correction of resistance measurement of large grounding systems

ETEP

Correction of Resistance Measurement of Large Grounding Systems S. BerberoviC, V. Boras

Abstract

A method f o r the systematic error correction which , vpears at the grounding system resistance measuremen by means of an auxiliary current electrode, when the distance between the edge of the grounding system and the center of the current electrode is shorter than the minimum distance required by the standard, is presented in thispape,: The systematic error increases with the ratio of the size of the grounding system to its distance from the current electrode. The method f o r the correction of the systematic error is based on modelling of grounding system and current electrode by hemispheres in uniform and two-layer soil model. The calculation results are presented by diagrams, which are useful f o r the evaluarion of systemaric error.

1 Introduction

A basic method, which is universally used for the measuring station ground resistance, is generally known as the “Fall-of-Potential” method and is described in [2]. For measuring resistance, the current source is connected between the station ground mat and the current electrode located at a great distance. The potential-measuring circuit is then connected between the station mat and the potential electrode, with measurements made at various locations of the electrode outside the station. The potential electrode is moved away from the grounding system under test in steps. In order to obtain a flat por- tion of the curve, where the measured resistance means the grounding system resistance, it is necessary for the current electrode to be effectively outside the “influence” of the grounding system to be tested (Fig. 1). This influence depends on the grounding system size and may be considered as the distance beyond which there is a negligible effect on the measured rise of ground voltage caused by ground current. Theoretically the influence extends to infinity, but practically the influence is limit- ed because it decreases with increasing of the distance from the tested grounding.

3.0 n 2.5 Potential electrode between the

test and current electrode

160 240 320 m 400

Fig. 1. Example of the ground impedance curves obtained by means of “Fall-of-Potential” method

d -

The IEEE Standard 8 1 - 1983 121 assumes that point electrodes are placed at a great distance, but it does not specify the error resulting from applying this method when relatively short distances are involved and which is caused by the mutual influence of the auxiliary electrode and the tested grounding system.

According to some standards in Tab. 1, the shortest distances between the current electrode and the tested grounding system are given, as well as the values of the systematic error, which should not be exceeded.

In this paper, systematic errors are analyzed, which occur by measuring at a much shorter distance than the one prescribed by the standards. According to the ANSMEEE Standard 8 1 - 1983 [2] and the resuits of the investigations of many researchers [5 , 6,9, 101, the as- sumption, that the current electrode and tested grounding system have a hemispherical shape, is made.

The mutual influence of close hemispherical ground electrodes can be taken into consideration by means of applying the images method. The current field of such ground electrodes can be solved as the field of the equivalent point current sources group. The apparent grounding resistance of the hemispherical ground electrodes, in which the mutual influence between the tested grounding system and- the current electrode is incorporated, is determined according to the ground electrode potential calculation. This resistance is compared with the theoretical grounding resistance, which is obtained when cur-

Standard Distance Systematic error

IEEE 8 1 - 1983 Large enough - IEEE 8 1 - Part I1 6.5 LD’ < 5 % NORMA CEI 1 1-8 5 LD‘ < 9 % CENELEC <5km** -

Tab. 1. Suggested shortest distances between the current electrode and the tested grounding system ( largest dimen- sion (LD) of the tested grounding system; ** distance regard- less of the tested grounding system dimensions)

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E TEP rent electrode is at an infinite distance, and afterwards the systematic error of the measurement is determined. From the numerical calculations in uniform and two-layer soil for various values of the hemispherical ground electrodes radii, and at distances much shorter than the minimum distances required by the standards, the obtained systematic errors are shown in the diagrams.

Thus determined systematic errors are compared with the results of the investigations of other authors, who have analyzed similar problems [5 ,6,9, 101. Using thus determined systematic errors, the measurement of the grounding system resistance can also be made for much shorter distances between the current and the tested grounding system than the one prescribed by the standards, and with the correction of the measured apparent grounding resistance.

2 Grounding Resistance

2.1 Hemispherical Ground Electrode

of a Hemispherical Ground Electrode

in Uniform Soil

We consider a hemispherical ground electrode of radius r, with the flat side level with the surface of uniform soil with resistivity pI as shown in Fig. 2, and assume that the current I flows through this electrode, spreading out radially in the soil. The streamlines are then exten- sions of the radii of the electrode and the equipotential surfaces are concentric half-spheres. On the air-ground boundary the condition is valid that the current I does not pass from the conductive ground half space into the air-filled non-conductive half space.

