Calculation of important design parameters for grounding systems in substations by Arun Balakrishnan A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Electrical Engineering Montana State University © Copyright by Arun Balakrishnan (1990) Abstract: An important aspect of substation grounding system design is the calculation of the ground resistance offered by the grounding grid, and the values of the mesh and step voltages on the surface. The values of these parameters have to be kept within certain established Emits, keeping in view the safety of the personnel present within and without, the substation area. Simple expressions for calculation of these parameters for square and rectangular grids are already available. Computer programs for designing substation grids, regardless of their shape, are available, but tend to be tedious in usage when the grids are not symmetrical, since a large amount of data has to be input to describe the grid configuration. Thus simplified expressions are preferred. Another important factor in grounding grid design, is the footing resistance, which depends on the resistivity and thickness of the surface rock layer. The footing resistance, at present, can be determined by use of an infinite series, which is not very easy to evaluate. There is hence need for a simplified finite expression for calculating the footing resistance. The value of footing resistance also affects the safe values of the mesh and step voltages. If two substation grounding grids are intertied by a bare conductor, when a fault occurs at one substation, it is important to determine, by a simple analytical procedure, the effect of this fault current on the other grounding grid, and the response of the system as a whole, to the fault current. The objectives of this thesis may be summarized as follows: (i) To develop a new set of expressions, which may be used for calculation of the ground resistance, and the mesh and step voltages, for all practical shapes of grounding grids, once information regarding the grid configuration and soil characteristics is available. (ii) To develop a finite expression, for calculating the footing resistance in a substation. (iii) To design amethod to calculate the total groundresistance of a system of two substation grounding grids intertied by a bare conductor, as seen from the station where the fault occurs, and the ground potential rise at both the substations. The development of the expressions for the calculation of the ground resistance, mesh and step voltages, footing resistance, and for evaluating the performance of intertied grids, is described in this thesis. Through a comparison with existing expressions for the calculation of these parameters, the considerable reduction in error through the use of the newly developed expressions is demonstrated. The scope for future work in this area is also discussed briefly.  CALCULATION OF IMPORTANT DESIGN PARAMETERS FOR GROUNDING SYSTEMS IN SUBSTATIONS by Arun Balakrishnan A thesis submitted in partial fulfillment of the requirements for the degree of Master o f Science in Electrical Engineering MONTANA STATE UNIVERSITY Bozeman* Montana June 1990 / I W / ii APPROVAL of a thesis submitted by Arun Balakrishnan This thesis has been read by each member o f the thesis committee and has been found to be satisfactory regarding content, English usage, format, citations, bibliographic style, and consistency, and is ready for submission to the College of Graduate Studies. Date Chairperson, Graduate Committee Approved for the Major Department Date ,/W AW Head, Major Department Approved for the College of Graduate Studies Date Graduate Dean iii STATEMENT OF PERMISSION TO USE In presenting this thesis in partial fulfillment o f the requirements for a master’s degree at Montana State University, I agree that the Library shall make it available to borrowers under rules o f the Library. Brief quotations from this thesis are allowable without special permission, provided accurate acknowledgement of source is made. Permission for extensive quotation from or reproduction of this thesis may be granted by my major professor, or in his absence, by the Dean of Libraries when, in the opinion o f either, the proposed use o f the material is for scholarly purposes. Any copying or use of the material in this thesis for financial gain shall not be allowed without my written permission. Signature Date 14-1 O . Z ACKNOWLEDGEMENTS Throughout the duration o f the author’s work at Montana State University, Dr. Baldev Thapar was a constant source o f inspiration and encouragement. His support, accessibility, and active participation, without which this work would not have been possible, are acknowledged with gratitude. The author is also grateful to Dr. Victor Gerez and Dr. Murari Kejariwal for their unstinting support and encouragement. Their valuable suggestions made during the time this thesis was written are appreciated. Mr. Prince Emmanuel is responsible for encoding parts o f an initial version o f the program RESIS. His help during development o f the later version of the program is acknowledged. The Department o f Electrical Engineering at Montana State University, and the Bonneville Power Administration are acknowledged for their financial support. VTABLE OF CONTENTS Page LIST OF TABLES vi LIST OF FIGURES vii ABSTRACT viii 1. INTRODUCTION I Background and Problem Definition I Literature Survey 7 2. THE PROGRAM RESIS 14 Program Description 14 Development of the Equations 16 Use o f the Program 20 3. GROUND RESISTANCE CALCULATIONS 23 Development of the Expression 23 Comparison with Existing Expressions 24 Comparison with Analog Model Tests 24 The Equation as Applied to Circular Plate and Strip Conductor 25 4. VOLTAGE CALCULATIONS 29 Development of the Expressions 29 Comparison with the Existing Methodology 32 5. CALCULATION OF THE FOOTING RESISTANCE 33 Development of the Finite Expression 33 Model I. Truncated Cone and Soil Model 35 Model 2. Trincated Cone Model 37 6. ANALYSIS OF SYSTEM OF INTERTTED GRIDS 41 Basic Considerations 41 Ladder Circuit Solution 42 An Approximate Solution to the Problem 44 Voltage Decay Along a Very Long Buried Conductor 46 7. CONCLUSIONS AND FUTURE WORK 50 REFERENCES CITED 53 vi LIST OF TABLES Table Page 1. Nomenclature for Self and Mutual Ground Resistance Expressions 17 2. Ground Resistance, Mesh and Step Voltages for Square Grids 20 3. Variation of Design Parameters with Conductor Size 22 4. Ground Resistance o f Grids o f Different Shapes 27 5. Ground Resistance Of Analog Models 28 6. Mesh and Step Voltages of Grids o f Different Shapes 31 7. Accuracy o f Finite Expression and Models 39 8. Voltage at the Two Grids 48 9. Current and Voltage Profile Along the Intertie 49 vii LIST OF FIGURES Figure Page 1. Variation of F(X) with h, for