Modeling of corona under positive lightning surges

Journal of Electrostatics 71 (2013) 848e853

Contents lists available at SciVerse ScienceDirect

Journal of Electrostatics journal homepage: www.elsevier.com/locate/elstat

Modeling of corona under positive lightning surges Caiwang Sheng*, Xiaoqing Zhang School of Electrical Engineering, Beijing Jiaotong University, Beijing 100044, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 12 November 2012 Received in revised form 21 May 2013 Accepted 17 June 2013 Available online 29 June 2013

A novel corona model for predicting the corona characteristics under positive lightning surges is proposed. The ionization process is described by a series of successive generations of electron avalanches. A simplified method is presented for computing the electric field in the ionization zone. With a discrete treatment for the ionization process in time and space, the proposed model computes the total charge of corona from the applied surge voltage and so the qeu curves can be obtained. A laboratory measurement of qeu curves is also made by a coaxial cylindrical electrode to check the validity of the proposed model. Ó 2013 Elsevier B.V. All rights reserved.

Keywords: Corona characteristics Electron avalanches Ionization zone qeu curves

1. Introduction Lightning surges traveling on the transmission lines are attenuated and distorted by several factors. Impulse corona is probably the most significant one of these factors. It plays an important role in determining the magnitude and waveform of lightning overvoltages occurring at substations and is relevant to both lightning protection design and insulation coordination of electric equipment. In calculating the lightning overvoltages, the behavior of impulse corona is described by the chargeevoltage characteristic, i.e. the qeu curve. Therefore, the need exists for prediction of qeu curves from the corona model. A number of corona models have been presented in literature [1e4]. Some of them are based on the physical mechanism of corona discharge for prediction of the qeu curves of high voltage transmission line experiencing impulse corona. However, these models involve rather complicated computations, especially the computation of the electric field in the ionization zone. The other models made an attempt to find the numerical solution of the continuity equations for the motion of electrons and ions for simulating the growth of corona discharge in time and space. Since the ionization is strongly dependent on the electric field, special care needs taking to choose a feasible numerical method for solving the electric field. The problem of high computational complexity and numerical instability are often

* Corresponding author. Tel.: þ86 15210576828. E-mail addresses: [email protected], (C. Sheng).

[email protected]

0304-3886/$ e see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.elstat.2013.06.008

encountered in the numerical solution process. In the light of the existing situation, a realistic corona model is proposed in this paper. The proposed model takes into account all the microcosmic evens, namely ionization, attachment, photoionization, etc. The ionization process is discretized in time and space. A simplified method is given for computing the electric field in the ionization zone. In terms of the applied surge voltage, the proposed model can quantitatively predict the qeu curves. Furthermore, an experimental setup is built and the qeu curves are measured in a coaxial cylindrical electrode. The comparison is made between computed and measured results to confirm the validity of the proposed model.

2. Corona model For a simplified description of the corona phenomenon around a high voltage transmission line, the corona model is set up on the basis of the coaxial cylindrical electrode since its geometry of electric field approximates to the high voltage transmission line. The radius of the inner electrode is R1 and that of the outer electrode is R2, as shown in Fig. 1. The corona discharge process is considered as a series of successive generations of electron avalanches which develop in the ionization zone around the inner electrode. The ionization zone is defined as the space where the resultant field strength is so high that the first coefficient of ionization (a) is greater than the coefficient of electron attachment (h); i.e. a
h [5]. Its boundary is initially cylindrical with radius rb.

C. Sheng, X. Zhang / Journal of Electrostatics 71 (2013) 848e853

849

Fig. 2. Schematic diagram of the initiatory electrons.

Zr

Fig. 1. The ionization zone of coaxial cylindrical electrode.

q1 ð1; iÞ ¼ q0 ð1; iÞ  exp 2.1. Initiatory electrons The applied surge voltage is represented by the double exponential function:

uðtÞ ¼ Um ½expðbtÞ  expðgtÞ

(1)

where Um, b and g are the constants determined by data-fitting of the surge voltage waveform. For a positive applied surge voltage, the initiatory electrons for starting the ionization process are those available in the atmosphere around the inner electrode. The number of the initiatory electrons N0 is expressed as [6]

N0 ¼ n0 

1 T0

ZT0 ZR1 0

ð1  h=aÞ2prdrdt

ða  hÞdr

i ¼ 1; 2; .; M

(3)

rb

where the subscript “” stands for the polarity of the line charge. When the first generation avalanches arrive at the surface of the inner electrode, they will end here. The positive ions left behind each electron avalanche are assumed to be a sphere of radius rs, as shown in Fig. 4. rs is approximately taken as the radius of the avalanche head rh [8]:

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u Zr u pffiffiffiffiffiffiffiffiffi DðrÞ u dr rs zrh ¼ 2 6Ds ¼ 2t6 Ve ðrÞ rb

(2)

rb

where T0 is the time corresponding to the coronal inception voltage U0, and n0 is the rate of primary electrons available per unit volume which depends on the steepness of the applied surge voltage [7]. These initiatory electrons are symmetrically distributed on a contour of radius rb and around the inner electrode, as shown in Fig. 2, and will start the first generation avalanche. 2.2. The first generation avalanche As the applied surge voltage exceeds the inception value (U0) slightly, the initiatory electrons are accelerated by the applied field from the boundary of the ionization zone to the inner electrode. Each initiatory electron (i ¼ k) will trigger an electron avalanche, in which more electrons and positive ions are created, as shown in Fig. 3. The charge of the initiatory electrons per unit length is represented as line charge q0(1, i) (i ¼ 1, 2, ., M). As the avalanches proceed toward the inner electrode, the produced charge q1(1, i) (i ¼ 1, 2, ., M) on each line can be calculated by

Fig. 3. Growth of first generation avalanche.

