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IEEE T r a n s a c t i o n s on Power ApparatusandSystems,
Vol. PAS-96, no. 1, January/February 1 9 7 7
RESISTANCE CALCULATION OF INTERCONNECTED GROUNDING ELECTRODES
Dinkar Mukhedkar, Senior Member IEEE
Ecole Polytechnique
Montreal, Quebec, Canada
Farid Dawalibi, Member IEEE
The Shawinigan Engineering Co. Ltd.
Montreal, Quebec, Canada
In previous publications [4] to [ 6 ] theAuthors determined the
general equations giving the potential caused by a complex electrode
buried in a two layer soil. Starting from these equations, the earth resistance of an electrode is calculated using the average potential method
in its ‘‘Integral” form.
Abstract-The calculation of earth resistances by the average potential method has proved to give accurateresults. Various publications
[ 1 ], [ 21 shows that current distributionin the electrode has little effect
on the calculated result assuming uniform currentdistribution.The
above method however, requires that integration be carried along the
electrode path [ 1 ] . For complex interconnected grounding electrodes
this introduces difficulties which usually have been avoided by taking
adequate points (Representative points) on the electrode and averaging
thepotentials calculated for these points [ 2 ] , [31, [4]. The average
potential obtained is then considered to be the potential rise of the
electrode. This however requires a judicious selection of points and/ora
large number of points in order to obtain an accurate result. For large
interconnected electrodes the consequence is large computing time or
difficulties to chose adequate representative points specially when uniform current distribution in the electrode is assumed. This paper introduces a method which permits the integration to be carried along the
path of aninterconnectedelectrode
buried in a two layer soil. The
general formula for the earth resistance calculation is derived and applied to typical cases. The reader should note that the method is also
applicable for the calculation of average potential values.
1. Analytical Study
1.1 Formulation of the problem
Reference [4] shows that the potential caused by agrounding
electrode can be written as follows.
LIST OF SYMBOLS
A parameter related to origin of conductor k
u, v, w are the modified coordinates of point M(x, y, x) where the
potential is calculated.
upk is the u coordinate of the extremity of conductor k
The electrode being broken down into m straight conductors.
The reference coordinate system being x,y,z and u,v,w being a
temporary rectilinear coordinates attached to conductor k such that the
conductor k is defined by the following.
A parameter related to extremityof conductor k
Coordinates with respect to the rectilinear reference system
Coordinates with respect to a temporary rectilinear system
First layer resistivity (Ohms - Meter)
Reflexion coefficient (u=p2-p 1/p2 + p 1)
First layer height (Meters)
Total buried length of the electrode (Meters)
Potentialinduced
by conductor k at point j (Volts)
Designate either u, v or w
Number of conductors in the electrode
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zy
The u,v,w being related to x,y,z by the following relations (figure
Total current injected in theelectrode(Amperes)
1):
Electrical impedance(Ohms)
U=(X-Xsk)COSAk.COSBk+(Y-Y,k)slnAk.cosBk
INTRODUCTION
( z-zSk)sinBk
In a companion paper [ 5 1 the Authors presented a method called
multi step analysis for the calculation of current distribution in interconnected grounding electrodes. The potential values used in the latter
could be determined from the formulas given in this paper, instead of
those given in reference [41. The advantage of doing so, is increased
accuracy of the results.
Provided that an adequate method is used it is possible to calculate
accurately earth resistances without previous knowledge of the current
distribution. When the average potential method is used in its “Integral”
form (mean value of the integral of the potential carried along the electrode contour), it turns out that the average value is, within a few percent, equal to the value estimated by successive approximations of the
integral equation.
v=(y-ysk)cosAk-(x-xsk)slnAk
w=( z-zsk)cosBk-(y-ysk)slnAk.slnBk
-(x-xsk)slnBk.cosAk
The expression of
1””“ Q(u,v,w,upk)
being:
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Paper F 75 521-5,recommended and approved by the IEEE Substations Committee of the IEEE Power Engineering Society for presentation at the IEEE PES
SummerMeeting, San Francisco, Calif., July 20-25, 1975. Manuscript submitted
January 31,1975; made available for printing May 5,1975.
