2.1. a-b-c model

The a-b-c model or the three-capacitor model is the original one to interpret the PD mechanism [3], and then it is usually employed to simulate the stochastic characters of PD [10, 11]. In the model, the dielectrics between electrodes, including the gas and solid insulation, are considered as capacitors, as in Figure 2. In detail, C<sup>1</sup> indicates cavity capacitance, C<sup>2</sup> is the capacitance of dielectric in series with the cavity, and C<sup>3</sup> is the capacitance of solid dielectric in parallel with the cavity. Besides, R1, R2, and R<sup>3</sup> indicate the resistance of corresponding part, respectively.

The occurrence and termination of PD depend on the potential difference across the cavity, U1. When U<sup>1</sup> exceeds the inception voltage, a discharge will take place and will stop when it is less than the extinction voltage. If a discharge occurs, C<sup>1</sup> is short-circuited, leading to a fast transient current to flow in the circuit due to a voltage difference between the voltage source and across C2. Based on the analysis of capacitor charging-discharging processes, the apparent charge magnitude, which reflects PD intensity, could be calculated.

It could be found that this model is very simple, but it can represent the transient process related to a discharge event and is often used to explain some experimental results. However, it could not describe the discharge process physically, and the concept, capacitor, is not strictly valid, because the interface between the cavity and the solid dielectric is not equipotential when a discharge takes place [22].

### 2.2. Pedersen's model

could be roughly divided into two categories: based on the point of view of circuit and based on the point of view of field. The former indicates a-b-c model and the latter consists of

The a-b-c model or the three-capacitor model is the original one to interpret the PD mechanism [3], and then it is usually employed to simulate the stochastic characters of PD [10, 11]. In the model, the dielectrics between electrodes, including the gas and solid insulation, are considered as capacitors, as in Figure 2. In detail, C<sup>1</sup> indicates cavity capacitance, C<sup>2</sup> is the capacitance of dielectric in series with the cavity, and C<sup>3</sup> is the capacitance of solid dielectric in parallel with the cavity. Besides, R1, R2, and R<sup>3</sup> indicate the resistance of corresponding part,

The occurrence and termination of PD depend on the potential difference across the cavity, U1. When U<sup>1</sup> exceeds the inception voltage, a discharge will take place and will stop when it is less than the extinction voltage. If a discharge occurs, C<sup>1</sup> is short-circuited, leading to a fast transient current to flow in the circuit due to a voltage difference between the voltage source and across C2. Based on the analysis of capacitor charging-discharging processes, the apparent

It could be found that this model is very simple, but it can represent the transient process related to a discharge event and is often used to explain some experimental results. However, it could not describe the discharge process physically, and the concept, capacitor, is not strictly valid, because the interface between the cavity and the solid dielectric is not equipotential

Pedersen's model, conductance model, and Niemeyer's model.

112 Plasma Science and Technology - Basic Fundamentals and Modern Applications

charge magnitude, which reflects PD intensity, could be calculated.

2.1. a-b-c model

respectively.

when a discharge takes place [22].

Figure 2. a-b-c model.

There are two important parameters of PD, that is, physical charges and apparent charges. The former indicate the charges generated during a discharge process, while the latter are measured charges through external circuit. In order to establish the link between physical charges and apparent charges, Pedersen proposed a model to describe the transient process [23]. Without considering the charge exchange between solid dielectric and the adjacent electrode, the amount of apparent charges equals the induced charges at an electrode surface due to charge generation, recombination, and movement during a discharge process. Therefore, if the physical charge distribution is known, the apparent charges could be calculated [24]

$$Q\_{\rm app} = -\iiint \lambda \rho \mathbf{d}V - \iint \lambda \sigma \mathbf{ds} \tag{1}$$

where r and σ indicate volume and surface charge density within the cavity, respectively. λ, a dimensionless function, depends on the charge location, which satisfies Laplace equation

$$\nabla \cdot \left( \varepsilon\_0 \varepsilon\_\mathbf{r} \nabla \lambda \right) = \mathbf{0} \tag{2}$$

where ε<sup>0</sup> is the vacuum permittivity, and ε<sup>r</sup> the relative permittivity.

Pedersen's model is helpful to understand the measured results by using the pulse current method. However, the apparent charges depend on physical charge distribution which results from the discharge process and keeps unknown in this model.

#### 2.3. Conductance model

When PD takes place, a plasma region with a high charge concentration in the cavity is formed, so the gas conductivity largely increases in comparison with the initial state. Based on this fact, the discharge process is simplified by the variation of gas conductivity [13], which can be described by the following equations:

$$
\nabla \cdot \mathbf{D} = \rho \tag{3}
$$

$$\nabla \cdot \mathbf{J} + \frac{\partial \rho}{\partial t} = 0 \tag{4}$$

where D is the electric displacement field, J the free current density. At the initial state, the gas conductivity is set to be zero. When a discharge takes place, it is set to be γgd and hence the electric field distribution within the cavity changes. In terms of the electric field evolution, some PD parameters are obtained, for example, apparent charges and physical charges.

Forssen compared the simulation results with the experimental data, and they were in general agreement but with a slight difference. Furthermore, Illias developed the simulation model by taking the surface emission and temperature variation during the discharge into account [14]. However, in any case, the increment of gas conductivity could not represent the PD process.

