1 Introduction

The requirements for electrical installations for the future power supply grid are becoming more important. These infrastructure facilities should function reliably, safely and efficiently for decades and also meet future requirements. When planning such often complex installations, the situation is further complicated by the fact that the choice of location may be limited or predetermined due to the grid topology or geography. This can result in unavoidable challenges when designing the electrical system. The correct design of the earthing system is therefore of great importance, since on the one hand it plays an important role in safe and reliable operation and on the other hand it is difficult to make changes to it after the system has been in operation. This shows that a high level of expertise in the electrical characteristics of earth electrodes is already important in the planning of electrical installations in order to fulfil the above-mentioned requirements as a whole. Basically, vertical earth rods should dissipate (fault) currents locally to deeper layers of the soil. Vertical earth rods offer the positive electrical property of dissipating locally high currents to the earth, but the local hazard due to partial fault voltages is high. To illustrate the properties of vertical earth rods in layered soil, calculations and measurements of the resistances to earth, the ohmic interference and the partial fault voltages were carried out for a 220/110 kV substation in construction.

2 Methodology

Two constellations of vertical earth rods at different locations are investigated by calculation, whereby the two locations differ in terms of the electric resistivity of the soil. The calculations were also supported and verified by measurements. The calculations are done with the developed programme OBEINFootnote 1, which works according to the calculation method of the potential coefficients with a fixed reference potential of 0 V at infinity (distant or reference earth). The current injected into the entire earthing arrangement (current to earth) is specified and distributed to the discrete earth electrodes on the basis of the calculated interference factors. The total current to earth for all calculations is set to IE = 1 kA. In order to determine the amperage dissipated to the soil by the vertical earth rods they are partitioned into 1 m segments. With the use of this program it is possible to calculate the currents dissipated per 1 m to the earth by the earth electrodes IEd, the partial fault voltages UFP, the fault voltage UF, the surface potential and the resistance to earth RE of the earthing arrangement. The fault voltage is defined as the multiplication of the current to earth by the resistance to earth (UF = IE ∙ RE) [1] as shown in Fig. 1.

Fig. 1
figure 1

Fault voltage UF and partial fault voltage UFP [1]

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The partial fault voltage is defined as the difference of the potential of two accessible points, which are separated by 1 m.

For verification, the resistances to earth of the vertical earth rods were measured using the fall-of-potential method [2,3,4].

2.1 Arrangement of the vertical earth rods constellation I

Constellation 1 shown in Fig. 2 consists of seven vertical earth rods (VER11Footnote 2–VER17), each with a conductor cross-section of 240 mm2 and a depth of 20 m.

Fig. 2
figure 2

Arrangement of vertical earth rods (VER)—constellation I

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The electric resistivities of the soil ρ1 = 120 Ωm (up to a depth of dρ1 = 1 m) and ρ2 = 160 Ωm have been determined by the Wenner’s method [5] and are homogeneous in the area of interest. As can be seen in Fig. 2, the distances of the vertical earth rods differ two dimensional. When measuring constellation I, one vertical earth rod after the other was installed, starting with VER11, and the resulting resistance to earth of the changing parallel circuit of the vertical earth rods was measured.

2.2 Arrangement of the vertical earth rods constellation II

The constellation 2 shown in Fig. 3 consists of four vertical earth rods (VER21–VER24) with a cross-section of 240 mm2 each and a depth of 10 m, as well as four vertical earth rods (VER25–VER28) with a depth of 20 m and the same cross-section of 240 mm2 each.

Fig. 3
figure 3

Arrangement of vertical earth rods (VER)—constellation II

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The electric resistivities of the soil ρ1 = 130 Ωm (up to a depth of dρ1 = 3 m) and ρ2 = 145 Ωm have been determined by the Wenner’s method and are homogeneous in the area of interest. As shown in Fig. 3, the constellation II vertical earth rods are in line and the distances between the 10 m vertical earth rods VER21 to VER24 are approx. 13 m and between the 20 m vertical earth rods VER25 to VER28 approx. 21 m. When measuring the resistances to earth of the constellation II vertical earth rods, all of them were already installed.

