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2150 IEEE TRANSACTIONS ON MAGNETICS. VO L 31 . NO 3. MAY 1995

A Finite Element Method for Calculating the Electromagnetic Fields

Generated by Substation Grounding Systems

B. Nekhoul, C. Guerin, P. Labie, G. Meunier and R. FeuilletLaboratoire d'Electrotechniquede Grenoble, LEG ,URA CNRS 355, ENSIEG, INPG,

BP 46, 38402 Saint Martin d'HSres Cedex France.

X . BnmotteCedrat Recherche, 10 chemin du Pri-cure, ZIRST 4301, 38943 Meylan Cedex France.

Abs t rac t - This paper develops a basic idea which

allows low frequency electromagnet ic f ie ldsgenerated by cylindrical conductors, with verysmall radii compared with their lengths, to becalculated. The proposed method can be applied toboth overhead conductors and those buried in aconducting environment. The paper presents adirect application for this concept which consistsin calculating electromagnetic fields created by asubs t at ion ground ing sys t em f o l low ing anaccidental short circuit or a stroke of lightning.The Finite Element Method (FEM) with severalformulations will be used. Currents in thecylindrical conductors which make up the earthnetwork grid are taken into account without volume

meshing; simple line discretization of these will becarried out. Electromagnetic fields generated byleakage currents and currents induced undergroundare taken into account. Furthermore, openboundaries of both air and earth environments areprocessed by introducing a spatial transformation.This application for an earth network enables us toshow the advantages of FEM processing comparedwith the method traditionally used in antennatheory for this kind of problem.

I. INTRODUCTION

During a fault on a power network, due to an accidentalshort circuit or stroke of lightning, after the transient statesome metal parts of Air Insulation Substations (AIS) maybecome live. In order to avoid this and to allow fast

discharge, most of these parts are linked to a safety systemcalled the "grounding system". This safety system is a groupof cylindrical conductors which for an AIS con sists of a gridburied underground at an average depth of about 0.5m. Thefault currents, which are generally homopolar, flow throughthe earth network. T he mass loop is often very widespread,leakage currents underground can be high (100 kA), theirelectromagnetic emission may cause disturbances inequipment in the substation itself (remote control,telemeasuremen ts and telephones) and stepup voltage (voltagebetween two human legs at l m ) may be dangerous for peoplenearby.

Quantification by calculating electromagnetic emissionfrom the grounding systems has already been approached byF. Dawalibi et al. [l]. The solution proposed consists insolving Maxwell equations numerically under their integral

forms; this method is better known as "integral formalism"and is often used in antenna theory [l ]. In this formalism,calculation is generally carried out in two steps : calculation.Manuscript received July 6, 1994.

of currents and then fields or vice-versa. When accou nting forthe two infinite semi-environments (air and earth), theauthors introduce boundary integrals and Sommerfeldintegrals which enable them to process change inenvironment [11 (air-earth or earth-air). The formalismproposed is therefore very difficult to implement andextremely heavy numerically. Furthermore, it only allowscalculation of fields generated by currents running throughconductors which are horizontal or vertical, overhead orunderground.

This work plans to solve formulations which will enablethe problem of electromagnetic emission from the earthnetwork to be dealt with by the E M . The advantage of thismodelling is that calculation in several steps can be avoided;a single set of equations is solved in only one step.Modelling with FEM is easier to implement and less CPU

time consuming because the two infinite semi-environmentsare accounted fo r with sp atial transformations rather than withboundary integrals and moving from one environment toanother is ensured by linking the different form ulations used.Furthermore, this solution is more accurate because the fieldscalculated are those caused by both currents in the con ductorsmaking up the earth network and leakage currents flowing inearth. Moreover, the problem can be dealt with whatever theposition of the conductor or its geometry, which means thatdifferent kinds of earth network (vertical pole, radiating armetc.) can be processed.

