freitas curocawa corona effect 2008

1

Corona Effect in Frequency DependentTransmission Line Models

M. A. Freitas, S. Kurokawa, Member, IEEE and J. Pissolato, Member, IEEE

Abstract - The inclusion of the corona effect in a frequencydependent transmission line model is proposed in this paper.The transmission line is represented through a cascade of 7t

circuits and the frequency dependence of the longitudinalparameters is synthesized with series and parallel resistors andinductors. The corona effect will be represented using the Garyand Skilling-Umoto models. The currents and voltages alongthe line are calculated by using state-space technique. Todemonstrate the accuracy and validity of the proposedfrequency dependent line model, time domain simulations of a10 km length single-phase line response under energizationprocedure will be presented.

Index Terms - transmission lines, time domain analysis,state space methods, corona.

I. INTRODUCTION

THE Corona effect is an electrostatic dischargemechanism which occurs due to ionization in an

insulation material subjected to electric field intensity over acritical level. This effect is observed on the surface of theconductors of an overhead transmission line when thegradient value of the existing potential exceeds the value ofthe air disruptive critical gradient. This gradient depends onthe pressure and the humidity of air where the conductor islocated and the form ofvoltage wave [1].

The electrical field divergent also influences the value ofthe disruptive gradient. In this way, the dust in air aroundthe conductor can be changed into a punctual source ofelectrostatic discharges. In transmission lines, theelectrostatic discharge due to the corona effect can result ina series of effects such as power loss, radio and televisioninterference, audible noise, electric field effect and changein capacitance [1], [2]. Considering the corona effect, it isalready evident that for lines operating at extra highvoltages, the energy losses can vary from a few kilowattsper unit length to hundreds of kilowatts per unit length,depending on the atmospheric conditions and the weather.The average losses can constitute only small parts of theJoule losses effect, but the maximum losses can have asignificant influence on the demands of the electricalsystem.

The corona effect can also occur due to overvoltagesurges resulting from a direct lightning stroke or a circuitbreaker operation in an overhead line. These surges cangenerate faults on lines and are of great concern to power

This work is supported by Coordenayao de Aperfeiyoamento de Pessoalde Nivel Superior (CAPES). M. Freitas and J. Pissolato are at UniversidadeEstadual de Campinas, Brazil ([email protected],pisso(~dsce.fee.llnicamp.br). S. Kurokawa is at Faculdade de Engenhariade Ilha Solteira, Unesp (Sao Paulo State University), Caixa Postal 31, IlhaSolteira, Postal Code 15385-000, SP, Brazil, fone/fax: 55 19 3788 3860.([email protected]).

978-1-4244-2218-0/08/$25.00 ©2008 IEEE.

systems engineers. In this case, if the overvoltage is largeenough to produce a corona, then its magnitude is attenuatedand its shape is distorted as it propagates along the line. Theknowledge of the peak values and the shapes of the surgewaveforms that are likely to occur on lines is very importantas a guide to provide proper protection and insulationcoordination for equipment connected to the line. This way,a good understanding of the corona effects and its inclusionin transmission line models is an important procedure.

Several models have been developed for the simulationof corona effects in power systems. Many of them arepresented to investigate overvoltages in high-voltage powertransmission lines due to lightning, energization and othersources of transients [2]-[9].

Recently, a transmission line model was shown [8] takinginto account the corona effect. In this paper the transmissionline was represented through a cascade of 1t circuits and thecorona effect was represented by the Gary and SkillingUmoto's models. The model proposed in [8], [10] does notconsider the frequency dependence of the longitudinalparameters of the line and is therefore restricted.

More recently [9] presented a frequency dependenttransmission line model with a corona. The model of theimpulse corona was incorporated into the ATP-EMTPprogram through its MODELS capability using a TACScontrolled type-91 time varying resistance and switcheswere used to insert the corona model on the line. In thismodel there is the possibility that numerical oscillations canoccur [9].

