1. Introduction
In recent years, several authors have studied the impact of the frequency dependence of electrical ground conductivity (
) and ground relative permittivity (
) in the calculation of electromagnetic transients, especially those developed by lightning strikes on overhead multiphase transmission lines (TLs) or on grounding systems [
1,
2,
3,
4,
5].
In 1930, Ratcliffe et al. introduced one of the first laboratory measurements of effective conductivity
and relative permittivity
of the soil in radio frequencies [
6]. In 1933, Smith-Rose presented a set of systematic measurements considering diverse soil samples [
7]. Decades later, between 1964 and 1967, Scott proposed the first set of mathematical expressions to calculate the electrical parameters of the soil (
and
) [
8]. In the 70s, based on Scott’s formulae, Longmire and Smith proposed a semi-theoretical model, the Universal Soil Model [
9]. At the same time, Messier presented an empirical model to calculate
and
[
10]. During the 1980s and 1990s, Visacro [
11] and Portela [
12] proposed a set of equations to calculate soil parameters between 100 Hz and 2 MHz based on laboratory measurements of diverse soil samples. More recently, Visacro and Alípio introduced two models based on field measurements. The first one uses fitting techniques to obtain formulae for the calculation of soil parameters [
13,
14]. The second model combines field measurements with electromagnetic theory and causality requirements into a physical model [
15].
Researches had evaluated the effects of the frequency dependence of
and
in grounding systems subject to lightning current [
5], transient responses on distribution networks near lightning strikes [
1], and overvoltages subject to lightning strikes [
2], respectively. Based on previous research, the International Council on Large Electric Systems (CIGRE) published Brochure 781 [
16] containing the main physical aspects related to the frequency dependence of
and
, its influence on the transient response of electrical systems due to lightning strikes, and recommendations to consider the above in practical engineering applications [
16].
Considering ground-return effect is crucial when evaluating the propagation of voltage and current waves in overhead TLs [
17]. In this regard, it is also important to include the frequency dependence of
and
. However, most electromagnetic transient (EMT)-type simulators compute the ground impedance using Carson’s formula, which assumes that the ground conduction current is much larger than the displacement current and excludes the frequency dependence of soil electrical parameters [
18]. These assumptions can generate errors in the simulation of soils with high resistivity and applications involving high-frequency phenomena, such as transients caused by lightning strikes on transmission lines. Some papers consider the frequency dependence of
and
during line energization [
18,
19,
20]. However, authors only consider one type of soil, which makes it difficult to generalize results.
Concerning the transient analysis of transmission lines located on frequency-dependent soils, we highlighted some works. In [
21], De Conti et al. analyze the transient voltages in a 3.6 km single and bi-phase transmission lines without ground wire located on soil of 10,000 Ω·m. In [
22], De Conti et al. present the transient voltages for two different single and bi-phase transmission lines without ground wires of 0.6 km and 1.8 km in length located on soils of 100, 1000 and 10,000 Ω·m. In [
23], Moura et al. analyze the influence of the frequency dependence of ground electrical parameters in terms of the p.u.l. longitudinal impedance and transversal admittance for a single-circuit three-phase transmission line with ground wire. In [
24], considering a set of measurements, He et al. investigate the impact of the frequency effect of
and
on the calculation of line parameters. However, their analysis does not extend to the time domain. In [
25], Papadopoulos et al. investigate the impact of frequency-dependent soil parameters on the propagation of transients in a 10 km overhead three-phase transmission line without ground wires. However, authors only analyze simple configurations corresponding to one or two ground wires because their focus is on the analysis of transients in overhead distribution networks. In [
4], Salarieh and Kordi show that the backflashover rate estimation of a transmission tower with the grounding electrodes buried in lossy ground is affected by the frequency-dependent soil electrical parameters.
