Electric Field Distribution on Zinc Oxide Pills in Gapless Surge Arresters Using Finite Element Method and Evolutionary Optimization Algorithms in HVAC Systems

Abstract

This article proposes a new method for the optimal design of zinc oxide arresters based on electric field distribution on the zinc oxide column using smart algorithms and finite element analysis. This design prevents premature aging of zinc oxide tablets, especially the initial and final tablets of the column, which have a higher voltage gradient than other parts of the arrester, and subsequently increases the sustainability of the network. The spacer height, fiberglass layer thickness, and grading ring diameter and its location were taken as the problem variables. The surge arrester was designed in AC/DC mode and two-dimensional symmetry using the COMSOL Multiphysics package. For the first time, rational Bezier curves were also used for the arrester design. This paper presents an optimization approach that combines and dynamically links the electrostatic modeling process and the particle swarm optimization (PSO) and differential evolution (DE) algorithms. The proposed approach is a general method that can be used in the design of high-voltage equipment. The results showed that the lifetime and reliability were increased by reducing voltage variations in the ZnO column. Therefore, maintenance cost and implementation of a surge arrester would be reduced. Finally, the external surface of the porcelain housing was placed under the influence of uniform distribution of electric field.

1. Introduction

Surge arrester is a protective device that damps overvoltages caused by lightning and prevents damage to the network and its equipment. On the one hand, damage to the system equipment imposes heavy costs on the electricity industry. On the other hand, it interrupts the supply of electrical energy needed by consumers and has an impact on the environment and sustainability of electricity delivery. Major disorders incurred by metal oxide surge arresters include electric and thermal disorders in static mode. An advanced type of surge arrester, known as metal oxide arrester (MOA), is made up of pills of oxide metals, such as zinc oxide (ZnO), which have nonlinear properties. MOA does not have an air gap and is always subject to the network voltage, so a leakage current passes through it and gradually heats it. If the arrester’s shape and size and the materials of its various parts are not designed properly, they will be isolated by internal or external circuit breakers in the case of overload on the varistors and the arrester will be removed from the circuit, otherwise, the sustainability of the network will be disrupted [1,2]. Designers are interested in identifying and considering important factors that impact the maximum electric field intensity of MOAs so that excessive inside and outside potential gradients can be avoided. Without this, serious damage to the insulation system of the arrester is possible.

While most studies have considered the design of the transient model of oxide pills [3,4,5,6,7], this study focuses on the electrostatic analysis of ZnO surge arresters.

Due to the height of high-voltage MOAs and the resulting stray capacitances, it is obvious that the voltage distribution along MOAs is not uniform. This means that the upper MO blocks are more stressed than the lower MO blocks. Additionally, there are permanent superimposed alternating voltages that have a higher frequency than the grid frequency. Both higher voltage and higher frequency lead to high stress on the MO blocks because of increased power dissipation. In these types of arresters, the pills located at the bottom and the top of the resistor column are subject to more wear and early aging because of the way they are located in the network and the concentration of electric field on them [8,9].

In order to achieve uniform distribution of the electric field along the arrester, the geometric shape of different parts of the ZnO surge arrester, including the spacers, fiberglass layer, grading ring, etc., must be considered [10]. The authors of [3,11] present suitable locations of grading rings in two 220 and 400 kV arresters.

This study aimed to develop an optimal design of ZnO surge arresters using evolutionary algorithms and finite element method (FEM) numerical analysis. For this purpose, parts of ZnO were optimally designed to have a great impact on the nonuniformity of the electric field in the ZnO length. Due to the nonlinearity of the problem space, it was necessary to use a software package to calculate electrical and magnetic parameters. Therefore, in this study, COMSOL software (Version 5.2) was used, which has the ability to link dynamically with MATLAB software (Version 2019b) [12]. The design parameters (P1 to P5) are shown in Figure 1.

The proposed method can be used as a general method for the optimal design of high-voltage equipment, such as bushings, switchgears, and electric machines, and even the design of other electrical equipment, such as photovoltaic systems with different purposes. In this study, the rational Bezier curves were used for contour modeling, and DE and PSO algorithms were used as optimization algorithms. The optimization process started with the generation of initial parameters by intelligent algorithms. Using these parameters, the electric field modeling and calculations were performed in the COMSOL Multiphysics environment, and the data required to calculate the objective function were sent to evolutionary algorithms in MATLAB. In the next step, the parameters were generated by the mentioned algorithms, and this process continued until the optimal contour was achieved.