In this paper we aim to show an alternative frequencydependent transmission line model taking into account thecorona effect. The line distributed parameters arerepresented by a cascade of 1t circuits and the frequencydependence of the longitudinal parameters are synthesizedwith series and parallel resistors and inductors [11]. Thecorona effect is represented by the Gary and Skilling-Umotomodels [8].

The first stage will show the development, accuracy andvalidity of the proposed model. After that the model used torepresent the corona effect will be shown. Finally, the linemodel with the corona effect will be used to simulate theenergization procedure of a 10 km single phase-line.

II. LINE REPRESENTED BY A CASCADE OF 1t CIRCUITS

Fig. 1 shows a cascade of 1t circuits that can be used torepresent a single-phase transmission line whose frequencydependence of the longitudinal parameters is disregarded[12]-[15].

(5)

(3)

(4)

[X] = [A][X] + [B]u

[X]T = [[Xl] [X2 ] ... [Xn ]]

[All] [A I2 ] [A ln ]

[A]=[A 21 ] [A 22 ] [A 2n ]

[AnI] [AnI] [Ann]

In (3) [X] is a vector of state variables, u is a vectorof inputs, and [A] and [B] are matrices.

In a single-phase line represented by connecting n short 1t

nominal sections in cascade circuits (1t nominal sections areshown in Fig. 2) we chose as state variables the capacitorsvoltage at the receiving terminal and the inductor current ineach 1t circuit. By using the state variables above mentioned,it was possible to write [X] and [A] as:

LRLR

R=R'~ L ,d (1)=L -n n

G=G'~ C=C'~ (2)n n

LR

where R', L', G' and C' are transmission line parametersper unit length, d is the line length and n is the quantity of 1t

circuits.The cascade of 1t circuits in Fig. 1 can be directly

implemented in an EMTP program [11]. This circuit can berepresented by its state equations. In this case a state modelcan be formulated using the capacitor voltages and inductorcurrents as the state variables [12-15]. If a state model isused, the state equations may be implemented on a personalcomputer and state variables can be solved using manytechniques [12-15].

It is known that the frequency dependence of thelongitudinal parameters of a transmission line can besynthesized with series and parallel resistors and inductors[16], and in [11] the use of a cascade of 1t circuits, isproposed as shown in Fig 2, to represent a frequencydependent transmission line.

In Fig. 1 Rand L are, respectively, the longitudinalresistance and inductance of the line segment represented bya 1t circuit and G and C are, respectively, the transversalconductance and capacitance. This way the R, L, G and Cparameters are written as being:

2III. PROPOSED MODEL FOR A FREQUENCY DEPENDENT LINE

· •. - A. State modelfor afrequency dependent transmission

~cline

C G C 2 Because a state model is an adequate representation of a• • • frequency independent line taking into account the corona

-F-ig'&-.-l.-S-in-gl-e--ph-a-"se-I-in-e-re-pr-es-en-t-ed-t-hr....l.ou-gh cascade of1t-c-ir-c-ui-ts-""":"""-- effect [8], we are proposing in this paper to use a similarmodel to represent a frequency dependent transmission linewith the corona effect.

Let us suppose that a single-phase line can berepresented by a cascade with n 1t circuits as shown in Fig.2. Let us also suppose that the frequency dependence of thelongitudinal parameters is synthesized by m resistors and minductors. If the state-space technique is used to representthis cascade, the state equations of the circuit will be writtenin the form [8]:

G

2

In Fig 2 Ro, R}, R2, ...Rmare constant resistances and La,Ll, L2, ...Lmare constant inductances. The series and parallelassociation of Fig. 2 results in the frequency dependentresistance and inductance of the line segment represented byonly a 1t circuit.

In [11] a cascade of 1t circuits, as shown in Fig. 2, wasimplemented in an EMTP program. The model was used tosimulate electromagnetic transients resulting in the normalswitching phenomenon.

Fig. 2. A 1t circuit unit with the frequency dependence.