Fulfilment of this boundary condition is achieved by applying the method of images [S]. According to this method, the hemispherical ground electrode and its image form spherical ground electrode of radius ro, through which the current 21 enters and leaves fictive space filled with uniform soil with resistivity pI which is unlimited in all directions.

The theoretical grounding resistance of a hemispherical ground electrode with radius ro is given in ab- solute value by:

PI 2 rox

RT= -.

+ Fig. 2. Hemispherical ground electrode in uniform soil

1 d0

V" / u A B

Fig. 3. Mutual reflection of two various radii conductive spheres with pA # q+, # 0

The surface soil potential distribution is expressed by the equation:

PI 2Tcx

rp(X) = -.

The method of images is used for the calculation of the current field of two conductive charged spheres, which are placed in an unlimited uniform space. Accord- ing to this method, the current field determines the group of the point current sources. The value and the position of the point current sources are determined by means of the recursive relations in [ 11.

Let us consider two conductive spheres of radii ro and rl placed do apart in the unlimited uniform space of resistivity p , , Fig. 3. Direct current is led to one of the spheres, whereas the same amount of the direct current is led from the other one. If spheres A and B are of the different radii, they will have the different potentials as well.

This problem can be solved by means of applying mutual reflections of the spheres, the results of which is an infinitive series of the reflected point current sources IAK and IBK (K = 0, 1,2, 3, . ..), placed at the respective distances bKA and bKB (K = 0, 1,2, ) from the center of the corresponding sphere.

The total currents flowing from the spheres A and B is obtained by addition of the currents of all reflected point current sources, as follows:

I A = IAO + I A I + I A 2 + l A 3 + * A * = - l B + (3)

ro do

UOA = 1, U I A =-, (7)

and where other factors aKA and am are obtained recur- sively.

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----. I , ,

Air ‘A0 ‘A I ‘A2IA3 ‘,

Fig. 4. Hemispherical ground electrodes of the various radii and with the indicated locations of the point current sources A I . . . A3 and B 1 . . . B3, which result from mutual reflection

Similarly, the current-field calculation of two hern- ispherical ground electrodes placed at the soil surface, as shown in Fig. 4, can be solved by means of superposition of two groups of point current sources obtained by the mutual reflection of two charged spheres placed in the uniform soil with the resistivity pI, and taking only one half of the current value, leading to the individual sphere.

The total potential at any point of the soil surface, which is formed by two current fields of two hemispherical ground electrodes A and B, Fig. 4, can be calculated as the sum of the potentials formed by two groups of reflected point current sources with values (IAo, I , , , ..., IA5 , ...) and (Iso, I , , , . ... I B S , ...) by means of the following equation:

( P = ( P A + % ,

where: (9)

So, respectively:

(PA = (PA0 + (PA I + (PA2 + * . * + (PA5 + .-.* and

(Pe= A(- IBO + IEJJ

( 1 1)

2X d o - x d 0 - b l g - x

+ I B 2 +...+ IB5 +...). (12) do - b2B - x

So, respectively:

do - bse - X

(PB = (PBO + ‘PSI (PB2 + .-. + (PSS + ... . (13) The first five factors uKA, urn, bKA and bm (K= 1,2,

..., 5) can be determined according to the equations given in Tab. 2 to Tab. 4.

If we insertx = ro into the eqs. (9), (10) and (12), we will have the potential yon the surface of the hemispherical ground electrode A.

The ratio between this potential and the total current n

= C a K A 9

K=O

alA= - mA

r a?A= -

m i - 1

‘i,m0 b2A= r- m,- 1 2

1 -7 1 m:-’tm,-r - m i + 1

mA-2m, - r m A + m A - r m A - m A - r mA -?( -? ) b5A= ‘i) 5 3 -2 3

Tab. 2. Equations of factors aKA and bKA (K = 1,2, . .., 5) for hemispherical electrode A

1 utg= -

m0

r a2,= - mA- I

-I

2

r, b lB= - m0

- 1 m A - m A - r 3 -2 mA b4B= r 5

m: - 2m: - r -2mi + 1

Tab. 3. Equations of factors am and 6 , ( K = 1,2, . . ., 5) for hemispherical electrode B

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%A = I

I UlB= -

mB 1 ~ I ~ K - I ) A ~ ~ ( z K - z ) B

r m~ ai z K - z IB - a( 2 K - I )A azm= - , K = l . 2 ...., 5 ....

