IEEE Transactions on Power Apparatus and Systems, Vol. PAS-99, No. 5 Sept/Oct 1980

CALCULATION OF SUBSTATION GROUNDING SYSTEM RESISTANCE USING MATRIX TECHNIQUES

D. L. Garrett, Member, IEEESouthern Company ServicesBirmingham, Alabama

Abstract - Recent papers have provided equationsfor calculating the self and mutual resistances of theindividual conductors in a grounding system. However,the overall grid resistance, or input resistance, isfound only after the currents in each component of thegrid are found. Since the input resistance is neces-sary in order to calculate the grid potential rise, itappears beneficial to develop a technique for directlycalculating the input resistance from the self andmutual resistances before solving for the currents.This paper develops such a technique. This techniqueis valid for both uniform or multi-layered soil models.

INTRODUCTION

With continuing increases in the cost of copperand other conductors commonly used in grounding sys-tems, it has become necessary to accurately model thegrounding system in order to design the most economicalgrounding system that will insure equipment and person-nel safety. Numerous past papers and texts (1, 2, 3)have developed simple equations for the resistance ofuniformly shaped grounding systems and usually describ-ed methods for applying these equations to irregularlyshaped grounding systems. While the accuracy of theseequations has been adequate in most cases for theuniformly shaped grounding systems, it has been foundthat calculated values for irregularly shaped groundingsystems have differed greatly in some cases from theactual values of resistance.

More recent papers (4, 5, 6) have developed equa-tions for the self and mutual resistances of the indi-vidual conductors in a grounding system. These equa-tions approximate the resistance of each conductorsegment by calculating the average of the potential ateach point along the conductor segment. The gridpotential rise is assumed to be a known quantity, thusallowing for the calculation of the current in eachconductor segment using the following matrix equation:

I== V (1)

where I is a vector consisting of the currents in each ofthe conductor segments, Vis the voltage on each of theconductor segments, and R is a matrix consisting ofthe self and mutual resistances of the conductorsegments which make up the grounding system. Then theinput resistance is found by dividing the grid potentialrise by the sum of the currents in each segment.

In actual practice, however, the grid potentialrise is found as the product of the total maximum faultcurrent in the grounding system and the input resistanceof the grounding system. Therefore, in order tocalculate the currents in the individual conductor

F 80 255-0 A paper recommended and approved by the

IEEE Substations Committee of the IEEE Power

Engineering Society for presentation at the IEEE PES

Winter Meeting, New York,, NY, February 3-8, 1980.

Manuscript submitted August 27, 1979; made available

for printing November 26, 1979.

Dr. H. J. Holley, Member, IEEEAl abama Power CompanyBirmingham, Alabama

segments from the self and mutual resistance matrix andthe grid potential rise, the input resistance of thegroundi ng system must fi rst be found.

One method of determining the input resistance ofthe grounding system from the resistance matrix involvesthe use of equation (1) under the assumption that thegrounding system is at the same potential everywhere.The current vector can be computed from equation (1) byequating each element of the voltage vector to unity.The input resistance is then obtained by taking thereciprocal of the sum of the elements of the currentvector. Once the input resistance is determined, thegrid potential rise can be calculated and used to scalethe elements of the current vector to their correctvalues.

In this paper, a direct method of calculating theinput resistance is described. The method does notrequire knowledge of either the voltage or the currenton the grounding system, other than making the assump-tion that the potential on the grounding system is thesame everywhere. The method manipulates the resistancematrix to reduce the matrix to a single quantity, theinput resistance. In general, manipulating the resist-ance matrix in this manner is more economical in compu-ter time than dealing with the complete matrix asrequired by the method described above.

BUILDING AND REDUCING THERESISTANCE MATRIX

The equations for the self and mutual resistancesof the components of a grounding system, including bothgrid conductors and ground rods, can be used to computethe elements of a resistance matrix of the groundingsystem. Since a number of publications present theequations necessary for computing these self and mutualresistances (3, 4, 5, 6, 9), the equations are not pre-sented here.

