Shielding Strategies for Electric Field Minimization in Current Transformers: A Finite Element Combined Design of Experiment Approach

Abstract

The internal electric field distribution is a key factor to characterize the partial discharge and voltage resistance of current transformers (CTs), and the conical current transformer has a special structure with complex electric field distribution. Therefore, it is more necessary to study, analyze and optimize the electric field distribution of CTs in the design process. In this paper, a new type of current transformer is taken as the research object. A simulation model is established by using finite element software, and the electric field distribution under the test of power-frequency withstand voltage is investigated. Then, a scheme is proposed, and the response surface is used to optimize the scheme. Experimental verification is conducted to validate the results. In this study, the finite element combined design of experiment approach can serve as a rapid and efficient approach for identifying an optimal solution to minimize the electric field strength within a current transformer.

1. Introduction

Partial discharge is a common issue with current transformers (CTs), which can lead to changes in the electric field distribution and molecular structure of the insulation. This deterioration in insulation performance can even result in breakdown [1]. To prevent such phenomena, it is important to analyze the electric field distribution and maximum electric field in the transformer, which can provide a theoretical basis for optimizing the structure of the transformer [2,3]. Recent years have seen an increase in the number of electric field analyses and studies conducted on $S{F}_6$ CTs with high voltage levels [4,5]. However, there has been a relative lack of studies on the electric field analysis of poured CTs. Conventional mathematical expressions is typically used to analyze very simple geometries [6,7]. This methodology necessitates the involvement of engineers with significant theoretical knowledge and practical experience in engineering, often resulting in challenges when designing and analyzing multiple operating points.

Recently, in order to determine the optimal conductive rod diameter for a poured CT, Dong utilized finite element simulation software to calculate the electro-thermal coupling field distribution characteristics of the CT under different currents, temperatures, and diameters of conductive rod. However, the study only simulated four groups of conductive rod diameters to derive the optimal diameter, without conducting a more comprehensive parameter search [8]. Jia conducted a simulation of the electric field distribution for a 400 kV DC CT using finite element analysis. The study also involved calculating the electric field distribution and dense areas at different voltages. The results obtained from the simulation were compared with those under $S{F}_6$ gas [9]. In order to address the issue of uneven E-field strength distribution in current transformers, Wu’s study proposed a method to enhance the weakness part of the insulation baffle inside the transformer [10]. This method relied on the electric field analysis of a pillar type current transformer using ANSOFT Maxwell software to obtain the distribution inside the transformer. Wu then identified the area of electric field concentration near the winding for local analysis and compared the magnitude of the improved local field strength to reach an optimized conclusion. However, the study did not provide the best method for achieving the desired outcome. Literature [11] determined the various characteristics of the load distribution and electric field distribution in the withstand voltage test through the finite element simulation of the ±1100 kV epoxy resin transformer and verified the results through experiments. In order to address the issue of the 500 kV substation failure in Peru, Ricardo conducted a finite element simulation of the transformer located within the substation. This simulation calculated the maximum electric field, concentrated electric field area, and inhomogeneous electric field distribution. Additionally, a brief analysis of the causes of the faults was conducted [12]. In order to optimize the insulation performance of current transformers and solve their flash or partial discharge problems, optimization schemes for the internal structure of the CT have been proposed in several studies, and the electric field distribution calculated by finite element method has been used to compare the results of several different optimization schemes with the tests of a realistic model. The optimized products performed well in tests [13,14,15,16]. Literature [17] conducted an electrostatic field simulation for various factors affecting transformer grounding current and obtained the grounding current under different factors. This paper presents the design and simulation of a conical current transformer with a voltage rating of 40.5 kV.

According to the client’s order, a special conical structure is needed to fit the installation. Therefore, a new conical CT was designed with the primary and secondary windings encapsulated into a single unit using epoxy resin. The secondary side core is typically rolled with strip silicon steel sheet. Compared to oil-immersed current transformers, this design offers higher mechanical strength, lighter weight, better insulation performance, and improved reliability performance [18,19,20]. Additionally, the internal medium of cast-resin type current transformers is epoxy resin, which is more environmentally friendly than $S{F}_6$ type current transformers. The global warming potential (GWP) of $S{F}_6$ is 23,900 times higher than that of $C{O}_2$, and the leakage of SF6 into the atmosphere will exacerbate the ongoing global warming phenomenon [21,22,23].

