Behavior of Overhead

IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 20, NO. 2, APRIL 2005 1669

Behavior of Overhead Transmission Line Parameterson the Presence of Ground Wires

Sérgio Kurokawa, Member, IEEE, José Pissolato Filho, Member, IEEE, Maria Cristina Tavares, Member, IEEE,Carlos M. Portela, Life Senior Member, IEEE, and Afonso J. Prado, Member, IEEE

Abstract—Initially this paper shows the ground wire reductionprocess for generic multiphase transmission lines and after, theground wire reduction process for a specific 440-kV three-phaseoverhead transmission line. Following this, the influence of theground wire reduction process considering two situations is shown:first, considering frequency independence and second, when theseparameters are considered as frequency dependent. This paperpresents analytical results for generic multiphase transmissionlines. For a specific 440-kV three-phase overhead transmissionline, analytical and graphic results are shown considering realdata for every frequency between 10 Hz and 1 MHz.

Index Terms—Admittance matrix, electromagnetic (EM) field,frequency-domain analysis, impedance matrix, multiconductortransmission lines, transmission-line matrix methods, transmis-sion-line theory.

I. INTRODUCTION

THE transmission-line ground wire system can be repre-sented explicitly or implicitly. In the explicit representa-

tion, the ground wires are considered as being additional con-ductors. This way, the longitudinal impedance matrix of a linewith phases and ground wire conductors is a

matrix.The implicit representation allows the effects of the ground

wires to be inserted in the phase conductors and after that, it ispossible to exclude the ground wires of the longitudinal per-unitimpedance matrix. This way, a transmission line with phasesand ground wires will have a matrix as being the longi-tudinal per-unit impedance matrix if the implicit representationof the ground wires is used.

In order to consider the implicit representation of groundwires, we can assume a zero transversal voltage of ground wiresalong the line. Such an assumption is representative of realityif ground wires are connected to earth in all structures, if eachstructure has a good grounding system, and if the distance be-tween consecutive grounding systems is reasonably shorter thana quarter wavelength for the frequency under analysis. For a typ-ical power transmission line, with a span of the order of 300 m,

Manuscript received September 22, 2003; revised February 2, 2004. Thiswork was supported in part by the Fundação de Amparo à Pesquisa do Estadode São Paulo. Paper no. TPWRD-00488-2003.

S. Kurokawa and A. J. Prado are with Faculdade de Engenharia de IlhaSolteira, Universidade Estadual Paulista, Ilha Soltiera 15385-000, Brazil(e-mail: [email protected]; [email protected]).

J. Pissolato Filho and M. C. Tavares are with Universidade Estadual deCampinas, Campinas 13081–970, Brazil (e-mail: [email protected];[email protected]).

C. M. Portela is with Universidade Federal do Rio de Janeiro, Rio de Janeiro21941–972, Brazil (e-mail: [email protected]).

Digital Object Identifier 10.1109/TPWRD.2004.833916

the span length is around a quarter wavelength for 250 kHz.So the assumption of zero transversal voltage along the groundwires may not be valid for frequencies above 0.2 MHz.

If the grounding conditions of the ground wires are welldefined, an implicit representation can be developed withoutassuming zero transversal voltage on the ground wires [1].However, in such conditions, the correct evaluation of lineparameters depends on a more detailed line representation ofline spans and grounding systems that are not dealt with in thispaper. Although the assumption of zero transversal voltage ofground wires is not applicable for frequencies above 0.2 MHz,some specific calculations using methodologies presented in [1]have shown that for high frequencies, the results obtained forequivalent three-phase parameters may be reasonably accuratein part due to the spread of span length that avoids the effect ofsimultaneous resonance conditions in all spans.

This considers that in implicit form, the ground wires are as-sumed to have transversal voltage. However, the accuracy ofresults obtained with the indicated simplifying assumption de-pends on specific line characteristics.

The basic advantage of implicit representation of the groundwires is to decrease the dimension of the impedance matrix.This reduction may be quite important and can be usefulwhen the transmission line is decoupled in its “quasi-modes”by using Clarke’s matrix as the single modal transformationmatrix. Clarke’s matrix is a 3 3 matrix and can be usedwith reasonable approximation to decompose a three-phasetransmission line with a vertical symmetry plane in its exactmodes [2], [3].

