Abstract—Three-dimensional finite element method has been used to calculate the electric field and voltage distribution along glass insulator string for 1000kV a.c. UHV transmission lines. Compared with the test results, it is verified that the calculation results are accurate. Various parametric designs of the grading rings of the glass insulator string have been analyzed. Based on the designs, the orthogonal experimental method has been employed to propose several optimum designs of the grading ring for reducing voltage distribution ratio along the glass insulator string. The optimization of grading rings indicates that the grading effect is preferable when tube diameter r equals to 60 mm, ring diameter R 550mm, height H 350mm and the number of insulators n 52.

I. INTRODUCTION or 1000kV a.c. UHV transmission lines, the field distribution is quite non-uniform because of the high

voltage and the long insulator string. High electric field intensity may cause strong corona, partial discharge, and premature aging of insulation. High voltage distribution ratio may breakdown insulators, so that the operation safety of lines can not be ensured [1-3]. Thus, it has important significance to calculate and improve the field distribution along the insulator string through well grading rings design.

Because of the distributed capacitance between the insulators and the iron tower, the voltage distribution is quite non-uniform along the insulator string. Various methods have been proposed to calculate the voltage distribution, such as equivalent capacitance method [4], charge simulation method [5] and finite element method. In recent years, some combined methods have been suggested, for example, the combined method based on the boundary element method and the moment method in the reference book [6]. However, there are a few problems among these methods: (1) Many of the methods are in two dimensions [7] which cannot simulate the practical situation very well; (2) Some of them cannot treat the infinite field region well which the insulators lie in. An effective method to solve the infinite field region is asymptotic boundary conditions method presented in the

1Corresponding author: Yanzhen Zhao(e-mail: zhaoyzh@mail.Xjtu. edu.cn ).

reference [8], but the method is limited to two dimensions. Another new method called domain-decomposition method was presented in the reference [9], but only can improve its boundary accuracy in the sub-domain boundaries. In this paper, the electric field and voltage distribution along glass insulator string for 1000kV a.c. UHV transmission lines is calculated using three-dimensional finite element method. The iron tower, cross arm, and grading rings are taken into account. The calculation results are verified to be accurate compared with the test results. Based on this, the field distributions with different design dimensions and position of the grading rings corresponding to the glass insulator string are calculated. The effects of the grading rings structural and location parameters, such as ring diameter R, tube diameter r, height H, and number of insulators n on the field distribution are investigated. The orthogonal experimental method is employed to propose several optimum designs of the grading ring to improve the field distribution.

II. THREE-DIMENSIONAL FINITE ELEMENT CALCULATION

MODEL In this paper, we consider a suspension II glass insulator

strings for 2×300kN wires for 1000kV a.c. UHV in Fig.1. In the operating UHV transmission system, the electric field of insulator string appears as three-dimensional field distribution because of the transmission tower and wires. We assume that no corona occurs, insulators are clean and dry, air moisture is low, leakage current and space current are negligible, and charges distribution on the metal caps of insulators keeps invariant. By suspension II glass insulator strings dimensions and structural symmetrical characteristic, the simplified finite element calculation model is established, as shown in Fig.2. Since both steel foot and cap between two insulators are metallic and contact with each other, they can be considered one solid. Insulator strings are hanged down on the tower with bulb suspension link, and are connected with bowl suspension plate and clamps. These hardware fittings structure is complicated, but has little effects on the whole electric field distribution, so that all can be simplified as cylinder solids. By our experience, the umbrellas and anti-pollution slots also have little effects on the whole field

Calculation of Electric Field and Voltage Distribution along Glass Insulator String and Optimum Design of its Grading Rings for

1000kV a.c. UHV Transmission Lines 1 Yanzhen Zhao, Hualiang Li, Xikui Ma

School of Electrical Engineering, Xi’an Jiaotong University, Xi’an 710049, China

F

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1-insulator 2-grading ring 3-yoke plate 4-wire 5-shading ring

Fig.1. Suspension II glass insulator strings for 2×300kN wires. distribution, so they can be discarded in order to save

memory and improve calculation speed. Both cross arm and iron tower can be simplified as finite grounding plane. The divided conductors are also considered as cylinder solids.

Fig.2. FEM calculation model. Boundary value problems corresponding to the finite

element model can be expressed as the following:

Where, type 1 boundary value is 770kV on the high voltage

terminal, the divided conductors, and 0V on the lower voltage

terminal and the iron tower; type 2 boundary includes air

surface and symmetrical interface. The floating conductors

include steel foot and cap of the insulators [10].

