FOURTH

INTERNATIONAL SYMPOSIUM ON HIGH VOLTAGE ENGINEERING

13.01

ATHENS • GREECE, 5 - 9 SEPTEMBER 1983

ON THE BOUNDARY CONDITIONS FOR UNIPOLAR DC CORONA FIELD CALCULATION

I.R. CIRIC, E. KUFFEL Department of Electrical Engineering

University of Manitoba Winnipeg, Canada

Abstract

In this paper effects of different boundary condi­tions at the discharging conductor surfaces on the unipola~ components of de corona field are presented and a method for evaluating these boundary condi­tions from ground level measurements is proposed. For voltage levels at which the Deutsch's (first) assumption is applicable, the analysis is based on the general integral expression of the voltage­current relation of an elementary flux tube. Numerical calculation is illustrated on the cylinder-to-plane configuration.

Keywords

Corona-Boundary Conditions

1. Introduction

Analysis of the ionized field associated with corona discharges is important for the design of high volt­age de lines because of the resulting power losses and the accompanying environmental effects. The main theoretical difficulty in analyzing corona ionized fields is due to the fact that no electrical quantity is accurately known at the discharging electrode surface except the potential. Attempts [l-10] to determine corona loss and field quantities for practical arrangements such as cylin­drical conductors above the ground plane involve various simplifying assumptions. Many researchers [2-4,6,8,10] have assumed that the electric field at the surface of a conductor in corona remains approx­imately equal to its onset value, independently of the conductor voltage. Corona losses have also been calculated by assuming a known charge density dis­tribution at the discharging conductor surface [5]. Recently, the field quantities have been computed by considering a known current density [7] or even a constant charge density distribution [9] at the surface of each coronating conductor. The results obtained by means of various methods have been com­pared with experimental data available for some test cases, but no attempt has been done so far to com­pare theoretical results corresponding to different boundary conditions at the discharging conductor surface. In this paper the impact of various boundary condi­tions on the unipolar component of the de corona field is considered. For normal voltage levels, for which the Deutsch's (first) assumption is applicable [11], the analysis is based on the voltage-current relationship of an arbitrary elementary flux tube. On the other hand, the unipolar field quantities at a coronating conductor surface are obtained in terms of the field quantities at the surface of the corre­sponding conductor without corona. This allows the determination of the actual boundary conditions on the discharging conductors from accurate measure­ments at the ground level.

2. Analyds

The general equations governing the steady-state

1

unipolar de corona field represent the Gauss's law, the relationship between the current density and field intensity and the current continuity equation, respectively:

j • kpE

V•j • 0

(1)

(2)

(3)

where the electric field is derived from a scalar potential

E • -Vlfl (4)

with E 0 being the permittivity of free space and k the ionic mobility. The following assumptions are made: 1) thermal diffusion of ions neglegible; 2) constant mobility of ions;

•

Figure 1: General configuration with n ~onductors.

,3) potential Ill is a function of space-charge-free potential + only, i.e. electric field has the same direction as the space-charge-free field (l]

(5)

4) space charge is of the same polarity for each elementary flux tube shown in Figure 1, between a discharging conductor e and a conductor i with­out corona. As necessary boundary conditions, the potential on the conductors must be known:

with the potential on one conductor chosen as refer­ence.

The above assumptions and boundary conditions are not sufficient to solve for the ionized field and a supplementary assumption - boundary condition must be added for each coronating conductor, aa shown below. From equa tiona ( 4), (5), (1) and (2), the electric field intensity, the charge density and the current density can be expressed aa (2,4]

E•E'dlll dt

p • -E (E')2 d2\ll 0 d¥

j • - Eok (E')3 .!_ (dlll/ 2 d4i d+

(7)

(8)

(9)

where E • is the space-charge-free electric field intensity

I' • -V+ (10)

The continuity equation (3) is equivalent to the condition of a constant current along any elementary flux tube k

jAS • - l (E' )3 .!_ (dlll) 2

t.S • Al • conat (11) 2 d+ dt

in which AS is tube. Integrating

dlll -d+

thto cross-sectional area along the tube yields

(c + _2_ Al F(+) )l/2 E0 k A11>

where C ia a constant of integration,

of the

(12)

A!j> • E' t.S (13)

is the flux of space-charge-free electric field intensity, constant along the tube, and

Ve F(4i) : j ~

4' (E') (14)

the integration being performed along the mean flux line of the tube. The general integral form of the voltage-current relation for any specified elementary flux tube (Figure 1) is derived from the equations (6) and (12)