In this paper, a 40.5 kV conical current transformer is modeled in a 3D structure. The rest of this paper is organized as follows. In Section 2, the transient simulation model is established by using finite element software. In Section 3, the electric field distribution under the power-frequency withstand voltage test (85 kV, 1 min) is studied to locate the point with the maximum value. Then, a scheme is proposed, and the maximum values before and after optimization are compared. In Section 4, the significance of three geometrical factors on the effect of the scheme is examined. Then, an optimized scheme is proposed by using response surface. In Section 5, a partial discharge detection platform is utilized for local discharge measurement. The results of the study can provide a reference for the design of new current transformer for factory production.

2. Establishment of Transient Field Finite Element Model

The Finite Element Method (FEM) is a versatile numerical analysis technique that can be applied to solve a wide range of engineering problems. It provides an approximation of the solution to boundary problems encountered in engineering [24]. FEM is used to dissect the field domain of the research object and divide it into numerous small and simple units, and cater to different accuracy requirements by reducing the physical prototype. In this study, all simulations are conducted through COMSOL Multiphysics software.

Considering the unsymmetrical structure of the current transformer, the traditional two-dimensional FEM cannot calculate the electric field distribution under unsymmetrical boundary condition. Therefore, this paper uses a refined 3D model to calculate the electric field characteristics of the current transformer. As illustrated in Figure 1, the body of the current transformer consists of two copper primary terminals ①, a copper conductive rod ② on the primary side, an iron bracket ③, and an iron secondary coil core ④. The primary side elements are soldered with metallic copper to ensure good electrical conductivity. In order to improve the electric field distribution, the internal conductor part of the CT is casted in epoxy resin after winding the semiconductor shield. Before casting, it is filled with $S{F}_6$ gas and pressurized to expel air. Additionally, the body and the cast-resin body are chamfered and polished to further improve the electric field distribution. The diameter of the conductive rod varies according to the rated current on the primary side, and the conductive rod with a diameter of 30 mm at 1250 A is the object of this paper.

Table 1. Material properties.

Components	Materials	Electrical Conductivity (S/m)	Relative Dielectric Constant
Primary terminals	Copper	$5.998\times {10}^7$	1
Conductive poles	Copper	$5.998\times {10}^7$	1
Brackets	Iron	$10.295\times {10}^6$	1
Secondary Coil Core	Iron	$10.295\times {10}^6$	1
Pouring body	Epoxy resin	$2\times {10}^{-14}$	3.5
	Oil	$0.3\times {10}^{-12}$	2.2

The properties are referred from [8].

shows the material and physical properties of each original component of the current transformer.

To reduce computational load during the simulation test, the size of the computational field under the AC/DC electrostatic field problem is limited. This is achieved by selecting an appropriate boundary to simulate infinity. In this paper, an outer box, the dimensions of which are five times of those of CT, is set to represent the computational field. The transformer is positioned at the center of the box. The outer surfaces of box is selected as zero electric potential surfaces. The simulation is conducted through one alternating current (AC) cycle by using COMSOL Multiphysics finite element software, and the criteria for the simulation termination are in accordance with the national standardization [25].

Physical Model

In the electrostatic field problem, the electric field can be solved for each point using Maxwell’s system of equations. Equation (1) is the system of Maxwell’s equations in a differential form.

$$\left\{\begin{array}{c}\nabla ·D=\rho \hfill \\ \nabla ·B=0\hfill \\ \nabla \times E=-\frac{\partial B}{\partial t}\hfill \\ \nabla \times H=J-\frac{\partial D}{\partial t}\hfill \end{array}\right..$$

(1)

In Equation (1), J is the conduction current density. For the dispersion equation $\nabla ·D=\rho$ of the electric field in the above equation, under each homogeneous dielectric condition ($\nabla \epsilon =0$), Equation (2) can be obtained by bringing in $D=\epsilon E=-\epsilon \nabla \phi$,

$$-{\nabla }^2\phi =\frac{\rho }{\epsilon }.$$

(2)

This equation is the Poisson equation for the electrostatic field. If there is no charge in the computational domain, when $\rho =0$, Equation (2) can be written as

$$-\epsilon {\nabla }^2\phi =0.$$

(3)

The equation is in the Laplace form for the electrostatic field. When solving the problem using the finite element method in the electrostatic field, in addition to the equations in the differential form described above, it is necessary to set boundary conditions to determine the uniqueness of the solution.