(6)

iLka current at inductor La;iLkl current at inductor Ll;iLk2 current at inductor L2;iLkm current at inductor Lm;Vk voltage at receiving end of the kth 1t circuit.

In (4) [X]T is the transposed matrix of [X]. It is possibleto observe that vector [X] has n elements and each elementis a vector with (m +2) elements. A generic element [Xk] of

vector [X] is written as being:

In (6) [Xk]T is the transposed matrix of [Xk] and k canvary from 1,2,··· to n . The elements of [Xkl are currents and

voltages of the kth 1t circuit. This way, we have:

Therefore, the vector of state variable [X] will have(m+ 2) n elements.

In (5) the matrix [A] is a tridiagonal matrix with thedimension (nx n). Each element in [A] is a matrix with the

dimension (m+2) x (m+2) .

C2

G

2

3A generic element [Ak,k] in (5) is written as:

j=m

LRj_ j=o .& R 2 R m 1--

Lo Lo Lo Lo Lo

.& _.& 0 0 0L1 L1

[Akkl = R 2 0_ R 2 0 0

L2 L20 0

R m 0 0_R m 0

Lm Lm1

0 0 0G

- --C C

where k can vary from 1, 2, ...to n.

A generic element [Ak-l,k] in (5) is written as:

t=O

/_---------

~2~:::~~sr~R!!R!~!R!~!R!!R!~!R!~!R!!::Fig. 3. Single-phase transmission line.

(7)where CT in Fig. 3 is a capacitance that represents a

transformer [12]. The value of CT is 6 llF.We are supposing that the line shown in Fig. 3 has its

longitudinal parameters synthesized by the series andparallel association shown in Fig. 4.

L'} L'2 L'3 L'4

Fig. 4. Circuit used to approximate unit parameters of the line.

[Ak-l,k l = [~ . . . ~]-If ... 0

(8)

where values of the resistors and inductors used in thecircuit shown in Fig. 4 are listed in Table I.

TABLE IPASSIVE ELEMENTS OF THE CIRCUIT SHOWN IN FIG. 4

where in (8) k can vary from 2 to n.

An element [Ak+l,k] in (5) has the following generalform:

Resistances (ohms)R'o 0.026

R'} 1.470

R' 2 2.354

R'3 20.149

R'4 111.111

Inductances (mH)L'o 2.209

L'} 0.740

L' 2 0.120

L'3 0.100

L'4 0.050

o

Therefore we can observe that [A] has thedimension (m + 2)n x (m + 2)n .

In (5) the vector [B] has a n (m + 2) x 1 dimension vector.

If u(1) is a voltage source connected at the sending end of theline, the vector [B] will have the following general form:

OL-..----L~~~~~~~~~~~~~~

101

102

103

10" 105

106

Frequency (Hz)Fig. 6. Unit inductance of the line synthesized with elements of Table I.

10·2L-..----L~~~~--'-'--'-'-'--~~~~~-'--'-'-'---~~~

101

102

103

10 105

106

Frequency (Hz)

Fig. 5. Unit resistance of the line synthesized with elements of Table I.

(9)

(10)

oo

[B]T =[Xo 0 ... 0]

where in (9) k can vary from 1 to (n-J)

Therefore, using (3)-(10) it is possible to describe acascade of 1t circuits that represents a frequency dependenttransmission line. The set of linear ordinary differentialequations can be transformed into a set of linear differenceequations using trapezoidal integration that can be evaluatedby using a personal computer [8,12].

B. Performance ofthe proposed model

To verify the performance of the proposed model, itwas used to represent a hypothetical 10 km single-phaseline. The line was energized with a 20 kV step voltage, as isshown in Fig. 3.

(11)

(13)

(14)

In (14), Oc is the corona loss constant, rand h isconductor height. The conductor radius is in centimeters.

where

B. The SkiUing-Umoto model

In the Skilling-Umoto model the corona capacitance (inF1m) is written as [8]:

11 = 0.22 r + 1.2 (12)

where r is the conductor radius in centimeters.

