IEEE TRANSACTIONS ON MAGNETICS, VOL. 26, NO. 2, MARCH 1990

EFFICIENT COMPUTATION OF ELECTRIC FIELD IN HIGH VOLTAGE EQUIPMENT

B.L. Qin, G.Z. Chen, Q. Qiao, J.N. Sheng, SM.,

Department of Electrical Engineering Xi'an Jiaotong University

Xi'an, 710049 People's Republic of China

ABSTRACT

The efficient indirect boundary element method is applied to the calculation of three dimensional electric field in high voltage devices. For solving this problem with sophisticated boundary conditions and with dielectric media, we adopt the real surface * elements, the fast and accurate computation approach and the thin-layer medium treatment techniques. Numerical results are given to verify the efficiency of the program and the accuracy of the calculation.

INTRODUCTION

In response to an increasing demand for electrical energy, operating transmission level voltages have increased considerably over the last two decades. Manufactures are therefore forced to reduce the size and weight of electrical equipment in order to remain trade competition. For designers, in power transformers, for example, they need a thorough understand of electric field and potential distribution in the insulation system. Figure I is a model taKen for study in this paper.

I

Fig. I The model

1-high voltage coil 2 -tapping coi 1 3-bushing 4-insulating cylinder 5-shaped board 6 -grounded met a1 t a m

It is Known model 1 ing the el simp1 e. However, for solving the

IEEE

that the

387

equation used t o r ctric fi ld is relatively there are some difficulties three dimensional problems

with sophisticated boundary conditions and with thin-layer media, if numerical methods are not suitably chosen for the actual problem t o be solved. The method in this Paper is based on the indirect boundary e 1 ement method with some improvements [ 1, 2, 3,4]. The efficiency is achieved by adopting - the " real surface elements, - the fast and accurate computation, - the thin-layer medium treatment techniques.

EFFICIENT STRATEGIES IN BEW

From the commonly used methods for solving electric fields, the most convenient one for the transformer problem, as we have found, is the boundary element method. The efficiency of it depends mostly on the aPProximation of entire boundary surface by a set of boundary elements, and on the approximation of charge distribution function. in the element.

1. Real Surface Elements

In power transformers, the shape of boundary surfaces can be classified broadly as two categories: rounded and plane. A s to match the former, the cy1 indrical element,

Fig.2 Real surface element

0018-9464/90/0300-0387$01.00 0 1990 IEEE

388

discal element, shown in figure 2 match the later, used ( figure 2(d

and circular element, as a-c), are available. As to the rectangular element is ) , These elements, with the

aid of proper co-ordinate transformation, can cover over the entire boundary surface of the transformer shown in figure I. Since these elements are chosen from the parts of the real surface in transformer configuration, the error caused by boundary surface approximation is minimized.

2. Charge Distribution Function

When analysing three dimensional field with rounded-type configuration, a Fourier- type surface charge is suitable. Along the 8 - direction, in figure 2(a-c), an unKnown charge density can be expressed as a series form

u(8)=~:ocmCos(m8)+EosmSin(m8) (1)

and along the €-direction, U is expressed as a linear function

o(€)-CJ+C2€ ( 2 )

In the rectangular element, shown in figure 2(d), a double linear-type surface charge is the most convenient one and with good accuracy. Thus, an unknown charge U is supposed to be the form

u(X,Y)=a1+a2X+a-jY+a4XY (3)

Using them the potential and electric field at any point can be solved analytically.

The potential and electric field formulation of the rounded surface with charge density of form ( 1 ) and (2) are given in [l] 141. In practical application, some time-consuming problems have been overcome by the fast computation techniques, such as fast toroidal function computation and high- speed integral.

A s to the plane rectangular element, as shown in Figure 2(d), the potential coefficients are formulated analytically in order to decrease the computing time. The potential 9 at a point P(Xo,Yo,Zo) due to a charge density u(X,Y) is represented as

(4)

After the integration equation(4) may be written in the form

2 XBB+HBB Q3 = m B L n A-B XM+HAB

THIN-LAYER DIELECTRIC PROBLEX ~~ ~

In the transformer, there exist thin- layer dielectric materials, such as insulating cylinder, snapped board, insulating barrier etc., as illustrated in figure 1. Using ordinary approach to handle this field problem leads to numerical difficulties because of the physical proximity of the two interfaces. A special finite element method, in which basic element contains the thin-layer medium, has been used[5]. However, the entire volume of the problem must be subdivided and this needs long set up time. An alternative method is the boundary element method. One can be described with reference to figure 3 as f 01 1 ows :

3 The thin-layer[2]

A thin ayer ( region 1 , between regions K and m ) is bounded by the interfaces SKI and Slm. The unit vector is labelled as nK1 and nIm respectively. The layer 1 is assumed to be of constant thiclmess At. In the method of thin-integral equations [23, unknowns are the sum and difference of surface charges at relative points on the two interfaces, i.e.

uls(PI<l)’ukl (%l)+Ulm(Plm)

389 Using some approximations, the difference equation and the sum equation for the charges in axially symmetric geometries was derived[2]. In the problem of transformer, although we meet up with three-dimensional fields, the thin-layer equations may be simple since E ~ = E ~ and is not much larger than E ~ . Actually, the sum equation is further simplified as

uls(%l)=o (6)

then the difference equation is expressed as

~ 1 ~ 1 d ( q c l ) - 2 ( ~ 1 - ~ ~ ) 1 ~ ~ a ~ / a n k l d s = o (7)

where p s is the surface charge density on Known potential surfaces and dielectric interfaces, which may be found by solving the conventional integral equations. Equations (6) and (7) constitute the complete set of equations for the thin 1 ayer charges. Obviously, the difference charges Old can be obtained very easily, therefore the electric field of the thin-layer dielectric can be solved.

