Efficient computation of rodbed grounding resistance in a homogeneous earth by Galerkin‘s moment method

Y. L. Chow M.M. Elsherbiny M.M.A.Salama

Indexing terms: Rodbed grounding resistance, Rodbed configurations, Moment method

Abstract: The paper computes the grounding resistance of rodbeds in a homogeneous earth by a moment method with the Galerkin’s approach. The self and mutual resistances matrix resulted is quite small. It shows that an N rod assembly requires only N x N matrix. The resistance of the rodbed grounding system calculated using this method has a percentage difference of less than 2% from the point matching moment method. The Galerkin’s moment method does not match the boundary potential at one point of a segment, but matches the boundary potential that is averaged over the whole segment. Such a simple change in the moment method makes the method variational and results in a very small error even with a substantial saving in computation effort. The effects of various system parameters on the rodbed parameters (such as rod sunken depth, rod length and radius, rod spacing, rodbed configuration, and number of the rods) are investigated and discussed in detail. The paper ends with a discussion on the accuracy of the surface ground potentials with the coarse segmentation of one segment per rod.

1 Introduction

One of the key requirements in substation earthing design is to calculate accurately the grounding resist- ance of the system. Simple empirical formulae for cal- culating the ground resistance have been proposed in the past [l-31. The resulted ground resistances obtained by these formulae are usually accurate for simple ground system configurations, however, these results deviate considerably from the measured ground resist- ances for the realistic grounding system, such as grounding grid in multiple-layer soils [4]. For such grounding systems finely segmented numerical analysis is frequently used to calculate the grounding resistance [5 ] . In most cases numerical analysis for the realistic 0 IEE. 1995 ZEE Proceedings online no. 19952103 Paper first recelved 25th November 1994 and In final revlsed form 26th

The authors are with the Electrical and Computer Engineering Depart- ment, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

.April 1995

IEE Proc.-Gener. Trunsm. Distrib., Vol. 142, No. 6, November 1995

grounding system resistance is done using the point matching moment method [6, 71. The analysis and the algorithm are discussed by many authors [4-71.

The calculation of the ground resistance using the moment method is straightforward with its numerical execution simple. The main drawback of this method is its computational burden, in other words, the require- ments of a large number of segments in the calculation. Joy et al. [8] had found that the accuracy of the solu- tion depended on the number of segments used to rep- resent either the horizontal conductors or the ground rod conductors. They gave guidelines for the maximum sizes of both the horizontal conductor segments and the ground rod segments [SI, as a function of the depth of the buried mesh, ground rod diameter and rod length. Although this treatment of the segmentation does reduce the size of the problem, hence the compu- tational time, it is still cumbersome and requires a knowledgeable user for effective use of the moment method.

The aim of this paper is to develop an accurate and efficient method for calculating the grounding resist- ance of rodbeds. This method utilises the Galerkin’s approach of the moment method [9]. Because of the variational nature of Galerkin’s method (summarised at the end of the Appendix) the self and mutual resist- ances of the system calculated from assumed uniform current distributions, produce small second order errors. Even with a uniform current distribution assumption, only one segment per grounding rod is needed to give an error <2% in the calculated resist- ance. This is a drastic reduction in computer resources.

To illustrate clearly the application of Galerkin’s method 191 to calculate the grounding resistance, only grounding rods in a homogeneous earth (i.e. a half space problem because of the air above) are considered in this paper. The use of Galerkin’s method for the cal- culation of the grounding grid is similar, but is not dis- cussed in this paper.

This paper is divided into five sections. Section 2 introduces the Galerkin’s method and application to a single rod in a homogeneous medium. Accurate for- mula for the resistance of a single ground rod in homo- geneous medium is derived in this section. In Section 3.1 the mutual resistance of two parallel ground rods in the same homogeneous medium is derived. In Section 3.2 the effect of the half space of air (on the top of a homogeneous earth) is introduced. In Section 4 the der- ivation of the resistances of multiple ground rods bur-

653

ied in homogeneous earth is obtained. Section 5 presents a discussion of the results.

