1. Introduction
As the main device of power systems, the power transformer is essential to the safe and reliable operation of the whole electrical network [
1].The insulation structure determines the electric field distribution; thus, the optimization of the main insulation structure is of great importance to ensure the safe and stable operation of the power transformer [
2,
3].With the increase of voltage levels, more and more attention has been paid to the electrical insulation problem of the power transformer. However, the insulation structure of a transformer is particularly complex, which makes the electric field distribution non-uniform. Therefore, to confirm the rationality and reliability of the main insulation structure of the transformer, it is necessary to test, calculate, and analyze the electric field distribution [
4].
In engineering applications, electric field analysis methods can be divided into two types, the analytical methods and the numerical methods. A few simple problems may be solved by the analytic method; but in the majority of cases, an analytical method simply does not exist or it is extremely difficult to obtain.As a numerical method, the FEM (finite element method) has been applied in theelectric field calculation [
5,
6,
7]. In the electrostatic ring structure optimization [
8], the traditional finite element method (FEM was used to analyze the influence of each variable on the electric field intensity and summarize its changing rule through several specific points. The FEM generally adopted the inference method [
8,
9], and the optimal values of the parameters were obtained by summarizing the changing law of each variable. Although it is possible to get the result, it cannot prove that the final result is the optimal solution, and it needs to constantly modify the model parameters for modeling analysis, which increases the calculation time.
In addition to the FEM, the structure optimization methods roughly fall into two types. One is to establish the function expression between the optimization objective and the variables by using sensitivity analysis methods [
10].Sometimes the selection of step length is involved in the sensitivity analysis methods and the accuracy cannot be guaranteed. Moreover, the sensitivity formulas are difficult to derive, and the process is rather complicated [
11]. When the analytical expression for the relationship between the optimization variables and optimization objectives cannot be found, the other type of methods, the global optimization methods, or the response surface methodology(RSM)combined with FEM can be used for the optimization design. Global optimization methods such as the genetic algorithm [
12,
13,
14,
15,
16] and the particle swarm optimization algorithm [
8,
17,
18,
19] need to call a large number of modeling and optimization programs repeatedly, which significantly affect the efficiency of optimization. Most importantly, without appropriate fitness functions the local search ability may become worse and the search efficiency may be reduced.
The RSM constructs an approximate function expression between the objective function and variables, and visualizes the objective function that is originally implicit. The repeated modeling and simulation can be avoided; thereby, the efficiency of the optimization can be significantly improved [
20]. As a mathematical and statistical optimization method [
21,
22], the RSM has been applied in many aspects. References [
23] and [
24] combined the RSM with the genetic algorithm and indicated that this method could reduce the number of experiments and solve the problem of fitness function selection. Reference [
25] used the RSM to optimize the armature structure, and proved that this method had high feasibility, which provided an effective method for armature structure improvement. References [
26,
27,
28] posed the methods for the optimization of an oil-immersed transformer cooling system and improved the design of the permanent magnet motor using RSM. In Reference [
29], the RSM was employed to optimize the structure of the motor and the experiment on the prototype was carried out to verify the efficiency and reliability of the RSM. Reference [
30] proposed a method by combining the RSM with the geometric feature charge simulated method to get the electric field; by using this method, the workload could be reduced and the computing time could be shortened as well. In addition, the RSM has been successfully applied in the mechanism of brake squeal [
31], biochemistry [
32], medicine [
33], ships [
34], machine design [
35], and other fields.
According to the research above, the RSM demonstrates the ability to deal with many optimization problems with good speed and convenience on the premise of ensuring accuracy and efficiency. However, there is no relevant research on its application to the electrostatic ring structure optimization design of the power transformer.
In this paper, the RSM was applied to optimize the electrostatic ring structure of a 500 kV power transformer. The power transformer model was built with ANSYS parametric design language (APDL) [
36,
37] and the response surface model was constructed according to the optimization parameters selected by the Taguchi method [
38,
39].Then, the fitting degree and prediction ability of the response surface model were analyzed to get the optimal result. Furthermore, we compared the optimization result with that of traditional FEM, and the accuracy and validity of the RSM were proved.
3. Experimental Process and Analysis
As can be seen in
Section 2.1, the maximum electric field intensity emerged at the upper right corner of the electrostatic ring. Therefore, we can select the parameters related to the electrostatic ring as the variables to optimize. The structure and parameters of the electrostatic ring are shown in
Figure 4.