2. Proposing Objective Function and Constraints of Problem

2.1. Objective Function of Problem

In this optimization problem, the aim is distribution of electric field on the surface of ZnO pill column, which can be described as follows:

$$Objective function=min\left(\raisebox{1ex}{$max\left(E_{tot}\right)$}\!\left/ \!\raisebox{-1ex}{$average\left(E_{tot}\right)$}\right.\right)$$

(1)

where $E_{tot}$ is the total electric field.

2.2. Constraints of Problem

The problem constraints can be divided into two categories: electrical and geometric. The electrical constraint includes the insulating strength of the air surrounding the arrester. If in the solution search process, the maximum value of the surface electric field of the lightning arrester exceeds the strength of the air, i.e., 2.4 kV/mm [13,14], the design parameters are unacceptable, where:

$$max\left(E_{tot}\right)\le 2.4$$

(2)

On the other hand, the geometric constraints are related to two limitations of the arrester height and the range of design parameters, which are as follows:

$$P1+\left|P2\right|\le 2400$$

(3)

$$P_{min}<P<P_{max}$$

(4)

Equation (3) shows the length of the arrester, which should not exceed the limit, while Equation (4) presents the range of each parameter. $P_{min}$ and $P_{max}$ are the lower and upper limits of the design parameters, respectively. It should be noted that variable P1 is used as the absolute value. The parameters P1, P2, P3, P4, and P5 are related to the thickness of the positive electrode, thickness of the negative electrode, thickness of the fiberglass layer, radius of the gradient ring, and height of the gradient ring, respectively.

3. Program Description and Model Evaluation

3.1. Electric Field Calculation

As the objects are unconventional geometric shapes, an analytical solution of the electric field is difficult or impossible. Therefore, the finite element method was utilized to calculate the electric field on the surface of ZnO pill column. There are many software to analyze problems with FEM. However, according to the nature of the problem described in Section 4 (utilizing software output in MATLAB version R2019b), the software should be linked to MATLAB m-file. Therefore, the COMSOL Multiphysics package (Version 5.2) was used.

In this study, the procedure outlined in Figure 2 was followed to calculate the total electric field, which is described as follows [15]:

Geometry: 2D axisymmetric geometry of the surge arrester can be drawn with a rational Bezier curve. The geometric shape is composed of a number of subdomains, which included a positive electrode, a negative electrode, a ZnO pill column, a fiberglass layer, spacers, porcelain housing, and external insulation air. Each subdomain was composed of coordinate (r, z) and weight coefficients (related to Bezier curve). This section analyzes the specified geometric objects and separates the resulting geometric object into the specified class.

Analyzed geometry: The geometry that COMSOL Multiphysics (Version 5.2) uses for the actual finite element analysis is called the analyzed geometry, where a name is determined for each subdomain. The geometry section defines the model’s analyzed geometry.

Initial mesh: This section creates a triangular free mesh on the analyzed geometry. It provides a number of input properties that can be used to control the simulated model. Some examples of the properties are as follows:

(1)

The maximum mesh size and curvature mesh size;

(2)

The growth rate for the mesh size (away from small details);

(3)

The resolution of narrow regions;

(4)

Geometry scaling before meshing;

(5)

The distribution of mesh elements on selected edges;

(6)

Determination of which domains of the geometry to mesh;

(7)

Mesh element type for each domain (available only for subdomains in 2D).

Many of these properties are available both globally and locally.

Refine mesh: This section can create a finer mesh. The refine mesh is only used in order to make the solver refine the longest edge of an element, to generate a completely new mesh, and to make the solver refine elements in a regular pattern. The finite element model contains 11,213 elements.

Application mode: The application modes contain predefined equations for a variety of physics applications, and it offers a convenient alternative way of setting up PDE problems. Thus, this section specifies the application mode class (electrostatics), the application module (AC/DC), and the application mode type (2D axisymmetric). According to the problem application, the structure specifies equations, boundary conditions, and relative permittivity in the FEM.

ODE setting: This specifies ordinary differential equation (ODE) and other scalar variables, equations, and initial conditions independent of the position.

Multiphysics: This section looks at the application structure within the FEM structure and updates the FEM structure. It is also used to combine the application modes to a composite PDE system.

Extended mesh: The extended mesh takes the model specification and puts it in a suitable form for the solver. This section converts the standard syntax to element syntax. The result is taken together with the specified element syntax. It also determines a partition of the domains, and each domain contains the same variables, equations, and constraints. Similarly, the mesh elements are partitioned. Within each mesh element group, the mesh elements are of the same type, and the same shape functions are used for the variables. The extended mesh part collects all the node points that are introduced when using higher-order elements and all defined variables and checks for name collisions. Generally, this section extends the finite element mesh to the desired element types.