3 Results for constellation I

Fig. 4 shows the amperage dissipated to the earth per a unit length of 1 m of only one vertical earth rod (VER11).

Fig. 4
figure 4

Dissipated currents to earth, VER11

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It can be seen that the amperage dissipated to the earth increases with the depth of the earth electrode and reaches its maximum at the deepest point.

The resistance to earth for a single rod with a given length of 20 m and a diameter of 17.48 mm is RE = 7.66 Ω.

If one neglects the top soil layer with a depth of 1 m and calculates the resistance to earth according to Eq. 1 [4], which is often used for vertical rods, the result is RE = 10.73 Ω.

$$R_{\mathrm{E}}=\frac{\rho }{2\uppi l}\ln \frac{4 l}{d}$$
(1)

The characters are defined as:

RE:

resistance to earth

ρ:

specific earth resistance

l:

length of the rod

d:

diameter of the rod

Using the method for this contribution, the resistivity to earth is RE = 9.63 Ω for a homogeneous soil with ρ = 160 Ωm. Calclulating the the resistance to earth using Eq. 1 delivers higher values than with the method of mirror charges used in this paper.

The main reason for the difference of the resistance to earth is different when using Eq. 1 [4] vs. the used calculation method in this work is following: Eq. 1 assumes that the electrical potential along the rod is the same over the entire length. In reality, according to the method of mirror charges, the rod behaves like a line electrode whose potential is the highest at the deepest point and has a nonlinear distribution. By segmenting the rod into single sections in a thought experiment, the resistance to earth of the sections decreases with increasing depth. Consequently, the dissipated amperage per unit length becomes higher at deeper segments (VER11 in Fig. 4).

For short rods up to some metres Eq. 1 usually provides sufficiently good results. With longer rod length, the difference between the results for the resistance to earth from Eq. 1 [4] and the used method of mirror charges increases. Equation 1 takes as basis a homogeneous soil and it is not useable for two-layered soils. But Eq. 1 [4] illustrates that the influence of the rod’s cross-section on the resistance to earth is minor: for example, a cross-section twice as large (2 × 240 mm2) decreases the resistance to earth by 0.44 Ω (steady-state, homogeneous soil, ρ = 160 Ωm). For technical reasons, the cross-section cannot be minimized to some square millimetres based on

  • the amperages in the kA range,

  • the transient demands,

  • the mechanical strength and

  • the corrosion resistance.

Fig. 5 shows the amperage dissipated to the earth per a unit length of 1 m of the earth electrode for the two vertical earth rods VER11 and VER12. The distance between VER11 and VER12 is 24.5 m and the resistance to earth of this arrangement is calculated as RE = 4.28 Ω.

Fig. 5
figure 5

Dissipated currents to earth, VER11–VER12

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It can be seen that the amperage of the dissipated current increases with depth at both vertical earth rods and is at its highest at the deepest point of the vertical earth rods. Furthermore, the current to earth is divided equally between the two vertical earth rods. The calculated partial fault voltages for this arrangement are shown in Figs. 6 and 7.

Fig. 6
figure 6

Partial fault voltages, VER11–VER12, 3d view

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Fig. 7
figure 7

Partial fault voltages, VER11–VER12, 2d view

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The calculated maximum partial fault voltages at both vertical earth rods are the same due to the identical earth electrode currents. Figure 7 shows that the partial fault voltages around the single vertical earth rods do not form concentric circles, indicating an electrical interference via the soil between them.

If the arrangement is extended by the vertical earth rod VER13, which is 16.6 m away from VER12 in absolute length, the calculated amperage per unit of length of the current dissipated to earth result as shown in Fig. 8.

Fig. 8
figure 8

Dissipated currents to earth, VER11–VER13

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The resistance to earth decreases to RE = 3.15 Ω compared to the arrangement with two vertical earth rods (RE = 4.28 Ω). The resistance to earth is not reduced by 1/3, the decrease is 26%.

