on overhead power line transients is developed based on a concept of time delay in corona formation. This concept gives a better understanding of many ...
IEEE Transactions on Power Delivery, Vol. 4, No. 2, April 1989
1145
A PRACTICAL MATHEMATICAL MODEL OF CORONA FOR CALCULATION OF TRANSIENTS ON TRANSMISSION LINES Xiao-rong Li Zhejiang University Hangzhou, P.R.C.
Keywords:
O.P. Malik, FIEEE University of Calgary Calgary, Alberta, Canada
Zhi-da Zhao, SMIEEE Zhejiang University Hangzhou, P.R.C.
Corona model, transmission l i n e s , transients.
ABSTRACT A practical mathematical model of corona e f f e c t on overhead power l i n e transients i s developed based on a concept of time delay i n corona formation. This concept gives a b e t t e r understanding o f many important experimental phenomena and improves the accuracy of the model. Good agreement i s shown between calculated and experimental q-v curves for many cases. The model has no l i m i t a t i o n i n application t o l i n e s with unknam q-v characteristics because a l l model parameters can be determined without knowing the experimental q-v characteristics of the l i n e i n question. INTRODUCTION Overvoltage protection and insulation coordination are based to a large degree on the knowledge of the magnitudes and waveforms of overvoltages. Two kinds of significant phenomena affect the attenuation and distortion of overvoltages propagating along overhead transmission lines: (a) variations of transversal parameters due to corona and (b) frequency dependence of longitudinal parameters caused by imperfect ground return and skin effect of conductors. Through many years' efforts, especially those in 1970's, satisfactory results have been achieved in the Owing to the study of frequency dependence [l]. complexity of the physical phenomenon of corona, a satisfactory approach to include corona effect in the calculation of 'travelling waves has not been developed, although many attempts have been made since the beginning of this century. The main problem is the lack of a sophisticated and practical corona model. In view of the fact that corona has probably a stronger influence on surges than frequency dependence, especially for surges with high peak values and short wavefronts, it is important to establish a good corona mode1. The techniques proposed in the literature to include corona effect in transmission line transients calculations may be classified into two methodologies. In the first methodology, a number of mathematicalphysical models have been developed on the basis of the physical mechanism of corona to obtain charge versus voltage characteristics in line transients computations [2-61. Taking both space and surface charges into account, the multilayer model presented in Ref. [6] is probably the best one of this type so far. However, the model contains a number of important numerical parameters to be chosen by a trialand-error method to match with the experimental results of the line in question. This is obviously impossible for those lines for which experimental data
8 8 SM 579-5 A paper recommended and approved by the IEEE Transmission and Distribution Committee of the IEEE Power Engineering Society for presentation at the IEEE/PES 1988 Summer Meeting, Portland, Oregon, July 24 - 29, 1988. Manuscript submitted January 22, 1988; available for printing May 13, 1988.
is not available. Other mathematical-physical models, described in Refs. [5, 7, 81, tend to be fairly complicated. Other papers published using the second methodology do not deal with the physical mechanism of corona. Earliest studies [9-111 calculated the corona effect on travelling waves by a so-called time-lag approach, in which the time lag of surge fronts due to corona was assumed to be constant per length of a line. This approach has low accuracy. More recently, a number of functions, such as polynomials, have been proposed to approximate the q-v curves [12-151 based on the analysis of the experimental relationship between the charge, q, stored by a line and the line to ground voltage, v . These approaches, however, have a common difficulty of predicting the q-v characteristics of the line beforehand. Moreover, it seems not quite proper to choose corona characteristic parameters by fitting calculated waveforms to those of experiments, as in Ref. [15]. Other authors introduced nonlinear branches of resistance and capacitance in the transmission line to represent the nonlinear effect of corona. The values of the elements in the branches are voltage dependent and are determined in most cases according to the energy relationship of corona [16-181 or the experimental q-v curves [191. Peek's quadratic law of AC corona loss is often applied to transient corona in this case, which is rather questionable. A practical model using the image method to calculate the dynamic capacitance of the line in corona is presented in this paper. It has the advantage that all parameters can be determined beforehand according to the condition of the line in question. As a result, even for those lines for which experimental data is unavailable, it is still applicable. A l s o , a concept of time delay in the formation of corona space charges is introduced to make the model more realistic and accurate both in simulation results and in understanding of many important characteristics of q-v curves. The model can be used to calculate corona effect on travelling waves.
