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Mutual interference of neighboring grounding systems and approximate formulation
Electric Power Systems Research 151 (2017) 166–173
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Mutual interference of neighboring grounding systems and approximate formulation J. Nahman ∗,1 , D. Salamon Faculty of Electrical Engineering, University of Belgrade, Serbia
a r t i c l e
i n f o
Article history: Received 3 December 2015 Received in revised form 8 March 2017 Accepted 23 May 2017 Keywords: Adjacent grounding systems Mathematical models Safety risks Approximate expressions
a b s t r a c t The paper develops mathematical models for the analysis of the mutual interference of closely positioned ground electrodes in order to assess the safety risks that might appear in various circumstances often characteristic for urban areas. The potentially hazardous situation that might arise in open pit mines at the equipment grounded by connection with the common ground electrode is also investigated. In the scope of the analysis of possible dangerous circumstances the maximum touch – and step – voltages that appear at the sites protected by the grounding grids are calculated for various distances between the grids and in case of uniform and nonuniform two – layer soil. Approximate expressions for assessing some interference effects are also provided and successfully checked for various cases. Based upon the results of the conducted series of calculations, the safety risks are determined for all investigated cases. It was shown that the highest safety risks can appear in the case when the neighboring grids are unintentionally connected as well as in the case when the neighboring grid is connected with some grounded objects, which is characteristic for urban areas and for open pit mines with common ground electrodes. © 2017 Elsevier B.V. All rights reserved.
1. Introduction The complexity of objects consuming electrical energy and their density in space is permanently increasing, which generates new problems concerning the safety of associated persons and equipment and needs for improving the methods for calculation and projecting of grounding systems. In the past, various mathematical models for a detailed analysis of complex grounding systems buried in uniform or two – layer soils have been developed based upon the mean potential concept in calculation of the mutual influences among the conductors of the complex ground electrode [1,2]. The application of the finite element modeling, that increases the computational burden to some extent, has been more recently used for more detailed calculation of various phenomena associated with ground electrodes and neighboring metallic structures linked with or buried in close proximity to them [3–5]. In Ref. [6] the analysis of the mutual influence between the equipment ground electrode on the high voltage side of 10 kV/0.4 kV distribution system transformer stations and the neutral ground electrodes on the low voltage side has been performed. The proximity effects for various spacing of these electrodes when alone and in the presence
∗ Corresponding author. E-mail addresses: [email protected], [email protected] (J. Nahman). 1 Freelance consultant for electrical power systems. http://dx.doi.org/10.1016/j.epsr.2017.05.029 0378-7796/© 2017 Elsevier B.V. All rights reserved.
of neighboring metallic pipes or other conducting structures have been also studied. The finite element modeling has been applied to analyze the potential distribution on the area surrounding ground electrodes as well as the potentials transferred by various metallic conductors buried in close proximity to the grounding area in order to identify possible hazardous situations for persons and equipment [4,7]. This analysis has been conducted both for uniform and two – layer soils. The interference phenomena between grounding systems have been recently detailed considered for assessing the grade of their independency. The matrix calculation method presented in Ref. [8] has been applied to study interactions between the substation grid and the safety ground bed in mining installations [9]. In a recent paper [10], the analysis of the effects of the interference between the ground electrode and fundaments of a transmission line tower and the ground electrode of the adjacent building has been performed and adequate protection measures elaborated. Reference [11] analyzes the effects of neighboring ground electrodes on the distribution of the potentials over an area around the interacting electrodes both in the vicinity of the active and the influenced ground electrode. This paper analyzes the potentially hazardous circumstances that can appear on the site protected by a grounding grid being in close proximity to another grounding grid dissipating the ground fault current into the surrounding soil. The maximum touch – and step – voltages that appear at the site of the neighboring ground-
J. Nahman, D. Salamon / Electric Power Systems Research 151 (2017) 166–173
ing grid are determined for various distances between the grids in the following characteristic cases that can appear in practice: a) the neighboring grid has no connections with other grounded facilities; b) the considered grounding grids are intentionally or unintentionally linked and c) the neighboring grid is connected to a grounded object. These analyses have been carried out both for uniform and two −layer soils. Approximate, comparatively simple expressions are also derived in the paper for assessing some interference effects and successfully checked for various cases. 2. Mathematical models 2.1. Relationships connecting two neighboring ground electrodes Let us consider two ground electrodes buried in close proximity. The ground electrodes are composed of sets of interconnected straight line linear elements. It is assumed that ground electrode 1 dissipates current J1 into the ground due to a ground fault in the installation associated with this electrode. General expression correlating the potentials of the electrodes elements and the currents leaking from them into the surrounding soil has the following form [2,6], [C1 ] · [I1 ] = J1
(1)
[r11 ] · [I1 ] + [r12 ] · [I2 ] = [C1 ] · E1
(2)
[r21 ] · [I1 ] + [r22 ] · [I2 ] = [C2 ] · E2
(3)
It is implied that the numbers of elements of ground electrodes 1 and 2 are N1 and N2 , respectively. [I1 ] and [I2 ] are column vectors of currents dissipating into the ground from the elements of the corresponding ground electrodes, whereas E1 and E2 are potentials of these elements. The elements of square matrices [r11 ] and [r22 ] are the self- and mutual resistances of the elements of electrodes alone. Elements of matrix [r12 ] are mutual resistances of the elements of ground electrode 1 with the elements of ground electrode 2. Matrix [r21 ] is transposed matrix [r12 ]. [C1 ] and [C2 ] are N1 and N2 -dimensional column vectors of units and [C1 ] is transposed vector [C1 ]. Resistances rjk being elements of resistance matrices in Eqs. (2) and (3) are calculated using the mean potential method. It is implied that the ground electrodes are composed of a set of connected straight-line thin round conductors. Each conductor is replaced by an ellipsoid of revolution with its foci being the ends of the conductor and its smaller axis being equal to the diameter of the conductor. This model of a conductor makes it possible to calculate the potential caused by the current discharging from the conductor at any point in the space using a simple formula based on the distances of this point from the conductor’s ends. By applying this expression formed for conductor j we can determine the mean potential of any conductor k caused by the current discharging from conductor j. By dividing this potential by conductor j current we obtain
Fig. 1. Grounding grids in close proximity.
167
the mutual resistance rjk between these two conductors in case when both conductors lie in a discontinues media with the same ground resistivity. For uniform soil the effect of discontinuity at the ground surface is taken by introducing the image of conductor j with respect to the ground surface. The potentials of conductor k are now calculated by summing the potentials generated by conductor j and its image. In the case of two horizontally stratified layers with different resistivities various images of conductor k are used depending in which of the layers conductors k and j are located. All the details of the calculations both for the uniform and the two-layer soil cases are presented in Ref. [2]. Some of the results obtained by the described calculation method have been successfully checked by modeling in electrolytic tank [6]. A more detailed description of the applied calculation method and a comparison with the exact analytical calculation model presented in Ref. [15] are given in Appendix A. As can be concluded from expressions (1)–(3), there are N1 + N2 unknown currents emanating from the elements of ground electrodes as well as unknown potentials of both ground electrodes. These expressions give N1 + N2 + 1 relationships among the considered variables. The missing relationship, completing the system of equations, depends on the circumstances to be analyzed, as will be discussed further on.
2.2. Electrode 2 is not connected to any other ground electrode In the case when ground electrode 2 is not connected to any other ground electrode the expression completing the system of equations given before is, Jc2 = [C2 ] · [I2 ] = 0
(4)
as current Jc 2 injected into the ground by electrode 2 is null in considered case because the currents from electrode 1 only pass through the elements of electrode 2 with the same total input and output values. The formed system of equations can be now solved for [I1 ], [I2 ], E1 and E2 . The touch – voltages over the site occupied by ground electrode 2 are caused by the difference of the potentials generated by electrode 1 on the ground surface above electrode 2 and the potential E2 . The potential on the ground surface at a standing point x can be calculated using the general expression:
Ex = [r1x ]
[r2x ] ·
[I1 ]
(5)
[I2 ]
with [r1x ] and [r2x ] designating N1 - and N2 -dimensional row vectors of mutual resistances of elements of ground electrodes 1 and 2 and the standing point at location x.
----- Route on the ground surface for calculating touch- and step-voltages.
