# Design and Multi-Objective Optimization of Fiber-Reinforced Polymer Composite Flywheel Rotors
## 5 Case Studies  
Two flywheel geometries, one from industry and one from academia, were analyzed using the developed optimization strategy. The considered geometries for the case studies are depicted in Table 6. The so-called ACME flywheel was designed for an LRT application [23]. This on-board flywheel design is intended to store a train's braking energy to increase the overall energy efficiency. In case of roof mounting on a train car, the construction volume is limited by the clearance to the catenary. If the flywheel is installed underneath the cabin floor, the height is limited by the required size for the passenger cabin. Therefore, the flywheel must be designed in a flat disk shape. The initial design, created in [23], was made of one aramid fiber and one carbon fiber polymer composite rim. In the second design case, the Rosesta T2 flywheel was considered, which is used for the LRT system in Zwickau, Germany, and is characterized by a disk shape design with a large diameter of 60 cm [24]. This type of flywheel is not mounted on the train but located in train stations (i.e., the flywheel is stationary). To obtain realistic results, a safety factor was introduced. A safety factor is considered useful to take into account effects that are not captured by the optimization algorithm, such as viscoelastic behavior, uncertainties with material properties, deviations from the plane stress assumption, and influences of complex hub geometry features. Since integrating the safety factor into the optimization process significantly changes the results, the safety factor was included by simply reducing the angular velocity. According to literature, it is a good approximation to assume that stresses increase quadratically with angular velocity [25]. Therefore, reducing the speed to 70% of the maximum allowable speed will lead to a safety factor of approximately 2. The fabrication equipment configurations presented in Section 3.2 were used. The employed material parameters are listed in Table 7. The hub is considered to be a ring-type hub and is therefore simply modeled as an additional rim. Since the energy contribution and stresses are comparatively low for lower radii, this simplification is considered sufficient [13]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Name** & **Energy [kWh]** & \begin{tabular}{c} **Rotational** \\ **Speed [rpm]** \\ \end{tabular} & \begin{tabular}{c} **Rotor** \\ **Height [mm]** \\ \end{tabular} & \
\begin{tabular}{c} **Inner Rotor** \\ **Radius [mm]** \\ \end{tabular} &
\begin{tabular}{c} **Outer Rotor** \\ **Radius [mm]** \\ \end{tabular} \\ \hline ACME flywheel [20] & 0.25 & 30,000 & 50 & 100 & 200 \\ Rosesta T2 [24] & 4.0 & 25,000 & 200 & 100 & 300 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Rotor specifications used for the case studies. The parameters for the GA and SQP are shown in Table 8 and were not changed for the case studies. The maximum number of generations is adjusted if the rim number becomes larger. Further, if the rim number is above one rim, the population size is also increased. While a high number of generations as well as a high population size may improve the solutions, they too increase the computation time. Using the proposed parameters, the outcome is a good compromise between performance and computational effort. The developed optimization algorithm was executed for both geometries. The two-dimensional and three-dimensional Pareto-fronts of the ACME-flywheel geometry are shown in Figures 3 and 4, respectively. The three-dimensional view in Figure 5 highlights the fact that productivity decreases for each additional rim. It has to be noted that the solution representing a rotor entirely made of aluminum alloy is not presented in the three-dimensional figure. This is due to the fact that the modeled productivity function can only return values for composite materials. Further, the manufacturing costs for the pure metal solution are equal to zero since a metal rim is neither filament-wound nor cured.