# A Review of Flywheel Energy Storage System Technologies and Their Applications
## 2 Description of Flywheel Energy Storage System
### Structure and Components of FESS
#### 2.2.1 Flywheel Rotor  
The stored energy in a flywheel is determined by the rotor shape and material. It is linearly proportional to the moment of inertia and the square of its angular velocity, as shown in Equation (1) [27; 34]:  
\[E=\frac{1}{2}I\omega^{2} \tag{1}\]  
where \(E\) is the stored kinetic energy, \(I\) is the moment of inertia, and \(\omega\) is the angular velocity. The useful energy of a flywheel within a speed range of minimum speed (\(\omega_{\text{min}}\)) and maximum speed (\(\omega_{\text{max}}\)) can be obtained by:  
\[E=\frac{1}{2}I({\omega_{\text{max}}}^{2}-{\omega_{\text{min}}}^{2})=\ \frac{1}{2}I{ \omega_{\text{max}}}^{2}\bigg{(}1-\frac{{\omega_{\text{min}}}^{2}}{{\omega_{ \text{max}}}^{2}}\bigg{)} \tag{2}\]  
Typically, an electrically driven flywheel normally operates between (\(\omega_{\text{min}}\)) and (\(\omega_{\text{max}}\)), to avoid too great a voltage variation and to limit the maximum MG torque for a given power rating. The moment of inertia is a function of the mass of the rotor and the rotor shape factor. Flywheels are often built as solid or hollow cylinders, ranging from short and disc-type, to long and drum-type [28; 35]. For a solid cylinder or disc-type flywheel, the moment of inertia is given by:  
\[I=\frac{1}{2}mr^{2} \tag{3}\]  
where \(m\) is the rotor mass and \(r\) is the outer radius. For a hollow cylinder flywheel of outer radius \(b\) and inner radius \(a\), as shown in Figure 2, the moment of inertia is:  
\[I=\frac{1}{2}m\Big{(}b^{2}-a^{2}\Big{)} \tag{4}\]  
For a flywheel with length \(h\) and mass density \(\rho\), the moment of inertia is determined by:  
\[I=\frac{1}{2}\pi\rho h\Big{(}b^{4}-a^{4}\Big{)} \tag{5}\]  
Thus:  
\[E=\frac{1}{4}\pi\rho h\omega^{2}\Big{(}b^{4}-a^{4}\Big{)} \tag{6}\]  
Figure 1: Structure and components of a flywheel. The maximum speed limit at which a flywheel may operate is determined by the strength of the rotor material, called tensile strength \(\sigma\)[18; 23]. A suitable safety margin must be maintained, to keep the stress experienced by the rotor below the strength of the rotor material. The maximum stress of a thin rotating ring is given by:  
\[\sigma_{\max}=\rho r^{2}\omega^{2} \tag{7}\]  
where \(\sigma\) is the maximum stress and \(\rho\) is the density of the flywheel material. More complex equations are available for different rotor geometries, but the maximum stress is always proportional to \(\rho\), and the square of peripheral speed, equal to \(r\omega\). The effect of rotor geometries can be accommodated by introducing a shape factor \(K\). The maximum specific energy and energy density are then given by:  
\[\frac{E}{m}=K\frac{\sigma_{\max}}{\rho}[\mathrm{J/kg}] \tag{8}\]  
\[\frac{E}{V}=K\sigma_{\max}\ \left[\mathrm{J/m^{3}}\right] \tag{9}\]  
Equations (8) and (9) indicate that the specific energy (energy per mass unit) and energy density (energy per volume unit) of the flywheel are dependent on its shape, expressed as shape factor \(K\). The shape of a flywheel is an important factor for determining the flywheel speed limit, and hence, the maximum energy that can be stored. The shape factor \(K\) is a measurement of flywheel material utilisation [18]. Figure 3 shows the values of \(K\) for the most common types of flywheel geometries. According to Equation (1), the stored energy of a flywheel can be optimised by either increasing the spinning speed (\(\omega\)) or increasing the moment of inertia (\(I\)). This allows two choices for FESS: low  
\[E=\frac{1}{2}I({\omega_{\text{max}}}^{2}-{\omega_{\text{min}}}^{2})=\ \frac{1}{2}I{ \omega_{\text{max}}}^{2}\bigg{(}1-\frac{{\omega_{\text{min}}}^{2}}{{\omega_{ \text{max}}}^{2}}\bigg{)} \tag{2}\]0 Different flywheel cross sections [18]. \[E=\frac{1}{2}I({\omega_{\text{max}}}^{2}-{\omega_{\text{min}}}^{2})=\ \frac{1}{2}I{ \omega_{\text{max}}}^{2}\bigg{(}1-\frac{{\omega_{\text{min}}}^{2}}{{\omega_{ \text{max}}}^{2}}\bigg{)} \tag{2}\]1 Hollow cylinder flywheel. speed FESS (typically up to 10,000 rpm) and high speed FESS (up to 100,000 rpm) [18; 23].