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arxiv: 2511.04356 · v2 · pith:PFTJSDIRnew · submitted 2025-11-06 · ⚛️ physics.plasm-ph

Stochastic simulation of partial discharge inception

Jannis Teunissen , Yuting Gao This is my paper

Pith reviewed 2026-05-25 07:22 UTC · model grok-4.3

classification ⚛️ physics.plasm-ph
keywords Monte Carlo simulationpartial discharge inceptionelectron avalancheinception probabilitytime laggas dischargephoton feedbackion feedback
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The pith

A Monte Carlo method estimates the probability of partial discharge inception per initial electron position and the associated time lag by simulating avalanches along field lines with feedback.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a Monte Carlo approach to model how electric discharges begin in gases by tracking the growth of electron avalanches from various starting points. It handles regions where the electric field is below the threshold for immediate breakdown by using a statistical description of avalanche sizes that works even with strong electron attachment. This allows computation of inception probabilities and time lags on unstructured grids of the electrostatic field, including photon and ion feedback effects. Such estimates matter because partial discharges can degrade insulation in high-voltage equipment over time.

Core claim

The method estimates the probability of discharge inception per initial electron position and the time lag by simulating avalanches that propagate along field lines, produce additional avalanches via photon and ion feedback, and use a statistical avalanche size distribution valid for gases with strong electron attachment.

What carries the argument

Monte Carlo simulation of electron avalanches propagating along field lines with photon and ion feedback, employing a statistical avalanche size distribution.

If this is right

  • Inception probability is estimated for initial electron positions throughout the domain, including below-critical-field regions.
  • Time lag between initial electron appearance and inception is estimated from avalanche statistics.
  • The approach is demonstrated in 2D Cartesian, 2D axisymmetric, and 3D electrode geometries.
  • The statistical avalanche size distribution is compared to and matches results from particle simulations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The position-dependent probability map could identify high-risk electron emission sites for electrode design adjustments.
  • Refining the input field grid would directly increase the spatial detail of the computed inception probabilities.
  • The method's use of a statistical size distribution may allow faster exploration of gas composition effects than full particle tracking.
  • Validation in time-varying fields would extend the current steady-field results to AC voltage cases.

Load-bearing premise

Avalanches are assumed to propagate strictly along field lines and an increasing number of avalanches over time is taken to mean that a discharge will form.

What would settle it

Systematic deviation between the model's predicted inception probabilities or time lags and direct experimental measurements in a controlled electrode geometry with known field distribution.

Figures

Figures reproduced from arXiv: 2511.04356 by Jannis Teunissen, Yuting Gao.

