## 3 Loss Calculation of IM Driven FESS
### Electrical Losses  
The electrical losses in FESS are mainly the due to the stator and rotor losses in the motor/generator set. The stator electrical losses are the combination of copper losses (\(P_{scl}\)) in the stator winding and core losses (\(P_{core}\)) in the stator core. The stator copper losses are proportional to the square of the RMS magnitude of stator currents \(I_{s}\) and is dependent on the value of the stator resistance \(R_{s}\). Core losses (eddy current and hysteresis losses) is the function of electrical angular frequency \(f_{e}\) and maximum \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline Stator Resistance (\(\mathbf{R_{s}}\)) & 11.85 m\(\Omega\) \\ Rotor Resistance (\(\mathbf{R_{r}}\)) & 9.29 m\(\Omega\) \\ Stator leakage inductance (\(\mathbf{L_{ls}}\)) & 0.2027 mH \\ Rotor leakage inductance (\(\mathbf{L_{lr}}\)) & 0.2027 mH \\ Mutual Inductance (\(\mathbf{L_{m}}\)) & 9.295 mH \\ Rated magnetic flux (\(\mathbf{\lambda_{m}}\)) & 0.75 Weber \\ Power rating & 100 kW \\ Nominal torque & 90 N\(\cdot\)m \\ Maximum speed & 20 krpm \\ Minimum speed & 10 krpm \\ Switching frequency & 20 kHz \\ DC bus voltage & 600 V \\ Air density (\(\mathbf{\rho}\)) & 0.0011 kg/m\({}^{3}\) \\ Air dynamic viscosity (\(\mu\)) & \(1.91\times 10^{-5}\) kg/m\(\cdot\)s \\ Flywheel rotor outer diameter (\(\mathbf{D_{r0}}\)) & 0.4 m \\ Shaft diameter (\(\mathbf{D_{s}}\)) & 0.025 m \\ Bearing design factor (\(\mathbf{f_{L}}\)) & 0.0005 \\ Kinematic oil viscosity (\(\mathbf{V_{0}}\)) & \(130\times 10^{6}\) m\({}^{2}\)/s \\ Lamination thickness (\(\mathbf{t}\)) & 0.15 mm \\ Conductivity of electrical steel (\(\sigma\)) & \(2.08\times 10^{6}\) Siemens/m \\ Density of electrical steel (\(\mathbf{\rho}\)) & 7650 kg/m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Values of coefficients and parameters of SCIM-FESS. magnetic flux density \(B_{m}\). Equation (10) presents the general relation of core losses in the electrical machine while the total electrical losses in the stator of IM are given by Equation (11). \[P_{core}=\left(K_{hys}f_{e}+K_{e}f_{e}^{2}\right)B_{m}^{2} \tag{10}\]  
\[P_{stator}=3l_{s}^{2}R_{s}+\left(K_{hys}f_{e}+K_{e}f_{e}^{2}\right)B_{m}^{2} \tag{11}\]  
where \(K_{hys}\) is the Steinmetz co-efficient, \(B_{m}\) is peak magnetic flux density and \(K_{e}\) is a constant factor as given by Equation (12) [64]. \[K_{e}=\frac{\pi^{2}t^{2}\sigma}{6\rho} \tag{12}\]  
where \(t\) is the thickness of laminations, \(\sigma\) is electrical conductivity and \(\rho\) is the density of the electrical steel. The rotor copper losses (\(P_{rcl}\)) are also a combination of copper and rotor iron losses (\(P_{core}\)) and are given by Equation (13). \[P_{rotor}=3l_{r}^{2}R_{r}+\left(K_{hys}f_{sl}+K_{e}f_{sl}^{2}\right)B_{m}^{2} \tag{13}\]  
where, \(f_{sl}=sf_{e}\), \(s\) is the slip, \(R_{r}\) is rotor resistance and \(l_{r}\) is the RMS magnitude of rotor currents in one phase of the rotor. For calculation of Steinmetz coefficient (\(K_{hys}\)), Equation (10) is divided by the term \(f_{e}B_{m}^{2}\) to create a linear relationship. \[\frac{P_{core}}{f_{e}B^{2}{}_{m}}=(x+yf_{e}) \tag{14}\]  
where, \(x=K_{hys}\) and \(y=K_{e}\) which can be simply calculated by substituting the values of parameters (\(t\), \(\rho\) and \(\sigma\)) into Equation (12) provided in Table 2 giving \(y=3.34\times 10^{-3}\) Hz. The value of \(x\) can be calculated by linear fitting of Equation (14) for the best approximation of the data. The stator and rotor core of the IM are assumed to be made of electric steel M470-50 HP. Manufacturer's core loss data for M470-50 HP is shown in Table 3 where the core loss data for specific electrical frequencies of 50 Hz, 100 Hz, 200 Hz and 400 Hz is provided. The electrical frequency of SCIM-FESS presented in this paper at 20 krpm is 333 Hz that lies between 200 Hz and 400 Hz. Therefore, the values of \(B_{m}\) at 333 Hz is approximated between 0.6-0.7 Tesla as calculated based on the methodology discussed in [65].