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The touch – voltage over bridged by standing on the ground surface at point x and touching an object connected to the ground electrode 2 equals, Etx = |Ex − E2 |
(6)
The voltage over bridged by a step from point x to a point 1 m apart is, Esx = |Ex − Ex+1 |
(7)
2.3. Ground electrodes are connected In some cases the neighboring grounding systems are connected unintentionally or intentionally. If the connections are made intentionally in order to decrease the resistance to ground, they should be done by insulated conductors to prevent the appearance of step voltages in the vicinity of these ties. The analysis that follows has been conducted for this case. In the considered case ground electrodes 1 and 2 form a common grounding system. Eq. (1) should now be replaced by the following one, [C1 ] · [I1 ] + [C2 ] · [I2 ] = J1
Table 1 Results of the calculations performed for = 100 m. d m
E1 V
E2 V
Et1 V
Et2 V
Es2 V
10 20 30 40 50 60 70 80 90 100 110 120
781.6 789.0 791.9 793.4 794.2 794.7 795.1 795.2 795.4 795.5 795.6 795.6
252.9 213.8 186.7 166.1 149.8 136.5 125.4 116.1 108.0 101.1 95.9 89.5
186.4 188.7 189.5 190.0 190.2 190.3 190.4 190.5 190.5 190.5 190.5 190.5
56.1 40.2 31.1 25.1 20.7 17.3 14.8 12.7 11.0 9.7 8.6 7.6
8.6 6.3 4.8 3.8 3.1 2.5 2.2 1.8 1.6 1.4 1.2 1.1
(8)
The relationship completing the mathematical model in the considered circumstances is: E1 = E2 = E
(9)
The formed system of equations should be solved for [I1 ], [I2 ] and E. Then, by applying expressions (5)–(7) we can calculate touch – and step – voltages over the area of interest. The total current leaking into the ground from electrode 2 is now,
Jc 2 = [C2 ] · [I2 ]
(10)
and the current dispersed from electrode 1, Jc1 = J1 − Jc2
(11)
2.4. Electrode 2 is connected to a grounded object In urban areas many of source transformer stations of the distribution systems are positioned in close vicinity to the buildings. The buildings are usually very solidly grounded via their fundament grounding systems and due to the mutual connections with the grounding systems of neighboring buildings through water and other common metallic installations and structures. Due to this fact, the resistances to ground of the grounding systems of buildings are very low. In such circumstances, the routes of currents flowing through the soil in case of a ground fault in the transformer station will be more oriented to the ground electrode of the neighboring building, which affects the potential distribution on the ground surface between the substation ground grid and the ground electrode of the building. The situations with the ground electrode 2 connected to a grounded object are also typical for open pit mines. The safety ground beds that are used as common protective ground for the objects on the pit, such as excavators for example, are grounded through the natural resistance to ground of these objects. To investigate the effect of the connections of the ground electrode 2 with other ground electrodes, Eqs. (1)–(3) should be complemented by relationships (10) and (12), E2 = −Jc2 · R
(12)
with R being the resistance to ground of grounding systems of adjacent objects to which the ground electrode 2 is connected. It is important to mention that in the considered case the sum of the
Fig. 2. Ground surface potential distribution along the straight line route halving the edge meshes of the grids in case d = 30 m and = 100 m.
currents flowing into the ground from electrode 2 is not null but equals to the current flowing to the third electrode with minus sign. The formed system of equations should be solved for [I1 ] and [I2 ]. By applying Eqs. (5)–(7) we can then calculate the touch – and step – voltages in the area of interest. 3. Application examples 3.1. Considered ground electrodes Two square grounding grids having the same shape and parameters are considered (Fig. 1). The side lengths are 60 m and depth of burial is 0.7 m. Elements of the grids are cylindrical conductors with 1 cm diameter. The adjacent edges of grids are spaced d meters. The calculated resistance to ground for ground electrode 1, when alone and = 100 m, is 0.796 . The maximum touch – and step – voltages calculated for J1 = 1 kA along the route halving the grid edge meshes are Et1 = 190.5 V and Es1 = 27.5 V. 3.2. Electrode 2 is not connected to any other ground electrode As discussed in Section 2.2, the touch – voltages in the area occupied by ground electrode 2 when current J1 is leaking into ground from ground electrode 1 arise because of the difference that appears between the ground surface potentials and the potential E2 of the ground electrode 2. We have calculated the maximum touch – voltages Et 1 and Et 2 at electrodes 1 and 2 as well as step-voltages Es2 at ground electrode 2 along the route halving the edge meshes of electrodes for various distances in case J1 = 1 kA and = 100 m. The results of these calculations are presented in Table 1.
J. Nahman, D. Salamon / Electric Power Systems Research 151 (2017) 166–173 Table 2 Results obtained in the case 1 = 100 m and 2 = 50 m.
169
Table 4 Results of the analysis for linked electrodes for = 100 m.
d m
E1 V
E2 V
Et1 V
Et2 V
Es2 V
d m
E1 , E2 V
Et1 , Et2 V
Es1 , Es2 V
10 20 30 40 50 60
472.1 474.4 475.3 475.8 476.1 476.2
125.5 106.6 93.2 83.0 74.9 68.2
163.3 164.2 164.6 164.8 164.8 164.9
35.1 25.1 19.4 15.5 12.8 10.7
5.6 4.1 3.2 2.1 1.8 1.6
10 20 30 40 50 60 70 80 90 100 110 120
517.2 501.4 489.3 479.7 472.0 465.6 460.3 455.7 451.7 448.3 445.3 442.6
117.7 112.3 110.1 107.8. 106.1 104.7 103.7 102.8 102.1 101.5 101.1 100.7
26.6 25.7 25.1 24.6 24.2 23.9 23.7 23.5 23.4 23.3 23.2 23.1
Table 3 Results for the case 1 = 50 m and 2 = 100 m. d m
E1 V
E2 V
Et1 V
Et2 V
Es2 V
10 20 30 40 50 60
667.3 678.0 682.3 684.4 685.6 686.3
254.4 214.2 186.7 166.0 149.7 136.4
107.7 109.9 110.8 111.2 111.4 111.6
42.3 30.4 23.6 19.3 15.9 13.5
6.4 4.5 3.4 2.9 2.2 1.8
For illustration, Fig. 2 presents the potentials on the ground surface above the grounding grids from Fig. 1 along the straight line route halving the edge meshes of the grids for d = 30 m (Fig. 2). It is important to mention that all the variables in Table 1 are proportional to the ground resistivity. Hence, the magnitudes of these variables in case, say = 50 m, are obtainable by halving the corresponding results quoted in Table 1. The same analysis has been also conducted for non uniform soil cases, for comparison. Two soil layers are considered with the thickness of the upper soil layer being 2 m. Table 2 presents the results obtained if the resistivity of the upper and bottom soil layers are respectively 1 = 100 m and 2 = 50 m. Table 3 quotes the results for 1 = 50 m and 2 = 100 m. By comparing the results obtained for various soil structures we can observe that potentials E1 depend considerably on the resistivity of the bottom soil layer. They are comparable with the values obtained for the corresponding uniform soil cases with soil resistivity being equal to the resistivity of the bottom soil layer. The same is the case with the potentials E2 where the difference between the uniform and the two soil layers case is very small. As can be seen, the touch and step voltages generated at the electrode 2 by electrode 1 are comparatively low compared to the potential E2 that can reach high magnitudes even in case d = 120 m. However, bearing in mind that the calculations are performed for J1 = 1000 A, for higher magnitudes of this current all considered potentials increase proportionally.
3.3. Ground electrodes are connected If the considered ground electrodes are connected, owing to the geometrical symmetry the fault currents leaking from the grids are the same for both grids as well as the generated potentially dangerous voltages. The grids can be intentionally or unintentionally linked through various metallic installations such as cables of different kinds, metallic pipes or rail road tracks. Further on we have considered the case when this link is realized by an insulated cable with negligible impedance. Table 4 quotes the calculated parameters for the considered case for different distances d between the grids for uniform soil with = 100 m. Fig. 3 displays the potential distribution on the ground surface along the route halving edge meshes of considered ground electrodes if linked, for d = 30 m.
Fig. 3. Ground surface potentials along the route halving the edge meshes of linked electrodes for = 100 m and d = 30 m.
In all considered cases the maximum magnitudes of investigated potentially dangerous voltages appear at the corner meshes of the grids on their sides facing the open area. It is evident that the linking of electrodes causes much higher touch voltages at electrode 2 when compared to the case with separated grids. This indicates that, before intentionally connecting grid 2 with grid 1, it would be necessary to check if the security measures applied in installations grounded by grid 2 can allow such an increment of dangerous voltages and ground currents. On the other side, the parallel connection of electrodes decreases ground currents and touch – voltages associated with grid 1. 3.4. Electrode 2 is connected to a grounded object As stated before, in urban areas the distribution source transformer stations are often located in a very close proximity to the surrounding buildings of various kinds. On the other side, the grounding systems of these buildings are usually, intentionally or not, interconnected and, therefore, have often very low resistance to ground. In such circumstances it is important to assess the potentially dangerous voltages that can appear at the ground electrode of an adjacent building nearest to the substation. By applying the mathematical model presented before in Subsections 2.1 and 2.4 we have calculated the potential distribution on the ground surface between the grounding grid 1, implying that it is the ground electrode of the source distribution station, and the grounding grid 2 representing the ground electrode of the building closest to the substation. These calculations have been performed under assumption that the grounding grid of the building is connected to the ground electrodes of surrounding buildings which equivalent resistance to ground is R = 0.1 . We have calculated the potential E1 of
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Table 5 Results for Situation 3 in case R = 0.1 .