p/p, = 0.005 to 0.5 12 2. Shapes and Sizes o f Grounding Grids 21 3. Model I. Truncated Cone and Soil Model 35 4. Variation in O1 with h, for p/p, = 0.005 to 0.5 37 5. Model 2. Truncated Cone Model 38 6. Variation in O2 with h, for p/p, = 0.005 to 0.5 40 Viii ABSTRACT An important aspect o f substation grounding system design is the calculation of the ground resistance offered by the grounding grid, and the values o f the mesh and step voltages on the surface. The values o f these parameters have to be kept within certain established Emits, keeping in view the safety o f the personnel present within and without, the substation area. Simple expressions for calculation of these parameters for square and rectangular grids are already available. Computer programs for designing substation grids, regardless o f their shape, are available, but tend to be tedious in usage when the grids are not symmetrical, since a large amount o f data has to be input to describe the grid configuration. Thus simplified expressions are preferred. Another important factor in grounding grid design, is the footing resistance, which depends on the resistivity and thickness o f the surface rock layer. The footing resistance, at present; can be determined by use of an infinite series, which is not very easy to evaluate. There is hence need for a simplified finite expression for calculating the footing resistance. The value of footing resistance also affects the safe values of the mesh and step voltages. U two substation grounding grids are intertied by a bare conductor, when a fault occurs at one substation, it is important to determine, by a simple analytical procedure, the effect of this fault current on the other grounding grid, and the response o f the system as a whole, to the fault current. The objectives o f this thesis may be summarized as follows: (i) To develop a new set of expressions, which may be used for calculation of the ground resistance, and the mesh and step voltages, for all practical shapes o f grounding grids, once information regarding the grid configuration and soil characteristics is available. (E) To develop a finite expression, for calculating the footing resistance in a substation; (Ei) To design amethod to calculate the total groundresistance o f a system of two substation grounding grids intertied by a bare conductor, as seen from the station where the fault occurs, and (lie ground potential rise at both the substations. The development o f the expressions for the calculation of the ground resistance, mesh and step voltages, footing resistance, and for evaluating the performance o f intertied grids, is described in this thesis. Through a comparison with existing expressions for the calculation of these parameters, the considerable reduction in error through the use o f the newly developed expressions is demonstrated. The scope for future work in this area is also discussed briefly. ICHAPTER I INTRODUCTION Background and Problem Definition A grounding system in a substation has two objectives: (1) To carry electric currents to ground under normal and fault conditions, without exceeding any operating and equipment limits, or adversely affecting continuity of service; (2) To ensure that a person in the vicinity o f the grounded facilities is not exposed to the danger o f critical electric shock. The ground return circuit is defined [1] as the circuit in which the earth or an equivalent conducting body is utilized to allow current circulation from or to its current source. The ground is a conducting connection by which an electric circuit or equipment is connected to the earth or some conducting body of relatively large extent that serves in place o f the earth. A system or apparatus is provided with ground for purposes o f establishing a ground return circuit and for maintaining its potential at approximately the potential of earth. Whenever a fault occurs in a substation, the earth becomes saturated with currents flowing to ground from the grounding grid and other ground electrodes buried below the surface o f the earth. The substation grounding grid is a system of horizontal, interconnected bare ground conductors, spaced a few meters apart, buried in the earth at a depth varying from 0.25 to 2.5 meters, which provides a common ground to all the equipment present in the substation. A ll grounding systems offer a finite resistance (to remote earth) to the fault currents. It is very important to have a very low value of this resistance, referred to as the ground resistance 2of a substation grid, which is preferably about I ohm for large substations, and between I o hm and 5 ohms for distribution substations. While under normal conditions, the grounded apparatus in a substation operates at near zero ground potential, under fault conditions, the portion o f the fault current that is conducted into the earth by the grounding grid causes the potential o f the grid to rise with respect to remote earth. This rise in voltage of the grounding grid is referred to as the ground potential rise (GPR), and is proportional to the magnitude o f the fault current, and the ground resistance o f the grounding grid. Two other important design parameters, which depend upon the value o f the ground potential rise and the ground resistance of the grounding grid are the mesh voltage and the step voltage. The conductors o f a grounding grid divide it into a number o f meshes, and the potential on the surface o f the earth, above the grounding grid, is not the same at all points. The touch voltage is defined as the potential difference between the ground potential rise, and the surface potential at the point where a person is standing, while being in contact with a grounded structure. The maximum value o f the touch voltage within the substation yard is known as the mesh voltage. The step voltage is defined as the difference in the surface potential experienced by a person, with his feet one meter apart, with no part o f his body being in contact with any grounded structure. The human body, for dc, and ac power frequencies (50 and 60Hz), can be represented as a finite noninductive resistance, of value in the range 500 to 5000 ohms, and is usually approximated by a value o f 1000 ohms [1]. There is hence a critical limit to the amount of shock energy that can be absorbed by a human being. Crossing this limit, which varies from person to person depending upon factors including the body weight, can prove to be dangerous. The most common physiological effects o f electric current on the body are perception, muscular contraction, unconsciousness, fibrillation o f the heart, respiratory nerve blockage, and burning. The fibrillation threshold, which can prove fatal, is reached when currents o f magnitude 60 - 3100 niA are reached. The duration for which