## 8 Discussion  
The DDR model described in Section 5 is a viable tool to describe the behavior of an insulating gas under general conditions. Indeed, once the transport parameters of gas ions (e.g., \(\mu^{+}\) and \(\mu^{-}\)) are obtained (e.g., in ad hoc experimental equipment [41]), the dynamic of the ions and conductivity can be determined. Directly specifying \(\sigma_{\text{G}}\) can be misleading especially in high-ionization conditions. The major differences between a standard EQS and the DDR approach are summarized in Table 3. This table can be used as a guideline to select the appropriate model for the specific case.  
It is worth mentioning that the major issue in using EQS with specified gas conductivity is the inability to reproduce the high-ionization regime since the selection of a unique value for \(\sigma_{\text{G}}\) is not trivial; thus, results are highly affected by this choice, as shown in Figure 7. Moreover, if ions are injected from the electrodes or defects [31], the conductivity has local variations which cannot be captured with simple conductivity models. For these reasons, the DDR model is preferable if a detailed description of the gas in terms of ion dynamics is required. Under specific conditions, a detailed description of the phenomena in the gas is mandatory to gain knowledge of the reliability and performance of HVDC-GIS. As mentioned in Section 2, the surface charge is one of the major fault causes in these systems. As can be seen in Figure 7, the \(\sigma_{\text{S}}\) distribution is strongly affected by ion dynamics in the high-ionization regime, motivating the use of detailed gas models in such conditions to prevent early fault of the component.  
The major obstacle in using the DDR model is the nonlinear time-dependent nature of the governing PDEs. The nonlinearity must be carefully addressed from a numerical viewpoint, e.g., by using the Newton-Raphson algorithm. This intrinsically increases the computational burden, solving the EQS with the DDR model for the gas results in two additional equations, i.e., Equations (12) and (13). A conductivity model \(\sigma_{\text{G}}(S)\) is preferable to simplify the numerical analysis as explained in Section 6; however, the simple relation expressed by Equation (17) does not apply to general geometries. Thus, the definition of a \(\sigma_{\text{G}}(S)\) from DDR results, extendable to general geometries, is a viable solution.  
Application of Equation (18) seems to be a better approach to catch RIC effects, accurately describing dielectric gas behavior, provided that the model conditions are such that high-ionic-concentration regions are not dominant inside the computational domain. Furthermore, its numerical performances are comparable to the solution of a traditional EQS simulation, allowing an easier investigation to be performed of large-scale and complex devices, as usually is the case in the field of nuclear fusion, whose solution can often require taking into account several hundreds of millions of degrees of freedom.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Approach** & **Advantages** & **Disadvantages** \\ \hline \multirow{5}{*}{EQS with specified \(\sigma_{\text{G}}\)} & \(\bullet\) Simple implementation & \(\bullet\) Does not account for ions’ distribution \\  & \(\bullet\) Effective tool for low-ionization regime & \(\bullet\) Ions’ injection cannot be considered \\  & & \(\bullet\) Not trivial selection of \(\sigma_{\text{G}}\) in high-ionization regime. Need for a functional dependence \\  & & \(\sigma_{\text{G}}=\sigma_{\text{G}}(S)\) \\ \hline \multirow{5}{*}{DDR} & \(\bullet\) Account for ions’ distribution & \(\bullet\) Intrinsically nonlinear model \\  & \(\bullet\) Effective tool for low- and high-ionization regimes & \(\bullet\) Requires additional Equations (12) and (13) to be solved \\ \cline{1-1}  & \(\bullet\) Ions’ injection can be considered & \\ \hline \hline \end{tabular}
\end{table}
Table 3: Main differences between standard EQS and DDR models for the gas.