Electric Power Systems Research, 7 (1984) 29 - 37 29

Calculation of Grounding Resistance

C. X. CHEN*

Department of Electrical Engineering, The Ohio State University, Columbus, Ohio 43210 (U.S.A.)

(Received 15 July 1983)

SUMMARY

This paper introduces an algorithm for the calculation o f grounding resistance based on the boundary element method (or current simulation method). A series o f computer results for different kinds o f grounding systems is presented. On the basis o f the computer results, an evaluation is made o f commonly used formulas for grounding resistance and an expression is given whereby tower grounding systems can be designed.

~ - d llll/lll~l[l/l/lll

Fig. 1. Vertical rod.

I / l l / ~ l l l l / l / l l l l l l l l l / " ¢.-

_.t.- " I

INTRODUCTION

Computation of grounding resistance is essentially a problem of solving an unvarying current field. Like all other field problems, it is quite difficult to obtain an accurate analytical solution, because of the boundary conditions. As a consequence, the derivation of formulas for the calculation of the ground- ing resistance is usually accompanied by some approximations. Using the approximation that the leakage current density is the same at all points along the electrode (uniform leakage current method), one Obtains [1]

p 41 R - In - - (1)

2~rl d

for a single vertical ground rod (see Fig. 1), and [2]

p 12 R = In - - (2)

21rl dh

*This work was completed while the author was a visiting scholar at Rensselaer Polytechnic Institute. Permanent address: Department of Electrical Power Engineering, Wuhan Institute of Hydraulic and Electric Engineering, Wuhan, Hubei, People's Republic of China.

Fig. 2. Horizontal rod.

for a buried horizontal rod (see Fig. 2). p is the soil resistivity, l the length of the rod, d the diameter of the rod, and h the depth of burial.

According to the average potential method, resistances of different horizontal grounding systems can be calculated by the formulas listed in Table 1 [3]. Neglecting all terms containing 2h/l in eqn. (9) of that Table, we obtain another general formula for the horizontal buried rod:

P ( I 21 1) R = 7r'--[ n (dh)l/2

- P 2 ~ l ( l n l 2 ~ - 0 " 6 1 ) (3)

The expression for the vertical ground rod derived by the average potential method will be [1]

) = n -- 0 .31 (4) 21rl -d

For practical cases, the difference of the results between (1) and (4) is about 5% and

0378-7796/84/$3.00 © Elsevier Sequoia/Printed in The Netherlands

30

the d i f fe rence be tween (2) and (3) can be as high as 9%.

With regard to the bur ied r ing-shaped e lec t rode , two formulas are c o m m o n l y used. One given in ref . 3 is

16D2 P ( l 1 2 ) R - P l n - - - - - n + 0 . 4 8

21r2D dh 2rl -~ (5)

t he o t h e r given in ref. 4 is

R P ln4?rD2 P (1 12 ) - - - - - - - + 0 .24

21r2D dh 27rI n dh (6)

where D is the d i ame te r o f the ring, l its c i r cumfe rence and o the r variables have been as de f ined previous ly . The d i f fe rence be tween these t w o equa t ions is a b o u t 2%.

In ref . 5, Schwarz suggests using a c o m m o n equa t ion for the ca lcula t ion o f g rounding resistance:

) = n 1 + N (7) lrL (dh) 1/2

where L is the to ta l length of the e lec t rodes in the grounding system. The addi t ional t e rm, N - - 1 , in the b racke t is --1 fo r a ho r i zon ta l rod accord ing to eqn. (3), - - 0 . 4 5 for a simple circular loop or ring [6] . Schwarz , however , did n o t give the value o f N - - 1 fo r the po in t star a r r angement which is c o m m o n l y used in t o w er grounding systems.

