11

Transmission System Transients: Grounding

William A. Chisholm

Kinectrics/Université du Quebec à Chicoutimi

11.1    General Concepts

11.2    Material Properties

11.3    Electrode Dimensions

11.4    Self-Capacitance of Electrodes

11.5    Initial Transient Response from Capacitance

11.6    Ground Electrode Impedance: Wire over Perfect Ground

11.7    Ground Electrode Impedance: Wire over Imperfect Ground

11.8    Analytical Treatment of Complex Electrode Shapes

11.9    Numerical Treatment of Complex Electrode Shapes

11.10  Treatment of Multilayer Soil Effects

11.11  Layer of Finite Thickness over Insulator

11.12  Treatment of Soil Ionization

11.13  Design Process

11.14  Design Recommendations

11.A   Appendix A: Relevant IEEE and IEC Standards in Lightning and Grounding

References

11.1  General Concepts

Electric power systems are often grounded, that is to say “intentionally connected to earth through a ground connection or connections of sufficiently low impedance and having sufficient current-carrying capacity to prevent the buildup of voltages which may result in undue hazard to connected equipment or to persons” (IEEE Std 100, 2000). Grounding affects the dynamic power-frequency voltages of unfaulted phases, and influences the choice of surge protection. Also, the tower footing impedance is an important specification for estimating the transient voltage across insulator strings for a lightning flash to an overhead ground wire, tower or phase conductor with surge arrester.

To mitigate ac fault conditions, systems can be grounded by any of three means (IEEE Std 100, 2000):

•  Inductance grounded, such that the system zero-sequence reactance is much higher than the positive-sequence reactance, and is also greater than the zero-sequence resistance. The ground-fault current then becomes more than 25% of the three-phase fault current.

•  Resistance grounded, either directly to ground, or indirectly through a transformer winding. The low-resistance-grounded system permits a higher ground-fault current (on the order of 25 A to several kiloamperes) for selective relay performance.

•  Resonant grounded, through a reactance with a value of inductive current that balances the power-frequency capacitive component of the ground-fault current during a single line-to-ground fault. With resonant grounding of a system, the net current is limited so that the fault arc will extinguish itself.

Power system transients have a variety of waveshapes, with spectral energy ranging from the power frequency harmonics up to broadband content in the 300-kHz range, associated with 1-μs rise and fall times of lightning currents and insulator breakdown voltages. With the wide frequency content of transient waveshapes, resonant grounding techniques offer no direct benefit in preventing arcing flashover from the grounding system to the insulated phases. Also, resistance grounds that may be effective for power frequency can have an additional inductive voltage rise (L dI/dt with dt = 1 μs) that dominates the transient response. Both resistive and inductive aspects must be considered in the selection of an appropriate ground electrode for adequate performance during transients such as lightning.

11.2  Material Properties

The earth resistivity (ρ, units of Ωm) dominates the potential rise on ground systems at low frequencies and currents. Near the surface, resistivity changes as a function of moisture, temperature, frequency, and electric field stress. Figure 11.1 shows that this variation can be quite large.

Soil moisture can change over periods of days or even hours, giving significant changes in resistivity especially in surface layers of soil. Reconnaissance of earth resistivity, from traditional four-terminal resistance measurements, is a classic tool in geological prospecting (Keller and Frischknecht, 1982). A current I(A) is injected at the outer two locations in a line of four equally spaced probes. A potential difference U(V) then appears between the inner two probes, which are separated by a distance a(m). The apparent resistivity ρ_a (Ωm) is then defined as

${\rho }_a=2\pi a\frac{U}{I}$

FIGURE 11.1  Change in resistivity with relative humidity and frequency for three typical surface materials. (From Visacro, S. and Portela, C.M., Soil permittivity and conductivity behavior on frequency range of transient phenomena in electric power systems, Proc. 5th ISH, Paper 93.06, August 1987.)

At a given location, several measurements of ρ_a are taken at geometrically spaced values of a, such as (a = 1, 3, 9, 27, 81 m or a = 2, 6, 18, 54 m) so that the outer probes from one measurement become the inner probes on the next. When ρ_a is constant with distance, the assumption of a uniform soil model is justified, and the effective resistivity ρ_e for any electrode size is simply ρ_a. However, in many cases, there are two or more layers of contrasting soil. The most difficult case tends to be a thin, conducting top layer (clay, till, sand) over a thick, poorly conducting rock layer (ρ₁ < ρ₂). This case will have an increasing value of ρ_a with distance. A simplified interpretation in this case can be given for a practical range of resistivity values: For flat electrodes, the effective resistivity, ρ_e, equals the value of ρ_a observed at a probe spacing of (a = 2s), where s is the maximum extent (e.g., the radius of a ring electrode), or a is the diagonal dimension of the legs of a tower.