Table 6 Calculation results for the open pit mine case.
d m
E1 V
Et1 V
E2 V
Et2 V
Es12 V
E1 V
E2 V
Et2 V
Ees V
10 20 30 40 50 60
709.1 737.6 752.9 762.6 769.2 773.9
214.9 206.1 202.0 199.7 198.2 197.3
28.7 24.1 20.9 18.6 16.8 15.3
111.7 86.5 71.2 60.6 52.6 46.4
56.5 49.8 47.6 46.5 45.9 45.6
793.7
47.0
10.1
45.6
Table 7 Maximum tolerable voltages. , ˝m
50
100
Et 70 , V Et 50 , V Es 70 , V Es 50 , V
337.5 249.4 408.2 301.6
361.1 266.8 502.4 371.2
shown that the natural value of the resistance to ground of an excavating shovel used in the mine “Kolubara”, owned by Serbian Power System Company, equals approximately 0.016 . Hence, the total resistance to ground of the considered connection including the resistance of the cable 3 × 25 mm2 copper ground conductors is R = 1.65 . Table 6 presents the results of the calculations obtained for J1 = 1 kA. Ees is the potential transferred to the excavating shovel. 3.5. Assessment of safety risks
Fig. 4. Potential distribution on the ground surface along the route halving the edge meshes of Grounding grids in case R = 0.1 for d = 30 m.
grounding grid 1, maximum touch – voltages Et1 and Et2 at grounding grids 1 and 2 as well as the maximum step voltage Es12 in the area between these grids for various spacing of grids. The results of the calculations conducted for the considered circumstances for = 100 m are presented in Table 5. As can be seen, both the touch voltages at the ground electrode 2 and the step voltages in the area between the electrodes have comparatively high values. For illustration, Fig. 4 displays the potential distribution on the ground surface along the straight line route halving the edge meshes of grounding grids in case R = 0.1 for distance d = 30 m. The circumstances considered here can also often happen in mines with surface exploitation. The potential that might appear at the common ground electrode for the mine facilities in case of ground faults in the neighboring source transformer station have been analyzed in Ref. [9]. Using the model presented in Subsections 2.1 and 2.4 we shall analyze also the touch potentials that might appear at the common ground electrode as well as the potential that will be transferred to the mobile electrical equipment spread over the pit by taking into account their resistance to ground. A typical situation that will be considered is displayed in Fig. 5. An excavating shovel in an open pit coal mine is supplied from the source substation over a heavy duty mining power cable having phase and ground conductors. To prevent the transfer of the potentials of source transformer station ground electrode 1 to the supplied equipment, the cable ground conductors are separated from electrode 1 and connected to electrode 2 serving as a common ground for this equipment. Electrodes 1 and 2 are the same as considered previously in the paper. The adjacent edges of electrodes are 170 m apart which is twice the length of the diagonal of the ground electrode 1. Such spacing between the electrodes is taken as satisfactory with regard to the potential transferred from source ground electrode 1 to electrode 2 in IC 8835, which regulates mine installations in U.S [9]. The ground resistivity is supposed to be = 100 m. The cable length between the common ground electrode and the shovel is taken to be 200 m. The performed field measurements [12] have
The maximum tolerable touch – and step – voltages according to the IEEE Std 80-2000 [13] can be determined by applying the following relationships, k Et = √ (1000 + 1.5) t k Es = √ (1000 + 6.0) t
(13)
with parameter k being 0.157 and 0.116 for persons with body weights equal to 70 kg and 50 kg, respectively, and t designating the fault clearing time. In Eq. (13) is the ground resistivity. In non uniform soil case for the resistivity of the upper soil layer will be used. Table 7 gives the maximum allowable voltages for cases considered in the preceding subsections if t = 0.25 s. In assessing the potential hazards in the considered cases we should remind that the calculated voltages are proportional to the ground fault current leaking from electrode 1. This makes it possible to simply determine the magnitudes of this current that cause higher touch- and step-voltages than those from Table 7 by using the results of previous calculations. The results of calculations presented in Subsection 3.2 show that in case when electrode 2 is not connected with any other ground electrode the touch – and particularly the step – voltages that can be generated at the area occupied by the electrode 2 are well below the values from Table 9 even for small distances between the electrodes if the ground fault current is not exceeding 5 kA. If only persons with body weight equal to or greater than 70 kg will be in the area occupied by electrode 2, critical for d = 10 m could be only currents higher than 6.5 kA. On the other side, the maximum touch – voltages at the area protected by ground electrode 1 are high and adequate measures for their reduction should be necessary, such as covering the surface of the site with gravel or crushed rocks. The analysis performed in Subsection 3.3 shows that an unintentional connection of electrode 2 to electrode 1 can cause serious safety problems in the installations grounded by electrode 2 if it is not designed for high ground fault currents. If only persons with body weight equal to or greater than 70 kg will be in the area occupied by electrode 2, critical for d = 10 m regarding the touch voltages will be ground fault currents higher than 3 kA.
J. Nahman, D. Salamon / Electric Power Systems Research 151 (2017) 166–173
171
Fig. 5. Supply and grounding of an excavating shovel. 1—source transformer station ground electrode, 2—common ground electrode, 3—power cable phase conductors, 4—power cable ground conductors, 5—excavating shovel.
Fig. 6. Grounding grid 2 in vertical position.
In the situation that might appear in urban areas, analyzed in Subsection 3.4, the maximum touch – voltage at neighboring building for d = 10 m is as high as in the case when the electrodes are connected, considered before in Subsection 3.3. However, the step – voltages in the area between the substation and the neighboring building are approximately two times higher than these voltages for connected electrodes in all analyzed cases. Hence, special measures should be undertaken to reduce these voltages as much as possible. In order to identify and avoid potentially dangerous circumstances, it would be necessary to perform some experimental investigations before putting in operation the transformer station. The analysis of a typical situation that might appear at open pit mines, performed in Subsection 3.4, has shown that the distance between the source substation grounding grid and the common ground electrode for the pit that is twice the diagonal of the substation grounding grid must not be sufficient for preventing the appearance of high potentials transferred to the pit facilities in case of high ground fault currents magnitudes. From the data given in Tables 6 and 7 we conclude that hazardous situations might appear for J1 > 7.9 kA. It is important to stress that the safety risk analysis performed above was based under assumption that the fault clearing time is relatively short that cannot be always achieved. 4. Approximate expressions 4.1. Potential of adjacent ground electrode and maximum touch – voltage in Subsection 3.2 case The ground grids can be approximated by round plates covering the same area as the real grids. The center of the plate should coincide with the geometrical center of the grid. For a rectangular grid the intersection of diagonals should be taken as this center.
----- Route on the ground surface for calculating touch – voltages.
The resistance to ground of the equivalent plate representing grid 1 equals [13,14], R1 =
2D1
D1 =
(14)
4 · A1
(15)
where D1 is the diameter of the plate and A1 is the area occupied by the ground grid which is approximated. Expressions (14) and (15) are strongly valid for the plates lying on the ground surface. However, the plate can quite satisfactorily model the grounding grids if their depth of burial is much smaller than their horizontal dimensions. The potential of the Plate 1 if the current emerging into the ground from it is J1 equals, E1 = R1 · J1 =
· J1 2D1
(16)
The potential on the ground surface at a radial distance x from the edge of the plate can be determined as [14]: Ex =
2 E1 · arcsin
D1 D1 + 2x)
(17)
The neighboring ground grid 2 is modeled also by an equivalent round plate covering the same area as the real grid. If the closest edges of the considered two grids are d m apart, the potential of grid 2 caused by current J‘1 emerging from grid 1 is, using the mean potential concept, U21 =
2E1 · M 2
x=M2
x=1
arcsin
D1 D1 + 2(d + x − 1)
(18)
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Fig. 7. Grounding grid 2 in horizontal position.
----- Route on the ground surface for calculating touch – voltages.
Table 8 Approximate results for Case B.II. and = 100 m.
