TítuloA general formulation based on the boundary element method for the analysis of grounded instalations in layered soils

Texto completo

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Preprint of the paper

"A General Formulation based on the Boundary Element Method for the Analysis

of Grounded Instalations in Layered Soils"

I. Colominas, F. Navarrina, J. Aneiros, M. Casteleiro (1998)

En "International Series on Advances in Boundary Elements", Vol. 4: Boundary Elements

XX, Sección 9: "Corrosion", 575---586. Brebbia C.A., Kassab A., Chopra M. (Editors);

Computational Mechanics Publications, Southampton, UK. (ISBN: 1-85312-592-X)

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element method for the analysis of grounded

instalations in layered soils

I.Colominas, F. Navarrina,J. Aneiros& M. Casteleiro

Dpto. de Metodos Matematicos y Representacion. E.T.S. de

Ingenieros de Caminos, Canales yPuertos. Universidad de La

Coru ~na. Campus de Elvi ~na, 15071 La Coru ~na. SPAIN

Email: colominas@iccp .udc.es

Abstract

Thedesignofsafegroundingsystemsrequirescomputingthep otential

level distribution on the earth surface for reasons of human security, as

well as theequivalentresistance of theearthing installationforreasons of

equipmentprotection(Sveraketal.[1],ANSI/IEEE[2]).

Inthelastthree decades several metho dsforgroundinganalysishave

b een prop osed, most of them based on practice and intuitive ideas.

Al-thoughthesetechniquesrepresente danimp ortantimprovementinthisarea,

someproblemssuchaslargecomputationalrequirements,unrealisticresults

whensegmentationofconductorsisincreased,anduncertaintyinthemargin

oferror,wererep orted(Sveraketal.[1],ANSI/IEEE[2],Garret&Pruitt[3]).

Navarrina etal.[4] andColominasetal.[5] have develop edinthelast

years a general b oundary element formulation for grounding analysis in

uniformsoils,inwhichtheseintuitivemetho dscanb eindentiedas

partic-ularcases. Furthermore, startingfromthis BEnumerical approach,more

ecientandaccurateformulationshaveb eendevelop edandsuccesfully

ap-plied(with avery reasonable computationalcost) to theanalysis of large

groundingsystems inelectricalsubstations.

In this pap er we present a new Boundary Element formulation for

substationgroundingsystems emb eddedinlayered soils. Thefeasibilityof

(3)

Physical phenomena offault currentsdissipation into theearth

canb edescrib edbymeansofMaxwell'sElectromagneticTheory

(Du-rand[6]). If one constrains the analysis to theobtention of the

elec-trokinetic steady-state resp onse and neglects the inner resistivity of

theearthingconductors|therefore,p otentialisassumedconstantin

everyp ointofthegroundingelectro desurface|,the3Dproblemcan

b ewrittenas

div

=0;

=0

gradV in E;

t

n n n n n

nn

n n n n n n n E

=0 in 0

E

; V =V

0

in 0; V 0!0; if jxxxxxxx

x x x

xx

x

xj!1;

(1)

where E is the earth,

its conductivity tensor, 0

E

the earth

sur-face, nnnnnnnnnnnnnn

E

its normal exterior unit eld and 0 the electro de surface

(Navarrina et al.[4], Colominas[7]). The solution to this problem

givesp otential V andcurrentdensity

atanarbitraryp oint xxxxxxx

x x x x x x

xwhen

theelectro de attains a voltage V

0

(GroundPotential Rise, orGPR)

relative toa distant groundingp oint.

Most of the metho ds prop osed are founded on the hyp othesis

that soil can b e considered homogeneous and isotropic. Therefore,

is substituted by an apparent scalar conductivity , that can b e

exp erimentally obtained (Sverak et al.[1]). In general, if the soil is

essentiallyuniform(horizontallyandvertically)inthesurrondingsof

thegroundinggrid,thisassumptiondo esnotintro ducesignicant

er-rors(ANSI/IEEE[2]). Nevertheless, since grounding design

parame-terscansignicantlychangeassoilconductivityvaries,itisnecessary

todevelopmoreaccuratemo delsthattakeintoaccountthevariation

ofsoil conductivity in thesurroundingsof thesubstationsite.

