## 5 Calculation of Losses in FESS
### Other Losses  
Calculating core losses and eddy current losses of the electrical MG is not easy as they are produced under load conditions and require experimental measurements. A theoretical measurement of the combination of eddy current losses in electrical machine conductors, iron core, and other metallic parts, due to flux leakages, is referred to as stray load losses [64]. It can be calculated in terms of stator \(dq\) currents \(i_{d}\) and \(i_{q}\), and angular speed \(\omega\), as shown in Equation (22) [65]. However, the stray loss constant \(C_{str}\) needs to be determined experimentally.  
\[P_{str}=C_{str}\omega^{2}\Big{(}{i_{d}}^{2}+{i_{q}}^{2}\Big{)} \tag{22}\]  
An approximation of the stray losses can be calculated based on IEEE standard expression in terms of the generated power \(P_{G}\) and rated power \(P_{R}\):  
\[P_{str}=0.005\frac{P_{G}{}^{2}}{P_{R}} \tag{23}\]  
As stray loss is produced in the electrical machine, the rated power \(P_{R}\) and electromagnetic torque \(T_{e}\) (N-m) of the PMSM is used to derive this expression in terms of the angular speed \(\omega\).  
For the calculation of losses in PMSM-Flywheel unit the following assumptions were made:  
* The clearances between stationary and rotating components is small, such that the boundary layers on stationary and rotating surfaces merge.
* The flywheel rotor is assumed to be a drum type steel laminated cylinder.
* The windage losses are calculated under maximum pressure conditions of 100 Pa at 40 \({}^{\circ}\)C and maximum speed of 20,000 rpm.
* Density values are calculated from temperature and pressure using ideal gas law.
* Windage on the motor-generator is neglected, since its rotor diameter is much smaller than that of the main rotor, and the power law on diameter for windage is 5.
* Switching loss and IGBT conduction loss in power converters are neglected as these losses are small compared to other losses in a FESS. Although, this might not be the case in some other storage systems, where they may need to be considered.
* It is assumed that the combined axial load on the bearing is 10% of the total flywheel rotor weight with the rest taken by the passive magnetic bearing.  
The parameters and coefficients of the flywheel unit are given in Table 5. Substituting the dynamic viscosity (\(\mu\)) and coolant density (\(\rho\)) in Equation (16) gives a Reynolds number of approximately 4890, and therefore Equation (14) can be used to calculate the torque coefficient (\(C_{M}\)).  
Substituting (14) and (16) in (13) gives windage losses as a function of the flywheel speed.  
\[P_{W}=0.121\frac{\sqrt{\rho\mu\omega^{5}}}{D_{ro}}\Big{(}{D_{ro}}^{5}-{D_{s}}^ {5}\Big{)} \tag{24}\]  
Equation (13) can be reduced further by substituting \(\rho,\ \mu,\ D_{ro}\) and \(D_{s}\) parameters from Table 5:  
\[P_{W}=4.51\times 10^{-7}\ \omega^{\frac{5}{2}} \tag{25}\]  
Power loss due to bearing friction is calculated by selecting a single row radial bearing with grease type of lubrication method. Substituting the relevant parameters from Table 5 into Equation (17) gives  
\[P_{B}=0.01\omega+2.67\times 10^{-4}\ \omega^{\frac{5}{5}} \tag{26}\]Similarly, copper and stray losses are calculated by substituting the parameters and coefficients in Equations (12) and (23):  
\[P_{cut}=3\times 10^{-5}\,\omega^{2} \tag{27}\]  
\[P_{str}=3.58\times 10^{-5}\,\omega^{2} \tag{28}\]  
The derived equations show that the windage loss is proportional to \(\omega^{\frac{5}{2}}\), the significant term in the bearing loss is proportional to \(\omega^{\frac{5}{3}}\), and copper and stray losses are proportional to the square of the angular speed.