Figure 1
Figure 1. Figure 1: Top: Ionization coefficient in N2 , using Biagi cross sections [5]. The curve 𝛼𝑇 ∕𝑁 was computed according to equation (18) with a Monte Carlo (MC) swarm code, while 𝛼spatial∕𝑁 was computed using BOLSIG+ with a spatial growth model. For comparison, 𝛼ef f corresponding to temporal growth is also shown. Bottom: the coefficients 𝜅 = 4𝐷̄ 𝐿 𝜈ef f∕𝑊 2 , 𝑐drift and 𝑐dif f from equations (17)– (20). One is subtrac… view at source ↗
Figure 2
Figure 2. Figure 2: Avalanche size distributions in uniform electric fields in N2 , for various gap sizes 𝑑, for an initial electron energy of 1 eV. The bars indicate results from 1000 particle simulations, and the curve shows the probability mass function according to equation (13). The probability 𝑃 ′ 1 (producing no additional ionization, see equation (11)) is given, and also estimated from the simulations. This is also do… view at source ↗
Figure 3
Figure 3. 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 2 (N2 , 𝐸 = 80 kV/cm, 𝑑 = 0.2 mm). The error bars indicate ± one standard deviation. 0 100 200 300 400 500 600 700 M (number of ionizations) 0.00 0.50 1.00 1.50 2.00 P(X = M) 1e 2 Msim = 6.39e+01 (4.3e+00) M = 6.62e+01 P 0 1, sim = 4.… view at source ↗
Figure 5
Figure 5. Figure 5: Avalanche size distribution for a conducting sphere of radius 𝑅 = 0.5 mm at a voltage 𝑉0 in air (80% N2 , 20% O2 ). Initial electrons are placed a distance 𝑑 from the sphere. criterion. By default, we use 𝑛inc = 1000 as the thresh￾old for the number of future avalanches. The smaller 𝑛inc is, the more likely it becomes that inception is ‘detected’ due to stochastic fluctuations, even though eventually all a… view at source ↗
Figure 4
Figure 4. Figure 4: Avalanche size distribution in uniform electric field in N2 containing 4% SF6 . The discrepancy for small avalanche sizes is expected, see the end of section 2.2. gas between the electrodes is artificial air (80% N2 , 20% O2 ) at 1 bar and 300 K. Photoionization is included as a secondary electron mechanism, using the default parameters given in section 2.6. Photoemission or ion secondary emis￾sion is not … view at source ↗
Figure 6
Figure 6. Figure 6: Inception electric field 𝐸inc = 𝑉inc∕𝑑 as a function of the inception threshold 𝑛inc used in equation (25). Three cases are shown: one in which the number of photoionization events produced by a single avalanche is limited to 𝑛inc∕2 (see section 2.6), one in which this number is not limited, and one without photoionization but with an ion secondary emission coefficient 𝛾𝑖 = 10−4. The simulations were perfo… view at source ↗
Figure 8
Figure 8. Figure 8: shows the inception probability 𝑝inc(𝐫) in the domain for 𝐸0 = 41.5 kV∕cm together with the estimated inception time 𝑡 inc. Since electrons drift up, the highest inception probability is found close to the bottom plate. The estimated inception times 𝑡 inc are a few tens of nanoseconds for electrons starting close to the bottom plate. For positions where inception is unlikely, 𝑡 inc is typically larger and … view at source ↗
Figure 10
Figure 10. Figure 10: Cartesian 2D domain (2 mm × 2 mm) with a conducting wire at the center. The electric potential is shown for an applied voltage 𝑉0 = 1 kV at the wire. The smaller circle on the left and the rectangle on the right are dielectric materials with a relative permittivity of one. The domain boundaries are grounded. 4.4. Inception with nearby objects To illustrate the effect of nearby objects on discharge incepti… view at source ↗
Figure 9
Figure 9. Figure 9: Simulation of inception in air around a spherical electrode at 𝑉0 = 4.8 kV, in an axisymmetric domain measuring 6 mm in the 𝑟-direction and 12 mm in the 𝑧-direction. Shown are the inception probability 𝑝inc, the ionization integral 𝐾∗ , the probability 𝑃 ′ 1 and the effective ionization coefficient 𝛼ef f . For 𝛼ef f only the negative range range is shown; the contour line corresponds to 𝛼ef f = 0. The numb… view at source ↗
Figure 12
Figure 12. Figure 12: Top: 3D domain measuring 20 mm× 15 mm× 15 mm with a conducting wire. The following boundary conditions are applied for the electric potential 𝜙: Neumann zero on the left and right sides, 𝜙 = 𝑉0 on the wire and 𝜙 = 0 on all other boundaries. Bottom: a central slice showing 𝜙 for 𝑉0 = 1 kV. photoionization as a secondary electron mechanism. The inception voltage corresponding to ̄𝑝inc = 10−6 is then about 𝑉… view at source ↗
Figure 11
Figure 11. Figure 11: Inception probabilities in air at 𝑉0 = 3.25 kV, using the domain shown in figure 10. Three cases are considered: a) only photoionization; b) photoionization with photoelectron emission from the rectangle with 𝛾surf = 1.0; and c) photoion￾ization with 𝛾surf = 1.0 on the rectangle and 𝛾𝑖 = 10−3 on the left dielectric. The number of runs per initial location was 𝑁runs = 400. gases without photoionization the… view at source ↗
Figure 13
Figure 13. Figure 13: Cross section showing results in 3D geometry with a conducting wire. Top: inception probability 𝑝inc and 𝐾∗ for a voltage 𝑉0 = +9.1 kV on the wire. Bottom: results for 𝑉0 = −10.5 kV. The number of runs per initial location was 𝑁runs = 100. rightmost one. This is due to the wider base of the leftmost protrusion, which leads to a slightly higher field near the wire. The noisy pattern in 𝑝inc seen for negati… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 0 minor

Summary. The paper presents a Monte Carlo method for estimating partial discharge inception probability per initial electron position and the associated time lag. It takes an unstructured grid of the electrostatic field as input and simulates electron avalanches that propagate along field lines, incorporating photon and ion feedback. A statistical avalanche-size distribution (valid for attaching gases) is used; inception is declared when the avalanche population increases over time. The distribution is compared to particle simulations, and the method is demonstrated on 2D Cartesian, 2D axisymmetric, and 3D electrode geometries.