Where N is an integer large enough to give the adequate accuracy.
59
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/"" zyx
zyxw
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zyxw
Ifu,v,w are the general coordinates of a point at the surface of the
electrode conductors. The resistance of the electrode can be calculated
by integrating equation (1) along the electrode.
We have:
The average value of the potential caused by the electrode on conductor j is therefore
1
V
P1I
'kJ'
pk @ ( u,
I,v j ,w J ,u Pk )du
Thus:
The mean value of the potential caused by the electrode on all the
conductors j=1,2.. ., m is consequently:
Where vkj is the potential induced by conductor k at point J .
Therefore equation 1 can be written as:
0
( A p j - Asj) Asj
javer.
x[[
Vj. d A j
P
If the total current injected by the electrode is I the electrode
earth resistance becomes:
& x
.. -
fz
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H
Bk
The remaining task is to calculate the integral of the potential vkj.
(rotation / 0 9
W
I
)
1.2 Derivation of the potential integral
Bk ( r o t a t i o n / o'v' )
'
vkj is expressed as a sum of inverse hyberbolic functions h a w
the following general form:
:,I
Figure 1
Rectilinear coordinate
(+-- )
systems
sinh-'
Where,
Therefore equation 1 can be written as:
vj =
2
1x +2blx+c
x = Aj
And (see appendix I)
vkj
k=l
a.
Assume now that j is any point at the surface of conductor j. The
coordinates of point j with respect to the rectilinear system attached to
conductor k being Uj, vj, Wj. Since j is a cylindrical conductor the u,v,w
are not independant and any two of these values can be written as a
linear function of the remaining one:
= -a I
bo = upk
(For series 1 )
-b
or,
a, = a
I
(For series 2)
bo = b I
J
Aj
111
wJ = a
+
al = (aI1 )2 + (a''
bl = .IIbII
.I11 ( b I I I
+
b~~~
c 1 = (b
I1 2
1
+
(bIII
+
+
kIII
kIII
1
2
With:
Let's calculate then,
Box + bo
+ 2blx + c1
60
)
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z
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zy
Integrating by parts, the following is obtained:
~ = a x + b
Where:
YO= a@ + bo
Y1= alx + bl
L
It is important to note that formula ( 1 1) is given for one value of
Where (see appendix 11)
dY
x-=
dx
with:
1) and one value of the series (%, bo) as shown by equations (IO).
b0
--
fi
When earth has a two layer structure, ( 1 1) should be used together
with (2). It is obvious that in such a case the use of a digital computer is
appropriate.
1.2.2 Interconnected electrodes
x = a 2 + 2bx + c
When a number t of electrodes are interconnected, ( 1 1) is still
applicable with however some modifications as explained hereafter:
The interconnected electrodes are subdivided in two groups:
- Remoteelectrodes
- Close electrodes
Assume that after the above classification, one obtains p group of
remoteinterconnectedelectrodes
(the mutual coupling between each
two arbitrary group can be neglected). Consequently, earth resistance of
each group i can be calculated separately, by use of formula (1 I ) , where
each group is considered as a unique electrode.
Let R1,R2,. . . Ri . . . Robe these calculated resistances. The
total earth resistance is then obtained by use of classical electrical circuit reducti 8n.
If Z 1,Z2,. . . , Zi, . . . , ZP are the equivalent impedances of the tie
circuit of each group, theequivalent electrical circuit to be used is shown
in figure 2.
