## 1 Introduction
A major drawback of this approach originates from the dependence of the material and geometrical parameter of the motor with respect to the model parameter calculation. White-box LPTNs are composed of several sub-nodes, which more accurately reflect the real thermal distribution in the motor. However, considerable nodes require more computation cost when adopting this method [28; 29]. In the case of the electric vehicle (EV) application, the dark gray-box LPTN has drawn more attention, since high demands are placed on cost and real-time computation in this industry. In addition, full material and geometric information of the motor is hard to access. The establishment of light gray-box LPTNs and white-box LPTNs is based on heat transfer theory. The accuracy of this method is difficult to guarantee due to the high dependency of the motor geometry model and the heat transfer characteristic of the material. Besides, the large differential-algebraic-equation system leads to complex calculation, which is not suitable for online temperature estimation. Thus, dark gray-box LPTNs are suitable to predict the temperature distribution in the PMSM of EVs. Three-node thermal models for accurate temperature estimation in permanent magnets, windings, and end windings are introduced in [20; 21; 23]. This modeling behavior is under some assumptions that the temperature distribution is homogeneous among each motor component and the other heat paths are ignored. In order to monitor the temperature of critical parts accurately, these components should be divided into several parts to get the precise heat distribution. With respect to the crucial part, the motor stator, [PERSON] divided the stator node into three nodes, i.e., the stator yoke, stator teeth, and stator winding [18]. The model robustness and accuracy were validated with a maximum temperature error of 8 \({}^{\circ}\)C. However, the above-mentioned studies only considered heat transfer in the radial direction neglecting the axial heat flow, and thus the real thermal flow among the PMSM cannot be reflected. However, the compromise between the precision and size of the reduced model must be made when employing the dark gray-box LPTN. Therefore, a simplified five-node thermal model is proposed in this paper considering both the real thermal flow and simplification of the model. We consider both the radial and axial heat transfer paths inside the motor and try to set up minimum thermal nodes to simulate the complete thermal circuit. In the existing literature, the specific parameters in the thermal model are identified instead of the data in the state equation applying a complex scheme [19; 20], which requires more computational resources and processing time. Thus, considering the good linear relationship of the state equation for the simplified thermal model, the multiple linear regression algorithm method is proposed in this paper to identify the thermal model parameter. This method does not depend on the structural parameters of the motor. It is unnecessary to identify each specific parameter in the thermal model, and only the parameters in the state equation need to be identified. In conclusion, the proposed five-node LPTN and parameter identification method are aimed at solving temperature estimation problems in automobile applications. Since the cost pressure and demand on real-time execution is very high in this industry, the proposed algorithm should not be complicated when applied in an embedded system, which has to deal with multiple tasks in the meantime. Under consideration of the specific application in the automobile industry, the proposed scheme is easy and convenient to implement. In addition, the temperature accuracy proves the effectiveness of the proposed method.