Table 10 Approximate results for R = 0.1 .
d, m
10
20
30
40
50
60
d, m
10
20
30
40
50
60
E2 , V Et2 , V d, m E2 , V Et2 , V
228.6 59.5 70 118.7 15.1
196.4 43.0 80 110.2 13.0
173.0 32.9 90 102.9 11.3
154.9 26.2 100 96.5 9.9
140.5 21.4 110 90.9 8.7
128.6 17.8 120 85.9 7.8
E2 ,V
27.4
23.5
20.7
18.5
16.8
15.4
Table 9 Results for the situations from Figs. 5 and 6. d, m
10
20
30
40
50
60
Horizontal position of electrode 2 E2 , V Et 2 , V
109.6 33.0
93.8 25.4
82.8 20.3
74.3 16.6
67.6 13.9
62.0 11.8
Vertical position of electrode 2 E2 , V Et 2, V
that in both analyzed cases electrode 2 is modeled by the identical equivalent round plate and that, therefore, for both cases the approximate approach gives the same results. The results obtained by complete and approximate modeling are presented in Table 9. The listed results show that the simple approximate approach presented in the paper provides again fair assessments of both the E2 and Et 2 magnitudes due to the interference between the electrodes for all considered distances between them. 4.2. Potential of adjacent ground electrode in Subsection 3.4 case
126.4 28.4
103.0 19.9
Approximation for both positions of electrode 2 105.8 91.7 E2 , V 30.0 22.3 Et 2, V
90.6 14.9
81.1 11.7
73.5 9.5
67.2 7.8
81.3 17.4
73.1 14.1
66.6 11.7
61.2 9.8
The following approximate relationships are relevant for the considered case by bearing in mind expression (3): U21 + Jc2 · R2 = E2 (21)
M2 is the number of 1 m spaced points along the straight line route crossing through the center of the area occupied by the equivalent Plate 2 going from edge to edge of this plate. U21 designates generally the potential of ground electrode 2 caused by the current dissipating into the ground from electrode 1. This is the approximation for the first member on the left hand side of the general expression (3). As, in the considered case, there is no current inserted into ground by electrode 2, i.e. Jc 2 = 0, the second member on the left hand side of expression (3) is null, which means that the potential of electrode 2 equals, E2 = U21
(19)
The touch potentials generated at the area occupied by grid 2 can be assessed by applying the expression, Et2 =
1 M2
x=M2
|Ex−1 − E2 |
(20)
x=1
Table 8 presents the results obtained using expressions (17)–(19) for cases analyzed in Subsection 3.2 for uniform soil. As can be seen, these results are close to the results from Table 1 obtained by complete modeling. This is particularly the case for touch – voltages that differ from the exact values less than 7%. in the extreme case. The described approach has been also checked for the case when electrode 2 has rectangular form and takes vertical (Fig. 6) and horizontal positions (Fig. 7) with respect to electrode 1. The complete and approximate modeling has been applied to determine the potentials E2 and Et2 that the current dispersed from electrode 1 generates at electrode 2 in case = 50 m. It is important to stress
E2 = −Jc2 · R with R2 being the resistance to ground of equivalent Plate 2, R2 =
2D2
(22)
Symbols Jc 2 and R have the same meaning as in Subsection 3.4. From Eq. (21) the following expression for the potential of electrode 2 is derived, E2 =
U21 R R + R2
(23)
Table 10 presents the results obtained by applying Eq. (22) to the example considered in Subsection 3.4 for R = 0.1 . As can be seen, the approximate results compare very well with the results from Table 5. By applying the derived approximate expressions to the open pit mine case with R = 1.65 , also considered in Subsection 3.4, we have obtained E2 = 46.6 V, which is practically the same value as calculated by the complete modeling. 5. Conclusions Mathematical models are presented for the analysis of the potentially hazardous situations that might appear by transferring the potentials generated by ground fault currents to the neighboring ground electrodes and equipment and personnel protected by them. Several characteristic interference situations have been detailed investigated and assessed depending on the magnitudes of the ground fault currents and the mutual positions of neighboring ground electrodes. The performed safety analysis has included the
J. Nahman, D. Salamon / Electric Power Systems Research 151 (2017) 166–173
calculation of grounding grid potentials, maximum touch – and step – voltages at the areas covered by these grids and in their vicinity. A detailed analysis of the case when the neighboring ground electrode is connected with other grounded objects is performed. These circumstances, that are important for urban areas, have not been considered so far. The paper has also proposed simple approximate relationships for assessing some effects of interference between the neighboring grounding grids that were successfully checked by comparing the results obtained by them to the results calculated by complete modeling. The analysis conducted in the paper has shown that an unintentional connection of ground grids can cause very high touch – and step – voltages at the neighboring grid in case of ground faults, that can lead to serious safety risks. In urban areas, where the neighboring grid can be connected with some grounded objects, both the touch – and step – voltages might significantly arise in ground fault cases. In the open pit mine case it is shown that the distance between the source substation grid and the common ground electrode for the pit facilities that is twice the diagonal of the substation grounding grid must not be sufficient for preventing potentially hazardous circumstances at pit facilities.
173
two steps are less than 0.1%, the last result obtained is taken as final. If this is not the case, the described procedure is continued by adding new points as before. For illustration, the calculation method described above was compared with the method that uses exact analytical expressions for determining the self- and mutual-resistances of parallel and perpendicular conductors of ground grids [15]. A quadratic symmetrical 20 m × 20 m grid with 64 meshes, buried in the soil with = 100 m at a depth of 0.5 m, was considered. Diameters of grid conductors are 9.45 mm. By applying the exact analytical model we obtained for the resistance to ground of the considered grid R = 2.35 . The semi-analytical method described above yielded R = 2.39 , which is a result being in good agreement with the exact one for practical real-life applications where soil resistivity and structure are not known with great certainty. The calculations were performed using MATLAB R12 code and computer: PC Pentium, 3.20 GHz, 4.00 GB RAM. The elapsed calculation time for the applied semi-analytical approach was 0.0930 s. The soil resistivity, diameters of ground conductors and the rectangular space coordinates of their end points are the necessary input data. It is assumed that the ground surface is the z = 0 plane. In the described semianalytical approach the ground conductors may be, generally, in any position to one other.
Appendix A. Mathematical model used in the analysis. References Let us consider a straight-line round conductor with length l, which is much greater than its diameter d, buried in uniform soil with resistivity . If a uniform density of the current I discharging from the conductor axis over its length is assumed, the equipotential surfaces will be ellipsoids of revolution about the conductor. The end-points of the conductor, say P and Q, will be the foci of these ellipsoids. The smaller axis of the ellipsoid modeling the conductor is equal to d. The potential at a point T caused by current I is equal, ϕT =
I ln 4l
D(P, T ) + D(Q, T ) + l D(P, T ) + D(Q, T ) − l
(A1)
with D(P,T) and D(P,T) designating the distances of point T from the conductor end-points. The self resistance of the considered conductor is, r=
ln 2l
2l
(A2)
d
The mutual resistance between conductor k and conductor j is determined as the mean potential of conductor j caused by the current discharging from conductor k per unit of this current, 1
ln 4lk N N
rkj =
m=1
D(P , T ) + D(Q , T ) + l m m k k k D(Pk , Tm ) + D(Qk , Tm ) − lk
(A3)
In Eq. (A3), N is the number of points Tm uniformly distributed along conductor j forming N + 1 segments of equal lengths. Parameter N is selected iteratively in the calculation process: the result obtained for a given N is compared by the result obtained after new N + 1 points are added halving the segments between previous points. If the difference between rkj values obtained in the described
[1] A.P.S. Meliopoulos, F. Xia, E.B. Joy, G.J. Cokkinides, An advanced computer model for grounding system analysis, IEEE Trans. Power Deliv. 8 (January (1)) (1993) 15–21. [2] J. Nahman, Digital calculation of earthing systems in nonuniform soil, Ärchiv fuer Elektrotechnik 62 (1) (1980) 10–24. [3] J. Nahman, I. Paunovic, Safety conditions in manholes in the vicinity of substations, IEEE Trans. Power Deliv. 18 (July (3)) (2003) 758–761. [4] I. Colominas, F. Navarrina, M. Casteleiro, Analysis of transferred earth potentials in grounding systems: a BEM numerical approach, IEEE Trans. Power Deliv. 20 (January (1)) (2005) 339–345. [5] E. Faleiro, F.J. Pazos, G. Asensio, G. Denche, D. Garcia, J. Moreno, Interaction between interconnected and isolated grounding systems: a case study of transferred potentials, IEEE Trans. Power Deliv. 30 (5) (2015) 2260–2267. [6] J. Nahman, D. Jelovac, High-voltage/medium (low) voltage substation earthing systems, IEE Proc. Pt. C 134 (January (1)) (1980) 75–80. [7] I. Colominas, F. Navarrina, M. Casteleiro, Numerical simulation of transferred potentials in earthing grids considering layered soil models, IEEE Trans. Power Deliv. 22 (July (3)) (2007) 1514–1522. [8] F. Freschi, M. Mitolo, M. Tartaglia, An effective semianalytical method for simulating grounding grids, IEEE Trans. Ind. Appl. 49 (January/February (1)) (2013) 256–263. [9] F. Freschi, M. Mitolo, M. Tartaglia, Interferences phenomena between separate grounding systems, IEEE Trans. Ind. Appl. 50 (July/August (4)) (2014) 2853–2860. [10] J. Ribiˇc, P. Maruˇsa, Protection of building in the vicinity of transmission towers against potential rise above the ground electrode—study case, IEEE Trans. Power Deliv. 31 (February (1)) (2016) 107–154. [11] G. Parise, L. Parise, L. Martirano, The interference of grounding systems, the floating behavior, IEEE Trans. Power Deliv. 51 (6) (2015) 5038–5043. [12] Transferred potentials in 6 kV and 20 kV networks of the open pit coal mines of Electrical Power System of Serbia (EPS) (in Serbian), in: Analytical and Experimental Study, Performed for EPS, Contract No. II-260/7-08, Nikola Tesla Institute, Belgrade, Serbia, 2009. [13] IEEE Guide for Safety in AC Substation Grounding, IEEE Std. 80-2000. [14] F. Ollendorf, Erdstroeme, Springer Verlag, Berlin, 1928. [15] J. Nahman, S. Skuletich, Irregularity correction factors for mesh and step voltages of grounding grids, IEEE Trans. PAS 99 (January/February (1)) (1980) 174–180.