BRIEF REVIEW OF EXPERIMENTAL Q-V CHARACTERISTICS It is difficult a s well as time consuming to determine experimentally charge vs voltage characteristics with good accuracy. Therefore, the results of only a few experiments are available in the literature 110, 20-231. More experiments with high accuracy are obviously needed since differences between these published results exist and some details of the q-v curves remain unclear. Nevertheless, it has the following general characteristics. 1) The q-v curve of corona follows a hysteresis loop as the voltage increases first and then decreases (Fig. 1). Mathematically, charge q is not a unique function of voltage v, and therefore it is necessary to distinguish the rising and the falling parts of the voltage. 2) The area enclosed by the hysteresis loop represents the energy loss during corona.
0885-8977/89/O400-1145$01 .WO1989 IEEE
1146 3) In the rising part of v, q increases proportionally provided that v is smaller than corona onset value, as shown in section OA of Fig. 1. The slope of the curve in this section is defined as the geometric (natural) capacitance (C of the line). g
PROPOSED MODEL Time-varying spacial electromagnetic fields propagating along fixed cross-section power lines are basically parallel plane fields even during corona. Maxwell's equations and boundary conditions take identical forms as those for static electric and magnetic fields. Therefore, the method of images in electrostatic solution is applicable, though small errors due to edge effect will arise if it is applied to finite lines. This must be borne in mind when comparing the calculated results to experimental data obtained from short sample conductors. The correct equivalent electric axis of a single conductor, b,
should
be
equal to
above
ground, where h is the average height of the line and r is the radius of the conductor (Fig. 3). In 0
practice, h is several orders of magnitude bigger than fig. 1 q-v hysteresis loop. 4) Corona onset value depends not only on conductor surface conditions, atmospheric condition, and line configurations, such as conductor sizes and number of subconductors, but also on the waveshapes and the polarities of the voltages. For instance, the onset value is greater for steep front voltages and for negative polarity. 5 ) After v reaches the onset value and continues to rise, corona explodes and develops. This results in q increasing in such a way that the derivative of the q-v curve, i.e. the dynamic capacitance Cd of the
This makes it ro, so b has a value very close to h. reasonable to assume the electric axis coincident with the geometric axis. Furthermore, b will change its value as corona develops. For simplicity, therefore, a continuous cylindrical symmetrical discharge is assumed and its electric axis is assumed to be coincident with the geometric axis of the conductor.
line, is kept increasing continuously. In other words, an increase in v will give rise to an increase in Cd. See section AB in Fig. 1. 6) As illustrated in Fig. 2 and in those q-v diagrams for impulse with similar waveshapes but different amplitudes, when two voltage waves having different steepness of wavefronts reach the same instantaneous value, the charge ql, produced by the
h
I
I i Fig. 3 Relation between h and b.
steeper voltage vl, is smaller than the charge, produced by
the less steep wave, v2.
92 Consequently,
the steeper voltage produces a narrower hysteresis loop, which results in a smaller loss of energy. 7) After v reaches its crest value (point B in Fig. 1) and begins to decrease, q continues to increase for a while until it reaches its own maximum (point C) , which is usually not far from point B and is related to the steepness of the voltage front. 8 ) As v drops from point C to point D or to zero as the case may be, q decreases followed by a quasilinear curve almost parallel to (or to be more accurate, just a little steeper than), section OA. 9) Point D can only appear in the case of thick conductors at the time that voltage reduces to a value close to corona threshold. 10) I f point D exists, q will decrease much more rapidly than before as v returns from point D to zero
Based on the above, the per unit capacitance and hence the q-v curves of an overhead line in corona can be obtained as follows. Corona Onset Values To calculate the dynamic capacitance, it necessary to determine corona onset values first.
is
The critical field strength E of air in the vicinity of a practical power line ctfn be expressed as = gom6f (1 + 0 . 3 / G 0 ) EC
where g is the critical strength of air in a uniform field, gsually equal to 30kV/cm m is the surface irregularity factor of the conductor and is commonly set to 0.75 6 is the relative air density, equal to 1.0 in the following examples f is the polarity factor of the voltage. The influence of rain can be introduced in factor m. Factor f j s given a higher value for negative polarity (1.0) than for positive polarity (0.51 as by other authors [71. The critical charge Q
bound on the conductor is
thus 2mr E o c Qc ~
Pig. 2 Effect of wavefronts.
