Ieee Transactions On Magnetics Volume 43 Issue 4 2007 Doi 10.1109 2ftmag.2.pdf

IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 4, APRIL 2007

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Calculation of Ion Flow Field Under HVdc Bipolar Transmission Lines by Integral Equation Method Bo Zhang, Jinliang He, Rong Zeng, Shanqiang Gu, and Lin Cao State Key Lab of Power Systems, Department of Electrical Engineering, Tsinghua University, Beijing 100084, China

The electric fields under high-voltage direct current (HVdc) transmission lines and the ions produced by corona affect each other greatly. This paper presents an iterative method to calculate ion flow fields under HVdc bipolar transmission lines in the presence of wind. Both the electric fields and the ion densities are calculated by integral equation method. The method is fast and stable. The results are in good agreement with the experimental data. The ion flow fields under a 800-kV bipolar direct current (dc) transmission line are analyzed. Index Terms—Electric fields, high-voltage direct current (HVdc) transmission lines, integral equations, ions.

I. INTRODUCTION

used to calculate the ion density. The method is efficient and stable.

HE high-voltage direct current (HVdc) power transmission constitutes an important technology in the development of large interconnected power networks, especially for the long-distance power transmission. Recently, China has built several 500-kV direct current (dc) transmission lines, and will build some 800-kV dc transmission lines. It is well known that when the electric field strength in the air is high enough coronas will occur and the ionization of air will generate space charges (ions and charged particles). The electric field and the space charges produced by HVdc lines will affect the natural balance of ions in the air, which might have some unknown biological and environmental effects. The electric field and ion current density on the ground are important factors in designing overhead HVdc transmission lines. Because the ion current density is affected by the product of electric field and ion density while the ion density determines the electric field, the ion flow field problems are nonlinear. Many iterative numerical methods have been presented to solve this difficult problem [1]–[5]. Most of them use differential equation methods such as finite element method (FEM) and finite difference method (FDM) to obtain the potential distribution from which the electric field is obtained by partial derivative. Then, the ion density is calculated from the electric field. Because the electric field is calculated from the derivative of potential, errors may be introduced and the iteration process may converge slowly [3], [5]. Based on the idea of integral equation methods, this paper presents a new numerical method of calculating ion flow fields under HVdc bipolar transmission lines in the presence of wind. The electric field is directly calculated by accumulating all the contribution from the ions in the air and the charges in the conductors. The integral form of the current continuity equation is

II. BASIC EQUATIONS

T

Digital Object Identifier 10.1109/TMAG.2007.892305

The problem is treated mathematically as a 2-D nonlinear field problem. The equations governing the bipolar ion field are [1] (1) (2) (3) (4) where electric field strength vector (V/m); absolute values of positive and negative space charge density (C/m ); positive ion current density vector (A/m ); negative ion current density vector (A/m ); total ion current density vector (A/m ); positive and negative ion mobilities (m /Vs); wind velocity vector (m/s); permittivity of air, 8.854 electron charge, 1.602

l0 l0

F/m C;

coefficient of recombination. III. ION FLOW FIELD CALCULATION Because the electric fields and the ions affect each other naturally, an iterative method is used to calculate the ion flow field. In any iterative step, calculations are divided into two distinct parts. The first part is to calculate the electric field from the charge density distribution. The second part is to calculate the

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Fig. 2. Analyzed

Fig. 1. Triangular elements surrounding node i.

6 400-kV bipolar dc transmission line.

C. Calculation of the Ion Densities

new charge density distribution based on the calculated electric field. At the same time, some assumptions are adopted. In this paper, the assumptions are as follows: 1) the magnitudes of electric field strength at coronating conductor surfaces remain constant at their onset values; 2) the thickness of the ionization layer around the conductors is negligible; 3) the mobility of ions is constant; 4) the diffusion of ions is neglected. With aforementioned assumptions, an iterative method is also presented based on integral equation method. First, the air region of interest is subdivided into small triangular elements. According to the idea of [6], the observation points of electric field are located on the middle points of the sides of the triangular elements, and the ion densities are evenly distributed around the trishown in Fig. 1. Because the observation angular vertices as points of electric field and the ion densities are disposed interlaced, the iteration process converges quickly. The main process of the method is as follows. A. Initial Step Simulating charges in the conductors are calculated based on the applied voltages at conductor surfaces by ignoring the existence of the ions in the air. The iterative process is started by and around the grid nodes. In assigning initial vales to and are zeros at initial. The more efficient this paper, and can be determined by [7]. B. Calculation of the Electric Fields If all the sources are known, the electric fields at the middle points of the sides of triangular elements can be calculated based on the Gauss’ law

(5) where is the area around node , and are the distances from the observation point to the source and to the source’s image, and is the simulating charge in the conductor. Because the electric field is obtained directly from the simulating charges and ion densities, the error introduced by FEM which is usually used to calculate the electric field from the derivative of potential is avoided [3], [5].

