A More Uniform Electric Field Distribution on Surge Arresters through the Optimal Design of Spacer and Fiber Glass Layer

In this study, the optimal design of spacers and fiber glass layer of a metal oxide surge arrester is presented in order to achieve a more uniform electric field distribution, inside and outside the arrester. This is done by using intelligent algorithms and numerical analysis, i.e., Finite Element Method (FEM). The introduced method can be used in order to determine the optimal dimensions of spacers and fiber glass layer so that the electric field distribution is optimized and the lifetime of highly stressed ZnO blocks in the vicinity of HV electrode is increased. In order to verify the results, Differential Evolution (DE) and Particle Swarm Optimization (PSO) algorithms are used.


INTRODUCTION
Metal Oxide Arresters (MOA) have been used extensively in high voltage power systems for providing protection for the insulation devices in power apparatus against dangerous over voltages.The lifetime of these arresters is dependent on their steady state performance.It has been observed in practice that the voltage distribution on the arrester is quite non-uniform.As a result, the discs at the top share a higher voltage and hence higher thermal stresses, than the remaining discs.This leas to a faster electric field aging of the discs at the top.To overcome this problem efforts are generally made to make the electric field distribution on the arrester as uniform as possible.
The transient analysis of MOA is investigated in some studies (He et al., 2003a, b;Csendes and Hamann, 1981;Han et al., 2005;Kumar and Mogaveera, 2002); but the present study focuses on electrostatic analysis of MOA.In MOA, the ZnO blocks placed near HV electrode, are highly stressed by excessive electric fields existed in the vicinity of the HV electrode.In other words, the electric field distribution is non-unifrom along the axis of the Metal Oxide Varistors (MOVs) and causes a faster aging of the varistors placed near the HV electrode.Spacers and Fiberglass Reinforce Plastic (FRP) layers are well known means to uniform the voltage distribution on the arresters.In order to gain a uniform distribution of electric field, proper geometry and size of Spacer and FRP layer is needed.
The electric field and voltage distribution on MOA are analyzed in studies (Vahidi et al., 2004a(Vahidi et al., , b, 2005)).Vahidi et al. (2005), the location of spacers and grading ring is studied also we presented optimal design of grading ring (Aghaebrahimi et al., 2012).However, in this study an algorithm for optimal design of the size and the geometry of Spacers and FRP in MOA to uniform the electric filed distribution along the axis of arrester is presented.The electric field is calculated by Finite Element Method (FEM) based software; then, Differential Evolution (DE) and Particle Swarm Optimization (PSO) algorithms are used to obtain the optimum size of Spacers and FRP layer for a better electric field distribution.As shown in Fig. 1, the design parameters which are to be optimized in order to uniform the electric field distribution, inside and outside of arrester, are P1, P2 and P3, in which P1 and P2 are the distance between up Spacer and down Spacer and the center of the arrester, respectively and P3 is the FRP layer's radius measured from the center of the arrester.
The aim of this study is to have the optimal design for these parts of MOA, using iteration theory and evolutionary algorithm.Electric field calculations are done using FEM and various geometric parameters of the insulation materials are considered in order to achieve optimal distribution of electric field, with a general default form of the surge arrester.In addition, the calculation of the electric field is done by COMSOL Multiphysics software package which has the ability to be connected to MATLAB software package (Farin, 1997).
This optimization method represents a universal method of optimization for various objective functions related to the arrester and other devices (Kitak et al., 2005(Kitak et al., , 2009;;Hesamzadeh et al., 2008).This approach, thus, is not only not limited to the purpose of optimizing the geometry of distribution of the electric field, but can also be performed by regarding the properties of the material used and various types of surrounding insulation materials.The results of Differential Evolution (DE) and Particle Swarm Optimization (PSO) algorithms applied to the case study optimization problems will be shown and it can be seen that the proposed method is effective and acceptable for an optimal field distribution on the column of ZnO pills.

PROBLEM 'S MODEL AND CONSTRAINTS
Objective function: The aim of the proposed optimization algorithm is to obtain the uniform electric field distribution along ZnO blocks.Therefore, the objective function is defined as follows: Objective function = min (E max /E mean ) where E max and E mean are the maximum and average of the electric field existed along arrester axis, respectively.