(15)

with C being a constant for a given tube. This constant ia to be determined by imposing an addi­tional boundary condition. Theoretically, this condition may be any given relationship involving E, p and j at the point on the conductor in corona or at the corresponding point on the conductor with­out corona. Once the constant C is determined, calculating the (Ve-Vi~ - AI characteristic (15) yields the local quantitiel! along the mean flux line of the tube, as functions of applied voltage Ve-Vi :

E • E'(C + 2-_ M_ F(+)]112

(16) t 0 k Aq,

j • E'~ , p • & (17)

The total ion current of a discharging conduc·tor, due to the unipolar c0111ponent of corona field, is obtained by integrating the current density (17) or, for a sufficient number of tubes, by adding all the tube currents related to the conductor.

3. Special cases of boundary conditions

We shall consider now the following three special types of boundary conditions at the surface of the conductor in corona, adopted so far by various researchers.

2

3.1. Electric field known on coronating conductor

If the electric field distribution at the coronating conductor surface is assumed to be known

El+•Ve •Ee

then from equations (7) and (12)

(18)

E 2 c • (...!.) , with E~ • E' l+-ve (19)

E~

Note that when the electric field at the surface of the conductor in corona is considered to remain con­stant at its onset value, the equations (16), (17) with (19) and (15) become those derived in [8] and are modified forms of the expressions previously obtained by Sarma and Janischewskyj {4). However the amount of computation associated with the formu­lation presented in this paper is substantially smaller, the knowledge of voltage-current relation (15) yielding directly the field quantities (16), (17), thus avoiding the usual iterative procedure £4 I.

3.2. Charge density known on corona.ting conductor

Let us assume now that the charge density distribu­tion at the surface of the conductoI in corona is known (5], [9]

(20)

Equations (2), (8), (11) and (12) yield in this case

c • (-1- AI)2 (21) k Pe A1!i

3.3. Current density known on coronating conductor

With (22)

where (AS)e is the cross-sectional area of the elementary flux tube at the coronating conductor surface, the current density and the electric field intensity along the mean flux line of the tube are given by the equations (17) and (16), wherein C is determined in terms of the applied voltage from the equation (15).

4. Evaluation of boundary conditions from measure­ments

On the basis of the analysis developed in Section 2, the quantities Ee , p and je at the discharging conductor surface can \e obtained in terms of those measured at the points on the surface of the conduc­tor without corona, for instance on ground in the case of HVDC transmission lines. Let ua denote by Ei , Pi and l1 the measured values of E, p and j , respectively, at the ground level. Only one of the distributions Ei , Pt and ji is necessary to be known in order to determine all the three quantities E , Pe and j • In effect, using the equations (12)-(?7) and (7)-(~) yields

E • Cl/2E' j • E' Al p e e•e et.1l>•e

(23)

where Al/A1ji is obtained from equation (15), with

Ei 2 2 Al C • (-) - - - F(Vi) when Ei ia known (24) Ej E0 k Alli

and 2 c • (-1- Al) - 2- Al F(V ) when Pi is known; (25) kpi A11> E 0 k A~ i

when j 111 known, C is determined from the equa­tion (15) with Al/Aq, replaced by ji/Ei •

5. Application to a cylinder-to-plane configuration

In order to illustrate quantitatively the influence of different boundary conditions at the surface of the conductors in corona upon the local quantities of the ionized field, we consider in this section two special cases relative to the well-known problem of a long circular cylindrical conductor parallel to ground.

x

+=o GROUND PLANE

Figure 2: Cylinder-to-plane configuration.

For this particular configuration the exact expres­sion of apace-charge-free field intensity is known (see for example (10]) and the integrals (14) can be calculated analytically. As a first special case we considered stant field intensity at the cylinder to its corona onset value E~ • E

0 ,

corona onset voltage V0

by tlO]

that of a con­surf ace, equal related to the

2 2 112

_H_. (H -r0 ) .ln{!!.... + [(!!....)2-l]l/2} Eoro Vo ro ro

(26)

The second special case is that of a constant charge density p0 at the cylinder surface which determines, for each voltage, the same total corona current as in the previous case [9]. In both cases the following normalized distributions of field intensity, current density and charge density E(N). ~E~,j(N).