11.3  Electrode Dimensions

Five dimensions are relevant for analysis of electrode response under steady-state and transient conditions. In order of decreasing importance, these are as follows:

s: The three-dimensional distance from the center of the electrode to its outermost point. For a spheroid in a conducting half-plane, s = MAX(a,b) where a is the maximum cross-section radius and b is the length in the axis of symmetry. Table 11.1 adapts equations for surface area of a cylinder, spheroid, and box to typical electrodes. Different dimensions dominate the s, g, and A_Total terms, depending on the electrode shape. Table 11.1 shows that the threedimensional extent s of cylinders and prisms are slightly larger than the s for a prolate spheroid of the same depth. The propagation time τ = s/c, calculated from speed-of-light propagation at c = 3 × 10⁸ m/s, is used to estimate transient electrode impedance.

g: The geometric radius of the electrode, $g={\left(R_x^2+R_y^2+R_z^2\right)}^{1/2}$, is used to estimate capacitance. For long, thin, or rectangular shapes, g = s; for a disk, g = √2s; for a hemisphere, g = √3s.

A_Total: The surface area in contact with soil. Electrodes with large surface area will have lower resistance, lower impedance, and less susceptibility to unpredictable effects of soil ionization. For objects with concave features, the area of the smallest convex body that can envelop is determined. With this model, a tube, a circular array of wires, or a collinear array of rings have the same area A_Total as a solid cylinder of the same dimensions. Buried horizontal wires expose area A_Total on both sides of the narrow trench. Table 11.1 provides further interpretations.

L: The total length of wire in a wire frame approximation to a three-dimensional solid, L, is used to correct for the contact resistance, typically with R_Contact ≤ ρ₁/L as described by Laurent (IEEE Standard 80, 1986). Contact resistance tends to be much smaller than the geometric resistance for most electrodes.

A_Wire: The total surface area of the wire of length L to be used to refine the calculation of R_Contact for wire frame electrodes (Table 11.1).

The transmission tower foundation tends to have the largest surface area A_total and greatest extent g compared to supplemental vertical rods and buried horizontal wires. Thus, a significant fraction of lightning current will flow into the foundation. As described in Grigsby (2011), the foundations may consist of the following:

•  One, two, or four reinforced concrete drilled shafts, each with a ratio length/diameter of 2:1 to 10:1

•  Directly embedded steel or concrete poles with a length/diameter ratio of 3:1 to 6:1

•  Spread foundations with steel lattice (grillage) beneath

•  Anchors, providing parallel grounding paths for current flow along tower guy wires, typically screwed into soil or grouted into drilled holes in rock to a length of at least 5 m

Unlike distribution lines, where the pole embedment can be a fixed (10%) fraction of line height, there is no single rule that relates the transmission tower height to the depth of its foundation. Also, there is a tendency to have foundations of reduced dimensions in rock of high electrical resistivity, while foundations need larger dimensions to ensure adequate transfer of mechanical loads in clay materials of low resistivity. Please refer to Grigsby (2011) for additional details.

TABLE 11.1 Values of A_Total s, g, L, and A_Wire for Typical Ground Electrodes in Half-Plane

11.4  Self-Capacitance of Electrodes

Electrode capacitance is easily calculated (Chow and Yovanovich, 1982; Chow and Srivastava, 1988) and offers elegant description of grounding response to transients. Also, the self-capacitance C_self to infinity of an arbitrary conducting object in full space has a useful dual relation to its steady-state resistance R in a half-space of conducting medium, given by (Weber, 1950)

$R=\frac{2{\epsilon }_o\rho }{C_{self}}$

(11.1)

where

ε_o is the permittivity constant, 8.854 × 10⁻¹² F/m

ρ is the earth resistivity, in Ωm

The transient impedance of the same arbitrary conducting object can be modeled using the time (τ) it takes to charge up its self-capacitance C_self. This time cannot be less than the maximum dimension of the electrode, s, divided by the speed of light. An average surge impedance Z, given by the ratio τ/C, can then be used to relate voltages and currents during any initial surge. The capacitance of an object is approximately (Chow and Yovanovich, 1982)

$C_{Self}={\epsilon }_o{c}_f\sqrt{4\pi A}$

(11.2)

where

A is the total surface area of the object, including both sides of disk-like objects

c_f is a correction factor between the capacitance of the object and the capacitance of a sphere with the same surface area

For a wide range of objects, 0.9< c_f < 1.2 using $c_f\approx \sqrt{2\gamma }/\ln \left(4\gamma \right)$, where γ is the ratio of length to width of the object.

A close estimate of c_f for spheroids is given by the following expression:

$c_f=\frac{4\pi g}{\sqrt{4\pi A}\ln \left(4\pi e^{\sqrt{3}}{g}^2/3A\right)}\cong \frac{3.54g}{\sqrt{A}\ln \left(23.7g^2/A\right)}$

(11.3)

Again, $g=\sqrt{r_x^2+r_y^2+r_z^2}$, the geometric radius. Equation 11.3 is exact for a sphere, and remains valid for a wide range of electrode shapes, from disk to rod.

A surprisingly large amount of the surface area of a solid can be removed without materially affecting its self-capacitance. For example, the capacitance of a solid box of dimensions 1 × 1 × 0.4 m and a surface area of 3.6 m² is 56 pF. The capacitance of two parallel loops filling out the same space is 50 pF, even though the loops have only 20% of the surface area of the box (Chow and Srivastava, 1988). The equivalent correction from wire frame approximations to solid objects in ground resistance calculations is known as a “Contact Resistance.”