The amperage of the currents dissipated to earth per unit length increase with rising depth for each vertical earth rod and have their maximum at the deepest point of the vertical earth rods. The same effect also occurred at the constellation of two vertical earth rods (constellation VER11–VER12). It can be seen that the amperages at the outer vertical earth rods VER11 and VER13 are higher than at the inner VER12. As a result the calculated partial fault voltages are the smallest at VER12 (Figs. 9 and 10).

Fig. 9
figure 9

Partial fault voltages, VER11–VER13, 3d view

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Fig. 10
figure 10

Partial fault voltages, VER11–VER13, 2d view

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The partial fault voltages in the area of the three vertical earth rods shows that the electrical interference from VER12 and VER13 is stronger than to VER11 in each case. This is due to the smaller distance between VER12 and VER13. This can also explain the reduction of the resistance to earth by 26%. The reduction depends, on the one hand on the number of vertical earth rods and on the other hand on the geometric arrangement and the length.

If four vertical earth rods (VER14–VER17) are added to the calculation model, the amperages of the dissipated currents to earth per unit length result as shown in Fig. 11.

Fig. 11
figure 11

Dissipated currents to earth, VER11–VER17

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The reduction of the resistance to earth is less than 50% (RE = 1.60 Ω) compared to the arrangement VER11–VER13. The thesis stated earlier can be confirmed that the resistance to earth does not decrease linearly with the number of vertical earth rods, but depends on their geometric arrangement. The calculation results of the partial fault voltages for the arrangement with seven vertical earth rods (VER11–VER17) in Figs. 12 and 13 show that the highest values at the vertical earth rods again occur at the outermost rods (VER11 and VER17).

Fig. 12
figure 12

Partial fault voltages, VER11–VER17, 3d view

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Fig. 13
figure 13

Partial fault voltages, VER11–VER17, 2d view

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At these outermost rods, the surface potential is steeper with increasing distance from the other rods, as the area of influence of the remaining vertical earth rods decreases. The partial fault voltages increase compared to the central rods. It can be seen that even small deviations from a straight alignment as well as changing distances between the rods lead to changes of the mutual interference and the partial fault voltages UFP. Although the partial fault voltage is lowest at the middle vertical earth rod VER14 (UFP = 788 V), in contrast it is equally high at VER15 and VER16 (UFP = 924 V). At VER13, the partial fault voltage is 901 V and even higher than at VER12, where the partial fault voltage is 866 V.

To verify the calculated resistances to earth RE,c, measurements were carried out using the fall-of-potential method. For the constellation I designated area, one vertical earth rod after the other has been installed and a measurement of the vertical earth rods galvanically connected for the measurement has been carried out in each case. The measurement was performed by injecting a sinusoidal current with 57 periods per second. A comparison of the calculated resistances to earth RE,c with the measured resistances to earth RE,m as well as the percentage, absolute deviation of the measured values from the calculated values Diff is provided in Table 1.

Table 1 Comparison of calculation and measurement results for constellation I
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The difference between the calculation and the measurement is less than 5%, from which the conclusion can be drawn that for electrical flow fields with typical power frequencies, a steady-state calculation method provides sufficiently high accuracy.

4 Results for constellation II

In constellation II, where two types of vertical earth rods with different lengths (10 and 20 m) are investigated, the calculated amperage of the dissipated currents to earth per unit length increases with depth, as in constellation I.

The amperages per unit length are in a similar range of values near the earth’s surface as well as at the deepest point of the vertical earth rods, but when considering the vertical earth rod’s amperages (per vertical earth rod), significant differences become visible: For example, the amperages per earth electrode for the group of 10 m vertical earth rods are in the range of approx. 80–90 A and are between for the 20 m deep vertical earth rods approx. 160 and 175 A. This means that, given the same earth conditions, longer vertical earth rods dissipate higher amperages to the earth than shorter ones. The calculation results of the partial fault voltages UFP are shown in Figs. 14 and 15. The maximum value of UFP is 837 V and occurs at VER28, its minimum value 758 V at VER26.