In this simulation, the secondary side of the current transformer is grounded, and the induced electric field strength ($\partial B/\partial t$) can be neglected. Therefore, the calculation can be regarded as under the electric quasi-static field (EQS). Moreover, under AC conditions, there is no space charge: $\rho =0$; thus, the control equation of the electric field can be expressed as

$$\left\{\begin{array}{c}divD=0\hfill \\ \\ E=-\nabla \phi \hfill \end{array}\right..$$

(4)

In order to simulate the power-frequency test, the primary side terminals and the conductive rod of the CT were used, 85 kV 50 Hz AC was applied, while secondary coil and bracket were grounded.

3. Simulation Results and Analysis

3.1. Electric-Field Distribution of Current Transformer

In order to observe the internal electric field distribution of the transformer, we set the longitudinal direction of the transformer to be the X-axis and the radial direction to be the Z-axis. The X-Z plane is selected as the observation plane. Figure 2 presents the cross-sectional view of the X-Z plane of the simulation. The electric field in the cross-sectional view is found to be mainly concentrated near the conductive rod, the surfaces of inserts, and the end of the terminals.

The transformer’s withstand voltage and partial discharge value can be impacted by the maximum electric field strength and its location. The maximum electric field strength and its location in the computational domain is shown in Figure 3. It shows that the maximum electric field strength in the transformer is 11.5561 kV/mm near the air gap. We define that point as $P_{max}$, the coordinates of which are x($-95.000$), y($-0.652$), z($-71.042$). A probe is positioned at $P_{max}$ to analyze the variations of electric field in two cyclicalities. The results in Figure 4 show that the electric field varies uniformly and reaches its maximum at 0.005 s, which corresponds to the peak moment of applied voltage.

In this paper, the bracket is installed with metal inserts and is directly exposed without being wrapped in epoxy resin. It is a weak part in the transformer for the electric field. Therefore, it is necessary to analyze the electric field in the vicinity of the insert. Figure 5 depicts the variation of electric field intensity along the x-axis near the insert, where the path is represented by the blue line within the diagram. The electric field strength gently varies from 0 to 4.6 mm along the X axis, while near the peak, it varies considerably, resulting in uneven distribution. The fluctuations are also considerable near 6.46 mm. The uneven distribution of the electric field can cause the transformer in the factory test to malfunction, leading to a discharge phenomenon. To balance the distribution, the transformer’s structure will be redesigned and the electric field distribution before and after optimization will be compared.

3.2. Optimization Scheme

The most common method of balancing the internal electric field distribution of a transformer is using a metal shield or a shielding net shown in Figure 6. A shield is typically employed at the end of capacitive insulation structures in vertical or inverted transformers. The shields are often designed in a columnar shape to balance the electric field exposed in the air and to reduce local discharge signals. A shielding net is often used in cast-resin type CTs, generally installed behind the transformer terminals, to allow the maximum electric field strength appearing on the shielding net to improve the transformer’s ability to withstand voltage. This paper proposes an optimization scheme for the transformer, which involves adding a shielding net with a radius of 42 mm and a wire diameter of 4 mm, positioned 70 mm from the primary terminal. The optimized internal electric field, circled in red, is primarily concentrated on the shielding net shown in Figure 7.

The maximum field strength in the optimized computational domain is on the inner surface of the shielding net with a value of 15.9492 kV/mm. Figure 8 displays the location of the maximum electric field in the computational domain, indicating a shift from the insert to the inner surface of the net, from the exposed part to the internal. Since the breakdown electric field strength of the epoxy resin is greater than 30 kV/mm, the internal electric field strength of the transformer remains within a safe value. Figure 9 shows the variation of electric field at the original maximum point $P_{max}$ during the simulation after improvement. The electric field strength at that point significantly declines throughout the calculation time, from the value of 11.5561 kV/mm to 5.0419 kV/mm. The improvement results are prominent.

