## 2 Design Parameters of Flywheel Energy Storage System
### Energy and Power Parameters  
In automotive applications, both volume and weight limitations are critical constraints. Therefore, the rotor diameter of FESS units designed for automobiles typically does not exceed 30 cm [4]. In tramway and bus applications, the rotor diameter can extend up to 40-50 cm. The restriction on rotor diameter necessitates an increase in rotational speed. For automotive-specific applications, rotor speeds generally range between 50,000 and 80,000 RPM [2]. The first example of the use of FESS for regenerative braking in vehicles dates back to the time when the FIA began to allow the use of kinetic energy recovery systems in Formula 1 cars. Later, two F1 teams were licensed to use this system [21; 22]. However, this system was not used effectively in the following years due to changing  
Figure 1: Flywheel energy storage system structure [20]. regulations. In the following years, Jaguar developed a KERS (kinetic energy recovery system)-based system for the XJ model vehicle. This vehicle also holds the title of being the first passenger sedan equipped with TOROTRAK. It is known that larger-scale versions of the system have been tested in public transport. It is claimed that the system provides an approximately 40 percent fuel efficiency improvement [23]. In the bus application in Figure 2, a FESS is integrated into the vehicle powertrain for regenerative braking. Considering the combined constraints of rotor diameter, speed, and weight, a carbon composite can be selected as the rotor material [11]. The rotor is fabricated through the filament winding of carbon fibers in successive layers, each of which is bonded with epoxy resin. Thus, the overall mass of the rotor remains below 30 kg [4]. In grid-scale energy storage systems, the rotor mass can reach up to 300 kg. For these applications, the rotational speed is limited to approximately 30,000 RPM [11]. In stationary applications, where mass is not a critical constraint and rotational speeds are comparatively lower, steel is often preferred as the rotor material. The magnitude of the stored energy is directly proportional to the weight of the rotor mass. FESS power plants are connected to the grid in parallel to avoid blackouts. There have been numerous FESS facilities operating in various regions of America since 2011. The installed power of these facilities is over 60 MW. Similarly, Canada established a storage facility in 2014 [24]. Another approach to enhance the stored energy is to increase the rotational speed rather than the rotor mass. Because the stored energy increases proportionally to the square of the rotation speed, the production of this type of high-speed FESS requires advanced technology. Considering its usage areas, it is generally seen in space and aviation, electric vehicle (EV) and defense industry applications. Equation (1) represents the stored energy in the flywheel energy storage unit. The formula represents the amount of stored kinetic energy 'E', the moment of inertia 'J', angular velocity 'w', mass'm', and rotor radius 'r'. \[E=\frac{1}{2}J\omega^{2} \tag{1}\]  
The moment of inertia depends on the rotor mass and geometry, as in Equation (2):  
\[J=\int r^{2}dm \tag{2}\]  
Figure 2: Bus application of FESS [23]. The energy stored in the flywheel must be calculated by taking into account its mass and material properties. Based on the strength characteristics of the material used, it is possible to determine the maximum velocity of the rotor. Equation (3) provides the value for the rotor's maximum angular velocity. In this equation, the following terms are defined:  
* \(\omega\) is the angular velocity;
* ro is the outer radius;
* ri is the inner radius (0 for a solid disc);
* \(\
u\) is Poisson's ratio;
* \(\rho\) is the material density;
* \(\sigma\)t, max is the yield stress. \[\omega=\sqrt{\frac{8\cdot\sigma_{t}}{(3+v)\ \cdot\ \rho\ \cdot\ (r_{O}^{2}-r_{I}^{2})}} \tag{3}\]  
Since FESS is used in a wide variety of areas, a precise framework cannot be drawn to determine its dimensions.