Refe rences 2 and 7 give a n o t h e r c o m m o n equa t ion for the calcula t ion o f grounding resistance. I t is

= n - - + A (8) 2rL dh

Values o f A are shown in Table 2. By rearranging eqns. (7) and (8), we dis-

cover tha t these tw o equa t ions give the same

TABLE 1

Formulas for different horizontal grounding systems [3 ]

b- A-~

R= ~ E~ + 1.0,1 0.20, ( ~ , 0.238 ( ~ 2 0.0~4 ( ~ . . . . ~ ,11~

2h '~ .=~,~,2912 , 0 7 1 ( ~ , 0 6 4 ~ 9 2 0 1 4 ~ T . . . . ~ (12~

2h ~ , R = I "~ . ['~'~h ~ + 6.851 - 3.128 (~-}+ 1.758 (~}2_ 0.480 (T) . . . . ] (13)

2h ~ R = ol-~x~. [ ~ ~ + i0.98 - 5.51 (~-) + 3.26 ~- 2_ 1.17 (~--) . . . . ] (14)

r 2~.-H

TABLE 2

Value of A in eqn. (8) [2, 7]

SHAPE OF

GROUNDING

SYSTEM

VALUE A 0.36 0.48

©

0.87

÷

1.169

D

8.81

result for the case of the simple circular loop only; for straight horizontal rods and square loops the results are quite different.

In this paper the accuracy of eqns. (1) - (8) is evaluated using a numerical method; a com-

31

puter program is developed for this purpose. A common expression is given for designing grounding systems under towers. The applica- tion to grounding grids of substations will be the subject of a subsequent paper.

COMPUTERPROGRAM

The program presented in this paper is based on the boundary element method [8]. The electrodes of the grounding system under consideration are conceptually divided into n straight elements. The leakage current density for each element is assumed to be constant but can vary from element to element. The voltage of each element is assumed to coincide with the value at the mid-point of the element. The natural division of the grounding system can be chosen as the elements. Then the computer program can subdivide the elements according to the accuracy required. For example, an eight-star system has eight elements by natural division (Fig. 3(a)). This will be increased to 16 after the first subdivision (Fig. 3(b)) and to 32 after the second subdivision (Fig. 3(c)).

The basic equations for this program are [6] :

~ R J j = V i = 1, 2 . . . . . n (15) i=1

Rg = V (16)

where Ij is the leakage current of element j, V the potential of the grounding system, Rg the grounding resistance, and n the total number of elements in the grid.

Ri~ , the mutual resistance between i and j, and R , , the self resistance of element i, are calcu- lated by the following equations:

Rij p tfftJ 2 dl - +

4~rL i . [(x --Xim) 2 + (y --Yim) 2 + (Z --Zirn)2] I/2

P)~ dl t P'jfl [(X--Xim) 2+ (y - -Yim) 2+ (Z +Zim) 2]1/2

Ri i -

p IfJ 2 dl

4 f f L i tPjl [(x --Xim) 2 + (y --yim) 2 + ( Z - Zim) 2 + ro 2] 1/2

+ f - dl P'jl [(X - -X im) 2 + (y - - y i m ) 2 + (Z + Zim)2] 1/2

+

(17)

(18)

(a) (b) (c)

Fig. 3. Division of an eight-point star system: (a) natural division; (b) first division; (c) second division.

32

I I r

L : !L . . . . . P~2_~ % (×i1" Yil" :iI )

" , I /~" P~2 (xi2' Yi2' " zi2) ", I / " Pj2

"',,71 Pil (Xjl' YjI' - Zjl) "v P'jl Pj2 (xj2' Yj2' " zj2)

/ l l l / l l l l l l l l / l l l l / I / ~ t x

P il (Xil' Yi l ' Zil) Pil Pim Pi2 Pi2 (xi2' Yi2' zi2) p , J

i F ~ - x ' i ' j ~ / ~ Pim (Xim' Yim' Zim) Y I ~ P j l PJ2[ Pjl (Xjl' YjI' Zjl)

Pj2 (xj2' Yj2' zj2)

Fig. 4. Diagram for the calculat ion o f mutua l and self resistances.

where r0 is the radius of the electrode, and Li and Lj axe the lengths of the individual elements . All o ther quant i t ies axe explained in Fig. 4. Integrat ing (17) and (18) gives:

- P [ln 2L2 + L l l + 2L(L2 + L l l + L212) 1/2 + In Ru

4 x L [ L l l + 2LL21

2L2L12 + 2L(L 2 + Ll2 + L222) 1/2 ]

J L12 + 2 L L 2 2 (19)

where

L = [(xi2 -- xil) 2 + (Yj2 - - Y/I) 2 + ( Z j 2 - - Zjl) 2] 1/2

L l l = 2[(xj2 -- xi l ) (x j l - - x i~ ) + (Y~2 - - Yil)(Yjl -- Yi~) + (z12 -- zjl)(zll -- z/m)]

