IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 18, NO. 3, JULY 2003 867

Calculation of Electric Field and PotentialDistributions Into Soil and Air Media for a Ground

Electrode of a HVDC SystemJos Eduardo Telles Villas and Carlos Medeiros Portela, Senior Member, IEEE

AbstractThe present article describes a methodology that al-lows suitable calculations of the electric field as well as the po-tentials and current densities due to current in the ground elec-trode of a HVDC system in any point of nonhomogeneous soilsand air media, taking into account the complexities of the prob-lems involved. Results of these variables are shown for the case ofa toroidal ground electrode of a HVDC system installed in homoge-nous and nonhomogenous soils.

Index TermsElectric field, ground electrode, heterogeneoussoil.

I. INTRODUCTION

Ahigher or lower complexity in the calculation of electricfields or potentials, as a result of current injection intothe soil through ground electrodes of a HVDC system (in amonopolar or homopolar modes), can be related to the followingaspects:

a) soil heterogeneous concerning the spatial distribution ofthe electrical resistivity;

b) occasional modifications of the electrical resistivity as aconsequence of natural causes or by the effect of soilheating due to the injected current and the associatedmodification of humidity distribution involving the elec-troosmosis phenomenon.

The first aspect involves higher depths and distances from theground electrodes (approximately 10 km). The latter, in general,affects only the ground electrode vicinities (until 10 m) [1].Additionally, the safety criteria concerning human beings mustbe satisfied for all unfavorable operating conditions. These par-ticularly include conditions leading to more pessimistic situa-tions related to the electrical resistivity, in the ground electrodevicinities.

II. SOIL MEDIUMThe soil represents a fundamental role concerning the ground

electrode behavior, since a current circulates in it continuously.As the soil does not behave as a well-defined material, it is nec-essary to consider a high number of aspects and parameters, withmore or less relative importance, according to the case, and to

Manuscript received June 18, 2002.J. E. T. Villas is with the Marte Engenharia Ltda./UERJ, State University of

Rio de Janeiro, 20.040-009, Brazil.C. M. Portela is with the COPPE/UFRJ, Federal University of Rio de Janeiro,

RJBrazil.Digital Object Identifier 10.1109/TPWRD.2003.809741

Fig. 1. Generic composition of earth layers and related typical values ofelectrical resistivity (in m).

analyze their characteristics, not only for higher distances fromthe ground electrodes, but also for significant depths. Fig. 1 dis-plays a composition of gross layering of the Earth and the orderof magnitude of the related electrical resistivities [2]. The cur-rent injected into the soil circulates between the ground elec-trodes through the earth crust and the mantle, being predomi-nant the earth crust concerning the characteristics and effectsthat rule the behavior and the design of ground electrodes.

III. METHODOLOGY PROPOSED TO ELECTRIC FIELD AND SOILPOTENTIAL CALCULATIONS

A. Homogeneous SoilsFor the hypothesis of homogeneous soil configuration (uni-

form electrical resistivity) and simple electrode geometries, theelectric field and potential calculations can be carried using an-alytical methods, as indicated in [1]. Fig. 2 shows a ground elec-trode with toroidal geometry, represented by a section conductorof radius , in a toroid of radius , in a depth of a homogenoussoil having electrical resistivity , being the horizontal dis-tance to the center of the electrode and , the distance referredto the center of a circular section of the conductor, in the groundelectrode boundary.

Under these conditions, the exact solution of the electricfield and potential can be expressed in terms of elliptic integrals.If and , two regions of interest can be assumed,where simple analytical solutions of potential distribution andelectric field can be obtained, as indicated in (1)(4).

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868 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 18, NO. 3, JULY 2003

Fig. 2. Schematic representation of a toroidal ground electrode.

1) Region 1 anda) Potential in a generic point

(1)

b) Electric field

(2)

c) Maximum electric field

(3)

2) Region 2a) Potential in a generic point

(4)

b) Electric field

(5)

Figs. 3 and 4 show, respectively, the potential and electricfield results on the soil surface , for a toroidal groundelectrode with the following data m, A,

m, m, and m.

B. Heterogeneous SoilsIn case of nonhomogeneous soils, if the soil can be assumed

homogeneous within regions separated by a few simple shapesurfaces (e.g., horizontal, vertical, or inclined planes, spheres,cuboids), it is also possible to obtain analytic solutions, consid-ering images or transmission-reflection routine procedures,in eventual series form. It is also possible to consider hetero-geneity of soil in vicinity of electrode, by example due to localheating and humidity changes, by means of an additive correc-tion procedure. For instance, let us consider a soil that can be as-sumed homogeneous within each of two layers separated by ahorizontal plane. Analytic solutions can be expressed in simpleseries form, corresponding to successive transmission-reflec-tion factors, in soil and in the horizontal plane that separatesthe two layers. The resulting formula for different positions of

Fig. 3. Potential U related to a remote earth and electric field E in directionx on soil (z = 0), in the earth electrode vicinity.

a point current source for a ground electrode located in a upperor the lower layer of a two layer soil is presented herein. Byintegration procedures, line or surface current sources formu-lation are easily obtained from point current source formula. Ifthe soil parameters have no simple distribution form, differentmethods can be justified [e.g., solutions expressed in series ofparticular solutions of electric field in cylindrical, spherical, orellipsoidal coordinates, and/or routine procedures of numericalsolution, within the global or integral concept (such as incharge simulation method and similar procedures)]; differen-tial-type methods, such finite differences and finite-elementsmethods.

