## 3 Results
### Results of the Electromagnetic Model  
The MC method was not applicable to the uncertainty propagation analysis in the EM model due to the increase in the computational time (\(\approx\)1 day/simulation) of the detailed model. For this reason, the PCE of quadrature order 3 has been used to build a surrogate model of the detailed one based on its results, since it proved to be sufficiently accurate to replace the detailed EL model too. The 3D-FOX calculates the evolution of the power \(P(t)\) deposited in the TF coil casing. The monitoring variables have been extracted from this evolution, considering the following:  
* The peak of the deposited power \(P_{peak}\); \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variable** & \(\overline{V}\pm 2\cdot\sigma\) & \(\overline{V}\pm 2\cdot\sigma\) & \(\overline{V}\pm 2\cdot\sigma\) \\  & **MC** & **PCE** & **UT** \\ \hline \(\tau\left[s\right]\) & \(14.77\pm 4.70\) & \(14.76\pm 4.69\) & \(14.77\pm 4.69\) \\ \(\Delta V_{FDU}\left[V\right]\) & \(5065.85\pm 1708.03\) & \(5067.26\pm 1708.40\) & \(5067.27\pm 1698.46\) \\ \(\Delta V_{TF}\left[V\right]\) & \(843.98\pm 284.56\) & \(844.21\pm 284.62\) & \(844.22\pm 282.97\) \\ \(l^{2}\cdot t\left[GA^{2}\cdot s\right]\) & \(4.67\pm 1.58\) & \(4.67\pm 1.57\) & \(4.67\pm 1.57\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison between the mean value and standard deviation of the monitored variables using the MC, PCE, and UT methods. Figure 5: Average evolution of (**a**) current and (**b**) \(l^{2}\cdot t\) with respective 1–99% confidence range evaluated with PCE. * The overall deposited energy \(E=\int_{0}^{t_{end}}P(t)dt\). The statistical distributions of the peak of the deposited power and of the energy deposited are reported in Figure 6; they highlight how the uncertainty in the inputs translates into huge uncertainty in terms of the peak power and deposited energy. Neglecting this uncertainty propagation may lead to disregarding the worst0 case scenarios connected to this transient, concerning, e.g., its TH effects. The average value and the standard deviation of the peak of the deposited power and of the deposited energy computed with the PCE have been benchmarked against those evaluated with the UT. The comparison of the results is shown in Table 2.  
The benchmark shows an excellent agreement between the PCE and the UT. This does not guarantee the accuracy of the statistical distribution obtained with PCE, but it suggests that the metamodel developed properly reproduces the main statistical data, namely,the mean and the variance. As already undertaken for the evolution of the current and of \(I^{2}\cdot t\) in the EL model analysis, here, the average power evolution and its 1-99% confidence range have been calculated using the PCE and are shown in Figure 7.  
The width of the 1-99% confidence range highlights the importance of considering the uncertainty on the input data to evaluate the worst-case scenario to be retained for the analysis of this transient. Moreover, the impact of the uncertainty is much larger in the first part of the transient (peak region). This is expected since the first part of the transient is driven by the large value of the time derivative of the current at the beginning of the FD, which is strongly affected by the \(K\) and \(\beta\) parameters of the varistors. On the contrary, the last part of the transient is less affected by the uncertainty of the inputs as the time derivative of the current is reduced, consequently reducing the power deposited. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Variable** & \(\overline{\mathbf{V}}\pm\mathbf{2\cdot\sigma}\) & \(\overline{\mathbf{V}}\pm\mathbf{2\cdot\sigma}\) \\  & **PCE** & **UT** \\ \hline \(P_{max}\left[MW\right]\) & \(1.01\pm 0.62\) & \(1.00\pm 0.61\) \\ \(E\left[M\right]\) & \(4.42\pm 1.40\) & \(4.42\pm 1.40\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison between the mean value and standard deviation of the peak power and the deposited energy evaluated with PCE and UT.