Considering the above works, it is noted that proper grounding modeling has been widely studied in the literature. However in these studies, the transient analyses are performed only for short transmission lines, typically related to distribution power systems. Thus, we understand that a more extensive analysis on the transient responses for commonly multiphase transmission lines (including the ground wires) located on the frequency-dependent soil electrical parameters are need. This objective of this paper is to evaluate the impact of frequency-dependent soil parameters in the transient responses of multiphase transmission lines generated by lightning strikes. First, we calculate the longitudinal impedance and transversal admittance matrices where the ground-return elements are computed by Nakagawa’s approach. The advantage of this approach consists of that the frequency dependence of soil parameters can be included for the analysis. We use the soil modeling proposed by Alípio and Visacro’s. We compare this results with those computed by Carson’s approach which assumes the frequency-constant soil parameters obtained for soils of moderate and high resistive soils. It is demonstrated that for high-resistive soils, a significant difference can be noted in the elements of the impedance and admittance matrices. We show the effects of both approaches in time-domain simulations of high-frequency transients (subsequent return stroke and 1 MHz Gaussian pulse) for diverse soil resistivity in single and double-circuits three-phase transmission lines of 5 km and 50 km in length considering two ground wires at the top of each the tower. Results demonstrated that an expressive difference at the transient peaks can be obtained when the soil is models by its frequency-dependent parameters. As shown, a reduction of 26.15% and of 42.75% in the voltage peaks in a single- and double-circuit lines located above a high-resistive soil are computed for the Gaussian pulse. As contribution of this paper, we analyse the transient responses generated by fast-front disturbances on single and double-circuit transmission lines with ground wires located above high resistive soils in a more realistic line configuration and length.We also show that the proper modeling of soil affects significantly the transient analysis.
This paper is organized as follows:
Section 2 presents the equations used to compute multiconductor line parameters.
Section 3 shows the Alípio and Visacro’s equations used in this paper to calculate the frequency-dependent soil electrical parameters.
Section 4 describes the subsequent return stroke and Gaussian pulse waveforms that we use to represent lightning strikes. In
Section 5, we present our methodology in steps to compute the frequency-domain responses. In
Section 6, we combine different TL topologies and ground parameters and calculate the line parameters of each combination. Then we simulate these TLs subject to the impulsive currents presented in
Section 4 and discuss the results obtained. Finally,
Section 7 presents the major conclusions of this paper.
3. Frequency-Dependent Soil Electrical Parameters
Soils consist of a very complex composition of compacted layers of earth made of organic and inorganic materials and rock that have disintegrated. The electrical soil parameters are dependent on the frequency and on the environmental factors such as the temperature, humidity and size of soil particles [
24]. The electromagnetic properties that characterize a soil are the magnetic permeability
, resistivity
(or conductivity
), and absolute dielectric permittivity
. The magnetic permeability is practically frequency independent and assumed to be equal to vacuum
. However, the last two,
and
, are greatly affected by the frequency specially for high-resistive soils [
37,
38,
39].
Based on the frequency-domain Ampere–Maxwell’s law, the total electric current density
(A/m
2) in a dielectric medium when an external electrical field
(V/m) is applied is give by [
15]
where
(A/m) is the magnetic field,
and
(A/m
2) are conductive and capacitive currents densities. The
(S/m) is the low-frequency conductivity, considered as a real number associated to the transport of electric charge and losses generated to the in the conducting process. However, the dielectric permittivity is a complex number which can be written as
, where
is related to the ability of the material to be polarized and store energy whereas
represents the losses generated due to the heat generated by the dipole frictions in the several polarization processes as detailed in [
15,
40]. Inserting
in (
21), yields
As shown by this equation, the effective conductivity
(S/m) increases with the frequency whereas the
decreases as the frequency of the applied field increases as detailed in [
40]. The physical explanation related to the electric resistivity
and dielectric permittivity
, and the different polarization processes that occurs in the ground molecules are explained in [
41].
As the frequency increases, the polarization processes are not able to follow the fast alternations of the electric field. This causes
to decrease as frequency increases. However, at high frequencies, there is an increase of losses per cycle causing
to increase its value. From these variations, as the frequency increases, the value of
increases while the value of
decreases. In [
41], Friedman thoroughly explains the polarization processes of ground molecules that affect electric resistivity
and dielectric permittivity
.