V. CORONA MODEL

A. The Gary model

If the corona capacitance is represented by the Garymodel, this capacitance is defined as [4,8]:

In (11), Ce is the corona capacitance (in F/m), C is thegeometric capacitance, V is the voltage across the coronaparameter, Ve is the corona inception voltage and '1 is acoefficient which for a single conductor, is given by thefollowing experimental formula [8]:

i) The voltage across the corona elements is greaterthan the corona inception voltage;

ii) The derivative of the voltage across the coronaelements with respect to time is positive.

If conditions i and ii above mentioned are notsimultaneously satisfied, the corona parameters Ce and Geare null.

When the conditions are satisfied it is possible tocalculate Ce and Ge as functions of the voltage across Ceand Ge.

In this way, if the corona effect is taken into account in afrequency dependent transmission line it is necessary toverify in which position of the line so that conditions i and iiare satisfied at each time instant. This way, the statematrices [A] and [B] must be calculated at each time instant.

There are several models to compute Ce and Ge and inthis paper two analytical models were used: the Gary modeland Skilling-Umoto model [8].

4In Fig. 8 Ce and Ge are, respectively, corona

capacitance and conductance of the line [4], C is geometriccapacitance and G is a conductance that can be usuallyneglected, except at very low frequencies [18].

Considering a dynamic model for the corona effect, thecorona capacitance and conductance are connected to thecircuit if the following conditions are simultaneouslysatisfied [4]:

+E~cc

0.50.40.1

> 40~Q)

m~ 300>

"Cc::Q)

m 20c:::~Q)uQ)

£t:10

0.2 0.3Time (ms)

Fig. 7. Receiving end voltage obtained with proposed model (1) and withEMTP program (2).

Figure 7 shows that, practically, there are norelevant differences between results obtained from the statespace equations and the EMTP program. Therefore, the stateequations developed in this paper represent a frequencydependent line.

IV. INSERTION OF THE CORONA EFFECT IN THE LINE MODEL

When the line voltage on the surface of the conductor ishigher than the corona voltage, the air around the conductorbecomes ionized and stores charge. The corona effect ontraveling waves result basically in power loss and change intransversal capacitance. Therefore, in a frequency dependentsingle-phase line represented by the model developed in thispaper, the corona effect can be represented as an additionalcapacitance and conductance between the conductor and theground in each 1t section that represents the line. Figure 8shows the inclusion of the corona effect in the kth 1t circuitof the cascade that represents a frequency dependent line (k> 1 and k < n).

Figures 5 and 6 show the resistance and inductanceresulting from the series and parallel association of thecircuit shown in Fig. 4. It is possible to verify in thesefigures that the resulting resistance and inductance of thecircuit shown in Fig. 4 have the behavior of the longitudinalparameters of a frequency dependent overhead transmissionline [17].

To ensure that state equations developed in thispaper to represent the frequency dependent line shown inFig. 3, we compared the results obtained from the solutionof these state equations with results obtained from theprocedure developed in [11], where the currents andvoltages along the cascade of 1t circuits were calculated withan EMTP program [17].

In Fig. 7 the voltage at the receiving end of the lineobtained from solutions of state equations and from anEMTP program, is shown.

5

Fig. 9. Single-phase transmission line with corona effect.

500

500

400

400100

100

1200

~ 9004)CJ)

~~ 600

1500 -----,--------,--------,---,---------------,

~ 9004)CJ

1¥~ 600

300

100 200 300 400 500Time (us)

Fig. 13. Simulated voltages at 2.5 km from the sending end of the line:Without corona effect (curve 1) and with corona effect (curve 2).

1200

300

300

>' 900~4)CJ

:!~ 600

1500

1200

[ 9004)CJ

~~ 600

1200

200 300Time (us)

Fig. 12. Simulated voltages at 5 km from the sending end of the line:Without corona effect (curve 1) and with corona effect (curve 2).

200 300Time (us)


oo 100 200 300 400 500Time (us)

Fig. 10. Simulated voltages at the receiving end of the line: Withoutcorona effect (curve 1) and with corona effect (curve 2).

