numerical simulation of earthing grids - wseas the authors in the last years for the computational...

Numerical simulation of earthing grids

IGNASI COLOMINAS, FERMIN NAVARRINA & MANUEL CASTELEIROUniversidade da Coruna

Group of Numerical Methods in Engineering, GMNICivil Engineering School, Campus de Elvina. 15071 LA CORUNA (SPAIN)

email→ [email protected] web→ http://caminos.udc.es/gmni/

Abstract: The paper presents the main numerical approaches based on the Boundary Element Method developedby the authors in the last years for the computational design of grounding systems of electrical installations inuniform and layered soils.

Key–Words:Grounding grids, Boundary Element Method, Layered soil model, Numerical simulation

1 Introduction

The “grounding” or “earthing” system of an electricalsubstation comprises all interconnected grounding fa-cilities of an specific area, being the “grounding grid”its main element. In general, it consists of a meshof interconnected cylindrical conductors, horizontallyburied and supplemented by ground rods verticallythrusted in specific places of the installation site [1].

The accurate design of a grounding system is es-sential to assure the safety of the persons, to protectthe equipment and to avoid interruptions in the powersupply. Thus, the apparent electrical resistance of thegrounding system must be low enough to guaranteethat fault currents dissipate mainly through the earth-ing electrode into the ground, while the values of elec-trical potentials between close points on the earth sur-face that can be connected by a person must be keptunder certain maximum safe limits [1, 2].

Obtaining the potential distribution on the earthsurface produced when a grounding grid is energizedduring a fault condition has been a challeging prob-lem since the early days of the industrial use of theelectricity. It is a well-known fact that the physicalphenomena of fault currents dissipation into the earthcan be modelled by means of Maxwell’s Electromag-netic Theory. However, its application and resolutionfor the computing of grounding grids of large installa-tions in practical cases present some severe difficul-ties. Obviously, no analytical solutions can be ob-tained in a real case. Moreover, the specific geometryof the grounding systems (a mesh of interconnectedbare conductors in which ratio diameter/length is rel-atively small) precludes the use of standard numericaltechniques (FEM or FDM), and obtaining sufficientlyaccurate results should imply unacceptable computingefforts in memory storage and CPU time [3, 4].

From a technical point of view, several methodsfor grounding design have been proposed. Most ofthem are founded on semiempirical works or on thebasis of intuitive ideas, such as superposition of punc-tual current sources and error averaging [1]. Althoughthese techniques represented a significant improve-ment in the earthing analysis area, also present someproblems related with their large computational re-quirements, the unrealistic results obtained when seg-mentation of conductors is increased, and the uncer-tainty in the margin of error [1, 2, 5, 6].

Taking into account all these facts, the authorshave proposed in the last years a general numeri-cal formulation for grounding analysis based on theBoundary Element Method. This BEM approach hasbeen successfully applied to the analysis and design ofreal grounding grids embedded in uniform [3, 4] andstratified soil models [7, 8].

In this paper we present the main highlights ofthis BEM numerical approach and we discuss its maincharacteristics, advantages and restrictions. Finally,an application example by using the real geometry ofa grounding system is presented.

g1

2g

g2

g1

a)

b)

Figure 1: Scheme of two-layer soil models: horizontal(a)) and vertical (b)).

Proceedings of the 6th WSEAS International Conference on Power Systems, Lisbon, Portugal, September 22-24, 2006 78

(89,143)

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Figure 2: Plan of the Grounding Grid.