Solve problem: This section detects whether a problem is linear or nonlinear and automatically selects the appropriate solution method. The default nonlinear solver strategy is a fully coupled damped Newton iteration. For this problem type, a segregated iteration gives faster and more reliable convergence and uses less memory.

Output: This part is used in order to compute the electric field value in the boundary of the ZnO pill column.

The proposed model is presented based on the electrostatic model. Therefore, the electrostatic field (E) can be defined using the gradient of the electrostatic potential (V) [16]:

$$E=-\nabla V$$

(5)

From Gauss’s law:

$$\nabla ·V=\nabla ·\epsilon E=-{\rho }_s$$

(6)

Substituting Equation (5) into (6) gives the following:

$$\nabla ·\mathsf{\epsilon }\nabla V=-{\rho }_s \to \mathsf{\epsilon }{\nabla }^{2}V+{\rho }_s=0$$

(7)

where ρ_s is free charge density, and ε is the permittivity. In general, free charges do not exist along the insulator surface (i.e., ρ_s =0); then, Equation (7) can be rewritten in the following form:

$${\nabla }^{2}V=0$$

(8)

The voltage is calculated by the Laplace Equation (8).

3.2. Optimization Algorithms

Because the problem space is large, it is necessary to use a search engine. In this problem, to obtain the relative optimal point, the DE algorithm was used, which has shown its ability in solving various problems of the power system [17,18,19,20]. In order to validate the presented method, the PSO algorithm was used, which is always known as a reliable algorithm [21,22]. The DE and PSO algorithms are explained below.

3.2.1. Differential Evolution Algorithm

The differential evolution (DE) algorithm has one of the highest speed and accuracy and at the same time is one of the simplest algorithms for solving mathematical problems. It was proposed by Storn and Price in 1995 [23]. In recent years, many studies have been carried out using DE [24,25,26,27]. 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). The DE steps are as follows:

(a)

Initial population generation

The initial population consists of NP members that are created randomly so that each member is in the feasible region. The structure of member “i” in the problem with dimension D is expressed as follows:

$$X_i=\left(x_{i,1},x_{i,2},\dots ,x_{i,D}\right)$$

(b)

Mutation

A new solution of Y_i in each iteration t is created as follows:

$$Y_i\left(t\right)=X_{r3}\left(t\right)+F\left(X_{r1}\left(t\right)-X_{r2}\left(t\right)\right) \hspace{1em}i=1,2,\dots ,NP$$

(9)

where $r_1,r_2,r_3\in [1, NP]$ are three random unequal integers, and F is a positive and real number, which is considered 0.5 in most problems.

(c)

Crossover

The new solution Z_i is created by combining X_i and Y_i as follows:

$$z_j\left(t\right)=\left\{\begin{array}{c}{y}_j\left(t\right) \hspace{1em}\hspace{1em}if rand\le CR or j=jrand\hfill \\ x_j\left(t\right) \hspace{1em}\hspace{1em}\hspace{1em}otherwise\hfill \end{array}\right.$$

(10)

(d)

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.

$$X_i\left(t+1\right)=\left\{\begin{array}{c}{Z}_i\left(t\right) \hspace{1em}\hspace{1em}if fit\left(X_i\left(t\right)\right)\ge fit\left(Z_i\left(t\right)\right)\hfill \\ X_i\left(t\right) \hspace{1em}\hspace{1em}\hspace{1em}otherwise\hfill \end{array}\right.$$

(11)

where $fit(.)$ shows the solution’s fitness.

(e)

Stop conditions

The searching process continues until the convergence criteria are satisfied. The iteration number is usually selected as the convergence criterion.

3.2.2. Particle Swarm Algorithm

Particle swarm optimization (PSO) is a population-based optimization algorithm proposed by Kennedy and Eberhart. The main idea in the 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 multidimensional search spaces [28,29].

