Overhead Lines
D. Dynamic models
2.5 Transmission Line Grounding .1 Introduction
2.5.3 Low-Frequency Models of Grounding Systems .1 Compact Grounding Systems
tance and remains constant and equal to its DC resistance. At high frequencies, the behavior is frequency dependent [83,94,95].
Some effects can affect the lightning performance of a power line: There is a
resid-•
ual inductance in any grounding arrangement that adds some stress to line insula-tion, while soil ionization and capacitive displacement currents tend to reduce the apparent impedance. Electrode inductance increases the grounding impedance, being this effect more signifi cant for lines with low grounding impedance. The ionization effects are important for many soil types, and decrease resistance by increasing the effective radius of the electrodes. The capacitive displacement cur-rent is only important in areas where soil resistivities are greater than 10 kΩ m.
Although much work has been done on the behavior of the ground impedance under lightning discharge, there is no consensus on how to apply this knowledge to the repre-sentation and the design of actual electrode systems, and the power-frequency grounding resistance is often used to predict the lightning performance of power lines.
The grounding system of a power line support can be broadly classifi ed as compact (con-centrated) or extended (distributed) [95,96]. A compact grounding can be represented by lumped-circuit elements. In an extended grounding system the travel time of the electro-magnetic fi elds along the electrodes is comparable with that along the support itself; this generally applies to grounding systems with physical dimensions exceeding 20 m.
Sections 2.5.3 and 2.5.4 provide a theoretical and practical base for the analysis of power line ground systems taking into account their frequency-dependent behavior, and show how to treat soil ionization.
2.5.3 Low-Frequency Models of Grounding Systems 2.5.3.1 Compact Grounding Systems
One of the simplest grounding confi gurations is a hemispherical electrode, see Figure 2.37.
Assuming uniform soil resistivity, a current fl owing from the hemisphere into the ground produces a current density in the surrounding soil given by the following expression:
Hemisphere electrode
a
x l
FIGURE 2.37 Hemisphere electrode.
=2π 2 J I
x (2.82)
where
J is the current density I is the total current
x is the distance from the center of the electrode
The electric fi eld strength that such current density produces in the soil is
= ρ = ρ π 2 2 E J I
x (2.83)
where ρ is the soil resistivity.
The voltage at any distance x is
ρ ρ ⎛ ⎞
=
∫
xa d = 2πI∫
xa 12d =2π ⎝I⎜1−1⎟⎠V E x x
x a x (2.84)
where a is the radius of the electrode.
The total voltage between the electrode and a far distance point (x ≅ ∞) is then
= ρ π 2 V I
a (2.85)
Finally, the total resistance of the electrode is obtained as follows:
= = ρ π 2 R V
I a (2.86)
This is the resistance experienced by current fl owing through the entire surrounding space. Most of this resistance is encountered in the region immediately around the elec-trode. From Equation 2.84, 50% of the total resistance is contained in x ≤ 2a, and 90% is contained in x ≤ 10a.
The rod is the most common type of ground electrode. Ground rods are generally made of galvanized steel, 2.5–3 m in length, less than 2.5 cm in diameter, and driven vertically down from the earth’s surface. When the length of the ground rod is much greater than its radius, the low-current, low-frequency impedance of a single ground rod is approximated by a resistance whose value may be obtained from the following expression:
ρ ⎛ ⎞
Some discrepancies can be found in the literature related to the expression given by some authors for the resistance of a driven ground rod; they are mainly due to the different approaches used to derive the above expression [26,81,83,97–100].
R0 decreases as either the buried length or the radius of the rod increase, but it does not decrease directly with length, so an increase in length above a certain limit will not sig-nifi cantly reduce the resistance.
Ground resistance can be reduced by connecting several rods in parallel. The resistance is inversely proportional to the number of parallel rods, provided the spacing between rods is large compared to their length. If the spacing is short and the n ground rods are arranged on a circle of diameter D, then Equation 2.87 is still valid if the radius a is replaced by an equivalent radius req[26]:
⎛ ⎞
That is, when the rods are closely spaced compared with their length, the whole ground arrangement behaves as one rod with a larger apparent diameter and a small reduction in resistance. As the rod spacing increases, the combined resistance decreases. When the spacing between adjacent rods, arranged on a circle of diameter D, is equal to or longer than the length of rods, the combined resistance of n ground rods can be approximated as follows [26]:
Tower
ZT
Rr
vc, ic
Earth surface Counterpoise v, i
FIGURE 2.38
Wave propagation along a counterpoise. (Adapted from Hileman, A.R., Insulation Coordination for Power Systems, Marcel Dekker, New York, 1999.)