It must be clarified that just the existence of a vertical sym-metry plane allows separation of the exact modes, includingground wires, in explicit or implicit form, into two groups: one“media and the other “antimedia.” One of the groups corre-sponds to Clarke’s components, the other corresponds toand zero Clarke’s components that are quasimodes. In explicitrepresentation of ground wires, there is no coupling between thetwo groups, one with two modes, the other with three modes. Inimplicit representation of ground wires, one group correspondsto Clarke’s component (exact mode), the other to and zeroClarke’s components (quasi-modes).

This paper shows a comparative study about implicit and ex-plicit representation of ground wires for frequencies between10 Hz and 1 MHz. First, the article shows analytic results forgeneric multiphase transmission lines. Then, these analytic re-sults are applied to a 440-kV three phase transmission line with500-km length. After that, using the original data of the 440-kVthree-phase transmission line, the paper shows the graphics re-

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1670 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 20, NO. 2, APRIL 2005

sults. The results presented in this paper were obtained with theassumption of zero transversal voltage of ground wires alongthe line.

II. FREQUENCY DEPENDENCE OF THE

TRANSMISSION-LINE PARAMETERS

It has long been recognized that one of the most important as-pects in the modeling of transmission lines for electromagnetic(EM) transient studies is to account for the frequency depen-dence of the parameters. Models which assume constant param-eters cannot adequately simulate the response of the line over thewide range of frequencies that are present in the signals duringtransient conditions. In most cases, the constant parameter rep-resentation produces a magnification of the higher harmonicsof the signals and, as a consequence, a general distortion of thewaveshapes and exaggerated magnitude peaks [4].

The parameters of transmission lines with ground return arehighly dependent on the frequency. Formulas to calculate theinfluence of the ground return were developed by Carson andPollaczek and these formulas can also be used for power lines.Both seem to give identical results for overhead lines, but Pol-laczek’s formula is more general inasmuch as it can also be usedfor underground conductors or pipes [5].

The inner impedance results from the EM field within theconductor. It only belongs to the self impedance and consistsof frequency-dependent resistance and inductance, which canbe calculated with good accuracy with formulas based onBessel functions of conductor geometric parameters, materialelectric conductivity and frequency, or with simplified formulasdepending on frequency range. Due to the skin effect, theresistance increases whereas the inductance decreases.

III. REDUCTION OF GROUND WIRES IN GENERIC

TRANSMISSION LINES

It is well known that for sinusoidal alternating electricalmagnitudes with complex representation of sinusoidal alter-nating electrical magnitudes, and with several approximationsand validity restrictions, the basic equations of a transmissionline are [1]

(1)

The basic equations of a transmission line are valid if somegeometric and EM field behavior simplifying assumptions canbe considered. The geometric simplifying assumptions consistof considering that the soil surface can be assumed plane, linecables are assumed horizontal and parallel among themselves,the distance between any pair of conductors is assumed muchhigher than the sum of their radii and the EM effects of structuresand insulators are neglected. It is also assumed that the EM fieldhas a quasi-stationary behavior in direction orthogonal to lineaxis [1].

In (1), and are per-unit length longitudinal impedanceand the shunt admittance matrices, respectively. The elements ofthe matrices and are frequency dependent. The vectors

and are, respectively, transversal voltages of the line ca-bles and longitudinal currents in the line cables.

Matrix is made up of self and mutual impedances. Theself impedance falls into three parts and the mutual impedanceinto two parts. These are listed in the following.

— The internal longitudinal impedance (per-unit length) isassociated with the EM field within the conductor. Withinthe assumption indicated above, such an EM field doesnot affect mutual terms, but only diagonal (self) termsof longitudinal impedance matrix. In general, the internalimpedance can be interpreted as a resistance and an in-ductance, both frequency dependent.

— The external longitudinal impedance is associated withthe EM field outside the conductors. Assuming a loss-less (infinite electric conductivity) ground and the otherassumptions indicated above, the external longitudinalimpedance (per-unit length) may be interpreted as cor-responding to an inductance matrix (per-unit length),frequency independent.

— For a lossy ground (finite electric conductivity), withinthe other assumptions indicated above, it is possibleto consider the effect in longitudinal per-unit lengthimpedance matrix of the EM field in soil by means of anadditional parcel of the external longitudinal impedancewith nonzero diagonal (self) and mutual elements that arefrequency dependent (that can be treated as frequency-de-pendent resistance and inductance matrices).