the fault current can be tolerated by most people is given as [1] (Jb) = (11) where I8 is the rms value o f current, in amperes, flowing through the body, ts is the duration of the current flow , in seconds, and S8 is the an empirical constant related to the electric shock energy tolerated by a certain percentage of a given population. Based on studies conducted by Dalziel [2], a current I8 that can be tolerated by 99.5% of all the population without ventricular fibrillation, has a magnitude given by Ifl — ^ SgIts (1.2) Here, ^[s^ has a value o f 0.116 based upon the fact that the shock energy that can be survived by 99.5% of people weighing 50 kg, results in a value o f S8 of 0.0135. Thus for a 50 kg body weight, /B = 0 .1 1 6 /^ (1.3) For a 70 kg body weight, the value of current I8 is obtained as 4= 0 .1 5 7 /^ 4 (IA) Since the above equations are based on tests for the 0.03-0.3 seconds time range, they are not valid for very short or long times. Ferris, e ta l [3], suggest a value o f IOOmAas the fibrillation threshold if duration of the shock is not specified. Beigelmeier [4] has suggested values o f 500 mA and 50 hiA respectively, for shock durations of less, and more, than one heartbeat duration. It is thus important that the grounding system be designed in a manner to keep the shock currents below the value mentioned above. The resistance o f the human body, R8, is assumed to be about 1000 ohms for defining the limits on the step and touch voltages. Substations usually have a layer of crushed rock spread on the soil. This usually provides a high resistivity layer below the feet of personnel in the substation. A highly simplified approach, which neglects the mutual ground resistance between the two feet o f the person, and assumes 4a very large depth of the crushed rock layer, gives the value o f the footing resistance (which is defined as the resistance o f the ground below the feet), Rfoon as 3ps, where ps is the resistivity of the crushed rock layer. This expression is derived by modelling the foot as a conducting disc of radius 8 cm. In accidental circuits for mesh and step voltages, the two feet are in parallel and series respectively; therefore, the total footing resistance is taken to be LSp5 and 6ps. The layer o f crushed rock greatly increases the contact resistance between the feet and the substation surface, which greatly reduces the current flowing through the body of a person present in the substation. The effect o f the layer o f crushed rock spread on the surface o f the substation area on the value o f the fault current flowing upwards into the body depends on its thickness, the relative resistivity o f the crushed rock layer and the lower soil, and the resistance o f the foot. The safety o f a person depends on prevention o f his absorbing the critical amount o f shock energy before the fault is cleared and the system re-energised. The driving voltage o f any accidental circuit should not exceed the limits defined below. The limit for the touch voltage is given as Etouch ~ (Rb + 0.5RfOOt)IB (L5) With Rb = 1000i2, and Rfoot = 3ps, equation (1.5), for a 50kg body weight, can be written as Efouch(SO) = (1000 + 1 .5 *C ,(W p ,)0 .1 1 6 /^ (1.6) The limit for the step voltage is given as Estep — (R-B + '^ Rfoot)!B (11T) Again, with Rb = 1CKXX2, and Rfoot = 3ps, equation (1.7), again for a 50kg body weight, can be written as Estepm = (1000 + 6*C^,*OpJ0.116A /4 (1.8) The factor C5 is included to account for the layer o f crushed rock which is spread on top of the soil in the substation area, and is of resistivity different from that o f the soil. It is a factor 5applied for compensating for the finite thickness o f the surface crushed rock layer. This factor and its calculation form an important aspect o f substation grounding grid design, and are dis­ cussed in detail later. For a uniform soil, with no upper crushed rock layer, the value of Ca equals unity. pa is then the resistivity o f the soil in iQ-m, and ta is the duration o f shock current in seconds. The calculated value of mesh voltage should not exceed the maximum allowable touch voltage, Etouch. Similarly, the calculated value o f the step voltage should not exceed the maximum allowable step voltage, Estep. The values OfEatep^ and Etouch(50) are calculated assuming the body weight o f the person experiencing these voltages to be 50kg. Calculation of the mesh and step voltages, and the ground resistance, as part o f a design methodology, can be made using different methods. A number of computer programs have been developed for this purpose, and give fairly accurate results. Unfortunately, programs are not always available in field sites. Also, preliminary design strategies do not require the high accuracy offered by these programs. It is hence important to have a simplified set o f expressions, which may be used to compute these design parameters. Most o f the work so far has concentrated on grids which are o f shapes approaching a square or a rectangle. Though simplified equations for making calculations o f the ground resistance, and mesh and step voltages for such grids have been developed, and have been in use for some time, not all grids can be approximated by a square or rectangle, and for the design of such grids, the use of the existing expressions leads to erroneous results. The following site parameters are known to have a substantial impact on the grid design: maximum fault current^), fault duration^), soil resistivity(p). The substation grid design depends on the area o f the grounding system, the conductor spacing, the depth of burial o f the grounding grid, and the shape o f the grid. The objectives of this thesis are to develop a new set of expressions, which can be applied to the design of substation grounding grids, o f all practical shapes. 