(15)

(16)

where

In (16) (Jo is the corona loss constant.

It was considered that per unit length geometriccapacitance C and conductance G of the line are constantand have a magnitude of C = 11.II11F/km and

G = 0.556 f.lS/km . The conductor radius is 2.54 cm and the

conductor height is 18.9 m. The capacitance CT in Fig. 9 isrepresenting a transformer and the value of the capacitanceCT is 6 llF [12].

The transmission line was represented through a cascadewith 110 1t circuits and its frequency dependent longitudinalparameters were fitted by the circuit shown in Fig. 4. Thevalues of the resistors and inductors used in the circuit ofFig. 4 are shown in Table I. The fitted parameters are shownin Figs. 5 and 6.

The corona effect was represented by the Gary modeland Skilling-Umoto model [4], [8]. It is considered that thecorona voltage is Vc = 550 kV and the Skiling-Umoto modelconstants are (jc =30 and (jG =10 7 [1].

The currents and voltages along the line were describedby using the state equations developed in (3)-(10) and thesestate equations were evaluated by using trapezoidalintegration [8], [12].

A. The Gary model results

Figures 10-13 show the voltage in four different positionsof the line during the energization procedure, with andwithout the inclusion of the corona effect. The corona effectwas represented by the Gary model.

VI. ApPLICATIONS AND RESULTS

To validate the model proposed we used it to simulate theopen circuit response of a hypothetical 10 km single-phaseline. In the simulation we energized the line with a 600 kVDC source as shown in Fig. 9.

t=O

/_---------~ 600kV 1CT

._~ .-.- _.-.- .-.-_.-.- _.-.- .-.-.-.-.- _.-.-.-.- _.-.- .-1.-.-_

.'!:.~:.'~:..'!:.~:..~'!8:..'!:.~:.'~:..'!8:.'~:..'!8,.'t:.~:..'!:.'!:.~:..'!:.'!:.~:..'!:.:~:.'~:..'!:..~:..'!:..'!:..'!,.'t:..~:.'!:..'!:..~

C. Corona loss

The corona attenuation loss is modeled by a resistivecurrent loss through conductance which is calculated asfollows [8]:

A

Highlight

6

1200

1500,-------,-----,-------------.-----

1500,------,--------,------,--------,---------,

1200

300

VII. CONCLUSIONS

This paper presents a frequency dependent line modeltaking into account the corona effect. In this model the lineis represented through a cascade of 1t circuits and thefrequency dependence of the longitudinal parameters aresynthesized with series and parallel resistors and inductors.

The proposed model for a frequency dependent line iswell known and it has already been used as a line model inEMTP programs [11]. However, in EMTP programs it isvery difficult to include an analytical corona model. Due tothis difficult, in this paper, the current and voltages alongthe cascade of 1t circuits were described by a state model.This representation is useful because it can be used tosimulate electromagnetic transients, considering thefrequency dependence of the transmission line parameters,directly in time domain without using inverse Fouriertransform. Simulation results obtained from the stateequation are practically identical to results obtained with aMicrotran that is an EMTP. However, after the first peakvalue, there is a small delay between the reference thatobtained with the proposed model. The delay presentedcould be due to the short length of the line analyzed and dueto rounding errors. This delay probably occurs becauseMicrotran limits the quantity of the significant numberduring the editing of the network that represents thefrequency dependence of the longitudinal parameters. Thesame does not occur when state space are valuated withMatlab. However, the authors believe that the delay time

300

~ 900Q)

en~~ 600

100 200 300 400 500Time (us)


~ 900Q)

en:l~ 600

O__--....L----~~----'----------L--------!o 100 200 300 400 500

Time (us)

Fig. 16. Simulated voltages at 5 km from the sending end of the line:Without corona effect (curve 1) and with corona effect (curve 2).