=
2h/(2h-ro) '
(2)
1147 and the critical voltage V 211
-
r
Vc = Ecro 2h -
is found to be 2h
? ( n a
-
r (3)
O).
For fast transient vol ages, it is not appropriate to neglect the differeke between the time that the applied voltage v(t) reaches a value and that all its resultant corona charge q'(t) is measured. In other words, at the time that all the corona charge q'(t) produced by v(t) is measured in a q-v test, the voltage has changed its value to v(t + T ) , where T C
2h
v(t) = aE
the value of T
(rc
C
q'(t) in terms of the corona physical mechanism. This concept is important in the establishment of corona model suitable for fast transient voltages, for which v(t + T ) could be quite different from v(t) because is in the order of 0.1 us 1241.
r
2h
-
r
C
Hence, the line-to-ground voltage is
-
2h
is the time delay in the formation of new corona charges. In experiments, therefore, v(t + Tc) and q'(t) are recorded at the same time as a related pair and plotted as a point in the q-v curves. As a matter that produces of fact, it is v(t) instead of v(t + T
-
2 Ln (C) r
=aEr c c
+ciErc c
ro) +
-
r 2hC
2h
-
r (7)
Consequently, the outer radius of the cylinder can be calculated iteratively using the following formula
If r (t) is known, the total charge q'(t) can be found fromceqn. (5) as 2h
In the application of this concept to the onset value in the q-v curves where dv/dt has a sudden change, an apparent corona onset voltage, VCa, is introduced:
q'(t) = 2nsaE r (t) c c
-
rc(t) (9)
2h
q'(t) consists of two parts, q (t) on the conduc1 Charge ql(t), equal to
tor and q2(t) in the cylinder. C in which t
is the critical
time of the
corona for voltage v(t), given by V
=
explosion of
v(tc).
9
~ ( t ) ,has no time delay in measurement, whereas at
least a part of q2(t), corresponding to the new corona charges, will be formed with a time delay and thus will not be measured until time (t
Geometric Capacitance Dividing eqn. (2) by eqn. (3) yields the geometric capacitance:
=
2h c c
.
With this assumption, the voltage drop on the corona cylinder is: r rc (4)
Therefore,
).
+ T
in q-v
ql(t + Tc) + q2 (t)
= 2nsaE r (t)
For simplicity, it is assumed that the electric field in the corona cylinder is constant and equal to cil: The introduction of coefficient ci is based on thg results of a direct field measurement described in Ref. [251 and the theoretical results calculated in Ref. 1261. Both of these results indicate that the surface yield of a conductor decreases to about 90% of critical value after corona occurs. The value of a is, therefore, set at 0.9 in the following examples, as is also used in Ref. [27].
T
experiments is actually q(t + Tc)
Dynamic Capacitance
+
the total charge measured at time (t
-
rc(t) 2h
+
Figure 4 is drawn for illustration. The actual experimental q-v characteristic is represented by curve A (solid line). Curve B, corresponding to the first term of the right hand of equation (101, is an ideal q-v curve under the circumstance of no time delay in corona charge formation. At the two differv interent points that a line parallel to q = C 9
-
sects curve B and qurve A , the corresponding voltages are v(t) and v(t + Tc), respectively. The difference between curves E! and A depends upon the steepness of the surge fronts. This difference is quite clear under fast transient voltages. When a slow transient voltage is applied, however, v(t + T ) is so close to C
where r
is the outer radius of the cylinder.
