## 4 Design and Control of SCIM-FESS  
The general schematic of the model discussed in this paper is presented in Figure 3. The flywheel rotor coupled with the SCIM is modelled as an inertia added to the shaft of the SCIM. The dynamics of SCIM-flywheel can be analysed and represented by Equation (28).  
\[J_{0}\frac{d\omega}{dt}=T_{e}-B\omega \tag{28}\]  
where \(T_{e}\) (N\(\cdot\)m) is the electromechanical torque, \(\omega\) (rad/s) is the angular velocity of the flywheel, \(J_{0}\) is the combined inertia (\(J_{0}=J_{SCIM}+J_{flywheel}\)) of the IM and flywheel rotor (kg\(\cdot\)m\({}^{2}\)) and \(B\) is the damping coefficient (N\(\cdot\)m\(\cdot\)s/rad).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Bearing Type** & **Nominal Contact Angle (Degrees)** & \(z\) & \(y\) \\ \hline Radial deep-groove & 0 & 0.0004–0.0006 & 0.55 \\ Thrust & 90 & 0.0008 & 0.33 \\ Double-row self-aligning & 10 & 0.0003 & 0.4 \\ Angular-contact & 30–40 & 0.001 & 0.33 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Values of \(z\) and \(y\). Adapted with permission from [69], Taylor and Francis Group LLC, 2019.  
The dynamic equations of the stator and rotor voltages of the SCIM are presented in Equations (29) and (30), respectively. For the ease of analysis and modelling of the control schemes, time varying sinusoidal quantities are transformed into \(d\)-\(q\) synchronous rotating frame by Park's transformation. A detailed analysis and description of the dynamic model of SCIM is discussed in [74].  
\[\left\{\begin{array}{l}v_{ds}=R_{s}i_{ds}-\omega_{e}\lambda_{qs}+\frac{d}{dt} \lambda_{ds}\\ v_{qs}=R_{s}i_{qs}+\omega_{e}\lambda_{ds}+\frac{d}{dt}\lambda_{ds}\end{array}\right. \tag{29}\]  
\[\left\{\begin{array}{l}0=R_{r}i_{dr}-(\omega_{e}-\omega_{r})\lambda_{qr}+ \frac{d}{dt}\lambda_{dr}\\ 0=R_{r}i_{qr}+(\omega_{e}-\omega_{r})\lambda_{dr}+\frac{d}{dt}\lambda_{qr} \end{array}\right. \tag{30}\]  
where, \(R_{r}\) and \(R_{s}\) are rotor and stator resistances (ohms), \(v_{dsg}\) is the stator voltage in \(d\)-\(q\) reference frame (V), \(\lambda_{dqr}\) and \(\lambda_{dsg}\) are rotor and stator magnetic flux linkages in \(d\)-\(q\) reference frame (Wb), \(i_{dqr}\) and \(i_{dsg}\) are rotor and stator currents in \(d\)-\(q\) reference frame (A), and \(\omega_{e}\) is the stator electrical speed (rad/s).  
The three-phase currents and the power flow from the electric grid are controlled by a vector-controlled grid side converter (GSC). The GSC acts as a rectifier or an inverter during acceleration and deceleration modes, respectively. Also, it regulates the DC-link voltage and the reactive currents exchanged with the electric grid. The speed and position of the flywheel are controlled by the machine side converter (MSC) that is connected to the GSC through a DC-link capacitor.  
The three-phase currents of the machine and grid, the DC-link voltage and the position and speed of flywheel shaft are measured quantities, which are inputs to the measurement and control block. The control block calculates the position of the shaft and compares the measured quantities with reference values to generate the gates pulses for GSC and MSC. During acceleration mode of the flywheel when power is consumed from the electric grid, the GSC acts as a rectifier and MSC acts as an inverter. When the flywheel reaches its reference speed value, no energy conversion occurs and it switches to standby mode, during which, the flywheel runs at an optimum speed except for the marginal speed decrease due to standby losses. In contrast, when a disturbance occurs in an electric grid, the flywheel starts to decelerate (discharge), the GSC acts as an inverter and MSC operates as a rectifier.