1200

1500,--------,------,------,--------,-------------,

:> 900eQ)

m~~ 600

300

O_~---L--------'--------'------L....---------'-----J

o 100 200 300 400 500Time (us)


1200

300

O__-....L--------'---...ll-----L.-..---------L-------'

o 100 200 300 400 500Time (us)

Fig. 14. Simulated voltages at the receiving end of the line: Without coronaeffect (curve 1) and with corona effect (curve 2).

1500

~ 900Q)mS~ 600

Figures 10-13 show that overvoltages, resulting from theenergization procedure which propagates along the line, areattenuated and distorted by the corona effect. These resultsagree with those shown in several papers [4,8]. Therefore,we can conclude that the proposed model for a frequencydependent line with the corona effect is correct and can beconsidered adequate to represent the line during transientsimulations.

B. Skilling-Umoto model results

Figures 14-17 show the voltage at four differentlocations on the line during the energization procedure, withand without the inclusion of the corona effect. The coronaeffect was represented by the Skilling-Umoto model.

In these figures it is possible to observe that when theSkilling-Umoto model is used to represent the corona effectin a frequency dependent line, the distortions andattenuations in the overvoltages along the line are lessaccentuated than when the Gary model is used. Theseresults agree with those obtained in [8].

needs to be carefully analyzed.The corona effect was represented with the Gary and

Skilling-Umoto models. The proposed model for afrequency dependent line with the corona effect was used tosimulate a transient resulting from the energization of asingle-phase line. The obtained results show thatovervoltages, resulting from the energization procedure,which propagates along the line, are attenuated and distortedby the corona effect. This results agree with those shown inseveral papers [4,8]. Therefore, we can conclude thatproposed model for a frequency dependent line with thecorona effect is correct and can be considered adequate torepresent the line during transient simulations.

The model was used to represent a single-phase line but itcan be easily extended to a non ideally transposed threephase line with a vertical symmetry plane if the Clarkematrix is used as being a modal transformation matrix [11].

We believe that the major contribution of the paper is topropose a friendly model to represent a frequency dependenttransmission line with the corona effect that can beimplemented directly in time domain without using anEMTP program. The model can be used to teach basicconcepts about wave propagations in transmission lines, toanalyze transient voltages and current distributions ontransmission lines and to compute electromagnetic transientson transmission lines

The proposed model utilizes the student's background innetwork theory and it can be easily implemented on apersonal computer.

VIII. ACKNOWLEDGEMENTS

To CAPES (Coordena9ao de Aperfei90amento de Pessoalde Nivel Superior) and FAPESP (Funda9ao de Amparo aPequisa do Estado de Sao Paulo) for the financial support.

7[7] A Ramirez, 1. L. Naredo, P. Moreno and L. Guardado,

"Electromagnetic transients in overhead lines consideringfrequency dependence and corona effect via the method ofcharacteristics", Int. J. Electrical Power and Energy Systems, vol.23 (3), pp. 179-188, 2001.

[8] M. S. Mamis, "State-space transient analysis of single-phasetransmission lines with corona", in International Conference onPower Systems Transients, New Orleans, 2003.

[9] T. 1. Gallagher, I. M. Dudurych, "Model of corona for an EMTPstudy of surge propagation along HV transmission lines", lEEProc. Gen. Transm. Distrib., vol. 151, nQ 1, pp. 61-66, January2004.

[10] 1. R. Marti, "Accurate modeling of frequency dependenttransmission line in electromagnetic transient simulations", IEEETrans. Power App. And Systems, vol. PAS-I01, nQ 1, pp. 147-155,Jan. 1982.

[11] M. C. Tavares, 1. Pissolato and C. M. Portela, "Mode domainmultiphase line model - Use in transients studies", IEEE Trans. onPower Delivery, vol. 14, nQ 4, pp. 1533-1544, Out. 1999.

[12] R. M. Nelms, G. B. Sheble, S. R. Newton and L. L. Grigsby,"Using a personal computer to teach power system transients",IEEE Trans. On Power Systems, vol. 4, nQ 3, pp. 1293-1297, Aug.1989.

