## 2 Background: Partial Discharge Inception Field Modeling and \"Three-Leg\" Approach
### Electrode Design and Field Calculation  
To validate the PD inception model and, thus, be able to carry out PD-free insulation design, the test object must be able to trigger exactly the type of PD (i.e., internal or surface), which is the potential issue in the designed insulation system. Focusing on electrodes and, e.g., surface discharges, the chosen electrode shape must be able to trigger PD on the surface, driven mostly by the tangential field rather than the orthogonal one, at the triple point (i.e., the point involving electrode contour, insulation surface, and surrounding gas). The focus here is on the test cell, represented in Figure 1, which is able to trigger surface discharges in order to characterize the PD surface resistance of insulating materials, which is of paramount importance to the design of insulation systems based on the redundancy concept. The purpose would be to allow an insulation system to withstand PD for all or part of its operation, still providing the specified life at a selected failure probability. The electrode system and test sample adopted in this study are sketched in Figure 2a. The contact between the electrode and insulation specimen is flat (i.e., perfectly adhering to the surface of the sample), and the curvature radius of the electrode is 750 \(\upmu\)m. A zoom of the electrode contour is highlighted in Figure 2b, being \(l\) the distance along the horizontal direction from the triple point and \(h\) the corresponding orthogonal distance from the sample to the electrode surface, so that in (1) \(d\equiv l\) or \(d\equiv h\) for surface and internal discharges, respectively. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Discharge Type** & \(\left(E\text{/}p\right)_{\sigma}\) & \(B\) & \(\beta\) & \(k_{s}\) \\ \hline Gas discharge & 25.2 & 8.6 & 2 & 1 \\ Surface discharge & 8.0 & 4.3 & 2 & \(\leq\)1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter values of the generalized partial discharge inception model (1), function of the type of discharge. Electric field calculations can be carried out, e.g., by the commercial FEM software COMSOL Multiphysics(r), for an AC sinusoidal (60 Hz) excitation voltage, \(V_{sim}\), of 1 kV. Let us note that field profiles do not change (for this test object) with increasing frequency (thus they would also hold for the PWM modulation waveform, referring to peak voltage). Typically, the orthogonal field is predominant over the tangential one at the same excitation voltage \(V_{sim}\). However, this does not mean that gas discharges are preferred over surface discharges, because the models for the inception of surface and gas PD have the same expression but different parameter values (see Table 1), leading to different inception field thresholds for gas and surface discharges. In order to estimate the PD inception field and voltage, field simulations (1 st leg) are correlated to the inception field models (2 nd leg). Indeed, the correspondence of the maximum field value (or close to it, see (2)) and relevant voltage with the inception field will provide the inception voltage, which can be calculated approximately by (3). To clarify this, Figure 3 shows an example of PDIV computation (exploiting the 1 st and 2 nd legs): PDIV is the voltage value at which the resulting field intercepts the PD inception field, obtained by (1). This holds for both surface (along the \(l\) direction) and gas discharge (along the \(h\) direction); see Figures 3a and 3b, respectively. Referring to Figure 3, intercepts occur at 9.1 kV/mm and 1.6 kV/mm for orthogonal and tangential fields, respectively, that correspond to PDIV = 3.3 kV for gas discharge and PDIV = 1.7 kV for surface discharge. Hence, surface PD is present at a lower voltage, which indicates that the considered electrode system is appropriate to test surface discharge phenomenology and to carry out PD-free design of the insulation system surface/interfaces. Figure 1: Photos of the test cell considered for the surface discharge testing experiments.