[Abstract] Several widespread intuitive techniques developed during the last two decades for substation grounding analysis, such as the Average Potential Method
(APM), have been recently identified as particular cases of a more general Boundary Element formulation [1]. In this approach, problems encountered with the
application of these methods [3] can be explained from a mathematically rigorous point of view, and innovative advanced and more eficient techniques can be
derived [2].
Numerical results obtained with low and medium levels of discretization
(equivalent resistance and leakage current density) seem to be reasonable. However, these solutions still have not been validated. Unrealistic results are obtained when domain discretization is increased, since no one procedure is yet
available to eliminate the above mentioned problems. Hence, numerical convergence analyses are precluded. The obtention of highly accurate numerical
results by means of standard techniques (FEM, Finite Differences) implies unapproachable computing requirements in practical cases. On the other side,
neither practical error estimates have been derived, nor analytical solutions are
known for practical cases, nor suficiently accurate experimental measurements
have been reported up to this point.
In this paper, we present a validation of the results obtained by the Boundary Element proposed formulation, including the classical methods. A highly
accurate solution to a specially designed test problem is obtained by means of a
2D FEM model, using up to 80; 000 degrees of freedom. Results are compared
with those carried out by Boundary Elements

Casteleiro, Manuel

Colominas, Ignasi

Navarrina, Fermín

Repositorio da Universidade da Coruña

Preprint of the paper  "A Validation of the Boundary Element Method for Grounding Grid Design and Computation" I. Colominas, F. Navarrina, M. Casteleiro (1992) En "Métodos Numéricos en Ingeniería y Ciencias Aplicadas. Numerical Methods in Engineering and Applied Sciencies", Sección VII: "Electromagnetics", pp. 1187--1196; H. Alder, J.C. Heinrich, S. Lavanchy, E. Oñate, B. Suárez (Editores); Centro Internacional de Métodos Numéricos en Ingeniería CIMNE, Barcelona (ISBN: 84-87867-16-2)   http://caminos.udc.es/gmni A VALIDATION OF THE BOUNDARY ELEMENTMETHOD FORGROUNDING GRID DESIGN AND COMPUTATIONI. Colominas (1), F. Navarrina (1), and M. Casteleiro (2)(1) E. T. S. de Ingenieros de Caminos, Canales y Puertos de La Coru~na.Depto. de Matematicas. Universidad de La Coru~na.Polgono de Sabon, P. 12{14. 15141 Arteixo (La Coru~na), SPAIN.(2) E. T. S. de Ingenieros de Caminos, Canales y Puertos de Barcelona.Depto. de Matematica Aplicada III. Universidad Politecnica de Catalu~na.C/ Gran Capitan S/N, Edicio C{2. 08034 Barcelona, SPAIN.SUMMARYSeveral widespread intuitive techniques developed during the last two de-cades for substation grounding analysis, such as the Average Potential Method(APM), have been recently identied as particular cases of a more general Bound-ary Element formulation [1]. In this approach, problems encountered with theapplication of these methods [3] can be explained from a mathematically rigor-ous point of view, and innovative advanced and more ecient techniques can bederived [2].Numerical results obtained with low and medium levels of discretization(equivalent resistance and leakage current density) seem to be reasonable. How-ever, these solutions still have not been validated. Unrealistic results are ob-tained when domain discretization is increased, since no one procedure is yetavailable to eliminate the above mentioned problems. Hence, numerical con-vergence analyses are precluded. The obtention of highly accurate numericalresults by means of standard techniques (FEM, Finite Dierences) implies un-approachable computing requirements in practical cases. On the other side,neither practical error estimates have been derived, nor analytical solutions areknown for practical cases, nor suciently accurate experimental measurementshave been reported up to this point.In this paper, we present a validation of the results obtained by the Bound-ary Element proposed formulation, including the classical methods. A highlyaccurate solution to a specially designed test problem is obtained by means of a2D FEM model, using up to 80; 000 degrees of freedom. Results are comparedwith those carried out by Boundary Elements.1. INTRODUCTIONThe electrokinetic stationary problem, related to fault electric currents dis-sipation into earth, can be written in the following form [1, 2, 8] = (    grad V ) ; div (  ) = 0 in E ;TnE= 0 in  E; V = 1 in   ; V  ! 