Problem constraints:
The constraint that must be considered in design of arrester is the dielectric strength of insulation housing.Therefore, the maximum electric field on surface of the porcelain housing must be equal or less than the electric strength of the air, i.e., 2.4 kV/mm.In the other words: kV/mm 4 .2 E max < (2) The rest constraints are defined considering the limits relative to arrester's dimensions.They can be defined as: while, (5) and max 3 3 min 3 where P 1-min and P 2-min are the height of ZnO block and P 3-min is the ZnO block's radius.Due to design considerations, P 1-max and P 2-max are arrester's height.Relationship (5) expressed by Hinrichsen (2001).Also, P 3-max is the radius of the arrester's porcelain housing.

OPTIMIZATION ALGORITHMS
In order to check the accuracy of the obtained results, two different optimization algorithms, Differential Evolution (DE) and Particle Swarm Optimization (PSO), are used.Each particle in PSO or each solution in the initial population of DE will be a candidate for the final solution of the problem.The general steps of the optimization process are as follows: Step 1 : Initializing the population A sample string in defining the initial population is as follows: Step 2 : Evaluating the initial population by FEM Step 3 : Creating the new solutions Step 4 : Constraint handling Step 5 : Selection Step 6 : If the convergence criterion is satisfied exit, else: go to step 3 The flowchart of above steps is given in Fig. 2.
The arrester is modeled and simulated in COMSOL software to calculate the electric field distribution.Considering the initial value of parameters and calculated quantities, new values for parameters are produced by the intelligent algorithm.This process continues until the optimization algorithm converges to the optimum values of Spacers and FRP layer parameters, i.e., P1 to P3.
Whenever the calculated values of P1, P2 and P3 exceed their relative limits, they will be fixed in bounds by a probability of 0.7 and a new feasible solution is created and is replaced by a probability of 0.3.

Differential Evolution algorithm: Differential
Evolution algorithm (DE) is one of the most high speed and accurate and at the same time simplest, algorithms for solving mathematical problems.It was proposed by Storn and Price (1995).In recent years, many studies have been done by DE (Coelho and Mariani, 2006;Chiou, 2009;Bhattacharya and Chattopadhyay, 2010;Duvvuru and Swarup, 2011).This evolutionary algorithm starts the search process by an initial random population.There are three operatives of crossover, mutation and selection and three controlling parameters of population size (NP), scale Factor (F) and Crossover Rate (CR).DE steps are as follows: • Initial population generation: The initial population consists of NP members, created randomly such that each member is in the feasible region.The structure of member i in the problem with dimension D is expressed by: where r 1 , r 2 , r 3 ε[1, NP] are three random unequal integers and F is a positive and real number, which is considered 0.5 in most problems.
• Crossover: The new solution Z i is created by combination of X i and Y i as follows: • Selection: If the fitness of the new solution is better than that of the previous solution, the new solution replaces the old one, otherwise, the previous solution is kept: where fit (.) shows the solution's fitness.
• Stop conditions: The searching process continues until the convergence criteria are satisfied.The iteration number is usually selected as the convergence criterion.
Particle Swarm Optimization: Particle Swarm Optimization (PSO) is a population-based optimization algorithm that proposed by Kennedy and Eberhart (1995).The main idea in PSO algorithm is the modeling and simulation of the group movement and behavior of birds in search for food.Each particle in PSO is considered as a candidate solution for solving the problem in multi-dimensional search spaces.Each particle has two components of X i (current position) and V i (current velocity) in n-dimensional problem search space as follows: where, n is the dimension of the problem and t is the iteration index.
The new position of each particle is created by its current position and its new velocity.Also, the new velocity is produced by four factors, i.e., current velocity, current position, best previous position of the particle (PR best R) and the best position among all of particles in all iterations (GR best R).Therefore, the new velocity is obtained as follows: In Eq. ( 11), pbesR i, j R is the jth dimension of particle i's best position and gbestR j R is the jth dimension of the best position among group's particles.Also, ω is the particle inertia coefficient, ωR max R and ωR min R are the final and initial values of inertia coefficient, respectively, iteris the current iteration, iterR max R is the number of all iterations, cR 1 R and cR 2 R are acceleration coefficients, rR 1 R and rR 2 R are random numbers between 0 and 1 and i and j are the particle and its dimension indices respectively.The new position of the particle is obtained by:

