## 2 FEM in Partial Discharge Simulation
### Simulation Framework
#### 2.1.3 Application of Physics Interfaces  
I. Electrostatic and Space Charge Modeling  
FEM simulations involve solving the governing equations of electrostatics and electrodynamics to capture the intricate behavior of PD phenomena. These equations include the following:  
1. Poisson's Equation  
This governs the electric potential (\(V\)) and electric field (**E**) distributions:  
\[\
abla^{2}V=-\frac{\rho}{\epsilon} \tag{1}\]  
where \(\rho\) is the charge density and \(\epsilon\) is the permittivity. 2. Charge Transport Equations  
These model the movement of charge carriers within the cavity:  
\[\frac{\partial\rho}{\partial t}=D\
abla^{2}\rho-R+S \tag{2}\]  
where \(D\) is the diffusion coefficient, \(R\) is the recombination rate, and \(S\) represents charge generation during PD events. These equations are solved iteratively using adaptive time-stepping to capture the rapid changes associated with PD events while maintaining numerical stability. II. Stochastic Modeling  
Stochastic models describe the probabilistic behavior of PD phenomena by incorporating random factors, such as electron availability, material defects, field fluctuations, and environmental conditions. Unlike deterministic models, stochastic methods emphasize the statistical distributions and variations in PD initiation and progression, offering insights into the intricate behavior of insulation systems. Figure 3 presents the deterministic PD model characteristics, highlighting a precise and consistent relationship between the applied AC voltage and cavity discharge events, including inception and extinction voltages. Discharge timings and magnitudes, such as time delays (\(T_{s}\)), are predictable and follow a set pattern within the AC cycle under ideal conditions. In contrast, Figure 4 showcases the stochastic PD model characteristics, which lack strict periodic behavior. Here, discharge timings and magnitudes vary across cycles due to stochastic influences like electron availability and material imperfections, introducing variability in inception/extinction voltages and discharge magnitudes. This model accounts for real-world randomness in PD behavior. The stochastic nature of partial discharge is driven by material defects, which introduce randomness through microscopic voids and impurities; electron availability, where random ionization events produce free electrons; and external noise, with unpredictable environmental conditions like temperature and humidity affecting PD behavior. Combining stochastic modeling with FEM enhances PD risk assessment, aiding in strategic decisions about insulation system reliability. PD events occur randomly, initiated under probabilistic conditions, mainly due to the presence of free electrons, which are essential for PD inception. These electrons may originate from cosmic rays, natural radioactivity [43], or previous PD incidents, making their occurrence unpredictable despite constant voltage. The stochastic nature of PD activity can be modeled using probability distributions and statistical methods to investigate its intrinsic uncertainty. (a) Exponential Model  
The exponential function characterizing the number of free electrons, \(N_{e}(t)\), is determined by the cavity voltage \(V_{\text{cav}}(t)\) relative to the critical voltage \(V_{\text{crit}}\). It highlights the rapid increase in free electrons as the voltage approaches \(V_{\text{crit}}\), according to [44]:  
\[N_{e}(t)=N_{e0}\exp\left(\left|\frac{V_{\text{cav}}(t)}{V_{\text{crit}}}\right|\right) \tag{3}\]  
where \(N_{e}(t)\) is the number of free electrons at time \(t\) and \(N_{e0}\) represents the initial number of free electrons. (b) CDF for PD Event  
Figure 4: Stochastic PD model. Figure 3: Deterministic PD model. The probability distribution of a PD event over time \(t\) is captured by a CDF, which accumulates free electron counts over a given time.