L21 = [(Xjl - - Xim) 2 + (Y11 - - Yim) 2 + ( Z j l - - Z/m) 2] 1/2

L l z = 2 [ ( x j 2 - - x j l ) ( X i l - - Xlm)- t - (Y/2 - - ~ j l ) ( Y / 2 - - Yim) -t- ( z j2 - - Z j l ) (Z j l ..I- Z / m ) ]

L z 2 = [ ( x i l - - Xim) 2 + (~./1 - - Yim) 2 -t- (Zjl -I- Z / m ) 2] 1/2

Xi m = (Xil "F X i 2 ) / 2

Y i m = (Yil + y i 2 ) / 2

Zim = (Zil q- Z i 2 ) / 2

Similarly,

- P [ln L + (L + 4ro2) 1/2 + In Rii

47rL [ - -L + (L + 4ro2) 1/2

2L 2 + L12 + 2 L ( L 2 + L,2 + L222) 1/2]

] L I 2 + 2 L L 2 2

where

L = [ ( x i 2 - - Xil) 2 + (Yi2 - - Yil) 2 + (zi2 - - g i l ) 2] 1/2

L 12 = 2 [ ( X i 2 - - X i l ) ( X i l - - X i m ) + (Yi2 - - Yil)(Yil - - Y/m) + ( Z i 2 - - Z i l ) ( Z i l + Z/m)]

L : : = [ ( x . - - xim) 2 + ( y . -- yim) 2 + (zil + zim) 2] ~/2

TABLE 3

Comparison with results of ref. 7

33

n x n Resistance (g)

R' R = - R

Potent ia l at the Center o f Corner Nesh (%)

U o 0 o '

54.215 54.306

72.676 72.240

79.805 79.326

gO' . U 0

2 x 2 64.249 64,160 0.001 -0.0017

4 x 4 20.472 20.600 -0.006 0.006

8 x 8 8,37 8.42 -0,006 0,006

The leakage currents are obtained by solving simultaneous equations (15). The grounding resistance can then be calculated from eqn. (16).

In symmetrical situations several elements may have the same current value, in which case the number of simultaneous equations in (15) can be reduced. A subroutine for reducing the number of simultaneous equa- tions in symmetrical cases is included in the program. Thus, for the eight-point star system shown in Fig. 3, there is only one equation when natural division is used. This increases to only 32 after the fifth division, even though the total number of elements is 256.

The accuracy of this program has been checked by comparing the results with those for 2 × 2, 4 X 4 and 8 × 8 grids of a previous paper [7] using the following parameters: Soil resistivity p 1000 ~2 m Radius of electrode r0 0.007 m Depth of burial h 0.5 m Length between two grid lines 8 m

The results are shown in Table 3, where R and U0 are the results of this program and

R ' and U0' axe taken from ref. 7. The errors are insignificant.

RESULTS AND DISCUSSION

In all the results given below a value of 100 ~2 m was assumed for the earth resistivity. Five subdivisions were made. This number of subdivisions was determined by comparing results for different numbers of subdivisions, shown in Table 4 for the eight-point star. We can see from Table 4 that the difference between the resistances obtained for the fourth and fifth subdivisions for an eight- point star is only 0.2%, even when its arms are 100 m long.

Buried horizontal rod Table 5 shows the computer results for the

buried horizontal rod and lists the results from eqns. (2) and (3) for comparison. The error of eqn. (2), which assumes uniform leakage currents, is in the range 4.39% - 7.56%. This is quite understandable from minimum

TABLE 4

Resistance ( ~ ) of the eight-point star for different subdivisions

. ~ d=O,O2M

~ 2 ~ h=O.SM

= 5 M

= 10 M

= I00 M

Number of Subdivisions

0 1 2 3 4 5

6,977 6.711 6.618 6. 577 6.557 6.546

3.856 3.726 3,672 3,65U 3.639 3.634

0.518 0.531 0.477 0.472 0.471 0.470

6

6. 544

3.631

34

TABLE 5 Resistance (~) of a buried horizontal rod

I--2~--I

d=O.O2M

h:O.8M

1.172

Computer Result

R

Equation (3)

