10th NATIONAL CONGRESS OF ELECTRICAL-ELECTRONICS-COMPUTER ENGINEERING CALCULATION OF POTENTIAL DISTRIBUTION AROUND A ROD GROUNDING STRUCTURE USING THE FINITE DIFFERENCE METHOD Özcan KALENDERLİ1 Ersan ŞENTÜRK2 Okan İhsan ÖZTÜRK3 1Istanbul Technical University, Faculty of Electrical-Electronics Engineering, Department of Electrical Engineering 2, 3Turkcell Communication Services Inc. 1e-mail: ozcan@elk.itu.edu.tr 2e-mail: ersan.senturk@turkcell.com.tr 3e-mail: okan.ozturk@turkcell.com.tr Keywords: Electrical Grounding, Potential Distribution, Finite Difference Method SUMMARY OF THE PROBLEM In this study, the numerical calculation of the potential distribution around rod ground electrodes, one of the commonly used types of ground electrodes in electrical grounding, was performed using a computer program method developed with two-dimensional finite difference calculations in cylindrical coordinates. The study provides the magnitude and variation of step and touch voltages, which are important for the safety of living beings, and offers a different approach option for calculating the potential electrical distribution around them. 1. INTRODUCTION Electrical grounding is an electrical installation used to protect dangerous electrical circuits, objects, elements, and living beings from the voltages we use and to provide the necessary ground potential for the operation of electrical systems. It consists of conductors called ground electrodes buried in the ground that connect the place to be grounded to the ground. It performs its protective function by safely diverting the currents generated under fault conditions to the ground without creating dangerous voltages for its surroundings. Simply put, the voltage generated on and around a grounding system depends closely on the type, dimensions, burial environment, and conditions of the grounding material, the ground electrode. High currents pass through the system during short circuits or electrical discharges such as lightning strikes, bursts, contacts, or connections. The voltage level generated in the grounding system and the structures connected to it by these currents depends on the grounding resistance, or more generally, the grounding impedance. Grounding resistance, in turn, depends on the properties of the soil and the ground electrode. This seemingly simple chain of dependencies has been the subject of many studies [1-4]. Determining the grounding resistance based on the geometry (rod, grounding strip, pipe, etc.), dimensions, plate, depth, and size of the ground electrode has been the focus of experimental studies [1-6]. Many theoretical and theoretical formulas and methods for calculating grounding resistance are given in the literature, regulations, and standards for grounding; Several measurement methods are described for determining grounding resistance by measuring it [7, 8]. Ultimately, when a current passes through a ground electrode with a certain resistance value, a voltage is generated in the ground electrode relative to the reference ground, and a potential distribution occurs between the ground electrode and the reference ground (Figure 1). Here, the concept of ground refers to the section of ground approximately 20 m away from the ground electrode, where the potential is theoretically considered zero in the ground electrode potential distribution. In practice, the ground around which the reference ground is formed is a U(V) axis. Figure 1. Potential distribution around a ground electrode. U(V): Voltage axis x(m): Distance axis Utk: Ground electrode voltage Ua: Step voltage 1. Potential distribution 2. Ground 3. Ground electrode 4. Reference ground The potential distribution around a ground electrode determines the magnitude of the touch and step voltages. The potential difference between the point of contact and a point 1 m away from it is called the touch voltage; the potential difference between two feet in a 1 m long step is called the step voltage. 197 ELECTRICAL - ELECTRONICS - COMPUTER ENGINEERING 10TH NATIONAL CONGRESS Both voltages must be within limits that are not dangerous. To be able to say something about these voltages, it is necessary to know the potential distribution. In practice, various formulas are used to obtain this information [7 - 8]. 