**1. Introduction**

For the safe operation of any power system network, it is important that key equipment such as transmission line hardware operate properly for many years. Transmission lines pass through several hundreds of kilometres, being subjected to various climatic conditions (sea salts, domestic pollution, natural pollution, dirt, and chemical residues in the industrial areas, etc.). Pollution of insulators is recognised as a major engineering concern since in pollutedareas,overheadlinesmayseetheirreliabilityandperformancedecline[1–4].

This is because when the contaminated surface is wet, the leakage currents then increase with potential flashover of the insulating surface, which means power outage [5]. Even though many investigations have been reported on the topic, there is still a dearth of information to gather. The physics of the flashover of polluted insulators is still not completely understood due to the complexity of the involved phenomenon resulting from the interaction of the electric field, pollution layer, and environmental conditions [6].

Due to fast-growing cities, industrialisation, and climate change, power grids are poised to experience more pollution. The study of the flashover of polluted insulators is therefore of interest due to potential power-related outages. Significant efforts have been made to better understand the evolution of insulator flashover. A large number of papers and books on the insulator surface flashover are available [7]. It is now well established that the flashover process of polluted insulators includes three steps: (i) arc initiation, (ii) arc

**Citation:** Amrani, M.L.; Bouazabia, S.; Fofana, I.; Meghnefi, F.; Jabbari, M.; Khelil, D.; Boudiaf, A. Modelling Surface Electric Discharge Propagation on Polluted Insulators under AC Voltage. *Energies* **2021**, *14*, 6653. https://doi.org/10.3390/ en14206653

Academic Editor: Abu-Siada Ahmed

Received: 10 September 2021 Accepted: 8 October 2021 Published: 14 October 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

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propagation, and (iii) flashover. The electric field distribution along the insulator, which depends on several factors, has a grea<sup>t</sup> influence on the arc initiation and propagation. The initiation of an arc starts in a region of high local electrical stress.

Theoretical models for simulating electrical discharge may allow for a better understanding of the physical processes involved in the flashover of polluted insulators. From the engineering point of view, the ultimate goal of the simulation/modelling is to predict the behaviour of polluted insulators from purely macroscopic data. Therefore, reliable models may considerably reduce the time factor required for laboratory tests. Thus far, several flashover models, mathematical, numerical, or experimental investigations, e.g., [7–12], have been reported to determine the critical characteristics of the electric arc.

Few mathematical or numerical studies have been conducted to predict the critical flashover voltage of polluted insulators considering that an arc behaves like an equipotential surface in contact with the pollution layer and assumes propagation in one dimension [13–15]. Some models describe the electric arc behaviour using various criteria: electric field (Hampton's criteria), leakage current variation (Hesketh's criteria), applied voltage and energy (Anjana and Lakshminarasimha's criteria), power variation (Wilkins' criteria), and equivalent impedance [16–20]. All these models are based on equations involving the static arc constants (*n* and *N*) [12]. A survey of the literature shows that their values vary over a wide range for different types of arcs [21]. These values depend not only on the arc medium but also on the electrolyte used to form the resistive layer. The fundamental shortcoming is related to the large range between these values when comes to the time to select values for a specific application. Slama et al. [2] proved that the parameters n and N are, in reality, not static and depend on the thermal characteristics of the arc along with the equivalent electrical circuit simulating the phenomena.

The present paper aimed at studying the inception and propagation of an electric arc discharge up to flashover under polluted conditions. The main features include the electric field distribution mapping and the fact that the propagation process is superficial (2D). The electric arc discharge model is based on finite element method (FEM). It is generally reported that in the case of uniform pollution, the discharge occurs in the dry bands generated by the circulation of the leakage current [11,22–24]. For non-uniform pollution, the arc develops in unpolluted areas [11,22–24]. These observations allow for confirmation that the electric arc starts with a discharge in the air in contact with the pollution layer. The model established in this paper is based on this result by exploiting Peek's work on electric discharges in gases.

The electrical arc discharges, initiated from the cylindrical electrode, propagate by stepping randomly along the polluted insulator plate once the field exceeds the threshold level computed at each time step. When the electric arc progresses, a new configuration is imposed on the system, represented by an increase of the arc length and changes in its characteristics. This increase in the arc length causes the inception field to change at each step. The electrical discharge is modelled by small branches in contact with the pollution layer. The proposed model is validated against experimental results. The comparison of the computed and measured flashover results indicates the validity of the proposed model for predicting the breakdown voltage. From this contribution, explicit information onto the influence of the electro-geometrical parameters on the electric field distribution that affects the flashover process of the polluted insulators is reported. Three aspects of the electrical discharge were analysed: arc initiation, discharge propagation law, and voltage drop. The influence of the electro-geometric parameters on the characteristics of the electric discharge is also discussed. In particular, the proposed model makes it possible to predict the flashover voltage of a polluted insulator according to the geometry of the electrodes, gap distance, and the conductivity of the pollution.

The rest of the paper is organised as follows. The laboratory model is described in Section 2. The experimental investigations are reported in Section 3. In Section 4, the mathematical model is described. The comparison of the simulated results with

experimental ones is reported in Section 5. Conclusions and future directions of research are provided in Section 6.

## **2. Laboratory Model**

The rod–plane arrangemen<sup>t</sup> constitutes one of the basic configurations in the investigation of non-uniform electric fields and insulation properties in high-voltage studies [25,26]. The physical laboratory model (Figure 1) is based on the plane model of Claverie and Porcheron [10,27]. It consists of a glass insulating plate on which a uniform pollution layer with specific conductivity (γ) is deposited. A cylindrical electrode with a radius (Rp) served for high voltage (U) application while a rectangular electrode of length (L) and width (a) served as the ground electrode. The cylindrical and rectangular electrodes are separated by a distance (D). It is also assumed that distance between the centre of the circular electrode and the end of the plane electrode is 'XL'.

**Figure 1.** Overview of the laboratory physical model. (**a**) Schematic illustration. (**b**) Photo.