Obviously,fromatechnical(andalsoeconomical) p oint ofview,

thedevelopmentofmo delstodescrib e allvariationsofsoil

conductiv-ityinthesurroundingsofagroundingsystemwouldb eunaordable.

Forthis reason, a morepractical and quiterealistic approach to

sit-uationswhereconductivityis notmarkedly uniformwithdepthisto

consider the earth stratied in a numb er of horizontal layers, each

one with anappropriate thickness and apparent scalar conductivity.

Infact,inmostcases, an equivalent two-layersoil mo delissucient

toobtainsafe designsof groundingsystems (ANSI/IEEE[2]).

When the groundingelectro de is buried in the upp er layer, (1)

can b ereduced totheNeumann Exterior Problem:

1V 1

=0 in E

1

; 1V

2

=0 in E

2

; V

1

=V

0

in 0;

dV 1 dn

=0 in 0

E

;

1 dV

1 dn

=

2 dV

2 dn

in 0

L ;

V =V in 0 ; V 0!0 and V 0!0 if jxxxxxxx

x x x

xx

x

xj!1;

(4)

1 2 L

interfaceb etween them,

1

and

2

theapparent scalarconductivities

ofb oth layers,andV

1

andV

2

thep otential ineachlayer(Aneiros[8]).

Inthis pap er,werestrict thefurtherdevelopment totheab ovecase.

Theanalogousstatementto(2)whenthegroundinggridis buriedin

thelowersoillayercanb efoundinAneiros[8]andColominasetal.[9].

2 Variational Statement of the Problem

Groundingsystems in mostof real electrical substationsconsist

of a grid of interconnected bare cylindrical conductors, horizontally

buried and supplemented by a numb er of vertical ro ds, which ratio

diameter/lenght uses to b e relatively small ( 10

03

). This

partic-ular geometry precludes to obtain analytical solutions, and the use

of standard numerical techniques (such as Finite Dierences or

Fi-nite Elements) that requires the discretization of domains E

1 and

E 2

,implies anoutofrangecomputationaleort. Atthisp oint,since

computation ofp otential is only required on 0

E

, and theequivalent

resistancecanb eeasilyobtainedin termsoftheleakagecurrent

den-sity =

t n n n n n n n n n n n n n n

on 0, b eing nnnnnnnnnn

n n n n

the normal exterior unit eld to 0, we

turnour attentiontoa b oundary element approach,which will only

require thediscretization of groundingsurface0 (Colominas[7]).

Onthe other hand, if one takesintoaccount that surroundings

of substation site are levelled during its construction, earth surface

0 E

and theinterface b etween thetwosoil layers 0

L

can b e assumed

horizontal(Aneiros[8]). Thus,application ofmetho dofimages

(sym-metry) and Green's Identity to problem (2) yield the following

ex-pressions for p otential V

1 (xxxxxxxxxx

x x x x 1

) and V

2 (xxxxxxxxxx

xx

x x 2

), at arbitrary p oints xxxxxxxxxx

x x x x 1 in E 1

and xxxxxxx

x x x x x x x 2 in E 2

, in terms of the unknown leakage current density

(

) atanyp oint

[ x ; y ; z

]on 0:

V 1

(xxxxxxx x x x x x x x 1 )= 1 4 1 ZZ 20 k 11

(xxxxxxx

xxx

x x x x 1 ; )(

)d0; 8xxxxxxx

x x x xx x x 1 2E 1 ; (3)

b eing k

11 (xxxxxxx

x x x xx x x 1 ;

)the weakly singular kernel

k 11

(xxxxxxxxxx x x x x 1 ; )= 1

r (xxxxxxxxxx x x x x 1 ;[ x ; y ; z ]) + 1

r (xxxxxxxxxx x x x x 1 ;[ x ; y ;0 z ]) + 1 X i=1 h i

r (xxxxxxx x x x x x x x 1 ;[ x ; y

;2iH+

z ])

