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TítuloA numerical formulation for grounding analysis in stratified soils

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Preprint of the paper

"A Numerical Formulation for Grounding Analysis in Stratified Soils" I. Colominas, F. Navarrina, M. Casteleiro (2002)

IEEE Transactions on Power Delivery.

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w) d, gD $t ‚„8R BP Y‚ j‚s,s‚ {Z„„‚t I‚tR$k wD s

(4)

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d

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b$Y „‚R}‚{ B Y‚ ‚s„Y RZ„Ps{‚ ostI B Y‚ $t‚„}YsR‚R

wd,DA‚b‚‚t jsk‚„R qY‚„‚PB„‚ Y$R ,‚„t‚j I‚}‚tIR Bt

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$t stk BY‚„ jsk‚„ w$‚ jsk‚„KD Y‚ s}}j${s$Bt BP p„‚‚t:R

yI‚t$k k$‚jIR B st ‚-}„‚RR$Bt Ys $R stsjBBZR B w;D $t ‚„8R BP Y‚ $t‚„sj ,‚„t‚j>Kw??????????????, DU

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] zs„$s$Btsj Rs‚8‚t BP Y‚ }„BAj‚8

!t Y‚ BY‚„ YstI Y‚ ‚-}„‚RR$Bt PB„ }B‚t$sj wxD sjRB YBjIR Bt Y‚ „BZtI$t ‚j‚{„BI‚ RZ„Ps{‚ bY‚„‚ }B‚tT

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qY‚„‚PB„‚ Y‚ j‚s,s‚ {Z„„‚t I‚tR$k 8ZR Rs$RPk Y‚

3„‚IYBj8 $t‚„sj ‚JZs$Bt BP Y‚ h„R ,$tI I‚ht‚I Bt U

d ;$Kd

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8 8

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wD

bY${Y 8ZR YBjI PB„ sjj 8‚8A‚„R jwD BP s RZ$sAj‚ {jsRR

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qY‚ ‚RR‚t$sj {Bt{‚} BP 8‚YBIR j$,‚ e wstI 3eD

$R JZ$‚ R$8}j‚U R$t{‚ Y‚ ‚-s{ RBjZ$Bt wD B BZ„ }„BAT

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YBjI PB„ stk b‚$Y PZt{$BtjwD <Bb‚%‚„ sR $ $R RYBbt

$t V}}‚tI$- yy b‚ {st PB„{‚ Y$R ‚JZs$Bt B YBjI PB„ s

R‚ BP ] RBT{sjj‚I ‚R PZt{$BtR \jiwDi qY$R s}}„Bs{Y

k$‚jIR s RkR‚8 Bt j$t‚s„ ‚JZs$BtR k}‚

]

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j‚8 qY‚ ‚j‚8‚tR pi7 BP Y‚ {B‚^{$‚tR 8s„$- stI Y‚

„$YTYstITR$I‚ ‚„8R "i 8ZR A‚ {B8}Z‚I stI I‚}‚tI

Bt Y‚ ‚B8‚„k BP Y‚ ‚j‚{„BI‚ stI Bt Y‚ Y${,t‚RR stI „‚R}‚{$%‚ {BtIZ{$%$$‚R BP Y‚ jsk‚„R qY‚ R‚j‚{$Bt BP I$&‚„‚t R‚R BP „$sj stI ‚R PZt{$BtR j‚sIR B R}‚{$h{ tZ8‚„${sj s}}„Bs{Y‚R qY‚ I$R{ZRR$Bt stI ‚-s8}j‚R }„‚T R‚t‚I $t Y$R }s}‚„ s„‚ „‚R„${‚I B psj‚„,$t k}‚ b‚$YT $t s}}„Bs{Y‚R w‚R PZt{$BtR s„‚ $I‚t${sj B „$sj PZt{T $BtRD qYZR Y‚ {B‚^{$‚tR 8s„$- $t j$t‚s„ RkR‚8 wD $R Rk88‚„${ stI }BR$$%‚ I‚ht$‚ >1dH

qV Ne y

6st'sC yy „CZtI't RkR‚9U _Ys„s{‚„'R'{R stI tZ9‚„'{sj 9CI‚j

*ss

WZ8A‚„ BP ‚j‚{„BI‚RU xn;

WZ8A‚„ BP „BZtI „BIRU 1;

*$s8‚‚„ BP ‚j‚{„BI‚RU dd1M 88

*$s8‚‚„ BP „BZtI „BIRU dx(( 88

*‚}Y BP Y‚ „$IU (x 8

N‚tY BP „BZtI „BIRU ; 8

s- I$8‚tR$BtR BP „$IU 1n(dx 81

p_iU d( ,z

e WZ8‚„${sj BI‚j

qk}‚ BP s}}„Bs{YU psj‚„,$t

qk}‚ BP d* ‚j‚8‚tU N$t‚s„

WZ8A‚„ BP ‚j‚8‚tRU xM1

*‚„‚‚R BP P„‚‚IB8U nME

(5)

yeee qiVW7V]qy!W7 !W _!Gei *eNyzeiO z!N  W! O !Wq< 1((d ;