KA 2 K - I )B a12K+I )B= , K = l , 2 ...., 5 ,...

ma K - I )B - 3 KA

b,= r, -%!- , K = 1 , 2 ...., 5 ,... 'I K-I )A

Tab. 4. General equations for factors aKA. om. b, and b, ( K = O , 1,2, _... 5 , ... )

which leaks from the hemispherical ground electrode A, gives the apparent resistance of this electrode, including the conductive influence of the other hemispherical ground electrode B:

R,=_. cp 1A

Comparing the apparent resistance of eq. (14) and theoretical resistance of eq. ( l ) , the equation of the relative systematic error a is obtained as follows:

RT - RA R T

a =

2.2 Hemispherical Ground Electrode in a Two-Layer Soil

If the hemispherical ground electrodes given in Fig. 2 and Fig. 4 are placed in the upper layer of the two- layer soil, then the boundary problems can be resolved by means of the method of images.

The point current source at the boundary between the layer and at the air-ground boundary is reflected in the infinite series of the point current sources.

The potential distribution at the points of the two- layer soil surface with the distance x from the point current source located in the upper soil layer, is given like this:

r 1

Let us assume that a hemispherical ground electrode is placed on the two-layer soil surface and that the upper

layer thickness is greater than the electrode radius. For determining the theoretical resistance of such a hemispherical ground electrode, i t can be taken that the same potential distribution gives point current source located in its center. Inserting x = ro and I = I. in eq. ( I6), we obtain the theoretical resistance of the hemispherical ground electrode in two-layer soil:

r 1

Similarly as in section 2.1, the current field of two hemispherical ground electrodes placed on the two- layer soil surface is solved using superposition of two point current sources groups obtained by the mutual reflection of the two spheres placed in uniform soil, with the difference that the potential distribution on the two- layer soil surface is in this case described by eq. ( 16) for each point current source. Using eqs. ( lo), (1 1) and (1 6) we determine first six potential terms of the first group of reflected point current sources of hemispherical ground electrode A. and they are given in Tab. 5. In the same way, that is using eqs. ( 12), (1 3) and ( 16), we determine first six potential terms of the second group of reflected point current sources of hemispherical ground electrode B, which are given in Tab. 6.

Inserting x = ro in the terms in Tab. 5 and 6, the total potential is calculated according to eqs. (9), ( 10) and ( I 3) as the sum of the potentials (qoA, qlA, .. ., q 5 A . . . .) and ( c p ~ ~ , VIB. ..., fp56, ...) ofthereflectedpointcurrentsourc- es. The ratio of this potential and of the total current

n

I,() a K A * K=O

r -3

r 1

I- 1

r 1

Tab. 5. First six potentid terms of the point current sources group of hemispherical grounding electrode A placed in two-layer soil

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p.u.

r 1

I I i I

rl Id, = 0.5 P.U. , I 0.4 p.u. 7

r 1

r 1

r 1

Tab. 6. First six potential terms of the point current sources group of hemispherical grounding electrode B placed in two- layer soil

leaking from the hemispherical ground electrode A, gives the apparent resistance of the electrode A. The conductive influence of the hemispherical ground electrode B is included in this resistance. The relative systematic error is obtained by means of eq. ( 15) including the values of the apparent and of the theoretical grounding resistance.

3 Analysis of Results and Correction of the Systematic Error

The systematic errors, which appear in the grounding system resistance measurement, are determined by means of the numerical calculation on the basis of the described model at various radii of the hemispherical ground electrodes. The results are shown in Fig. 5.

If we take the current electrode B as the point current source (curve “0.00001 p. u.” in Fig. 5 ) , we can see that the values of systematic measurement errors occur in the range of (0.8 . . . 0.9) ro/do. In the case of setting the point current electrode B at the distance 2.5 times greater than that of the diameter of the tested electrode A, the systematic error sums up to 20 %, as it is shown by the curve “O.OOO0 1 p. u.” in Fig. 5. If the distance of the point current electrode is equal to 5 diameters of the tested electrode A (according to the Italian Standard), the systematic error will approxi- mately be 8.8 %. The theoretic grounding resistance RT can be determined by means of the measured apparent resistance R, and of the already determined systematic error in the equation:

0.6

I 0.5

$ 0.4 0.3

0.2

0. I

0

A

h

v

0 0. I 0.2 0.3 0.4 p.u. 0.5 fo ld , --c

Fig. 5. Systematic errors a(roldo) of the apparent resistance of a hemisphere placed in a homogeneous soil, as a function of r l Id,

The numerical calculation, made according to section 2.2 of this paper, gives the values of the systematic error for various reflection factors and for various radii of the hemispherical ground electrodes A and B, which are placed in upper layer of the two-layer soil at the depth of h =do.