11. F I NI TE ELEMEN TORMULATIONS

In the problem we are dealing with three elements(materials) to come into play : cylindrical copper conductors,ground and air. For this reason, different formulations areused to describe the physical phenomena in each domain (lowconducting volume "ground" area, line conducting "copper"area and non-co nducting volume "air" area). Physical analysisof the problem of electromagnetic field generated by buriednetwork shows that it can be solved with an AV formulationin conducting areas, this being justified by the presence ofcurrents in the buried conductors and in the ground and bychange in conductivity when moving from the conductor tothe ground and vice-versa. A vector potential A or a totalmagnetic scalar potential cp formulation may b e used in theair. This modelling of earth emission owes a great deal to 0.

Biro's et al. work [ 2 ] . n this study they highlight the use ofmagnetic vector potential A when dealing w ith eddy currents.

For conducting volume areas, the AV volume formulationis generally used. This formulation is translated, for an

isotropic, homogeneous and linear environment by the twofollowing equations :

0018-9464/95$04.00 0 995 IEEE

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4 4 - 4 4 -(1rot w).(rot A) +I divw) (divA)

P4 4 4 -

+ 0jjoW.A + oW.gradV) d R = 0 (1 )

(2)

w: weighting function of a scalar potential for a volumeelement.W: weighting function of a vector potential for a volumeelement.

For non-conducting areas (air) with source current, themagnetic vector potential A formulation is generally used.This formulation is translated by the following equation :

(3)

Numerical processing with FEM is often CP U timeconsuming. In order to optimise this , the number ofunknowns has to be reduced by using as few vectorialmagnitudes as possible. In the air, this CPU time is saved byintroducing the formulation called total magnetic scalar

I , II

IQ(O&&w.(jjoA + gradV)) dQ = 0

jn(k(z.(zi> +I divw) (divA) dR = 0

4 4

P + -7

' potential, given by the following equation :

JQ(p(G ) . k Z 9)) R = 0

cp : total magnetic scalar potential.

(4)

111. LINEAV FORMULATION

In finite element approximation, the domain to be studiedis discretized into elements (volume, surface and lines).Sometimes the geometry of an area of the problem to bemodelled shows a particular disproportion where dimensionsare concerned (e.g. thin plate, thin cylindrical conductor, etc.).This type of geometry introduces drawbacks during 3Dmeshing using an automated mesh generator. Such type ofgeometry leads to a large number of elements which canincrease the size of the problem considerably and upsetsnumerical processing. This geometrical constraint leads us tosuggest, for a conductor with a very small radius compared toits length, another form for the AV formulation which willbe set out in the following.

Fo r a cylindrical conductor, with low frequency harmoniccurrent, with a diameter small enough compared to its length(thin wire structure) it can easily be shown that vector A andscalar V potentials are constant on a surface perpendicular tothe axis of the conductor. This has been show n for the case ofthin conducting shells in [3].

Fig. 1. Line device.

On this condition, energy in the conductor is preserved by

translating the volume integral as follows.

S: section of conductor, h: length of conductor and R:

volume of conductor.This way of writing the volume integral means that

volume meshing of the cylindrical conductors can be avoided.Only line discretization of the conductor is needed and thenthe AV formulation has to be written in a "line" form. Theequation system giving the AV formu lation then becomes:

- 4--. -4S (1r o t W M o t A ) +1divWh)(divA)

P- 4 - -+ 0 joWh.A + oWh.gradV)dX= 0 (5)l A

4 -S ( ~ g G w ~ . ( j j o Agradv)) dh = 0 (6)

Jh

W A:weighting function of a scalar potential for a lineelement.WA: weighting function of a vector potential for a lineelement.

It should however be noticed that this way of writing isonly valid on condition that the thickness of the skin isgreater than the diame ter of the conductor.

IV. OPENBOUNDARY OM AIN ODELLING

A simple physical analysis of the problem of substationgrounding system shows the need for including openboundaries of semi-environm ents (air and ground) andparticularly that of the ground. It is obvious that the earthresistance, therefore the flow of current in the grid andunderground, depends on the "ground" volume that is takeninto account. Indeed, considering that the bare conductors arein direct contact with the low conducting ground, it isobvious that the equivalent resistance of grid and grounddepends on the amount of "ground" included.