Each particle has two components, namely, X_i (current position) and V_i (current velocity), in the n-dimensional problem search space as follows:

$$\begin{array}{c}{X}_i\left(t\right)=\left(x_i^1\left(t\right),x_i^2\left(t\right),\dots ,x_i^n\left(t\right)\right)\\ V_i\left(t\right)=\left(v_i^1\left(t\right),v_i^2\left(t\right),\dots ,v_i^n\left(t\right)\right)\end{array}$$

(12)

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. In addition, the new velocity is produced by four factors, i.e., current velocity, current position, best previous position of the particle ($P_{best}$), and the best position among all of the particles in all iterations ($G_{best}$). Therefore, the new velocity is obtained as follows:

$$v_{i,j}=\omega .v_{i,j}+c_1{r}_1\left(\bullet \right)\left[{pbest}_{i,j}-x_{i,j}\right]+c_2{r}_2\left(\bullet \right)\left[{gbest}_j-x_{i,j}\right]$$

(13)

$$\omega ={\omega }_{max}-\frac{{\omega }_{max}-{\omega }_{min}}{{iter}_{max}}\times iter$$

(14)

In Equation (13), $pbes{t}_{i,j}$ is the jth dimension of particle i’s best position; $gbes{t}_j$ is the jth dimension of the best position among the group’s particles; $\omega$ is the particle inertia coefficient; ${\omega }_{\max }$ and ${\omega }_{\min }$ are the final and initial values of inertia coefficient, respectively; iter is the current iteration, $ite{r}_{\max }$ is the number of all iterations, C1 and C2 are acceleration coefficients; r₁ and r₂ 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 as follows:

$$x_{i,j+1}=x_{i,j}+v_{i,j}$$

(15)

3.3. Problem Steps

Each particle or each solution in the initial population in PSO and DE, respectively, has the potential to be a final solution of the problem. The program process is as follows:

Step 1: Production of the initial population

The initial population is defined as a simple string as follows:

P1

P2

P3

P4

P5

Step 2: Evaluation of the initial population by COMSOL Multiphysics software package;

Step 3: Creation of the new population;

Step 4: Satisfying constraints;

Step 5: Election;

Step 6: If the convergence condition is established, exit, otherwise, go to step 3.

The program flowchart is seen in more detail in Figure 3.

A model was simulated through linking MATLAB and COMSOL Multiphysics in COMSOL, and the electric field characters were calculated by finite element analysis. The required parameters for objective function calculation were sent to MATLAB. In the next step, the optimization algorithms were used to calculate the new parameters using the objective function, the initial values of the random variables, and the calculated parameters resulting from it. This process was repeated until an optimal model was obtained to satisfy the objective function.

3.4. Constraint Handling

P1 was created randomly, and P2 was generated according to P1 for satisfying the maximum MOA length.

$$P2=rand\left(P\mathrm{1,2.4}\right)$$

(16)

To satisfy Equation (4), the solutions that violate the up/down bounds were fixed in bounds by a probability of 0.7, and a novel possible solution was developed and exchanged by a probability of 0.3.

4. Numerical Results

4.1. The Studied System

Simulation of the design of a sample arrester was carried out to show the abilities of the proposed method. The studied arrester was a 230 kV ZnO arrester, the data of which are presented in [30].

Table 1. Relative permittivity of various parts of MOA used in this design [31].

Relative Permittivity	ZnO Pills	Porcelain Housing	Spacer	FRP	Electrode	Grading Ring
εr	150	3.6	103	4.6	106	103

shows constants of the relative permittivity of different parts of the ZnO surge arrester. All the field magnitudes were based on a one per unit applied voltage, and negative electrode was considered to be the ground. The different values of design parameters, including initial, min, and max, and the optimum values of DE and PSO algorithms for each of them with a population (NP) of 150 and 100 iterations are shown in

Table 2. Optimized values of design parameters.

Parameter	Min (mm)	Max (mm)	Init. (mm)	Optim. DE (mm)	Optim. PSO (mm)
P1	861	1539	900	1163.6	1351.6
P2	861	1539	900	1228.1	1392.1
P3	40	79	50	71.3	73.0
P4	221	500	260	221.1	344.6
P5	500	1539	830	1091.6	860.2

. All sizes are in millimeters. As shown in Table 2, sufficient and permissible domains existed for search algorithms to achieve the absolute optimum. The numerical studies were implemented in a PIV computer with dual-core E2180 CPU with 2 GB RAM.

The minimum and maximum values of the variables shown in Table 2 were selected as follows:

• Based on 40% tolerance of the true values;
• No overlap in the control area with surrounding areas.

In other words, the initial aim was to achieve almost 40% tolerance of the selected range of variables. The control area should not enter the surrounding areas because the electric field cannot be calculated in this state. If the subdomains overlap and the bond of every subdomain enters the surrounding subdomain, the extended mesh part will incur an error and the PDE equation cannot be solved to obtain the electric field. Thus, the minimum or maximum values were selected in such a way that it did not overlap with the other areas. For example, the minimum of variable P1 from the spacer was selected as the maximum value of the length of the positive part of the ZnO pill column.

