e}30}{\pi t_{w}}\ast i_{m}\dot{f}} \end{array} \right. \tag{22}\]  
where \(v_{a}\) is the actual velocity of the tractor in m/s. In summary, there were only two independent variables of the proposed EMS: the motor speed and the working mode. However, the selection of working mode in this study was based on the maximum overall efficiency principle, and the efficient mode was selected among the three driving modes. When both EM_S and EM_R can provide the required power of the DMET, the mode with the least power losses is automatically selected according to the optimal solution. Therefore, the speed of EM_S was chosen as the control variable, the optimization model can be rewritten as Equation (23) when the target acceleration \(a_{t}\) and speed \(v_{t}\) are decided by the working cycle. \[\min_{\omega_{m1}}\!\!P_{e}(v_{t},a_{t},\omega_{m1}) \tag{23}\]  
To ensure the safe and reasonable operation of the motors, the following inequality constraints were necessary. \[\left\{ \begin{array}{l}0\leq\omega_{m1}(t)\leq\omega_{\max 1}\\ 0\leq\omega_{m2}(t)\leq\omega_{\max 2}\\ 0\leq\left|T_{m1}(\omega_{m1}(t))\right|\leq T_{max1}(\omega_{m1}(t))\\ 0\leq\left|T_{m2}(\omega_{m2}(t))\right|\leq T_{max2}(\omega_{m2}(t))\end{array} \right. \tag{24}\]Since the design space contained many local optimal points, it was difficult to find the global optimal point using traditional optimization algorithms. Here, we used the quadratic programming algorithm to find the approximate optimal solution and calculate the instantaneous overall efficiency. The flow chart of power distribution is shown in Figure 4.  
Figure 5 shows the curve of the objective function in the feasible area under the case that PDR = 0.65, \(k\) = 2.6, SOC = 60, and the shape of the objective function changes according to the driving conditions.