+

i

r (xxxxxxx x x x xx x x 1 ;[ x ; y

;2iH0

z ])

+

i

r (xxxxxxxxxxxxxx ;[ ; ;02iH+ ])

+

i

r (xxxxxxxxxxxxxx ;[ ; ;02iH0 ])

i

(5)

V 2

(xxxxxxx x x x x x x x 2 )= 1 4 1 ZZ 20 k 12

(xxxxxxx

xxx

x x x x 2 ; )(

)d0; 8xxxxxxx

x x x xx x x 2 2E 2 ; (5)

withweakly singular kernel k

12 (xxxxxxxxxxxxxx

2 ;)

k 12

(xxxxxxxxxxxxxx 2

;)=

1+

r (xxxxxxx x x x xx x x 2 ;[ x ; y ; z ]) + 1+

r (xxxxxxx

xxx

x x x x 2 ;[ x ; y ;0 z ]) + 1 X i=1 h

(1+)

i

r (xxxxxxx x x x x x x x 2 ;[ x ; y

;2iH+

z ])

+

(1+)

i

r (xxxxxxx x x x xx x x 2 ;[ x ; y

;2iH0

z ])

i

; (6)

wherer (xxxxxxxxxx

xx x x;[ x ; y ; z

])indicates thedistance fromxxxxxxxxxx

x x x x to [ x ; y ; z ]

|and to symmetric p oints of with resp ect to 0

E

and 0

L

, which

distances app earin dierentterms in kernels(4)and (6)|,H isthe

heigth oftheupp ersoil layer,and isa relation b etween

conductiv-itiesof b oth layers: =(

1 0 2 )=( 1 + 2

) |Colominaset al.[9]|.

Now,the application of b oundary condition V

1

() =1,820

(since V and

are prop ortional tothe GPR value, we can use the

normalizedb oundarycondition V

0

=1)leads toa Fredholmintegral

equationoftherstkinddenedon0,intermsoftheleakagecurrent

density (Colominas[5]). Aweakervariationalformof thisequation

can b ewrittenas:

ZZ 20

w ()

1 4 1 ZZ 20 k 11

(; ) () d001

!

d0=0; (7)

which must hold for all memb ers w (

) of a suitable class of test

fuctions dened on 0. Obviously, a Boundary Element approach

seemstob ethe rightchoice tosolve equation (7).

3 Boundary Element Formulation

For a given set of N trial functions fN

i (

)g dened on 0, and

foragivenset ofM2Db oundaryelementsf0

g,theunknown

leak-agecurrent density and the grounding electro de surface 0 can b e

discretized in theform,

( )= N X i=1 i N i (

); 0=

M [ =1 0 ; (8)

and expressions(3)and (5)can b e approximatedas

V 1

(xxxxxxx x x x xx x x 1 )= N X i V 1i

(xxxxxxx x x x x x x x 1 ); V 1i (xxxxxxx

x x x xx x x 1 )= M X V 1i

(xxxxxxx x x x xx x x 1

); 8xxxxxxx

xxx

(6)

V 2

(xxxxxxxxxxxxxx 2 )= X i=1 i V 2i (xxxxxxxxxxxxxx

2

); V

2i (xxxxxxxxxxxxxx

2 )= X =1 V 2i

(xxxxxxxxxxxxxx 2

); 8xxxxxxxxxxxxxx

2

2E

2 ;

(10)

b eing p otential co ecients V

1i and V 2i , V 1i

(xxxxxxxxxx x x x x 1 )= 1 4 1 Z Z 20 k 11 (xxxxxxxxxx

x x x x 1 ; )N i ( )d0

; 8xxxxxxxxxx

x x x x 1 2E 1 ; (11) V 2i

(xxxxxxxxxx x x x x 2 )= 1 4 1 Z Z 20 k 12 (xxxxxxxxxx

x x x x 2 ; )N i ( )d0

; 8xxxxxxxxxx

x x x x 2 2E 2 : (12)