Contents lists available at ScienceDirect
Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
Mutual interference of neighboring grounding systems and approximate formulation J. Nahman ∗,1 , D. Salamon Faculty of Electrical Engineering, University of Belgrade, Serbia
a r t i c l e
i n f o
Article history: Received 3 December 2015 Received in revised form 8 March 2017 Accepted 23 May 2017 Keywords: Adjacent grounding systems Mathematical models Safety risks Approximate expressions
a b s t r a c t The paper develops mathematical models for the analysis of the mutual interference of closely positioned ground electrodes in order to assess the safety risks that might appear in various circumstances often characteristic for urban areas. The potentially hazardous situation that might arise in open pit mines at the equipment grounded by connection with the common ground electrode is also investigated. In the scope of the analysis of possible dangerous circumstances the maximum touch – and step – voltages that appear at the sites protected by the grounding grids are calculated for various distances between the grids and in case of uniform and nonuniform two – layer soil. Approximate expressions for assessing some interference effects are also provided and successfully checked for various cases. Based upon the results of the conducted series of calculations, the safety risks are determined for all investigated cases. It was shown that the highest safety risks can appear in the case when the neighboring grids are unintentionally connected as well as in the case when the neighboring grid is connected with some grounded objects, which is characteristic for urban areas and for open pit mines with common ground electrodes. © 2017 Elsevier B.V. All rights reserved.
1. Introduction The complexity of objects consuming electrical energy and their density in space is permanently increasing, which generates new problems concerning the safety of associated persons and equipment and needs for improving the methods for calculation and projecting of grounding systems. In the past, various mathematical models for a detailed analysis of complex grounding systems buried in uniform or two – layer soils have been developed based upon the mean potential concept in calculation of the mutual influences among the conductors of the complex ground electrode [1,2]. The application of the finite element modeling, that increases the computational burden to some extent, has been more recently used for more detailed calculation of various phenomena associated with ground electrodes and neighboring metallic structures linked with or buried in close proximity to them [3–5]. In Ref. [6] the analysis of the mutual influence between the equipment ground electrode on the high voltage side of 10 kV/0.4 kV distribution system transformer stations and the neutral ground electrodes on the low voltage side has been performed. The proximity effects for various spacing of these electrodes when alone and in the presence
∗ Corresponding author. E-mail addresses: [email protected], [email protected] (J. Nahman). 1 Freelance consultant for electrical power systems. http://dx.doi.org/10.1016/j.epsr.2017.05.029 0378-7796/© 2017 Elsevier B.V. All rights reserved.
of neighboring metallic pipes or other conducting structures have been also studied. The finite element modeling has been applied to analyze the potential distribution on the area surrounding ground electrodes as well as the potentials transferred by various metallic conductors buried in close proximity to the grounding area in order to identify possible hazardous situations for persons and equipment [4,7]. This analysis has been conducted both for uniform and two – layer soils. The interference phenomena between grounding systems have been recently detailed considered for assessing the grade of their independency. The matrix calculation method presented in Ref. [8] has been applied to study interactions between the substation grid and the safety ground bed in mining installations [9]. In a recent paper [10], the analysis of the effects of the interference between the ground electrode and fundaments of a transmission line tower and the ground electrode of the adjacent building has been performed and adequate protection measures elaborated. Reference [11] analyzes the effects of neighboring ground electrodes on the distribution of the potentials over an area around the interacting electrodes both in the vicinity of the active and the influenced ground electrode. This paper analyzes the potentially hazardous circumstances that can appear on the site protected by a grounding grid being in close proximity to another grounding grid dissipating the ground fault current into the surrounding soil. The maximum touch – and step – voltages that appear at the site of the neighboring ground-
J. Nahman, D. Salamon / Electric Power Systems Research 151 (2017) 166–173
ing grid are determined for various distances between the grids in the following characteristic cases that can appear in practice: a) the neighboring grid has no connections with other grounded facilities; b) the considered grounding grids are intentionally or unintentionally linked and c) the neighboring grid is connected to a grounded object. These analyses have been carried out both for uniform and two −layer soils. Approximate, comparatively simple expressions are also derived in the paper for assessing some interference effects and successfully checked for various cases. 2. Mathematical models 2.1. Relationships connecting two neighboring ground electrodes Let us consider two ground electrodes buried in close proximity. The ground electrodes are composed of sets of interconnected straight line linear elements. It is assumed that ground electrode 1 dissipates current J1 into the ground due to a ground fault in the installation associated with this electrode. General expression correlating the potentials of the electrodes elements and the currents leaking from them into the surrounding soil has the following form [2,6], [C1 ] · [I1 ] = J1
(1)
[r11 ] · [I1 ] + [r12 ] · [I2 ] = [C1 ] · E1
(2)
[r21 ] · [I1 ] + [r22 ] · [I2 ] = [C2 ] · E2
(3)
It is implied that the numbers of elements of ground electrodes 1 and 2 are N1 and N2 , respectively. [I1 ] and [I2 ] are column vectors of currents dissipating into the ground from the elements of the corresponding ground electrodes, whereas E1 and E2 are potentials of these elements. The elements of square matrices [r11 ] and [r22 ] are the self- and mutual resistances of the elements of electrodes alone. Elements of matrix [r12 ] are mutual resistances of the elements of ground electrode 1 with the elements of ground electrode 2. Matrix [r21 ] is transposed matrix [r12 ]. [C1 ] and [C2 ] are N1 and N2 -dimensional column vectors of units and [C1 ] is transposed vector [C1 ]. Resistances rjk being elements of resistance matrices in Eqs. (2) and (3) are calculated using the mean potential method. It is implied that the ground electrodes are composed of a set of connected straight-line thin round conductors. Each conductor is replaced by an ellipsoid of revolution with its foci being the ends of the conductor and its smaller axis being equal to the diameter of the conductor. This model of a conductor makes it possible to calculate the potential caused by the current discharging from the conductor at any point in the space using a simple formula based on the distances of this point from the conductor’s ends. By applying this expression formed for conductor j we can determine the mean potential of any conductor k caused by the current discharging from conductor j. By dividing this potential by conductor j current we obtain
Fig. 1. Grounding grids in close proximity.
167
the mutual resistance rjk between these two conductors in case when both conductors lie in a discontinues media with the same ground resistivity. For uniform soil the effect of discontinuity at the ground surface is taken by introducing the image of conductor j with respect to the ground surface. The potentials of conductor k are now calculated by summing the potentials generated by conductor j and its image. In the case of two horizontally stratified layers with different resistivities various images of conductor k are used depending in which of the layers conductors k and j are located. All the details of the calculations both for the uniform and the two-layer soil cases are presented in Ref. [2]. Some of the results obtained by the described calculation method have been successfully checked by modeling in electrolytic tank [6]. A more detailed description of the applied calculation method and a comparison with the exact analytical calculation model presented in Ref. [15] are given in Appendix A. As can be concluded from expressions (1)–(3), there are N1 + N2 unknown currents emanating from the elements of ground electrodes as well as unknown potentials of both ground electrodes. These expressions give N1 + N2 + 1 relationships among the considered variables. The missing relationship, completing the system of equations, depends on the circumstances to be analyzed, as will be discussed further on.
