## 2 Materials and Methods
### Parameter Matching of DICPS
#### 2.3.2. Gear Ratio Matching  
Since tractors often work in heavy-duty working conditions, such as plowing, requiring a large amount of power over a long duration, gear ratios should be reasonably designed to satisfy the demand power of DMETs within sufficient speed ranges. As can be seen from Equations (7)-(9), the dynamic performance of the DICPS mainly depends on \(k\), \(i_{r}\), \(i_{s}\), \(i_{m}\), and \(i_{f}\), where \(i_{m}\) and \(i_{f}\) are the same as those in SMPS. Since the values of \(i_{r}\) and \(i_{s}\) are closely related to the size of \(k\), and to verify the effect of \(k\) on the economy of DICPS, we set the feasible range of \(k\) as (1.6, 7.6). The maximum output torque of the DICPS in the three driving modes was  
\[\left\{ \begin{array}{ll}T_{\text{cmax},s}=(1+k)i_{s}T_{\text{max}}(\omega_{m1 })&M1S\\ T_{\text{cmax},r}=(1+k)i_{r}T_{\text{max}}(\omega_{m2})/k&M2S\\ T_{\text{cmax},D}=\min(T_{\text{cmax},s},T_{\text{cmax},r})&DMC\end{array} \right. \tag{13}\]  
where \(T_{\text{cmax},s}\), \(T_{\text{cmax},r}\), and \(T_{\text{cmax},D}\) denote the maximum output torque in the M1S mode, the M2S mode, and the DMC mode, respectively, and \(T_{\text{max1}}\) and \(T_{\text{max2}}\) denote the maximum output torque of EM_S and EM_R. In the DMC mode, both motors are active at the same time. Although their speed can be independently controlled, their torque depends entirely on the required output torque, as shown in Equation (2). As a result, the branch with the smallest torque capacity limits the maximum torque output of this mode. In order to maximize the operating range of the dual-motor driving mode, the gear ratio \(i_{r}\) and \(i_{s}\) must be reasonably selected so that \(T_{\text{cmax},s}=T_{\text{cmax},r}\). \[\frac{i_{s}}{i_{r}}=\frac{1}{k}\cdot\frac{T_{\text{max2}}}{T_{\text{max1}}} \tag{14}\]  
As can be seen from Figure 3, the available torque of the motor varies with speed and the motor can provide a constant maximum torque from the stationary point to the rated speed. The available torque then decreases with the increase of the motor speed. Since tractors have the characteristics of low speed and high torque in field operations, the speed range is mainly 0.8-10 km/h. Therefore, to provide the maximum available torque for the DICPS in heavy-duty operations, both motors should run at less than the rated speed. The restriction be expressed as  
\[\frac{\omega_{\text{c1}}}{i_{s}}+\frac{\omega_{\text{c2}}}{i_{r}}k\geq\frac{ 10}{3.6\cdot r_{q}}\cdot i_{inf}(1+k) \tag{15}\]  
Figure 3: Efficiency map of EM_T in SMPS. where \(\omega_{e1}\) and \(\omega_{e2}\) are the rated speed of EM_S and EM_R, respectively. To ensure that the DICPS can provide sufficient power for plowing and other heavy-duty operations, the maximum torque limit for plowing, at the maximum operating speed of 10 km/h, can be expressed as  
\[\left\{ \begin{array}{l}i_{m}i_{f}T_{\text{cmax}}(v)\geq F_{total}\cdot r_{q}\\ F_{total}=F_{T\text{max}}+mgf\\ F_{T\text{max}}=1.2\cdot(Zb_{1}h_{s}s_{r})\end{array} \right. \tag{16}\]  
where \(F_{total}\) is the total resistance on the tractor, \(F_{Tmax}\) refers to the maximum tractive force, \(Z\) is the number of plowshares, \(b_{1}\) is the ploughshare width, \(h_{s}\) is the tillage depth, and \(s_{r}\) is the soil-specific resistance. To ensure that the DMET can reach a maximum speed of 30 km/h, the relationship between the maximum output speed of the DICPS and gear ratios needs to be satisfied as follows. \[\left\{ \begin{array}{l}\frac{(\omega_{\text{max1}}/i_{k}+k\omega_{\text{ max2}}/i_{k})}{1+k}=\omega_{\text{max},c}\\ \omega_{\text{max},c}\geq\frac{V_{\text{max1}}i_{m}}{3.6r_{w}}\end{array} \right. \tag{17}\]  
where \(\omega_{\text{max1}}\), \(\omega_{\text{max2}}\), and \(\omega_{max,c}\) represent the maximum rotational speed for EM_S, EM_R, and the carrier, respectively, and \(V_{max}\) is the maximum speed of the tractor. Tractors need to meet the maximum gradient requirements in road transport conditions.