NUMRICAL RISULTS
The proposed method has been applied to the design of the grading ring of a real arrester.The voltage rating of the arrester is 220 kV.All other data are as presented by Hinrichsen (2001).
The dielectric constants of different parts of the arrester are presented in Table 1.All of the electric field values are in the form of per unit, based on the values of the applied voltage.The initial values of design parameters, the range of permissible variation of each parameter and the resultant optimum values, calculated by PSO and DE optimization methods, are presented in Table 2.The initial dimensions of Spacers and Fiber glass layer are those used in conventional design and presented in the catalog.Based on the results, in order to uniform the electric field on ZnO block, the optimal height for the upper spacer and lower spacer is 350 mm (1200-850) and 349.7 mm (1199.7-850)respectively.The 850 mm is the height of ZnO block from the origin of the vertical axis in Fig. 1.Also, the optimum value for P3, which is the radius of the Spacer and is directly related to the FRP layer, is found to be equal to 69 mm and its optimum thickness is found to be equal to 39 mm.
The optimization parameters of DE method, which are the Number of Population (NP), the scale Factor (F) and the crossing constant (CR), are equal to 100, 0.5 and 0.9, respectively.The parameters of PSO method, which are the Number of Particles (NP), the individual learning coefficient C1 and the group learning coefficient C2, are equal to 100, 1 and 1, respectively.Figure 3 shows the convergence processes of the two mentioned optimization algorithms during 100 iterations.
Table 3 presents the minimum values of the objective function (1) calculated by DE and PSO methods and the maximum electric field existing along the arrester axis with the initial dimensions of3 T Spacers and FRP layer and with the optimized dimensions calculated by the proposed method.According to Table 3, the maximum electric field on ZnO varistors with initial parameters is equal to 1.4002 p.u. which can be decreased to 1.1410 p.u. (for DE) by using the dimensions determined for Spacers and FRP layer by the proposed optimization method.Figure 4 shows the electric field contour on the surface of ZnO block for initial parameters and with DE and PSO optimizations.The decrease in the maximum value of the electric field, as a result of the optimization procedure, is quite clear.

CONCLUTION
In this study, the optimal design of the surge arrester's Spacers and Fiber glass layer, regarding the field distribution on the zinc oxide tablet and using PSO and DE algorithms, was studied.The simultaneous use of geometry parameters display, producing a newly developed mesh and numerical calculation methods, with the help of finite element analysis and intelligent algorithms, is found to be reliable.Also, the major task in the arrester's design is its optimization process.Regarding the studied test sample, it has been proved that both DE and PSO algorithms are reliable and completely applicable for modeling the arrester elements and other fast electromagnetic components.
The numerical results show that arrester elements have a direct effect on the voltage gradient and with suitable design, the lowest concentration of electric field on the surface of considered parts (here: ZnO block) can be achieved.Also, the proposed method is a general-purpose method of modeling not only for the surge arrester, but also for other power system insulation elements.

Fig. 2 :
Fig. 2: The flowchart of optimal design of a 220 kV MOAThen, the obtained results are transferred to MATLAB in order to evaluate the objective function (1).Considering the initial value of parameters and calculated quantities, new values for parameters are produced by the intelligent algorithm.This process continues until the optimization algorithm converges to the optimum values of Spacers and FRP layer parameters, i.e., P1 to P3.Whenever the calculated values of P1, P2 and P3 exceed their relative limits, they will be fixed in bounds by a probability of 0.7 and a new feasible solution is created and is replaced by a probability of 0.3.

Table 1 :
Relative permittivity of various parts of arrester Relative permi.

Table 2 :
Optimization values of design parameters, calculated by optimization algorithms