Applying the above assumptions to r
=
r
, gives
v(t) that the difference is no longer noticeable (the former tends to the latter). It should also be noted in Fig. 4 that it is not q2(t + Tc), but q3(t+Tc), the charge difference
(5)
between curves E! and A for the same instantaneous voltage value, that cannot be measured at time (t + T but will be measured at time (t + 2T 1.
in which q' is the total charge on the conductor and in the cylinder. Thus, the voltage between the outer radius of the cylinder and ground is given by
From Fig. 4, many important characteristics of q-v curves can be explained easily by the proposed model, such as a) The steeper the surge front is, the higher
1148 the observed onset values. But for slow transient voltages, the difference is negligible. b) A fast transient voltage produces a narrower and lower q-v loop than a slow one, as in Fig. 2.
corresponding computed results of the proposed model under identical conditions are also given in these figures. Results for positive fast impulse, negative fast impulse, positive slow impulse and negative slow impulse are shown in Figs. 5, 6, 7 and 8 respectively. Results for a different line geometry are shown in Fig. 9. It is evident from Figs. 5-9 that the proposed model has good accuracy for both fast and slow transient voltages under both positive and negative polarities. m
Fig. 4
c ca Effect of time delay in corona.
-~
Actual experimental q-v curve q-v curve without time delay in corona --q(t) = c e v(t) g c) The charge produced during corona will not reach its maximum until the crest value of the applied fast transient voltage is over for a short time depending mainly on the steepness of the front as described in the preceding section.
-------
After the total charge q(t) is calculated, the computation of the dynamic capacitance becomes very easy:
i.
13.
&
ru. I -
Experimental Results
10
When v(t) begins to decrease, q(t), still continues to increase. According to eqn. (U), C (t) has d negative values, as also observed in experimental results (section CD of Fig. 1). It is thus shown that eqn. (11) holds true until q(t) reaches its maximum *ma As v(t) reduces after q(t) reaches Qm, in addi-
IQ
tion to the decrease in charge bound on the conductor, given by Aq (t) = C Av(t), corona space charge will 1 g also decrease slowly. As a result, the decreasing speeds of the total charge q(t) are always slightly higher than C until v(t) reduces to corona extinction 4 value (point D in Fig. 1). Considering that corona effect on surge wavetails is small and that wavetails have less importance than wavefronts in overvoltage protection and insulation coordination, a constant decreasing speed of q(t) equal to C is assumed in this paper for convenience. g
lif
(b) Computed results Fig. 5
RESULTS OF THE MODEL To check the validity of the model, it is desirable to compare the computed results based on this model and the experimental data. Unfortunately, among the very limited number of experimental results available in the literature, few are suitable for this purpose. Results given in Ref. [ 2 3 1 are probably the best for this purpose. The experimental q-v diagrams in Fig. S-9 are reproduced from Ref. [231. The
Comparison for r
= 13.2mm,
+1.2/50!ls.
1149
(a) Experimental Results
"t (a) Experimental Results
(b) Computed results Fig. 6
Comparison for r
= 13.2m, -1.2/5OW.
(b) Computed results
(a) Experimental Results
Fig. 8
Comparison for r
= 13.2",
-10/75ps.
1150 REFERENCES
(a) Experimental Results
't
(b) Computed Results Fig. 9 Comparison for r
=
5 m , -1.2/50~~.
CONCLUSIONS 1. A fairly simple model of corona effect for calculation of transients on transmission lines has been developed using the method of images. 2. Results computed by this model agree well with experimental q-v characteristics for both fast and slow impulse voltages under both polarities. 3. The introduction of a concept of time delay in the formation of corona space charges in the model leads to better results and better understanding of many important characteristics of q-v curves, especially for fast transient voltages. 4. The model is applicable for transmission lines with either known or unknown q-v characteristics, because all the model parameters can be determined beforehand according to the conditions of the line in question. 5. Though only single-conductor single-phase lines are considered in this paper, the model can be extended to multi-conductor and multi-phase cases without much difficulty. 6. The model developed can be used to calculate the propagation of surges on overhead lines including corona effect.