A similar analysis also applies to matrix . The maindifference concerning simplifying assumptions is that foreffects, it is “reasonably accurate” in typical conditions toassume ideal conductors and ground (infinite electric conduc-tivity). Assuming the validity of assuming zero transversalvoltage of grounding conductors, the validity restrictions for

are similar to those applicable to .Considering (1) represents a multiphase transmission line

with phases and ground wires, it is possible to write theand matrices as being

(2)

(3)

In (2) and (3), the submatrix indices and refer to the phaseand ground conductors set, respectively.

If zero voltage can be assumed all along the ground wires,these cables can be reduced. The reduction of the andmatrices results in

(4)

(5)

In (4) and (5), and matrices are longitudinalimpedance matrix and shunt admittance matrix, respectively,considering the ground wires are reduced. A more detailedanalysis of the reduction of and matrices to obtainand matrices is presented in Appendix A.

The reduction process used in (4) and (5) assumes that thetransversal voltage of ground wires is zero along the line. The

KUROKAWA et al.: BEHAVIOR OF OVERHEAD TRANSMISSION LINE PARAMETERS ON THE PRESENCE OF GROUND WIRES 1671

validity constraints and errors of this assumption have been dis-cussed above as potential ways to avoid such a restriction aswell.

IV. REDUCTION OF CONTINUOUS GROUND WIRES IN A

THREE-PHASE TRANSMISSION LINE

Fig. 1 shows a 440-kV three-phase transmission line with avertical symmetry plane.

Each one of the phases 1, 2, and 3 is constituted by four Gros-beak subconductors. The conductors 4 and 5 are ground wiresand they are conductors. The soil resistivityis 1000 m and it is assumed that the phase conductors areuntransposed.

The longitudinal impedance per-unit length matrix of the lineshown in Fig. 1 is

(6)

Because the transmission line shown in Fig. 1 has a verticalsymmetry plane, the matrix can be written as being

(7)

Comparing (2) and (7), it is possible to observe that the ,, , and matrices are written as being

(8)

(9)

(10)

(11)

If zero voltage is assumed along the ground wires, and sub-stituting (8)–(11) in (4), it is possible to obtain the longitudinalimpedance matrix that represents the line when the groundwires are implicitly represented. The matrix is written as

(12)

where the elements of (12) are

(13)

(14)

(15)

Fig. 1. A 440-kV three-phase transmission line.

Fig. 2. Three phase with ground wires implicitly represented.

(16)

(17)

(18)

(19)

Observing (13)–(19), it is possible to observe that a genericself element and a generic mutual element of thematrix can be written as being

(20)

(21)

After the reduction of the ground wires, the line shown inFig. 1 is now represented as being shown in Fig. 2.

It is possible to observe that explicit representation of theground wires of the line shown in Fig. 1 requires a 5 5 ma-trix to represent the longitudinal impedance matrix while im-plicit representation of the ground wires requires a 3 3 matrix.Therefore, the implicit representation of ground wires is usefulif the line shown in Fig. 1 needs to be separated into its “quasi-modes” by using Clarke’s matrix.

A more detailed analysis of the modal representation ofthe three-phase transmission lines using Clarke’s matrix ispresented in Appendix B.

V. INFLUENCE OF THE GROUND WIRES IN A THREE-PHASE

TRANSMISSION LINE WITH CONSTANT PARAMETERS

Consider, in a hypothetical situation, in the transmission lineshown in Fig. 1, the following occur.


— Ground has infinite conductivity.— The longitudinal internal impedance, per-unit length, of

cables (phases and ground wires) is equivalent to resis-tance and inductance, both frequency independent. Al-though this assumption does not correspond to real phys-ical cables, it is physically consistent for the purposes ofcalculations used in the paper. Namely, this behavior canbe ideally approached assuming thin cylindrical conduc-tors of high (but finite) electrical conductivity with an ex-ternal radius equal to the radius of the real correspondingline conductors, and making the conductivity increase

and the conductor depth decrease ( 0), butmaintaining the product constant.

The hypotheses previously mentioned do not apply to a realtransmission line, but can be used with several restrictions toshow the influence of the ground wires implicitly represented.