11.5  Initial Transient Response from Capacitance

Once the capacitance of a conducting electrode and its minimum charging time τ have been estimated, its average transient impedance can be computed from the relation Z = τ/C_self, obtained from the use of open-circuit stub transmission lines for tuning radio antennas. The charging time τ of the conducting electrode in free space is the maximum three-dimensional extent, s, divided by the velocity of light in free space, c. This transient impedance of the ground electrode will be seen only during the charging time from the center of the electrode to its full extent, and the impedance will reduce with increasing time as the electromagnetic fields start to interact with the surroundings, including soil, towers, and overhead ground wires.

The following graph (Figure 11.2) gives the initial transient response of conducting spheroid electrodes (see Table 11.2).

FIGURE 11.2  Relation between transient impedance and aspect ratio: spheroid electrodes in half space.

TABLE 11.2 Transient Impedance of Conducting Electrodes

The main observation from Figure 11.2 is that wide, flat electrodes will have inherently better initial transient response than long, thin electrodes. This includes electrodes in both horizontal and vertical planes, giving a physical reason why long leads to remote ground electrodes are not effective under transient conditions.

Chisholm and Janischewskyj (1989) confirmed an apparent initial transient impedance of about 60 Ω for a perfectly conducting ground plane (infinite circular disk) that reduced as a function of time, with a rate of reduction depends on the electromagnetic travel time from tower top to tower base. Baba and Rakov (2007) validated the imperfect reflection from a ground plane using advanced FDTD numerical methods and offered more detailed interpretation. For compact electrodes, the response can be lumped into an inductance element (L_Self = τ²/C_Self). The potential rise on the earth electrode can then be estimated from the simple circuit model:

$U_{electrode}=RI+\left(\frac{{\tau }^2}{C_{Self}}\right)\frac{dI}{dt}$

(11.4)

A numerical example for Equation 11.4 is useful. For ρ = 100 Ωm and a disk electrode of 5 m radius, the resistance to remote ground will be 3.2 Ω and τ²/C_self = 0.8 μH. For a typical (median) lightning stroke with I = 30 kA and dI/dt = 24 kA/μs at the peak, the two terms of the peak potential rise V_Pk will be

$V_{Pk}=RI+L\frac{dI}{dt}=\left(30\text{kA}\right)\left(3.2\Omega \right)+\frac{24\text{kA}}{\mu s}\left(0.8\text{μH}\right)=96\text{kV}+19\text{kV}$

The inductive term is desirably low in the example, but it can dominate the response of long, thin electrodes in low-resistivity soil. For distributed electrodes, however, the circuit approximation in Equation 11.4 eventually fails as rate of current rise (dI/dt) increases and surge impedance models must be used as developed next.

11.6  Ground Electrode Impedance: Wire over Perfect Ground

The inductance L per unit length of a distributed grounding connection over a conducting plane can be calculated from the surge impedance Z of the wire and its travel time τ:

$L=Z\tau =\left(\frac{{\tau }^2}{C_{Self}}\right)Z=60\ln \left(\frac{2h}{r}\right)$

(11.5)

where

Z is the surge impedance (Ω) of a wire of radius r(m) over a ground plane at a height h(m)

τ = s/c, where s is the wire length or electrode extent (m) divided by the speed of light c (3.0 × 10⁸ m/s)

11.7  Ground Electrode Impedance: Wire over Imperfect Ground

When the electrode is placed over imperfect ground, the effective return depth of current will increase. Bewley (1963) suggests that the plane for image currents, in his tests of buried horizontal wires, was 61 m (200’) below the earth surface. Some analytical indication of the increase in return depth is given by the normal skin depth, $\delta =1/\sqrt{\pi f{\mu }_o/\rho }$, which decreases from 460 m (60 Hz) to 11 m (100 kHz) for a resistivity of ρ = 50 Ωm. Deri et al (1981) propose a more complete approach, replacing the height h in Equation 11.6 with (h + p) where with the depth p to an equivalent perfectly conducting plane is a complex number given by

$p=\sqrt{\frac{\rho }{j\omega {\mu }_o}}$

(11.6)

where

ω is 2π times the frequency (Hz)

μ_o is 4π × 10⁻⁷ H/m

ρ is the resistivity in Ωm

For good soil with a resistivity of ρ = 50 Ωm, the complex depth is 230 (1 – j) m at 60 Hz and 5.6 (1 – j) m at 100 kHz. The complex depth is related to the normal skin depth by the relation 1/p = (1 + j) 1/δ. The velocity of propagation will also reduce as the wire is brought close to, or buried in, imperfectly conducting ground. Darveniza (2007) simplifies the approach and recommends the following expression for effective height above ground h_eff, to be substituted into Equation 11.5:

$h_{eff}=h+0.15\sqrt{\rho }$

(11.7)

where

ρ is the soil resistivity (Ωm)

h_eff and h are in m

11.8  Analytical Treatment of Complex Electrode Shapes

Simple analytical expressions are documented for a variety of regular electrode shapes (see, e.g., Sunde, 1949; Smythe, 1950; Weber, 1950; Keller and Frischknecht, 1982). However, grounding of electrical systems often consists of several interconnected components, making estimation of footing resistance more difficult. The tower foundation can be a single or (more typically) four concrete cylinders, often reinforced with steel. In the preferred case, the steel is bonded electrically, and a grounding connection is brought out of the form before the concrete is poured. In areas with low soil resistivity, four concrete footings can often provide a low tower resistance without supplemental electrodes.