Moreover, for a given set of N test functions fw

j ( )g dened

on0,thevariational statement(7)isreduced tothesystemof linear

equations N X i=1 R ji i = j

; j =1;:::;N; (13)

R ji = M X =1 M X =1 R ji ; j = M X =1 j ; (14) R ji = 1 4 1 Z Z 20 w j ( ) Z Z 20 k 11 ( ; )N i ( )d0 d0 (15) j = Z Z 20 w j ( ) d0 : (16)

It is imp ortant to notice that expression (15)is satised when

electro des and areburied intheupp erlayer. Obviously,if apart

oftheearthinggridisin thelowerlayer(

2E 2

),theintegralkernel

mustb esubstituted byk

12 ( ;

) in (6) |Aneiros[8]|.

In real problems, thediscretization required tosolve the ab ove

equationswouldimplyalargenumb erofdegreesof freedom. Onthe

otherhand,theco ecientmatrixin(13)isfullandeachcontribution

(15)requires an extremely high numb er of evaluations of the kernel

and double integration on a 2D domain. For these reasons, some

additional simplications in the BEMapproachmustb e intro duced

toreducethe computationalcost.

4 Approximated 1D BE Formulation

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elec-thecross sectionof thecylindrical electro de (ANSI/IEEE[2],

Navar-rinaet al.[4]).

Theintro ductionofthishyp othesisofcircumferentialuniformity

allows toobtain approximated expressions of p otential (3) and (5).

Thus, b eing L the whole set of axial lines of the electro des,

b the

orthogonalprojectionovertheaxisofagivengenericp oint

20,(

b )

theconductor diameter, and b (

b

) the approximatedleakage current

density at this p oint (assumed uniform around the cross section),

expressions(3)and (5)result in

b V 1

(xxxxxxxxxx

xx x x 1 )= 1 4 1 Z b 2L ( b ) k 11 (xxxxxxxxxx

x x x x; b

)b(

b

) dL; 8xxxxxxxxxx

x x x x 1 2E 1 ; (17) b V 2

(xxxxxxx x x x xx x x 2 )= 1 4 1 Z b 2L ( b ) k 12 (xxxxxxx

x x x x x x x; b

)b(

b

) dL; 8xxxxxxx

xxx

x x x x 2 2E 2 : (18) where k 11 (xxxxxxx

xxx

x x x x; b ) and k 12 (xxxxxxx

x x x x x x x; b

) are the average of kernels (4) and (6)

aroundthecross sectionat

b (Colominas[7],Aneiros[8]).

Now, since the leakage current is not exactly uniform around

the cross section, b oundary condition V

1

(

) = 1;

2 0 will not b e

strictly satised at every p oint

on the electro de surface 0, and

variational form (7) will not verifyanymore. However, if we restrict

the class of trial functions to those with circumferential uniformity,

forall memb ersw (bb

) of a suitable class of test functions dened on

L,itmustholdtheweakervariational form

Z b 2L (b

)w(b b

) 1 4 1 Z b 2L ( b ) k 11 (b

; b

) (b

b

)dL01

!

dL=0; (19)

where k

11 (b;

b

) is the average of kernel k

11 (b;

b

) in (4) around the

crosssections atp oints b and

b

(Colominas[7],Aneiros[8]).

Resolution of variational statement(19)requires the

discretiza-tion of the domain: the axial lines of electro des of the grounding

grid. Thus, for given sets of 1D b oundary elements and trial

func-tionsdened on L,thewhole setof theaxial lines and theunknown

approximatedleakagecurrent density b can b ediscretized.

Further-more,wecan obtain adiscretized versionforapproximatedp otential

(8)

func-implies integrationona1Ddomain(Aneiros[8],Colominaset al.[9]).

Thecomputingeortofthisapproximated1Dapproachis

dras-tically reduced in comparison with the 2D one, since the 1D

dis-cretization(thesetofaxiallines)willb emuchmoresimple.