The matrix of this hypothetical line is shown in (6). Con-sidering the hypotheses mentioned above, a generic self element

and a generic mutual element of the matrix are ex-pressed as being

(22)

(23)

In (22) and (23), is the angular frequency and andare, respectively, the self resistance and the self inductance ofthe phase . The term is the mutual inductance betweenphases and . The terms , , and were calculated fora specific frequency equal to 60 Hz, considering the magneticfield in the conductors (skin effect) and, also, the magnetic fieldin air. These terms are frequency independent.

Considering (22) and (23) and using (13)–(19), it is possibleto obtain the equations of the elements of the matrix. As anexample, equations for and are shown

(24)

(25)

where

(26)

(27)

(28)

(29)

Other elements of the matrix can be obtained with a similarprocedure.

In (26)–(29), and are, respectively, the self re-sistance and self inductance of phase when ground wires areimplicitly represented. The elements and are, re-spectively, the mutual resistance and mutual inductance betweenphases and . It can be observed in (26)–(29) that parametersare frequency dependent. The mutual resistances are real com-ponents that are principal off-diagonal in as is defined in[5].

Fig. 3. Self resistances R and R .

Fig. 4. Self inductances L and L .

Observing (26)–(29), it is possible to conclude that implicitrepresentation of the ground wires transforms the frequencyindependent parameters into frequency-dependent parameters.The implicit representation of the ground wires also producesfrequency-dependent mutual resistances.

Fig. 3 shows the self resistance , of the primitive matrix[Z], and of the reduced matrix .

Fig. 3 shows that after the reduction of the ground wires, theself resistance of phase 1 is not altered in low frequencies but ithas become frequency dependent for an intermediate frequencyrange. In the high frequencies, the self resistance takes on a con-stant value and this value is larger than the value of the self re-sistance before reduction of the ground wires.

Fig. 4 shows the self inductance of the reduced matrixand of the reduced matrix . This figure shows that afterreduction of the ground wires, the self inductance of phase 1 isnot altered at low frequencies but it has become frequency de-pendent for an intermediary frequency range. At high frequen-cies, the self inductance takes on a constant value and this valueis lower than the value of the self inductance before reductionof the ground wires.


Fig. 5. Mutual resistances R and R .

Fig. 6. Mutual inductances L and L .

Fig. 5 shows the mutual resistance of the primitive matrixand of the reduced matrix .

Fig. 5 shows that if it is considered that the transmission linehas a conductor whose internal impedance, per-unit length, isequivalent to a resistance and an inductance, both frequency in-dependent, and is over a ground with infinite conductivity, themutual resistances are nil when the ground wires are explic-itly represented. When the ground wires are implicitly repre-sented, the matrix has mutual resistances that are frequencydependent.

Fig. 6 shows the mutual inductance of the primitive ma-trix and of the reduced matrix .

Fig. 6 shows that after the reduction of the ground wires, themutual inductances of the reduced longitudinal impedance arewritten as being frequency dependent.

Fig. 7. Self resistances R and R .

VI. INFLUENCE OF THE GROUND WIRES IN

THREE-PHASE TRANSMISSION LINES WITH

FREQUENCY-DEPENDENT PARAMETERS

Let us consider now that the overhead transmission lineshown in Fig. 1 is over nonideal ground with finite conductivityand that the cables are not perfect conductors and do not behavein the ideal limit conditions assumed above (resistance andinductance, per-unit length, frequency independent). In thisreal situation, the self and mutual parameters are frequencydependent and are calculated considering the ground return andskin effect [5], [6]. Therefore, the elements of matrix arewritten as being

(30)

(31)

In (30) and (31), and are, respectively, the selfresistance and self inductance of the phase and and

are, respectively, the mutual resistance and mutual in-ductance between the phases and .

After that, using the procedure that was shown in Section III,it is possible to obtain the implicit representation of groundwires. Because elements of matrix are calculated by usingcomplex formulas [5], [6], a computational routine was used tocalculate matrix and the algebraic equations of the elementsof the and matrices will not be shown. Only the graphicresults will be shown.

Fig. 7 shows the self resistance of the primitive matrixand of the reduced matrix .