R Er ro r (%)

Z : 2.5 M 22.253 21.454 - 3.59 23.409

= 5 M 13.082 12.933 - 1.13 13.911

£ = i 0 M 7.578 7.570 - 0. I I 8.059

Z = 50 M 2.018 2.026 0,40 2.124

= i00 M 1.120 1,123 0.27

Equation (2)

R Error (%)

5.19

7.56

6.45

4.84

4.39

energy considerations. The charge distribution on the conductor should result in the smallest energy storage (Q2/C) for the system. The assumption of a uniform charge distribution (or current distribution in an unvarying current field) instead of a nonuniform one will cause a decrease in capacitance (or an increase in resistance).

The results from eqn. (3) are closer to the computer results. The increasing error with decrease of I probably results from the decision

to neglect all 2h/l terms in the derivation of eqn. (3).

Vertical grounding rod The computer results and the results from

analytical equations (1) and (4) for the vertical grounding rod are given in Table 6. These results again show the coincidence of the average potential method (eqn. (4)) with the numerical method as performed on the com-

TABLE 6

Resistance (~) of a vertical grounding rod

d=O.O2M Computer Result

R

Equation (4} Equation ( i }

R Error (%) R Error (%)

: 2.5 M 37.447 37.610 0.43 39.563 5.19

Z : 5 M 21.920 21.011 0.43 21.981 4.62

Z = 10 M 11.588 11.609 0.18 12.097 4.20

TABLE 7

Resistance (~) of a ring-shaped electrode

Equation (6) Equation (5) ( ~ d:O.O2M

h:O.8M

Computer Re sul t

R R Er ro r (%) R Er ror (%)

D= 5M 10.236 10.014 - 2.17 10.257 0.2

D: IOM 5.830 5.709 - 2,08 5.831 - 0,2

D: 50M 1.494 1.468 - 1.74 1.492 - 0.13

D=IOOM 0.817 0.804 - 1.59 0.816 - 0.12

puter and the error of the uniform leakage current method (eqn. (1)).

Buried ring-shaped electrode A sixty-four-side polygon is used to approxi-

mate a circle for the numerical calculation. The results are shown in Table 7. Comparison of these results with those given by eqns. (5) and (6) reveals that eqn. (5) is the more accurate.

Different star connected electrodes Tables 8 and 9 show the results for the

four-point star system and eight-point star system. On comparing the computer results with the results obtained from the average potential method, i.e., eqns. (12) and (14), we can see that the error caused by the average potential method increases as the number of points of the star increases. Hence, it can be concluded that, although the result from the average potential method is close to the actual value for the single grounding rod,

35

this method should not be extended indis- criminately to all cases, especially for those having a relatively large number of electrodes such as the multiple-point star system, ground- ing grid or plate.

The computer results for various point star systems with different arm lengths, for circular loops with different diameters and for square loops with different perimeters for cases of different electrode radius and depth of burial are shown in Table 10. A common expression can be developed based on these results. For a constant total length of the electrode, the straight electrode provides the lowest resistance, and eqn. (3) gives quite accurate results for the resistance of a straight horizontal rod. It is reasonable, therefore, to establish the common equation on the basis of eqn. (3) by adding to it a shielding coeffi- cient K, i.e.,

= n - - -- 0.6 + K (20) 2~rL dh

TABLE 8 Resistance ( ~ ) of a four-point star

d=O.O2M

h:O.SM

Computer Result

R

Equation (12)

~--2 {-~ R Error (%)

: 2.5 M 15.041 15.680 4.25

: 5 M 8.702 9.052 4.02

= 10 M 4.934 5.126 3.90

= 50 M 1.254 1.291 2.96

= 100 M 0.684 0.701 2.53

TABLE 9 Resistance ( ~ ) of an eight-point star

dfO.O2M

h=O.SM F-2~H

Z : 5 M

x= IOM

Computer Re su]t

R

6.546

3.631

7.728

4.040

Equation (14)

Error (%)