In the 1930s, Smith carried out the first study regarding frequency dependence of soil electrical parameters [
42]. Later, in the 1970s, several authors developed theoretical and experimental models based on field experiments as summarized in [
5,
16]. Recently, in the electromagnetic software tools that solve the frequency-domain Maxwell’s equations include the frequency dependence of soil electrical parameters to carry out the lightning performance on grounding systems and on transmission lines such as in [
22,
37,
38,
39,
43].
Based on field measurements of different soils in Brazil, Visacro and Alípio proposed curve-fitted expressions to calculate the frequency dependence of soil resistivity
and relative permittivity
as a function of frequency [
14]. Later, in 2014, they presented a semi-theoretical causal model to obtain frequency-dependent soil electrical parameters based on various measurements in the 100 Hz–4 MHz frequency range.
Alípio and Visacro’s model adopts three levels of conservativeness to take the dispersion of the frequency dependence of the soil and eventual uncertainties into account. This model proposes the following equations [
15]
where
(mS/m) is the low-frequency (at 100 Hz) conductivity of the soil and
is the relative permittivity at higher frequencies. The values employed in this simulation for
,
, and
are:
= 1.26
,
= 0.54 and
= 12 [
15].
To show the impact of frequency in the soil electrical parameters, we calculated the electrical parameters of soils with low, medium, and high resistivity from 100 Hz to 10 MHz using (
23) and (
24).
Figure 2 shows the resistivity
and relative permittivity
of soils that have a 100 Hz resistivity
of 500, 1000, 5000, and 10,000 Ω·m which are related to several types of soil according to [
16].
Figure 2 shows that high resistivity soils (
= 5000 and 10,000 Ω·m are much more affected by the frequency effect than soils with moderate and low resistivity. As
increases, the relative permittivity ε
rg decreases asymptotically to values between 15 and 30 (typical values of soils represented by frequency-independent parameters). The soils of 500, 1000, 5000 and 10,000 Ω·m will be considered in the transient analysis of this work. Another reason for choosing high resistive soils is based on the recent works considering the same resistive range in their works [
21,
22,
23].
According to CIGRE [
16], practical engineering studies or EMT analysis of TLs involving the lightning performance of grounding systems buried on soils with resistivity above of 700 Ω·m must consider the frequency dependence of
and
. For transmission lines, the CIGRE brochure [
16] recommends that lines located on soils of resistivity above 700 Ω·m, the frequency dependence of the soil must be considered for a precise transient analysis. The recommendations about considering the frequency dependence of the soil for studies on transmission lines are summarized as follows [
16].
< 300 Ω·m, the recommendation is to ignore the frequency-dependency of soil;
300 ≤ < 700 Ω·m, it is recommended to include the frequency-dependency of soil;
≥ 700 Ω·m, it is mandatory to consider the frequency-dependency of soil for a precise transient analysis.
5. Methodology
The methodology for compute the frequency- and time-domain responses are presented in this section. In our analysis, we disregard the tower surge impedance, tower-footing grounding electrodes and the ionization of the soil.
Figure 5 shows the steps performed to obtain frequency and time-domain results, which are summarized as follows:
Step 1: Define transmission line configuration: Tower geometry and conductor parameters such as bundle and radius of the phase and ground wires conductors. Choose the electrical parameters (conductivity, relative permittivity, and magnetic permeability) of air and soil. Select the frequency range, varying from 100 Hz up to 10 MHz. To include the frequency dependence of soil parameters, we calculate the frequency-dependent relative permittivity and conductivity of the soil from the soil conductivity measured at 100 Hz using (
23) and (
24) [
15].