Formula ( 1 1 ) however isvalid only for the general cases where
a ] # 0. In most practical cases a ] however is zero (horizontal or vertical
conductors)
a = a1 + ao2
b = b l + aobo
c = c1 + bo2
And :
+ c1
x 1 = a1x2+2ablx
dy
If a1 # o then x 5 can be written as follows (see appendix 11)
With:
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A = alcl-bl
fi =
a,bl
2
- alb,
1.2.2 Particular cases
Polynomial X1 can be written as the product:
I)al=O;%#O
The applicable formula reduces to:
Where:
'bl
x 1 = -+
a1
a1
j-
A
'bl
A
J-
a1
a1
x2 = - -
- xs .In
Long and tedious calculations given briefly in appendix 111 shows
that the final expression for the integral can be written as follows:
61
aoxs + bo + f i s
i
Referent: group
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zyxwv
zyxw
A
Ak = 0
We have:
R5
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zyxw
Therefore:
Thus:
a l = 0
, bl=0
And the reduced formula should be used
A)
'
4
q
For series 1
-
J
Common e a r t h
,
ao= -1
a = l
,
b=-L
,
,
%= 1
,
bo
= L
c
= ~2 + r2
bo
c
= O
For series 2
Figure 2
E q u i v ae l e cnttcriirccaul i t
11) a1 = 0 ; a.
c l = r2
c l = r2
a = l
,
b = O
= r2
= 0
The applicable formula reduces to:
(I4)
For series 2
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2. Application to simple cases
Since
# 0, equation(13) should be used. Aftersimplification
and reduction the following is obtained for each case A and B
2.1 Horizontal cylindrical conductor
Consider an horizontal cylindrical conductor of length L and radius
r buried at a depth e in a uniform soil (v = 0 only), as shown in figure 3
v,
-1
. ( L l n[ L + F J
=P I
4TL
{I z
;w
2
L
If e = o the resistance reduces to:
Figure 3
Horizontal cylindrical conductor
62
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Which can also be written as:
These values when replaced in (2) give:
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zyxwvutsrqp
zyxwvu
zyxw
zyxwvuts
zyxwvutsrq
upk = L
US^ = L/2 ;
Vsj = -L/2
; WS~= r
upj = L/2 ;
vpj = L/2
; wsj= r
This is twice the resistance given by sunde [ 1] for a straight conductor imbedded in a uniform media of infinite extent.
Therefore:
X,= Vj
L/2)
(?pj=
And:
2.2 Four point star horizontal electrode
This electrode is shown in Figure 4. It consists of two horizontal
wires perpendicular at their center point. It is assumed that the two
conductors are of the Same length L. The resistance of the electrode is
according to (9):
a1 = 0
;
a11 = 1
; a111 = 0
bI= L/2
;
bII = 0
; bIII = r
bl = 0
; c l = (r + 2qe)2
Thus:
a1 = 1
;
A = r+2qe ;
21
p
=-bo
Because bI = upk/2 and a1 = 0, series 1 and 2 are identical.
However, by symetrical considerations the above can be reduced
A) q =O
into:
For series 1 and 2
; A = r
c l = r2 ; ao= 0 ; bo=
L/2
Where:
Jlkk is the average potentialvalue induced by conductor k on itself
and is given by (15). (Uniform soil).
Jlkj is the average potential value induced by conductor k on conductor j and is calculated in the following:
a
= 1
; b= 0
; c = L2/4+r2
; p = -L/2
B) q = 1
For series 1 and 2
e
W
;
=,a
a = 1
;
b = 0
Four p o i n ts t a rh o r i z o n t a le l e c t r o d e
; A = r+2e
. ; c = L2/4+r2+4e2; p = -L/2
If all ratios 4r2/L2 and 16e2/L2 are neglected with respect to 1
the following final expression is obtained (for cases A
and B):
For simplicity, it is assumed that earth is uniform (7 is equal to 0
only.) We have with respect to
axis:
the x,y,z
Xsk = 0
;
Ysk = 0
;
zsk= e
xpk = L
;
Ypk = O
;
Zpk'
Zsj = e
Zpj= e + r
xsj = L/2
;
ysj = -L/2
;
xpj = L/2
;
ypj = L/2
;
With:
; bo= L/2
Equation (1 1) reduces to (for case A or B respectively, when A2 is
neglected)
%.
