*4.1. Basic Concept*

The backbone of the model originated from the Obenaus concept [29]. This concept was firstly developed to describe the flashover under DC conditions. Furthermore, some researchers such as Claverie [10] and Rizk [12,30,31] provided the essential criteria to adapt the Obenaus concept to AC voltages. The AC flashover on a polluted surface is considered as an arc in series with a residual resistance consisting of a pollution layer that is not bridged by the arc (Figure 3). The circuit equation then reads as follows:

$$
\mathcal{U} = \mathcal{V}\boldsymbol{\rmarc}\mathbf{c} + \mathcal{R}p \; (\mathcal{l}\_{\rm arc}) \, \mathcal{I} \tag{1}
$$

$$\text{Varc} = \text{N}l\_{\text{arc}}I^{-n} \tag{2}$$

where

> *U*: applied voltage (V); *N* and *n*: arc constants; *larc*: arc length (cm); *I*: leakage current (A); *Varc*: arc voltage (V); *Rp* (*larc*): residual resistance of the pollution layer (Ω).

**Figure 3.** Model of flashover on a polluted insulator [29].

The arc tends to extinguish at every half-cycle as the current passes through zero. The re-ignition can occur when the applied voltage reaches the critical value (depending on leakage current *I* and arc length *larc*) according to Claverie and Porcheron [10,27]:

$$
\mathcal{U}\_{\rm rei} \ge \frac{800 \, l\_{\rm arc}}{\sqrt{I}} \tag{3}
$$

On the basis of some experimental results, Hampton established that the arc cannot propagate unless the voltage gradient in the pollution layer *Ep* exceeds the voltage gradient in the arc column *Earc* [16]. Therefore, the propagation condition yields

$$Ep \text{ > Error} \tag{4}$$

On the basis of these equations, we propose a two-dimensional model. The fundamentally new feature of the model is its use of a field mapping criterion to describe the arc propagation. The angle between the arc discharge step and the axis of the gap is random with an assumed form of uniform probability distribution. The initiation and propagation criteria of the electric discharge are estimated by the modified Peek's formula. The main features are described in the next section.

#### *4.2. Simulation Model*

Computations of the model were performed using the scientific library MATLAB®. To reproduce the arc discharge progression forms, we established a circular mesh (Figure 4), made up of (nmesh × mmesh) elements with angular (Δθ) and radial (Δr) steps. The nodes of this mesh represent the targeted points by the arc discharge.

The development of a superficial model of electrical arc discharge propagation is reproduced under the following assumptions:


In the computer program, the system is reproduced by a matrix of dimension mmesh × nmesh where the centre of the circular electrode constitutes the element (1, 1) with the coordinates (0, 0). The other points are defined geometrically (x, y) or as a matrix (i, j) abscissas, as defined by Equations (5) and (6).

The difficulty of an electric arc rises as it takes place on a polluted surface. The conductivity and the distribution of pollution layers not only influences the inception/propagation of a surface electric discharge but is itself influenced by this electric discharge. The established model concerns two steps of discharge evolution: inception and propagation.

**Figure 4.** Progression discharge grid. (**a**) Mesh. (**b**) Target points.

The coordinates of the points are calculated by the following relationships:

$$\propto = \text{ i } \Delta \mathbf{r} \cos(\mathbf{j} \,\Delta \theta) \tag{5}$$

$$\mathbf{y} = \text{ i } \Delta \mathbf{r} \sin(\mathbf{j} \,\Delta \theta) \tag{6}$$

with

> Δθ, Δr: angular and radial steps, respectively; j: index varying between 1 and mmesh = 2π/Δθ; mmesh: number of angular divisions; i: index varying between 1 and nmesh = (X/Δr) + 1; nmesh: number of radial divisions.

## 4.2.1. Discharge Inception

When the applied voltage stress (electric field) between two electrodes in an insulation system reaches a threshold value, a first corona discharge of initial length L0 is initiated. If the conditions for the discharge propagation are satisfied, the discharge channel fed by the arc discharge current lengthens. Otherwise, the channel switches off and disappears unless the applied voltage reaches the threshold re-ignition value [32,33].