0 if jxj  ! 1(1)where V and  are the potential and current density at an arbitrary point x;domain E is the earth,  its conductivity (assumed constant),  Eits surface (as-sumed horizontal) and nEits normal exterior unit eld; and   is the groundingelectrode surface.Another important magnitude is the equivalent resistance of the groundingelectrode-earth systemReq=Z Z  d  1(2)being  = Tn in  , the leakage current density and n the normal exterior uniteld to the grounding electrode surface  .In the last decades some intuitive techniques for grounding grid analysis,such as the Average Potential Method (APM), have been developed. However,signicant problems have been encountered in the application of these meth-ods [3].A new Boundary Element approach has been recently presented [1, 6] thatincludes the above mentioned intuitive techniques as particular cases. In thiskind of formulation the unknown quantity is the leakage current density , whilethe potential at an arbitrary point and the equivalent resistance Reqmust becomputed subsequently.Although new more ecient techniques can be derived from this approach,it has not been possible to test the accuracy of the results (that seem reasonablefor low and medium levels of discretization) up to this point. Large numericalinstabilities appear when discretization is rened, giving unrealistic results [1, 2,3]. On the other hand, to obtain highly accurate solutions by means of standardtechniques (FEM, Finite Dierences) should imply unapproachable computingrequirements; analytical solutions are not available for real problems; neitheraccurate experimental measurements nor practical error estimates have beenobtained; and, nally, the leakage current density is a dicult magnitude tocompare, which real distribution is not obvious.All these aspects are specially important because the Boundary ElementMethod could become a powerful numerical computing tool for the systematicanalysis of this kind of problems.A validation of BEM results is presented in this paper. A test has beenspecically designed in order to be solved by a 2D Finite Element model and aBoundary Element technique in order to compare the results.2. TEST PROBLEM STATEMENTLet the grounding electrode be a single cylindrical conductor bar, whichradius of the cross section a is small in comparison with its length L (' 10 3times smaller). This seems to be an adecquate geometry for our purposes, sincegrounding electrodes in practice consist of a number of interconnected bare cylin-drical conductors (grounding grids).Let the electrode be buried so deep that the Neumann boundary conditionin (1) can be neglected, domain E can be considered innite, and the problembecomes symmetric with respect to the axis of the cylinder.Although an analytical solution for this test problem is available in termsof Bessel expansions [8], we turn our attention to a FEM axial symmetry model,because computing the leakage current density throughout the bar length shouldbe extremely time-consuming by this kind of series.Taking advantage of the symmetries, we can write our test problems incylindrical coordinates (Figure 1.a)@2V@r2+1r@V@r+@2V@z2= 0 in V = 1 in  @V@n= 0 in  qV  ! 0 if jxj  ! 1(3)being r and z the radial and latitude coordinates.Problem (3) can now be solved by the Finite Element Method. Results ob-tained will be potential values at selected nodal points in the discretized domain. In order to validate BEM solutions, the leakage current density  must becomputed from potential values at nodal points close to the cylinder boundary . Because of our high accuracy requirements in computing , the nite elementmesh arrangement should be carefully designed.3. NUMERICAL MODELA weighted residuals statement for problem (3) can be written asZWr@2V@r2+@V@r+ r@2V@z2d = 0 (4)Applying Green's identity and taking into account the Neumann boundaryconditions we obtain the weaker formZ rW@V@nd   Zr@W@r@V@r+@W@z@V@zd = 0 (5)for all members W of a suitable class of weight functions on , with boundaryconditionsV = 1 in   ; V  ! 