6They may be summarized as: (i) To develop a new set o f expressions, which may be used for calculation of the ground resistance, and the mesh and step voltages, for all practical shapes o f grounding grids. (ii) To develop a finite expression, for calculating the footing resistance in a substation, which is easy to use, and gives fairly accurate results. (iii) To design amethod to calculate the total ground resistance o f a system of two substation grounding grids intertied by a bare conductor, as seen from the station where the fault occurs, and the ground potential at both the substations. The results o f the research covered in this thesis have been presented in references [5], [6], [7], and [8]. Chapter I has thus far given a brief introduction to the areas o f substation grounding with which the topics covered in this thesis are related. Some work regarding calculation of the ground resistance, mesh voltage, step voltage and footing resistance is available in literature, and is summarized in the next section of this chapter. Chapter 2 covers the algorithm of the program RESTS which has been used as the primary development reference for the work involving the calculation o f ground resistance and mesh and step voltages reported in this thesis. Relevant equations for developing the resistance matrix are also discussed. In Chapter 3, the calculations involving the grounding resistance have been described, and a comparison of the new expression for ground resistance, presented in this thesis, has been made with expressions already available in literature. The accuracy o f the simplified expressions is also discussed for various shapes o f grounding grids. Chapter 4 covers the development of the expressions for the calculation of mesh and step voltages for grids o f irregular shapes. A comparison of the new expressions with the methodology recommended by the IEEE Std. 80 [1] is also made. 7Chapter 5 describes the development o f the expression for the calculation o f the footing resistance. In addition, two fairly accurate analytical models to calculate the footing resistance have also been presented. A comparison of the new expression with the expressions recom­ mended by the IEEE Std. 80-1986 is made, and the accuracy o f the new expression and the two models demonstrated. Chapter 6 provides details about the method for determining the performance o f two substation grids interbed by a bare, buried conductor, when a fault occurs in one of the sub- stadons. From the suggested method, the total ground resistance o f the system, as seen from the stabon where the fault occurs, and the ground potential rise at both substabons can be calculated. A simplified method, derived using the transmission line equabons, is discussed in detail. The thesis is concluded with a brief summary o f the work done, and its benefits, in Chapter 7. Possible future work is also discussed. This section gives a brief summary o f the expressions already available for the calculabon of the ground resistance, mesh and step voltages; and the footing resistance, and the performance of interconnected grounding systems. A minimum value o f the ground resistance offered by a grid buried in soil o f uniform resisdvity, can be obtained from the expression for the resistance o f a circular plate, which may be applied to square grids. This expression is given as [I]: where Rg is the resistance to ground of the substabon grid in ohms, p is the resisdvity of soil in ohm-meters, and A is the area o f the grounding grid in square meters. Literature Survey (1.9) 8The upper limit to the ground resistance o f substation grids with square shapes can be obtained by use o f the formula given by Laurent [9], and Niemann [10]: where L is the total length o f grid conductors in meters. The factor p/L accounts for the fact that the resistance o f a solid plate is always lower than from the formula, the difference in resistance reduces, and approaches zero as the total length of the conductors gets infinitely large. While these expressions may be used with reasonable accuracy for square grids buried in the soil at depths less than 0.25 meters, for grids buried between a minimum of 0.25 meters and a maximum of 2.5 meters, Sverak [11], introduces a correction factor to account for the variation in the depth o f burial, and develops the expression for the calculation of ground resistance as where h is the depth of burial o f the grid in meters; The resistance of a grid, embedded in a soil o f uniform resistivity, can also be calculated by use o f the expression presented by Schwarz [12]. Schwarz’ expression is valid for multi-grid systems, and may be simplified for a single grid, without ground rods, in a soil of uniform resistivity p, as where a * = ^ 2rh, for conductors buried at a depth o f h meters, and r is the radius o f the conductors in meters. (1.10) that o f a grid o f the same shape, composed o f a finite number of conductors. As can be seen ( 1 .11) (1.12) A limitation o f the expression proposed by Schwarz is that the factors K1 and K2 can be obtained only by use o f graphs, which prevents an analytical solution using a computer. This 9deficiency may be overcome by use o f the expressions for K1 and K2 derived by Kercel [13]. These expressions are given as: K1 =1.84- ab A - 2 [a a +^a2+b2 I 1+—In b +^a2+b2I b J b II 4 J 3b2 3c2 3 a V (1.13) 4(4 +b) , (a +b) (a +^a2+ {bHf) (M ) 1 „ (b/2) + ^ ja2 + (b/2f 2 -(b/2)+^a2+(b/2f (1.14) where b is the length of the long side o f the grid, and a is the length o f the short side of the grid in meters. These expressions, and the graphs, can only be applied to a limited set o f rectangles, with a maximum length to width ratio of 8:1. Simplified equations have also been proposed for the calculation o f the mesh and step voltages. The best expressions available so far are those developed by Sverak, and recommended by the IEEE Std. 80. The mesh voltage, for square grids with square meshes, is calculated as plaKmK-i L (1.15) where Ig is the maximum grid current that flows from the grid to ground in amperes. The spacing factor for the mesh voltage, Km is given as I 2n D2 (D+2hf h 16M + SDd Ad +— In-----------Kh n(2n - 1) (1.16) D is the spacing between parallel conductors in meters, h is the depth of burial of grid conductors in meters, and d is the diameter of grid conductors in meters, and n is the number of parallel conductors in any one dirction. K1, Kji, and Kh are correction factors, used to compensate for the 10 fact that the actual grids are different from the model, which is based on a system of n parallel conductors in one direction. The actual grids have conductors in two directions, and these conductors are connected at the intersections. The correction factor for grid geometry, Ki is given as Ki = 0.656 + 0.172*« (LI?) The corrective weighting factor, Kii, that adjusts the effects o f inner conductors on the comer mesh is given as Kil = 1/(2« f ” (1.18) Kh, the corrective factor that emphasizes the effects o f grid depth, is given as Kh= -^ l+h (1.19) The step voltage may be calculated as n P^aKsKi ' L ( 1.20) The spacing factor for step voltage, Ks, is given as s + z r b + ^ 1- ° - 5“" ) (1.21) For rectangular grids with square meshes, the value o f n is modified as follows: For mesh voltage calculations, n is given as n = V«a«b (1.22) where nA is the number o f parallel conductors along one o f the coordinate axis, and nB the number along the other the axis. For step voltage calculations, n is given as n =max(«A, ziB) (1.23) 11 These equations are recommended for use within the following limits: 0.25m < h < 2.5m (1.24a) d < 0.25/1 (1.246) D > 2.5m (1.24c) A ll the expressions given above are recommended for use with grids with shapes which may be approximated by a square, or a rectangle with length