If the electric field distribution is known, the ion densities can also be calculated. In this paper, the ion densities are obtained based on the integral form of the current continuity equation. That is, by substituting (2) into (4), the following can be derived: (6) The corresponding integral forms in 2-D space will be (7) where and are the boundary and area of region , respecand are the correspondent components pertively, and pendicular to . According to the sketch map in Fig. 1, the numerical equation of (7) for node will be (8) where is the sequence number of the triangular elements, is the triangular side opposite to node and are the components of electric field and wind velocity perpendicular to is the area of element and are the average ion determined by the ion densities around the verdensities on and are the average ion densitices connected to , and ties of the corresponding triangle determined by the charge densities around the triangular vertices. For example, and will be (9) (10) For the nodes on the transmission conductors with positive voltage, the negative ion densities around them are always zeros, while the positive ones are equal to the components of electric fields perpendicular to the conductor’s surface. For the nodes on the transmission conductors with negative voltage, the situation is reversed. Both the positive and the negative ion densities around the nodes on the boundary of the air region are also zeros. be the same with the previous one, and In (8), first let can be worked out. Then, the new is obtained the new . This process can be repeated two to three from the new times to accelerate the total convergent speed according to our experiences.

ZHANG et al.: CALCULATION OF ION FLOW FIELD UNDER HVDC BIPOLAR TRANSMISSION LINES

Fig. 3. Computed and measured ground-level profiles of electric field strength. (b) Ion current density.

E and J . (a) Total

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Fig. 4. Effect of number of subconductors on ground-level profiles of E and

J . (a) Total electric field strength. (b) Ion current density. V. APPLICATION

D. Repetition Process Steps explained in Sections III-B and III-C will be repeated until the ion densities obtained in two consecutive iterations differ from each other by amounts smaller than specified tolerances. IV. VALIDATION In order to verify the validity of the method, the ground-level and for a 400-kV bipolar dc transmission profiles of 1.2 cm V s, line shown in Fig. 2 are computed, where 1.5 cm V s, cm /s, 0 m/s, the 23.99 kV/cm. The measured corona onset electric field data are also presented by [8]. Both results are shown in Fig. 3. Measured results are subject to many types of errors and vary with changing atmospheric conditions [5], [9]. Nevertheless, it can be seen that the computed results are in agreement with the experimental data, which shows the validity of the method.

As an application, the ground-level profiles of and for 800-kV bipolar dc transmission line are computed, where and have the same values as in Section IV. The area of the cross section of each subconductor is 630 mm . The 18.45 kV/cm. corona onset of the electric field First, the effect of the number of subconductors at intervals of 0.45 m on the ground-level profiles of and is analyzed when the height of the lines is 18 m and the distance between the two poles is 22 m. The results are shown in Fig. 4. It can be seen that when the number of subconductors increases, both and decrease. This is because the electric field strength on the conductor’s surface decreases when the number of subconductors increases, which makes the ion difficult to escape from the conductor. The effect of the height of the lines on the ground-level proand is also analyzed when there are six subconfiles of ductors with intervals of 0.45 m at each polar and the distance between the two poles is 22 m. The results are shown in Fig. 5, a

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The effect of the distance between the two poles on the and is also analyzed. However, ground-level profiles of this kind of effect is very small compared with the previous two effects. VI. CONCLUSION An iterative method to calculate ion flow fields under HVdc bipolar transmission lines in the presence of wind is presented in this paper. Both the electric fields and the ion densities are calculated by integral equation method. The results are in agreement with the experimental data. By analyzing the ion flow fields under a 800-kV bipolar dc transmission line, it can be seen that both and decrease when the number of subconductors and the height of the lines increase, while the effect of the distance between the two poles is very small. REFERENCES

Fig. 5. Effect of height of lines on ground-level profiles of E and J . (a) Total electric field strength. (b) Ion current density.

from which it can be seen that both and number of subconductors increases.

[1] M. Yu, E. Kuffel, and J. Polk, “A new algorithm for calculating HVDC corona with the presence of wind,” IEEE Trans. Magn., vol. 28, no. 5, pp. 2802–2804, Sep. 1992. [2] W. Janischewskyj and G. Gela, “Finite element solution for electric fields of coronating dc transmission lines,” IEEE Trans. Power Appl. Syst., vol. PAS-98, no. 3, pp. 1000–1012, May 1979. [3] T. Takuma and T. Kawamoto, “A very stable calculation method for ion flow field of HVDC transmission lines,” IEEE Trans. Power Del., vol. PD-2, no. 1, pp. 189–198, Jan. 1987. [4] Z. M. Al-Hamouz and M. Abdel-Salam, “Improved calculation of finite-element analysis of bipolar corona including ion diffusion,” IEEE Trans. Ind. Appl., vol. 34, no. 2, pp. 301–309, Mar. 1998. [5] B. Qin, J. Sheng, Z. Yan, and G. Gela, “Accurate calculation of ion flow field under HVDC bipolar transmission lines,” IEEE Trans. Power Del., vol. 3, no. 1, pp. 368–376, Jan. 1988. [6] E. Tonti, “Finite formulation of electromagnetic field,” IEEE Trans. Magn., vol. 38, no. 2, pp. 333–336, Mar. 2002. [7] M. Sarma and W. Janischewskyj, “Analysis of corona losses on dc transmission lines Pt. II—Bipolar lines,” IEEE Trans. Power Appl. Syst., vol. PAS-88, no. 5, pp. 1476–1491, Oct. 1969. [8] G. B. Johnson, “Electric fields and ion currents of a 400 kV HVDC test line,” IEEE Trans. Power Appl. Syst., vol. PAS-102, no. 4, pp. 2559–2568, Aug. 1983. [9] Electric Power Research Institute (EPRI), “Transmission line reference book HVDC to 600 kV,” Palo Alto, CA.

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decrease when the Manuscript received April 24, 2006 (e-mail: [email protected]).