0 if (r; z)  ! 1 (6)Potential V can now be discretizedV (r; z) =j=NXj=1VjNj(r; z) (7)being Vjthe nodal potential values and Njthe shape functions.Using Galerkin formulation (W = Ni), functional equation (5) results in alinear equations systemj=NXj=1KijVj= qi; i = 1; :::;NVj= 1 ; 8 j 2 Qqi= 0 ; 8 i 62 Q(8)being Q the set of nodal points on the boundary  , andKij=Zr@Ni@r@Nj@r+@Ni@z@Nj@zd (9)Leakage current density  can be computed subsequently, either by inter-polation over potential values on nodal points close to boundary  , or by meansof the ux quantities qiin equations system (8). Equivalent resistance Reqcanbe computed next by means of expression (2).Since we are analyzing a Dirichlet Exterior Problem, the Finite Elementmesh should be large enough (in extension) to adecquately simulate the nullpotential boundary condition in the innite, by prescribing null potential valuesat suciently remote nodal points. On the other hand, our purpose is to obtainhighly accurate results for leakage current and equivalent resistance. Therefore,we need to design highly dense and regular meshes only in the proximity of theelectrode boundary.For these reasons, an specic mesh generator for this kind of test problemshas been developed. Several layers of small (in comparison with the electroderadius a) regular quadrilateral elements are generated around boundary  , inorder to capture high precision results in this area. Next, elements size growsexponentially as distance to boundary increases, by alternating layers of trianglesand quadrilaterals. Since the number of elements in consecutive layers decreasesexponentially, the null potential boundary condition is prescribed in relativelyfew nodal points. Special care has been taken to preserve regularity of elementsthrough the growing process (see Figures 1.a and 1.b).In the structured grids that we have generated by means of this strategy,most of elements (around 75% of total) are placed in the proximity of the elec-trode surface. Therefore, the greater part of the computing eort is devoted toensure a suciently high level of accuracy in this area.Dierent techniques have been tested for solving the linear system of equa-tions (8). Although matrices are symmetric, positive denite and banded, directmethods are out of range due to the size of the problems (up to eighty thou-sand degrees of freedom). Several iterative methods have been tested for a setof medium size problems. Namely: Jacobi and Gauss-Seidel (with and withoutoverrelaxation and preconditioning) and Conjugate Gradients. The very bestresults have been obtained by an element-by-element preconditioned conjugategradients algorithm [7] without assembly of the global matrix. This techniqueturned out to be extremely ecient, and it was nally used for solving the largescale problems that are presented in this paper.4. NUMERICAL RESULTSThe test problem characteristics presented are : bar length = 2.0 m; radiusof the bar cross section = 0.005 m; and earth resistivity = 1.0 Ohm  m.Figures 2 and 3 present dierent leakage current density distributions throu-ghout the bar length obtained by a BEM formulation [2] for several type ofelements (constant, linear and parabolic leakage current) with results obtainedby FEM with large structured grids (12441 d.o.f./13410 elements in case 3.b,24987 d.o.f./26914 elements in case 3.d, and 80533 d.o.f./84528 elements in case3.f). Figure 4 presents a summary of the equivalent resistance values in all thesecases.With respect to the equivalent resistance, it can be shown that BEM resultsagree signicantly with those obtained by FEM. The relative error between theBEM solution for one single parabolic element and the FEM solution for morethan eighty thousand degrees of freedom is less than 1:25%. On the other hand,no signicant improvement seems to be achieved by increasing discretizationlevel, since