to width ratio limited to less than 8:1. Expressions are also available for the calculation of footing resistance [I]. They may be summarized as: £Ii ' d.25) F(K) = 1+ 2 £ Q M = I (1.26) Q = K n l [ \ + Q n X f f 2 (1.27) ^ = (P-PJ/(P + PJ (1.28) where p3 is the resistivity o f the surface crushed rock layer in ohm-meters, p is the resistivity of soil in ohm-meters, X = h/b, h is the thickness o f the crushed rock layer in meters, b is the equivalent radius o f a foot equalling 8 centimeters, and K is the reflection factor. Figure I shows the variation o f F(X) with h, with p/ps varying from 0.005 to 0.5. Though the value o f footing resistance may be calculated from the above expressions, due to the infinite nature o f equation (1.26), they are difficult to use without the help o f a computer or programmable calculator. Seedhar, Arora and Thapar [14] have proposed a finite expression to compute the potential at any point due to a point current source located anywhere in a two layer soil. A finite expression for calculating the footing resistance may be derived from this expression. 12 Sverak has also proposed a simplified formula for calculating the footing resistance, which may be given as where C(X) is given as (1.29) C (X )= 1 -0 .1 0 6 l ~p. 2/1,+ 0.106 where h, is the thickness of the top crushed rock layer. (1.30) S 0.6 - Figure I. Variation of F(X) with h, for p/p, = 0.005 to 0.5 13 Sobralj e ta l [15] have developed a method for studying interconnected grounding systems, connected by overhead ground wires. In their analysis, the mutual effect o f the interconnected grids are neglected, and the system is assumed to comprise o f a number of grids, and overhead ground wires, which transfer the potential o f the energised grid to other connected grids. 14 CHAPTER 2 THE PROGRAM RESIS Program Description With a view to have an available reference, with which any newly developed expressions may be compared, a computer program RESTS, based on the finite element analysis, was developed. This program can handle grid configurations composed of straight linear conductors, laid in three mutually perpendicular directions. For a given grid configuration, the grid con­ ductors are divided into small straight segments, each about 5 meters in length. The self and mutual resistances of these segments are then calculated to obtain the ground resistance o f the grid. For grids with a very large number of parallel conductors, the voltage drop may be neglected due to the lower value of current flowing per conductor. For large grids with a small number o f conductors, it becomes necessary to express the voltage on each segment as a function of the applied voltage, and the voltage drops due to the leakage currents flowing through the resistance and inductance, and mutual inductances o f the conductor segments. Since practical grids are rarely o f this type, in the program RESTS, the conductors are assumed to be at the same potential. This is a valid assumption, as at power frequencies, the resistance and inductance of the segments are negligible when compared to the resistance between the segments and the earth. A t power frequencies, the voltage drop along the conductors becomes important only if the length o f the conductors becomes very large. 15 Whenever a fault occurs in a substation, a part o f the fault current flows to the ground through the grounding grid. This fault current flowing through the grounding grid causes the potential o f the grid, with respect to the remote ground, to rise. This potential rise of the conductor segments is also known as the ground potential rise (GPR) of the grid. A part o f the fault current flowing through the grid, is dissipated by each o f the conductor segments. It is thus important to determine the current distribution from the segments o f the grid conductors. Once the current distribution has been determined, the voltage at any point on the surface o f the earth is determined by obtaining the sum of voltages at that point due to the current dissipated by each o f the segments. In this fashion, the voltage profile in any direction on the surface o f the earth, can be calculated. Once the voltage profile is available, the mesh and step voltages can be computed. For a given design, based on the configuration of the grid conductors, the approximate area within which the point where the minimum potential is likely to occur, can usually be identified. Within the specified area, the potentials at a finite number of points are determined, and the minimum of these values, and its location, are identified. The difference between the GPR and this minimum voltage, gives the value of mesh voltage for the specific configuration.For grids with irregular shapes, it is likely that the point of minimum potential may not be readily identifiable, and a larger area, within which this point is likely to occur, may have to be chosen. The step voltage is calculated as the difference in voltage between the two points on the surface o f the ground, one directly above a comer o f the grid, and the other one meter away in the direction of the diagonal o f the comer mesh extended outwards. In case o f grids with irregular shapes, the step voltage at all the comer points may be determined, and the highest of these values recorded as the maximum step voltage of the grid. 16 The equation for the calculation of potential at a point, and subsequently the equations for the calculation o f the self and mutual resistances o f the various conductor segments are derived on the basis o f the method of images [15]. This is done to account for the boundary between the earth and the air. Development of the Equations The potential at any point (x,y,z), produced by a linear conductor m parallel to the x-axis, discharging a current I in an infinite medium o f resistivity p, is given by [16]: p/ ^ j(x -xm +L/2)2 + (y - y m)2 + (z - z m)2 +x -Xm +L/2 -r~r*ln I -- ----------------- y (x—xm -L U )2 + (y — y,„f + (z — zm)2+x - x m —L/2 (2.1) Similar equations can be written for conductors oriented along the y-, and z-, coordinate axes. The nomenclature used in equation (2.1) above, and in equations (2.2), (2.3), and (2.4) below are explained in Table I. The self ground resistance Rmm, o f a conductor segment, is defined as the ratio o f the voltage on the segment to the current flowing out o f the segment, neglecting the effect o f other conductor segments, and is determined by calculating the average potential on the surface of the conductor, when it discharges a unit current. Rmm is given as * - = ^ * < 2‘ toy - 2 + to i ^ r + f - T > (2.2) This equation includes the effect o f the image of the conductor segment under consideration. The mutual ground resistance, Rmn, between two linear conductor segments m and n, is determined by