1(1j,iV1j,iV ++ +− + j,1iV) + + h r2 h r2 1( −+ j,iV4j,1Vi ) − − = 0 (2) In this study, the potential distribution around the grounding rod, which is commonly used in practice, was calculated using the finite difference method. It will be one of the types of grounding rods [9]. In this equation, written for each node of the network whose potential is unknown, h is the mesh size or step size of the network, and r is the coordinate of the node where the equation is written. 3. ROD GROUNDING AND FINITE DIFFERENCE MODEL A rod grounding electrode with a diameter of 16 mm and a length of 2 m was considered as a model to calculate the potential distribution around a grounding electrode. In accordance with the concept of reference ground, it was assumed that the potential value is zero volts at every point 20 m away from this grounding electrode. The model created with this in mind is shown in Figure 3. The side of each square mesh was taken as h = 2 meters. In this way, the solution region where the problem is examined consists of 90 square meshes and 112 nodes. 20 m 2. FINITE DIFFERENCE METHOD The finite difference method (FDM) is a numerical method also used in potential distribution calculations [9]. Its principle is based on calculating the potential distribution in a closed region whose potential distribution is given by the Laplace or Poisson equation using finite difference equations for the derivatives known from the numerical derivative topic in numerical analysis. For this, for example, the study region is divided into a network with square, rectangular or triangular meshes (Figure 2). In two-dimensional problems, y j+2 yj+1 yj yj-1 yj-2 h V i, j+1 2 Vi-1,j 3 Vi,j Vi+1,j 0 1 Vi,j-1 4 h x i-2 x i-1 xi x i+1 xi+2 Figure 3. Example of a two-dimensional, square-mesh finite difference mesh in Cartesian coordinates. Finite difference equations are written in place of Laplace or Poisson equations at the nodes of the known mesh. Thus, a linear system of equations containing the nodes and their potentials is obtained. In these equations, the linear system of equations is solved using boundary conditions or known node potentials, and the unknown node potentials are found. In this study, a mesh structure with square meshes is used. Finite difference equations are written in cylindrical coordinates in accordance with the geometry of the problem under investigation. The two-dimensional Laplace equation in cylindrical coordinates is; 2 V ∂ 2 r ∂ + 1 r V ∂ r ∂ + 2 V ∂ 2 z ∂ = 0 (1) Figure 3. Rod ground electrode SFY model. The potential values at nodes 1, 12, 23, 34, 35, 45, 55, 56, 65, 74, 83, 84, 91, 92, 93, 94, 98, 99, 100, 101 and 112 in the square-mesh section of the model shown in Figure 3 and with node numbers given in Figure 4 are zero volts because each of these nodes is 20 meters away from the 2-meter long copper rod electrode ground electrode placed between nodes 11 and 102. The potential values of nodes other than those numbered 1, 11, 12, 23, 34, 35, 45, 55, 56, 65, 74, 83, 84, 91, 92, 93, 94, 98, 99, 100, 101, 102 and 112, whose potential values are known, are unknown. Here, r and z are cylindrical coordinates, V = V(r, z) is the potential. The finite difference expression of equation (1) is: In this study, it is aimed to calculate the potential values of the nodes whose potential values are unknown using the Finite Difference Method (FDM). The finite difference equation for each node was written separately as in expression (2), and the resulting set of equations was used to find the potential values of the nodes using linear equations in the MATLAB 6.0 program. Considering the creation and arrangement of the equations used in the calculations, a Visual Basic-based program was prepared in MS EXCEL, and after the arrangement of the equations, the resulting matrix was obtained without errors and suitable for use. The solution of the MATLAB 6.0 program set was used for this equation. In the created set, when the coefficient matrix [A], which is formed by the coefficients at the beginning of the potential expressions of the nodes with unknown potential values, the matrix [x], which is formed by the potentials of the nodes with unknown potential values, and the matrix [B], which is formed by the potential values of the nodes with known potential values, a relationship of [A] . [x] = [B] is obtained between these matrices. Since the number of unknown nodes is 88, the size of the coefficient matrix that needs to be created will be 88 x 88. 