(4)

850

C. Sheng, X. Zhang / Journal of Electrostatics 71 (2013) 848e853

charges. The field component generated by the applied surge voltage is expressed by

E1 ¼

uðtÞ r  lnðR2 =R1 Þ

(6)

The number of the line charges on a circumference is taken to be large enough and the line charges are highly concentrated on the circumference. Therefore, the field distribution generated by the line charges approximates to that by the cylindrical shell with charge density. For simplifying field calculation, the space line charges are confined a cylindrical shell of thickness rh given by (4), as shown in Fig. 5. The charge density lþ is given by

lþ ¼

Qþ1

h

p ðr þ s þ 4rh Þ2  ðr þ s þ 2rh Þ2

i (7)

Qþ1 ¼ 4prh ðr þ s þ 3rh Þ where Qþ1 is given by

Qþ1 ¼

M X i¼1

qþ1 ð1; iÞ

(8)

The charge density le is given by

Fig. 4. Schematic diagram depicting the location of line charge.

Q1

where D is the electron diffusion coefficient, s the life of the avalanche and Ve the electron drift velocity. For the positive ion spheres, the related charge on each line per unit length is represented as line charge qþ1(1, i) (i ¼ 1, 2, ., M) and determined as

l ¼

qþ1 ð1; iÞ ¼ q1 ð1; iÞ  q0 ð1; iÞ

while Q1 is calculated by replacing qþ1(1, i) with q1(1, i) in (8). Based on this simplified treatment, the field component contributed by the space charges can be evaluated by the Gauss theorem. Thus, the resultant field can be determined by summing of these two field components.

(5)

The line charges qþ1(1, i) per unit length are located around the inner electrode with a distance s ðs ¼ 1=ða  hÞÞ from the line charges q1(1, i). The line charges q1(1, i) are accelerated by the resultant field consisting of the applied field and the field due to the space charge. With time the line charges q1(1, i) and, subsequently, qþ1(1, i) accelerate radially until the electrons q1(1, i) are absorbed by the inner electrode. Thereafter, the positive line charges qþ1(1, i) adjacentpffiffiffiffiffiffiffiffiffiffiffi to the inner electrode with the thickness of r1 ðr1 ¼ 2 6Ds1 Þ. s1 is the lifetime of the first generation avalanche. 2.3. Calculation of resultant field During the growth of the first generation avalanche, the resultant field is generated by the applied surge voltage and the space

h

p ðr þ 2rh Þ2  r2

i (9)

Q1 ¼ 4prh ðr þ rh Þ

2.4. The second generation avalanche The photons are emitted to trigger the second avalanches during the growth of the first generation avalanche, as shown in Fig. 6. In fact, the photons are emitted in all directions. For the sake of simplification, the photoelectrons produced by photoionization are assumed to be uniformly distributed in the radial direction. Therefore, the ionization zone is divided into cylindrical shell contours of radii rc(j) (j ¼ 1, 2, ., N) and line charges q02(j, i) can be located for starting the second generation avalanches, as shown in Fig. 7. The equation describing the photoionization phenomena has

Fig. 5. Schematic diagram of calculating the field produced by space charge.

C. Sheng, X. Zhang / Journal of Electrostatics 71 (2013) 848e853

851

charge, f1 is the photoionization coefficient determined experimentally as a function of pressure times distance, f2 is a dimensionless function evaluated numerically [9], and p is the pressure of the air surrounding the inner electrode. As shown in Fig. 7, there are a series of line charges q02(j, i) (j ¼ 1, 2, ., N) distributed radially. As for the first generation avalanche, the lifetime of the second generation avalanche s2 is the time required for the line charge q02(1, i) located close to the boundary of the ionization zone to grow until it comes closer to the positive space charge left by the first generation avalanche. The photoelectrons produced during the growth of each avalanche in the second generation are calculated in the same way stated above. The positive ions produced in second generation extend p the space charge region around the inner electrode by r2 ffiffiffiffiffiffiffiffiffiffiffi ðr2 ¼ 2 6Ds2 Þ. The photoelectrons are produced in the growth of the second generation avalanches which start at the different contours of the ionization zone, i.e., at the radii rc(j) (j ¼ 1, 2, ., N). They will be emitted to trigger the third generation. 2.5. The subsequent generations avalanches

Fig. 6. Schematic diagram depicting the process of photoionization.

been given in Ref. [9]. The number of photoelectrons generated at a cylindrical surface is expressed by

Mq02 ðj; iÞ ¼ e

rZ¼ R1

r ¼ rc ðjÞ

M X ðqe1 ð1; iÞ=eÞ  f1 p i¼1

2prc ðjÞ dr 2prc ðjÞ pffiffiffiffiffiffiffiffiffiffi rc ðjÞ  r þ f2 r1  p  i (10)

where q02(j, i) (j ¼ 1, 2, ., N and i ¼ 1, 2, ., M) is line charge initiating the second avalanche at the jth contour, e is electron

Fig. 7. Growth of second generation avalanche in the positive corona.