III. VERIFICATION To obtain confidence that the three-dimensional finite

element calculation model would provide accurate results, a comparison was made between the calculation results and the measurement results, as shown in Fig. 3. From Fig. 3, we see that the calculation results are satisfied. The first ten insulators bear about 45 percent of the total voltage. The maximal voltage of per insulator reaches a value about 47kV which almost approaches wet tolerant voltage of insulator, 50kV. Therefore, we need to optimize the grading rings structural and location parameters and other parameter such as number of insulators to reduce voltage distribution ratio and the maximal field intensity.

0 10 20 30 40 50 600

1

2

3

4

5

6

7

8

9

10 Calculation result Test result

Vol

tage

dis

tribu

tion

ratio

/ %

Number of insulators( from high voltage terminal)

Fig.3. Comparison between the calculation results and the test results.

IV. OPTIMUM DESIGN OF THE GRADING RINGS OF THE GLASS

INSULATOR STRING We already know that the voltage distribution ratio of

insulators depends on R, r, H and n. We can write this map function F as Rv = F(R, r, H, n), which is a nonlinear mathematical expression and can hardly expressed in explicit function, where Rv denotes the voltage ratio of insulators. Our optimal object is to find best combination of R, r, H, n so that the voltage distribution ratio of insulators become as low as possible. In general, we can simply change each parameter in its definition domain to obtain several different design plans. Then, by comparing the design plans, we may choose the best one of them. However, in this method, we always need lots of computation amount. Especially when the practical structure is complicated or there are many variables, the method is hardly to use. In this paper, we will employ the orthogonal experimental method. This is a simple and effective method

2 0ϕ∇ =

1 1| ( )S f pϕ =

2 2| ( )s f pnϕ∂ =

∂

k kCϕ =

Whole field region Type 1 boundary conditions Type 2 boundary conditions The surface of floating conductors

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[11,12]. The essence of the orthogonal experimental method is a multi-factor, experimental and optimization method. This method can help you to reach step by step the optimum solution and can be used whether there are interaction effects in the practical problems or not. The method includes two contents: arranging experimental plans and analyzing experimental results. The detailed procedure is as the following:

1. Determine the number and level of factors. 2. Arrange experiments by selecting a suitable orthogonal

experimental table. 3. Set top of the table. 4. List the experimental plans. 5. Analyze the experimental results. In our project, we select four factors such as ring Diameter

R, tube diameter r, height H, and number of insulators n. The level of the factors is shown in TABLEⅠ.

With the factors and their level, we select L9(34) orthogonal experimental table to arrange experimental plans without consideration of interaction effects between the factors. TABLE 2 shows the experiments arrangement and its visual analysis, where S1, S2 and S3 are different optimum objects which mean standard deviation of the voltage distribution ratio of all insulators, the forward twenty insulators by magnitude and the forward twenty insulators from the high voltage terminal, respectively. The standard deviation can be found from the definition expression:

1

( )

1

m

ii

X XS

m=

−=

−

∑

Where X , iX and m are the average voltage distribution ratio of insulators, the voltage distribution ratio of per insulator and the number of the insulators, respectively.

In TABLE 2 , A 1 , A 2 and A 3 denote the average value of S1, S2 and S3 , E 1 , E 2 and E 3 denote the extreme difference S1, S2 and S3.

The smaller the standard deviation is, the more uniform the voltage distribution ratio is. So we see that the voltage distribution of experiment 9 is the most uniform in all the experiments. The bigger the extreme difference is, the greater the factor effect is on object. Based on this, we conclude three optimum parameter combinations which are also shown in TABLE Ⅱ. By optimum object S1, the best combination is H3R3r3n3，in which h 350mm, R 550mm、r 60mm and n 55. In a similar way, we forecast the other two best combinations by optimum object S2 and S3 .We note that the result from plan 2 is exactly same as from plan 3. Fig. 4 shows the comparison of the results from experiment 9 and from the three optimum combinations. We note that both of the maximum and the minimum voltage distribution ratio by plan 1 and plan 2(3) are smaller than those by experiment 9. The specific numerical results are listed in TABLE Ⅲ.