2 Mathematical Model

Equations of Maxwell’s Electromagnetic Theory arethe starting point for studying the physical phenom-ena of the fault current dissipation into the ground [9].Constraining the analysis to the obtention of the elec-trokinetic steady-state response and neglecting the in-ner resistivity of the grounding electrode (therefore,potential is assumed constant in every point of its sur-face), the 3D problem can be stated as

div(σσσσσσσσσσσσσσ) = 0, σσσσσσσσσσσσσσ = −γγγγγγγγγγγγγγ grad(V ) in E ;σσσσσσσσσσσσσσtnnnnnnnnnnnnnnE = 0 in ΓE ; V = VΓ in Γ ;

V → 0, if |xxxxxxxxxxxxxx| → ∞; (1)

beingE the earth,γγγγγγγγγγγγγγ its conductivity tensor,ΓE theearth surface,nnnnnnnnnnnnnnE its normal exterior unit field andΓthe electrode surface [4]. Therefore, when the earth-ing electrode attains a known voltageVΓ (the so-called“Ground Potential Rise”, or GPR) relative to a dis-tant grounding point, the solution to (1) gives poten-tial V and current densityσσσσσσσσσσσσσσ at an arbitrary pointxxxxxxxxxxxxxx.On the other hand, as well as obtaining the potentialdistribution, other important parameters for groundinganalysis can be easily computed from (1), such as theleakage current densityσ at an arbitrary point of theelectrode surface, the total surge currentIΓ that flowsinto the ground, and the equivalent resistance of theearthing systemReq:

σ = σσσσσσσσσσσσσσtnnnnnnnnnnnnnn, IΓ =∫ ∫

Γσ dΓ, Req =

VΓ

IΓ, (2)

wherennnnnnnnnnnnnn is the normal exterior unit field toΓ. Finally,sinceV andσσσσσσσσσσσσσσ are proportional to the GPR value, it isgenerally used the non-restrictive normalized bound-ary conditionVΓ = 1 [4].

The two main problems that appear when onestries to solve problem (1) in the grounding analysisfield are the definition of the kind of soil and the ge-ometry of the earthing electrode.

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Figure 3: Potential distribution (×10 kV) on groundsurface obtained by considering the soil model A.

With regard to the soil, the development of soilmodels describing all variations of the soil conduc-tivity around the grounding system would be obvi-ously unaffordable, neither from the economical norfrom the technical point of view. For these reasons, inmost of methods for grounding analysis proposed upto this moment, the soil is considered homogeneousand isotropic. Therefore, the conductivity tensorγγγγγγγγγγγγγγ issubstituted by an “apparent” scalar conductivityγ thatcan be experimentally obtained [1, 2]. This hypothe-sis can be accepted and does not introduce significanterrors if the soil is basically uniform (in the horizontaland the vertical direction) up to a distance of approx-imately 3 to 5 times the diagonal dimension of thegrounding grid, measured from its edge. And it canalso be used but with less accuracy, if the resistivityvaries slightly with depth [1].

Nevertheless, since design parameters involved ingrounding analysis can significantly change as soilconductivity varies, more accurate models have beenproposed to take into account its variations in thevicinity of the grounding site. Hence, a more prac-tical and quite realistic approach when conductivity isnot markedly uniform with depth consists of consid-ering the soil stratified in a number ofL horizontal


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Figure 4: Potential distribution (×10 kV) on groundsurface obtained by considering the soil model B.

layers (normally, two or three), each one defined byan appropriate thickness and an apparent scalar con-ductivity [1] as it is shown in Fig. 1a). With this soilmodel, if the grounding electrode is buried in layerb,the mathematical problem (1) can be written in termsof the following Neumann exterior problem

∆Vc = 0 in Ec ; c = 1, L;Vc−1 = Vc, in Γ(c−1,c) ; c = 2, L;

γc−1dVc−1

dn = γcdVcdn in Γ(c−1,c) ; c = 2, L;

dV1dn = 0 in ΓE ; Vb = 1 in Γ;

Vc −→ 0, if |xxxxxxxxxxxxxx| → ∞ ; c = 1, L; (3)

beingEc each one of the soil layers(c = 1, . . . , L),Γ(c−1,c) the interphase between two layers (c− 1 andc), γc the scalar conductivity of layerc, andVc thepotential at every point of layerc [8, 10]. Obviously,the grounding analysis with a uniform soil model is aparticular case of (3) with one layer [4].