solution obtained by BEM for one single and fty parabolic elementsdiers in less than 0:6%. Actually, the solution obtained by BEM for one singleconstant element (equivalent to the Average Potential Method with one segmen-tation [2, 3, 5]) could be considered accurate enough for practical purposes.However, signicant dierences are found with respect to the leakage currentdensity distributions throughout the bar length. FEM results (Figures 3.b, 3.dand 3.f) show that real distribution is quite smooth in the center of the bar, butvaries sharply near both free ends. All the BEM models give an accurate average(the equivalent resistance diers slightly, as pointed out before), but the leakagecurrent density distribution is substantially dierent. Obviously, is dicult forthe BEM models to adjust the boundary condition V = 1 on the electrodesurface near the free ends [1, 2]. Nevertheless, this eect is less pronounced ininterconnected bars within grounding grids, and has no special importance inpractical cases.Anyhow, it seems important to obtain an accurate enough distribution ofthe leakage current. In fact, the computed potential level at a certain point onthe earth surface can vary signicantly if the leakage current distribution in aclose electrode is erroneous [3]. After all, a 2:0% relative error for a 50; 000Vfault condition represents a 1; 000V absolute error.METHOD Discretization Equiv. Resist. ()BEM { 3 Constant Elements 0.451706BEM { 10 Constant Elements 0.449737BEM { 100 Constant Elements 0.447752BEM { 2 Linear Elements 0.450381BEM { 10 Linear Elements 0.448693BEM { 100 Linear Elements 0.447327BEM { 1 Parabolic Element 0.449764BEM { 5 Parabolic Elements 0.448595BEM { 50 Parabolic Elements 0.447297FEM { 12441 Degrees of Freedom 0.439689FEM { 24987 Degrees of Freedom 0.454940FEM { 80533 Degrees of Freedom 0.444223Figure 4.|Equivalent resistances obtained by BEM and FEM for dierent levels ofdiscretization.Figure 3 shows that actual BEM models (including the classical methods)fail to provide an accurate leakage current distribution when discretization isincreased, due to numerical instabilities which origin can now be explained [2].For higher orders of discretization, instabilities can strecht out the whole barlength, giving unrealistic results in subsequent computation of potential levelson the earth surface [3]. However, results obtained for low levels of discretization(Figure 2) are accurate enough for most practical purposes. In fact, the solutionsobtained for one single parabolic element (Figure 2.e), and two linear elements(Figure 2.c) seem to be quite reasonable. On the other hand, results obtainedby BEM for higher levels of discretization (Figures 3.a, 3.c and 3.e) could beconsidered extremely accurate if oscillations around the real solution could beavoided or eliminated by smoothing.Passing now to other matters, some additional tests have also been carriedout with the FEM model. No signicant variations have been observed whenthe internal electrode resistivity is considered. On the other hand, the currentdensity leaving the ends of the conductor has been computed. Obtained valuesare negligible, in the oreder of magnitude predicted by other authors [3].0. 50. 100. 150. 200. 250. 300. 350. 400.Number of iterations0.1E-050.1E-040.1E-030.1E-020.1E-010.1E+00Error normFigure 5.|Evolution of Logarithmic Error Norm versus the Number of Iterationsin the algorithm of Preconditioned Conjugate Gradient Method.5. CONCLUSIONSIn this paper, a numerical validation of a general BEM formulation for sub-station grounding analysis has been presented. The proposed BEM formulationincludes several widespread intuitive techniques (such as the Average PotentialMethod) as particular cases.A highly accurate solution to a specially designed test problem has beenobtained by means of a 2D FEM model, using up to 80; 000 degrees of free-dom. Results have been compared with those carried out by the proposed BEMformulation [1, 2].Numerical results show that, for practical purposes, BEM formulations pro-vide accurate enough results with low/medium levels of discretization, whilehigher order elements (linear or parabolics) seem