While the emphasis of this paper is on grounding resistance with a coarse segmentation of one segment per rod, the accuracy of the earth surface voltages does not suffer. In Section 6 it is shown that because the rods are driven vertically into the ground even with such coarse segmentation, the errors of earth surface voltage are no more than 5%. With two and three seg- ments used the errors in surface voltage are further reduced to 3% and 2%, respectively. Section 7 gives the conclusion on resistance and surface voltage.

m f n

4

__y I I

i +2 E9

a segment is magnified z=o reference I ---2

Fig. 1 Two parallel conductors (need not to be equal) divided into N number of segments from 1 to N Fig. l a is for the mutual resistance calculation (m = n), while Fig. lb is for the self resistance calculation (m = n). rmn is the distance between the centres of two segments (m and it), d is the perpendicular distance between the two paral- lel conductors, and a i s the radius of a conductor. The reference plane (z=O) may be placed anywhere

2 The Galerkin's moment method

Assume there are two conducting bodies in a homoge- neous medium, as shown in Fig. 1, it is required to cal- culate the resistance to infinity of the two conductors. In the moment method, the conducting surface is divided into segments each diffuses a current Ii, the relationship between voltages and current can be writ- ten as:

. . .

R 1 1 R I 2 . . . . . . R 1 N

R2l R 2 2 . . . ... R 2 N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

R N ~ R N ~ . . . . . . RNN

where Ii is the current of segment i (i = 1, 2, ..., N), 6 is the voltage of segment i (i = 1, 2, ..., N), and Rmn is resistance element, i.e. resistance between segment number n and segment number m.

For conducting surfaces, VI = V, = .... = V (equipo- tential surface). If the resistance elements are known, the current can be calculated by the inversion of the matrix of eqn. 1 . After this the resistance of the system is

V (2) R = - 5 I ,

2= 1

2. I Point matching method - small segments From the point matching method [9], the segments are small and the resistance elements of the above matrix may be expressed as follows:

654

P 4.irrmn

Rmn

(4)

where Rnn is the self resistance element of a segment n, and rmn is the distance between the centres of segment m and segment n.

2.2 Galerkin's method - with large segments and error reduction property With the point matching method, the segment size has to be small (i.e. large number of segments) so that the 1

error in calculating l/rmn in eqn. 3 becomes small. This ti

difficulty is avoided when, in Galerkin's method [9], the 1hmn term is averaged over the segment (i.e. harmonic mean). Such averaging makes the term variational [lo], i.e. with error reduction property which can be proven analytically [lo] and shown numerically at the end of this section. With an assumed uniform current distribu- tion the Appendix shows that the resistance element (mutual and self) becomes:

R,, = E D - ' 47T

(5) where D-' can be interpreted as the reciprocal of the effective distance between two segments. In detail this reciprocal distance function can be written in terms of its variables as: D-l(b,, a2; bl, al; d) which means that

X 1 d d D-I = -___- d b? - a 2 bl - a 1

(6) where

(7 ) f(x) = z sinh-I IL: - dl + z2

and shown in Fig. la: al and bl are the z co-ordinates of one segment (i.e. zml and z,, for segment m in Fig. la) a2 and b2 are the z co-ordinates of the other segment (i.e. znI and zn2 for segment n in Fig. la) d is the perpendicular distance between the two seg- ments. In the case of self terms (m = n) the perpendicu- lar distance d is not zero. In this case the segments are considered as two current lines, one of the two lines is taken along the cylinder axis while the other is on the surface, i.e. d is equal to the radius of the conductor, a, shown in Fig. lb.

z

z=o

Fig.2

The driven rod is divided into N segments (each of length Al), so is its image

Single rod is buried at depth hb from the earth surface in a homo-,'

, i

geneous soil of resistivity p i

IEE Proc.-Gener. Transm. Distrib., Vol. 142, No. 6, November 1995

To demonstrate the accuracy of the Galerkin's moment method, the following example is used: Fig. 2 shows a segmented ground rod driven vertically in a homogeneous earth and the image due to the air-earth interface. Let there be N segments. The self resistance element of the nth segment and its image (from the air- earth interface), using eqns. 5 and 6, can be derived as:

2a --B

homogenous medium of resistivity i~ ohmm

where

D,itl = D-yal , 0; Al, 0; a ) (9) for the self resistance of the actual segment, and

for the mutual between the segment n and its image. The rod radius a, and the co-ordinates Al, z,+A1/2, etc. can be identified in Fig. 2. The mutual resistance between segments m and n is