The threshold electric field is estimated using empirical Peek's law [33–38]:

$$E\_c = E\_{ai} \delta \left( 1 + \frac{K}{\sqrt{\delta R\_p}} \right) m\_1.m\_2 \tag{7}$$

The threshold voltage of the discharge inception for the considered configuration is [7,37,38]

$$Llc = E\_{ai} \delta \left( 1 + \frac{K}{\sqrt{\delta R p}} \right) \frac{\left( \ln \left( \frac{2D + Rp}{Rp} \right) \right)}{\left( \frac{1}{Rp} + \frac{1}{2D + Rp} \right)} \begin{array}{c} m\_1 \ m\_2 \end{array} \tag{8}$$

where

> *Eai* = 21.2 kVrms/cm, *K* = 0.3;

*Rp*: radius of rod electrode in cm;

 *m*1: geometric correction coefficient (between 0 and 1) being equal to

$$m\_1 = \frac{\log(X\_L)}{2\text{ }\pi} \tag{9}$$

*XL*: distance between the centre of the circular electrode and the end of the plane electrode;

*m*2: correction coefficient of the pollution (0.3 to 0.9). For a clean surface, *m*2 = 0.9, whereas for highly polluted one, *m*2 = 0.3.

The surrounding condition is also taken into account through the density:

$$\boldsymbol{\delta} = (\mathbf{3}.92 \,\mathrm{P})/(\mathbf{273} + \mathrm{t})\tag{10}$$

P: air pressure in cm Hg;

t: the surrounding temperature in ◦C.

The Equations (8)–(10) are used to determine the threshold voltage according to the electro-geometrical parameters and climatic conditions [7,37,38].

#### 4.2.2. Discharge Propagation Criteria

A discharge initiation on a polluted insulator may not lead to a breakdown process unless two conditions are satisfied. First, the required conditions for discharge propagation must be satisfied. Second, the discharge should not stop, and therefore the first condition remains valid during the progression of the arc up to the flashover (otherwise the voltage has to be increased).

A mandatory condition for an electric discharge to be initiated is related to the distribution of electric fields, which greatly depend on geometrical dimensions and electric parameters of pollution.

The propagation of electric discharge being the succession of avalanches of critical size and the last phase is called the final jump, which corresponds to the establishment of the flashover between the electrodes.

In this work, the electric discharge starts from the energised electrode (cylindrical electrode) where the electric field value is high enough to sustain the discharge propagation to the grounded electrodes. Indeed, the electric discharge is considered to evolve randomly when the field at the target point exceeds the threshold field of corona appearance.

The proposed multi-arc FEM model uses the inception field calculated from Equation (7) as condition to initiate the discharge.

At each evolution step, the extreme point of the discharge M, a three-options random probability propagation is offered (Figure 4b): L (left), C (centre), and R (right).

Considering the target point with electric field E, the discharge evolution criterion (Hampton criteria) towards this point is given by

$$\mathcal{E} \succ \theta \to\_{\mathbb{C}} \tag{11}$$

where

> EC: the critical field (Equation (7);

*ϑ*: random variable generated at each step and each branch by a uniform probability distribution in the model.

The uniform law allows for pseudo-random number generation algorithms, to generate random variables with the same probability of occurrence over an interval.

If the likelihood is sufficiently high enough, a new segmen<sup>t</sup> connects the extremity M to the targeted point, which becomes itself a new discharge extremity.

After each propagation step, the electric field distribution is modified by the newly added branches. The electric field distribution is therefore computed.

The voltage at the new extremity referred to as *Varc* is given by the following equation:

$$V\_{\rm arc} = \mathcal{U} - \Delta \upsilon \tag{12}$$

with Δ*v* [V] representing the linear voltage drop along the discharge channel.

When the electric arcs progress, a new configuration is imposed on the system by the arc length and its voltage drop. This new configuration affects the critical field at each step of the evolution. The inception field is recalculated at each step of the electric discharge evolution (Equation (13)).

$$E\_{\rm c}(l\_{\rm arc}) = E\_{\rm ai} \cdot \delta \left( 1 + \frac{K}{\sqrt{\delta \, R \, (l\_{\rm arc})}} \right) m\_1 \, m\_2 \tag{13}$$

where

$$R(l\_{\rm arc}) = R\_p + l\_{\rm arc} \tag{14}$$

*larc*: the maximum radial length of the discharge

$$l\_{\text{arc}} = \Delta \text{r. N}\_{\text{bv}} \tag{15}$$

*Nbe*: number of steps;

Δr: radial steps.

