## Appendix A Parametrizing the Thermal Model  
From a thermal point of view, the motor is modeled with four nodes: stator winding (node 1), stator core (node 2), rotor winding (node 3), and rotor core (node 4). The thermal equivalent network is illustrated in Figure 11 with thermal capacitances; it is connected to a power source and has thermal conductance among the nodes and to the cooling air. Values of the thermal capacitance C\({}_{18}\), C\({}_{28}\), C\({}_{3r}\), and C\({}_{4r}\) are calculated analytically from the geometric and material information of the motor, where the indices 1s, 2s, 3r, and 4r refer to Node 1-Stator, Node 2-Stator, Node 3-Rotor, and Node 4-Rotor respectively. The capacitance for stator yoke C\({}_{18}\) is the sum of the capacitance of stator housing, stator back iron, stator tooth, and flange mounting [36]. The stator winding capacitance C\({}_{28}\) includes the capacitance for the stator winding and the end winding capacitances. The capacitance for stator yoke C\({}_{1r}\) is the sum of the capacitance of the rotor yoke and rotor bars. The rotor winding capacitance C\({}_{2r}\) includes the capacitance for the rotor winding and the end winding capacitances. The thermal conductance \(\lambda_{18}\), \(\lambda_{28}\), \(\lambda_{3r}\), and \(\lambda_{4r}\) vary with the airflow due to the convection. The model shown in Figure 11 can be represented mathematically by the following first-order differential system:  
\[P_{1}=\text{C}_{18}\frac{dT_{1}}{\text{dt}}+\lambda_{18}(T_{1}-T_{\text{env}} )+\lambda_{128}(T_{1}-T_{2}), \tag{11}\]  
\[P_{2}=\text{C}_{28}\frac{dT_{2}}{\text{dt}}+\lambda_{28}(T_{2}-T_{\text{env}} )+\lambda_{128}(T_{2}-T_{1}), \tag{12}\]  
\[P_{3}=\text{C}_{3r}\frac{dT_{3}}{\text{dt}}+\lambda_{3r}(T_{3}-T_{\text{env}} )+\lambda_{34r}(T_{3}-T_{4}), \tag{13}\]  
\[P_{4}=\text{C}_{4r}\frac{dT_{4}}{\text{dt}}+\lambda_{4r}(T_{4}-T_{\text{env}} )+\lambda_{34r}(T_{4}-T_{3}), \tag{14}\]  
where T\({}_{i}\) is the temperature at the corresponding node i. The temperatures of the cooling air at the four nodes (marked as SW, SC, RW, and RC in Figure 11) are assigned to the environmental (or ambient) temperature T\({}_{\text{env}}\). The losses at the four nodes in Figure 11 are distributed as shown in Table 11 and they can be calculated as follows:  
\[P_{1}=K_{stemp}P_{cut1}+K_{stray}P_{stray}+K_{harm}P_{harm}, \tag{15}\]  
\[P_{2}=K_{pfe}P_{fe}, \tag{16}\]  
\[P_{3}=K_{temp}P_{cut2}+\left(1-K_{stray}\right)P_{stray}+(1-K_{harm})P_{harm}, \tag{17}\]  
\[P_{4}=\left(1-K_{pfe}\right)P_{fe}, \tag{18}\]where \(\rm P_{cul}\), \(\rm P_{cul}\), \(\rm P_{stray}\), \(\rm P_{harm}\), and \(\rm P_{fe}\) are the stator copper loss, rotor copper loss, stray loss, harmonic loss, and iron loss, respectively. The coefficients \(\rm K_{stemp}\), \(\rm K_{rtemp}\), \(\rm K_{stray}\), \(\rm K_{harm}\) and \(\rm K_{Pfe}\) are the corresponding loss coefficients. The losses in Equations (A5)-(A8) can be calculated as follows [8; 37; 38; 39]:  
\[P_{cul}=R_{1}\ I_{1}^{2},\] (A9)  
\[P_{cul}=R_{21}\ I_{21}^{2},\] (A10)  
\[P_{2}=\text{C}_{28}\frac{dT_{2}}{\text{dt}}+\lambda_{28}(T_{2}-T_{\text{env}} )+\lambda_{128}(T_{2}-T_{1}), \tag{12}\]0 (A11)  
\[P_{2}=\text{C}_{28}\frac{dT_{2}}{\text{dt}}+\lambda_{28}(T_{2}-T_{\text{env}} )+\lambda_{128}(T_{2}-T_{1}), \tag{12}\]1 (A12)  
where \(\rm I_{1}\), and \(\rm I_{21}\) are the stator and rotor currents, respectively, and \(\rm R_{1}\) and \(\rm R_{2}\) are the stator and rotor winding resistances, respectively, which depend on the temperature according to following equations:  
\[P_{2}=\text{C}_{28}\frac{dT_{2}}{\text{dt}}+\lambda_{28}(T_{2}-T_{\text{env}} )+\lambda_{128}(T_{2}-T_{1}), \tag{12}\]2 (A13)  
\[P_{2}=\text{C}_{28}\frac{dT_{2}}{\text{dt}}+\lambda_{28}(T_{2}-T_{\text{env}} )+\lambda_{128}(T_{2}-T_{1}), \tag{12}\]3 (A14)  
where \(\rm R_{1,20}\) and \(\rm R_{21,20}\) are the stator and rotor winding resistances at 20 \({}^{\circ}\)C, and \(\rm\alpha_{R1}\) and \(\rm\alpha_{R21}\) are the temperature coefficients of the stator and rotor, respectively.