#### 2.4. Niemeyer's model

Niemeyer considered PD within the cavity as a streamer-type discharge, because only this type could be detected and has engineering significance [9]. After analyzing the physical processes of PD, he proposed several equations to describe PD, as follows:

$$\int\_{0}^{\chi\_{\rm cr}} \overline{\alpha}[E(\mathbf{x})]d\mathbf{x} \ge K\_{\rm cr} \tag{5}$$

free electrons for the next PD occurrence. The interaction between adjacent PDs could not be represented by singe PD. Therefore, it is necessary to establish a simulation model which could present the discharge development process and take the memory effect into account to obtain

Numerical Modeling of Partial Discharge Development Process

http://dx.doi.org/10.5772/intechopen.79215

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As for the PD simulation, on one hand, the model should reflect physical processes as much as possible, and on the other hand, a large number of data should be obtained to get the statistical parameters of repetitive PDs due to the stochastic characters. There is a contraction that taking too much physical processes into account must result in the model complexity and a large calculation consumption which is not beneficial to statistical analysis. Therefore, some important processes should be considered in the simulation model, while others are abandoned.

By reviewing the PD simulation models, it is found that two processes are crucial to cavity PD characters, that is, streamer development and surface process. Obviously, the apparent charges that could be detected by pulse current method are determined by streamer development in the cavity. Surface process mainly consists of charge accumulation on the interface and surface emission of charge. After the streamer lands on the dielectric surface, charges accumulate and will affect the subsequent PD behavior. Besides, surface emission could provide free electrons for the next PD. It should be noted that the distribution of surface charges generated by previous discharge does not keep unchanged until subsequent one takes place. Due to the surface or bulk conductivity of dielectric, the accumulated charges may decay. To sum it up, the streamer development and surface charge accumulation reflect a single PD process, while surface charge accumulation, decay, and emission represent the interaction of adjacent dis-

charges during a PD sequence, which should be considered in the simulation model.

The streamer development is quantitatively described by fluid equations, as follows:

<sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>N</sup>eαj j� <sup>W</sup><sup>e</sup> <sup>N</sup>eNp<sup>β</sup> � <sup>N</sup>nNp<sup>β</sup> � <sup>∇</sup> � <sup>N</sup>pW<sup>p</sup>

<sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>N</sup>eαj j <sup>W</sup><sup>e</sup> � <sup>N</sup>eηj j <sup>W</sup><sup>e</sup> � <sup>N</sup>eNp<sup>β</sup> � <sup>∇</sup> � ð Þ <sup>N</sup>eW<sup>e</sup> � <sup>D</sup>∇N<sup>e</sup> (8)

(9)

Because sandwich-type samples are widely used in the experimental researches on PD, a cylindrical cavity with a diameter of 2 mm and a height of 0.25 mm is employed in our simulation model, as in Figure 3. The cavity, full of atmospheric pressure air, is embedded within the solid dielectric, of which the relative permittivity equals 2.3. The thickness of dielectric barriers is set to be identical to the cavity height. Although during the discharge process, the temperature of cavity may slightly increase due to the joule heating from discharges, the temperature variation is neglected in our model, which means that the pressure in

3. Numerical modeling of PD sequences using fluid equations

the stochastic characters of PD sequences.

3.1. Simulation model construction

the cavity keeps unchanged.

∂N<sup>e</sup>

∂N<sup>p</sup>

$$
\Delta U\_{\rm res} \approx E\_{\rm ch} l\_{\rm str} \tag{6}
$$

$$
\mathfrak{q} = \pm \mathfrak{g}\pi \varepsilon\_0 l \Delta \mathfrak{U} l\_{\rm FD} \tag{7}
$$

Eq. (5) is actually the well-known critical avalanche criterion, in which α, the function of electric field, indicates the effective ionization coefficient, Kcr the logarithm of a critical number of electrons that has to accumulate in the avalanche head to make the avalanche self-propagating by its own space charge field, and xcr the distance within α which exceeds zero. In terms of it, the inception field of PD occurrence could be obtained. Eq. (6) simply describes the streamer propagation, where Ech is the electric field in the discharge channel, Ures the residual voltage instantaneously after discharge, and lstr the distance to which streamer could propagate. Eq. (7) establishes the relationship between physical charges and potential difference before and after a PD, in which g is a dimensionless proportionality factor and l the cavity scale.

Based on the model, Niemeyer simulated PD behaviors within a spherical cavity by considering the stochastic supply of free electrons, which agreed with experimental data qualitatively and quantitatively although there was a slight disagreement in the phase and magnitude distributions of PD. However, there is a significant shortcoming that the electric field distribution was assumed to be uniform within the cavity. Considering this point, Illias developed the simulation model in which the deployed charges were not uniform and Poisson's equation was employed to calculate PD parameters [16, 17].

#### 2.5. Plasma model

In terms of physical processes, a cavity PD is similar to the filamentary dielectric barrier discharge (DBD) [25]. As for the latter, fluid equations are widely used to simulate gas discharge process [26, 27], which describe the impact ionization, charge drift, diffusion, recombination, and some secondary effects. In recent years, several researchers employed them to simulate the PD occurring in a cavity [18–20]. For example, Novak and Bartnikas established a two-dimensional breakdown model based on the continuity equations for electrons and ions to examine the influence of surface charges upon the partial discharge behavior [19]. In terms of it, the evolution of electric field and charge concentration distribution within the cavity during the discharge process was obtained, as well as the discharge current pulse.

However, the behaviors of single PD could not represent that of continuous PDs due to the memory effect. On one hand, residual charges generated by previous discharge land on the cavity surface and affect the electric field distribution within the cavity, leading to the change of subsequent PD characters. On the other hand, the accumulated surface charges may provide free electrons for the next PD occurrence. The interaction between adjacent PDs could not be represented by singe PD. Therefore, it is necessary to establish a simulation model which could present the discharge development process and take the memory effect into account to obtain the stochastic characters of PD sequences.
