## 3 Flywheel Electrical Model  
### Flywheel System  
The flywheel system consists of a 20 kW-rated module based on a steel rotor coupled to a 4-poles brushless electrical machine (EM) with surface permanent magnets (PMs). The power rating complies with commercially available components; higher power requirement can be fulfilled by combining more units. Installing the EM outside the flywheel housing, usually kept at low pressure to reduce aerodynamical drag losses, simplifies both the EM-AC/DC power converter connection and the EM cooling [23]. A sketch of the flywheel system is provided in Figure 10, showing also the DC bus circuit shared with the battery modules. The main system operational data and sizes are reported in Table 2. The speed and temperature range identify the minimum and maximum allowable values during the operating conditions. The maximum \(\Omega_{max}\) and minimum \(\Omega_{min}\) speed values are defined as a trade-off between deliverable energy per cycle and the electromechanical constraints which remarkably affect the system design. In fact, both the maximum allowable stress for the rotary components and the torque/voltage/current limits for the electromagnetic and electronic appliances must be considered. The maximum temperature is constrained by insulation and PM thermal limits. The flywheel rotor (FWR) is designed assuming an AISI 4340 steel (ultimate strength \(\sigma_{u}=1790\) MPa) truncated De Laval disc with an outer ballast [24]. The resulting shape and velocity factors \(K=0.96\) and \(\chi=2.51\) yield the mass values 69.3 kg and 3.7 kg for the disc and the ballast, respectively. PM EMs are generally preferred for their higher efficiency and power density. Coreless stator configurations can be particularly convenient due to negligible no-load electromagnetic losses and low coil inductance, the latter aspect profitable the dynamic torque control promptness [25]. On the other side, they rely on high grade PMs and complicated winding assembly, so in this paper the more \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{**Operational Data**} & \multicolumn{2}{c}{**Sizes**} \\ \hline Rated power & 20 kW & Active length & 120 mm (EM) \\ Speed range & 10,833 rpm\(\div\)32,500 rpm & Airgap width & 2 mm (EM) \\ Max current rms value & 223 A & Outer diameter & 400 mm (EM) \\ Max torque (@40 \({}^{\circ}\)C) & 17.62 Nm & Flywheel disc height & 91 mm \\ Max phase voltage (@40 \({}^{\circ}\)C) & 90 V & Rotating mass & 9 kg (EM) \\ Temperature range & 25 \({}^{\circ}\)C \(\div\) 130 \({}^{\circ}\)C & Rotational inertia & 1.17 kgm\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Main operational FWR an EM data and sizes. Figure 10: Sketch of the flywheel energy storage system. conventional iron cored EM configuration is considered, benefiting from reduced ohmic losses during frequent charge/discharge operations--generally required for grid frequency regulation--and the chance to adopt ferrite PMs, with negligible eddy current losses and operating temperature (300 \({}^{\circ}\)C) remarkably higher than rare earth PMs, even if with lower magnetic performances. The magnetic properties at 40 \({}^{\circ}\)C (retentivity \(B_{r}\) = 0.384 T, coercivity \(H_{c}\) = 272 kA/m) fulfill the voltage and torque ratings at \(\Omega_{max}\) and \(\Omega_{min}\) respectively, provided that a 3-mm gap (including the airgap width and the PM retainment sleeve thickness) is held between the PM and the inner stator diameters. The developed flywheel system model enables us to simulate the mechanical dynamics assuming the EM steady-state operation, by virtue of significantly smaller electrical time constants than mechanical ones and fast response of the electronic converter regulation system. However, most of the system parameters depend (often nonlinearly) on various process variables, such as temperature, speed and current, and therefore reliable models taking into account such phenomena are essential to effectively assess the power exchange and the charge/discharge time intervals.