The diagrams in Fig. 6 to Fig. 10 give the calculated values of the systematic error. The figures show that the systematic error increases with the increase of the reflection factor k. In Fig. 11, two curves of resistance characteristic are given, existing between the grounding system of a large plant and the auxiliary electrode. Curve 2 in Fig. 1 1 represents the experimental curve obtained in [ 5 ] , where a large grounding system with the grounding area of 25 500 m2 and the radius of 92 m is equivalent to the hemispherical grounding electrode with a radius of ro = 46 m. The auxiliary electrode con- sists of seven rods fixed at 0.5 m into the soil. The distance between the center of the test electrode and the auxiliary electrode was, when measured, do = 259 m. Curve I in Fig. 11 represents the theoretical curve ob-

0.6

I 0.5 A -

E”

‘ 0.3

0.2

0. I

0

‘o 0.4 L -

0.4 p.u. 0.5 0 0.1 0.2 0.3 roldo -

Fig. 6. Systematic errors a(ro/do) of the apparent resistance of a hemisphere placed in two-layer soil and with k = 0.1. as a function of rl Id,

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I I I 0.3 D . u . ~

ro/do -* Fig. 7. Systematic errors a(ro/do) of the apparent resistance of a hemisphere placed in two-layer soil and with k = 0.2, as a function of r l Ido

p.u. 0.4 p.u. 1

0.2 p.u. A

. p 0.4

0.3

0.2

0.1

v

0 0. I 0.2 0.3 0.4 pa. 0.5

Fig. 8. Systematic errors a(ro/do) of the apparent resistance of a hemisphere placed in two-layer soil and with k = 0.667. as a function of r l Id0

ro/& -*

r l Ido = 0.5 P.U.

/A 0.1 p.u. 1

I I i

ro/do - Fig. 9. Systematic errors a(ro/do) of the apparent resistance of a hemisphere placed in two-layer soil and with k = 0.9, as a function of rl Ido

-71 - ---I.--- 7- -- I !

I i r l /do = 0.5 p.u. I

roldo --c

Fig. 10. Systematic errors a(ro/do) of the apparent resistance of a hemisphere placed in two-layer soil and with k = -0.6, as a function of r l /do I

ro = 0.178 P.U. -0.8

-1.2 r , = 0.002 p.u.

I

~ I I

0.2 0.4 0.6 0.8 p.u. 1.0 0 -2.0 '

do --t

Fig. 11. Resistance characteristic between the grounding system of a large plant and the auxiliary electrode (I theoretical curve, 2 experimental curve)

tained by means of the grounding resistance numerical calculation in a uniform soil on the basis of the equation given in section 2.1. Fig. 11 shows a good agreement of both curves.

4 Conclusions

In this paper, we have presented a method for the correction of the systematic error which results from measuring the grounding resistance in a uniform and two-layer soil by means of an auxiliary current electrode, when the distance between the edge of the tested grounding system and the center of the auxiliary electrode is shorter than the minimum distance required by the standards.

The values of the systematic error are given by means of the analytical expressions and the diagrams. Measuring the grounding resistance in the case of a shorter distance, the real grounding resistance can be calculated on the basis of the detected measurement

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systematic error and the measured apparent grounding resistance. For each measurement and corresponding reflection factor the systematic errors are determined by the equivalent radii of the grounding system and by the auxiliary electrode as well as by their mutual distance. Thus the measurement of the grounding resistance can be performed as well for very short distances between the auxiliary electrode and the test electrode, with correction of the measured results with systematic errors given in this paper.

The proposed model can also be successfully ap- plied to the large grounding systems of irregular shape, by using the equivalent hemispherical electrode.

Also, systematic errors are computed for the case of a two-layer soil, which represents a more realistic model of a composed soil structure.