In some cases on both transmission and distribution systems, a metal grillage (or pole butt-wrap) is installed at the base after excavation. This deep electrode is more effective than a surface electrode of the same area. Also, grillage and pole-wrap electrodes are protected from vandalism and frost damage.

Supplemental grounding electrodes are often installed during line construction or upgrade. The following approaches are used:

•  Horizontal conductors are bonded to the tower, and then buried at a practical depth.

•  Vertical rods are driven into the soil at some distance from the tower, and then bonded to the tower base, again using bare wires, buried at a practical depth.

•  Supplemental guy wires are added to the tower (often for higher mechanical rating) and then grounded using rock or soil anchors at some distance away.

Supplemental grounding should be considered to have a finite lifetime of 5–20 years, especially in areas where the soil freezes in winter. Also, auxiliary electrodes such as rock anchors should be designed to carry their share of impulse current, and to withstand the associated traverse forces.

The resistance of an electrode that envelops all contacts can be used to obtain a good estimate of the combined resistance of a complex, interconnected electrode. From Chisholm and Janischewskyj (1989), the resistance of a solid rectangular electrode is approximately

$R_{Geometric}=\frac{\rho }{2\pi s}\ln \left(\frac{2\pi e{s}^2}{A}\right)\cong \frac{\rho }{2\pi s}\ln \left(\frac{17s^2}{A}\right)$

(11.8)

where

s is the three-dimensional distance from the center to the furthest point on the electrode

A is the convex surface area that would be exposed if the electrode were excavated

e is the exponential constant, 2.718

The resistance can also be estimated using the geometric radius g rather than the maximum dimension s:

$R_{Geometric}\cong \frac{\rho }{2\pi g}\ln \left(\frac{11.8g^2}{A}\right)$

(11.9)

where

$g=\sqrt{r_x^2+r_y^2+r_z^2}$ is the geometric radius of the electrode, r_x,y,z = maximum x, y, and z dimensions 11.8 = (2πe√³)/3

If the electrode is a wire frame, rather than a solid, then a correction for contact resistance should be added to the geometric resistance:

$R_{WireFrame}=R_{Geometric}+R_{Contact}=R_{Geometric}+K\frac{{\rho }_1}{L}$

(11.10)

where

L is the total length of the wire frame

ρ₁ is the resistivity of the upper layer of soil (the layer next to the wire)

K is a constant that varies from 0.5 ≤ K ≤ 1.3 for most electrode shapes depending on fill ratio

Many examples of contact resistance can be found in the literature on grounding (e.g., Sunde 1949; IEEE Standard 80, 1986). Consider for example the resistance of a surface disk of radius s compared to a ring of the same radius, made from wire with a diameter d:

$\begin{array}{l}{R}_{Disc}=\frac{\rho }{4s};R_{Ring}=\frac{\rho }{2{\pi }^{2}s}\ln \left(\frac{4s}{d}\right)\hfill \\ g=\sqrt{s^2+s^2+d^2}\approx \sqrt{2}s;A=\pi s^2;R_{Geometric}=\frac{\rho }{2\pi g}\ln \left(\frac{11.8g^2}{4.4s}\right)\hfill \\ R_{Contact}=R_{Ring}-R_{Disc}\hfill \end{array}$

(11.11)

The difference between the resistance of a ring of d = 13 mm wire and a disk of the same radius s can be described using a value of K in Equation 11.10 that ranges from K = 0.48 for s = 2 m, to K = 0.99 for s = 10 m ring, to K = 1.33 for s = 30 m, depending on the logarithm of the ratio of overall to wire surface areas as follows:

$K=\frac{1}{2\pi }\ln \left(\frac{A_{Total}}{2A_{Wire}}\right),K>0$

(11.12)

Equation 11.12 is valid for most electrode shapes, such as combinations of rings, grids, and rods. When several (N) radial wires meet at a point, the contact resistance coefficient K in Equation 11.10 should use

$K=\frac{1}{2\pi }\ln \left(\frac{N^2{A}_{Total}}{8A_{Wire}}\right),K>0$

(11.13)

where

N is the number of wires meeting at a point

A_Total is the surface area of the pattern made by the radial array

A_Wire is the surface area of the wires themselves

11.9  Numerical Treatment of Complex Electrode Shapes

Solving for the combined resistance of a number of individual electrode elements, such as four foundations of a transmission tower along with some horizontal wires and ground rods, can take several approaches.

The geometric resistance of the entire electrode can be computed, and the contact resistance correction for a wire frame approximation can be added as in Equation 11.11. However, in some cases such as solid cylinders of concrete, the “length” of wire to use is not obvious, and also, the value of K varies depending on whether the electrode is tightly meshed or open.