(11) P

477 Rmn = -(Dk:i + D,',,)

where

A1 A1

for the mutual between actual segment m and actual segment n, and

' -2 +-,-z,----;a A1 A1 D-' = D-l z,+--,z,-- a1 2 1 A1 mn2 ( 2 2 ' 2

(13)

for the mutual resistance between actual segment m and the image of segment a. The co-ordinates are iden- tified in Fig. 2. Once the self and mutual resistances of all the N segments are defined, the rod resistance is cal- culated by eqn. 2 after the inversion of the resistance matrix of eqn. 1. With one segment, the matrix of eqn. 1 is reduced to 1 x 1, i.e. single formula of R = R,, in eqn. 8.

To investigate the accuracy of Galerkin's moment method on the computed ground resistance of the ground rod, due to changing the number of segments, the above technique is used with N varied from 1 to 100. For a rod of 10m length and radius of .Olm bur- ied in lOOQ-m earth, at zero sunken depth, the com- puted resistance is 11.70Q with one segment and 11.6OQ with 100 segments. In fact, for number of seg- ments greater than 30, the calculated resistance remains almost constant. This is the power of the variational principle in error reduction, despite the approximate assumption of uniform current distribution. Since the difference in the calculated resistance, using one seg- ment, is less than l%, from this point on the rod is rep- resented by one segment in the analysis.

3

3. I Single-space problem For simplicity, let us first consider a pair of parallel rods in a homogeneous medium, as shown in Fig. 3. Due to the small error observed in the last section, each rod is taken as one segment and is represented by

Mutual resistances of two parallel rods

IEE Proc -Gener Transm Dastrib., Vol 142, No 6, November 1995

I

T I

I

I

I

I

I I

I

I

I

T

I

I

I

I

2a I- -b

I

T

I

I

I

I

I

I I

I

I

I

T

I

I

I

I t-----d-----+ I I I I

1 1

Fig.3 geneous medium of resistivity p SZ m

Set of two parallel rods of equal length ( I ) surrounded by a homo-

an equivalent current source of a uniform current den- sity along the centre line of the rod. The mutual resist- ance of the system is calculated, based on eqn. 6, to give

r 7

R12 = I sinh-lG 1 - {I + ( F)2+ (F) 1 (14) 2T1 L J

where 1 is the rod length, d12 the perpendicular distance between the two rods, p the resistivity of the surround- ing space, and R12 the mutual resistance between the two rods. Similar to the resistance of eqn. 8 Rl l of each rod can be considered as the mutual resistance between the central filament current and the rod cylin- drical surface. This is obtained by replacing 'd12', the rods separation, in eqn. 14 by 'a', the radius of the rod. Therefore, R1, is given by,

r 1

RI1 = ~ I sinh-' - d1+ (;)2+ (;)I (15) 2Tl a L -I

In most of the grounding applications the length I of the rod is much larger than its radius a, then eqn. 15 is reduced to

If the distance between the two'rods is greater than the length of each rod, eqn. 13 is reduced to

3.2 Two half-space problem In this section the more realistic case, as shown in Fig. 4, is considered. There are two half spaces. The upper half space is air and has infinite resistivity. The rods are buried in the lower half space, homogeneous earth with resistivity p. Because of the air-earth inter- face, there are rod images as shown in Fig. 4. Assum- ing the top of the rod touches the interface, because of the image, each rod appears to have a length 21. This means that the rod is twice as long, but the current comes from only the lower half of the doubled rod. Thus eqns. 14 and 15 become

r _I

L

and

655

For rod length 1 much greater than its radius a, eqn. 19 reduces to

Eqn. 20 is a well known IEEE-80 standard formula [2]. For rod separation d12 much larger than the rod length I, eqn. 18 becomes

In the above equations the depth of the rods, hb, is taken as zero. For a sunken rods (i.e. h, > 0), the orig- inal formula of eqn. 6 for the effective distance has to be used.

n n I / I I images

I / I I I I I / I1 I I I / I I

air

earth with resistivity p ohmm

'//////////////////////////////i

L

I 'I

Fig.4 Two buried rods and their images

4 homogeneous soil

For N number of parallel driven rods in a homogene- ous soil, the rods are assumed to carry the same poten- tial V, each rod has its own uniform current, 11, 12, ..., IN, and the voltage resistance matrix equation becomes identical to eqn. 1 except that VI = V2 = ... = V and the Rmn elements are given by eqns. 18-21. The seg- mented current Ii are the unknowns. Eqn. 1 is solved by matrix inversion and then the resistance calculated by eqn. 2.