Fig. 14
figure 14

Dissipated currents to earth, VER21–VER28

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Fig. 15
figure 15

Partial fault voltages, VER21–VER28, 3d view

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The 2D view (Fig. 16) shows that the electrical interference between of the group of 10 m vertical earth rods (VER21–VER24) are similarly as the group of 20 m vertical earth rods (VER25–VER28). This is due to the geometrical arrangement. Although the 20 m vertical earth rods are twice as long, the distance between them is also almost twice as large (approx. 21 m) as between the 10 m vertical earth rods (approx. 13 m). If one would compare the partial fault voltages UFP at a single 10 m rod with a 20 m rod at the same current to earth, one would expect at least twice as high a value for the 10 m vertical earth rod.

Fig. 16
figure 16

Partial fault voltages, VER21–VER28, 2d view

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This consideration does not apply to constellation II for the following reasons: the total currents dissipated to earth to a depth of 10 m are higher for the 10 m vertical earth rod, which is the total earth electrode amperage for them, than for the 20 m vertical earth rods (approx. 75 A). Also, the deeper the currents are dissipated to earth, the smaller is their contribution for the surface potential increase. In summary, the geometrical arrangement, where the distance between the 20 m vertical earth rods is almost double that of the 10 m vertical earth rods, as well as the amperage dissipated to earth, show that UFP for the 10 m vertical earth rods is in a similar value range as for the 20 m vertical earth rods. To verify the calculation results for the calculated resistance to earth RE,c, the resistance to earth RE,m has been measured at 57 Hz using the fall-of-potential method, as was done for constellation I. The results of the calculation and the measurement as well as the percentage, absolute deviation of the measured values from the calculated values Diff are compared in Table 2.

Table 2 Comparison of calculation and measurement results for constellation II
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When comparing the calculated and measured values in Table 2, it is noticeable that the measured values deviates more strongly from the calculated values than in constellation I in Table 1. This is due to the fact that, in contrast to constellation I, all vertical earth rods were already installed at the time of the measurement. There were also other installations in the earth in the vicinity, resulting in lower measured values than calculated values due to the interference between the vertical earth rods and other installations. However, it can be seen that the deviation decreases with the number of vertical earth rods connected galvanically in parallel. For example, it is 19% for the VER21–VER22 measurement and only 10% for the VER21–VER28 measurement.

5 Conclusion/Summary

Calculations of the (partial) earth currents dissipated to earth, partial fault voltages as well as resistances to earth are carried out for vertical earth rods in different geometrical arrangements, soil layers and rod lengths. In particular, measurements in a 220/110 kV substation in construction have been carried out on the vertical earth rods to verify the calculated resistances to earth.

The calculation results of the ohmic interference show that the vertical earth rods interfere to different intents via the electric flow field depending on their geometric dimensions and arrangement. As a consequence, even with homogeneous soil layers, the partial fault voltages as well as the currents dissipated to earth differ at each vertical earth rod. In general, it can be positively determined as an important property of vertical earth rods that the amperage of the partial earth electrode currents dissipated to earth per unit length increases with increasing depth, whereby the resistance of the soil of the considered depth has to be taken into account. The higher the resistance of the soil, the lower the amperage dissipated to earth per unit length. Irrespective of this and the length of the vertical earth rods, the highest amperage is dissipated to earth at the lowest point of the vertical earth rods. For the in practice occurring frequent case of soil layers with poorer conductivity at the earth’s surface than in deeper layers, this has hardly any influence. Regarding to the resulting partial fault voltages, since the physical relationship between electrical current and electrical resistance, described by Ohm’s law is also valid here. If the electrical current decreases and the electrical resistance increases, the voltage hardly changes. In order to achieve a significant reduction in the partial fault voltages on the surface in the area of vertical earth rods, one solution is applying material with very poor electrical conductivity like tarmac, so that the dissipation of currents to earth close to the surface is prevented.

Verified by the measurements, it is shown that there is a non-linear correlation between the number of vertical earth rods and the resistance to earth in the arrangements investigated. According to this, the ohmic interference between the vertical earth rods depends on their distance and their length. As can be seen from the present work, vertical earth rods have the positive property of locally dissipating currents into deep layers of the earth. The good correlation between the presented steady-state calculation and the undisturbed measurement of the resistances to earth of the vertical earth rods confirms that steady-state calculation methods are appropriated for designing earthing systems for typical power frequencies.