For taking into account of open boundary problems, FEMprovides several modelling methods [41:

- truncation: the domain to be modelled is surrounded by abox, as big as poss ible , and boundary condi t ionscorresponding to infinity (potentials are set to zero) or to a

tangential field or a normal field [2] are imposed on the facesof this box. The bigger the box, the smaller the error.- inclusion of infinity with open boundaries by boundary

- infinite elements- use of spatial transformation s [ 5 -6 ] .- et c ..

For substation grounding systems, the dimensions of theburied grid may be some hundred metres. It is then clear thatcorrect processing of emission from the earth network meansmobilising huge calculation capacities where finite elementprocessing is carried out by truncation. In order to modelinfinity correctly without increasing the size of the problem,we have chosen to use the method based on spatialtransformation [S-61.

A .Principle of the Method

The method consists of converting the initial openboundary problem into a closed boundary problem usingspatial transformation. Consider that the open boundary

integrals

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domain E is the sum of a closed internal subdomain Eint,treated using FEM and of an extemal open su bdomain, E,,[.

We then-have: E = Eint U Eext. The real external domain

is linked to the closed image domain, Eext-image. by meansof a bijective transformation as it appears in figure 2.  Thisdomain E&t-image is meshed by classical finite element andused to solve the problem with the finite element m ethod.

The best method for taking into account the transformationis to consider that the elements of the image domain (whichis meshed with classical elements) allows to define on thereal extemal a new type of elements (w e called mapped finiteelements), as shown in figure 2. As a consequence, thismethod may be ap plied to any finite element form ulations.

In paper [6], this method was applied to model air formagnetostatic scalar problem. In this study, we take intoaccount the two infinite semi-environments, the air (A or cpformulation) and the earth including eddy currents in ground(A V formulation).

Real open boundary

, domain ,image domain

Closed and meshed

mapped element image element element of referenceFig. 2. Mapped, image and.reference elements.

B. Appl icat ion fo r Model l ing the Two Infinite Senzi-Environments Air and Ground

Compared to [5], the interesting point for this infinitemodelling method come from the two infinite semi-environmen ts, air and earth including eddy currents in ground.The physical properties of these two regions are different, aswell as formulations (A V formulation for the ground witheddy currents, A or cp formulation for the air),

Geometrically speaking to obtain the two infinite semi-environments, we used the parallelepiped shell mapping [6]cut into two parts in y equal to zero plane (figure 3) . From anumerical point of view, nothing special would be necessaryto do in the softwa re when dealing w ith air using A

formulation and ground with AV formulation, because thecoupling is then implicit (continuity of the normalcomponent of the flux density is ensured as A is continuousbetween two nodal elements and the continuity of thetangential component of the m agnetic field is ensured in a

weakly sense from the governing equations). In this study,we preferred the use of scalar formulation ( c p ) to take intoaccount the semi-environment air. It is then necessary toapply the mapping on the volume elements as well as on the

boundary elements between the air region and the earth toensure field continuity.

V . FINITEELEMENT ODELLING

Figure  3 recapitulates FEM modelling of the problem ofelectromagnetic fields generated by substation groundingsystem.

an cp: total magnehc

Fig. 3. Regions, boundary sections and interfaces.

The problem described on the above diagram is notcomplete if interface conditions are not determined . In order toensure continuity of the field between the areas modelled,continuity conditions must be introduced on the interfaces.Use of the vector potential A has been studied in paper [21

who explains the different couplings which can ensureconditions of continuity between fo rmulations.

VI. APPLICATION

Figure 4.a represents a 60mx60m grid buried 0.5m fromthe air-ground interface, giving a rough idea of the substationgrounding system. A 1 kA current at a lOHz frequency isinjected in the middle. Results of calculation for this typicalexample are given by F.Dawalibi et al. [l].All the results ofsimulations which are given in this paragraph are thoseobtained on an 80mx80m working matrix placed on the air-

ground interface.

Fig 4 a. A square grounding grid subject IS a lightning stroke.

- Characteristics of the ground are:

- Characteristics of the grid are:ps= po ; ps = 1000Rm ; = 5 E o .

pC=po; ; p c = 1.7 1 0 8 R m ; =EO .

rc = 0.5 cm (radius of the conductors).