A second possibility is to develop a matrix of self- and mutual-resistance values of each solid, conducting element. The geometric resistances defined in Equation 11.10 give the self-resistance R_ii. The mutual resistance between the centroids of two objects is calculated from their separation d_ij:

$R_{ij}=\frac{\rho }{2\pi d_{ij}}$

(11.14)

A symmetrical matrix of self- and mutual resistances is multiplied by a vector of current values to obtain the potentials at each electrode. The matrix can be built, inverted, and solved in Excel as follows.

FIGURE 11.3  Numerical solutions of resistance for 10 m rods at 5 m grid spacing (IEEE Guide 80/2000).

In our case, we will consider five 3 m long rods of 5 mm radius in ρ = 320 Ωm soil, on a 5 m × 5 m grid as shown in Figure 11.3. The self-resistance is calculated from g = 10 m, A = 2πrl = 0.4 m² in Equation 11.10 as 119 Ω. The distance between the center rod and all others is d_1j = 7.07 m, d₂₃ = d₃₄ = d₄₅ = d₂₅ = 10 m, and d₂₄ = d₃₅ = 14.1 m. The mutual resistances are given by R_ij = ρ/(2πd_ij) from Equation 11.14 as follows:

In Excel, a square matrix can be inverted by highlighting a blank space of the same size (5 × 5 cells in this case), typing in the formula = MINVERSE(A1:E5) where the original R_ij matrix resides in cells A1:E5, and pressing the F2 key and then <Ctrl><Shift><Enter> at the same time.

The inverted matrix can be multiplied by a vector of unit potentials (1s) to obtain the currents in each electrode. The unit potential divided by the sum of these currents gives the resistance of the combined electrode, including self and mutual effects. The following table completes the numerical example.

To multiply the inverted resistance matrix (say in locations A10:E15) by a vector of unit potentials (say in locations G10:G15), the cells I10:I15 would be highlighted, the formula = MMULT(A10:E15,G10:G15) would be entered, and the F2 key would be pressed and then <Ctrl><Shift><Enter> pressed at the same time.

The currents are summed to a total of 35.27 mA, giving a resistance of 28.4 Ω. The five resistances in parallel, ignoring mutual effects, would have a resistance of 23.9 Ω, so the effect at a separation of 5 m is still significant.

A series of numerical calculations are found for a similar set of electrodes in (IEEE Guide 80, 2000, p. 183) for 10 m rods on 5 m grid spacing in 100 Ωm soil.

The relative reduction in resistance is largest when the second electrode is added, and additional nearby rods are seen to be less effective.

11.10  Treatment of Multilayer Soil Effects

Generally, the treatment of footing resistance in lightning calculations considers a homogeneous soil with a finite conductivity. This treatment, however, seldom matches field observations, particularly in areas where grounding is difficult. Under these conditions, a thin “overburden” layer of conducting clay, till, or gravel often rests on top of insulating rock. The distribution of resistivity values for a particular overburden material and condition can be narrow, with standard deviations usually less than 10%. However, the variation of overburden depth with distance can be large. Airborne electromagnetic survey techniques at multiple frequencies in the 10–100 kHz range offer a cost-effective method of reconnaissance of the overburden parameters of resistivity and depth.

FIGURE 11.4  Relative effective resistivity versus ratio of electrode radius to upper-layer depth, with resistivity ratio as a parameter.

Once a resistivity survey has established an upper-layer resistivity ρ₁, a layer depth d, and a lower-layer resistivity ρ₂, the equivalent resistivity ρ_e can be computed. For a disk-like electrode buried just below the surface, ρ_e from the elliptic-integral solutions of (Zaborsky, 1955) can approximated with better than 5% accuracy by the following empirical equations:

$\begin{array}{l}{\rho }_e={\rho }_1\frac{1+C\left({\rho }_2/{\rho }_1\right)\left(r/d\right)}{1+C\left(r/d\right)}\hfill \\ C=\{\begin{array}{ll}{\rho }_1\ge {\rho }_2,\hfill & \frac{1}{1.4+{\left({\rho }_2/{\rho }_1\right)}^{0.8}}\hfill \\ {\rho }_1<{\rho }_2,\hfill & \frac{1}{1.4+{\left({\rho }_2/{\rho }_1\right)}^{0.8}+{\left(\left({\rho }_2/{\rho }_1\right)\left(r/d\right)\right)}^{0.5}}\hfill \end{array}\hfill \end{array}$

(11.15)

The ratio of effective resistivity to upper-layer resistivity varies with the ratio of electrode radius s to upper-layer depth d, with a small ratio of s/d giving a ratio of unity as shown in Figure 11.4.

Normally, electrode penetration through an upper layer would only be desirable in extreme examples of Case 1 (ρ₁ ≫ ρ₂). Rather than recomputing the effective resistivity with revised image locations, the effects of the upper layer can be neglected, with the connection through ρ₁ providing only series self-inductance.