Fig. 8 shows the self inductance of the primitive matrixand of the reduced matrix and Fig. 9 shows the

mutual resistance of the primitive matrix and ofthe reduced matrix .

Fig. 10 shows the mutual inductance of the primitive ma-trix and of the reduced matrix .

Figs. 7–10 show that the longitudinal parameters of the lineare modified if ground wires are represented implicitly. In this


Fig. 8. Self inductances L and L .

Fig. 9. Mutual resistances R and R .

Fig. 10. Mutual inductances L and L .

case, the inductances of the reduced matrix are lower than in-ductances of the primitive matrix. The resistances have a dis-tinct behavior according to the frequency. In an initial frequencyrange, the implicit representation of the ground wires increasesthe value of the resistances and in a last frequency range, thevalue of the resistances are reduced due to implicit representa-tion of the ground wires.

VII. CONCLUSION

This paper has shown that the presence of continuous groundwires alters the longitudinal parameters of overhead transmis-sion lines. The analysis has been completed using the reductionof the ground wires.

The influence of the ground wires has been initially pre-sented for a generic multiphase transmission line assuming zerovoltage all along the ground wires. The longitudinal per-unitimpedance matrix obtained after the reduction of the groundwires considers an implicit representation of the ground wiresand is denominated reduced impedance matrix. The mathemat-ical results have shown that the implicit representation of theground wires alter the longitudinal per-unit impedance matrixwhere initially the ground wires were explicitly represented.

The calculation of the reduced matrix of a hypothetical trans-mission line, where the ground has an infinite conductivity andthe phase and ground wire conductors are ideal conductors, hasshown that the implicit representation of the ground wires trans-forms frequency independent parameters into frequency-depen-dent parameters. In this case, there is also the appearance of themutual resistances that were not present when the explicit rep-resentation of the ground wires was considered.

In a more general situation where the longitudinal per-unit pa-rameters are considered frequency dependent, the implicit rep-resentation of the ground wires alters these parameters. The in-ductances are reduced when the implicit representation of theground wires is considered. In a first frequency range, the resis-tances are increased and, in a final frequency range, the resis-tances are decreased.

APPENDIX A

Consider the equations of a transmission line, within the as-sumptions indicated in I and III, with the form (1)

(A.1)

(A.2)

In (A.1) and (A.2), the vectors [V] and [I] are written as being

(A3)

(A.4)

In (A.3), the vector contains the transversal voltagesof the phase conductors, and the vector contains thetransversal voltages of the ground wire cable conductors. Sim-ilarly, in (A.4), the vectors and contain, respectively,


the longitudinal currents in the phase conductors and in theground wire cables.

Substituting (A.3) and (A.4) in (A.1), it is possible to write

(A.5)

The development of (A.5) results in

(A6)

(A.7)

With the assumptions indicated in I and III (with the conse-quent error, discussed above), with the ground wire cables con-nected to earth in all structures, is assumed nil. Therefore,from (A.7), it is possible to obtain

(A.8)

Substituting (A.8) in (A.6)

(A.9)

From (A.9), it is possible to write

(A.10)

where

(A.11)

In (A.11), is the longitudinal impedance matrix when theground wire cables are implicitly represented.

Using a similar procedure, in the indicated assumptions, it ispossible to write the shunt admittance matrix, when the groundwire cables are implicitly represented as being

(A.12)

APPENDIX B

Assume a nontransposed three-phase transmission line with avertical symmetry plane with the ground wires already reduced,as shown in Fig. 11.

The per-unit length longitudinal impedance matrix of theline shown in Fig. 11 is

(B.1)

If it is proposed to work with modal components, or good ap-proximation, the per-unit length longitudinal impedance matrixin modal domain is written as being [3]

(B.2)

In (B.2), is Clark’s matrix and it is written as being [3]

(B.3)

Fig. 11. Schematic representation of a single three-phase line.

Substituting (B.1) and (B.3) in (B.2), it is possible to writeas being [3]

(B.4)

where

(B.5)

(B.6)

(B.7)

(B.8)

Note in (B.4) that Clark’s matrix separates two groups ofmodes: the and other two modes represented by compo-nents. From longitudinal impedance in mode domain , thefollowing can be seen.

— is an exact mode because there is no coupling betweenit and the others.

— The same is not true for and zero components becausethere is a coupling term .