1141
1151
161
171
J.R. Marti, "Accurate Modelling of FrequencyDependent Transmission Lines in Electromagnetic Transient Simulations", IEEE Trans. on Power Apparatus and Systems, Vol. PAS-101, No. 1, 1982, pp. 147-157. J.J. Clade, C.H. Gary, and C.A. Lefevre, "Calculation of Corona Losses Beyond the Critical Gradient in Alternating Voltages", IEEE Trans. on Power Apparatus and Systems, Vol. PAS-88, No. 5, May 1969, pp. 695-703. M. Afghahi and R.J. Harrington, "Charge Model for Studying Corona during Surges on Overhead Transmission Lines", IEE Proc., Vol. 130, Pt. C, No. 1, Jan. 1983, pp. 16-21. R.J. Harrington and M. Afghahi, "Implementation of a Computer Model to Include the Effect of Corona in Transient Overvoltage Calculations", IEEE Trans. on Power Apparatus and Systems, Vol. PAS-102, No. 4, April 1983, pp. 902-910. Zhi-ying Wang and Yu-yi Zhao, "A Model of Corona on Transmission Lines", Paper presented at the Annual Meeting on High Voltage Engineering, Yichang, China, Nov. 1985, pp. 1-14. A. Semlyen and Huang Wei-Gang , "Corona Modelling for the Calculation of Transients on Transmission Lines", IEEE Trans., Vol. PWKD-1, No. 3, July 1986, pp. 228-239. N.L. Ouick and G.L. Kusic, "Including Corona Effects for Travelling Waves on Transmission Lines", IEEE Trans. on Power Apparatus and Systems, Vol. PAS-103, No. 12, Dec. 1984, pp. 3643-3650. M. Khalifa, R. Radwan, A. Zeitoun and A. AbdelFattah, "Computation of Corona Current and Its Effect on Travelling Surges. Part I - Case of Positive Lightning Surges". IEEE Trans. on Power Apparatus and Systems, Vol. PAS-89, No. 8, Nov./Dec. 1970, pp. 1816-1825. H.H. Skilling and P. de K. Dykes, "Distortion of Travelling Waves by Corona", Electrical Engineering, Vol. 56, July 1937, pp. 850-857. C.F. Wagner and B.L. Lloyd, "Effects of Corona on Traveling Waves" , AIEE Trans. on Power Apparatus and Systems, Vol. 74, Part 111, Oct. 1955, PP. 858-872. N. Hylten-Cavallius and P. Gjerlov, "Distortion of Travelling Waves in High-Voltage Power Lines", ASEA Research, No. 2, 1959, pp. 147-180. G. Gela and W. Janischewskyj, "Surges on SingleConductor Transmission Lines Exhibiting Effects of Frequency and Corona", Proc. of International Symposium on Circuits and Systems, IEEE, 1976, pp. 614-617. M. Mihailescu-Suliciu and I. Suliciu, "A Rate Type Constitutive Equation for the Description Of the Corona Effect", IEEE Trans. on Power Apparatus and Systems, Vol. PAS-100, No. 8, Aug. 1981, -pp. - 3681-3685. C. Gary, A. Timotin and D. Cristescu, "Prediction of Surge Propagation Influenced by Corona and Skin Effect", IEE Proc., Vol. 130, Pt. A, No. 5, July 1983, pp. 264-272. A. Inoue, "Propagation Analysis of Overvoltage Surges with Corona Based Upon Charge Versus Voltage Curve", IEEE Trans. on Power Apparatus and Systems, Vol. PAS-104, No. 3, March 1985, pp. 655-662. S. Hayashi, J. Umoto and E. Nakamura, "Surge Analyser to Analyse Attenuation and Distortion of Surges on Single-Conductor Systems Caused by Corona Loss", Journal of Institute of Electrical Engineers in Japan, Vol. 86-1, No. 928, Jan. 1966, pp. 108-114. K.C. Lee, "Non-Linear Corona Models in an EMTP", IEEE Trans. on Power Apparatus and Systems, Vol. PAS-102, NO. 9, Sept. 1983, pp. 2936-2942.
