Stochastic simulation of partial discharge inception

TL;DR

This work tackles the stochastic prediction of partial discharge inception in gases by introducing a Monte Carlo framework that operates on unstructured electrostatic fields. It combines a Kendall birth-death avalanche model with spatial growth, multi-avalanche generation, photoionization, and secondary emission to estimate inception probability $p_\mathrm{inc}$ and inception time $t_\mathrm{inc}$, including PDIV when $\bar{p}_\mathrm{inc}$ crosses a threshold. The method precomputes avalanche statistics on the grid, enables fast field-line sampling, and validates against particle simulations across 2D and 3D geometries, demonstrating accurate avalanche-size distributions and inception patterns. The results enable efficient PD inception assessment, including the influence of surfaces and nearby objects, with practical implications for insulation design and PDIV estimation. The authors also outline extensions to improve field-inhomogeneity handling, diffusion, and experimental validation.

Abstract

We present a Monte Carlo method for simulating the inception of electric discharges in gases. The input consists of an unstructured grid containing the electrostatic field. The output of the model is the estimated probability of discharge inception per initial electron position, as well as the estimated time lag between the appearance of the initial electron and discharge inception. To obtain these quantities electron avalanches are simulated for initial electron positions throughout the whole domain, also including regions below the critical electric field. Avalanches are assumed to propagate along field lines, and they can produce additional avalanches due to photon and ion feedback. If the number of avalanches keeps increasing over time we assume that an electric discharge will eventually form. A statistical distribution for the electron avalanche size is used, which is also valid for gases with strong electron attachment. We compare this distribution against the results of particle simulations. Furthermore, we demonstrate examples of inception simulations in 2D Cartesian, 2D axisymmetric and 3D electrode geometries.

Figures (13)

• Figure 1: Top: Ionization coefficient in N$_2$, using Biagi cross sections Biagi_lxcat. The curve $\alpha_T/N$ was computed according to equation \ref{['eq:alpha-T']} with a Monte Carlo (MC) swarm code, while $\alpha_\mathrm{spatial}/N$ was computed using BOLSIG+ with a spatial growth model. For comparison, $\alpha_\mathrm{eff}$ corresponding to temporal growth is also shown. Bottom: the coefficients $\kappa = 4 \bar{D}_{L} \nu_\mathrm{eff}/W^2$, $c_\mathrm{drift}$ and $c_\mathrm{diff}$ from equations \ref{['eq:ne-x']}-- \ref{['eq:c-diff']}.
• Figure 2: Avalanche size distributions in uniform electric fields in N$_2$, for various gap sizes $d$, for an initial electron energy of $1 \, \textrm{eV}$. The bars indicate results from 1000 particle simulations, and the curve shows the probability mass function according to equation \ref{['eq:P-M']}. The probability $P_1'$ (producing no additional ionization, see equation \ref{['eq:P-M1']}) is given, and also estimated from the simulations. This is also done for the mean number of ionizations ($\bar{M}$), see equation \ref{['eq:M']}. Estimated standard deviations are indicated between parentheses.
• Figure 3: Effect of initial electron energy on the avalanche size, determined using 4000 particle simulations per initial energy. The conditions correspond to the bottom case of figure \ref{['fig:uniform-N2']} (N$_2$, $E = 80 \, \textrm{kV/cm}$, $d = 0.2 \, \textrm{mm}$). The error bars indicate $\pm$ one standard deviation.
• Figure 4: Avalanche size distribution in uniform electric field in N$_2$ containing 4% SF$_6$. The discrepancy for small avalanche sizes is expected, see the end of section \ref{['sec:numb-ioniz-avalanche']}.
• Figure 5: Avalanche size distribution for a conducting sphere of radius $R = 0.5 \, \textrm{mm}$ at a voltage $V_0$ in air (80% N$_2$, 20% O$_2$). Initial electrons are placed a distance $d$ from the sphere.