For nontransposed lines, the self-impedance terms are almostthe same. The mutual impedance terms, although different, arealso similar, and the difference is small in the frequency rangeof transient analysis. Therefore, the coupling term canbe discarded and the components and zero can be treated as“quasi-modes” for nontransposed lines with the restriction of avertical symmetry plane [3].

REFERENCES

[1] C. M. Portela and M. C. Tavares, “Modeling, simulation and optimiza-tion of transmission lines. Applicability and limitations of some usedprocedures,” in Proc. IEEE T&D Latin America, São Paulo, Brazil,2003.

[2] M. C. Tavares, J. Pissolato, and C. M. Portela, “Quasimodes multiphasetransmission line model,” Electr. Power Syst. Res., vol. 49, pp. 159–167,1999.

[3] , “Mode domain multiphase transmission line model—use in tran-sient studies,” IEEE Trans. Power Del., vol. 14, no. 4, pp. 1533–1544,Oct. 1999.

[4] J. R. Marti, “Accurate modeling of frequency-dependent transmissionlines in electromagnetic transient simulations,” IEEE Trans. Power App.Syst., vol. PAS-101, no. 1, pp. 147–155, Jan. 1982.

[5] H. W. Dommel, EMTP Theory Book Portland, OR, 1986.[6] L. Marti, “Low-order approximation of transmission line parameters for

frequency-dependent models,” IEEE Trans. Power App. Syst., vol. PAS102, no. 11, pp. 3582–3589, Nov. 1983.


Sérgio Kurokawa (S’01–M’04) was born in Umuarama, Paraná, Brazil, in1966. He received the D.Sc. degree in electrical engineering from UniversidadeEstadual de Campinas, Campinas, Brazil, in 2003.

Currently, he is Assistant Professor of Universidade Estadual Paulista, SãoPaulo, Brazil. His research interests are EM transients in power electric systemsand models of long transmission lines used in studies of EM transients.

José Pissolato Filho (M’95) was born in Campinas, São Paulo, Brazil, in 1951.He received the D.Sc. degree in electrical engineering from Université PaulSabatier, Toulouse, France, in 1986.

Currently, he is with the Universidade Estadual de Campinas (Departamentode Sistemas e Controle de energia), Campinas, Brazil, where he has been since1979. His research interests include high-voltage engineering, EM transients,and EM compatibility (EMC).

Maria Cristina Tavares (M’98) received the electrical engineering degree in1984 from the Federal University of Rio de Janeiro (UFRJ), Rio de Janeiro,Brazil, the M.Sc. degree in 1991 from COPPE/UFRJ, and the Ph.D. degreein 1998 from the State University of Campinas (UNICAMP), Campinas, SP,Brazil.

She was a Consulting Engineer in consulting firms, working in power systemanalysis, HVDC studies (developed at ABB Power Systems, Sweden), modeldevelopment (at EMTP), and electrical transmission planning. She developedDESTRO, a graphical preprocessor for ATP. Presently, she is an Assistant Pro-fessor at UNICAMP. Her main research interests are power system analysis,long-distance transmission, and computer applications for analysis of powersystem transients.

Carlos M. Portela (SM’68–LSM’02) received the electrical engineering degreein 1958 and the D.Sc. degree in 1963, both from IST, Lisbon Technical Univer-sity (IST-UTL), Portugal.

Since 1972, he was a Cathedratic Professor at IST-UTL, where he was re-sponsible for Portuguese electrical network studies and planning and electricalnetwork operation. He has been responsible for major studies and projects inthe electric power and industry sectors in Portugal, Brazil, and other countries.Presently, he is a Titular Professor at the Federal University of Rio de Janeiro,Rio de Janeiro, Brazil, working on research projects in transmission systemsand equipment.

Afonso J. Prado (S’95–M’03) received the M.Sc. degree from Faculdade de En-genharia de Ilha Solteira, Universidade Estadual Paulista (FEIS/UNESP), SãoPaulo, Brazil, in 1995, and the Ph.D. degree from UNICAMP—Campinas StateUniversity, Campinas, Brazil, in 2002.

Currently, he is a Researcher with FEIS/UNESP (sponsored by the Fundaçãode Amparo à Pesquisa do Estado de São Paulo). His main research interestsinclude EM transients of transmission lines.

Behavior of Overhead

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