Figure 10 displays a comparison of the electric field on the surface of the insert. The line graph illustrates a smoother distribution of values near the peak, where the maximum value of the electric field distribution on the axial distance is reduced from 6.46 kV/mm to 6.34 kV/mm.

The location of the maximum electric field of the current transformer shifts from the external insert to the internal net surface after the shielding net is added. The electric field strength at $P_{max}$ decreases from 11.5561 kV/mm to 5.0419 kV/mm, and the distribution near the insert over the axial distance is smoother. The comparison results show that the addition of a shielding mesh to the transformer enacts a certain improvement on the electric field.

4. Design of Experiment and Response Surface

Design of experiments (DOE) is a statistical theory and methodology used to observe how changing input factors affects the output response. By analyzing the effect of each input factor on the output response, DOE helps to optimize the input factors for the desired output response. This process involves the use of mathematical statistics to obtain information [26]. For the experimental design to the above scheme, three factors of the shielding net, radius, wire diameter, and the location of installation, were selected as the variable input factors, and the maximum electric field a point $P_{max}$ was selected as the response result. In this study, DOE analysis was conducted through Minitab software.

4.1. Selection of Factors

To design the response surface, it is necessary to determine the input factor and its range of values. For the electrical design, it is important that the shielding net is at least 15 mm away from the surface of the casting body to ensure proper insulation. Therefore, the maximum radius of the shielding net can be up to 42 mm. However, if the minimum radius is less than 35 mm, the simulation results, as shown in Figure 11 circled in red, indicate that the maximum electric field appears at the end of the terminal rather than on the shielding net. As a result, the minimum radius of the shielding net must be 35 mm.

The wire diameter level of the outer ring of the shielding net is limited to 1–4 mm due to production constraints. Additionally, the installation position is restricted to be after the primary terminal and before the secondary coil, while also ensuring a distance of at least 15 mm from the surface of the casting body. As a result, the installation distance ranges from 55 to 85 mm to the surface of the primary terminal.

Table 2. Range of factor values.

Factor	Low Level	High Level
Radius (mm)	35	42
Wire diameter (mm)	1	4
Distance (mm)	55	85

displays the selected impact factors and their corresponding value ranges.

4.2. Establishment of Response Surface

In this paper, a Central Composite Experimental Design (CCED) including three-time center point experiments was conducted to analyze the effects of the factors above. The generated randomized trial sequence table was as depicted in

Table 3. Experimental sequence.

Std Order 1	Run Order	Pt Type 2	Radius (mm)	Wire Diameter (mm)	Distance (mm)	E-F Strength (kV/mm)
10	1	$-1$	1	0	0	4.56949
7	2	1	$-1$	1	1	5.27290
9	3	$-1$	$-1$	0	0	4.45780
3	4	1	$-1$	1	$-1$	4.80860
6	5	1	1	$-1$	1	5.67650
8	6	1	1	1	1	6.09160
4	7	1	1	1	$-1$	4.60530
2	8	1	1	$-1$	$-1$	4.29080
12	9	$-1$	0	1	0	5.40000
15	10	0	0	0	0	4.91300
17	11	0	0	0	0	4.90000
11	12	$-1$	0	$-1$	0	5.04776
13	13	$-1$	0	0	$-1$	4.63900
5	14	1	$-1$	$-1$	1	4.87040
1	15	1	$-1$	$-1$	$-1$	4.97230
16	16	0	0	0	0	4.90410
14	17	$-1$	0	0	1	5.31000

¹ Standard Order. ² Point Type.

. In Table 3, StdOrder is the Standard Order of CCED which must run randomly. PtType is the Point Type of CCED, where $-1$ means the center point of surface, 0 means the center point of body, and 1 means the vertex. All values of factors are shown in a digital format, where $-1$ means a low level, 0 means a medium level, and 1 means a high level. A total of 17 simulation trials were conducted in accordance with the Run Order. The simulation results were assigned to their experimental groups, respectively, for factor analysis. To achieve a more accurate fit, a second-order model should be considered, which incorporates the interaction between the three factors and their squared term effects. Figure 12 shows the Pareto diagram of the influencing factors obtained by using Minitab software. From the diagram, the factors and interaction effects on the response are $distance>distanceandradiusinteraction>wirediamete{r}^2>radiu{s}^2>wirediameter>radius>wirediameteranddistanceinteraction>distanc{e}^2$. The interaction effect between radius and wire diameter is insignificant and is removed in the subsequent analysis process.