Step 2: Use (
3) and (
4) to calculate the p.u.l. impedance matrix
and p.u.l. admittance matrix
of the TL defined in Step 1. This paper compares:
the parameters computed by MatLab using the powersys/power_lineparam routine, that uses Carson’s Equations (
8) and (
9). In this Matlab tool, the ground-return admittance
is neglected, resulting in
=
. Furthermore, the Carson’s approach considers that the soil is modeled by frequency-independent parameters and the displacement currents are ignored.
the parameters calculated using Nakagawa’s formulae, where
,
,
,
,
are respectively given by (
5), (
6), (
7), (
10), (
11), (
17) and (
20). We use Alípio’s Equation (
23) and (
24) to include the frequency dependence of
and
.
Step 3: Define simulation parameters e.g., time step, simulation time. Select the type of lightning strike waveform connected to the TL. In this paper, we model lightning strikes as ideal current sources that contain the waveforms presented in
Section 4. Set the loads connected to the receiving end (open-circuit, short-circuit or load).
Step 4: Calculate the line’s two-port equations (y-parameters) that relate terminals’ currents to voltages using the line parameters calculated in step 2. The two-port equations are described in [
53]. Compute voltages and currents at line terminals in the frequency domain.
Step 5: Transform voltages and currents to the time domain using the Numerical Laplace Transform [
54]. Then, we compare the time-domain responses.
All the calculations are carried out in MatLab.
6. Numerical Results and Discussion
We present our results in three sections:
Section 6.1 shows the longitudinal impedance and transversal admittance of a single-circuit three-phase TL and a double-circuit three-phase TL, including the ground wires at the top, located above four homogeneous soils.
Section 6.2 shows the transient responses of the single-circuit three-phase TL with ground wires subject to lightning currents modeled as the subsequent return stroke of
Section 4.1 and the Gaussian pulse of
Section 4.2.
Section 6.3 shows the transient responses of the double-circuit three-phase TL with ground wires, subject to lightning currents modeled as the subsequent return stroke of
Section 4.1 and the Gaussian pulse of
Section 4.2.
We computed the transient responses of the single-circuit three-phase TL of
Figure 6a and the double-circuit three-phase TL of
Figure 6b. We considered two line lengths: 5 km and 50 km. Both TLs are located above four types of homogeneous high-resistive soils that have a 100 Hz resistivity of 500 Ω·m, 1000 Ω·m, 5000 Ω·m and 10,000 Ω·m.
6.1. Longitudinal Impedance and Transversal Admittance
We computed the longitudinal impedance and transversal admittance matrices of the single-circuit three-phase TL of
Figure 6a and the double-circuit three-phase TL of
Figure 6b, both with ground wires at the top of the towers, using the following approaches:
Carson’s approach–frequency constant soil parameters: Equations (
5)–(
7) allow the calculation of the internal and external impedances in (
3). MatLab’s powersys/power_lineparam routine estimates the ground-return impedance by solving Carson’s Formula (
8) and (
9) considering frequency-constant resistivities of 500 Ω·m, 1000 Ω·m, 5000 Ω·m and 10,000 Ω·m. Carson’s approach considers the soil as a perfect conductor for the calculation of the shunt admittance. This approach neglects the displacement currents dispersed in the ground. Therefore
is neglected (
=
).
Nakagawa’s approach—frequency dependent soil parameters: Equations (
5)–(
7) allow the calculation of the internal and external impedances in (
3). Nakagawa’s Formulae (
10), (
11), (
18) and (
19) in combination with Alípio and Visacro’s Equation (
23) and (
24) allow the calculation of
and
matrices considering frequency-dependent soil electrical parameters. The frequency-dependent soil resistivity
and permittivity
are computed for soils that have a 100 Hz resistivity of 500 Ω·m, 1000 Ω·m, 5000 Ω·m and 10,000 Ω·m.
For both approaches, we compare in
Figure 7 and
Figure 8 the elements in the first row of
and
, respectively. We chose to show only the elements in the first row of each matrix because
and
are symmetrical. To better show the differences in the magnitudes of
and
, results presented in
Figure 7 and
Figure 8 are calculated for the 10 kHz–10MHz frequency range.
