## 2 Description of Flywheel Energy Storage Systems
### Structure and Components of FESS: Priciples and Components of FESS
#### 2.2.1 The Rotor  
The energy stored in a flywheel is a function of its material and rotor shape. It is proportional to the square of the angular velocity and is linear to the moment of inertia, which is determined by the rotor shape and material as shown in Equation (1) [27; 34]:  
\[E=\frac{1}{2}I\omega^{2} \tag{1}\]  
where \(E\) is the stored energy, and \(I\) and \(\omega\) are the moment of inertia and angular velocity of the flywheel rotor, respectively. This allows optimization of the stored energy of a flywheel by either increasing \(I\) or \(\omega\). Typically, a flywheel driven by an electrical machine operates between a minimum speed ( \(\omega\) min) and maximum speed ( \(\omega_{\max}\)) to limit maximum torque on the MG for a given power rating, and  
Figure 1: Structure and components of flywheel. also avoid abundant voltage variations. The useful stored energy of the flywheel between (\(\omega_{\text{min}}\)) and (\(\omega_{\text{max}}\)) can be obtained by  
\[E=\frac{1}{2}I\Big{(}{\omega_{\text{max}}}^{2}-{\omega_{\text{min}}}^{2}\Big{)} = \frac{1}{2}I{\omega_{\text{max}}}^{2}\bigg{(}1-\frac{{\omega_{\text{min}}}^{2} }{{\omega_{\text{max}}}^{2}}\bigg{)} \tag{2}\]  
The moment of inertia is determined by the shape factor and mass of the flywheel rotor. The flywheel rotor can be solid or a hollow cylinder-shape, ranging from short and disc-type to long and drum type [28; 34]. The moment of inertia of a solid cylinder or disc type flywheel is given by  
\[I=\frac{1}{2}mr^{2} \tag{3}\]  
where \(m\) and \(r\) represent the mass and radius of the rotor, respectively. The moment of inertia for hollow cylinder flywheel with inner radius \(a\) and outer radius \(b\) is  
\[I=\frac{1}{2}m\Big{(}b^{2}+a^{2}\Big{)} \tag{4}\]  
The moment of inertia as a function of mass density \(\rho\) and length \(h\) is obtained by  
\[I=\frac{1}{2}\pi\rho h\Big{(}b^{4}-a^{4}\Big{)} \tag{5}\]  
Therefore, using Equations (1) and (5),  
\[E=\frac{1}{4}\pi\rho h\omega^{2}\Big{(}b^{4}-a^{4}\Big{)} \tag{6}\]  
The maximum speed of the flywheel rotor is limited by the tensile strength \(\sigma\) of the rotor material given by [24; 35]  
\[\sigma_{\text{max}}=\rho r^{2}\omega^{2} \tag{7}\]  
Equation (7) indicates the maximum tensile stress of a thin rotating ring with material density \(\rho\). The geometric effect of the rotor on the maximum stored energy is introduced by the shape factor \(K\). The \(K\) value ranges between 0 and 1, depending on the geometry of the flywheel rotor. Flywheel specific energy and energy density as a function of the shape factor \(K\) are presented in Equations (8) and (9):  
\[\frac{E}{m}=K\frac{\sigma_{\text{max}}}{\rho}\ [J/\text{kg}] \tag{8}\]  
\[\frac{\text{E}}{\text{V}}=K\sigma_{\text{max}}\ \Big{[}J/\text{m}^{3}\Big{]} \tag{9}\]  
Intermediate Speed Flywheels  
A new class of intermediate speed flywheel is under development based on the use of steel laminates with a construction that eliminates the stress raising feature of a central hole normally required for composite flywheel. The peripheral speed of the flywheel can be much higher than what would be considered safe for a solid monolithic structure [37]. The catastrophic fracture in the rotor by fatigue crack is restricted to a single laminate which will only release a small fraction of the energy contained. It is relatively simple to compare the steel laminated flywheel rotor with a composite for an assumed value of peripheral speed based on realistic maximum operating stresses in the two materials. The moment of inertia is a geometric value, and the energy stored is then proportional to material density and the square of the speed (Equation (6)). From this analysis, the rotor mass and volume ratio of steel relative to composite can be found as shown in Figure 1:0  
Interestingly the mass of the steel rotor is over 4 times higher, but its volume is halved. It should be noted that the volume of the carbon fibre rotor is considered based on its external dimensions which govern the space taken up by the rotor.