Regarding the geometry of the earthing systems,in most of real electrical substations the groundinggrid consists of a mesh of hundreds of interconnectedcylindrical conductors horizontally buried and supple-mented by ground rods, with a ratio between the di-ameter and the length of the electrodes very small(∼ 10−3). Evidently, no analytical solutions existto problem (3) with this geometry, and the use ofwidespread numerical techniques (i.e., finite differ-ences or finite elements) should imply a completely

out of range computing effort due to the cost of mesh-ing (with a 3D discretization) in the vicinity of theelectrodes in an effective way to impose the essentialboundary conditionVb = 1 on their surfaces [4].

For these reasons, we work to achieve an equiv-alent expression to problem (3) in terms of the un-known leakage current densityσ defined on theboundaryΓ. In this way, a boundary element ap-proach for this equivalent problem would only requirethe discretization of the grounding surfaceΓ, avoidingthe discretization of the domainsEc [4].

The deduction of this integral expression for thepotential can be performed if we introduce in themodel one fact related with the engineering problem,that is, the levelling and regularization processes ofthe surroundings of the substation site during the con-struction process of the electrical installation. Thus,the earth surfaceΓE and the interfacesΓ(c−1,c) be-tween layers can be assumed horizontal; consequentlywe will adopt an “horizontally layered soil model” inour mathematical model in order to catch the varia-tions of the soil conductivity with depth (Fig. 1a)).In the case of variations of the soil conductivity nearthe grounding system, the simplest model we can state(a “vertically layered soil model”) consist of asumingthat the earth surfaceΓE is horizontal, while the in-terfacesΓ(c−1,c) are parallel one to another and per-pendicular toΓE (Fig. 1b)). Now, the subsequent ap-plication of the “method of images” and the Green’sIdentity to (3) allows to obtain the following expres-sion for potentialVc(xxxxxxxxxxxxxxc) at an arbitrary pointxxxxxxxxxxxxxxc ∈ Ec

(c = 1, . . . , L), in terms of the leakage current densityσ(ξξξξξξξξξξξξξξ) at every pointξξξξξξξξξξξξξξ of the electrode surfaceΓ, whichis buried in the layerb:

Vc(xxxxxxxxxxxxxxc) =1

4πγb

∫ ∫

Γkbc(xxxxxxxxxxxxxxc, ξξξξξξξξξξξξξξ) σ(ξξξξξξξξξξξξξξ)dΓ, (4)

where the integral kernelkbc(xxxxxxxxxxxxxxc, ξξξξξξξξξξξξξξ) is formed by aninfinite series of terms corresponding to the resultantimages [8, 10]. This weakly singular kernel dependson the inverse of the distances from the pointxxxxxxxxxxxxxxc tothe pointξξξξξξξξξξξξξξ and to all the symmetric points ofξξξξξξξξξξξξξξ (its“images”) with respect to the earth surfaceΓE andto the interphasesΓ(c−1,c) between layers. Therefore,this kernel depends on the thickness and conductiv-ity of each layer. It is important to remark that, ina general case, the expression of the integral kernelskbc(xxxxxxxxxxxxxxc, ξξξξξξξξξξξξξξ) can be very complicated, despite the gen-eration of electrical images is a conceptually simplewell-known process [11], and its evaluation in prac-tice may require a high computing effort (explicit ex-pressions of the integral kernels for a two-layer soilmodel can be found in references [8, 10, 12]).

Expression (4) is very important for the solutionof the problem, because if the leakage current den-


sity σ is known, then it is possible to obtain the valueof the electrical potential at an arbitrary pointxxxxxxxxxxxxxxc, andby using (2) it is also possible to compute the totalsurge current that flows from the grounding system, itsequivalent resistance and most of the remaining safetyand design parameters of a grounding grid [4].