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1 Unit = 10 m

3$ 1 7st$sB yy „BZtI$t RkR‚8U _jst BP Y‚ ‚s„Y$t „$I wY‚ R$Zs$Bt BP Y‚ „BZtI „BIR $R 8s„,‚I b$Y Ajs{, }B$tRD

qY‚ htsj jskBZ BP Y‚ tZ8‚„${sj }„B{‚IZ„‚ B A‚ $8T }j‚8‚t‚I „‚8$tIR Y‚ RBT{sjj‚I X{B8}Z‚„ 8‚YBIR" PB„

„BZtI$t stsjkR$R >EH bY‚„‚pi7{B‚^{$‚tR {B„„‚R}BtI B

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!A%$BZRjk Y$R e PB„8Zjs$Bt {BZjI A‚ s}}j$‚I B BY‚„ {sR‚R b$Y s Y$Y‚„ tZ8A‚„ BP jsk‚„R <Bb‚%‚„ $ $R $8}B„st B s,‚ $tB s{{BZt Ys Y‚ ]_= $8‚ 8sk $t{„‚sR‚ {BtR$I‚„sAjk 8s$tjk A‚{sZR‚ BP bB Ps{R Ys s„‚ tB R„${jk „‚js‚I B Y‚ ABZtIs„k ‚j‚8‚t s}}„Bs{YU h„R Y‚ {B8}j‚-$k BP Y‚ ,‚„t‚jR $t Y‚ $t‚„sj ‚-}„‚RT R$Bt BP }B‚t$sj w;D BAs$t‚I Ak s}}j${s$Bt BP Y‚ 8‚YBI BP $8s‚R wZRZsjjk s 8Zj$}j‚ $tht$‚ R‚„$‚RD stI R‚{BtI Y‚ tZ8A‚„ BP ‚„8R BP Y‚R‚ R‚„$‚R Ys $ $R t‚{‚RRs„k B

3$ n 7st$sB yy „BZtI$t RkR‚8U _B‚t$sj I$R„$AZ$Bt wd( ,zD Bt „BZtI RZ„Ps{‚ BAs$t‚I b$Y s YB8B‚t‚BZR stI $RB„B}${ RB$j 8BI‚j

3$ ; 7st$sB yy „BZtI$t RkR‚8U _B‚t$sj I$R„$AZ$Bt wd( ,zD Bt „BZtI RZ„Ps{‚ BAs$t‚I b$Y s bB jsk‚„ RB$j 8BI‚j

‚%sjZs‚ R}‚{$sjjk bY‚t {BtIZ{$%$$‚R A‚b‚‚t jsk‚„R s„‚ %‚„k I$&‚„‚t

yyy! V}}j'{s'Ct CP Y‚ #h s}}„Cs{Y C s „‚sj „CZtI't RkR‚9U h0s9}j‚R stI -'R{ZRR'Ct

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$sB yy RZARs$Bt {jBR‚ B Y‚ {$k BP 7st$sB I‚ ]B8T

(6)

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6st'sC yy „CZtI't RkR‚9U i‚RZjR CBs't‚I PC„ I'PP‚„‚t RC'j 9CI‚jR

!t‚ jsk‚„ RB$j 8BI‚j

es„Y „‚R$R$%$kU E( 8

qBsj {Z„„‚tU En ,V

eJZ$%sj‚t „‚R$Rst{‚U (d;

]_= $8‚ wV_ ;(((DU  R

qbB jsk‚„ RB$j 8BI‚j

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NBb‚„ jsk‚„ „‚R$R$%$kU E( 8

qY${,t‚RR Z}}‚„ jsk‚„U d1 8

qBsj {Z„„‚tU xEd ,V

eJZ$%sj‚t „‚R$Rst{‚U (dM

]_= $8‚ wV_ ;(((DU dnnx 8$t

qY‚ „BZtI$t RkR‚8 }„B‚{R s Bsj s„‚s BP nM((( 81

qY‚ RZI$‚I s„‚s $R s b$I‚„ RZ}‚„$8}BR‚I „‚{stZjs„ 9Bt‚

BP n((1E( 81 w$‚ M((( 81D qY‚ p„BZtI _B‚t$sj

i$R‚ wp_iD {BtR$I‚„‚I $t Y$R RZIk $R d( ,z qY‚ }jst BP Y‚ ‚s„Y$t „$I wR‚‚ hZ„‚ 1D stI Y‚ ‚t‚„sj Iss wR‚‚ sAj‚ yD b‚„‚ BAs$t‚I P„B8 Y‚ „BZtI$t }jstR stI R}‚{$T h{s$BtR BP Y‚ RZARs$Bt }„B%$I‚I Ak Y‚ }Bb‚„ {B8}stk qY‚ {Ys„s{‚„$R${R BP Y‚ tZ8‚„${sj 8BI‚j Ys YsR A‚‚t ZR‚I $t Y$R ‚-s8}j‚ {st A‚ PBZtI $t sAj‚ y

qsAj‚ yy {B8}s„‚R Y‚ tZ8‚„${sj „‚RZjR wY‚ ‚JZ$%sj‚t „‚R$Rst{‚ stI Y‚ Bsj ‚j‚{„${sj {Z„„‚t j‚s,‚I $tB Y‚ „BZtID BP Y‚ stsjkR$R BP Y‚ 7st$sB yy „BZtI$t RkRT ‚8 BAs$t‚I Ak ZR$t Y‚ YB8B‚t‚BZR stI $RB„B}${ RB$j 8BI‚j wXBt‚ jsk‚„ RB$j 8BI‚j"D stI Y‚ }„B}BR‚I bBTjsk‚„ RB$j 8BI‚j