11.11  Layer of Finite thickness over Insulator

A simpler two-layer soil treatment is appropriate for Case 2 when ρ₂ ≫ ρ₁, or equivalently the reflection coefficient from upper to lower layer Γ₁₂ approaches 1. Under these conditions, the following summation in Equation 11.16 describes the resistance of a single hemisphere of radius s in a finitely conducting slab with resistivity ρ₁ and thickness h:

FIGURE 11.5  Asymptotic behavior of hemispherical electrode in conducting layer over insulator.

$\begin{array}{l}{\Gamma }_{12}=\frac{{\rho }_2-{\rho }_1}{{\rho }_2+{\rho }_1}\approx 1\text{for}{\rho }_2\gg {\rho }_1{R}_{Keller}=\frac{{\rho }_1}{2\pi s}\left(1+2\cdot {\displaystyle \sum _{n=1}^{\infty }\frac{{\Gamma }_{12}^n}{\sqrt{1+{\left(2nh/s\right)}^2}}}\right)\hfill \\ R_{Loyka}\approx \frac{{\rho }_1}{2\pi s}\left[1+\frac{s}{h}\ln \left(\frac{1+\sqrt{1+{\left({10}^{5}h/2h\right)}^2}}{1+\sqrt{1+{\left(s/2h\right)}^2}}\right)\right]\approx \frac{{\rho }_1}{2\pi s}\left[1+\frac{s}{h}\ln \left(\frac{50,000}{1+\sqrt{1+{\left(s/2h\right)}^2}}\right)\right]\hfill \end{array}$

(11.16)

For a reflection coefficient Γ₁₂ = 1, the Keller series in Equation 11.16 becomes the harmonic series and converges slowly. Loyka (1999) offers a convenient approximation to the sum, developed using the potential from a second hemisphere, located sufficiently far away (e.g., 10⁵d, where d is the layer thickness) to have no influence. Figure 11.5 shows that the Loyka approximation to the Keller series is good for a wide range of transmission line grounding applications, and that finite upper layer depth has a strong influence on the resistance of the electrodes.

11.12  Treatment of Soil Ionization

Under high electric fields, air will ionize and become effectively a conductor. The transient electric fields needed to ionize air in small volumes of soil, or to flashover across the soil surface, are typically between 100 and 1000 kV/m (Korsuncev, 1958; Liew and Darveniza, 1974; Oettle, 1988). Considering that the potential rise on a small ground electrode can reach 1 MV, the origins of 10 m furrows around small (inadequate) ground electrodes after lightning strikes become clear. Surface arcing activity is unpredictable and may transfer lightning surge currents to unprotected facilities. Thus, power system ground electrodes for lightning protection should have sufficient area and multiplicity to limit ionization.

Korsuncev used similarity analysis to relate dimensionless ratios of s, ρ, resistance R, current I, and critical breakdown gradient E_o as follows:

${\Pi }_1=\frac{Rs}{\rho }\text{with}{\Pi }_1^O=\frac{1}{2\pi }\ln \left(\frac{2\pi e{s}^2}{A}\right)$

(11.17)

${\Pi }_2=\frac{\rho I}{E_o{s}^2}$

(11.18)

${\Pi }_1=\text{MIN}\left({\Pi }_1^O,0.26\cdot {\Pi }_2^{-0.31}\right)$

(11.19)

$R=\frac{\rho {\Pi }_1}{s}$

(11.20)

where

s is the three-dimensional distance from the center to the furthest point on the electrode, m

I is the electrode current, A

ρ is the resistivity, Ωm

E_o is the critical breakdown gradient of the soil, usually 300–1000 kV/m

R is the resistance of the ground electrode under ionized conditions

The calculation of ionized electrode resistance proceeds as follows:

•  A value of ${\Pi }_1^O$ is calculated from Equation 11.17. This un-ionized value will range from ${\Pi }_1^O=0.159$ (for a hemisphere) to 1.07 for a 3 m long, 0.01 m radius cylinder.

•  A value of Π₂ is calculated from Equation 11.18. For a 3 m rod at 100 kA in 100 Ωm soil, with E_o of 300 kV/m, the value of Π₂ = 3.7 is obtained.

•  A value of Π₁ is computed from Π₂ using 0.26 ${\Pi }_2^{-0.31}$. This value, from Equation 11.19, represents the fully ionized sphere with the gradient of E_o at the injected current; for the rod example, Π₁ = 0.173.

•  If the ionized value of Π₁ is not greater than ${\Pi }_1^O$, then there is not enough current to ionize the footing, so the unionized resistance from ${\Pi }_1^O$ will be seen for the calculation of resistance in Equation 11.20. For the 3 m rod, ionization reduces the 36 Ω low-current resistance to 6 Ω at 100 kA.

In two-layer soils with sparse electrodes, ionization effects will tend to reduce the contact resistance (from Equation 11.10) without altering the surface area A or characteristic dimension s. This will tend to reduce the influence of ionization, since the geometric resistance of the electrode is unchanged.

11.13  Design Process

The inputs to the design process are the prospective lightning surge current, the local soil resistivity, and the dimensions of the tower base or trial electrode. The low-frequency geometric resistance of the electrode is computed from Equation 11.9 and double-checked with the estimate from Equations 11.15 and 11.20.