1151
and C.H. Shih, " A Nonlinear Circuit for Transmission L,ines in Corona", IEEF: Traris. on Power Apparatus arid Systems, Vol. PAS-lGC, No. 3, March 1981, pp. 1420-1430. D.F. Oakeshott arid R.T. Waters, "Non-Uniform Field Breakdown: Engineering Models", Gaseous Dielectrics 111, edited by L.G. Christophorou, Perqamon, 1982, pp. 103-118. K. Davis and R.W.G. Cook, "The Surge Corona Discharge", IEEE Proc. Vol. 108, Pdrt C, 1961, pp. 230-239. G.N. Aleksandrov arid G.A. Shcherhakova, " C o r o n a dur ing Sw itch ing D isch S. rge Characte 1- 1 stic s Surqes", Electric Technology USSR, Vol. 4 , 1968, pp. 117-128. P.S. Maruvada, H . Merlemerilis ar:d R. Malewski, "Corona Characteristics of Conductor Bundles Under Impulse Vo3taqes", IEEE l'rans. on Power Apparatus dIld Systems, Vol. PAS-96, NO. 1, Jan./Feb., 1977, pp. 102-115. C. Gary, G. Draqan and D. Critescu, "Attenuation of Travelling Waves Caused hy Corona", CIGRE Paper, No. 33-13, 1978, pp. 1-38. L . 1 1 . Fisher sild R. Bederson, "Formative Time Lags o f Spark Breakdown 1 1 1 Air in U n i f c r r i Fields at Low Overvoltaqes", Physical Review, 2r.d Series, Vol. 81, Jan. 1 3951, pp. 109-114. R.T. Waters, T.E.S. Rickard arld b:.k. Stark, "Direct Measurement. of Electric F l c l d at Lire Cocductors during A . C . Corana", Frc:. :Et, 701. 119, No. 6, June 1972, pp. 717-7;:. PI. Khalifa and M. Abdel-Salar, "Calc,::it;F.q the Surface Fields of Ccr6uctors lr 2c lEF:, Vol. 1 3 0 , No. 12, Dec. 1973, CI;. 15-4-13-j. M. Abdel-Salam, M. Farqhaly a n d S. &5e:-;3::a:, "Monopolar Coroiia cr Bundle Co:.s~:ctcrz", ~EEE Trans. on Power Apparatus 3 PAS-101, NO. 10, Oct. 1CIF2, p s . H.M.
Kudyan
Model
Xiao-ronr Li .*.*:asbo:-:-. 1:-. Fuzhou, P . R . of China on Se?:t..rhir 2 1 , 1959. He received his B . S . and !.:.S. degrees in E1ec:riczl Enginiarir.6 from Zhejiang Universit:;, E a n g z h o u , Zhej iang, P.R . of Ct1ir.a. in 1982 and 1984, re sp ect i7.e 1y . Since 1985 he has been a Ph.D. student in Electrical Engineering at . Zhejiang University. From September 1986 to August 1987 he spent one year doing research in power transmission at the University of Calgary, Calgary, Alberta, Canada. He is now doing research in Electrical Engineering and Materials Science at the University of Connecticut, Storrs, Connecticut.
O . P . Malik (M'66 - SM'69 - F'87) graduated in electrical engineering from Delhi Polytechnic, India, in 1952 and obtained the M . E . degree in electrical machine design from the University of Roorkee, India, in 1962. In 1965 he received the Ph.D. degree from the University of London, London, England, and D.I.C. from the Imperial College of Science and Technology, London. From 1952 to 1961 he worked with electric utilities in India on
various aspects of design, construction, and operation of power systems. For one year he was a Confederation of British Industries scholar in the United Kingdom. In 1965 he worked with the English Electric Company in England. He is now in Canada, where he taught for two years at the University of Windsor and is at present at The University of Calgary. Dr. Malik is a fellow of the Institution of Electrical Engineers (London), a member of the Canadian Electrical Association, and the American Society for Engineering Education. He is a registered Professional Engineer in the Provinces of Alberta arid Ontario, Canada,
Zhi-da Zhao (SM'87) was born in Zhejiang, China on August 1, 1930. He received the B.S. degree in Electrical Engineering from Zhej iang University, Hangzhou, China in 1952. From 1952 to 1955 he studied as a postgraduate student in Harbin Polytechnic Institute, Harbin, China. From 1956 to 1977 he worked at the Electrical Engineering Department of Zhejiang University as a lecturer. He became an associate professor in 1978 and is presently a full professor at the same University. His areas of specialization are Electric Power Engineering and High Voltage Engineering. His current interests are EHVAC and EHVDC transmission of electric power, especially HV probleins in DC transmission. He is the author or coauthor of o-:rr 33 technical articles and 6 books. Trof. Zhao is a member of the CSEE (Chinese so=:--.. of Electrical Engineering). H e is currently . c - c"1, ~-c.t i : - . ~ zof n the subcommittee on HVDC Transmission ..icof :?.e C S E , council r.einber of the Zhejiang Electric Po.;er Society and its senior consultant in HV Engineering, member of All-China Higher Electric Power Educarian Committee and member of the Chinese National Committee of CIGRE.
