## 6 Analysis and Control of FESS  
### System Configuration and Mathematical Model  
The general structure of FESS studied in this paper is as shown in Figure 1. The flywheel rotor is integrated with a MG to support the electrical energy conversion. The integrated MG is controlled by a bidirectional converter, which can operate as an inverter to charge the flywheel, or as a rectifier to discharge the flywheel. The flywheel and the MG are connected coaxially, thus controlling the MG, and the converter enables control of the FESS. The MG is a PMSM which is widely used in motion control applications due to its high performance, compact structure, high power density, high steady state torque density, and high efficiency. It maximises the performance in applications where variable speed is required [66]. For the same rating, PMSM has a higher power density to that of an induction motor. Due to its simple method of control, lower mass and low losses, PMSM has been broadly used in numerous applications in recent years [67; 68; 69; 70; 71]. The operation of the PMSM-Flywheel is controlled using proportional-integral (PI) cascaded control structure. The controller is designed to follow the \begin{table}
\begin{tabular}{l l} \hline \hline \multicolumn{1}{l}{Stator winding resistance (\(R_{s}\))} & 0.20 \(\Omega\) \\ \multicolumn{1}{l}{Armature inductance (\(I_{d}=I_{u}\))} & 0.834 mH \\ \multicolumn{1}{l}{Combined inertia (PMSM and flywheel) (\(I\))} & 12 kg m\({}^{2}\) \\ \multicolumn{1}{l}{Friction coefficient (\(B_{m}\))} & 0.8 \(\times\) 10\({}^{-4}\) N\(\cdot\)m\(\cdot\)s \\ \multicolumn{1}{l}{Permanent magnet flux (\(\lambda\))} & 0.175 Wb \\ \multicolumn{1}{l}{Number of poles} & 2 \\ \multicolumn{1}{l}{Power rating} & 10 kW \\ \multicolumn{1}{l}{Energy storage rating} & 5 kWh \\ \multicolumn{1}{l}{Maximum torque} & 12 N\(\cdot\)m \\ \multicolumn{1}{l}{Maximum output power} & 15 kW \\ \multicolumn{1}{l}{Rated torque} & 8 N\(\cdot\)m \\ \multicolumn{1}{l}{Maximum angular speed} & 20,000 rpm \\ \multicolumn{1}{l}{Minimum angular speed} & 10,000 rpm \\ \multicolumn{1}{l}{Machine rated over speed} & 25,000 rpm \\ \multicolumn{1}{l}{Switching frequency} & 10 kHz \\ \multicolumn{1}{l}{Filter damping resistance} & 0.75 \(\Omega\) \\ \multicolumn{1}{l}{Filter capacitance} & 4 \(\times\) 10\({}^{-5}\) F \\ \multicolumn{1}{l}{Filter inductance} & 640 \(\times\) 10\({}^{-6}\) H \\ \multicolumn{1}{l}{DC-bus voltage} & 600 V \\ \multicolumn{1}{l}{Voltage constant (\(K_{e}\))} & 2 \(\times\) 10\({}^{-3}\) V\(\cdot\)s/rad \\ \multicolumn{1}{l}{Air density (\(\rho\))} & 0.0011 kg/m\({}^{3}\) \\ \multicolumn{1}{l}{Air dynamic viscosity (\(\mu\))} & 1.91 \(\times\) 10\({}^{-5}\) kg/m\(\cdot\)s \\ \multicolumn{1}{l}{Flywheel rotor outer diameter (\(D_{ro}\))} & 0.4 m \\ \multicolumn{1}{l}{Shaft diameter (\(D_{s}\))} & 0.025 m \\ \multicolumn{1}{l}{Torque coefficient (\(C_{M}\))} & 0.0554 \\ \multicolumn{1}{l}{Bearing design factor (\(f_{L}\))} & 0.0007 \\ \multicolumn{1}{l}{Kinematics oil viscosity (\(V_{o}\))} & 130 mm\({}^{2}\)/s (40 \({}^{\circ}\)C) \\ \hline \hline \end{tabular}  
* PMSM parameters obtained from real data of the motor manufacturer.  
\end{table} 
Table 5: Parameters and coefficients of a combined surface-mounted permanent magnet synchronous motor (PMSM) 1 and flywheel energy storage systems (FESS). changes in position, speed, and torque of the machine to a set of reference values. Due to its flexibility, it is widely used in industry to control the position and velocity of AC motor drives. It generally consists of segregated control loops with the outermost position loop followed by the inner speed loop, and the innermost current (torque) loop as shown in Figure 3. This technique requires the response time of the current loop to be the fastest compared to the speed and position loops [72]. Flywheel applications do not require the position loop, since the main objective is to control the state of charge based on information from the speed feedback loop.