2.2. Electrode 2 is not connected to any other ground electrode In the case when ground electrode 2 is not connected to any other ground electrode the expression completing the system of equations given before is, Jc2 = [C2 ] · [I2 ] = 0
(4)
as current Jc 2 injected into the ground by electrode 2 is null in considered case because the currents from electrode 1 only pass through the elements of electrode 2 with the same total input and output values. The formed system of equations can be now solved for [I1 ], [I2 ], E1 and E2 . The touch – voltages over the site occupied by ground electrode 2 are caused by the difference of the potentials generated by electrode 1 on the ground surface above electrode 2 and the potential E2 . The potential on the ground surface at a standing point x can be calculated using the general expression:
Ex = [r1x ]
[r2x ] ·
[I1 ]
(5)
[I2 ]
with [r1x ] and [r2x ] designating N1 - and N2 -dimensional row vectors of mutual resistances of elements of ground electrodes 1 and 2 and the standing point at location x.
----- Route on the ground surface for calculating touch- and step-voltages.
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The touch – voltage over bridged by standing on the ground surface at point x and touching an object connected to the ground electrode 2 equals, Etx = |Ex − E2 |
(6)
The voltage over bridged by a step from point x to a point 1 m apart is, Esx = |Ex − Ex+1 |
(7)
2.3. Ground electrodes are connected In some cases the neighboring grounding systems are connected unintentionally or intentionally. If the connections are made intentionally in order to decrease the resistance to ground, they should be done by insulated conductors to prevent the appearance of step voltages in the vicinity of these ties. The analysis that follows has been conducted for this case. In the considered case ground electrodes 1 and 2 form a common grounding system. Eq. (1) should now be replaced by the following one, [C1 ] · [I1 ] + [C2 ] · [I2 ] = J1
Table 1 Results of the calculations performed for = 100 m. d m
E1 V
E2 V
Et1 V
Et2 V
Es2 V
10 20 30 40 50 60 70 80 90 100 110 120
781.6 789.0 791.9 793.4 794.2 794.7 795.1 795.2 795.4 795.5 795.6 795.6
252.9 213.8 186.7 166.1 149.8 136.5 125.4 116.1 108.0 101.1 95.9 89.5
186.4 188.7 189.5 190.0 190.2 190.3 190.4 190.5 190.5 190.5 190.5 190.5
56.1 40.2 31.1 25.1 20.7 17.3 14.8 12.7 11.0 9.7 8.6 7.6
8.6 6.3 4.8 3.8 3.1 2.5 2.2 1.8 1.6 1.4 1.2 1.1
(8)
The relationship completing the mathematical model in the considered circumstances is: E1 = E2 = E
(9)
The formed system of equations should be solved for [I1 ], [I2 ] and E. Then, by applying expressions (5)–(7) we can calculate touch – and step – voltages over the area of interest. The total current leaking into the ground from electrode 2 is now,
Jc 2 = [C2 ] · [I2 ]
(10)
and the current dispersed from electrode 1, Jc1 = J1 − Jc2
(11)
2.4. Electrode 2 is connected to a grounded object In urban areas many of source transformer stations of the distribution systems are positioned in close vicinity to the buildings. The buildings are usually very solidly grounded via their fundament grounding systems and due to the mutual connections with the grounding systems of neighboring buildings through water and other common metallic installations and structures. Due to this fact, the resistances to ground of the grounding systems of buildings are very low. In such circumstances, the routes of currents flowing through the soil in case of a ground fault in the transformer station will be more oriented to the ground electrode of the neighboring building, which affects the potential distribution on the ground surface between the substation ground grid and the ground electrode of the building. The situations with the ground electrode 2 connected to a grounded object are also typical for open pit mines. The safety ground beds that are used as common protective ground for the objects on the pit, such as excavators for example, are grounded through the natural resistance to ground of these objects. To investigate the effect of the connections of the ground electrode 2 with other ground electrodes, Eqs. (1)–(3) should be complemented by relationships (10) and (12), E2 = −Jc2 · R
(12)
with R being the resistance to ground of grounding systems of adjacent objects to which the ground electrode 2 is connected. It is important to mention that in the considered case the sum of the
Fig. 2. Ground surface potential distribution along the straight line route halving the edge meshes of the grids in case d = 30 m and = 100 m.
currents flowing into the ground from electrode 2 is not null but equals to the current flowing to the third electrode with minus sign. The formed system of equations should be solved for [I1 ] and [I2 ]. By applying Eqs. (5)–(7) we can then calculate the touch – and step – voltages in the area of interest. 3. Application examples 3.1. Considered ground electrodes Two square grounding grids having the same shape and parameters are considered (Fig. 1). The side lengths are 60 m and depth of burial is 0.7 m. Elements of the grids are cylindrical conductors with 1 cm diameter. The adjacent edges of grids are spaced d meters. The calculated resistance to ground for ground electrode 1, when alone and = 100 m, is 0.796 . The maximum touch – and step – voltages calculated for J1 = 1 kA along the route halving the grid edge meshes are Et1 = 190.5 V and Es1 = 27.5 V. 3.2. Electrode 2 is not connected to any other ground electrode As discussed in Section 2.2, the touch – voltages in the area occupied by ground electrode 2 when current J1 is leaking into ground from ground electrode 1 arise because of the difference that appears between the ground surface potentials and the potential E2 of the ground electrode 2. We have calculated the maximum touch – voltages Et 1 and Et 2 at electrodes 1 and 2 as well as step-voltages Es2 at ground electrode 2 along the route halving the edge meshes of electrodes for various distances in case J1 = 1 kA and = 100 m. The results of these calculations are presented in Table 1.
J. Nahman, D. Salamon / Electric Power Systems Research 151 (2017) 166–173 Table 2 Results obtained in the case 1 = 100 m and 2 = 50 m.
169
Table 4 Results of the analysis for linked electrodes for = 100 m.
d m
E1 V
E2 V
Et1 V
Et2 V
Es2 V
d m
E1 , E2 V
Et1 , Et2 V
Es1 , Es2 V
10 20 30 40 50 60
472.1 474.4 475.3 475.8 476.1 476.2
125.5 106.6 93.2 83.0 74.9 68.2
163.3 164.2 164.6 164.8 164.8 164.9
35.1 25.1 19.4 15.5 12.8 10.7
5.6 4.1 3.2 2.1 1.8 1.6
10 20 30 40 50 60 70 80 90 100 110 120
517.2 501.4 489.3 479.7 472.0 465.6 460.3 455.7 451.7 448.3 445.3 442.6
117.7 112.3 110.1 107.8. 106.1 104.7 103.7 102.8 102.1 101.5 101.1 100.7
26.6 25.7 25.1 24.6 24.2 23.9 23.7 23.5 23.4 23.3 23.2 23.1
Table 3 Results for the case 1 = 50 m and 2 = 100 m. d m
E1 V
E2 V
Et1 V
Et2 V
Es2 V
10 20 30 40 50 60
667.3 678.0 682.3 684.4 685.6 686.3
254.4 214.2 186.7 166.0 149.7 136.4
107.7 109.9 110.8 111.2 111.4 111.6
42.3 30.4 23.6 19.3 15.9 13.5
6.4 4.5 3.4 2.9 2.2 1.8
For illustration, Fig. 2 presents the potentials on the ground surface above the grounding grids from Fig. 1 along the straight line route halving the edge meshes of the grids for d = 30 m (Fig. 2). It is important to mention that all the variables in Table 1 are proportional to the ground resistivity. Hence, the magnitudes of these variables in case, say = 50 m, are obtainable by halving the corresponding results quoted in Table 1. The same analysis has been also conducted for non uniform soil cases, for comparison. Two soil layers are considered with the thickness of the upper soil layer being 2 m. Table 2 presents the results obtained if the resistivity of the upper and bottom soil layers are respectively 1 = 100 m and 2 = 50 m. Table 3 quotes the results for 1 = 50 m and 2 = 100 m. By comparing the results obtained for various soil structures we can observe that potentials E1 depend considerably on the resistivity of the bottom soil layer. They are comparable with the values obtained for the corresponding uniform soil cases with soil resistivity being equal to the resistivity of the bottom soil layer. The same is the case with the potentials E2 where the difference between the uniform and the two soil layers case is very small. As can be seen, the touch and step voltages generated at the electrode 2 by electrode 1 are comparatively low compared to the potential E2 that can reach high magnitudes even in case d = 120 m. However, bearing in mind that the calculations are performed for J1 = 1000 A, for higher magnitudes of this current all considered potentials increase proportionally.