The dimension of the resistance matrix is equal tothe number of individual components of the groundingsystem. For a large substation, the resistance matrixcan be of very high dimension. Therefore, it is bene-ficial to use techniques which are economical in compu-ter time and storage when evaluating a number of alter-nate grounding systems for a large substation. Sinceall the components of a grounding system are intercon-nected, and since the conductivity of earth is much lessthan that of the components of the grounding system, theassumption that the components are all at the samepotential is valid.

A system of four mutually coupled conductors isused to explain how this assumption can be used toefficiently determine the input resistance. Equation(2) gives the relationship between the four conductorcurrents and the four conductor voltages of the ground-ing system. The resistance matrix of equation (2) isactually symmetrical, but for clarity in the followinganalysis, it is dealt with as an unsymmetrical matrix.

e2R2e3 - R31e4 R41

R12R22''32R42

14 1IR24

4343

(2)

0018-9510/80/0900-2008$00.75© 1980 IEEE

2008

It is helpful to express equation (2) as four individualequations. That is,

e = R1 i + R12 i2 +R13 i3 + R14 i4

e2 R21 i1 +R22 i2 +R23 i3 R24 i4 (3)

2009

e1 R11 R12 R13 R14- R13 i1

2 21 R23 R23 R23 i2 (10)

e3 R31 R32 R33 R34- R33 i3

0 R41-R31 R42-R32 R43-R33 R44+R33

-R34 R43 i4

e3 R31 i1+ R32 i2 +R33 i3 +R34 i4

e4 R41 i 1 R42 i 2 +R43 i 3 R44 i 4

At this point the fact that all the conductors of thesystem are interconnected has not been taken intoaccount. Suppose that conductor #4 is connected toconductor #3 as shown in Figure 1. The connection isaccounted for mathematically by equating the voltage e4to the voltage e3 in equations (3).

From Figure 1 it is seen that

i3 i3 i4 (4)

Substituting equation (4) into equations (3), with e4equated to e3, gives

e1=R 1i1+R12 i2+R13i3+(R 14-R13)4i )

eC,=R,llil+R9,i +R91i z+(R9A-R9')iA (6)

e3=R31i1+R32i2+R33i3+(R34-R33 )i4 (7)

e3=R 41i +R42i2+R43i3 (R44 R43)i4 (8)

If equation (7) is subtracted from equation (8), theresult is

0=(R41-R31)i1+(R42-R32)i2+(R43-R33)i3

(R33 R44 R34 R43)i4 (9)

Equations (5), (6), (7), and (9)matrix form as

can be expressed

Note that the resistance matrix of,equation (10) isobtained by first subtracting the third column of theoriginal matrix of equation (2) from the fourth column,and then subtracting the third row of the resultingmatrix from the fourth row. -This is easily programmableon a digital computer. In general, for an n x n matrix,the (n-1)th column of the original matrix is subtractedfrom the nth column. Then the (n-1)th row is subtractedfrom the nth row of the resulting matrix.

Note also that equation (10) is in a suitable formfor eliminating the last row and column by Kron's methodfor eliminating short-circuited meshes (7, 8). Thisprocedure is best explained by writing equation (10) incompound matrix form to obtain

ea R R ia aa ab a

where ba bb b

where

F11Raa = R

R31

Rab R14 - R

R2 - R2R34 - R33

Rba = 41-R31

Rbb = LR33 + 44I in

R12 R13R22 R23R32 R33

R42-R32 R43-R37

- R3 - R4

#2 \ I

#2

+ + + Lel 2

#3 3

#4 4

* >

Figure 1. Circuit with four mutually coupled conductorswith conductors #3 and #4 interconnected.