[13] M. S. Mamis and A. Nacaroglu, "Transient voltage and currentdistributions on transmission lines", IEE.Proc. Gener. Transm.Distrib., vol. 149, nQ 6, pp. 705-712, Nov.2002

[14] M. S. Mamis, "Computation of electromagnetic transients ontransmission lines with nonlinear components", IEE.Proc. Gener.Transm. Distrib., vol. 150, nQ 2, pp. 200-203, March 2003.

[15] 1. A. R. Macias, A. G. Exposito and A. Alfonso, "A comparison oftechniques for state-space transient analysis of transmission lines",IEEE Trans. Power Delivery, vol. 20, nQ 2, pp. 894-903, April2005.

[16] M. S. Sarto, A. Scarlatti and C. L. Holloway, "On the use of fittingmodels for the time-domain analysis on problems with frequencydependent parameters", in Proc. IEEE Int. Symp. OnElectromagnetic Compatibility, 2002, vol. 1, pp. 588-593.

[17]H. W. Dommel, "EMTP theory book", Vancouver, 1986.[18]1. A. Martinez, B. Gustavsen and D. Durbak, "Parameters

determination for modeling system transients - Part I: Overheadlines", IEEE Trans. Power Delivery, vol. 20, nQ 3, pp. 2038-2044,July 2005.

[1]

[2]

[3]

[4]

[5]

[6]

IX. REFERENCES

M. Khalifa, High-voltage Engineering: Theory and Practice. NewYork: Marcel Dekker Inc., 1990.X-R Li, O. P. Malik and Z-D. Zhao, "Computation of transmissionline transients including corona effects", IEEE Trans. on PowerDelivery, vol. 4, nQ 3, pp. 1816-1822, July 1989.S. Carneiro, H. W. Dommel, 1. R. Marti and H. M. Barros, "Anefficient procedure for the implementation of corona models inelectromagnetic transients programs", IEEE Trans. Power Deliv.,vol 9 nQ 2, pp. 849-855, April 1994.1. L. Naredo, A. C. Soudack and 1. R. Marti, "Simulation oftransients on transmission lines with corona via the method ofcharacteristics", lEE Proc. Gener. Transm. and Distrib., vol. 142(1), pp. 81-87, January 1995.M. T. C. Barros, "Identification of the capacitance coefficients ofmultiphase transmission lines exhibiting corona under transientconditions", IEEE Trans. on Power Delivery, vol. 10, nQ 3, pp.1642-1648, July 1995.1. F. Guillier, M. Poloujadoff, M. Rioual, "Damping model oftraveling waves by corona effect along extra high voltage threephase lines", IEEE Trans. on Power Delivery, vol. 4, nQ 3, pp.1851-1861, October 1995.

X. BIOGRAPHIES

Michelly Alcantara de Freitas. She graduated in electrical engineeringat Unesp - Sao Paulo State University in Ilha Solteira, Brazil (2004). Atpresent she is developing her D.Sc degree in electrical engineeringUniversidade Estadual de Campinas, Campinas, SP. Her main researchinterests are transmission line models used to simulate electromagnetictransients in power electric systems.

Sergio Kurokawa (S'Ol-M'04). He graduated in electrical engineering(1990). Since 1994 he has been Professor of Unesp - Sao Paulo StateUniversity in Ilha Solteira, Brazil. He received his D.Sc. degree in electricalengineering from Universidade Estadual de Campinas, Campinas, SP. Hismain research interests are electromagnetic transients in power electricsystems and models of long transmission lines used in studies ofelectromagnetic transients.

Jose Pissolato Filho (M'95). He received the D.Sc. degree in electricalengineering from Universite Paul Sabatier, Toulouse, France, 1986. Since1979 he has been with Universidade Estadual de Campinas (Departamentode Sistemas e Controle de energia), Brazil. His main research interests arein high voltage engineering, electromagnetic transients and electromagneticcompatibility.

freitas curocawa corona effect 2008

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