5 Results and discussion

Resistance of multiple driven rods buried in a

Section 2 shows that a ground rod can be considered as one segment and yet the calculated resistance is accu- rate within 1% error. Based on this result a rodbed of N parallel rods can be represented by only N segments. Therefore, the ground resistance of a rodbed can be calculated by solving an N x N matrix. This represents a considerable savings in computational effort and computer memory. However, in this case, the error for N rods may not be as small as 1% because of matrix solution.

To compare the results of the present technique with the published methods, the rodbed configurations and the system parameters used in [4, 51 are selected. Figs. 5 and 6 show the results of these comparisons. It is

656

observed that the present method is very efficient in computer resources. It, however, yields differences in resistance up to 2% from the results of the detailed techniques [4, 51.

11.6 6.31 4.43 3 41 r -I- i r -I- t r - I - t T - I - t t -+- -I F -+- -I I- -+- -i I- - I - 4 L _ I _ J L - I_ J 1-1- J c _ I _ 4 C11.851 C6.431 C4.521 c- -1

2.14 2.24 2.64 2.96

c -+- -+ +- - 1 - + +- - I - + t -+- i C2.161 c--1 c- -1 C3.011

r-7-3 +-T-t 7 - 1 - 7 7 - 1 - 7

L -A- 4 L -A- 4 L - 1 - 4 c - 1 - 4

Fig. 5 Example for comparison between the resistances of multiple rods in a homogeneous earth calculated from the Galerkins technique and the detailed numerical analysis [4] The resistance value in square brackets are those quoted from [4], while the top value of each configuration represents the Galerkin's method values. The rod length is 10 m, its radius is .01 m, and the soil resistivity is 100 P m 11

6cm x 6cm 6cm x 2cm 6cm x 6cm 6cm x 6cm I

I I I Galerkink I results I 0.28 0.35 0.33 0 35

I I I I

results I 027 1 0.34 1 0.32 1 0.35 I 1::; I

Fig. 6 For configurations shown, the resistances based on the Galerkin's method are compared with those in 151 The rod length is 3 cm and its radius is 3 mm. The displayed numbers are for the ratio of the resistance of the rodbed to the resistance of a single rod. The listed dimensions are the overall dimensions of rodbed configurations

e . . 8.5

0 0 2 0 4 0 6 0 8 1 0

Fig.7 Variation of resistance of sunken rods (R) normalised to the unsunken one (Ro) Three configurations are selected single rod, 9 rods with 1 m spacing between the adjacent rods, and 9 rods with 10 m spacing The rod diameter is 318" Its length is 3 m, and spacing d = 10 m

sunken depth, m

Because of the efficiency of Galerkin's method it can be used to rapidly study the effects of different ground- , ing system parameters on the rodbed grounding resist- ; ance. Fig. 7 shows the effect on resistance through ' sinking the top of the rodbed below the earth surface. , It is clear from Fig. 7 that by increasing the sunken I

depth from 0 metre to 1 metre for the 3 m rods resist- +

ance reduction is only 10%. The reduction lessens with \ a decrease in number of rods or increases in rod sepa-

i IEE Proc -Gener Transm Distvib Vol 142 No 6, November 1995

ration. This effect is due to the reduction of the mutual resistance between rods.