5 List of Symbols

distance between the centers ground electrodes upper layer thickness

of hemispherical

discharging current of a spherical ground electrode A, when there is no influence of the other electrode B discharging current of a spherical ground electrode B, when there is no influence of the other electrode A values of the reflected point current sources for the spherical ground electrode A caused by mutual reflections between spheres A and B; K = 1,2,3, ... values of the reflected point current sources for the spherical ground electrode B caused by mutual reflections between spheres A and B; K = 1,2,3, ... the reflection factor; the following expression

inverse relative value of the radius of electrode A; the following expression holds: mA = d/ro inverse relative value of the radius of electrode B; the following expression holds: mB = d / r , radius of the hemispherical ground electrode A radius of the hemispherical ground electrode B relative value of the hemispherical ground electrodes radii; the following expression holds: r0hI = r unknown theoretical grounding resistance of a hemispherical ground electrode apparent measured grounding resistance of a hemispherical ground electrode including conductive influence of the other electrode systematic error, which occurs when the grounding resistance is measured; the following expression holds: a= (RT - RJR, upper soil layer resistivity bottom soil layer resistivity

holds: k = (p2 - P I )/(pz + P I )

References [ I ] Bosanac. T : Electromagnetic Field Theory (in Croa-

tian). ZagrebICroatia: Technical Book, 1973 [2] IEEE-Stand. 81-1983: Guide for measuring earth resis-

tivity, ground impedance and earth surface potentials of ground system. New York/USA IEEE-Guide 81, Part I1 (8-23-89 Draft 21): Guide for measurement impedance and safety characteristics of large extended or interconnected grounding systems. New YorkAJSh

[4] CENELEC Stand. (Draft prEN50179): Powerinstallation exceeding 1 kV AC. BrusseldBelgium: IEC TC99,1994

[5] Corbellini, G.: Corbellini. U.: Definition and Measure- ment of the Earth Resistance of an Electrode Covering a Large Area. ETEP Eur. Trans. on Electr. Power 5 ( 1 995) no.3,pp. 173-180 Dawulibi. E : Mukhedkac D.: Ground electrode resistance measurements in non uniform soils. IEEE Trans. on Power Appar. a. Syst. (1974) no. 1, pp. 109- 116 Takahashi, T; Kawase, T : Analysis of apparent resistivity in a multi-layer earth structure. IEEETnns. on Power Delivery PWRD-5 (1990) no. 2, pp. 604-612

[8] Oifenitor$ E : Erdstrome. Basel/Switzerland: Birk- hauser. 1969

[9] curdfs. E. B.: Some of the fundamental aspects of grounding resistance measurements. AIEE Trans. 77 (1958) P. I, pp. 760

[ 101 Tagg, C. E : Measurement ofearthelectrode resistance with particular reference to earth-electrode systems covering a largearea,Proc.IEE 111 (1964)no. 12,pp.2118-2130

[3]

[6]

[7]

Manuscript received on October 9, 1998

The Authors Dr.-Ing. Sead Berberovif (1953) received his MSc degree from the Facul- ty of Electrical Engineering and Com- puting of the University in Zagreb/ Croatia in 1983. He received his PhD degree from the same Faculty in 1991. His major field of interest is numerical computation of electromagnetic field and particularly grounding system analyses. He IS professor at the Faculty of Electrical Engineering and Comput-

ingof the University in Zagreb. (Faculty of Electrical Engineer- ing andcomputing, University Zagreb, Unska3,10000Zagreb/ Croatia, Phone: + 3 85 1 6 1 29-6 29, Fax: + 3 85 1 6 1 29-6 I 6, E-mail: sead.berberovic @fer.hr)

Dr.-Ing. Vedran Boras (1956) received his MSc degree from the Faculty of Electrical Engineering of the Univer- sity in Sarajevo in 1990. He received his PhD degree from the Faculty of Electrical Engineering and Computing of the University in ZagreblCroatia in 1997. His major fields of interest in- clude grounding systems and power system analyses. Presently he is work- ing with Power Utility “EP HZ Herceg

Bosne” as an assistant general manager for development. (Power Utility “EP HZ Herceg Bosne”, Research and Devel- opment Department, Dr. Mile Budaka IMa, 88000 Mostad Bosnia and Herzegovina, Phone: + 38 78 83 10-8 47, Fax: + 38 78 83 17-1 57, E-mail: [email protected])

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correction of resistance measurement of large grounding systems

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