Due to the existence of the pressboard, for easy analysis,
H is simplified as
h, the distance from the top surface of the electrostatic ring (without insulation layer) to the first pressboard above the electrostatic ring. The relationship between the two arcs of the electrostatic ring can be expressed by Equation (6).
Among them, the coordinates of the center of the two arcs are (x1, y1) and (x2, y2). Values T1 and T2 are the ordinate values of the upper and lower surfaces of the electrostatic ring (excluding the insulating layer). As can be seen from the above equation, R2 and θ2 can be derived from R1 and θ1. Therefore, the four optimization variables R1, θ1, R2, and θ2 can be simplified into two variables, namely R1 and θ1.
The optimization process is demonstrated in
Figure 5.
3.1. Taguchi Experiment
Through the above analysis, the Taguchi experiment was carried out with
R1,
θ1,
h,
s, and
w as variables to be optimized. Taking the median values of the variables on both sides and their adjacent values as the impact factors, the impact factors of the optimized parameters are listed in
Table 2.
The orthogonal table
L9(3
5) is obtained by the Taguchi method, where
L represents an orthogonal matrix; 9 stands for the number of trials; 3 is the number of times each variable is taken, and 5 symbolizes the number of optimization parameters. According to the orthogonal table, the maximum electric field intensity can be achieved by the mutual call between MATLAB and ANSYS, as shown in the
Table 3.
From the results obtained by
Table 3, the proportion of each parameter is calculated by Equation (7).
where
xi represents
R1,
θ1,
h,
s, and
w;
EE is the proportion of each optimization parameter;
mxi(
Ei) is the average value of
Emax under the
ith impact factor of x; and
m(
E) is the average value of
Emax. The proportion of each optimization parameter is listed in
Table 4.
It can be seen from
Table 5 that
R1,
h,
s, and
w have greater influence on the maximum electric field intensity,while
θ1 can be eliminated from the impact variables. The Taguchi method effectively reduced the workload of the numerical analysis of the subsequent response surface. In the following experiment, for the convenience of analysis, the electrostatic ring was divided into two arcs; both of them were 90 degrees. The values
R1,
h,
s, and
w were selected as experimental variables.
3.2. Experimental Data
After determining the experimental variables, the appropriate range of values was selected for the response surface experiment. There are four optimization variables in this experiment, so α=2. The design and combination of the experimental sample points were carried out by the CCD method, as shown in
Table 5.
Table 6 is the experimental data results of the sample points which were obtained by calling each other between ANSYS and MATLAB.
The last two columns of
Table 6 are the number of elements and nodes (degrees of freedom) of the local finite element model in
Figure 1b. According to statistics, it took only 61.8 seconds to complete the above 30 CCD experiments. Without the CCD method, 625 RSM experiments would be required. Thus, by using CCD method, the optimization efficiency was greatly improved.
Referring to the above data, the coefficients of the response surface model were obtained by Equation (5). We can get the initial response surface model from
Table 7.
3.3. Analysis of Variance
The ANOVA table summarizes the results of the variance analysis, as shown in
Table 8.
As shown in
Table 8, the
p-value of the regression model was less than 0.0001, which means that the choice of the quadratic polynomial model is reasonable, and the relationship between the maximum electric field intensity value,
y, and the regression equation is extremely significant. Thus, the null hypothesis,
H0, is not valid and can be rejected. The
p-value of all linear terms were less than 0.0001, indicating that each variable in
R1,
h,
s, and
w is an extremely significant factor affecting the optimization objective. Except that the interaction between
R1 and
s was significant, the
p-value of each of the other two was greater than 0.05, demonstrating the interaction between the two has little effect on the response. The
p-values of the quadratic terms
R12 and
s2were less than 0.0001, and the significance was higher than the other two, which is an extremely significant factor. In addition,
is 0.9841, which is very close to 1. The model can explain 98.41% of the test changes and better reflect the law of data changes with less experimental error and better fitting. Furthermore, from the confidence interval listed in
Table 8, the significant items are also
R1,
h,
s,
w,
R1s,
R12, and
s2, for the upper and lower bounds of their confidence intervals do not contain zero.
3.4. Diagnostic Analysis
To verify the applicability of the model, the diagnostic analysis was generally made by making the regression analysis and giving some diagnostic plots.
