Hybrid Grey Wolf and Finite Element Method (GWO-FEM) Algorithm for Enhancing High Voltage Insulator String Performance in Wet Pollution Conditions

The presence of wet pollution on the upper surface of a string insulator increases the electric field on the insulator surface, especially at the triple junction (pin-cement, cementporcelain) as well as at the surrounding air of the insulator. The rise of the electric field leads to the ionization of the air surrounding the insulator. This phenomenon, called corona discharge, is accompanied by several consequences that are harmful to the electricity transmission network, such as electromagnetic interferences, energy losses, visible light, audible noise, and the destruction of materials by erosion. If favorable conditions are gathered, it may even cause the flashover of the insulator. Designing an optimal insulator shape that reduces this electric field value at the triple junction will be an important achievement in enhancing the performance of electrical grids. The objective of this paper is to evolve a hybrid algorithm based on the GWO-FEM for optimizing the shape and the electrical performance of a string insulator. To achieve this purpose, this work is structured in four parts. First, modeling of the insulator string geometry is conducted in Comsol-multiphysics, then FEM computation of the electric field on the polluted surface of the string insulator is completed, and the maximum electric field value at the triple junction is saved as the fitness function that will be sent to the GWO algorithm to be optimized (minimized). The third part of the work is devoted to the coding of the constrained (electrical and geometrical constraints) GWO algorithm in a Matlab interface, and finally, coupling the GWO code with the FEM code. This study is achieved in wet polluted conditions. The results are given in both 2D and 3D representations. From the obtained results we can confirm that the developed GWO-FEM hybrid algorithm for optimizing insulator strings is a very promising tool for designing and enhancing the shape and the electrical performance of insulators. Keywords-insulator design; grey wolf optimisation; finite element method; 3D electric field

INTRODUCTION Transmission and distribution lines constitute the link between production and consumers. These lines are held by insulating strings [1]. Their performance is crucial for the good functioning of the electrical network. It is well established by experimental studies and confirmed by simulations [2][3][4][5][6][7][8], that the highest electric field value around an insulating string is found on the last pin in contact with the high voltage conductor, more exactly at the triple junction (pin-cement, cement-porcelain), which also corresponds to the highest thermal point [9], therefore, these areas are very critical in high voltage lines. The presence of high electric field leads to corona discharges by ionizing the surrounding air of the insulator, resulting in energy losses, electromagnetic interferences, visible light, audible noise, and the destruction of materials by erosion [10][11][12]. When the electric field stress exceeds a threshold value, especially in the presence of pollution conditions [13], flashover of the insulator string may occur. Thus, finding the insulating string design that can reduce the critical value of the electric field at the triple junction region, and at the same time reduce the weight of the insulator constitutes, a considerable advantage for improving the performance of electrical networks. It is here where the contribution of the present paper is registered.
Designing an improved shape of the insulator string involves taking into account many constraints, such as electrical, mechanical, environmental and economic, so using simulations prior to real testing and manufacturing is money and time saving. Recent works carried out concerning the improvement of the high voltage insulators performance based on optimization methods, are divided into two main categories: (i) optimizing the shape of the chain elements [14][15][16][17][18][19][20][21][22][23] and (ii) grading ring optimization [24][25]. This research aims to study a hybrid algorithm based on the Grey Wolf Optimization and the Finite Element Method (GWO-FEM) for designing an insulator sting and enhancing its electrical and geometrical performance. The current paper brings a complementarity to fill some existing gaps in the field of designing high voltage insulators. Its main contributions are: • The proposal of the new hybrid GWO-FEM algorithm for insulator optimization.
• Corona ring optimization and one-element insulator optimization have been already done in previous studies, but, to the best of our knowledge, total string optimization is achieved for the first time in this paper.
• By using optimization constraints, two goals are achieved, namely reduction of the electric field value at the triple junction and of the weight of the insulator string.
• 3D geometry representation and electric field distribution were conducted.
• The most significant contribution of this work is the pollution case study which is conducted for the first time in this paper.
To accomplish the above contributions, this work is divided into four main parts: • Modeling of the insulator string geometry in Comsolmultiphysics.
• FEM computation of the electric field on the insulation surface of the insulator and saving the maximum electric field value at the triple junction as the fitness function that will be sent to the GWO algorithm to be optimized.
• Coupling of the GWO code with the FEM code. Figure 1 shows one element of the studied 6-element string insulator. The parameters of the insulators are given in Table I. The model is a string insulator of 6 elements of an U400B glass insulator commonly used in the Algerian power grid. The modeling of the insulator geometry in Comsol-multiphysics is represented in 2D and 3D in Figure 2.   A wet pollution layer ( Figure 3) is put on the upper surface of each element of the string. The thickness of the pollution layer is 0.5mm. The pollution layer is in contact with the earthed cap. The material parameters used for the simulation are given in Table II.

