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

Texto completo

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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)

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GROUNDING GRID DESIGN AND COMPUTATION

I. Colominas (1), F. Navarrina (1), and M. Castelei ro (2)

(1) E. T. S. de Ingenieros de Caminos, Canales y Puertos de La Coru ~na. Depto. de Matematicas. Universidad de La Coru ~na.

Polgono de Sabon, P. 12{14. 15141 Arteixo (La Coru ~na), SPAIN.

(2) E. T. S. de Ingenieros de Caminos, Canales y Puertos de Barcelona. Depto. de Matematica Aplicada III. Universidad Politecnica de Catalu ~na. C/ Gran Capitan S/N, Edicio C{2. 08034 Barcelona, SPAIN.

SUMMARY

Several widespread intuitive techniques develop ed during the last two de-cades for substation groundinganalysis, such as the Average Potential Metho d (APM),haveb eenrecentlyidentiedasparticularcasesofamoregeneral Bound-ary Element formulation [1]. In this approach, problems encountered with the application of these metho ds[3] can b e explainedfrom a mathematically rigor-ousp ointof view,and innovativeadvanced andmoreecienttechniquescan b e derived [2].

Numerical results obtained with low and medium levels of discretization (equivalent resistanceandleakage currentdensity) seemto b e reasonable. How-ever, these solutions still have not b een validated. Unrealistic results are ob-tained when domain discretization is increased, since no one pro cedure is yet available to eliminate the ab ove mentioned problems. Hence, numerical con-vergence analyses are precluded. The obtention of highly accurate numerical results by means of standard techniques (FEM, Finite Dierences) implies un-approachable computing requirements in practical cases. On the other side, neither practical errorestimates haveb een derived, nor analytical solutions are known for practical cases, nor suciently accurate exp erimental measurements have b een rep ortedup to this p oint.

Inthispap er, wepresenta validation ofthe resultsobtained by the Bound-ary Element prop osed formulation, including the classical metho ds. A highly accuratesolution toa sp eciallydesigned testproblem isobtained bymeans of a 2D FEM mo del, using up to 80;000 degrees of freedom. Results are compared with those carried outby BoundaryElements.

1. INTRODUCTION

The electrokinetic stationaryproblem,related to fault electric currents dis-sipation intoearth, can b e written in the following form[1, 2, 8]

= ( 0 gradV ) ; div ( ) = 0 in E ;

T

n E

= 0 in 0

E

; V = 1 in 0 ; V 0!0 if jxj0!1

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where V and are the p otential and current density at an arbitrary p oint x; domainE istheearth, itsconductivity (assumedconstant),0

E

itssurface (as-sumed horizontal)andn

E

(3)

electro de-earth system

R eq

=

Z Z

0

d0

01

(2)

b eing = T

nin0, theleakagecurrentdensityandnthenormalexterior unit eld to the groundingelectro de surface0.

In the last decades some intuitive techniques for grounding grid analysis, such as the Average PotentialMetho d (APM), have b een develop ed. However, signicant problems have b een encountered in the application of these meth-o ds [3].

A newBoundary Element approach has b een recently presented[1, 6] that includes the ab ove mentioned intuitive techniques as particular cases. In this kindof formulationtheunknownquantityistheleakagecurrentdensity,while the p otential at an arbitrary p oint and the equivalent resistance R

eq

must b e computed subsequently.

Although new more ecienttechniquescan b e derivedfrom this approach, ithasnot b eenp ossibletotesttheaccuracy of theresults (thatseemreasonable for low and medium levels of discretization) up to this p oint. Large numerical instabilitiesapp earwhen discretizationisrened,givingunrealisticresults[1,2, 3]. Ontheotherhand,toobtain highlyaccuratesolutionsbymeansof standard techniques (FEM, Finite Dierences) should imply unapproachable computing requirements; analytical solutions are not available for real problems; neither accurate exp erimental measurements nor practical error estimates have b een obtained; and, nally, the leakage current density is a dicult magnitude to compare, which real distributionis not obvious.

All these asp ects are sp ecially imp ortant b ecause the Boundary Element Metho d could b ecome a p owerful numerical computing to ol for the systematic analysisof this kindof problems.

A validation of BEM results is presented in this pap er. A test has b een sp ecically designed in order to b e solved by a 2D Finite Elementmo del and a Boundary Elementtechniquein order to compare the results.

2. TEST PROBLEM STATEMENT

Let the grounding electro de b e a single cylindrical conductor bar, which radiusof the cross section a is small in comparisonwith its length L (' 10

03

times smaller). This seems tob e an adecquate geometry forour purp oses,since groundingelectro desinpracticeconsistofanumberofinterconnectedbare cylin-drical conductors (groundinggrids).