10 I \ \

'I 1 length of rod(s), m 10

Fig. 8 Normalisedpercentage resistance of driven rodjs) versus the length of the rodjs) The resistances are normalised to the resistance of a driven rod(s) of 1 m length (Rl). Curves from the top are for: 8 rods, 5 mm radius each; single rod, 5 cm radius single rod, 5 mm radius; and a single rod with extremely small radius (a + lo-' m) are included for comparison. In the figure d = 10 m

Fig. 8 shows the effect of increasing the length I of the vertical rod in a rodbed system. In this study, a sin- gle rod of radius 5cm, a single rod of a small radius (a = 0.5cm), and a rodbed of 8 rods (a = 0.5cm and rod- bed configuration is shown in the figure) are used in the calculations. The resistances of the rod(s) are nor- malised to the resistance of one metre rod (of the same radius) to facilitate the comparison. The results are plotted against the rod length for the three selected configurations mentioned above. The plot is log-log scale to show the approximate 111 relation between the resistance R of the rod and its length 1. This approxi- mation deteriorates with increasing rod radius of a sin- gle rod and with increased number of rods. Fig. 9 shows the effect of varying the rodbed configuration from a square one to a rectangular configuration. The length of the configuration perimeter and number of rods per side, are kept constant. The ratio of the lengths of the two sides of the configuration is varied from 1: 1 to 2: 1. It is clear from the figure that this var- iation does not change the resistance of the rodbed too much (<2%).

configuration used

a c l 1.04 1 b- o . .

c 1 0 3 1 $ 0

a 0 . .

I S 7 5 8 8 a ~ ' I ' ' ~ ~ ~ ~ ~ ~ ~ 1 " - ~ ~ * * 1 1 '

1 .o 1.2 1 .4 1.6 1.8 2.0 ratio of two sides, (olb)

Fig.9 Ratio of the resistances of a rectangular configuration rodbed ( R ) and of a square rodbed (RI ) of the same number of rods ( S ) , the rodbeds

length (3m) have the same perimetev length (SOm), same rod radius ( l c m ) , and vod

Finally, the grounding resistance is calculated for a rodbed configuration with and without a central rod.

IEE Proc.-Gener. Trunsm. Distrib., Vol. 142, No. 6, November 1995

Fig. 10 shows the rodbed configuration used, the dis- tance d between the rods is varied with respect to rod length according to dll = 1, dll < 1, and dll > 1. It is found that the central rod has little effect on the grounding resistance of the rodbed if the spacings between the other rods are small or the number of rods is large.

. . . . e

. . . . .

changing in resistance by adding a central conductor

- 0.5"Io

3 - 1 .a%

10 - 4 .I "lo

I I Fig.10 over rod length (1) (i.e. junction of d/l)

Effect in adding a central rod as a function of rod spacing ( d )

6 driven into a homogeneous soil

This paper shows that good resistance (error <3%) for a rodbed can be obtained with a very coarse segmenta- tion of only one segment per rod with uniform leakage current. To complement, this section shows that a nearly as good (error 4%) earth surface voltage can be obtained with such a coarse segmentation.

We shall choose a very stringent case of only one rod (in the rodbed) and compute the rod surface potential with the above coarse segmentation of one segment per rod. This is chosen for two reasons: (i) without the mutual potentials from other rods in the rodbed to even out the potential, it is expected that the error in potential along the single rod will be most severe; and (ii) if the earth surface potential at the rod has a low error, the error on the nearby earth surface potential will be even lower, because the error at a distance approaches the error of grounding resistance of the rod.

The one driven rod is as shown in Fig. 11. The reflection of the nonconductive air above the earth sur- face makes the rod appear to be twice as long (i.e. 2L) and in a homogeneous medium of resistivity p.

The earth surface voltage from a simple rod

U i Fig. 11 Single driven rod in a homogeneous soil

To satisfy the curiosity in what improvement can one get in having more than one segment per rod Fig. 12 shows three rod-surface voltage distributions: solid curve for actual constant potential along the rod, dashed curve for a single segment with uniform current along the whole rod, and the curve with circles for two segments on the rod: as shown in Fig. 12 a large one

651

of length 49/50 of the rod length and a small one at the tip with a length of 1/50 of the rod length, each seg- ment has its own uniform current distribution. It is shown that the error in the rod-surface voltage at the earth surface is 5% for the one segment rod, and 4% for the two segments rod. The error at the lower tip of the rod reaches 32% but this rod tip is very distant from the earth surface. Therefore, the error does not have much effect on the voltage at the earth surface. It is mentioned in the last paragraph that this error decreases on the earth surface away from the rod say, 1%, the error in the grounding resistance of the rod. This error also decreases with the averaging influence of other rods, i.e. there could be more than one rod in the rodbed.

