## 2 Proposed Thermal Model  
### Simplified Lumped Parameter Thermal Model  
[PERSON] [30] analyzed the structure of a fully enclosed air-cooled induction motor and proposed a 10-node LPTN-based on heat transfer theory, as shown in Figure 1. According to the motor structure, this model is divided into 10 nodes: The casing, stator core, stator teeth, stator coil, air gap, end winding, end cover air gap, rotor coil, rotor core, and shaft. It reflects the heat transfer process inside the motor objectively and is therefore widely applied to offline temperature analysis of motors. Figure 1: Complete thermal model of the induction motor. However, the accuracy of the thermal model based on heat transfer theory is difficult to guarantee. The reasons are listed as follows:  
1. Ignoring the contact resistance: In LPTN, the internal thermal resistance of the solid is mainly considered, and the contact thermal resistance between the solids is neglected. In fact, due to the influence of factors, such as installation, this part of the thermal resistance has a great impact on the radial heat flow. 2. High dependence on the accuracy of the motor geometry model and the heat transfer characteristics of the material: In order to obtain an accurate motor model, a considerable number of motor dimensions and material-related parameters are required. In actual engineering practice, full material and geometric information is not available. 3. The fluid model accuracy is not ideal: The model established for the internal air of the motor is based on the empirical formula of heat transfer theory. However, these formulas are sensitive to the size and shape of the air gap. 4. Complex calculation, not suitable for online estimation: It can be found that the model order is too high due to the modeling of each part of the motor, leading to the large differential-algebraic-equation system. Based on the above analysis, a simplification model has been proposed in this paper to predict the temperatures in PMSMs. According to the structure of the motor, PMSM can mainly be divided into several parts: The cooling jackets, casing, stator core, wingdings, permanent magnet, rotor core, and shaft. The cross-section of the modelled motor is shown in Figure 2. With respect to the simplified thermal model, the coolant, casing, stator, rotor, and end cap are chosen as the thermal nodes. Several reasons are taken into consideration as follows:  
1. The heat exchange between the coolant and the motor is much greater than the heat exchange between the casing and the environment. The coolant is the main way to exchange heat with the outside world in PMSMs, so the influence of the outside temperature is ignored. At the same time, the coolant is an important heat source outside the motor, which is collected by the motor controller under real vehicle conditions. 2. The motor stator node consists of the stator core and stator windings. These parts are closely connected and have a complicated geometry. If it is treated as a node, the complexity of the model can be greatly simplified. Besides, the stator loss is also an important heat source inside the motor. 3. The motor rotor node comprises the rotor core, permanent magnet, and motor shaft. This part is separated from the stator node with the air gap. The rotor iron loss is also an important internal heat source. 4. The end cap and the casing are considered as two important nodes in connecting the axial and radial heat paths, making the thermal flow in a complete circuit. 5. In water-cooled PMSMs, there is no rib on the motor rotor to increase the convection heat dissipation effect, and the heat transfer effect of the end air is greatly weakened. Thus, the axial heat transfer path through the end air to the end cap is neglected and the topology of the model is thus simplified. The heat path in the axial direction has been considered only in the shaft. Therefore, this proposed model is optimized according to the structural characteristics of PMSMs for vehicles.