Let the electro de b e buried so deep that the Neumannb oundary condition in (1) can b e neglected, domain E can b e considered innite, and the problem b ecomessymmetric with resp ect to the axisof the cylinder.

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cylindricalco ordinates (Figure1.a) @ 2 V @r 2 + 1 r @V @r + @ 2 V @z 2

= 0 in

V = 1 in 0

@V

@n

= 0 in 0

q

V 0! 0 if jxj 0! 1

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b eing r and z the radial andlatitude co ordinates.

Problem(3) can now b e solved by the FiniteElementMetho d. Results ob-tainedwillb e p otentialvaluesatselectedno dalp ointsinthediscretizeddomain . In order to validate BEM solutions, the leakage current density must b e computed from p otential values at no dal p oints close to the cylinder b oundary 0. Becauseofourhighaccuracy requirementsincomputing,theniteelement mesh arrangementshould b e carefullydesigned.

3. NUMERICAL MODEL

A weightedresidualsstatement forproblem (3) canb e written as

Z W r @ 2 V @r 2 + @V @r + r @ 2 V @z 2

d = 0 (4)

Applying Green's identity and taking into account the Neumannb oundary conditions we obtain the weaker form

Z 0 rW @V @n d0 0 Z r @W @r @V @r + @W @z @V @z

d = 0 (5)

for all memb ersW of a suitable class of weight functionson , with b oundary conditions

V = 1 in 0; V 0! 0 if (r;z) 0! 1 (6)

Potential V can now b e discretized

V(r;z) = j=N X j=1 V j N j

(r;z) (7)

b eing V j

the no dal p otentialvalues and N j

the shap efunctions.

UsingGalerkin formulation (W = N i

), functional equation (5) results in a linear equationssystem

j=N X j=1 K ij V j = q i

; i=1;:::;N

V j

= 1; 8j 2 Q

q = 0; 8i 62 Q

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K ij

= Z

r

@N

i @r

@N j @r

+ @N

i @z

@N j @z

d (9)

Leakage current density can b e computed subsequently, either by inter-p olation over p otentialvalues onno dalp ointsclose tob oundary0, orby means of the ux quantities q

i

in equations system (8). Equivalent resistance R eq

can b e computednext by means of expression(2).

Since we are analyzing a Dirichlet Exterior Problem, the Finite Element mesh should b e large enough (in extension) to adecquately simulate the null p otentialb oundarycondition inthe innite,by prescribingnullp otentialvalues atsuciently remoteno dalp oints. Onthe otherhand, ourpurp oseis toobtain highlyaccurate results forleakage currentand equivalent resistance. Therefore, weneed to design highlydense and regular meshes only in the proximity of the electro de b oundary.

For these reasons,an sp ecic mesh generator forthis kind of test problems has b een develop ed. Several layers of small (in comparison with the electro de radius a) regular quadrilateral elements are generated around b oundary 0, in order to capture high precision results in this area. Next, elements size grows exp onentiallyasdistancetob oundaryincreases,byalternatinglayersoftriangles andquadrilaterals. Sincethenumb erofelementsinconsecutivelayersdecreases exp onentially, the null p otential b oundary condition is prescrib ed in relatively few no dalp oints. Sp ecial care hasb een taken topreserveregularityof elements through the growingpro cess(see Figures 1.a and 1.b).

In the structured grids that we have generated by means of this strategy, most of elements (around75% of total) are placedin the proximity of the elec-tro de surface. Therefore,the greater partof the computing eort is devoted to ensure a suciently high levelof accuracy in this area.

Dierenttechniqueshaveb een testedfor solvingthe linear systemof equa-tions(8). Althoughmatricesaresymmetric,p ositivedeniteandbanded,direct metho ds are out of range due to the size of the problems (up to eighty thou-sand degrees of freedom). Several iterative metho dshave b een tested for a set of medium size problems. Namely: Jacobi and Gauss-Seidel (with and without overrelaxation and preconditioning) and Conjugate Gradients. The very b est results have b een obtained by an element-by-element preconditioned conjugate gradients algorithm [7] without assembly of the global matrix. This technique turnedout tob e extremelyecient,and itwasnallyused forsolvingthelarge scale problemsthat are presentedin this pap er.

4. NUMERICAL RESULTS

The testproblem characteristicspresentedare : bar length= 2.0m; radius of the bar crosssection = 0.005 m; and earth resistivity = 1.0 Ohm1m.

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3.f). Figure4presentsasummaryof theequivalentresistancevalues inallthese cases.