.20 3

> o,80-i 5- 0.60

0-1 , , , , I , 8 , , , , , , , , , , I , , I I I I , , I I I I I I I b I 1 I I I I I I I I I I I I I I

0 2 4 6 8 10 z -distance on rod from ground, m

Fig. 12 Three voltage distributions (one exact and two approximate) along a driven rod from the earth surface to the lower tip Solid line: exact, dotted line: one segment uniform distribution throughout, and circled line with circles: two segments (i.e. two uniform current distributions). The segments are marked along the rod

lines: solid

al U

._ E

% soi I

dash circles Fig.13 Key to Figs. I 2 and 14

Fig. 14 shows three different rod surface voltage dis- tributions for comparison, two are the same as in Fig. 12 the third curve with circles is for three seg- ments rod as: 44/50, 5/50, and 1/50 of the rod length. As shown in the figure this three-segments configura- tion makes the error in earth-surface potential at the rod to drop to 2%. The figure also shows that the rod- surface voltage error near the lower rod tip is also

658

reduced. With more segments the grounding resistance error also decreases to less than 2%.

1 2

1 0 I

I 0 8

>

5 0 6

0 4

0 2

>

~

0 , , , , , , , , , ~ , , , , , , , , , , , , , , I , ~ , , , , I , , , , , I , , , , , , , , 1 , , 1 I

0 2 4 6 8 10 z-dis tance on rod from ground, m

Fig. 14 Three voltage distributions (one exact and two approximate) along a driven rod from the earth surface to the lower tip Solid line: exact, dotted line: one segment uniform distribution throughout, and circled line with circles: three segments (i.e. three uniform current distribu- tions). The segments are marked along the rod

The above discussion shows that even with a coarse segmentation of one segment per rod the surface volt- age error is less than 5%. If the rod is sunken below the earth surface, there will be an upper rod tip near the earth surface and facing the corresponding tip of the image. This may increase the error of the earth surface voltage. However, as shown in Figs. 12-14 this error is easily corrected by adding one or two segments at the upper rod tip.

It may be noted that such a desirable error property with coarse segmentation does not occur for horizontal rods (e.g. grid rods) because both rod tips are near the earth surface.

9 Conclusion

The use of Galerkin's moment method to calculate the ground resistance, results in a substantial saving in computation effort for evaluating of the matrix ele- ments for both rodbeds and a single rod. In fact, for the single rod the matrix is reduced to one element, i.e. a simple formula.

This method is based on the variational Galerkin's formulation in deriving the analytical formulas of the self and mutual resistances. The derived formulas are function of rod length, rod diameter, distance between rods and the soil resistivity. Because of the variational property of error reduction even with an error in the assumption of uniform current distribution, the number of segments used to represent each rod can be 1 reduced to one segment without introducing greater , error (say less than 2%) in an N rod rodbed. The ' present technique is proven to be very accurate and efficient when it is compared with the published meth- ods.

The present results give rise to the following conclu- 1 sions: (i) The resistance of the rodbed slightly decreases with ' the sunken depth from the ground surface, however, this amount of resistance reduction decreases for large spacing between adjacent rods and small number of rods. (ii) The relationship between the resistance of a single rod is approximately inversely proportional to the rod

!

IEE Proc -Gener Transm Distrcb , Vol 142, No 6, November 1995

length if the rod is thin. The approximation deterio- rates for a large diameter rod or for rodbed. (iii) High precision in placing the rods in site (con- structing) is not important if the total length of perime- ter of the rodbed is identical. (iv) The central rod does not reduce the resistance of a crowded rodbed by much if the average distance between the adjacent rods in the perimeter, is equal or less than the length of the rod. Using coarse segmenta- tion, one segment per rod, results in a low error for earth surface potential calculations for the case of non- sunken rod (error <5%). For sunken rods one needs three segments at the most per rod to achieve such high accuracy.

8 Acknowledgments

This work is supported by a strategic grant 134197 of the National Science and Engineering Research Coun- cil of Canada.