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Considering the size of the grid, we have used the spatialtransformation for modelling the two open boundary semi-environments. This means the size of the problem can bereduced. The ground is taken to b e an isotropic, homogeneousand linear environment.

Fig. 4.b. Magnetic field on Earth surface (Results obtained with FEM).

?. I

4 ~ l 4 i n ~Disunceflma 9 hs (m)

Fig. 4.c. Magnetic field on E arth surfac e (Results published in.[l]).

Fig. 4.d. Electric field on Earth surface (Results obtained with E M ) .

1' = {

I 1

Disuncs A b u l X Axis (m)

Fig. 4.e. Electric field on Earth surface (R esults published in [ I].

The FEM can be applied to determine the potential in earthand the resistance of grounding network for any complexelectrode. In table 1 are confronted the results obtained by F.Dawalibi et al. [7] and the proposed method.

TABLE 1. Comparison of the results.

Variable I FEM I Ref [7]Resistance (Q) I 6 7.33

Results of calculation obtained with FE M compare wellwith those obtained by F. Dawalibi et al. both from the pointof view of general shape and amplitude. Amplitude obtainedwith FEM are slightly higher (in the magnetic field and inthe electric field); This may be explained by the fact thatfields emitted by leakage current underground are included inthe formalism, which is not the case for the method proposedby F. Dawalibi et al. where only currents running throughconductors intervene.

VII. CONCLUSION

Comparison between these results of FEM calculation andthose published by F. Dawalibi et al. has shown that FEMmodelling is more than acceptable. FEM has severaladvantages which have already been mentioned, the most

important being easier and cheaper implementation forincluding open boundary semi-environments, conductivityvariation, calculating the magnetic field at any poin t in space(air and ground), the electric field underground and to avoidthe use of Sommerfeld integrals [l],which are very difficultto solve exactly and are very cpu time consum ing, contrary tothe E M . The general shape of results obtained with E Mcan be improved by refining the mesh. This should increasethe size of the problem and inevitably cp u ime. This has notalready been verified .

It would be interesting for further work to process the sameproblem at very high frequency. A finite element solution ofthe problem would certainly be advantageous.

I t is possiblc to take into account several layers of theground with different properties with the mapped boundaryelements.

REFERENCES

[ I ] W . Xiong , F. P. Dawalibi and A. Selby, "Frequency Response ofSubstat ion Ground Systems Subject to Lightning Strikes", GIGREI n t e r n a t i o n a l S y m p o s i u m , P o w er S y s t em E l ec t r o m ag n e t i cCompatibility. Switzerland, Lausanne 93. 100-06.

[2 ] 0.Biro and K. Preis, "On the Use of the Magnetic Vector Potential inthe Finite Element Analysis of the Three Dimensional Eddy Current",IEEE Trans. on Magnetics, Vol. 25. No 4. July 1989, pp 3145-3159.

[3 ] T. Nakata, N. Takahasi, IC. Fujiwara and Y. hiraki, "3D MagneticField Analysis Using Special Elements", IEEE Trans. on Magnetics,vol . 26, No 5, eptember 1990 , pp. 2379-2381.

[ 4 ] C.R.I . Emson, "Methods for the so lu t ion of open boundaryelectromagnetic field problems", IEE Proc, Vol. 135, Pt. A, N"3,March 1988.

[SI X. Brunotte, G. Meunier and J.P. Bonigiraud, "Ship magnetizationsModelling by the Finite Element Method", IEEE. T rans. on magnetics,Vol. 29 , No , pp. 1970-19 75, March 1993.

[6 ] X. Brunotte, G. Meunier and J. F. Imhoff, "Finite element solution ofunbounded prob lems using transformation: a rigorous powerful and

emy solution", IEEE Trans. on magnetics, Vol. 25 , No 2, March 1992,pp. 1663-1666,.

[7] F. Dawalibi and D.Mukhadar, "Optimum Design of Substation in TwoLayer Earth Structure, Part 1- Analytical Study , IEEE Trans on PAS,Vol. PAS-94, No 2. pp. 252-261, March / April 1975.