The leakage current densityσ can be obtained bysolving a Fredholm integral equation of the first kinddefined onΓ [4]. This integral equation results by im-posing the boundary conditionVb(χχχχχχχχχχχχχχ) = 1 ∀χχχχχχχχχχχχχχ ∈ Γ inthe expression for potential (4), which also holds onthe electrode surfaceΓ. Finally, we can obtain a vari-ational form of this integral expression imposing thatit is verified in the sense of weighted residuals, that is,the following integral identity

∫ ∫

Γw(χχχχχχχχχχχχχχ)

[1− 1

4πγb

∫ ∫

Γkbb(χχχχχχχχχχχχχχ, ξξξξξξξξξξξξξξ)σ(ξξξξξξξξξξξξξξ) dΓ

]dΓ = 0

(5)which must hold for all membersw(χχχχχχχχχχχχχχ) of a suitableclass of weight functions defined on the surfaceΓ[4]. Now, since the unknown of the problem (that is,the leakage current densityσ) is only defined on theelectrode surfaceΓ, a boundary element formulationseems to be the right choice to solve (5) [4].

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Figure 5: Potential distribution (×10 kV) on groundsurface obtained by considering the soil model C.

3 BEM Numerical Models

The development of a general numerical approachbased on the BEM for solving the equation (5) re-quires the discretization of the unknown leakage cur-rent densityσ and the electrode surfaceΓ, in terms ofa given set ofN trial functionsNi(ξξξξξξξξξξξξξξ) defined onΓand a given set ofM 2D boundary elementsΓα:

σ(ξξξξξξξξξξξξξξ) ≈ σh(ξξξξξξξξξξξξξξ) =N∑

i=1

Ni(ξξξξξξξξξξξξξξ) σhi , Γ =

M⋃

α=1

Γα (6)

Likewise, it is possible to obtain a discretized ver-sion of the integral expression (4) for potentialVc(xxxxxxxxxxxxxxc)

Vc(xxxxxxxxxxxxxxc) =N∑

i=1

Vci(xxxxxxxxxxxxxxc) σh

i , ∀xxxxxxxxxxxxxxc ∈ Ec; c = 1, L; (7)

being

Vci(xxxxxxxxxxxxxxc) =M∑

α=1

V αci

(xxxxxxxxxxxxxxc), (8)

V αci

(xxxxxxxxxxxxxxc) =1

4πγb

∫ ∫

Γαkbc(xxxxxxxxxxxxxxc, ξξξξξξξξξξξξξξ) Ni(ξξξξξξξξξξξξξξ) dΓα. (9)

Finally, for a given set ofN test functionswj(χχχχχχχχχχχχχχ) defined onΓ, the variational form (5) is re-duced to the following system of linear equations

N∑

i=1

[Rji] σhi = νj , j = 1, . . . , N, (10)

being

Rji =M∑

β=1

M∑

α=1

Rβαji ,

Rβαji =

14πγb

∫ ∫

Γβwj(χχχχχχχχχχχχχχ)

∫ ∫

Γαkbb(χχχχχχχχχχχχχχ, ξξξξξξξξξξξξξξ)Ni(ξξξξξξξξξξξξξξ)dΓαdΓβ

νj =M∑

β=1

νβj , νβ

j =∫ ∫

Γβwj(χχχχχχχχχχχχχχ)dΓβ. (11)

Solution of this linear system provides the val-ues of the current densitiesσh

i (i = 1, . . . , N) leak-ing from the nodes of the grid. This general BEMapproach can be applied in cases of grounding gridwith a few number of electrodes since the statementof linear system (10) requires the discretization of thewhole surfaceΓ of the grounding electrodes. In prac-tical cases, with hundreds of cylindrical electrodes,this fact implies a large number of degrees of freedom.Besides, the coefficient matrix is full and the computa-tion of terms (11) requires double integration on a 2Ddomain [4]. Apart from these numerical drawbacks,


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Figure 6: Potential distribution (×10 kV) on groundsurface obtained by considering the soil model D.

in the case of kernelskbb(·, ·) defined by infinite se-ries (i.e., in the two-layer soil models), the computa-tion of coefficientsRβα

ji may require the evaluation ofan extremely high number of terms of the series [8].