2010

Rab R14 R13

R34 R33

Rba = [R41-R31 R42-R32 R43-R33]

Rbb = 33 + R44 R34 R4

ea = [0] la{:]2 lb [i4]

Equation (11) may be written as

ea = Raa ia + Ra b

e Rbai bbib

Solving equation (13) for ib and substitutinginto equation (12) gives

ea = (Raa Rab Rbb Rba)i a

The resistance matrix of equation (14) completelyrepresents the system of Figure 1 even though itsdimension is one less than the dimension of the originalresistance matrix. Since R is a one-dimensionalmatrix, no matrix inversion i's required. Furthermore,the elimination of the row and column of the resistancematrix is easy to program on a digital computer. Let R..represent any element of the n x n resistance matrix.Let R'.. represent any element of the (n-1) xt n-1)resista1n6e matrix obtained by eliminating the n rowand column of the original matrix. Equation (14) showsthat

R'. = R.. - R. R A/R (15)ij ij in nj nn

where

i = 1, 2, 3, ...n-1

j = 1, 2, 3, ...n-1

After the nth row and column of the original resistancematrix is eliminated, the procedure is repeated on theremaining (n-1) x (n-1) matrix to obtain an (n-2) x (n-2)matrix. This procedure is repeated until the n x nmatrix is reduced to a 1 x 1 matrix. The element of this1 x 1 matrix is the input resistance of the groundingsystem.

The original resistance matrix of the groundi'system is symmett'cal. After subtracting e (n-i)columqhfrom the n column and then the (n-1) row fromthe n row of the resulting matrix, the result is asymmetrical matrix. Symetry is also maintained in theelimination of the n row and column. Therefore,advantage can be taken of this symmetry to reduce thecomputer storage requirements to approximately half ofthe storage required if symmetry is not considered.

Sample Calculation

To illustrate the technique described above, a3 X 3 matrix consisting of random numbers will bereduced to a single element. If a grounding system canbe described by this 3 X 3 matrix of self and mutual

resistances, the reduction to a single element repre-sents the reduction of these resistances to the inputresistance of the grounding system.

Let

R an 3 8 5 (16)L4 6 9

Then subtracting the first row and column from the thirdrow and column, respectively, results in the following:

7 1 2-7 7 1 -5

3 8 5-3 = 3 8 2 (17)

L4-7 6-1 9-4-2+7 -3 5 10

(12) Eliminating the resulting third row and column yields

3) F7 ~(-3)(-5) 1 -(5)(5)(13) 7 (3 5 1 - 1 5.5 3.5-(18)the result -3 (-3)(2) 8 - (2)(5) L3.6 75]

Repeating the procedure on the resulting matrix of (18)(14) gives

5.5 3.z-.5+.5 5. 5-2.L0

3.6-5.5 7-3.6-3.5+5.5 -1.9 5.4

and

[. 5- (-2.0)(-1.9)] = [4. 796]

(19)

= Ruinput

Comparison of Results

It is beyond the scope of this paper to derive equationsfor the self and mutual resistances of the components ofthe grounding system. These equations have beenpresented in the literature. However, for comparisonpurposes, the technique described in this paper iscompared to the resistance of various grounding systemsas determined by previous methods, and the resultssummarized in Table I. These grounding systems areshown in Figures 2-5.

Table I. Comparison of Test Cases - Matrix Techniquevs. Existing Methods

Case Matrix Dwight Schwartz Laurent Gross &No. Technique (1) (2) (10) Associates (12)

1 2.12Q 2.12Q 1.66Q 1.85Q -

2 2.01Q 2.01Q 1.83Q 2.27Q -

3 .76Q - 0.77Q 0.72Q 0.73Q

4 0.64Q - 0.68Q 0.63Q 0.62Q

30. 48m

30. 48m

30.48m

30.48m

diameter = 1.27cmdepth = .457m

p = 100ohm-m

Figure 2. Case 1 - Four Point Star

2011

60.96mA diameter = 1.27cm

15.24m depth = .457mv p = 100ohm-m

Figure 3. Case 2 - Two Parallel Buried Conductors

96.38mdiameter = 1. 27cmdepth = .457mp = 100ohm-m

96.38m

Figure 4. Case 3 - One Mesh Grid

48. 19m

48. 19m

diameter = 1. 27cmdept = .457mP = 100ohm-m

8. D. W. Coleman, F. Watts, and R. B. Shipley, "Digi-tal Calculation of Overhead Transmission - Line -Constants." AIEE Transactions (Power Appartus andSystems), vol. 77, pp. 1266-68, Feb. 1959.