The third generation avalanche grows as previously discussed. The line charges q03(j, i) (j ¼ 1, 2, ., N and i ¼ 1, 2, ., M) starting their growth are accelerated by the resultant field which is calculated by the above method. Thus the growth of avalanche generation will continue until the ionization zone is filled with space charge. 3. Experimental verification In order to verify the validity of the corona model proposed above, an experimental setup is built to measure the qeu curves in high voltage laboratory of our university, as shown in Fig. 8. The inner electrode of the coaxial cylindrical electrode system is a copper tube of radius 2.5 mm, while the outer coaxial cylindrical electrode consists of three sections of radius 0.5 m. The capacitor connected between the main cylinder (the middle section) and earth is used to measure the charge due to corona discharge. For reduction of the edge effect on electric field two grounded guard (0.5 m) cylinders are set on both sides of the main cylinder (1 m). The surge voltages generated by an impulse generator are applied to the inner electrode to generate corona discharge. The waveform of u is measured through a voltage divider (consisting of C1 and C2) and the waveform of q is measured by the integration of the voltage across Cq. The signals u and q are recorded by the digital oscilloscope and so the qeu cures can be obtained. The qeu cures

Fig. 8. Experimental setup.

852

C. Sheng, X. Zhang / Journal of Electrostatics 71 (2013) 848e853

Fig. 9. Computed and measured qeu curves for coaxial cylindrical electrode under different applied surge voltages, (a) 84/130 kV, 1.2/50 ms surge voltage, (b) 72/130 kV, 5.7/50 ms surge voltage, (c) 84/130 kV, 1.2/90 ms surge voltage and (d) 84/130 kV, 1.2/20 ms surge voltage.

measured at several values of waveform parameters of the applied surge voltages, as shown in Fig. 9, where the corresponding computed cures are given simultaneously for comparison. It can be seen from Fig. 9 that a better agreement appears between computed and measured results.

4. Conclusion A corona model has been proposed for predicting the qeu curves of high voltage transmission lines under the positive surge voltages. The model has the capability of taking into account the main processes and events of corona, namely ionization, attachment, photoionization, etc. In the model, the simplified computation has been performed for the electric field generated by the space charge. The discretization procedure in time and space has been given for a quantitative description of the ionization process. The qeu curves can be obtained by using a MATLAB program developed on the basis of the discretization procedure. The laboratory measurement of the qeu curves has also been carried out by a coaxial cylindrical electrode system. The measured qeu curves

agree reasonably with the computed ones, which confirm the validity of the proposed model.

Acknowledgment The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China under contract no. 50977002.

References [1] M. Khalifa, R. Radwan, A. Zeitoun, A. Abdel-Fattah, Computation of corona current and its effect on travelling surges. Part I e case of positive lightning surge, IEEE Trans. Power App. Syst. 89 (1970) 1816e1825. [2] S. El-Debeiky, M. Khalifa, Calculating the corona pulse characteristics and its radio interference, IEEE Trans. Power App. Syst. 90 (1971) 165e179. [3] A.J. Davies, C.S. Davies, C.J. Evans, Computer simulation of rapidly developing gaseous discharges, Proc. Inst. Electr. Eng. 118 (1971) 816e823. [4] I.P. Vereshchagin, I.V. Pashinin, A.A. Beloglovskii, Mathematical Modelling of Processes in a Pulsed Streamer Corona: Methods of Solving the Equations of Continuity of a Flow of Particles. Vestnik MEI, No. 2 (2004).

C. Sheng, X. Zhang / Journal of Electrostatics 71 (2013) 848e853 [5] M. Khalifa, M. Abdel-Salam, Improved calculation of corona pulse characteristics, IEEE Trans. Power App. Syst. 93 (1974) 1693e1699. [6] M. Abdel-Salam, E. Keith Stanek, Mathematicalephysical model of corona from surges on high-voltage lines, IEEE Trans. Ind. Appl. 23 (1987) 481e489. [7] Klaus Ragaller, Surges in High Voltage Networks, Plenum, New York, 1980.

853

[8] R.C. Fletcher, Impulse breakdown in the 109 sec. range of air at atmospheric pressure, Phys. Rev. 76 (1949) 1501e1511. [9] G.L. Smith, Distributed source corona photoionization calculations applicable to finite element computer models, in: Proceedings of IEEE-IAS Annual Meeting, Mexico City, Mexico, 1983, pp. 1204e1209.