The model used to compute the breakdown voltage is based on Obenaus [29] and Claverie-Porcheron [10,27] models. The electric field distribution is obtained by the Finite Element Method Magnetics software (FEMM) [38]. To do this, a rectangular domain defines the calculation limits with a Neumann condition. MATLAB is used to introduce the electrogeometric data in the form of an executable program and the exploitation of the FEMM results.

During each calculation step, if the propagation condition is satisfied, the discharge lengthens. Otherwise, the peak value of the applied voltage (U) has to be increased and the parameters initialised.

The flashover voltage is obtained when the discharge length exceeds the inter electrode gap D. Figure 5 presents the general flowchart of simulation.

**Figure 5.** General flowchart of the simulation.

#### **5. Simulation Results and Validation against Experiments**

The flashover of the polluted insulator was measured and simulated while considering the effect of various electro-geometric parameters including conductivity, inter-electrode distance, HV electrode radius, and ground electrode width. For this purpose, we considered various electro-geometrical constraints as follows


A comparison between simulated and experimental data to validate the proposed model is reported hereafter. The variation of the flashover voltage as a function of the above-mentioned electro-geometric parameters under AC voltage is given in Figures 6–13.

**Figure 6.** Flashover vs. pollution layer conductivity (Rp = 1.5 cm, L = 25.5 cm).

**Figure 7.** Flashover vs. pollution layer conductivity (Rp = 2.5 cm, L = 25.5 cm).

**Figure 8.** Flashover vs. pollution layer conductivity (Rp = 1.5 cm, L = 15.5 cm).

**Figure 9.** Flashover vs. pollution layer conductivity (Rp = 2.5 cm, L = 15.5 cm).

**Figure 10.** Flashover vs. inter-electrode distance for electrode radius (γ = 900 μS/cm, L = 25.5 cm).

**Figure 11.** Flashover vs. inter-electrode distance for electrode radius (γ = 900 μS/cm, L = 15.5 cm).

**Figure 12.** Flashover vs. inter-electrode distance for different electrode radius (γ = 80 μS/cm, L = 25.5 cm).

**Figure 13.** Flashover vs. inter-electrode distance for electrode radius (γ = 80 μS/cm, L = 15.5 cm).

It can be observed from Figures 6–9 that the flashover voltage decreased with the pollution layer's conductivity. In other words, increasing the pollution conductivity caused the electrical performance to worsen. It can also be seen that increasing inter-electrode distance and energised electrode radius caused the disruptive voltage to increase, regardless of the pollution layer's conductivity and the length of the grounded electrode.

The electric field decreased with increasing D or Rp, thus requiring a higher level of applied voltage to generate a conductive current and dry bands that stimulate generation of arcs, leading to flashover.

The breakdown voltage was reduced with increasing pollution conductivity. The presence of pollution favours a conductive current in the regions of high electrical field stress, which create dry bands and facilitate arc generations.

The variation of the geometrical parameters such as the radius and the length of the ground electrode, as reported in Figures 10–13, impacts the flashover by affecting the distribution of the electric field. For a given inter-electrode distance, with any increase in the electrode radius or the length of grounded electrodes, the electric field tends to be uniform. However, the length effect is less important than that of the inter-electrode distance.

For a given radius and inter-electrode distance, it should be noticed that the experimental data have the same order of magnitude as those predicted with the established numerical model (less than 20% error). This difference can be traced to the possible displacement of the wet part of the pollution, the difficulty in applying uniformly the pollution in real life, and/or the residues from the previous discharges.

It can be seen that the simulation results were close to the experimental ones for small radius, large gap electrodes, and large plane electrode. Furthermore, the difference between the simulation and experimental results was reduced when the HV electrode radius decreased and the ground electrode length increased. The random variable generated at each step and each branch by a uniform probability distribution in the model may be seen as a clear limitation of the model in its actual form.
