# Design and Multi-Objective Optimization of Fiber-Reinforced Polymer Composite Flywheel Rotors
## 5 Case Studies
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Material Property** & **Aluminum Alloy** & **Glass/Epoxy** & **Aramid/Epoxy** & **Carbon/Epoxy** & **Unit** \\ \hline \(E_{1}\) & 71.7 & 41.0 & 66.7 & 170 & GPa \\ \(E_{2}\) & 71.7 & 10.4 & 5.5 & 16.36 & GPa \\ \(G_{12}\) & 26.9 & 4.3 & 2.2 & 7.8 & GPa \\ \(\
u_{12}\) & 0.33 & 0.28 & 0.34 & 0.32 & - \\ \(\
u_{23}\) & 0.33 & 0.49 & 0.5 & 0.41 & - \\ \(\rho\) & 2810 & 1970 & 1343 & 1568 & kg/m\({}^{3}\) \\ \(X^{\text{T}}\) & 503 & 1140 & 1400 & 2720 & MPa \\ \(X^{\text{C}}\) & 503 & 620 & 335 & 1689 & MPa \\ \(Y^{\text{T}}\) & 503 & 39 & 30 & 64 & MPa \\ \(Y^{\text{C}}\) & 503 & 39 & 30 & 64 & MPa \\ \(S\) & 331 & 128 & 158 & 307 & MPa \\
**Specific cost** & 5 & 10 & 22 & 90 & CAD/kg \\
**Longitudinal CTE** & 23.6 & 8.6 & \(-\)3.5 & \(-\)0.16 & \(10^{-6}\)/K \\
**Transverse CTE** & 23.6 & 22.1 & 35 & 18.9 & \(10^{-6}\)/K \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Material properties used for the case studies. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline GA: population size (\(N_{\text{rim}}=1\)) & 50 & - \\ GA: population size (\(N_{\text{rim}}\geq 2\)) & 200 & - \\ GA: minimum number of generations & 100 & - \\ GA: maximum number of generations & 400 + 200 \(\times\)\(N_{\text{rim}}\) & - \\ GA: relative tolerance & \(10^{-5}\) & - \\ EA: crossover rate & 20\% & - \\ SQP: objective tolerance & \(10^{-4}\) & - \\ SQP: constraint tolerance & \(10^{-5}\) & - \\ Temperature difference & \(-150\) & K \\ Maximum number of rims & 5 & - \\ Thinnest rim width & 1 & mm \\ Minimum angular velocity & 0 & rpm \\ Maximum angular velocity & 80,000 & rpm \\ Minimum interference between rims & 0 & mm \\ Maximum interference between rims & 0.8 & mm \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Algorithm parameters for the case studies. In the two-dimensional Pareto-front in Figure 4, SED\({}_{100\%}\) denotes the solution with the highest SED found. Further, SED\({}_{85\%}\) represents the solution which achieves 85% of the highest SED possible. Due to the discrete material properties and different numbers of rims, the Pareto-front shows several cost-jumps between the solutions. Comparing the marked solutions, SED\({}_{100\%}\) achieves a 17% higher stored energy than SED\({}_{85\%}\). However, costs also increase by 36%. The Pareto-front creates the possibility to analyze trade-offs, such as the SED-cost trade-off, that are required. Knowing the Pareto-front, the design decision may differ, as the associated trade-offs may be considered too large. This highlights the advantage of a multi-objective optimization approach, which provides more information for decision-makers. For each geometry the solution with the highest energy as well as the composite rotor that is most cost-efficient, and the cheapest solution are herein discussed (see Table 9). In each case, the most cost-efficient design was a rotor entirely made of aluminum. (However, recall that the model does not adequately capture manufacturing costs for a pure metal solution). It has to be noted that the high-energy solution does not necessarily provide the highest energy that is attainable for a certain geometry. Pareto-fronts are useful to highlight the trade-offs that must be made to achieve higher  
Figure 4: Two-dimensional Pareto-front of the ACME geometry, AL: aluminum, A: aramid, G: glass, C: carbon. Figure 5: Three-dimensional Pareto-front of the ACME geometry. energy. To answer questions about which solution has the highest energy or which solution is the most cost-efficient, single-objective optimizations may be more appropriate. As shown in Table 9, flywheel energy capacities differ between 90 Wh to 2.96 kWh, while costs vary between 66 CAD to 5976 CAD. Manufacturing costs tend to be negligible for the higher-capacity (Rosetta T2) composite flywheel rotor using more expensive materials such as carbon fibers. For the lowest-capacity composite flywheel (ACME), the manufacturing costs constitute the main cost-driver with a cost-share of 57%.