3.3. Ground electrodes are connected If the considered ground electrodes are connected, owing to the geometrical symmetry the fault currents leaking from the grids are the same for both grids as well as the generated potentially dangerous voltages. The grids can be intentionally or unintentionally linked through various metallic installations such as cables of different kinds, metallic pipes or rail road tracks. Further on we have considered the case when this link is realized by an insulated cable with negligible impedance. Table 4 quotes the calculated parameters for the considered case for different distances d between the grids for uniform soil with = 100 m. Fig. 3 displays the potential distribution on the ground surface along the route halving edge meshes of considered ground electrodes if linked, for d = 30 m.
Fig. 3. Ground surface potentials along the route halving the edge meshes of linked electrodes for = 100 m and d = 30 m.
In all considered cases the maximum magnitudes of investigated potentially dangerous voltages appear at the corner meshes of the grids on their sides facing the open area. It is evident that the linking of electrodes causes much higher touch voltages at electrode 2 when compared to the case with separated grids. This indicates that, before intentionally connecting grid 2 with grid 1, it would be necessary to check if the security measures applied in installations grounded by grid 2 can allow such an increment of dangerous voltages and ground currents. On the other side, the parallel connection of electrodes decreases ground currents and touch – voltages associated with grid 1. 3.4. Electrode 2 is connected to a grounded object As stated before, in urban areas the distribution source transformer stations are often located in a very close proximity to the surrounding buildings of various kinds. On the other side, the grounding systems of these buildings are usually, intentionally or not, interconnected and, therefore, have often very low resistance to ground. In such circumstances it is important to assess the potentially dangerous voltages that can appear at the ground electrode of an adjacent building nearest to the substation. By applying the mathematical model presented before in Subsections 2.1 and 2.4 we have calculated the potential distribution on the ground surface between the grounding grid 1, implying that it is the ground electrode of the source distribution station, and the grounding grid 2 representing the ground electrode of the building closest to the substation. These calculations have been performed under assumption that the grounding grid of the building is connected to the ground electrodes of surrounding buildings which equivalent resistance to ground is R = 0.1 . We have calculated the potential E1 of
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Table 5 Results for Situation 3 in case R = 0.1 .
Table 6 Calculation results for the open pit mine case.
d m
E1 V
Et1 V
E2 V
Et2 V
Es12 V
E1 V
E2 V
Et2 V
Ees V
10 20 30 40 50 60
709.1 737.6 752.9 762.6 769.2 773.9
214.9 206.1 202.0 199.7 198.2 197.3
28.7 24.1 20.9 18.6 16.8 15.3
111.7 86.5 71.2 60.6 52.6 46.4
56.5 49.8 47.6 46.5 45.9 45.6
793.7
47.0
10.1
45.6
Table 7 Maximum tolerable voltages. , ˝m
50
100
Et 70 , V Et 50 , V Es 70 , V Es 50 , V
337.5 249.4 408.2 301.6
361.1 266.8 502.4 371.2
shown that the natural value of the resistance to ground of an excavating shovel used in the mine “Kolubara”, owned by Serbian Power System Company, equals approximately 0.016 . Hence, the total resistance to ground of the considered connection including the resistance of the cable 3 × 25 mm2 copper ground conductors is R = 1.65 . Table 6 presents the results of the calculations obtained for J1 = 1 kA. Ees is the potential transferred to the excavating shovel. 3.5. Assessment of safety risks
Fig. 4. Potential distribution on the ground surface along the route halving the edge meshes of Grounding grids in case R = 0.1 for d = 30 m.
grounding grid 1, maximum touch – voltages Et1 and Et2 at grounding grids 1 and 2 as well as the maximum step voltage Es12 in the area between these grids for various spacing of grids. The results of the calculations conducted for the considered circumstances for = 100 m are presented in Table 5. As can be seen, both the touch voltages at the ground electrode 2 and the step voltages in the area between the electrodes have comparatively high values. For illustration, Fig. 4 displays the potential distribution on the ground surface along the straight line route halving the edge meshes of grounding grids in case R = 0.1 for distance d = 30 m. The circumstances considered here can also often happen in mines with surface exploitation. The potential that might appear at the common ground electrode for the mine facilities in case of ground faults in the neighboring source transformer station have been analyzed in Ref. [9]. Using the model presented in Subsections 2.1 and 2.4 we shall analyze also the touch potentials that might appear at the common ground electrode as well as the potential that will be transferred to the mobile electrical equipment spread over the pit by taking into account their resistance to ground. A typical situation that will be considered is displayed in Fig. 5. An excavating shovel in an open pit coal mine is supplied from the source substation over a heavy duty mining power cable having phase and ground conductors. To prevent the transfer of the potentials of source transformer station ground electrode 1 to the supplied equipment, the cable ground conductors are separated from electrode 1 and connected to electrode 2 serving as a common ground for this equipment. Electrodes 1 and 2 are the same as considered previously in the paper. The adjacent edges of electrodes are 170 m apart which is twice the length of the diagonal of the ground electrode 1. Such spacing between the electrodes is taken as satisfactory with regard to the potential transferred from source ground electrode 1 to electrode 2 in IC 8835, which regulates mine installations in U.S [9]. The ground resistivity is supposed to be = 100 m. The cable length between the common ground electrode and the shovel is taken to be 200 m. The performed field measurements [12] have
The maximum tolerable touch – and step – voltages according to the IEEE Std 80-2000 [13] can be determined by applying the following relationships, k Et = √ (1000 + 1.5) t k Es = √ (1000 + 6.0) t
(13)
with parameter k being 0.157 and 0.116 for persons with body weights equal to 70 kg and 50 kg, respectively, and t designating the fault clearing time. In Eq. (13) is the ground resistivity. In non uniform soil case for the resistivity of the upper soil layer will be used. Table 7 gives the maximum allowable voltages for cases considered in the preceding subsections if t = 0.25 s. In assessing the potential hazards in the considered cases we should remind that the calculated voltages are proportional to the ground fault current leaking from electrode 1. This makes it possible to simply determine the magnitudes of this current that cause higher touch- and step-voltages than those from Table 7 by using the results of previous calculations. The results of calculations presented in Subsection 3.2 show that in case when electrode 2 is not connected with any other ground electrode the touch – and particularly the step – voltages that can be generated at the area occupied by the electrode 2 are well below the values from Table 9 even for small distances between the electrodes if the ground fault current is not exceeding 5 kA. If only persons with body weight equal to or greater than 70 kg will be in the area occupied by electrode 2, critical for d = 10 m could be only currents higher than 6.5 kA. On the other side, the maximum touch – voltages at the area protected by ground electrode 1 are high and adequate measures for their reduction should be necessary, such as covering the surface of the site with gravel or crushed rocks. The analysis performed in Subsection 3.3 shows that an unintentional connection of electrode 2 to electrode 1 can cause serious safety problems in the installations grounded by electrode 2 if it is not designed for high ground fault currents. If only persons with body weight equal to or greater than 70 kg will be in the area occupied by electrode 2, critical for d = 10 m regarding the touch voltages will be ground fault currents higher than 3 kA.
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171
Fig. 5. Supply and grounding of an excavating shovel. 1—source transformer station ground electrode, 2—common ground electrode, 3—power cable phase conductors, 4—power cable ground conductors, 5—excavating shovel.
Fig. 6. Grounding grid 2 in vertical position.
In the situation that might appear in urban areas, analyzed in Subsection 3.4, the maximum touch – voltage at neighboring building for d = 10 m is as high as in the case when the electrodes are connected, considered before in Subsection 3.3. However, the step – voltages in the area between the substation and the neighboring building are approximately two times higher than these voltages for connected electrodes in all analyzed cases. Hence, special measures should be undertaken to reduce these voltages as much as possible. In order to identify and avoid potentially dangerous circumstances, it would be necessary to perform some experimental investigations before putting in operation the transformer station. The analysis of a typical situation that might appear at open pit mines, performed in Subsection 3.4, has shown that the distance between the source substation grounding grid and the common ground electrode for the pit that is twice the diagonal of the substation grounding grid must not be sufficient for preventing the appearance of high potentials transferred to the pit facilities in case of high ground fault currents magnitudes. From the data given in Tables 6 and 7 we conclude that hazardous situations might appear for J1 > 7.9 kA. It is important to stress that the safety risk analysis performed above was based under assumption that the fault clearing time is relatively short that cannot be always achieved. 4. Approximate expressions 4.1. Potential of adjacent ground electrode and maximum touch – voltage in Subsection 3.2 case The ground grids can be approximated by round plates covering the same area as the real grids. The center of the plate should coincide with the geometrical center of the grid. For a rectangular grid the intersection of diagonals should be taken as this center.
----- Route on the ground surface for calculating touch – voltages.