calculating the average potential on the surface of conductor segment m, when conductor segment n is discharging a unit current. To account for the effect of the image seg­ ments, the computer is programmed such that the equation is applied twice, once for the two 17 segments, and once for one o f the segments and the image o f the other. The sum of these values gives us the total mutual ground resistance between the two conductor segments. The mutual resistance o f segment m, as seen by the segment n is equal to the mutual resistance of segment n as seen by the segment m (ie), Rmn = Rnm. TABLE I. NOMENCLATURE FOR SELF AND MUTUAL GROUND RESISTANCE EXPRESSIONS L = length of the conductor segments r = radius of the conductors h = depth of burial of the grid Xm5YmlZm = x,y,z coordinates of the mth segment Xn5YntZn = x5y,z coordinates of the nth segment \ Xm-X j= a \yb-y„i=i> i L2+M 2=L1 a2+ b2 = Ci ^a2+b2+4h2 = c2 i ( a + L f + b2 = d1 V(o +L)2 + b2+4h2 = d2 i ( a - L ) 2 + b2 = e1 V (4 - L f + b2+4h2 = e2 d-y+a +L =Z1 d2+a +L =f2 Cj+a-L =g, e2+ a - L = g 2 (C1 +ti) =J1 (c,+ 412=Jz L L La +—= +h tan G1)] (5.8) (5.9) A(X) = [4X+7t(p/pJ]/[7t(l FtanG1)] G1 = I20x[e-X\ - K - ln(l - X)}] + 52 (5.11) (5.10) 37 65 - 64 - 62 - 59 - 57 - 56 - 55 - 54 - Figure 4 - Variation in 0, with h, for p/p, = 0.005 to 0.5 Model 2. Truncated Cone Model The truncated cone model is shown in Figure 5. The soil and the crushed rock layer are represented by a truncated cone. The angle of the cone in this case is different from the truncated cone and soil model, and is given by: G2 = (7 — 2.6//1) ( I + 1.7 /0 + 48 (5 .12) Again, the empirical formula for G2 is obtained by comparing the expression for B(X) given in equation (5.14), with F(X), so that the footing resistance is the same in both cases. 38 2b L I h surface layer o f crushed rock Figure 5 - Model 2. Truncated Cone Model The footing resistance in this case is determined by the resistance o f the cone, and is given as Figure 6 shows the variation in G2 with h, for various ratios o f p/p,. The values o f B (X ) computed for the range o f parameters specified in equation (5.7), are given in Table 5. The computed values are once again not very different from the values obtained through the infinite series, and the errors are not more than 4%. The two models have been tested to be quite accurate for the range specified in equation (5.7) for h and p/p,. Though the truncated cone and soil model may be used beyond this range, the truncated cone model gives erroneous results for values not in this range. (5.13) B(X) = 4X/[n(l +X IanG2)] (5.14) 39 TABLE 7. ACCURACY OF FINITE EXPRESSION AND MODELS F(X) - From summation of infinite series H(X) - From finite expression (equation 5.11) A(X) - From Model I B(X) - From Model 2 R. - p /p FUNCTION %ERROR h(cm) R, F(X) H(X) A(X) B(X) H(X) A(X) B(X) 7.5 2 0.733 0.734 0.737 0.709 0.14 0.55 -3.27 7.5 5 0.532 0.538 0:540 0.545 1.13 1.50 2.44 7.5 10 0.455 0.464 0:464 0.468 1:98 1.98 2.86 7.5 20 0.414 0.426 0.423 0.421 2.90 2.17 1.69 7.5 50 0.389 0.402 0.397 0.390 3.34 2.06 0.26 7.5 100 0.380 0.394 0.389 0.379 3.68 2.37 -0.26 7.5 200 0.376 0.390 0.384 0.373 3.72 2.13 -0.80 12.5 2 0.825 0.826 0.815 0.826 0.12 -1.21 0.12 12.5 5 0.691 0.693 0.685 0.697 0.29 -0.87 0.87 12.5 10 0.640 0.642 0.636 0.641 0.31 -0.63 0.16 12.5 20 0.612 0.615 0.610 0.608 0.49 -0.33 -0.65 12.5 50 . 0.595 0.598 0.594 0.588 0.50 -0.17 -1.18 12.5 100 0.589 0.592 0.589 0.581 0.51 0.00 -1.36 12.5 200 0.586 0.589 0.586 0.577 0:51 0.00 -1.54 17.5 2 0.872 0.872 0.862 0.882 0.00 -1.15 1.15 17.5 5 0.773 0.774 0.778 0.791 0.13 0.65 2.33 17.5 10 0.735 0.736 0.749 0.751 0.14 1.90 2.18 17.5 20 0.714 0.715 0.734 0.728 0.14 2.80 1.96 17.5 50 0.702 0.703 0.725 0.714 0.14 3.28 1.71 17.5 100 0.697 0.698 0.722 0.709 0.14 3.59 1.72 17.5 200 0.695 0.696 0.721 0.706 0.14 3.74 1.58 20.0 2 0.887 0.887 0.876 0.900 0.00 -1.24 1.47 20.0 5 0.800 0.800 0.803 0.825 0.00 0.38 3.13 20.0 10 0.766 0.767 0.778 0.792 0.13 1.57 3.39 20.0 20 0.748 0.749 0.766 0.773 0.13 2.41 3.34 20.0 50 0.737 0.738 0.759 0.762 0.14 2.99 3.39 20.0 100 0.733 0.734 0.756 0.757 0.14 3.14 3.27 20.0 200 0.731 0.732 0.755 0.755 0.14 3.28 3.28 40 66 — 64 - 62 - 60 - 58 - 56 - 54 - 52 - 50 - 46 - 44 - 40 38 - h (cm) Figure 6 - Variation in O2 with h ,for p/p, = 0.005 to 0.5 41 CHAPTER 6 ANALYSIS OF SYSTEM OF INTERTIED GRIDS Basic Considerations The grounding grids o f two substations are likely to get connected unintentionally through overhead ground wires on transmission lines, underground cables, metallic pipes etc. going from one substation to the other. Sometimes, the interconnection may be intentional in the form of a bare grounding conductor, connecting the two grids. When a fault occurs in one of the inter­ connected substations, it is important to determine the ground resistance o f the system of two gnds, and the interconnecting conductor. A lso o f interest is the ground potential rise at the two substations, and the voltage profile along the interconnecting conductor. For an analysis to determine the potential rise, system ground resistance, and the voltage profile, the following data are considered necessary: (i) The resistivity of the soil, p, which is considered to be uniform. (ii) The ground resistance o f the two grids, R1 and R2, at the two substations. (iii) The number n, length X, and radius a, o f the underground bare conductors going from one station to the other. A lso required is the resistivity pc o f the conductors. It is assumed that all the conductors are o f the same length, radius, and are made of the same material. The current injected at one substation gets divided over the interconnecting conductors. It is thus logical to ignore the voltage drop along the conductors of the grid. The grounding grids at the two substations are represented by equivalent circular plates of radius T1 and r2 respectively, such that Ri — p/(4r j) Rz — p/ (4*2) (6.1) 42 Underground conductors are usually buried at a shallow depth, and their length is usually much longer than the spacing between them. A system of interconnecting conductors can thus be represented by a single equivalent conductor, having distributed parameters comprising the internal impedance Z, which may be represented as resistance R, and inductance L; and the external admittance Y comprising conductance G, and capacitance C. At power frequencies, the shunt current drawn by the capacitance is negligible as compared to the current drawn by the conductance. Thus, the shunt capacitance may be ignored for all practical purposes. The impedance and admittance may be hence defined as Z =R+J(Inf )L (6 .2) and Y = G (6.3) where nna n ohms/m P 2 t o ? - iA to?