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References

NAHMAN, J., and SKULETICH, S.: ‘Irregularity correction factors for mesh and step voltages grounding grids’, ZEEE Trans., 1980, PAS-99, (I), pp. 174-180 Substation Committee Working Group 78.1: ‘Safe substation grounding - Part 11’, ZEEE Trans., 1982, PAS-101, pp. 40064023 ‘IEEE guide for safety in AC substation grounding’, ANSUIEEE Std. 8011986. DAWALIBI, F., and MUKHEDAR, D.: ‘Influence of grounding rods on grounding grids’, ZEEE Trans., 1979, PAS-98, (6), pp. 2089-2097 TAGG, G.F.: ‘Multiple-driven-rod earth connections’, ZEE Proc. C, 1980, 127, (4), pp. 240-247

B, R.P., and MELIOPOULOS, A.P.: ‘Graphical data for ground grid analysis’, ZEEE Trans., 1983, PAS-102, (9), pp. 3038-3048 JOY, E.B., MELIOPOULOS, A.P., WEBB, R.P.: ‘User’s manual for analysis techniques for power substation grounding systems’, Manual for Research Report on Electric Power Research Insti- tute, Project 1494-2, p. 104, 1983 JOY, E.B., and WILSON, R.E.: ‘Accuracy study of the ground grid analysis algorithm’, ZEEE Trans., 1986, PWRD-1, (3), pp. 97-1 03

JOY, E.B., PAIK, N., BREWER, T.E., WILSON, R.E., WEB-

HARRINGTON, R.F.: ‘Field computation by moment methods’ (The Macmillan Company NY, 1968)

IO COLLIN, R.E.: ‘Field theory of guided waves’ (IEE Press, 1991) 11 CHOW. Y.L., SRIVASTAVA, K.D.: ‘Non-uniform electric field induced voltage calculations’, Canadian Electrical Association Report 117 T 317, 1988

10 Appendix

IO. I Harmonic mean distance between two parallel current lines Each resistance element in the matrix of Galerkin’s moment method is given by (p/4zdefl) where de# is the harmonic mean distance between current lines. Fig. 15 shows a magnified drawing of two parallel segments IZ and m. The two segments are a part of two parallel conductors. The effective distance, deff between the two segments can derived in terms of the co-ordinates of the segments and their lengths, i.e. zml, zm2, znl, zn2, AIm, and AIn as shown in Fig. 15. The reciprocal of deg can be calculated as follows [ll]:

where r = 4 ( z m - z,)2 + d2 (23)

Substitution of eqn. 23 into eqn. 22 and integrate with respect to z, to get

(24) The indefinite integration of any one term in the inte- grand gives

~ ( I I : ) = / sinh-’ 2 dx = II: sinh-I II: - d g (25)

With two terms in the integrand and two limits of the integration, one gets for eqn. 24

I

[ f (%$) - f (!?j!L) - f (7) + f ( 3 3 1 I

It is shown by R. Collin [lo] that eqn. 22 is a varia- tional functional. Such a functional reduces the first order error in the argument (i.e. the integrand) to a sec- ond order error in the value of the functional. The first

i vu vu

4 Fig. 15 Two parallel line current segments

Fig. 16 Two parallel line current of same height and co-ordinates

order error in the integrand comes from the fact, that we have assumed that the current density is uniform and not increasing at the ends of the line. Despite this first order error, the second order error in lldCflis very small. For a line current the error is only 1%. It is also shown by Collin [lo] that the functional D1 of eqn. 22 is a minimum. This means that (being proportional to

659 IEE Proc.-Gener. Transm. Distrib., Vol. 142, No. 6, November 1995

0.’) the approximate grounding resistance obtained from Galerkin’s method must be an upper bound. This further implies that any published value that is higher than the resistance obtained by Galerkin’s method, must have more error than the later one. This assump- tion requires that no approximation (other than the original assumption of uniform current distribution in a segment) is used in Galerkin’s derivation. This is the case in this paper.

660

10.2 Harmonic mean distance between two current lines of the same height For the same height, shown in Fig. 16, the limits of integration for eqn. 22 become bl = b, = 1 and ul = u2 = 0. Substituting these limits into eqn. 26 gives

r 1

IEE Proc.-Gener. Transm. Distrib., Vol. 142, No. 6, November 1995