## 2 Theoretical Model of Loss Calculation and Thermal Analysis
### Advanced Thermal Modeling Methods  
Thermal modeling of high-speed magnetic field-modulated motors in flywheel energy storage systems necessitates sophisticated methodologies to accurately predict temperature distributions under dynamic operating conditions [27]. The evolution of thermal modeling approaches has progressed from simplified analytical methods to advanced multiphysics frameworks that capture the complex thermal behavior of these systems [11]. Lumped parameter thermal networks (LPTNs) remain foundational in thermal analysis due to their computational efficiency and physical interpretability [28; 29]. In this approach, the motor structure is discretized into thermal nodes connected by thermal resistances and capacitances, with the heat transfer described by the following:  
\[C_{i}\frac{dT_{i}}{dt}=\sum_{j=1}^{n}\frac{T_{j}-T_{i}}{R_{ij}}+P_{i} \tag{4}\]  
where \(C_{i}\) represents the thermal capacitance of node \(i\), \(T_{i}\) is the temperature of node \(i\), \(R_{ij}\) is the thermal resistance between nodes \(i\) and \(j\), and \(P_{i}\) is the heat generated at node \(i\). For high-speed applications, these parameters exhibit speed dependence, particularly the convective heat transfer coefficients that scale with rotational velocity according to the following:  
\[h_{conv}=k_{1}+k_{2}\cdot Re^{k_{3}}\cdot Pr^{k_{4}} \tag{5}\]  
where \(Re\) and \(Pr\) represent the Reynolds and Prandtl numbers, respectively, with coefficients \(k_{1}\) through \(k_{4}\) determined empirically or through numerical analysis. Computational fluid dynamics (CFD) approaches provide more detailed thermal analyses by solving the governing conservation equations for mass, momentum, and energy. For vacuum-sealed flywheel systems, the Navier-Stokes equations reduce to the following:  
\[ \begin{array}{l}\rho\frac{\partial v}{\partial t}+\rho(v\cdot\
abla)v=-\
abla p +\mu\
abla^{2}v+\rho g\\ \rho c_{p}\frac{\partial T}{\partial t}+\rho c_{p}\mathbf{v}\cdot\
abla T= \
abla\cdot(k\
abla T)+\dot{q}\end{array}  \tag{6}\]  
where \(\rho\), \(v\), \(p\), \(\mu\), \(c_{p}\), \(k\), and \(\dot{q}\) denote fluid density, velocity vector, pressure, dynamic viscosity, specific heat capacity, thermal conductivity, and volumetric heat generation, respectively. These equations are typically solved using finite volume or finite element methods with appropriate boundary conditions and turbulence models. The coupled electromagnetic-thermal analysis methodology represents a state-of-the-art approach, wherein electromagnetic field solutions provide spatially and temporally resolved loss distributions that serve as thermal sources in the heat transfer analysis. This bi-directional coupling accounts for temperature-dependent material properties through iterative solution procedures:  
\[ \begin{array}{l}\sigma(T)=\sigma_{0}/[1+\alpha(T-T_{0})]\\ B_{r}(T)=B_{r0}[1+\alpha_{B}(T-T_{0})]\end{array}  \tag{7}\]  
where \(\sigma(T)\) represents temperature-dependent electrical conductivity and \(B_{r}(T)\) denotes the temperature-dependent remanent flux density of permanent magnets. This integrated approach enables accurate prediction of thermal hotspots, particularly in critical components like permanent magnets and rotor structures where cooling is limited. Recentadvancements incorporate machine learning techniques to accelerate thermal model parameterization and reduce computational demands while maintaining prediction accuracy. The thermal boundary conditions and convection coefficients used in this analysis were carefully selected based on experimental calibration. For water cooling interfaces, the convection coefficient was determined through experimental testing using the Wilson plot method, resulting in the correlation h = 2350\(\cdot\)v\({}^{0.8}\)W/m\({}^{2}\)K (where v is the coolant velocity in m/s), with an uncertainty of \(\pm\)12% across the flow rate range of 2-10 L/min.