Consequently, the authors have developed an ap-proximated approach to this general BEM formulationby introducing some additional hypotheses in order toreduce the computational effort [4]. Basically, this ap-proximated approach lies in taking into account thereal geometry of grounding systems in practice, byassuming that the leakage current densityσ is con-stant around the cross section of the conductors of thegrid (now, only the axial lines of the electrodes haveto be discretized). Besides, it is possible to developa highly efficient analytical integration technique tocompute the coefficient terms of the linear system ofequations so that they can be computed by means ofexplicit formulae [4].

The resulting numerical approach reminds the so-called “computer methods” for grounding analysis[6], whereRji coefficients correspond to “mutual andself resistances” between segments of conductors. Infact, some particular cases of our BEM approach canbe identified with any of the very early intuitive meth-ods that were proposed in the 60’s on the basis of re-placing each segment of electrode by an ’imaginarysphere’. In the case of a Galerkin type weighting withconstant leakage current elements, the numerical ap-proach can be identified with a kind of more recent

methods, like the APM [5], in which each segmentof electrode is substituted by ’a line of point sourcesover the length of the conductor’ [4, 6, 13]. Now, ourBEM approach allows to explain from a mathematicalpoint of view the problems encountered by other au-thors with the application of these widespread meth-ods [6], while new more efficient and accurate numer-ical formulations can be derived [4, 13].

The example presented in this paper correspondsto a case of a grounding grid of a real substation com-puted by using different soil models with one and twolayers. Obviously, this BEM formulation can be ap-plied to any other case with a higher number of lay-ers. However, CPU time may increase exponentiallydue to the poor rate of convergence of the underlyingseries expansions. We have dealt this problem by twodifferent ways: the improvement of the convergenceby using new extrapolation techniques, and the de-velopment of a massively parallel numerical approachbased on the previous ones for grounding analysis inlayered soils. In this way, the proposed BEM tech-nique has been implemented in a CAD system devel-oped by the authors for grounding substation design,extending the original capabilities of this tool initiallydeveloped for uniform soil models [4].

Recently, we have developed a new numerical ap-proach from the previous one in order to tackle a com-mon and very important engineering problem in thegrounding field: the analysis of transferred potentials.In this case, the objective is to compute the potentialscan be transferred to other grounded conductors in thevicinity of the earthing installation, and subsequentlythey could affect distant points through communica-tion or signal circuits, neutral wires, pipes, rails, ormetallic fences. This effect could produce serioussafety problems that should be estimated somehow[1]. Our BEM numerical approach allows the analy-sis of transferred potentials in grounding installationsembedded in uniform and layered soils, being possi-ble to check the safety parameters of the groundingsystem and its surroundings [12].

4 Example & Conclusions

In this section a complete application example of theBoundary Element numerical formulation will be pre-sented. The numerical approach proposed up to herehas been implemented in a Computer Aided Designsystem. This tool performs grounding analysis for allthe soil models exposed in the previous sections.

The example consists of the application of theCAD system to the analysis of a large real ground-ing grid: this earthing system is formed by a grid of408 segments of cylindrical conductor of the same di-


ameter (12.85 mm) buried to a depth of 80 cm. Thegrounding system has a right-angled triangle shape of143×89 m2 and protects a total area of 6,600 m2 (Fig.2). The grid has been discretized in 408 linear leakagecurrent elements which implies 238 degrees of free-dom. The Ground Potential Rise (GPR) considered inthis study has been 10 kV.