The other technique proposed here for treating the thin-layer dielectric problems can be described as this: both single-layer charges and double-layer charges, or only double-layer charges are used as unknowns and they are located on the middle surface S between SKI and S i m If E~ is not equal to Em both single-layer and double-layer charges exist; on the other hand. if ER is the same as Em, only the double-layer charges exist.

Let us imagine that positive charges are located on surface Slm, while negative charges with an equal amount are located on Ski, and they are separated by a very small distance At. This charges constitute a double-layer charges (or dipole layer). Their dipole moment per unit area 7, or surface density, is defined as follows

7:nlim(uAt) At -0

U-0

where Q is the charge density of the single- layer charges; n is the unit vector, its direction is from negative charge to positive charge.

The potentials on two sides of the double-layer surface are not continuous, their potential difference is

(9)

If the double-layer surface is divided into a number of elements and the point matching method with zero order basic function is used, the potential and field strength at the contour point i due to the source point j are expressed as follows, respectively,

RESULTS AND DISCUSSION

To demonstrate the efficiency and the accuracy of the method, we have applied the computer code developed by using above techniques to calculate several test problems, which have their own analytical solutions, respectively. For example, coaxial cylinder, which has a length of 20 cm, inner radius of 5 cm, outer radius of 10 cm, and a thin-layer dielectric with thickness of 0.05 cm located on the radius of 7.5 cm, is calculated. The numerical results are shown in Table 1.

For the practical problems as shown in Figure 1 , it is hard to evaluate the error of the results because there is no analytical solution and the error range of the potentical cannot represent that of the field intensity. Therefore, in this paper, we use

Et

En AEtn:-Y 100%

to evaluate the error of calculated field intensity on the surface of conductor, where Et is the tangential component and En is the normal component at the same point on the conductor surface.

In the calculation of the power transformer the errors AEtn are limited within 10% . Finally , in the concerned region, it does not exceed 3.6% . Figure 4 shown a set of equipotentical lines which is taken out from Figure 1. The location is between the high voltage coil and the tapping coil near the high voltage lead.

It should be pointed out that the existence of the thin-layer dielectric in AC power transformer does not obviously affect the field distribution. The computing results of the quoted example show that the maximum relative error between the numerical results with thin-layer dielectric and that without it is less than 3.8% for potential and 2.1% for field strength. However, as a further research we intend to study the space charge and field distribution in transformer oil and to consider the electrization of insulating oil. In this case,the thin-layer

390

5.0

7.5

10.0

Table 1 ,

100.00 29.03 100.00 28.76 0.03056 0.106 0.930

41.75 19.35 41.58 19.24 0.05378 0.280 0.568

0.00 14.52 0.00 14.42 0.03292 0.228 0.689

Radius I Analytical Res. 1 Numerical Res. I Error

Where AEl=Et/Enrioo% AE2.I(Ea-En)/Ealrloo%

dielectic must be carefully treated.

CONCLUSION

When usina boundary element me h solve a complex electric field,

d to the

requirements of large memory and computer time are reduced successfully with a number of techniques, such as the real surface '' element, the fast and accurate computation, and the thin-layer medium treatment. The computer program based on this efficient boundary element method is an important tool for the transformer insulation design. It has been used to identify regions of high field strength where electrical breakdown might occur.

REFERENCES

[ I 1 S. Sato, W.S. Zaeng1:'Effective 3- dimensional electric field calculation by surface charge simulation method", IEE Proc.

[2] R.G. Olsen: Integral equations for electrostatics problems with thin dielectric or conducting layers", IEEE Trans on Electrical Insulation, Vol.EI-21, No.4, 1986, PP.565-573. [31 B. KrStaJiC, Z.AndJellC, Z. Mi19 JKOVl C : "An improvement in 3d electrostatic field calculationn, 5th Int. Symp. on High Voltage Eng. Aug.24-28, 1987, F.R. Germany, 31.02 [4] Yang Xile,Sheng Jianni:"A high accuracy annular element interpolation method", BrlJing Int. Symp. on Electromagnetic Fields in Electrical Eng , Oct.19-21, 1988, P.R. mina, ~ ~ 2 . 3 0 I51 Yuan Jinsha, Fei Zengyao: "A new method for computing electric field involving thin-layer media with its application for EHV transformer design", ibid, Ps2.14

VOl 133, Pt.A. No.2, 1986, pp.77-83.

Fig.4 Equipotientical lines