Further-more,averagedkernels

k

11

(bx;

b ),

k

12

(bx;

b

) and

k

11 (b

;

b

)can b e

evalu-atedbyusing thesuitable unexp ensi ve approximations develop ed by

Colominas[7],forthecomputationof average kernels involved in the

groundinganalysisin uniformsoil mo dels.

Ontheotherhand,sincecomputationofremaininglineintegrals

isnotobviousandstandardquadraturescannotb eusedduetothe

ill-conditioning of integrands,we can p erformedsuitable arrangements

inthenalexpressionsofthematrixco ecients(Aneiros[8]),sothat

we can use the highly ecient analytical integration techniques

de-rived by Navarrina et al.[4] and Colominas et al.[7] for grounding

systemsin uniform soils.

5 Application to a Real Case

TheBoundary Element formulationpresentedin this pap er has

b eenincluded in thecomputeraided design system\TOTBEM"

de-velop ed byauthorsin recentyearsforthegroundinganalysis

(Caste-leiro et al.[11]). With this system, it has b een p ossible to compute

accurately earthinggrids in uniform soil mo dels of electrical

substa-tions of medium/big sizes, with acceptable computing requirements

in memorystorageand CPU time(Colominas etal.[10]).

In the resolution of real grounding systems emb edded in two

layeredsoils,thecomputingeortrequiredcanb everyhigh,sp eciall y

whenconductivitie sofsoil layersareverydierent(jj1),b ecause

therateofconvergenceinthecomputationofaveragekernels

k

11 ,

k

12

and k

11

isverylow,and itisnecessarytoevaluatealargenumb erof

termsof these series in order toobtain accurate results (Aneiros[8]).

In this pap er, we present the analysis of the Balaidos I I

sub-station grounding (close to the city of Vigo in Spain), by using a

uniform soil mo del and a two layer one. The earthing system

(g-ure 1) is formed by a grid of 107 cylindrical conductors (diameter:

11.28mm)buriedtoadepthof80cm,supplemented with67vertical

ro ds(eachonehasalengthof2.5mandadiameterof14.0mm). The

GroundPotentialRise considered hasb een 10kV.Characteristicsof

thesoil mo delsarepresentedin tableI.Thenumericalmo delusedin

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and p otential proles on the earth surface along two dierent lines

obtained with the BEM approach by using uniform and two layer

soil mo dels can b e found in table I and gure 1. As it is shown,

results noticeably vary when dierent soil mo dels are used, and in

consequence thegroundingdesignparameters(equivalent resistance,

thetouch,stepandmeshvoltages,etc.) maysignicantlychange. For

this reason, it will b e essential to p erform the analysis of a system

with this BEM technique although the computing cost increases, in

cases where conductivity changes markedly with depth, in order to

assurethesafetyof theinstallation.

It is imp ortant to notice that,in this case, the grounding

anal-ysis in the two layer soil mo del is very complicated b ecause part of

thegrid is buried in theupp er layer and other in the lower.

There-fore, implementation of the numerical approach must b e carefully

p erformed, in order to consider the dierent arrangements of

elec-tro desinthesoil. Furthermore,since conductivitesoflayersarevery

dierent (=00:94), theanalysis requires an imp ortant computing

eort.

TableI.| BalaidosI ISubstation: Resultsbyusingdierentsoilmo dels

TwoLayer SoilMo del UniformSoilMo del

Upp erLayerResistivity:2000m |

Lower LayerResistivity:60m |

HeightofUpp er Layer: 1.2m EarthResistivity: 60 m

FaultCurrent: 14.7kA FaultCurrent: 24.9kA

EquivalentResistance: 0.681 EquivalentResistance : 0.401

CPUTime(AXP4000): 144.5sec. CPUTime(AXP4000): 1.5 sec.