If the electrode is compact (e.g., a driven rod), then the ionization effects should be estimated using Equations 11.18 and 11.19, with the reduced resistance under high-current conditions being given by Equation 11.20.

If the electrode is distributed (e.g., a buried wire), then the ionized resistance will only reduce the contact resistance term in Equation 11.10, and for practical purposes it will be sufficient to disregard this term and consider only the geometric resistance estimated from Equation 11.8 or Equation 11.9.

In each case, an apparent transient impedance of the ground electrode should be modeled as an equivalent inductance in series with the resistive rise associated with the geometric resistance.

The fraction of surge current absorbed by a given electrode will change as its size is adjusted, making the design process iterative.

11.14  Design Recommendations

It is prudent where possible to select an electrode size that does not rely on ionization for adequate transient performance. This can be achieved by inverting the design process as follows.

•  Establish ${\Pi }_1^O$, the shape coefficient of the electrode (0.16 for a hemisphere, 0.26 for a cube, and 0.27 for a disk).

•  Establish the value of ${\Pi }_2^O$ that will just cause ionization (Chisholm and Janischewskyj 1989), inverting Equation 11.19 to obtain

${\Pi }_2^O=0.0131{\left({\Pi }_1^O\right)}^{-3.24}$

(11.21)

For the example of a disk electrode, the value of ${\Pi }_2^O=0.92$ is obtained.

•  The extent (in this case radius) of the electrode s needed to prevent ionization in soil of resistivity ρ in Ωm with current I in kA and voltage gradient E_o in kV/m is given by

$s=\sqrt{\frac{\rho I}{E_o{\Pi }_2^O}}$

(11.22)

For ρ = 300 Ωm, I = 30 kA, and E_o of 400 kV/m, a disk or ring radius of s = 9.1 m will be sufficient to prevent ionization. The geometric resistance of this electrode from Equation 11.20 is 8.9 Ω.

 The following advice is especially relevant for transmission towers or other tall structures, where a low-impedance ground is needed to limit lightning-transient overvoltages:

•  Choose a wide, flat electrode shape rather than a long, thin shape. Four radial counterpoise wires of 60 m will be two to eight times more effective than a single counterpoise of 240 m under lightning surge conditions.

•  Take advantage of natural elements in the structure grounding, such as foundations and guy anchors, by planning for electrical connections and by extending radial wires outward from these points.

•  In rocky areas, use modern airborne techniques to survey resistivity and layer depth using several frequencies up to 100 kHz. Place towers where conductive covering material is deep.

•  Provide grounding staff with the tools and techniques to preestimate the amount of wire required for target footing impedance values, using simple interpretation of two-layer soil data.

•  Near areas where transferred lightning potentials could be dangerous to adjacent objects or systems, use sufficient electrode dimensions to limit ionization, that is to remain on the ${\Pi }_1^O$ characteristic.

11.A  Appendix A: relevant IEEE and IEC Standards in Lightning and Grounding

11.A.1    ANSI/IEEE Std 80-1986: IEEE Guide for Safety in AC Substation Grounding

Presents essential guidelines for assuring safety through proper grounding at ac substations at all voltage levels. Provides design criteria to establish safe limits for potential differences within a station, under fault conditions, between possible points of contact. Uses a step-by-step format to describe test methods, design and testing of grounding systems. Provides English translations of fundamental papers on grounding by such as Rüdenberg and Laurent that are not widely available.

11.A.2    ANSI/IEEE Std 80-2000: IEEE Guide for Safety in AC Substation Grounding

Provides an improved methodology for interpreting two-layer soil resistivity and using the values in the design of ac substations. Provides methods for determining the maximum grid current at substations, some of which also predict the maximum fault currents available on lines close by. Provides a number of new worked examples in appendices.

11.A.3    IEC 62305: Protection against Lightning, Edition 2.0, 2010–12

The European Technical Committee 81 of the International Electrotechnical Commission (IEC, www.iec.ch) prepared an authoritative and comprehensive lightning protection standard in five parts as follows:

•  Part 1, General principles

•  Part 2, Risk management

•  Part 3, Physical damage to structures and life hazard

•  Part 4, Electrical and electronic systems within structures

•  Part 5, Services (Edition 1)

The rationale in these reference documents forms a sound basis for national lightning protection standards of structures and wind turbines.

11.A.4    IEEE Std 81-1983: IEEE Guide for Measuring Earth Resistivity, Ground Impedance, and Earth Surface Potentials of a Ground System

The present state of the technique of measuring ground resistance and impedance, earth resistivity, and potential gradients from currents in the earth, and the prediction of the magnitude of ground resistance and potential gradients from scale-model tests are described and discussed. Factors influencing the choice of instruments and the techniques for various types of measurements are covered. These include the purpose of the measurement, the accuracy required, the type of instruments available, possible sources of error, and the nature of the ground or grounding system under test. The intent is to assist the engineer or technician in obtaining and interpreting accurate, reliable data. The test procedures described promote the safety of personnel and property and prevent interference with the operation of neighboring facilities. The standard is under revision as of September 2010.