V/H

were calculated for the range of H/r0

< 5000 and 1 < V/V

0 < 3 , at the cylinder surface and at the

ground level. Some typic~l distributions of the ratio of current density at ground obtained in the latter special case and in the previous one are presented in Figure 3. On the other hand, the quantities in (27) were cal­culated at the cylinder surface from their values at the ground level following the procedure developed in Section 4. A sample of these results is present­ed in Figure 4. All the calculations were performed with an accuracy of 10-2%,

6. Conclusions

The unipolar de corona field analysis presented, baaed on the voltage-current relationship of an elementary flux tube, is applicable to any geometri­cal configuration and any type of boundary condi­tions at the discharging conductor surface. For the first time, a method for evaluating the boundary conditions at the coronating conductor sur­faces from accurate ground level measurements has been proposed. Thie method is applicable to any geometrical configuration. The general formulation was applied to the cylinder­to-plane configuration. Comparison between numerical results obtained for two special cases of boundary

3

~ Cl) z

4.5

~ 3.5 ~ z LU

~ 2.5

0

LU > 1.5 ti ..J LU 0::

0· 5 _ .... 5..__.....__......_-'---'---o'---.._....__-'-_,_~5

Figure 3:

.-

25

>-t:: 20 Cl) z LU 0

~ 15 a: <l :c 0

~ 10 !:::! -' <l ~

~ 5 z

Figure 4:

y/H

Ratio of current densities on ground corresponding to the assumptions of a constant charge density and of a constant field intensity on corona ting cylinder, for H/r

0 • 1000 •

li.5000 ro 3000

1000 500 250 100 50 10

45 90 135 180 8 IN DEGREES

Charge density distribution (27) on cylinder, for V/V

0 • 2, calculated from a

ground level distribution corresponding to a constant field intensity on coronat­ing conductor.

conditions shows that the conditions adopted at the surface of conductor in corona have a considerable influence on the field quantities. For instance, as shown in Figure 3, the current density at the ground level, calculated on the hypothesis of a constant charge density at the discharging conductor surface, is a few times greater - for y/H > 1 - and smaller -for y/H ~ 1 - than the values corresponding to the assumption of a constant electric field at the conductor surface.

The authors intend to extend the study presented to bundle conductors and without using the Deutsch's (first) assumption, by applying the finite element method, Available experimental data at the ground level will be used in order to predetermine the actual boundary conditions at the coronating conductor sur­faces.

7. References

(1) Deutsch, w.: "Uber die Dichteverteilung Unipolarer lonestrome", Ann. Physik, Vol. 5, PP• 589-613, 1933.

[2] Popkov, V.I.: "On the Theory of Unipolar DC Corona". Elektrichestvo, No. 1, PP• 33-48, 1949, NRC Techn. Transl. Tr-1093.

(3) Tsyrlin, L.E.: "Some Questions in the Mathemati­cal Theory of the Corons Discharge at Constant Potential", J, Techn. Phys., Vol. 26, No. 11, pp. 2439-2453, 1956.

(4) Sarma, M.P., Janischewskyj, w.: "Analysis of Corona Losses on DC Transmission Lines: !­Unipolar Lines". IEEE Trans. Power Apparatus and Systems, Vol. PAS-88, PP• 718-731, 1969.

(5) Khalifa, M., Abdel-Salam, M.: "Improved Method for Calculating DC Corona Losses". IEEE Trans. Power Apparatus and Systems, Vol. PAS-93, PP• 720-726, 1974.

4

(6) Janischewskyj, W., Gela, G.: "Finite Element Solution for Electric Fields of Coronating DC Transmission Lines", IEEE Trans. Power Apparatus and Systems, Vol.~ PAS-98, PP• 1000-J.012, 1979.

(7) Sunaga, Y., Sawada, Y.: "Method of Calculating Ionized Field of HVDC Transmission Lines and Analysis of Space Charge Effects on RI". IEEE Trans. Power Apparatus and Systems, Vol. PAS-99, PP• 605-615, 1980.

(8) Parekh, H., Chow, Y.L., Srivastava, K.D.: "A Simple Method of Calculating Corona Loss ou Unipolar DC Transmission Lines", IEEE Trans. Electrical Insulation, Vol. El·· 15, No. 6, December 1980, pp. 455-460.

(9) Takuma, T., Ikeda, T., Kawamoto, T.: "Calculation of Ion Flow Fields of HVDC Transmission Lines". IEEE Trans. Power Apparatus and Systems, Vol. PAS-100, PP• 4802-4810, 1981.

(10] Ciric, l.R., Kuffel, E.: "New Analytical Expres­sions for Calculating Unipolar DC Corona Losses", IEEE Trans, Power Apparatus and Systems, Vol. PAS-101, PP• 2988-2994, 1982.

[11) Janischewskyj, w., Maruvada, P.S., Gela, G.: "Corona Losses and Ionized Fields of HVDC Transmission Lines". CIGRE-Session, Paris, France, 1-9 September 1982, Paper 36-09.