102 103 104 105 106 107 108 109 110 111 112 11 22 33 44 54 64 73 82 90 97 101 10 21 32 43 53 63 72 81 89 96 100 9 8 7 6 5 4 3 2 1 20 31 42 52 62 71 80 88 95 99 19 30 41 51 61 70 79 87 94 98 18 29 40 50 60 69 78 86 93 17 28 39 49 59 68 77 85 92 Figure 4. Grounding system SFY solution network. After solving the equation system, the potential values obtained at the nodal points of the finite difference network are shown in Figure 5. As seen in Figure 5, the potentials in terms of nodes 11 and 102 were accepted as 100 Volts in order to evaluate and facilitate the transition to other voltage values by normalizing them. Solution Percentage (%) 100 81.0671 67.7645 56.7853 47.0328 38.0457 29.6045 21.6069 14.0187 6.8363 0 100 82.771 68.9501 57.5002 47.48 38.3541 29.8448 21.8161 14.2058 6.9696 0 79.2723 72.7284 63.734 54.5486 45.6359 37.0624 28.8178 20.8966 13.339 6.2889 0 68.0944 65.0695 58.9175 51.4519 43.513 35.4673 27.4736 19.6018 11.9199 4.7487 0 60.6368 58.7962 54.226 48.0115 40.9054 33.3571 25.6061 17.72 9.5115 0 0 54.9952 53.6233 49.8955 44.4699 37.9494 30.7839 23.2767 15.5893 7.8219 0 50.3966 49.2151 45.8764 40.8313 34.5894 27.5954 20.2344 12.7042 5.5226 0 46.4521 45.2994 41.9806 36.8693 30.4896 23.2715 15.8827 7.9407 0 0 42.9517 41.6615 37.8777 31.9569 24.7747 16.2924 9.3672 0 0 39.828 38.1322 32.8824 24.0289 15.2977 0 0 0 37.2401 34.6522 25.1043 0 0 0 0 0 0 Figure 5. Potential values of the nodes in volts obtained by SFY. The potential distribution on the ground surface to be drawn with these obtained potential values is of great importance in terms of calculating step and touch voltages. Figure 6 shows the potential distribution on the ground surface obtained by the SFY analysis. U (V) 120 100 80 60 40 20 0 1 2 3 4 5 6 7 8 9 10 11 Distance (m) Figure 6. Potential distribution on the ground surface around a rod ground electrode. 4. RESULTS As seen from these calculations, the potential distribution around a rod ground electrode, which is a type of electrical ground electrode commonly used in grounding, is finite ground electrode. 199 ELECTRICAL - ELECTRONICS - COMPUTER ENGINEERING 10th NATIONAL CONGRESS [2] Takahashi T., Kawase T., "Calculation of Earth Resistance for a Deep-Driven Rod in a Multi-Layer Structure", IEEE Transactions on Power Delivery, Vol. 6, No. 2, April 1991. [3] Meliopoulos APS, Xia F., Joy EB, Cokkinides GJ, ’An Advanced Computer Model for Grounding System Analysis,’ IEEE Transactions on Power Delivery, Vol.8, No. 1, April 1993. [4] Dawalibi F., Mukhedkar D., ’Influence of Ground Rods on Grounding Grids,’ IEEE Transactions on Power Apparatus and Systems, Vol. PAS-98, No.6, Nov./Dec. 1979, pp. 2089-2098. [5] Yıldırım, H., Kalenderli, Ö., Türkay, B., Çelikyay, M., "Computer-aided analysis of grounding networks", Electrical Engineering 6th National Congress, Bursa, pp. 130-133, 11-17 September 1995. [6] Hasse, P., Overvoltage Protection of Low Voltage Systems, IEE Power and Energy Series 33, United Kingdom, 2000, [7] ANSI/IEEE Std 80-1986, IEEE Guide for Safety in AC Substation Grounding, 1986. [8] Regulation on Grounding in Electrical Installations, TMMOB, Chamber of Electrical Engineers Bursa Branch, Bursa, 2001 (Published in the Official Gazette dated 21 August 2001 and numbered 24500). [9] Kalenderli, Ö., Finite Element Method Lecture Notes in Electrical Engineering, İ.T.Ü., 2003. The difference method can be easily found numerically. With these calculations made on the computer, it is possible to determine the potential distribution in terms of both touch and step voltage both accurately and quickly. When the calculated results are examined, it is seen that the potential and the potential difference per unit length decrease as the distance from the ground electrode increases. The effect of the ground electrode length can be seen by making calculations for different ground electrode lengths, and the effect of the burial depth can be seen by making calculations for different burial depths. It is known that this has effects that reduce the spreading resistance and step voltage. The advantage of this study is that it will be possible to obtain and see the change of the potential distribution and potential differences with the mentioned magnitudes. In conclusion, the potential calculation in these dimensions and depths can be evaluated, and safe and accurate design can be made. REFERENCES [1] Bogensperger JH, Frei J., Pack S., "Resistance of Grounding Systems Stationary and Transient Behaviour", Ninth International Symposium on High Voltage Engineering, August 28- September 1, 1995. 200
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