This grounding system has been calculated byusing the following soil models: Model A is anhomogeneous soil model with conductivityγ =0.016 (Ωm)−1, corresponding to the case of an uni-form terrain made, for example, of clay (Fig. 3).Model B is a vertically layered soil (Fig. 1b)), beingthe conductivity of the part of the soil that containsthe earthing gridγ1 = 0.016 (Ωm)−1, and the othersideγ2 = 0.005 (Ωm)−1 representing a less conduc-tive soil (Fig. 4). Model C is similar to model B butwith γ2 = 0 (Ωm)−1; this corresponds to the case oftaking into account the presence of a retaining wallfor example, quite frequent in substations placed intourban areas (Fig. 5), and Model D is an horizontallylayered soil (Fig. 1a)), beingγ1 = 0.005 (Ωm)−1 theconductivity of the upper layer,γ2 = 0.016 (Ωm)−1

the conductivity of the lower layer, andh = 1 m thethickness of the upper layer (Fig. 6). The equivalentresistance obtained for each soil model has been: A:0.3128Ω, B: 0.3712Ω, C: 0.4163Ω and D: 0.3705Ω.

As it is shown in this example, the results ob-tained by using a multiple-layer soil model can benoticeably different from those obtained by using auniform soil model, and the computed design param-eters of the grounding system do significantly vary.Therefore, it could be advisable or even mandatory touse efficient multi-layer soil formulations (such as theproposed approach) to analyze grounding systems asa general rule, in spite of the increase in the computa-tional cost, specially when the conductivity of the soilchanges markedly with depth.

Acknowledgments

This work has been partially financed by the “Minis-terio de Educacion y Ciencia” of the Spanish Govern-ment (project #DPI2004-05156).

References:

[1] ANSI/IEEE Std.80,IEEE Guide for safety in ACSubstation Grounding, New York, (2000).

[2] J.G. Sveraket al., Safe substation grounding,Part I: IEEE T. on Power Apparatus and Sys-tems,100, 4281-4290, (1981); Part II: IEEE T.

on Power Apparatus and Systems,101, 4006-4023, (1982)

[3] F. Navarrina, I. Colominas, M. Casteleiro,Ana-lytical Integration Techniques for Earthing GridComputation by BEM, Num. Met. in Eng. andAppl. Sci., 1197–1206, CIMNE, Barcelona,(1992).

[4] I. Colominas, F. Navarrina, M. Casteleiro,A boundary element numerical approach forgrounding grid computation, Comput. MethodsAppl. Mech. Engrg.,174, 73-90, (1999).

[5] R.J. Heppe,Computation of potential at sur-face above an energized grid or other elec-trode, allowing for non-uniform current distribu-tion, IEEE T. Power Apparatus and Systems,98,1978-1988, (1979).

[6] D.L. Garrett and J.G. Pruitt,Problems encoun-tered with the Average Potential Method of ana-lyzing substation grounding systems, IEEE T. onPower Apparatus and Systems,104, 3586-3596,(1985).

[7] I. Colominas, J. Gomez-Calvino, F. Navarrina,M. Casteleiro,Computer analysis of earthingsystems in horizontally or vertically layeredsoils, Electric Power Systems Research,59, 149-156, (2001).

[8] I. Colominas, F. Navarrina, M. Casteleiro,A nu-merical formulation for grounding analysis instratified soils, IEEE Transactions on Power De-livery, 17, 587-595, (2002).

[9] E. Durand, Electrostatique, Masson, Paris,(1966).

[10] G.F. Tagg,Earth Resistances, Pitman Pub. Co.,New York, (1964).

[11] O.D. Kellog, Foundations of potential theory,Springer Verlag, Berlın, (1967).

[12] I. Colominas, F. Navarrina, M. Casteleiro,Anal-ysis of transferred earth potentials in ground-ing systems: A BEM numerical approach, IEEETransactions on Power Delivery,20, 339-345,(2005).

[13] F. Navarrina, I. Colominas, M. CasteleiroWhy do computer methods for grounding pro-duce anomalous results?, IEEE Transactions onPower Delivery,18, 1192-1202, (2003).


numerical simulation of earthing grids - wseas the authors in the last years for the computational...

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