to be quite more ecient thantraditional constant elements.However, further research will be necessary to avoid or eliminate numericalinstabilities (which origin can now be explained [2]), that produce unrealisticresults when discretization level is increased. These powerful numerical toolsshould probably become widespread techniques for the systematic analysis ofsubstation groundings in a near future.6. ACKNOWLEDGMENTSThis work has been partially supported by the \Plan Nacional de Investi-gacion y Desarrollo Electrico del Ministerio de Industria y Energa" of the Span-ish Government, through the OCIDE Research Project \Dise~no de Tomas deTierra Asistido por Ordenador", (Grant Number TC0772 UPC-FECSA), andby a research fellowship of \Xunta de Galicia", through the University of LaCoru~na.REFERENCES[1]. NAVARRINA, F., MORENO, L., BENDITO, E., ENCINAS, A., LEDES-MA, A., and CASTELEIRO, M. { \Computer Aided Design of GroundingGrids: A Boundary Element Approach", Mathematics in Industry, KluwerAcademic Publishers, Netherlands, pp. 307{314, (1991).[2]. NAVARRINA, F., COLOMINAS, I., and CASTELEIRO, M. { \Analyti-cal Integration Techniques for Earthing Grid Computation by BoundaryElement Methods", International Congress on Numerical Methods in Engi-neering and Applied Sciences, Concepcion, (1992).[3]. GARRETT, D.L. and PRUITT, J.G. { \Problems Encountered with theAverage Potential Method of Analyzing Substation Grounding Systems",IEEE Trans. on Power App. and Systems, 104, No. 12, pp. 3586{3596,(1985).[4]. DAUTRAY, R. and LIONS, J.L. { \Analyse Mathematique et Calcul Nume-rique pour les Sciences et les Techniques", Vol. 6, Masson, Paris, (1988).[5]. HEPPE, R.J. { \Computation of Potential at Surface above an energizedgrid or other electrode, allowing for Non-Uniform Current Distribution",IEEE Trans. on Power App. and Systems, 98, No. 6, pp. 1978{1989, (1979).[6]. MORENO, Ll. { \Disseny Assistit per Ordinador de Postes a Terra enInstallacions Electriques", Tesina de Especialidad, E.T.S. de Ingenieros deCaminos, Canales y Puertos. Universitat Politecnica de Catalunya. Barce-lona, (1989).[7]. PINI, G., GAMBOLATI, G. { \Is a simple diagonal scaling the best precon-ditioner for conjugate gradients on supercomputers ? ", Advances on WaterResources , 13, No. 3, pp. 147{153, (1990).[8]. TAGG, G. { \Earth Resistances", The Whitefriars Press Ltd, London,(1964).LEAKAGE CURRENT DENSITY0.00 0.50 1.00 1.50 2.00 a.-BEM (3 constant elements)0.0025.0050.0075.00100.00 LEAKAGE CURRENT DENSITY0.00 0.50 1.00 1.50 2.00 b.-BEM (10 constant elements)0.0025.0050.0075.00100.000.00 0.50 1.00 1.50 2.00 c.-BEM (2 linear elements)0.0025.0050.0075.00100.000.00 0.50 1.00 1.50 2.00 d.-BEM (10 linear elements)0.0025.0050.0075.00100.000.00 0.50 1.00 1.50 2.00 e.-BEM (1 parabolic element)0.0025.0050.0075.00100.000.00 0.50 1.00 1.50 2.00 f.-BEM (5 parabolic elements)0.0025.0050.0075.00100.00Figure 2.|Comparison between results obtained by BEMwith dierent type and num-ber of elements. Leakage Current Density throughout the bar length.LEAKAGE CURRENT DENSITY0.00 0.50 1.00 1.50 2.00 a.-BEM (100 constant elements)0.0025.0050.0075.00100.00 LEAKAGE CURRENT DENSITY0.00 0.50 1.00 1.50 2.00 b.-FEM (12441 d.o.f.)0.0025.0050.0075.00100.000.00 0.50 1.00 1.50 2.00 c.-BEM (100 linear elements)0.0025.0050.0075.00100.000.00 0.50 1.00 1.50 2.00 d.-FEM (24987 d.o.f.)0.0025.0050.0075.00100.000.00 0.50 1.00 1.50 2.00 e.-BEM (50 parabolic elements)0.0025.0050.0075.00100.000.00 0.50 1.00 1.50 2.00 f.-FEM (80533 d.o.f.)0.0025.0050.0075.00100.00Figure 3.|Comparison between results obtained by BEMwith dierent type and num-ber of elements and results obtained by FEM with three meshes. LeakageCurrent Density throughout the bar length.Figure 1.a.|Test Problem Schematic Representation. Bar Boundary marked withthick line.Figure 1.b.|Structured Grid used in FEM. Detail of zone close to point A.

A validation of the boundary element method for grounding grid design and
computation

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A validation of the boundary element method for grounding grid design and computation

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