The resistance to ground of the equivalent plate representing grid 1 equals [13,14], R1 =
2D1
D1 =
(14)
4 · A1
(15)
where D1 is the diameter of the plate and A1 is the area occupied by the ground grid which is approximated. Expressions (14) and (15) are strongly valid for the plates lying on the ground surface. However, the plate can quite satisfactorily model the grounding grids if their depth of burial is much smaller than their horizontal dimensions. The potential of the Plate 1 if the current emerging into the ground from it is J1 equals, E1 = R1 · J1 =
· J1 2D1
(16)
The potential on the ground surface at a radial distance x from the edge of the plate can be determined as [14]: Ex =
2 E1 · arcsin
D1 D1 + 2x)
(17)
The neighboring ground grid 2 is modeled also by an equivalent round plate covering the same area as the real grid. If the closest edges of the considered two grids are d m apart, the potential of grid 2 caused by current J‘1 emerging from grid 1 is, using the mean potential concept, U21 =
2E1 · M 2
x=M2
x=1
arcsin
D1 D1 + 2(d + x − 1)
(18)
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Fig. 7. Grounding grid 2 in horizontal position.
----- Route on the ground surface for calculating touch – voltages.
Table 8 Approximate results for Case B.II. and = 100 m.
Table 10 Approximate results for R = 0.1 .
d, m
10
20
30
40
50
60
d, m
10
20
30
40
50
60
E2 , V Et2 , V d, m E2 , V Et2 , V
228.6 59.5 70 118.7 15.1
196.4 43.0 80 110.2 13.0
173.0 32.9 90 102.9 11.3
154.9 26.2 100 96.5 9.9
140.5 21.4 110 90.9 8.7
128.6 17.8 120 85.9 7.8
E2 ,V
27.4
23.5
20.7
18.5
16.8
15.4
Table 9 Results for the situations from Figs. 5 and 6. d, m
10
20
30
40
50
60
Horizontal position of electrode 2 E2 , V Et 2 , V
109.6 33.0
93.8 25.4
82.8 20.3
74.3 16.6
67.6 13.9
62.0 11.8
Vertical position of electrode 2 E2 , V Et 2, V
that in both analyzed cases electrode 2 is modeled by the identical equivalent round plate and that, therefore, for both cases the approximate approach gives the same results. The results obtained by complete and approximate modeling are presented in Table 9. The listed results show that the simple approximate approach presented in the paper provides again fair assessments of both the E2 and Et 2 magnitudes due to the interference between the electrodes for all considered distances between them. 4.2. Potential of adjacent ground electrode in Subsection 3.4 case
126.4 28.4
103.0 19.9
Approximation for both positions of electrode 2 105.8 91.7 E2 , V 30.0 22.3 Et 2, V
90.6 14.9
81.1 11.7
73.5 9.5
67.2 7.8
81.3 17.4
73.1 14.1
66.6 11.7
61.2 9.8
The following approximate relationships are relevant for the considered case by bearing in mind expression (3): U21 + Jc2 · R2 = E2 (21)
M2 is the number of 1 m spaced points along the straight line route crossing through the center of the area occupied by the equivalent Plate 2 going from edge to edge of this plate. U21 designates generally the potential of ground electrode 2 caused by the current dissipating into the ground from electrode 1. This is the approximation for the first member on the left hand side of the general expression (3). As, in the considered case, there is no current inserted into ground by electrode 2, i.e. Jc 2 = 0, the second member on the left hand side of expression (3) is null, which means that the potential of electrode 2 equals, E2 = U21
(19)
The touch potentials generated at the area occupied by grid 2 can be assessed by applying the expression, Et2 =
1 M2
x=M2
|Ex−1 − E2 |
(20)
x=1
Table 8 presents the results obtained using expressions (17)–(19) for cases analyzed in Subsection 3.2 for uniform soil. As can be seen, these results are close to the results from Table 1 obtained by complete modeling. This is particularly the case for touch – voltages that differ from the exact values less than 7%. in the extreme case. The described approach has been also checked for the case when electrode 2 has rectangular form and takes vertical (Fig. 6) and horizontal positions (Fig. 7) with respect to electrode 1. The complete and approximate modeling has been applied to determine the potentials E2 and Et2 that the current dispersed from electrode 1 generates at electrode 2 in case = 50 m. It is important to stress
E2 = −Jc2 · R with R2 being the resistance to ground of equivalent Plate 2, R2 =
2D2
(22)
Symbols Jc 2 and R have the same meaning as in Subsection 3.4. From Eq. (21) the following expression for the potential of electrode 2 is derived, E2 =
U21 R R + R2
(23)
Table 10 presents the results obtained by applying Eq. (22) to the example considered in Subsection 3.4 for R = 0.1 . As can be seen, the approximate results compare very well with the results from Table 5. By applying the derived approximate expressions to the open pit mine case with R = 1.65 , also considered in Subsection 3.4, we have obtained E2 = 46.6 V, which is practically the same value as calculated by the complete modeling. 5. Conclusions Mathematical models are presented for the analysis of the potentially hazardous situations that might appear by transferring the potentials generated by ground fault currents to the neighboring ground electrodes and equipment and personnel protected by them. Several characteristic interference situations have been detailed investigated and assessed depending on the magnitudes of the ground fault currents and the mutual positions of neighboring ground electrodes. The performed safety analysis has included the
J. Nahman, D. Salamon / Electric Power Systems Research 151 (2017) 166–173
calculation of grounding grid potentials, maximum touch – and step – voltages at the areas covered by these grids and in their vicinity. A detailed analysis of the case when the neighboring ground electrode is connected with other grounded objects is performed. These circumstances, that are important for urban areas, have not been considered so far. The paper has also proposed simple approximate relationships for assessing some effects of interference between the neighboring grounding grids that were successfully checked by comparing the results obtained by them to the results calculated by complete modeling. The analysis conducted in the paper has shown that an unintentional connection of ground grids can cause very high touch – and step – voltages at the neighboring grid in case of ground faults, that can lead to serious safety risks. In urban areas, where the neighboring grid can be connected with some grounded objects, both the touch – and step – voltages might significantly arise in ground fault cases. In the open pit mine case it is shown that the distance between the source substation grid and the common ground electrode for the pit facilities that is twice the diagonal of the substation grounding grid must not be sufficient for preventing potentially hazardous circumstances at pit facilities.
173
two steps are less than 0.1%, the last result obtained is taken as final. If this is not the case, the described procedure is continued by adding new points as before. For illustration, the calculation method described above was compared with the method that uses exact analytical expressions for determining the self- and mutual-resistances of parallel and perpendicular conductors of ground grids [15]. A quadratic symmetrical 20 m × 20 m grid with 64 meshes, buried in the soil with = 100 m at a depth of 0.5 m, was considered. Diameters of grid conductors are 9.45 mm. By applying the exact analytical model we obtained for the resistance to ground of the considered grid R = 2.35 . The semi-analytical method described above yielded R = 2.39 , which is a result being in good agreement with the exact one for practical real-life applications where soil resistivity and structure are not known with great certainty. The calculations were performed using MATLAB R12 code and computer: PC Pentium, 3.20 GHz, 4.00 GB RAM. The elapsed calculation time for the applied semi-analytical approach was 0.0930 s. The soil resistivity, diameters of ground conductors and the rectangular space coordinates of their end points are the necessary input data. It is assumed that the ground surface is the z = 0 plane. In the described semianalytical approach the ground conductors may be, generally, in any position to one other.
Appendix A. Mathematical model used in the analysis. References Let us consider a straight-line round conductor with length l, which is much greater than its diameter d, buried in uniform soil with resistivity . If a uniform density of the current I discharging from the conductor axis over its length is assumed, the equipotential surfaces will be ellipsoids of revolution about the conductor. The end-points of the conductor, say P and Q, will be the foci of these ellipsoids. The smaller axis of the ellipsoid modeling the conductor is equal to d. The potential at a point T caused by current I is equal, ϕT =
I ln 4l
D(P, T ) + D(Q, T ) + l D(P, T ) + D(Q, T ) − l
(A1)
with D(P,T) and D(P,T) designating the distances of point T from the conductor end-points. The self resistance of the considered conductor is, r=
ln 2l
2l
(A2)
d
The mutual resistance between conductor k and conductor j is determined as the mean potential of conductor j caused by the current discharging from conductor k per unit of this current, 1
ln 4lk N N
rkj =
m=1
D(P , T ) + D(Q , T ) + l m m k k k D(Pk , Tm ) + D(Qk , Tm ) − lk
(A3)
In Eq. (A3), N is the number of points Tm uniformly distributed along conductor j forming N + 1 segments of equal lengths. Parameter N is selected iteratively in the calculation process: the result obtained for a given N is compared by the result obtained after new N + 1 points are added halving the segments between previous points. If the difference between rkj values obtained in the described
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