11.18

12.26

= 50 M 0.877 0.961 9.58

= 100 M 0.470 0.510 8.47

36

TABLE 10

Resistance of different shapes of grounding systems

SHAPE OF GROUNDING SYSTEM

~(M) ~(M) d(M}

2.5 3.8 0.02

).8 0.02

).6 0.02 5

1.6 0.01

).4 0.02

).8 0.02

).6 0.02 10

).6 O.Ol

).4 0.02

].8 0.02

).6 0.02 50

i0.6 0.01

0.4 0.02

i00 ).8 0.02

RC(Q) ,RA(O) 5(%) RC(Q) RA(Q) 6(%) RC(Q) RA(Q) 6 (%) RC(Q) RA(Q) 5(%)

22.253 21.498 -3.39 23.030 22.836 8.84 17.421 17.327 -0.54

13.082 12.956 -0.97 13.539 13.624 0.62 10.146 10.134 -0.12 10.236 10.257 -0.2

13.469 13.414 -0.41 13.948 14.082 0.96 I0.436 10.440 0.34 10.540 10.548 -0.07

14.584 14.517-0.46 15.066 15.185 0.79 11.197 11.175 -0.20 11.237 11.251 -0.12

14.039 14.059 0.14 14.541 14.727 1.28 10.850 10.869 0.18 10.962 10.960 0.02

7.578 7.581 0.04 7.831 7.91611.08 5.807 5.802 -0.07 5.830 5.831 -0.02

7.787 7.810 0.30 8.047 8.144 1.21 5.956 5.955 -0.02 5.978 5.977 0.02

8.346 8.361 0.19 8.607 8.695 1.03 6.335 6.323 -0.19 6,329 6.328 0.02

8.090 8.133 0.53 8.356 8.467 1.39 6.167 6.170 0,05 6.186 6.182 0.06

2,029 0.52 I 2,074 2.095 1,03 1.506 1.502 -0.27 1.494 1.492 0.13 2.018

2.064 2.074 0.50 2.118 2.141 1.09 1.537 1.533 -0.29 1.523 1.521 0.13

2.173 2.185 0.53 2.230 2.251 0.96 1.611 1.606 -0.31 1.593 1.592 0.06

2.127 2.139 0.56 2.183 2.206 1.04 1,581 1.576 -0.34 1.564 1.562 0.13

1.120 1.124 0.36 1.148 1.158 0.87 0.828 0,825 -0.41 0.817 0.816 0.12

SHAPE OF _ _ ~ F ~ /

A SYSTEM ..g.~ _~ ~-JZ ~ ~'- ~#. -

~(M) h(M) d(M) RC(O) RA(~) 5(%) RC(~) RA(C~) 5(%) RC(~) RA(~) 5(%) RC(•) RA(Q) 5(%) .I

2.5 0.8 0.02 15.041 15.327 1.89 15.208 15.5021 1.93

0.8 0.02 8.702 8.767 0.75 8.789 8.854 0.75 7.267 7.410 1.97 6.546 6.829 4.32

0.6 0.02 8.943 8.995 0.59 9.045 9.083 0.42 7.459 7.563 1.38 6.716 6.944 3.38 5

0.6 0.01 9.532 9.547 0 .16 9.597 9.634 0.39 7.878 7.930 0.66 7.049 7.219 2.41

0.4 0 .02 9.278 9.319 0 .47 9.399 9.406 0.08 7.712 7.777 0.85 6.930 7.105 2.52

0.8 0.02 4.934 4.9351 0.02 4.976 4.979 0.06 4.068 4.073 0.12 3.631 3.690 1.63

D.6 0 .02 5.052 5.050 -0.05 5.099 5.093 -0.11 4.155 4.149 -0.14 3.703 3.748 1.20 10

0.6 0 .01 5.345 5.325 -0.37 5.375 5.369 -0.11 4.362 4.333 -0.66 3.870 3.885 0.40

D.4 0 .02 5.218 5.211 -0.14 5.269 5.255 -0.28 4.271 4.257 -0.32 3.799 3.828 0.77

D.8 0.02 1.254 1.234 -0.86 1.257 1.252 -0.40 1.003 0.985 - I .76 0.877 0.866 -1.24

D.6 0.02 1.277 1.272 -0.41 1.280 1.275 -0.41' 1.018 1.001 -1.71 0.889 0.875 -1.28

50 0.6 0.011 1,334 1.321 -0.95 1.335 1.330 -0.38 1.058 1.037 -1.94 0.920 0.905 -1.61 H

D.4 0.02| 1.311 1.298 -0.97 1.313 1.307 -0.45 1.041 1.022 -I.81 0.907 0.894 -1.47

1 ~ 10.8 0.021 | 0.684 0.677 -1.06 0.684 0.681 -0.44 0.541 0.529 -2.14 0.470 0.461 -1.99

where R C is the computer result. R A is the result of equation (20) and 5 is the error between RC and R A.

where L is t he to t a l length o f t he e lec t rodes . Obv ious ly K has a value o f zero fo r the s t ra ight rod and a g rea te r value for all o t h e r cases because o f t he shielding ef fec t .