The design of the normal probability plot made the cumulative normal distribution a straight line. In
Figure 6a, the points on the normal probability plot were approximately distributed in a straight line, which indicates the accuracy of the data for calculating the electric field intensity.
As can be seen from the residuals versus run plot in
Figure 6b, the residuals of runs were distributed within the prescribed appropriate range, which implies that choosing the quadratic polynomial as the response surface function is particularly suitable. In
Figure 6c and
Table 9, there was a good correlation between the predicted value and the actual value, which indicates that the polynomial model has a good predictive ability. In a word, the response surface model adopted in this study is acceptable.
3.5. 3-D Response Surface Analysis
For a better demonstration of the influence of the interaction between two variables, the 3-D plots were adopted. If the interaction has a significant impact on the response surface, the variation of 3-D plot will be accordingly large.
Figure 7a shows that when
R1 and
s interact, if we fix one of the variables and increase the other variable, the maximum electric field intensity will decrease. As can be seen from the trend of the surface graph, the maximum electric field intensity decreased faster with
R1 than with
s, indicating that
R1 has a greater influence on the response value. Comparing
Figure 7a with
Figure 7b, the response surface of
Figure 7b fluctuated slightly, which means that the change of the interaction term
hw has no significant effect on the maximum electric field intensity. Similarly,
hs and
sw are not significant factors either.
As can be seen from
Figure 7c,d, compared with
s and
w, the change of the maximum electric field intensity was larger with the change of
R1. The maximum electric field intensity increases with the increase of
w, which has negative effects on the model. Therefore, in the structural design process, a smaller starting position
w should be selected as far as possible within a reasonable range. From the variation range of the two figures, the interactions between
R1 and
h,
R1, and
w may be a significant factor affecting the maximum electric field intensity. However, since their
p-values in
Table 8 are little larger than 0.05,
R1h and
R1w can be considered insignificant factors.
Through the above analysis, the insignificant factors, such as
R1h,
R1w,
hs,
hw,
sw,
h2, and
w2, are removed, and the final response surface equation with a higher prediction ability is shown in Equation (8).
4. Results and Discussion
It is an important goal of the response surface optimization to get the optimal results with the appropriate values assigned to each variable. Equation (8) is the quadratic response surface model with single objective optimization, which can be solved quickly by quadratic programming. By comparing the maximum electric field intensities on the electrostatic ring before and after optimization in
Figure 8, the decrease of the maximum electric field intensity can be observed more intuitively. The result shows that the maximum electric field intensity has been significantly reduced.
In order to verify the validity of the RSM, we compared the result of RSM with that of traditional FEM. The FEM taking much time and requiring many experiments, to ensure the fairness of the experiment and save the calculation time, thirty experiments were carried out by using the FEM. As can be seen from
Table 10, the results obtained by RSM and FEM were consistent, while the time the of RSM experiments was much less, which demonstrates the effectiveness of RSM and also shows the high optimization efficiency and good prediction ability of this method.
In
Figure 9, a comparison of the variations of variables predicted by the FEM and those actually obtained by the RSM was made. It can be observed that the variations of variables in this paper are monotonic, so FEM can obtain the optimal results by inference in this special case. However, this method is no longer applicable when the variables are not monotonic. The RSM has a wider range for applications and the result obtained by it is more rigorous and convincing.
5. Conclusions
In this paper, the response surface methodology, Taguchi method, and ANSYS parametric modeling language were combined to optimize the electrostatic ring of a 500 kV power transformer. The APDL extracted the maximum electric field intensity automatically and the mutual calls between MATLAB and ANSYS solved the problem of repeated manual modeling and meshing in traditional methods. To avoid the increased calculating workload of developing experiments aroused by irrelevant variables, the Taguchi method was utilized to filter the variables before making response surface experiments, by which the optimization process was effectively simplified. The experimental points were constructed by the experimental design method of CCD, and the response surface model was then obtained. In order to ensure the accuracy and predictability of the model, the response surface model was analyzed by ANOVA, diagnostic analysis, and the significance test of regression, and the final response surface equation was determined. Comparing the optimization results with the traditional finite element method, on the premise of ensuring accuracy, the proposed method in this paper has a higher optimization efficiency.
In general, the structure of the electrostatic ring is optimized more systematically and completely through the Taguchi method and response surface experiments. The feasibility and efficiency of this method were verified, which provides a new and more systematic, fast, and efficient way for the optimization of the transformer insulation structure.