6 Conclusions

A numerical approach based on the BEM for the analysis of

substationearthingsystems emb edded in layered soils hasb een

pre-sented. This formulation hasb een applied tothe practicalcase of a

groundedsysteminanequivalenttwolayersoil. Takingintoaccount

thereal geometry of these systems, thegeneral 2D approach can b e

rewrittenin terms of an approximated 1D version. Moreover, since

suitablearrangementscanb edoneinthediscretizedexpressions,itis

p ossible to usethesameanalytical integrationtechniques develop ed

by the authors forgrounding analysis in uniform soils. Finally, the

BEMformulationprop osed isa general metho dology that allows to

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Thiswork has b een partially supp orted bythe p owercompany

\Union Fenosa", byresearch fellowships of theR&D General

Secre-taryofthe\Xuntade Galicia"andtheUniversityofLaCoru ~na,and

by thecompany\Fecsa".

References

[1] Sverak J.G.,Dick W.K.,Do dds T.H.,Hepp e R.H., Safe

Substa-tionsGrounding,IEEETPAS,100,4281{4290,1981

[2] ANSI/IEEEStd.80,Guide forSafetyinACSubstation

Ground-ing,IEEE Inc., New York,1986

[3] GarretD.L.,PruittJ.G.,ProblemsEncounteredwiththeAPMof

AnalyzingGroundingSystems,IEEETPAS,104,4006{4023,1985

[4] NavarrinaF.,ColominasI.,CasteleiroM.,AnalyticalIntegration

TechniquesforEarthingGridComputationbyBEM,Num. Met.

in Eng. and Appl. Sci.,1197{1206,CIMNE, Barcelona,1992

[5] Colominas I., Navarrina F., Casteleiro M., Una Formulacion

Numerica Generalparael Calculo yDise ~nodeTomasdeTierra

enGrandesInstalacionesElectricas,Rev. Int. Met. Num. Calc.

yDis. Ing., 13, 383{401,1997

[6] DurandE.,

Electrostatique,MassonEd.,Paris,1966

[7] ColominasI.,Cal. yDis. porOrdenadordeTomasdeTierraen

InstalacionesElectricas: Una Formulacion Num. basada en el

MetodoIntegralde Elementosde Contorno,Ph.D.Thesis,

ETS-ICCP,ULC,1995

[8] Aneiros J.M., Una Formulacion Num. para Calc. y Dis. de

TomasdeTierradeSubestacionesElec. conModelosdeTerreno

de2 Capas,ResearchRep ort, ETSICCP,1996

[9] Colominas I., Navarrina F., Casteleiro M., Aneiros J., A BEM

Formulation for CAD of GroundingSystems in Stratied Soils,

IVWorldCong. onComputational Mech., Buenos Aires, 1998

[10] ColominasI.,NavarrinaF.,CasteleiroM.,A BoundaryElement

Formulation for the Substation Grounding Design, Advances in

Engineering Software,Elsevier, [inpress]

[11] CasteleiroM.,HernandezL.A.,ColominasI.,NavarrinaF.,User

Guideof system\TOTBEM" forthe AnalysisandCADof

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1 Unit=10 m

0.

10.

20.

30.

40.

50.

60.

70.

80.

90.

0.0

2.0

4.0

6.0

8.0

10.0

Potential (kV)

0.

120.

0.

90.

0.

10.

20.

30.

40.

50.

60.

70.

80.

90.

100.

110.

120.

Distance (m)

0.0

2.0

4.0

6.0

8.0

10.0

Potential (kV)

Fig. 1.|E.R. BaladosI I : Plan of the Grounding(verticalro ds

markedwith black p oints), and Potential proles alongtwo

dier-ent lines (in discontinuous line: results of the uniform mo del; in

Figure

Table I.| Balaidos I I Substation: Results by using dierent soil mo dels

Table I.|

Balaidos I I Substation: Results by using di erent soil mo dels p.9
Fig. 1.| E.R. Bala  dos I I : Plan of the Grounding (vertical ro ds
Fig. 1.| E.R. Bala  dos I I : Plan of the Grounding (vertical ro ds p.11

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