Values fo r the coe f f i c i en t K der ived f r o m the c o m p u t e r resul ts are s u m m a r i z e d in Tab le 11. St r ic t ly speaking , K is a f f ec t ed by the length o f the e l ec t rode as well as the

shape o f the g round ing sys t em. The longer the e l ec t rode , the s t ronger the shielding e f f ec t will be. Hence a range o f values has been assigned to the coe f f i c i en t K. The K values in Tab le 11 can be used fo r sy s t ems having a rm lengths o f 2.5 - 100 m, wi th the cons t r a in t t h a t a rm length be l imi ted to 5 - 100 m fo r the s ix -po in t s tar and 10 - 100 m for the eight-

TABLE 11

Coefficient K (eqn. (20)) and A (eqn. (8))

37

SHAPE OF

K 0 0.42 0.60 1.08 1.49 1.60 3.63 6.25

A (ffiK-0.6) -U.60 -0.18 0 0.48 0.89 1 3.03 5.65

point star. This range is sufficient for practical purposes.

If the last two terms in the bracket of eqn. (20) are combined and designated by A the common equation will have the same form as eqn. (8). However, the value A should be taken from Table 11 and not from Table 3.

Table 11 shows that the shielding effect for a square loop is greater than that for a four- point star and the shielding effect for a circular loop is greater than that for a three-point star. Table 2 leads to the opposite conclusion. The error arises as a consequence of using the average potential method.

From Table 10 we can see that the resis- tances based on coefficient K (orA) in Table 11 are very close to the computer results; the error is within +2%.

(6) A common expression for designing tower grounding systems can be adopted from eqn. (8), but the value A should be taken from Table 11.

(7) The errors between the results obtained by the expression suggested in this paper and the computer results are within +2%.

ACKNOWLEDGEMENTS

The author wishes to thank Professor Allan Greenwood and Professor Keith Nelson at Rensselaer Polytechnic Institute for their support in this work. Thanks are also due to Professor Donald Kasten at The Ohio State University for profitable suggestions.

CONCLUSIONS

(1) The resistance determined by the uniform leakage current method is on the high side.

(2) For a single rod the resistance given by the average potential method is close to the computed result. This is not true for ground- ing systems having a relatively large number of electrodes such as multiple-point stars, grounding grids and plates.

(3) There are almost no errors between the computer results and the results from eqn. (5) derived by the average potential method for the circular loop, because of the symmetriza- tion of the circle.

(4) The shielding effect for a square loop is greater than that for a four-point star.

(5) The shielding effect for a circular loop is greater than that for a three-point star.

REFERENCES

1 E. D. Sunde, Earth Conduction Effects in Trans- mission Systems, Dover, New York, 1968.

2 A. B. Oslon, Calculation of different complex grounding systems, Elektrichestvo, (4) (1968) 58 - 61.

3 H. B. Dwight, Calculation of resistances to ground, AIEE Trans., 55 (1936) 1319 - 1328.

4 Xie Kuangrun, High Voltage Static Electrical Field, Shanghai, China, 1962 (in Chinese).

5 S. J. Schwarz, Analytical expressions for the resistance of grounding systems, AIEE Trans., 73 (1954) 1011 - 1016.

6 E. T. B. Gross, B. V. Chitnis and L. J. Stration, Grounding grid for high-voltage stations, AIEE Trans., 72 (1953) 709 - 810.

7 R. J. Heppe, Computation of potential at surface above an energized grid or other electrode, allow- ing for non-uniform current distribution, IEEE Trans., PAS-98 (1979) 1978 - 1985.

8 C. A. Bribhia, The Boundary Element Method for Engineers, New York, 1978.