1) Ground Electrode (Current Source) Located in theUpper Layer of a Two-Layer Ground: Assume a groundelectrode (source current ) located in point , with coordinates

, situated in an upper layer of a two layer soil (elec-trical resistivities and upper layer depth see Fig. 5).The resulting electric field, as an immediate consequence of aperfect reflection condition reflection coefficient inthe soil, is equivalent, in media 1 and 2, to two punctual currentsources and , under the conditions described.

Taking into account successive reflections in the andplanes of waves from and , the potential and the elec-tric field and current density in a generic point located inmedium (1), are identical to those corresponding to a succes-sion of punctual current sources, symmetrical in relation to thesoil, in the vertical of , for the following distances from thesoil (in algebraic means, countered positively from the soil, indownward direction): ; ; ; ;

; , in a uniform medium ofresistivity (see Fig. 6), and the currents, in those sources, of

VILLAS AND PORTELA: CALCULATION OF ELECTRIC FIELD AND POTENTIAL DISTRIBUTIONS INTO SOIL AND AIR MEDIA 869

Fig. 4. Electric fieldE in direction x on the soil surface (z = 0) and potentialU related to a remote earth. approximated solution for x re ; X R, approximated solution for x re ; X R out of validity domain.. correct values.

the form . The electric field in medium (2) is equivalentto one created in an uniform soil of electrical resistivity , bya succession of punctual current sources, in a vertical of , forthe following soil distances: ; ; ;

; ; ; and the currents, in thesesources, of the form (see Fig. 7).

The electric field above the soil surface is identical to one ina homogeneous medium of electrical resistivity associated

Fig. 5. Current source located in the upper layer of a two layer soil.

Fig. 6. Point in an upper layerground electrode in the upper layer.

Fig. 7. Point in a lower layerground electrode in the upper layer.

870 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 18, NO. 3, JULY 2003

Fig. 8. Point above soil surfaceground electrode in the upper layer.

with a succession of punctual current sources, in the vertical of, for the soil distances ; ; ; ;

; ; and with currents in thesesources of the type (see Fig. 8).

The expressions for calculating the potential for genericpoints located in medium (1), medium (2), and above the soilsurface are indicated in (6)(8), as well as the related elec-tric and density fields and .

1) Electric Potential in a Point Locateda) In medium 1

(6)

Fig. 9. Current source located in a lower layer of a two layer soil.

b) In medium 2

(7)

c) Above soil surface

(8)

2) Earth Electrode (Current Source) Located in the LowerLayer of a Two Layer Ground: In a similar way, from the con-ditions of Fig. 9, the series of punctual current sources as a re-sult of successive reflections in and planes is placed, re-sulting in the succession of punctual current sources for differentlocations of the point of interest for voltage and electric fieldcalculations. The calculation of the electric and density cur-rent fields, and the potential , for generic points located in

VILLAS AND PORTELA: CALCULATION OF ELECTRIC FIELD AND POTENTIAL DISTRIBUTIONS INTO SOIL AND AIR MEDIA 871

Fig. 10. Point in an upper layerground electrode in the lower layer.

media (1), (2) and above soil surface (see Figs. 1012), is doneby means of (10)(12) (in stationary conditions).

1) Electric Potential in a point locateda) In medium 1

(9)b) In medium 2

(10)

Fig. 11. Point in a lower layerground electrode in the lower layer.

Fig. 12. Point above soil surfaceground electrode in the lower layer.

c) Above soil surface

(11)grad (12)

872 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 18, NO. 3, JULY 2003

C. Current DensityThe current density components in axes , , and in

any point of interest can be calculated through the expressionsbelow, using the electric field obtained for that point as a func-tion of the electrical conductivity and the ground electrodegeometry under analysis

(13)

D. Resistance, Potential Rise, and Dissipated PowerThe resistance of a ground electrode depends on soil char-

acteristics (spatial distribution of electrical resistivity) and itsgeometry and may vary in time as a result of the following phe-nomena affecting the electrical resistivity

meteorological conditions (rain, solar radiation, temper-ature and air humidity) that originate variations of theelectrical resistivity, especially for lower depths, resultingin soil humidity modification, including temperatureheterogeneity;

climatic and hydrologic effects, causing occasional varia-tion of the freathic level;

effects from the current injected into the soil, through theground electrode, resulting in soil heating, electroosmosisand change of humidity distribution and electrochemicaleffects;

eventual humidification or drainage in the soil, caused byexternal factors, in the ground electrode vicinities.

