## 3 Impact of the Electrode Shape on the Electric Field Computation and PD Inception Field Estimation  
### Influence of Electrode Curvature Radius  
Inaccurate evaluations of electrode shape (i.e., their curvature radius and contour shape) can have a significant impact on the computation of electric field in the space surrounding electrode contours, which could affect the estimation of the PD inception field, the typology of incepted PD, and, due to their influence on \(k_{s}\) and \(d\), the accuracy of model (1).  
In this section, parametric simulations quantify the effect of small geometrical variations of the HV electrode contour of Figure 2b on the estimation of electric field profiles (both orthogonal and tangential to the insulation surfaces) and on surface and gas PD inception field and voltage. Practically, such variations (shape and curvature radius of the  
Figure 3: Simulated electric field profiles, relevant to the electrode configuration of Figure 1, intersect the PD inception fields estimated according to (1) and Table 1. The \(x\)-axis zero is the triple point P (see Figure 1). (**a**) Simulated orthogonal field (in gas) intersecting the PD inception field curve at 0.38 mm from the triple point with an inception field equal to 9.1 kV/mm. (**b**) Simulated tangent field (on the specimen surface), with \(ks=0.0018\), intersecting the PD inception field of 1.60 kV/mm at about 0.3 mm from the triple point.  
HV electrode) could be due to manufacturing defects, usage deterioration, or electrode surface machining to remove tracks from previous discharges.  
A 0.6 mm single-layer XLPE specimen is used in the simulation (and validation experiments described in the next section). The reference curvature radius, \(R\), of the high voltage electrode (zoomed in Figure 2b), i.e., 750 \(\upmu\)m, is varied by steps of 25 \(\upmu\)m, from a minimum of 600 \(\upmu\)m to a maximum of 900 \(\upmu\)m. For each curvature radius, both the surface and gas inception fields (see (1)) and the relevant inception voltages (see (3)), are computed. The results are summarized in Table 2 and graphically represented in Figure 4.  
Regarding surface discharges, Table 2 shows a slight decrease in the value of the inception field with the increment of the curvature radius, \(R\). This is due to the increase in the scale parameter \(k_{s}\) (see (2)) which in turn depends on the tangent field profile shape (see Figure 4a). On the contrary, surface PDIV increases with the curvature radius (\(R\)), since the peak of the field profile decreases slightly with the increment of \(R\) (see Figure 4a). Hence, Table 2 shows that a variation of 300 \(\upmu\)m in the HV curvature radius causes a difference of 0.07 kV/mm for the inception field of the tangential component and a difference of 0.2 kV in the surface PDIV estimation.  
Referring to gas discharges, Figure 4b highlights that changes in the electrode curvature radius shift the orthogonal field curves, thus varying the intersection point between the computed and PD inception fields. However, Table 2 indicates that gas PDIV values are always very close to 3.3 kV, thus being reasonably stable with electrode curvature radius.  
This quantitative discussion holds as simulations and inception field models are considered (first and second legs, see Section 2), showing that only slight variations of PDIV values come from a variation of the curvature radius <300 \(\upmu\)m. Regarding PD detection, it can therefore be argued that such curvature radius variations do not noticeably affect measurement accuracy.