The proximity effect between the ground rods tends to increase the combined resistance, thus diminishing the advantage of multiple rods.
2.5.3.2 Extended Grounding Systems
The contact area of a grounding system with the earth can be increased by installing a coun-terpoise, which is a conductor buried in the ground at a depth of about 1 m. Common arrange-ments include one or more radial wires extending out from each tower base, single or mul-tiple continuous wires from tower to tower, or combinations of radial and continuous wires.
Counterpoises may be also combined with driven rods. In such case, a wave traveling down the tower impinges on the combination of the vertically driven rod and the counterpoise, and results in a wave which travels out along the counterpoises at about 1/3 of the speed of light, see Figure 2.38 [3]. The current wave initially meets the surge impedance of the conductor, whose value decreases with time and reaches a steady-state value when the current is distrib-uted through the counterpoise length. After a few round-trip travel times, the impedance is reduced to the total leakage resistance of the counterpoise (Section 2.5.2.2).
When the length of the conductor is much greater than its burial depth, the low-current low-frequency resistance of a horizontal conductor buried in the soil can be given by [26]
ρ ⎛ ⎞
=π ⎝⎜ − ⎟⎠
0
ln 2 1
R 2
ad (2.91)
where
ℓ is the length of the buried conductor, in m a is the conductor radius, in m
d is the burial depth of the conductor, in m ρ is the soil resistivity, in Ω m
The steady-state resistance is not greatly infl uenced by either a or d.
Several short wires, arranged radially, may be more effective than a single long wire even if the total length and contact resistance of both arrangements are the same. The initial surge impedance of several wires is lower and the steady-state contact resistance is reached faster. The low-current low-frequency resistance of n radial conductors is [26]
−
As n becomes very large, the last term between brackets approaches 1.22n [95], and in the limit
→ ρ
0
R 2.57 (2.93)
Table 2.8 shows the resistance of several simple electrodes. It is important to remember that several different equations have been proposed for the calculation of ground elec-trodes, and each one gives a slightly different result. See for instance Ref. [83].
2.5.3.3 Grounding Resistance in Nonhomogeneous Soils
The expressions presented for calculating the grounding resistance use only one resistivity value, since a homogeneous soil is considered. Several methods have been developed to deal with nonhomogeneous soils. One of the methods consists in stratifying the soil in two lay-ers, which could represent the effect of multiple layers. The fi rst layer reaches a depth d and is characterized by a resistivity ρ1, while the second layer has an infi nite depth and a resis-tivity ρ2. For a great majority of lines, it is possible to determine a two-layer soil structure that can represent typical soils for grounding purposes. In some cases it might not be pos-sible to defi ne a two-layer soil structure, then a three-layer soil model must be considered.
The computation of the grounding resistance depends on the type of ground electrode and the depth of the fi rst layer with respect to the length of the grounding electrode. For a rod driven into the upper layer only, the ground resistance is deduced as follows [89,100]:
∞
is the refl ection coeffi cient.
The fi rst term is the resistance of a rod having length ℓ driven into soil of resistivity ρ1, and the second term represents the additional resistance due to the second layer.
If the rod penetrates both layers, then the ground resistance is deduced as follows [89]:
( )
∞TABLE 2.8
Ground Resistance of Elementary Electrodes
Electrode Schematic Representation Dimensions Grounding Resistance
Hemisphere a Radius a
where
F is the penetration factor given by
= + Γ
− Γ + Γ 1
1 2 /
F d (2.97)
R1 is the resistance of the rod driven in uniform soil of resistivity ρ1
Ra is the additional resistance due to the second layer
A simpler two-layer soil treatment is appropriate when ρ2 >> ρ1; that is, when the refl ec-tion coeffi cient approaches unity. Under this condiec-tion, the resistance of a single hemi-sphere of radius a in a layer soil with resistivity ρ1 and thickness d may be approximated by the following expression [96,101]:
( )
For a refl ection coeffi cient Γ = 1, the series in Equation 2.98 becomes the harmonic series and converges slowly. Loyka proposes the following approximation [96,102]:
( )
This approximation is accurate enough for a wide range of transmission line grounding applications, and proves that a fi nite upper layer depth has a strong infl uence on the elec-trode resistance [96].