The ground electrode resistance means, for a given currentinjected through it, the potential rise of a ground electrodein relation to a remote earth. The dissipated power in the soilby the ground electrode is given by

(14)(15)

The potential in the ground electrode represents an upperlimit of the touch, step, and transferred potentials. The dissi-pated power in the soil, even for lower values of the groundelectrode resistance, may be not negligible in terms of lossesof the converters system.

E. Safety Restraints for Human BeingsConcerning the direct current, human beings have a higher

threshold current when compared with the alternating current inindustrial frequency (50 or 60 Hz) with tolerable voltage limitsindicated in [5]. A safe electrode design must take into accountthese aspects due to the risk of electrical shocks involved.

IV. EXAMPLE CASESThe main characteristics of the ground electrode and the soil

considered in the simulations under analysis will be given. Theresulting profiles of electric fields calculated for homogeneousand nonhomogeneous soils are indicated in Fig. 13, in theboundaries of the ground electrode in its plane of installation

m . The profile of current densities around the groundelectrode in its vicinities as well as the potentials are indicated,

Fig. 13. Electric field profiles (Case 1, in m).

Fig. 14. Current densities.

Fig. 15. Potential profile (Case 2).

respectively, in Figs. 14 and 15 for different depths of interestfor a homogeneous soil (Case 2). Fig. 16 show the spatialdistribution of the electric field related to the results shown inFig. 13.

These results fit well with those reported in [6], allowing agood conclusion of the effectiveness of the proposed technique.

a) Ground electrode geometry: ring (toroidal section); electrode diameter: 300 m; conductor diameter: 0.6 m; depth: 2.7 m;

VILLAS AND PORTELA: CALCULATION OF ELECTRIC FIELD AND POTENTIAL DISTRIBUTIONS INTO SOIL AND AIR MEDIA 873

Fig. 16. Electric field spatial distribution.

number of segments used for ground electrode mod-eling: 144;

injected current: 2000 A.b) Soil

Case 1) nonhomogeneous soil m;m; m;

Case 2) homogeneous soil m.

V. CONCLUSION The equations presented in this article and the method-

ology described allow sufficiently accurate calculations ofelectric field, density current, and potential in any pointof the soil medium for a HVDC system ground electrodeconfigurations immersed in homogeneous and nonhomo-geneous soils;

When symmetries in ground electrodes are present,like spherical and cylindrical geometries, the use oftwo-dimensional (2-D) approximations is valid and ad-visable, aiming memory storage and time computer lessprocessing;

For a proper analysis of the electroosmosis and soilheating phenomena, the correct calculation of the electricfield in generic points of the soil and above it (air) isessential. The electric field variation as a function of thetemporal changes of soil electrical resistivity by migrationof the humidity present in the medium is also a mandatoryaspect to be observed.

REFERENCES[1] C. Portela, Interaction among electric, thermal and electroosmotic

phenomena in electrodes of direct current systems (in Portuguese), inII FrenchBrazilian Symposium in Electric and Magnetic Fields, SoPaulo, 1989.

[2] EPRI EL-2020, Project 1467-1, in HVDC Ground ElectrodeDesign. San Francisco, CA: IEC, 1981, pp. 4.14.6.

[3] J. E. T. Villas et al., Computation of electric fields using ground gridperformance equations, IEEE Trans. Power Delivery, vol. PWRD-2,pp. 682690, July 1987.

[4] R. J. Heppe, Computation of potential at surface above energized gridor other electrode, allowing for nonuniform current distribution, IEEETrans. Power Apparat. Syst., vol. PAS-98, pp. 19781989, Nov./Dec.1979.

[5] E. W. Kimbark, Direct Current Transmission (Book). New York:Wiley, 1971, pp. 450455.

[6] R. J. Heppe, Computer aided design of a toroidal ground electrode ina two-layer soil, IEEE Trans. Power Apparat. Syst., vol. PWRD-2, pp.744749, July 1987.

Jos Eduardo Telles Villas received the electricalengineer degree from the UERJ - State University ofRio de Janeiro (UERJ), Brazil, in 1976, and the M.Sc.and D.Sc. degrees from the Federal University of Riode Janeiro, Brazil (COPPE/UFRJ) in 1980 and 2000,respectively.

Currently, he is Marte Engenharia Technical Di-rector and UERJ teacher. Since 1977, he has workedin the major consulting companies in Brazil, beinginvolved with ITAIPU Power Plant System Studies.His research interests include electromagnetic tran-

sients phenomena, HVDC studies, and grounding area.

Carlos Medeiros Portela received the electrical en-gineering degree from IST in 1958, UTLLisbonTechnical University, Portugal, and the D.Sc. degreefrom UTL in 1963.

Currently, he is Titular Professor at FederalUniversity of Rio de Janeiro, Brazil (COPPE/UFRJ),working on research projects in transmission systemsand equipment. He was Cathedratic Professor atUTL since 1972. He was responsible for Portugueseelectrical network studies and planning and forPortuguese electrical network operation, and for

several other studies and projects in Portugal/Lisbon electrical. He was alsoresponsible for some major studies and projects in the electric power andindustry sectors in Brazil and other countries.

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