Significance. If the inception criterion can be rigorously justified, the approach would offer a computationally lighter alternative to full particle-in-cell tracking for mapping inception probabilities across complex domains, including sub-critical regions. The statistical avalanche model and multi-geometry demonstrations are practical strengths for high-voltage insulation design. The field-line approximation and heuristic growth criterion, however, limit the immediate quantitative reliability of the outputs.

major comments (3)
  1. [Abstract] Abstract: the inception decision rule ('If the number of avalanches keeps increasing over time we assume that an electric discharge will eventually form') is presented as a heuristic without derivation, validation against Townsend/streamer criteria, or demonstration that unbounded growth is equivalent to gap-bridging self-sustained discharge rather than bounded multiplication. This rule directly determines both reported output quantities.
  2. [Abstract] Abstract: avalanches are restricted to propagation strictly along field lines, omitting transverse diffusion. No error estimate or sensitivity study is supplied for how this approximation affects photon/ion feedback probabilities, especially in the 3D demonstration cases.
  3. [Abstract] Abstract: while the statistical avalanche-size distribution is stated to have been compared against particle simulations, the manuscript provides no quantitative metrics (e.g., goodness-of-fit statistics, attachment-coefficient range, or error bars), leaving the claimed validity for inception calculations unverified at the level needed to support the central results.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive feedback on our Monte Carlo method for partial discharge inception simulation. We address each major comment below and outline planned revisions where appropriate.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the inception decision rule ('If the number of avalanches keeps increasing over time we assume that an electric discharge will eventually form') is presented as a heuristic without derivation, validation against Townsend/streamer criteria, or demonstration that unbounded growth is equivalent to gap-bridging self-sustained discharge rather than bounded multiplication. This rule directly determines both reported output quantities.

    Authors: We acknowledge that the criterion is presented as a modeling assumption rather than a rigorously derived threshold. It rests on the observation that sustained growth in avalanche count signals the onset of feedback-dominated multiplication leading to breakdown. In revision we will expand the methods section with a brief derivation linking the rule to the classical Townsend integral exceeding unity and will add a short discussion of its relation to streamer criteria, including a note on regimes where bounded multiplication could occur. This addresses the direct dependence of the reported probabilities and time lags. revision: yes

  2. Referee: [Abstract] Abstract: avalanches are restricted to propagation strictly along field lines, omitting transverse diffusion. No error estimate or sensitivity study is supplied for how this approximation affects photon/ion feedback probabilities, especially in the 3D demonstration cases.

    Authors: The field-line propagation is an explicit modeling choice to enable efficient sampling over large unstructured grids. Transverse diffusion is neglected because, near inception, the drift velocity dominates and the mean free path is short relative to field-line curvature. We agree that a quantitative sensitivity assessment is missing. In the revised manuscript we will add a dedicated subsection that reports the effect of a small transverse perturbation (implemented via a limited set of off-axis particle tracks) on the computed inception probabilities for the 3D geometry, thereby providing the requested error estimate. revision: yes

  3. Referee: [Abstract] Abstract: while the statistical avalanche-size distribution is stated to have been compared against particle simulations, the manuscript provides no quantitative metrics (e.g., goodness-of-fit statistics, attachment-coefficient range, or error bars), leaving the claimed validity for inception calculations unverified at the level needed to support the central results.

    Authors: The comparison is shown only qualitatively in the current manuscript. We will augment the results section with quantitative measures: Kolmogorov-Smirnov distances, mean absolute percentage error, and 95 % confidence intervals on the fitted parameters, all evaluated across the attachment-coefficient range used in the inception examples. These metrics will be tabulated and discussed to substantiate the distribution's suitability for the reported probabilities. revision: yes

Circularity Check

0 steps flagged

No circularity: forward Monte Carlo simulation with external benchmarking

full rationale

The paper presents a Monte Carlo method that simulates electron avalanches along field lines with photon/ion feedback and applies a statistical avalanche size distribution that is validated by direct comparison to separate particle simulations. The inception criterion (increasing avalanche count over time) is introduced as an explicit modeling assumption rather than derived from equations within the paper. No load-bearing steps reduce by construction to fitted parameters, self-citations, or renamed inputs; the central outputs are produced by forward simulation whose statistical components are checked against independent particle results. This is a standard non-circular forward-modeling approach.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The method rests on domain assumptions about avalanche propagation and inception criteria rather than new physical entities or fitted constants visible in the abstract.

axioms (3)
  • domain assumption Avalanches propagate along field lines
    Explicitly stated as the propagation model in the abstract.
  • domain assumption If the number of avalanches keeps increasing over time then a discharge will eventually form
    Core decision rule for converting avalanche statistics into an inception probability.
  • domain assumption Statistical distribution for avalanche size remains valid for gases with strong electron attachment
    Invoked to justify use of the distribution across the domain including attaching conditions.

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