Withresp ect totheequivalentresistance, itcan b eshownthatBEM results agree signicantly withthose obtained by FEM.The relative errorb etween the BEM solution for one single parab olic element and the FEM solution for more thaneighty thousanddegreesof freedomisless than1:25%. Ontheotherhand, no signicant improvement seems to b e achieved by increasing discretization level,sincesolution obtainedby BEMforonesingleandfty parab olicelements diersin lessthan 0:6%. Actually,the solutionobtainedby BEM foronesingle constantelement(equivalenttotheAveragePotentialMetho dwithone segmen-tation [2, 3, 5]) couldb e considered accurate enoughforpractical purp oses.

However,signicantdierencesarefoundwithresp ecttotheleakagecurrent density distributions throughout the bar length. FEMresults (Figures 3.b, 3.d and3.f)showthat realdistributionis quitesmo othinthe center ofthe bar,but variessharplynearb othfreeends. AlltheBEMmo delsgiveanaccurateaverage (theequivalentresistancediersslightly,asp ointed outb efore),but theleakage current density distribution is substantially dierent. Obviously, is dicult for the BEM mo dels to adjust the b oundary condition V = 1 on the electro de surface near the free ends [1, 2]. Nevertheless, this eect is less pronounced in interconnected bars within grounding grids, and has no sp ecial imp ortance in practical cases.

Anyhow, it seems imp ortant to obtain an accurate enough distribution of the leakage current. In fact, the computed p otential level at a certain p oint on the earth surface can vary signicantly if the leakage current distribution in a close electro de is erroneous [3]. After all, a 2:0% relative error for a 50;000V fault condition represents a1;000V absolute error.

METHOD Discretizati on Equiv. Resist. ()

BEM { 3 ConstantElements 0.451706

BEM { 10 ConstantElements 0.449737

BEM { 100 ConstantElements 0.447752

BEM { 2 Linear Elements 0.450381

BEM { 10 Linear Elements 0.448693

BEM { 100 Linear Elements 0.447327

BEM { 1 Parab olicElement 0.449764

BEM { 5 Parab olicElements 0.448595

BEM { 50 Parab olicElements 0.447297

FEM { 12441 Degreesof Freedom 0.439689

FEM { 24987 Degreesof Freedom 0.454940

FEM { 80533 Degreesof Freedom 0.444223

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fail to provide an accurate leakage current distribution when discretization is increased, due to numerical instabilities which origin can now b e explained[2]. For higher orders of discretization, instabilities can strecht out the whole bar length, giving unrealistic results in subsequent computation of p otential levels ontheearthsurface[3]. However,resultsobtainedforlowlevelsofdiscretization (Figure2)areaccurateenoughformostpracticalpurp oses. Infact, thesolutions obtained for one single parab olicelement (Figure 2.e), and two linear elements (Figure 2.c) seem to b e quite reasonable. On the other hand, results obtained by BEM for higher levels of discretization (Figures 3.a, 3.c and 3.e) could b e considered extremely accurate if oscillations around the real solution could b e avoidedor eliminated by smo othing.

Passingnow to other matters, some additional tests havealso b een carried out with the FEM mo del. No signicant variations have b een observed when the internal electro de resistivity is considered. On the other hand, the current density leaving the ends of the conductor has b een computed. Obtainedvalues are negligible, inthe oreder of magnitude predictedby other authors [3].

0.

50.

100.

150.

200.

250.

300.

350.

400.

Number of iterations

0.1E-05

0.1E-04

0.1E-03

0.1E-02

0.1E-01

0.1E+00

Error norm

Figure 5.|EvolutionofLogarithmicErrorNormversustheNumb erofIterations in the algorithmof Preconditioned Conjugate Gradient Metho d.

5. CONCLUSIONS

Inthispap er, anumericalvalidationof a generalBEM formulationfor sub-station groundinganalysishas b een presented. The prop osedBEM formulation includes several widespread intuitive techniques (such as the Average Potential Metho d) as particular cases.

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formulation [1,2].

Numericalresultsshowthat, forpracticalpurp oses,BEM formulations pro-vide accurate enough results with low/medium levels of discretization, while higherorder elements(linearorparab olics)seem tob e quite moreecientthan traditional constant elements.

However, furtherresearch willb e necessaryto avoidor eliminate numerical instabilities (which origin can now b e explained [2]), that pro duce unrealistic results when discretization level is increased. These p owerful numerical to ols should probably b ecome widespread techniques for the systematic analysis of substation groundingsin a near future.

6. ACKNOWLEDGMENTS

This work has b een partially supp orted by the \Plan Nacional de Investi-gacionyDesarrolloElectricodelMinisteriodeIndustriayEnerga"ofthe Span-ish Government, through the OCIDE Research Project \Dise ~no de Tomas de Tierra Asistido p or Ordenador", (Grant Numb er TC0772 UPC-FECSA), and by a research fellowship of \Xunta de Galicia", through the University of La Coru ~na.