1152 Discussion
P. Sarma Maruvada (IREQ, Varennes, Qukbec, Canada): This paper follows the recent trend in attemptingto obtain purely analytical solutions to what is inherently a very complex problem namely the attenuation of surge voltages due to corona. Although such attempts are commendable,there is also the danger of over simplification. During the course of our own extensive studies on the subject [1,2], we have realised that the q-v diagram may not be adequate to model corona attenuation and that a satisfactory model should also be able to reproduce the current waveform in the time domaine. Experimental studies till now have concentrated on the q-v diagrams, probably because they are easier to mesure, but it is becoming clear that further experimental studies are required before satisfactory models can be developped. The model in this paper is based on rather simplistic assumptions, the most important being that the electric field remains constant at (or slightly below)the onset value, not only at the conductor surface but throughout the corona layer. There is experimental and theoretical justification for the first part but no justification at all for the second part [3, 41. The concept of time delay introduced in this paper does not appear to be justified on the basis of the physical mechanisms involved. The charge being the integral of the corona current from the conductor, it can continue to increase even after the voltage maximum since corona current can flow after this instant. Experimental q-v curves show that this phenomenon is much more evident with switching rather than with lightning impulses, which contradicts the explanation given in the paper. References Reference 22 of the paper. P. S . Maruvada, D. H. Nguyen, H. Hamadani-Zadeh, “Studies on Modeling Corona Attenuation of Dynamic Overvoltages” IEEE Paper no. 88 SM 583-7. R. T. Waters, T. E. S . Rickard, W. B. Stark, “Direct Measurement of electric field at line conductors during ac corona” hoc. LEE, 119, 1972, pp 717-723. Maruvada P. Sarma, W. Janischewskyj, “Dc corona on smooth conductors in air” Proc IEE, 116, 1969, 161-166.
of corona and therefore h a w been widely uscd as an expcrimental basis for the modelling of corona for many years. That is why the authors established thcir modcl based on a q-v diagram analysis. Ncvcrthcless, as pointed out by the discusser, it is worthwhile trying to make an investigation with respect to corona current waveforms, which might be helpful both in undcrstanding the phenomenon and in improving the model accuracy. The physical process of corona discharges is so complicated that many of its properties still remain unclear. To simplify thc problcm, assumptions are obt4ously necesqary. Since the field distribution in the corona layer has not becn well studied, the constant ficld assumption niadc in the papcr as a first order approximation appcars to be reasonable in view of the property that corona space charges tend to achieve such a distribution that will makc the field distribution more uniform. Differcnt assumptions may also be made, such as that given in Ref. [SI of the paper. The comparison between the cxperimental q-v diagrams and the calculated diagrams based on the model offcrs a measure for the goodness of fit of the assumption at this stagc. Thc model has also been checked by serving as a subroutine in an algorithm for computation of transmission line transients including corona effects [ A I ] . The authors apprcciatc the discussion of the time discrepency bctwecn the voltage and charge maxima. It was noted before from the publishcd q-v diagrams, including Ref. [ 2 2 ] of the papcr (a paper written by the discusser and others), that the phenomenon is more evident for fast impulses than for slow ones. Now this secms to be questionable since the discusser has contradictory experience. The explanation given in the papcr does not fit to the expericncc indeed if it is true. However, the authors cannot explain this expericncc in terms of corona current, either.
Manuscript received August 17, 1988.
References Xiao-rong Li: The authors thank Dr. Maruvada for his interest and commcnts concerning the papcr. The authors agrec with the discusscr that the q-v diagrams may not be adequate to modcl the transmission line corona if very high accuracy is required. However, the q-v diagrams do carry information of the major charactcristics
[AI]
Xiao-rong Li. O.P.Malik, and Zhi-da Zhao, “Computation of Transmission Line Transients Including Corona Effects”, Paper submittcd to IEEE PES 1989 Wintcr Mccting Manuscript received September 19,1988.