TABLE I

THE LEVEL OF THE FACTORS

R/mm H/mm n r/mm

450 220 50 50 500 300 52 55 550 350 55 60

TABLE Ⅱ

EXPERIMENT ARRANGEMENT AND ITS SIMPLE ANALYSIS

Column 1 2 3 4 Optimum objects

Factor R/mm H/mm n r/mm 100 S1 100S2 100S3 Experiment 1 450 220 50 50 165.403 178.419 185.119 Experiment 2 450 300 52 55 156.585 162.319 169.618 Experiment 3 450 350 55 60 154.280 160.818 168.075 Experiment 4 500 220 52 60 159.501 169.209 176.231 Experiment 5 500 300 55 50 155.041 161.884 168.456 Experiment 6 500 350 50 55 153.812 153.531 159.917 Experiment 7 550 220 55 55 157.010 169.359 176.722 Experiment 8 550 300 50 60 152.335 152.727 161.032 Experiment 9 550 350 52 50 151.305 151.518 158.848

A 1 158.756 160.638 157.183 157.250 A 2 156.118 154.654 155.797 155.802 A 3 153.550 153.132 155.444 155.372 by S1

E1 5.206 7.506 1.739 1.878

The optimum parameter combination is H3R3r3n3 (plan 1).

A 1 167.185 172.329 161.559 163.940 A 2 161.541 158.977 161.015 161.736 A 3 157.868 155.289 164.020 160.918 by S2

E2 9.317 17.040 3.005 3.022

The optimum parameter combination is H3R3r3n2 (plan 2)

A 1 174.271 179.357 168.689 170.808 A 2 168.201 166.369 168.232 168.752 A 3 165.534 162.280 171.084 168.446 by S3

E3 8.737 17.077 2.852 2.362

The optimum parameter combination is H3R3r3n2 (plan 3)

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0 10 20 30 40 50 600

1

2

3

4

5

6

7

8

9

10

Volta

ge d

istri

butio

n ra

tio /

%

Number of insulators( from high voltage terminal)

Experiment 9 plan 1 plan 2(3)

Fig.4. Comparison of the results from experiment 9 and from the three

optimum combinations. Where S, Vmax, Rvmax and Rvmin are the standard deviation of the voltage distribution ratio, the maximum voltage, the maximum and the minimum voltage distribution ratio of insulators respectively. In TABLE Ⅲ, it is obvious that the results from plan 1 and plan 2(3) are both better than that from experiment 9. We would like to adopt plan 2(3) because this can also save insulators.

V. CONCLUSION

(1) The electric field and voltage distribution along glass insulator string for 1000kV a.c. UHV transmission lines have been calculated by using Three-dimensional finite element method. Because the iron tower, cross arm, and grading rings are taken into account, we can obtain acceptability numerical results.

TABLE Ⅲ SPECIFIC NUMERICAL RESULTS

Experiment 9 Plan 1 Plan2(or 3)

S 151.305 147.816 148.638 Vmax/kV 47.477 43.908 43.826 Rvmax/％ 6.17 5.70 5.66 Rvmin/％ 0.72 0.66 0.72

(2) The voltage distribution is quite non-uniform along the

insulator string for 1000kV a.c. UHV transmission lines. The first ten insulators bear about 45 percent of the total voltage. The maximal voltage of per insulator reaches a value about 47kV which almost approaches wet tolerant voltage of insulator.

(3) The orthogonal experimental method has been employed in optimizing design of the grading ring. An optimum parameter combination has been suggested. The results show that the plan we adopt here can reduce the maximum voltage distribution ratio from 6.17% down to 5.66% along the glass insulator string, improve the electric field distribution and satisfy the security requirement of insulators.

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[3] Liu Zhenya. Ultra-High Voltage Grid. China Economic Publishing House, 2005.

[4] Institute of Electrical Engineers of Japan, Insulator . China Machine Press, 1990.

[5] Gu Leguan, Zhang Jianhui, Sun Caixin, “Partially Optimized Charge Simulation Method for Calculating Electrical Field Distribution along Polluted Insulator,” Proceedings of the Chinese Society for Electrical Engineering, S1, pp.65-71, 1993.

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[8] Sima Wenxia, Yang Qing, Sun Caixin, “Optimization of Corona Ring Design for EHV Composite Insulator Using Finite Element and Neural Network Method,” Proceedings of the Chinese Society for Electrical Engineering, Vol,25(17), pp.115-120, 2005.

[9] Imre Sebestyen, Electric-Field calculation for HV insulators using Domain-Decomposition method[J]. IEEE Transactions on magnetics，2002，38(2)：1213-1216.

[10] Zhang Ziyang, Zhao Yanzhen, Li Hualiang, Ma Xikui, “Voltage distribution along glass insulator strings for 1000 kV ultra high voltage transmission lines,” East China Electric Power, Vol.35, pp.29-31, June 2007.

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