z
0
crl2=+ 4 d
+
e
+
+r
-rL tan -1 3.6568r
L
(r + e )
tan
-1 3.6568(r+e)
L
Where:
I
i =
(linear current density in the electrode)
2L
-
63
L
resistance
The
of the electrode is therefore: (Replace
in
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(16) $&
APPENDIX I
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Where N(4) = 2.45
If e/L and r/L can be neglected then ( 18) is identical to ( 19)
Q
= Oorl
u = n or -n (if n=O u is applied only once in the formula. When the
soil is uniform the sum given in equation (3) reduces to two terms,
Conclusions
3.
two while
a for
layer structure., there will be twoforterms
n positive
and two others for n negative)
The above two examples c o n f m that it is possible to calculate the
resistance of electrodes of various forms by using one unique formula.
The advantages of the method proposed by the Authors are the following.
APPENDIX I1
1) The method avoids theuse of various formulas for each form of
electrode
We have
2) The method is free of approximations
y = sinh-l (z)
3) It can be applied to uniform or two layer earth structure
, therefore:
4) The method is applicable to curvilinear electrodes as well. The
electrode is simply divided into reasonably straight conductors
5) It can be easily programmed. The computing time will be
proportional to thesquare of the number of conductors.
With
6) Finally, the method is not restricted to resistance calculations
but can be used when the distribution of linear current density in the
electrode has to be calculated with great accuracy
ACKNOWLEDGMENTS
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After derivation and simplification:
Mr. F. Dawalibi whoisaregisteredDoctoral
Student wishes to
thank Ecole Polytechnique for the facilities offered. Also the Authors
express their appreciation t o the National Research Council of Canada
and the Department of Energy, Mines and Resources of Government of
Canada for providingthe. necessary financial supportof thework.
The Authors would like to thank the management of the Shawinigan
Engineering Company Limited, Montreal, Quebec,for their cooperation.
With
REFERENCES
[ 11E.
D. Sunde,Earthconduction effects in transmissionsystems,
Dover Publications, New York, 1968 (Book).
[2] T. N. Giao, M. P. Sarma, “Effect of two layer earth on the electric
field near HVDC electrodes”, IEEE Transactions, Vol. PAS-91,
No. 6, November 1972, pp.2356-65.
[3] B. Thapar, E. T.B . Gross, “Grounding grids for high voltage
stations, part IV,resistanceofgroundinggrids
in non uniform
soil”, IEEE Transactions, PAS, Vol. 82, Oct. 1963, pp. 782-7823
[41F.
Dawalibi, D. Mukhedkar,“Optimum
design ofsubstation
grounding in atwo layer earthstructure”, PartI - Analytical
study. IEEE Transactions paper No. T 74-191-3.
[ 51 F. Dawalibi, D. Mukhedkar, “Multi Step Analysis of Interconnected
electrodes” Paper submitted simultaneously to IEEE Power Summer Meeting.1975.
[6] F. Dawalibi, D. Mukhedkar, “Sur les conditions de simulation des
mises a la terre”, RevueCeneraled’Electn’cife, tome 83, No. 3,
March 74, Paris.
[7] H. B. Dwight, ‘‘Calculationof resistance to ground”, AIEE Trans.,
p. 1319, Dec. 1936
a = a1 + a2
0
b = b l + %bo
c = cl + b i
This expression reduces after simplification to therelation given in
page 3.
APPENDIX 111
x%
dx when integrated by parts can be written as:
( A x + B) dx
x-x 1)(x-x2)
64
Jrr
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Where x ] and x2 are given in page 3. The first integral is easily determined. The second however could be broken in two:
After long calculations it can be shown that these integralsreduces
to
T-
65
zyxwv
= A (ao -&I