-iA siemens/m henries/m (6.4) (6.5) (6 6) where A is the geometric mean radius of the conductors o f the intertie in meters, and X is the length o f the inter tie in meters. Ladder Circuit Solution A solution to the problem of evaluating the potential rise, ground resistance and voltage profile, can be obtained by dividing the equivalent intertie conductor into a number of sections, o f equal length, each of which may be represented as api section, cascaded to form the conductor. 43 X/m, has an internal impedance o f ZX/m, and an external admittance o f YX/(2m) at each leg of the pi-sections. These m pi sections, and the two circular plates representing the grounding grids, form the system under consideration. The mutual resistance between the two grids is given as -r sin" X + i'i + A12 (6.7) where r — T1, if r1>r2, and r — r2, if r2>r ,^ and p is the resistivity o f the soil in ohm-meters: The mutual resistance between one grid and any node point p o f the ladder network is given by P sin"1 2nr r +x (6 8) where r is the radius o f the equivalent plate representing the grid, in meters and x is the distance of the node p from the grid, in meters. The mutual resistance between two nodes p and q, which are located at the end of the p111 and qth segments of the intertie, each of length X/m, is determined as vq InXIm p__ j f Xpq +X/2m Xna-X I lm (6.9) where Xpq is the distance between the two nodes p and q. When current I is injected at grid I, it is dissipated to ground from each grid, and by each node of the pi-section. A computer program INTEG has been developed to determine the current dissipated and the voltage at the two grids and each node o f the pi-sections forming the intertie. For a ladder network composed of N nodes, the current dissipated and voltage, at each o f the N nodes, form 2N unknowns that are to be determined. The following 2N equations are available: I. N equations giving the voltages at the nodes in terms of the currents dissipated at each node, and the self and mutual ground resistances of the nodes. 44 2; N -I equations giving the difference in the voltage o f adjoining nodes in terms o f the series impedance and the current flowing between the two nodes. 3. One equation given by the sum of all currents dissipated at the nodes, which is equal to the known fault current injected into the system at the first grid. The above 2N equations may be solved for the 2N unknown variables. Using the program INTEG, more than 1000 cases were studied for systems in which the following variations were observed in the resistance o f the two grids, soil resistivity, and characteristics o f the intertie. R1 and R2 = I to 20 ohms, p = 50 to 3000 ohm-meters Interconnecting conductor: Length = 100 to 3000 meters Radius = 5.84 mm (4/0 AWG solid conductor) Number = 2 to 6 Spacing = 0.25 to 5 meters The voltage at the two grids, as obtained for a few representative cases are given in Table 8 . The current dissipated and voltage at the grids and nodes, for a sample case, is given in Table 9. The resistance o f the first grid is I ohms. For a fault current o f 1000 amperes, the ground potential rise o f grid I, would be 1000volts. Due to the intertie and the second grid, the effective resistance o f the system, as seen by the fault.current falls to 178/1000 = 0.178 ohms. An Approximate Solution to the Problem The solution of the ladder circuit is simplified if the mutual resistances between the nodes of the pi-sections are neglected. The transmission line equations can then be applied to the 45 intertie. The characteristic impedance Z0 and the propagation constant y o f the intertie are then given by Z0 = (ZIY)112 Y=(ZT)1'2 (6.11) (6 .12) The ladder circuit may then be solved by adopting the foUowing methodology: I-The ground resistance o f the intertie alone is determined by calculating the input impedance o f the intertie at end I. This is given as [18] Zi =Z 0Ztanh(YX) (6.13) 2-The ground resistance of grid 2, R2, is then connected at the far end of the intertie. The ground impedance o f the iritertie and grid 2 as seen from end I is given as [18] Zi2=Z l R2+X0 Ianh(YX)) (6.14)Z0+R2Ianh(YX) J 3.The ground resistance o f grid I, R1, is in parallel with Zi2. There is also mutual ground resistance between grid I, and the ground system formed by the intertie and grid 2. An approximate value o f the mutual ground resistance Rm, is obtained as P sin"1 i\ +X /2 i f I Zi - Z 12 |<| R2 - Z i21 -Rn from equation(6.1) i f \Zi - Z i2 \>\R2- Z i21 (6.15) (6.16) 4. The ground impedance of the complete system comprising the two grids and the intertie is then given by Zsys = (RiZi2 -RmV(Ri +Z i2 - 2Rm) (6.17) 5. The voltage at the first grid is then given by V1=IpZsys (6.18) where If is the injected fault current. 46 6 -The current carried by the intertie at end I, I, is then calculated as 1 Ri+Zi2-TRm <6-19) 7 .The voltage at the second grid, V2, is obtained as V2 = V1 coshCyX) -Z iZ0Sinh(YX) (6.20) 8 .The voltage at any point on the intertie, can be determined by replacing X b yx in equation (6 .20 ), where x is the distance o f the point under consideration from grid I. A comparison o f the voltages at the two grids, obtained from this method, for a few representative cases, is made with the voltages as obtained from the ladder circuit method in Table 8 . The results may be observed to be quite accurate. Voltage Decay Along a Very Long Buried Conductor For an infinitely long conductor buried in the ground near the surface, the shunt con­ ductance and series inductance of the conductor are so related that a simple approximate expression for the propagation constant is obtained. Equations (6.4), (6.5), and (6 .6 ) can be simplified for a copper conductor to give 1.7 x KT 2 , .74X + l 2 3 7 Mtrln (6.21) Under practical conditions, the real part of ZY varies from 2.38 to 93.15, and may hence be neglected as compared to the imaginary part. Thusi Z Y = J - I Q - * P (6.22) 47 which implies Y=VzF (6.23) (6.24) (6.25) Now, Y=CH-Jp, where a and pare the attenuation constant andphase constant respectively. The attenuation constant a may be simplified to give a=l/(100Vp) (6.26) a is a measure o f the rate at which the voltage decays along the length o f the conductor. The length at which the voltage drops to 36.8% of its initial value is defined as the length constant, X, and is given by X = 1/a = lOOVp (6.27) The length constant may be observed to depend primarily on the resistivity of the soil. At a distance of 2X, the voltage on the intertie will drop to 13.5% of its initial value. Therefore, the two station grids at a distance o f more than 2X ihtertied with a buried bare copper conductor, will have negligible effect on each other. 48 TABLE 8. VOLTAGE AT THE TWO GRIDS, CALCULATED BY APPROXIMATE METHOD Radius of each conductor = 0.00584 m.