Figure 7 shows that
,
, and
increase rapidly after 200 kHz. At 10 MHz,
reaches a value of 60,000 Ω/km for the single-circuit TL and 40,000 Ω/km for the double-circuit TL. Carson’s approach estimates higher values than Nakagawa’s approach with frequency-dependent soil resistivity and relative permittivity. This occurs because Carson’s formula neglects displacement currents by assuming that the relative permittivity of the soil is equal to 1, a constant value in a quasi-static transverse electromagnetic mode.
On the other hand, Nakagawa’s approach includes the frequency-dependent soil relative permittivity in the propagation function
in (
14). For both TLs, the low-frequency resistivity did not cause significant differences on
,
, and
. As can be seen, for the soils of 500 Ω·m and 1000 Ω·m, they present similar values for the elements
,
, and
in all frequency range. However, as the frequency increases, the difference between the two approach are pronounced for the
,
, and
computed for the 5000 Ω·m and 10,000 Ω·m. The difference is more significant for the double-circuit transmission line. The difference between the values occur due to higher variation in the soil resistivity as the frequency increases which is more pronounced for high resistive soils, as seen in the
Figure 2.
Figure 8 shows that the transversal admittance also increases rapidly after 200 kHz. Carson’s approach neglects
, whereas Nakagawa’s approach includes the frequency-dependent soil electrical parameters in the calculation of
. However,
Figure 8 shows that both approaches generate similar admittances, which means that
does not affect
substantially. In addition, note that variation of
has almost no effect in the calculation of
. The maximum value of the transversal admittance element Y
11 for the double-circuit transmission line reaches approximately 1.5 S/km, whereas for the single-circuit line, the admittance element Y
11 is equal to 0.8 S/km.
We observe that both approaches have lead to similar results for transversal admittance and the frequency dependence of the soil parameters has no significant impact on these elements Y11, Y12, and Y13.
Based on the behavior of the ground-return impedances and admittances, the time-domain transient responses on transmission lines located above high-resistive soils are influenced by fast-front disturbances such as the lightning discharges, characterized by high-frequency spectrum as detailed in the following sections.
6.2. Single-Circuit Transmission Line
We place the single-circuit three-phase TL with ground wires of
Figure 6a located above four high-resistive soils, each having a resistivity of 500 (Ω·m), 1,000 (Ω·m), 5000 (Ω·m) and 10,000 (Ω·m).
Figure 7 and
Figure 8 show the p.u.l. longitudinal impedance and transversal admittance of each of these lines, calculated using Carson’s approach and Nakagawa’s approach, respectively. For each soil, we consider TL with lengths of 5 km and 50 km. We leave one of the line terminals open-circuited and connect the other terminal to two distinct current sources, detailed as follows:
Figure 10 and
Figure 11 contain several voltage peaks caused by reflections from the receiving end due to traveling waves within the TL. The line’s open-circuited terminal causes voltage peaks to be more intense. The number and intensity of each reflection depend on the lightning current waveform and the length of the TL. For instance, the 5 km TL has more intense voltage peaks than the line that is 50 km long. The short duration of the Gaussian pulses causes their reflections to be more distinguishable from each other.
Figure 10 and
Figure 11 also show shift between time-domain responses that increases with time. This behaviour can be seen in the detailed in
Figure 11.
In order to investigate the diffenreces in the time-domain responses, we present in
Table 4 the percent deviation between voltage peaks simulated considering frequency-dependent soil parameters (FD-Nakagawa’s Approach) and frequency-independent soil parameters (FI-Carson’s approach). The percentage deviation is calculated in the following equation
where the voltage peak
is obtained considering the frequency-independent (FI) or the frequency-dependent soil model (FD). We calculate the percent deviation between the first three peaks for each simulation.
Table 4 shows that the deviation is more pronounced in the simulations where the lightning current source is represented by the 1MHz - FWHM Gaussian pulse of
Section 4.2.