9. G. F. Tagge, Earth Resistances, Pitman PublishingCorp., N.Y., 1964.

10. IEEE Standard 80 - 1976, IEEE Guide for Safetyin Substation Grounding, June 1976.

11. D. L. Garrett, "The Average Potential Method ofCalculating the Resistance of Power SubstationGrounding Systems," M. S. Thesis, University ofAlabama in Birmingham, 1978.

12. E. T. B. Gross, B. V. Chitnis, and L. J. Stratton,"Grounding Grids for High-Voltage Stations." AIEETransactions, Vol. 72, pt. III, pp. 799-810, Aug.1953.

48.19 m 48.19m

Figure 5. Case 4 - Four Mesh Grid

SUMMARY AND CONCLUSIONS

The resistance matrix of a large grounding systemis likely to be of large dimension. Since it can beassumed (without introducing significant error) that allthe components of the grounding system are at the samepotential, a procedure that is easy to program on adigital computer can be employed to systematicallyreduce the size of the matrix when computing the inputresistance of the grounding system. This systematicprocedure requires less computer time than when workingwith the entire matrix and using a matrix inversionprocedure. Since symmetrical matrices are dealt with inall cases, the property of symmetry further reduces thecomputer time and storage requirements.

REFERENCES

1. H. B. Dwight, "Calculation of Resistances toGround." AIEE Transactions (Electrical Engineer-ing), vol. 55, pp. 1319-28, Dec. 1936.

2. S. J. Schwarz, "Analytical Expressions for theResistance of Grounding Systems." AIEE Transac-tions, Vol. 73, pt. III-B, pp. 1011-16, Aug. 1954.

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

4. F. Dawalibi and D. Mukhecdkar, "Optimum Design ofSubstation Grounding in a Two Layer Earth Struc-ture, Part I - Analytical Study." IEEE Transac-tions, PAS-94, pp. 252-259, March/April 1975.

5. R. J. Heppe, "Computation of Potential at SurfaceAbove an Energized Grid or Other Electrode, Allow-ing for Non-Uniform Current Distribution." PaperF79-274-2 presented at IEEE P.E.S. Winter Meeting,New York, N. Y., Feb. 1979.

6. E. B. Joy, A. P. Meliopoulos, and R. P. Webb, "Touchand Step Calculation for Substation GroundingSystems." Paper A 79-052-2 presented at IEEEP.E.S. Winter Meeting, New York, NY, Feb. 1979.

7. G. Kron, Tensor Analysis of Networks, John Wiley &Sons, 1939.

David Lane Garrett (S'75-M'76) was born in Birmingham,AL, on January 10, 1954. He received the B.S. and M.S.degrees in Engineering from the University of Alabama inBirmingham (UAB) in 1976 and 1978, respectively.

In 1974 he joined Southern Company Services as aco-operative education student and as an engineer in1976. He is presently employed in the Technical StudiesSection of the System Planning Technical ServicesDepartment.

Mr. Garrett is a member of the IEEE Power Engi-neering Society, Omicron Delta Kappa, and the UABEngineering Honor Society. He is a registered Engineer-In-Training in the State of Alabama and is presently amember of the IEEE Working Group to revise the Guide forSafety in A-C Substations (IEEE 80).

Henry J. Holley was born in Jones Mill, TN, on Au-gust 1, 1932. He obtained the BSEE degree from theMissouri School of Mines and Metallurgy, the MSEE degreefrom the Illinois Institute of Technology, and the PhDdegree from the University of Tennessee. He worked as anelectrical engineer for the Tennessee Valley Authorityfrom 1959 to 1968. He has taught electrical engineeringat the University of Tennnessee, the University ofKentucky, Texas A&M University, and the Univirsity ofAlabama in Birmingham. He is presently the Manager ofEconomic and Special Studies at the Alabama PowerCompany in Birmingham. He is a past member of the IEEEWorking Group on Transient Recovery Voltages and thePower Circuit Breaker Subcommittee. Mr. Holley is aregistered professional engineer in the State of Alabamaand is a member of Tau Beta Pi, Eta Kappa Nu, Phi KappaPhi, and Sigma Xi.