(4/0) Depth of burial = 0.5 m. Current injected = 1 kA. Resistivity of soil = P ohm-m. Resistance of first grid = R1 ohms. Resistance of second grid = R2 ohms. Length of conductors = L m. Number of conductors = n Spacing between conductors = S m. Voltage at Grid I = V1 volts. Voltage at Grid 2 = V2 volts. Ladder Ckt Approx Meth P R, R2 L n S V1 V2 V1 V2 50 I I 700 2 5.00 191 123 215 111 50 I I 700 6 0.25 178 125 204 114 50 I I 700 6 5.00 119 91 146 85 50 i I 1000 2 0.25 232 90 265 73 50 I I 1000 2 5.00 197 80 232 65 200 5 5 700 2 5.00 615 574 621 566 200 5 5 700 6 0.25 584 561 590 554 200 I I 1000 2 0.25 395 278 422 242 200 I I 1000 2 5.00 356 262 382 229 200 I I 1000 6 0.25 338 268 367 234 500 I 5 1000 2 5.00 598 562 597 546 500 I 5 1000 6 0.25 585 565 585 546 500 I 5 1000 6 5.00 501 486 505 475 500 I 10 1000 2 0.25 653 622 653 610 500 I 10 1000 2 5.00 618 588 619 578 1000 10 10 300 2 5.00 2854 2841 3397 3384 1000 10 10 300 6 0.25 3493 3489 3330 3325 1000 10 10 300 6 5.00 1965 1961 2710 2705 1000 5 5 700 2 0.25 1847 1810 1782 1735 1000 5 5 700 2 5.00 1741 1707 1671 1628 3000 10 5 3000 6 5.00 1529 1404 1556 1308 3000 10 10 3000 2 0.25 2416 2145 2429 2047 3000 10 10 3000 2 5.00 2196 1964 2209 1877 3000 10 10 3000 6 0.25 2103 1952 2120 1867 3000 10 10 3000 6 5.00 1667 1570 1694 1515 49 TABLE 9. CURRENT AND VOLTAGE PROFILE ALONG THE INTERTIE Resistivity of soil Length of connecting conductors Radius of each conductor Number of conductors Spacing between conductors Current injected at grid I Resistance of grid I Resistance of grid 2 Current dissipated at node p Voltage at node p = 50 = 700 = 0.00584 = 6 = 0.25 = 1000 = 1 = 1 ohm-meters meters meters (AWG 4/0 solid conductor) meters . amperes ohm ohm amperes volts node ip . v p I 154 178 a 63 178 b 97 165 C 88 154 d 81 146 e 77 139 f 74 134 g 71 128 h 70 127 i 71 126 k 42 125 2 103 125 50 CHAPTER 7 CONCLUSIONS ANDPUTURE WORK The expressions presented in this thesis, for the calculation of ground resistance, and mesh and step voltages, are easy to use, and should be useful in the design o f substation grids. Existing formulas for the calculation of these parameters, are applicable only to square grids, or rect­ angular grids with a limited length to width ratio. When these formulas are used for the design of grids with other shapes, the results could be highly erroneous. Since the errors from the use of existing formulas are usually on the positive side, they lead to substantial wastage of material. Since the new expressions reduce these errors, they should lead to considerable savings in material, and labour. The validity o f the expression for ground resistance has been verified for grids whose shapes and sizes are given in Figure 2. The use o f this expression has also been verified for the two extreme cases o f a strip conductor, and a circular plate, and through the use of scale models. It is thus expected that the equation should be valid for all intermediate shapes and sizes o f grids. All the grid shapes given in Figure 2 have a common characteristic in that they use the factor VA ILp as a measure o f the length over which the area is spread. This is only valid for grid shapes for which a straight line drawn perpendicular to the periphery at any point, when extended outside the area o f the grid, always remains outside. The suggested equation, thus, is not valid for grid shapes which do not satisfy this criteria. Rarely are such grids encountered. 51 A similar limitation exists for the expressions developed for the calculation o f step and mesh voltages. It may have been observed that the errors in the calculation o f the step voltage are higher than those for mesh voltage calculations. As mentioned earlier, this is usually acceptable. In case higher accuracy is required, new expressions for step voltage calculations may be warranted. The purpose o f developing a new expression for calculating the footing resistance within a substation yard, has not been reduction o f error, but the ease o f use. The expression developed on the basis o f the two-layer soil model, and recommended by the TF.P.F. Std. 80, is not easy to use, and requires the help o f a computer or a programmable calculator. The finite formula suggested by Sverak, as mentioned earlier, gives errors as high as 50% in some cases, and is thus not readily applicable. The new expression, and the models developed in this thesis, do not give errors o f more than 4%, for any o f the cases considered, which cover the usual practical situations. The two models presented give information regarding the region of crushed rock below tire foot that effectively contributes to the footing resistance. This information is useful for carrying out field investigations. The method to evaluate the performance o f intertied grids is easy to use, and gives an improved idea o f the likely behaviour of such a system. The simplified method, which is based on the transmission line equations, should be used to make a quick analysis o f the system, and should not be used as a final analysis tool. For comprehensive and acurate calculations, the program INTEG is preferred. It is interesting to note that the length constant depends primarily on the resistivity o f the soil. At a distance o f 2X, the voltage on the intertie will drop to 13.5% of its initial value. Thus, two substation grids located a distance more than 2X apart will have negligible effect on each other. 52 The expressions for the calculation of step voltage have been adapted from the calculations carried out for the mesh voltage. As Suchi the errors for the step voltage calculations are higher. More work needs to be done in this area. One possibility is the development o f a separate current irregularity factor Kj for step voltage as a function o f n, and possibly, the depth o f burial. The equations as presented in this paper cannot be applied to the practical design o f a grid, as rarely is the grid configuration known ahead o f time. It is thus necessary to formulate a design strategy, where, once the safety limits for the step and maximum touch voltages are established, the addition o f grid conductors can be done on an incremental basis. Such a formulation can incorporate the equations presented in this paper, and be applied to grids o f all practical shapes. The results presented here have been collected on the basis o f computer runs and analog model tests. It is very important to collect data on the basis o f measurements made in actual substation sites, and make a comparison of the measured and calculated data. This would also give a good indication of the direction in which future work should proceed. 53 REFERENCES CITED 54 [1] "IEEE Guide for Safety in AC Substation Grounding" ANST/TF.F.F. SM Rn" 1986. [2] Dalziel,C.F„ F.P.Massoglia, "Let-Go Currents and Voltages". ATRF Transactions Vol.75, Part H, 1956, pp.49-56. --------------------------- . 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