REFERENCES

[1]. NAVA RRINA, F., MORENO, L., BENDITO, E., ENCINAS, A.,

LEDES-MA, A., and CASTELEIRO,M. { \Computer Aided Design of Grounding

Grids: A Boundary Element Approach", Mathematics in Industry, Kluwer Academic Publishers, Netherlands,pp. 307{314,(1991).

[2]. NAVA RRINA, F., COLOMINAS, I., and CASTELEIRO, M. {

\Analyti-cal Integration Techniques for Earthing Grid Computation by Boundary ElementMetho ds", International Congress on Numerical Methods in Engi-neering and Applied Sciences, Concep cion, (1992).

[3]. GARRETT, D.L. and PRUITT, J.G. { \Problems Encountered with the Average Potential Metho d of Analyzing Substation Grounding Systems", IEEE Trans. on Power App. and Systems, 104, No. 12, pp. 3586{3596, (1985).

[4]. DAUTRAY,R.andLIONS,J.L.{\AnalyseMathematiqueetCalcul Num e-rique pour les Sciences et les Techniques", Vol.6, Masson, Paris,(1988).

[5]. HEPPE, R.J. { \Computation of Potential at Surface ab ove an energized grid or other electro de, allowing for Non-Uniform Current Distribution", IEEETrans.on PowerApp. andSystems,98,No.6,pp.1978{1989,(1979).

[6]. MORENO, Ll. { \Disseny Assistit per Ordinad or de Postes a Terra en Instal1lacions Electriques", Tesina de Esp ecialidad,E.T.S. de Ingenierosde Caminos, Canales y Puertos. Universitat Politecnica de Catalunya. Barce-lona, (1989).

[7]. PINI,G.,GAMBOLATI, G.{\Isasimplediagonalscalingtheb est precon-ditionerforconjugate gradientsonsup ercomputers?",Advances on Water Resources, 13, No. 3,pp. 147{153, (1990).

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LEAKAGE CURRENT DENSITY

0.00

0.50

1.00

1.50

2.00

a.-BEM (3 constant elements)

0.00

25.00

50.00

75.00

100.00

LEAKAGE CURRENT DENSITY

0.00

0.50

1.00

1.50

2.00

b.-BEM (10 constant elements)

0.00

25.00

50.00

75.00

100.00

0.00

0.50

1.00

1.50

2.00

c.-BEM (2 linear elements)

0.00

25.00

50.00

75.00

100.00

0.00

0.50

1.00

1.50

2.00

d.-BEM (10 linear elements)

0.00

25.00

50.00

75.00

100.00

0.00

0.50

1.00

1.50

2.00

e.-BEM (1 parabolic element)

0.00

25.00

50.00

75.00

100.00

0.00

0.50

1.00

1.50

2.00

f.-BEM (5 parabolic elements)

0.00

25.00

50.00

75.00

100.00

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LEAKAGE CURRENT DENSITY

0.00

0.50

1.00

1.50

2.00

a.-BEM (100 constant elements)

0.00

25.00

50.00

75.00

100.00

LEAKAGE CURRENT DENSITY

0.00

0.50

1.00

1.50

2.00

b.-FEM (12441 d.o.f.)

0.00

25.00

50.00

75.00

100.00

0.00

0.50

1.00

1.50

2.00

c.-BEM (100 linear elements)

0.00

25.00

50.00

75.00

100.00

0.00

0.50

1.00

1.50

2.00

d.-FEM (24987 d.o.f.)

0.00

25.00

50.00

75.00

100.00

0.00

0.50

1.00

1.50

2.00

e.-BEM (50 parabolic elements)

0.00

25.00

50.00

75.00

100.00

0.00

0.50

1.00

1.50

2.00

f.-FEM (80533 d.o.f.)

0.00

25.00

50.00

75.00

100.00

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thickline.

Figure

Figure 4.|Equivalent resistances obtained by BEM and FEM for dierent levels of

Figure 4.|Equivalent

resistances obtained by BEM and FEM for di erent levels of p.6
Figure 5.|Evolution of Logarithmic Error Norm versus the Numb er of Iterations

Figure 5.|Evolution

of Logarithmic Error Norm versus the Numb er of Iterations p.7
Figure 2.|Comparison b etween results obtained by BEM with dierent typ e and num-

Figure 2.|Comparison

b etween results obtained by BEM with di erent typ e and num- p.9
Figure 3.|Comparison b etween results obtained by BEM with dierent typ e and num-

Figure 3.|Comparison

b etween results obtained by BEM with di erent typ e and num- p.10
Figure 1.b.|Structured Grid used in FEM. Detail of zone close to p oint A.

Figure 1.b.|Structured

Grid used in FEM. Detail of zone close to p oint A. p.11

Referencias

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