4.2. Sensitivity Analysis

To illustrate the effects of DE and PSO parameters on the results, sensitivity analysis was performed.

Table 3. Comparison of results for different values of optimized parameters for DE.

NP	F	CR	Objective Function Value	Iteration
150	0.5	0.9	1.8476	100
125	0.4	1	1.8476	100
75	0.6	0.6	1.8495	100
25	0.3	1	1.8472	100
10	0.5	0.8	1.8483	100
8	0.7	0.9	1.8543	100

shows different optimum values of the objective function for the DE algorithm for different values of the crossing constant (CR), the scale factor (F), and the initial number of population (NP).

Table 4. Comparison of results for different values of optimized parameters for PSO.

NP	F	CR	Objective Function Value	Iteration
150	1	1	1.8477	100
135	0.8	1	1.8493	100
100	0.9	0.9	1.8476	100
95	0.9	0.7	1.8488	100
35	0.7	0.8	1.8520	100
9	1	0.9	1.8595	100

shows the PSO results with different values for the learning coefficient of individual (C1), group (C2), and number of particles (NP). According to these results, a run with NP = 25, F = 0.3, and CR = 1 with objective function value of 1.8472 gave the best solution (Table 3). The objective function value of 1.8595 with NP = 9, C1 = 1, and C2 = 0.9 did not have a global optimum value (Table 4). Figure 3 and Figure 4 show the convergence graphs for Table 3 and Table 4, respectively. Figure 5 displays the DE and PSO convergence graphs with NP = 25, F = 0.3, and CR = 1 for DE and NP = 100, C1 = 0.9, and C2 = 0.9 for PSO. As can be seen from Figure 5, the objective function value in the DE algorithm was better than that of PSO.

As shown in Figure 6, the maximum voltage gradient, which occurred at the beginning and end of the arrester oxide tablets, with the initial values of the parameters was 1.2850. The optimization with NP = 25, F = 0.3, and CR = 1 for the DE algorithm and NP = 100, C1 = 0.9, and C2 = 0.9 for the PSO algorithm led to 1.1237 and 1.0783, respectively. With this design approach, the premature wear of the pills exposed to electric field concentration can be avoided. The CPU time of the DE and PSO algorithms were 17 and 18 h, respectively, for population size 150 and iteration number 100.

In addition to electric field distribution on the surface of the ZnO column, field distribution on the external surface of the porcelain housing was analyzed. Figure 7 and Figure 8 shows electric distribution of good uniformity along the leakage path for porcelain housing before and after optimal design.

5. Conclusions

In this study, using the DE algorithm, the optimal design for an ZnO surge arrester was developed. The performance of this algorithm was compared with the PSO algorithm in different examples. The optimization of different parts of the arrester, including the thickness of the fiberglass layer, length of the arrester, thickness of the spacers, and position of the grading ring, were investigated. In order to prevent premature aging of the end and initial ZnO pills due to high-voltage gradient, the distribution of the electric field on the surface of the pill column was considered as the design goal. In the optimal contour design, the contour between two points was replaced by a number of Bezier curves. The number of curves and the degree of curvature were selected according to the degree of change between the two points in the given problem and the required accuracy. In addition, the DE and PSO algorithms were used to optimize and modify parameters of the rational Bézier curves. The obtained results showed the optimal performance of the presented method.

The simulation results illustrated that the maximum field on the surface of ZnO pills changed from 1.285 to 1.0988 p.u., while the average field changed from 0.6337 to 0.5946 p.u. Therefore, with this design, the maximum and average fields were reduced by 14.49 and 6.17%, respectively. It was observed that different parts of the arrester had a significant effect on the sudden change of the electric field, and it was possible to achieve the minimum voltage gradient on the column of ZnO with an accurate and optimal design. The maximum field on the external surface of the porcelain housing also changed from 3.833 to 2.598, representing a decrease of 32.22%.

This study found an optimal value of arrester design parameters to achieve well-distributed electric stress with the lowest possible maximum value. With this design, high electric fields near the terminal electrodes will be suppressed by the arrester, thus reducing premature aging and improving the reliability and sustainability of the power system. This method can be used for the optimal design of other high-voltage equipment as well as for new designs in other fields, such as antenna design, mechanical equipment, and civil engineering.

Finally, in future, our research team intends to present an analysis of electrical aging and the network outage rate caused by it under pollution conditions in the form of mathematical relations and laboratory tests.