The maximum percent deviation is 26.15% which is obtained between the Carson’s approach and Nakagawa’s approach for the third peak for the 50 km single-circuit TL located soil of 5000 Ω·m generated by the Gaussian pulse. This difference occurs to due the attenuation constant is higher frequencies associated with Carson’s approach is larger than that calculated by the Nakagawa’s approach. However, for the 50 km single-circuit TL located soils of 500 Ω·m and 1000 Ω·m, the percentage deviation is negative associated to the Gaussian pulse. This occur due to the larger frequency content of the Gaussian pulse associated with soils of moderate resistivity.
The difference between the voltage peaks for the Carson’s approach with frequency-independent and Nakagawa’s approach with frequency-dependent soil models are more pronounced for those computed by the Gaussian pulse due to its higher energy at the high frequencies compared to subsequent return stroke.
6.3. Double-Circuit Transmission Line
Similar to the single-circuit TL, we place the double-circuit three-phase TL including the ground wires of
Figure 6b above four high-resistive soils, each having a resistivity of 500 Ω·m, 1000 Ω·m, 5000 Ω·m and 10,000 Ω·m.
Figure 7 and
Figure 8 show, respectively, the p.u.l. longitudinal impedance and transversal admittance of each of these lines calculated using Carson’s approach and Nakagawa’s approach. For each soil, we considered two line lengths: 5 km and 50 km. We leave one of the line terminals open-circuited and connect the other terminal to a subsequent return stroke waveform and a 1 MHz—FWHM Gaussian pulse.
We present in:
Table 5 the plots obtained for each simulation,
Figure 12 the circuit employed for the simulations,
Figure 13 and
Figure 14 the transient voltages at the receiving end of the TL for each of the scenarios mentioned above,
Table 6 the voltage peak deviation between Carson’s approach (frequency-constant soil parameters) and Nakagawa’s approach (frequency-dependent soil parameters).
Similar to the single-circuit TL, the percentage deviation is higher for the simulations that include the Gaussian pulse.
The percentage deviation increases with the length of the line, reaching a maximum value of 42.75% in the third peak of the double-circuit line that is located above a 10,000 Ω·mm soil. Furthermore, the percentage deviation of voltage peaks is more pronounced for the double circuit TL than those computed for the single-circuit TL using the Gaussian pulse. The electromagnetic coupling between conductors causes the induced voltages of phases 2 and 3, which have significant amplitudes. The frequency dependence of soil parameters also has an impact on the voltage peaks in phases 2 and 3 where can present a significant difference between the two approaches.
The percentage deviations increase as the soil resistivity increases for the voltages on the double-circuit transmission lines for both types of lightning currents as seen in the
Table 6. The first peaks of the transient responses calculated by Nakagawa’s approach is slightly higher than those computed by Carson’s approach, using the Gaussian pulse on the 5 km double-circuit line. The difference in voltage peaks calculated with both approaches increases with soil resistivity. This difference is also dependent on the type of lightning current which affects significantly in the reduction of the voltage peaks. It is demonstrated that when the frequency dependence of the soil is considered, this reduction can be around 42% for the double-circuit line above the 10,000 Ω·mm soil subject to the Gaussian pulse. This result can be explained due to the fact that the Gaussian pulse contains a large frequency content associated with high energy im comparison with the subsequent stroke.
Results demonstrated that accurate computation of electromagnetic transients requires the proper modeling of the soil. In that case, the frequency-dependent soil electrical parameters must be included to compute the ground-return impedance and admittance of the transmission lines. The importance in the calculation of accurate responses results in the correct sizing of insulation for many components in power systems, e.g., insulator strings, pre-insertion resistors, surge arresters, and transformers [
30,
55]. By an accurate evaluation of voltage peaks, insulation can be estimated more accurately and thus reduce costs.
Inaccurate estimation of voltage peaks could result in malfunctions of the protective devices, inadequate installation of the surge arrester in lines that can lead to outages in power systems while degrading the quality of supplied energy. Furthermore, most of the available Electromagnetic transient (EMT)-type simulators compute the ground impedance using Carson’s approach which can result in high errors when the transmission line is located above high-resistive soils. This paper highlights the importance of combining the Nakagawa’s approach to the computation of the ground-return impedance and ground-return admittance including the frequency-dependent soil parameters.