11.A.5    IEEE Std 81.2-1991: IEEE Guide for Measurement of Impedance and Safety Characteristics of Large, Extended, or Interconnected Grounding Systems

Practical instrumentation methods are presented for measuring the ac characteristics of large, extended, or interconnected grounding systems. Measurements of impedance to remote earth, step and touch potentials, and current distributions are covered for grounding systems ranging in complexity from small grids (less than 900 m²) with only a few connected overhead or direct-burial bare concentric neutrals, to large grids (greater than 20,000 m²) with many connected neutrals, overhead ground wires (sky wires), counterpoises, grid tie conductors, cable shields, and metallic pipes. This standard addresses measurement safety; earth-return mutual errors; low-current measurements; power-system staged faults; communication and control cable transfer impedance; current distribution (current splits) in the grounding system; step, touch, mesh, and profile measurements; the foot-equivalent electrode earth resistance; and instrumentation characteristics and limitations.

11.A.6    IEEE Std 367-1996: IEEE Recommended Practice for Determining the Electric Power Station Ground Potential Rise and Induced Voltage from a Power Fault

Information for the determination of the appropriate values of fault-produced power station ground potential rise (GPR) and induction for use in the design of protection systems is provided. Included are the determination of the appropriate value of fault current to be used in the GPR calculation; taking into account the waveform, probability, and duration of the fault current; the determination of inducing currents, the mutual impedance between power and telephone facilities, and shield factors; the vectorial summation of GPR and induction; considerations regarding the power station GPR zone of influence; and communications channel time requirements for noninterruptible services. Guidance for the calculation of power station GPR and longitudinal induction (LI) voltages is provided, as well as guidance for their appropriate reduction from worst-case values, for use in metallic telecommunication protection design.

11.A.7    IEEE Std 524a-1993: IEEE Guide to Grounding during the Installation of Overhead Transmission Line Conductors—Supplement to IEEE Guide to the Installation of Overhead Transmission Line Conductors

General recommendations for the selection of methods and equipment found to be effective and practical for grounding during the stringing of overhead transmission line conductors and overhead ground wires are provided. The guide is directed to transmission voltages only. The aim is to present in one document sufficient details of present day grounding practices and equipment used in effective grounding and to provide electrical theory and considerations necessary to safeguard personnel during the stringing operations of transmission lines.

11.A.8    IEEE Std 837-2002: IEEE Standard for Qualifying Permanent Connections Used in Substation Grounding

Directions and methods for qualifying permanent connections used for substation grounding are provided. Particular attention is given to the connectors used within the grid system, connectors used to join ground leads to the grid system, and connectors used to join the ground leads to equipment and structures. The purpose is to give assurance to the user that connectors meeting the requirements of this standard will perform in a satisfactory manner over the lifetime of the installation provided, that the proper connectors are selected for the application, and that they are installed correctly. Parameters for testing grounding connections on aluminum, copper, steel, copper-clad steel, galvanized steel, stainless steel, and stainless-clad steel are addressed. Performance criteria are established, test procedures are provided, and mechanical, current–temperature cycling, freeze-thaw, corrosion, and fault-current tests are specified.

11.A.9    IEEE Std 1048-2003: IEEE Guide for Protective Grounding of Power Lines

Guidelines are provided for safe protective grounding methods for persons engaged in deenergized overhead transmission and distribution line maintenance. They comprise state-of-the-art information on protective grounding as currently practiced by power utilities in North America. The principles of protective grounding are discussed. Grounding practices and equipment, power-line construction, and ground electrodes are covered.

11.A.10  IEEE Std 1050-2004: IEEE Guide for Instrumentation and Control Equipment Grounding in Generating Stations

Information about grounding methods for generating station instrumentation and control (I & C) equipment is provided. The identification of I & C equipment grounding methods to achieve both a suitable level of protection for personnel and equipment is included, as well as suitable noise immunity for signal ground references in generating stations. Both ideal theoretical methods and accepted practices in the electric utility industry are presented.

11.A.11  IEEE Std 1243-1997: IEEE Guide for Improving the Lightning Performance of Transmission Lines

Procedures for evaluating the lightning outage rate of overhead transmission lines at voltage levels of 69 kV or higher are described. Effects of improved insulation, shielding, coupling and grounding on back-flashover, and shielding failure rates of transmission lines are then discussed. Finally, a description of the IEEE FLASH program to predict lighting outage rates from www.ieee.org/pes-lightning is provided.

11.A.12  IEEE Std 1313.1-1996: IEEE Standard for Insulation Coordination—Definitions, Principles, and Rules

The procedure for selection of the withstand voltages for equipment phase-to-ground and phase-to-phase insulation systems is specified. A list of standard insulation levels, based on the voltage stress to which the equipment is being exposed, is also identified. This standard applies to three-phase ac systems above 1 kV.

11.A.13  IEEE Std 1410-2010: IEEE Guide for Improving the Lightning Performance of Distribution Lines

Procedures for evaluating the lightning outage rate of overhead distribution lines at voltage levels below 69 kV are described. The guide then identifies factors that contribute to lightning-caused faults on the line insulation of overhead distribution lines and suggested improvements to existing and new constructions, including evaluation of the effect of wood or fiberglass materials in series with standard polymer or ceramic insulators.

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