After removing the insignificant interaction effect, the model’s response surfaces were established, and are shown in Figure 13. The response surfaces between the parameters reveal that in the simple model, the interaction effect between some parameters is evident. For instance, in Figure 13a, the surface of radius and wire diameter interaction is ‘saddle-shaped’, indicating that the interaction effect is not prominent. In Figure 13b, the contour line of radius and distance interaction is ‘stepped’, indicating that the interaction effect has the most significant impact on the electric field. In Figure 13c, the contour line of the interaction of line diameter and distance is circular, suggesting that the interaction effect is significant.

Table 4. Predicted result.

Radius (mm)	Wire Diameter (mm)	Distance (mm)	Prediction of Electric Field Size (kV/mm)
42	2.2727	55	4.053

shows that the minimum predicted electric field strength, which is the result of response surface, is 4.053 kV/mm when the radius is 42 mm, the wire diameter is 2.2727 mm, and the installation distance is 55 mm. To verify this optimal response, the simulation model parameters were modified, and the maximum electric field strength obtained from the point $P_{max}$ was found to be 4.16982 kV/mm, which is only 2.89% different from the DOE analysis. This is illustrated in Figure 14.

5. Partial Discharge Test

Figure 15 presents a schematic diagram of the primary side circuit, established according to the (PRC) National Standard of CT partial discharge test [25]. The circuit comprises the test transformer $T_r$, the power-frequency test transformer $T_x$, the protective resistor $R_0$, the partial discharge measuring instrument $C_k$, the detecting impedance $Z_d$, and the CT S that requires testing. The system generates a pulse current partial discharge signal, which is obtained through the coupling capacitor and directly sent to the local discharge measuring instrument. A photo of the test station is shown in Figure 16.

The partial discharge meter measured a background signal of 1.70 pC after the conectining was completed without applying voltage. The partial discharge signal of the current transformer before and after improvement is shown in Figure 17. Before improvement, the meter measured 14.21 pC in the low-voltage state, 48.57 pC in the high-voltage state, and the comparison result was 5.72 pC at a low voltage and 36.4 pC at a high voltage, which was 60% lower after improvement. The optimization effect is obvious. The CT withstand voltage test at 62 kV showed an obvious discharge phenomenon without the addition of a shielding net. However, after optimization, it could withstand up to 85 kV in 1 min without any discharge. Figure 18 shows the power-frequency withstand voltage experiment data. When there is a discharge, the current signal significantly surges to zero.

6. Conclusions

In this paper, the electric field distribution of a new type of CT during voltage withstand test is investigated. The simulation model of the CT is established in this paper, and an improvement scheme is proposed. The DOE is conducted on the effects of radius, wire diameter and shielding position on the E-field distribution to obtain the optimal solution. Finally, the factory voltage withstand test is conducted for verification. The main conclusions of the study are as follows:

• The weak area of the electric field distribution in the CT is found near the metal insert of the bracket. When the input voltage is at its maximum, the electric field strength is 11.5561 kV/mm. At $P_{max}$, the electric field on the surface of the insert distributes unevenly along the axial direction.
• After optimization, the maximum electric field in the current transformer is located on the shielding net, with a value of 15.9492 kV/mm. The electric field strength at $P_{max}$ is reduced to 5.0419 kV/mm, which is a decrease of 56.37%.
• The response surfaces predict the minimum electric field strength is 4.053 kV/mm when the radius is 42 mm, the wire diameter is 2.2727 mm, and the position is 55 mm. The simulation result is 4.16982 kV/mm.
• The optimization effect is obvious when the improved CT is subjected to partial discharge test. Without a shielding net, the CT withstand voltage test at 62 kV appears obviously discharging. After optimization, it can withstand up to 85 kV in 1 min without discharging.
• In this paper, a finite element combined design of experiment approach is used for optimization. Future research could try to use a more systematic algorithm, such as neural network combined with genetic algorithm, to select more factors that may affect the electric field for optimization and compare it with the method in this paper.