3$Z„‚R n stI ; RYBb Y‚ }B‚t$sj I$R„$AZ$BtR Bt Y‚ ‚s„Y RZ„Ps{‚ wbY‚t Y‚ „BZtI$t ‚j‚{„BI‚ ss$tR Y‚ p_i %Bjs‚D BAs$t‚I Ak ZR$t Y‚ YB8B‚t‚BZR stI $RB„B}${ RB$j 8BI‚j stI Y‚ }„B}BR‚I bBTjsk‚„ RB$j 8BI‚j y $R BA%$BZR Ys ABY }B‚t$sj I$R„$AZ$BtR I$&‚„ <BbT ‚%‚„ $ $R ,tBbt Ys tB${‚sAjk I$&‚„‚t {BtBZ„ j$t‚ I„sb$tR IB tB t‚{‚RRs„$jk {B„„‚R}BtI B R$t$h{st I$PT P‚„‚t{‚R A‚b‚‚t Y‚ }jB‚I „‚RZjR 3B„ Y$R „‚sRBt $t hZ„‚ x b‚ {B8}s„‚ Y‚ }B‚t$sj }„Bhj‚R {B8}Z‚I b$Y Y‚ bB RB$j 8BI‚jR sjBt bB I$&‚„‚t j$t‚R Bt Y‚ „BZtI RZ„Ps{‚

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0.0

0.0

360.

280.

1 Unit = 10 m

Distance (m)

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3‚„8?t Ws%s„„$ts bsR AB„t $t 7st$sB I‚ ]B8}BR‚js wNs ]B„ZKtsD 7}s$t $t dE( <‚ „‚{‚$%‚I s I‚„‚‚ $t ]$%$j et$t‚‚„$t P„B8 Y‚ =t$%‚„R$IsI _Bj$?‚{t${s I‚ sI„$I $t dMn stI s _Y* I‚„‚‚ $t ]$%$j et$t‚‚„$t P„B8 Y‚ =t$%‚„R$IsI _Bj$?‚{t${s I‚ s„{‚jBts $t dM ‚b‚‚t dM stI d( Y‚ R‚„%‚I sR sRT RB{$s‚ }„BP‚RRB„ stI PZjj }„BP‚RRB„ BP V}}j$‚I sY‚8s${R s Y‚ ]$%$j et$t‚‚„$t 7{YBBj BP Y‚ =t$%‚„R$IsI _Bj$?‚{t${s I‚ ]ssjZKts bY‚„‚ Y‚ sjRB bsR Y‚sI BP Y‚ ]B8}Z‚„ ]‚t‚„ stI z${‚T*‚st yt dd Y‚ PBZtI‚I Y‚ ]$%$j et$t‚‚„$t 7{YBBj BP Y‚ =t$%‚„R$IsI I‚ Ns ]B„ZKts bY‚„‚ Y‚ bsR *‚st A‚b‚‚t dd stI d stI R‚„%‚R sR _„BP‚RRB„ BP V}}j$‚I sY‚8s${R stI ]$%$j etT $t‚‚„$t _„BP Ws%s„„$ts $%‚R {BZ„R‚R Bt tZ8‚„${sj 8‚YBIR stI B}$8$9s$Bt PB„ ‚t$t‚‚„R R$t{‚ 8B„‚ Yst dx k‚s„R <$R „‚R‚s„{Y h‚jIR s„‚ „‚js‚I B Y‚ I‚%‚jB}8‚t stI s}}j${s$Bt BP tZ8‚„${sj ‚{Yt$JZ‚R PB„ stsjkR$R stI I‚R$t B}$8$9s$Bt $t ‚t$t‚‚„$t <‚ YsR I$„‚{‚I stI\B„ {BjjsAB„s‚I $t s tZ8A‚„ BP iL* }„B`‚{R ABY RZ}}B„‚I Ak }ZAj${ stI }„$%s‚ PZtIR stI $R Y‚ sZYB„ BP 8B„‚ Yst E( }s}‚„R $t R{$‚t$h{ `BZ„tsjR ABB,R stI }„B{‚‚I$tR BP $t‚„T ts$Btsj {Bt„‚RR‚R <‚ $R sjRB 8‚8A‚„ BP Y‚ ‚-‚{Z$%‚ {BZt{$j BP 7eWy stI 8‚8A‚„ BP yV] w‚T8s$jU ts%s„„$tso${{}ZI{‚RD

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