III. FEM COMPUTATION OF POTENTIAL AND ELECTRIC FIELD
After the meshing process, potential and electric field distributions are computed by solving (1) and (2) by using FEM.
where E represents the electric field and V is the electric potential.
The electric potential varies from 100kV from the pin to 0V on the cap as shown in Figure 4 giving the potential repartition along the creepage distance of the insulator string. Electric potential repartition on the creepage path of the insulator string.
From the electric field repartition ( Figure 5), starting from the pin towards the cap, we record a maximum value of 190kV/cm at the last pin. This obtained result is in adequacy with the results from the literature review [2][3][4][5][6][7][8][9], which attest that the highest electric field value is located at the last pin of the string connected to the high voltage and exactly at the triple junction. So, for the optimization purpose, the maximum electric field E max recorded at this triple junction is saved as the fitness function and is sent to the GWO cod to be optimized (minimized). The potential and electric field distribution in 2D and 3D are given in Figures 6 and 7 respectively. A zoom in the last pin shows clearly the triple junction with the highest electric field value.

IV. OPTIMIZATION PROCESS USING GWO
The principle of the GWO algorithm [28] is based on reproducing the hunting technique of grey wolves and modeling mathematically their social hierarchy. Four categories can be noticed in wolf population. The α (alpha), β (beta), and δ (delta) wolfs represent the optimal solutions for the optimization problem. The last category is the ω (omega) wolves, which are the followers [28,30]. The mathematical modeling of the steps of the hunting process is given below.
• Encircling: (surrounding the prey while hunting), is modeled by (3) and (4): where t is the current iteration, Xp and X are the positions of the prey and the grey wolf respectively, C and A are the coefficient vectors calculated by (5) and (6): The elements of the vector a linearly decrease from 2 to 0, and r 1 , r 2 are random vectors in [0,1].
where X α , Xߚ, and X δ are the first three best solutions at iteration t, A 1 , A 2 , and A 3 are obtained from (5), and D α , Dߚ, and D δ are obtained from (7).
• Attacking: when the prey representing the optimal solution, can no longer move, the wolves launch the attack, which marks the end of the hunt. This process is mathematically reproduced by decreasing the value of a linearly from 2 to 0.

V. GWO AND FEM COUPLING
The hybrid GWO-FEM algorithm is obtained by following the different steps given in the flowchart of Figure 8. industrial model in order to obtain a shape with a reduced creeapage distance and as a consequence reduced weight of the insulator. The stopping criterion of the hybrid algorithm is 150 iterations [15]. The obtained shape of the optimized insulator string is given in Figure 10. A comparative analysis of the results before and after optimization is established in Table III.   The potential and the electric field distributions after optimization are given in Figures 11 and 12 respectively. The potential and electric field repartition along the creepage distance (from the pin to the cap) on the optimized insulator string are registered in Figures 13 and 14 respectively.

VII. DISCUSSION
From the comparative analysis between the cases before and after optimization (Table III), we can clearly notice the efficiency of the proposed optimization method since the two objectives set in this work, namely the reduction of the electric field and the creepage distance, have been correctly achieved. The developed GWO-FEM hybrid algorithm for optimizing insulator strings is a promissing technique for designing and enhancing the shape and the electrical performance of insulators. The comparative analysis carried out between the reference string and the optimized one in the prensence of a wet pollution shows the following findings : • A decrease of 8% in the electric field at the triple junction of the last insulator connected to the high voltage was recorded. This reduction constitutes a major advantage with regard to the reduction of corona discharges at the surface of the insulator, and consequently a decline of the noise, the light, and the electromagnetic interferences issued from the ionization phenomenon of the air particles surounding the insulator sting. Other positive consequences are the elongation of the insulator lifetime following the reduction of the corrosive effect of the crona discharges and the reduced risk of insulator flashover.
• Creepage length decreased by 25% and the diametral surface of the string insulator by 15.14%, which represent a huge advantage resulting in an important weight decrease and thus improving the economic aspect of the manufacturing of the insulators.
• The 3D study gives a more realistic aspect to the achieved results.
Comparing the obtained results with [23,24], where the objective function (the electric field at the pin region) is predicted using Artificial Neural Networks, big similarity in the obtained shape after optimization is noticed.

VIII. CONCLUSION
The developed GWO-FEM hybrid algorithm provides very satisfactory results, therefore it can be used as a powerful tool for desinging high voltage insulators in general. With this hybrid algorithm, optimal design of the insolator string is reached, satisfying two constraints: an electrical constraint (minimizing the maxium electric field value) and a geometrial constraint (minimizing the weight of the insulator string). The main advantage of the proposed hybrid algorithm is the possibility of combining other constaints that are involved in the flashover of high voltage insulators, such as the dry band formation, NSDD encapsulation, water draining, and also mechanical and thermal stresses can be taken into account, which is possible with the use of the computation interface of Comsol-muliphysics that allows multiphysic studies at the same time and thus permitting the obtainment of an optimal design encompassing all the stresses undergone by the insulator string.