## 3 Modeling Parameters
### Flywheel rotor Simulation Parameters  
Two cases were considered to investigate the effects of viscoelastic behavior on the flywheel rotor. The first case simulates a worst-case scenario for creep and viscoelastic stress relaxation in the flywheel rotor. The second case more closely simulates a realistic scenario of an FESS experiencing daily charge/discharge cycles. The FESS capacity and flywheel rotor dimensions are identical between the two cases. Recent studies on appropriate sizing of FESS have identified various values ranging between 3 kWh and 20 kWh for residential applications, light rail transit, electric vehicle charging, and frequency regulation for microgrid applications [16; 31; 32]. For the present study, a capacity of 10 kWh was chosen as it is situated in the middle of the range for the applications mentioned above. Note that energy storage capacity scales linearly with rotor height (axial dimension), and scaling is not expected to affect creep behavior appreciably, so the chosen rotor configuration can easily be scaled up or down to adjust for a given application. This scaling could be done, as suggested in [33], by stacking individual composite disks on top of one another to form the rotor, in which case, the analysis for each individual disk is performed as described herein while capacity may be increased or decreased as needed. To illustrate the chosen capacity, a recent study [34] on residential photovoltaic (PV) potential in Lethbridge, Alberta, Canada, identified that the majority of residential homes had roof space for up to 10 kW of solar PV, meaning the FESS in this study could reasonably be expected to reach full capacity throughout the day even under less than ideal irradiation conditions, in order to provide power during high demand times such as in the evening. The simulated flywheel rotor dimensions and energy capacity used in this study are given in Figure 1 and Table 3. Note that changing power demand would necessarily require accelerating or decelerating the flywheel rotor, imposing shear stresses, which is not included in the current model, hence justifying the aforementioned biaxial stress condition. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Material** & \(\mathbf{S_{11}}\) [Pa\({}^{-1}\)] & \(\mathbf{S_{22}}\) [Pa\({}^{-1}\)] & \(\mathbf{S_{33}}\) [Pa\({}^{-1}\)] & \(\mathbf{S_{66}}\) [Pa\({}^{-1}\)] & \(\mathbf{\
u}\) \\ \hline Aluminum & \(1.39\times 10^{-11}\) & \(1.39\times 10^{-11}\) & \(1.39\times 10^{-11}\) & \(3.72\times 10^{-11}\) & 0.33 \\ CFRP & \(S_{11}^{0}(t)^{0.01}\) & \(S_{22}^{0}(t)^{0.03}\) & \(S_{33}^{0}(t)^{0.03}\) & \(S_{66}^{0}(t)^{0.03}\) & 0.31 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties for aluminum 7075-T6 [29] and viscoelastic equations for CFRP [30] used in the present study. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Material** & \(\mathbf{\sigma_{1t}}\) [MPa] & \(\mathbf{\sigma_{1c}}\) [MPa] & \(\mathbf{\sigma_{3t}}\) [MPa] & \(\mathbf{\sigma_{3c}}\) [MPa] & \(\mathbf{\tau}\) [MPa] \\ \hline CFRP & 2720 & 1689 & 64.1 & 307 & 137 \\ Aluminum & 572 & 572 & 572 & 572 & 331 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Directionally dependent strengths of CFRP and yield strength of aluminum used to find \(SR\) from the Tsai-Wu failure criterion. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Aluminum Hub** & **CFRP Rim** & **Complete Rotor** \\ \hline Inner radius & 160 mm & 200 mm & - \\ Outer radius & 200 mm & 330 mm & - \\ Press-fit interference & - & - & 0.8 mm \\ Rotor height & - & - & 430 mm \\ Energy capacity & - & - & 11.19 kWh \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Flywheel rotor rim dimensions, press-fit interference, and energy capacity. Case 1: The worst-case scenario for viscoelastic effects is simulated by assuming the flywheel rotor to operate at its limit load at all times. The model was used to simulate 10 years of operation.