Choosing configurations of transmission line tower grounding by back flashover probability value

Abstract

There is a considerable number of works devoted to electrical characteristics of grounding. These characteristics are important in general. However, in application to grounding of transmission line towers they are not enough to determine what grounding construction is preferable in some particular case, because these characteristics are calculated or measured apart from the grounded object, and only limited number of current (or voltage) source waveforms is used. This paper indicates reasons in favor of the fact that to choose the optimum design of grounding, the calculation model should include the tower as it is. The probability of back flashover, which provides both qualitative and quantitative estimate of the grounding structure efficiency, can be taken as the criterion for the grounding design. The insulation flashover probability is calculated on the basis of engineering method, which evaluates breakdown strength of insulation for nonstandard waveshapes, and probability data on lightning currents. Different approaches are examined for identifying the back flashover probability, as not only amplitudes but also other parameters can be taken into account. Finite-difference time-domain method is used for calculations of transients. It is found that lightning current waveform can greatly influence calculated back flashover probability value.

Keywords

grounding / transmission line tower / back flashover probability / FDTD method

Introduction

At present, there exist numerous methods for the calculation of electric characteristics of the groundings. However, there is no unanimous opinion what method is the most appropriate for the given objective, as each method has its own pros and cons. The existing calculation methods are usually divided into three groups: quasi-static methods [ 1, 2], the methods using telegrapher’s equations [ 3], and the methods using the equations of electrodynamics [ 4, 5].

The methods in the first group are relatively easy to apply. They are not demanding for considerable computer resources, but they do not take into account the delay in electromagnetic field propagation. The methods in the second group correctly consider the field propagation along separate conductors, but not in the whole volume. The methods in the third group accurately consider the particulars of the field propagation, but they require lengthy calculations. It is important to point out that even the methods in the third group have their own particular features and limitations: some of them are more accurate about modeling thin conductors, and some of them manage inhomogeneous medium better. Reviews and descriptions of the methods can be also found in many papers handling the calculations of electric characteristics for groundings and towers [ 3, 6– 8].

The finite-difference time-domain (FDTD) method has been selected, as, besides accurate consideration of the behavior of the electromagnetic field, it can take into account inhomogeneous soil. Besides, there exists the possibility of modeling various kinds of non-linear phenomena (such as soil ionization [ 9], and corona [ 10]).

Need for including towers in the calculation model

As a rule, electric characteristics of groundings are calculated apart from tower. As the main electric characteristics of the grounding, its resistance is usually taken (In this paper, the so-called transient grounding resistance is used. However, this term may be arguable). In the event of direct current and voltage, grounding resistance is unambiguously determined via the grounding potential ratio to its current. However, in the event of high frequency processes, the resistance (or impedance) of the grounding can be calculated in several manners [ 11]. Besides, there appear some more problems.

For example, as transient grounding resistance is calculated, it is not clear how the current lead wire shall be located and how the potential shall be calculated. In addition, the transient resistance value is affected by the form of pulse of the current or voltage source.

The methods using retarded scalar and vector potentials do not require the use of current lead wire, rather—energizing a concrete node of the circuit of calculation [ 7] (that is, generally speaking, different from a real situation). However, this is seemingly not possible for the FDTD method. Therefore, the location of the current lead wire will affect the calculated transient resistance.

If the potential is calculated via the electric field integral, it will also contain a path-dep6endent component, specified by a vector potential, as

(1) E=-\nabla U-{\displaystyle \frac{\partial A}{\partial t}},

where E is the electric field vector, U is the scalar potential, and A is the vector potential.

Even that the locations for current lead wire and calculation routes for the potential are determined, there also exists a problem about the comparison of groundings. For different grounding designs with equal quantity of metal consumed, one often has to compare “more inductive” grounding (with higher transient resistance within small periods of time, but lower stationary resistance) and “less inductive” grounding (with lower transient resistance in the beginning, but higher stationary resistance). Let us review, for example, a three- and four-point star groundings [ 1] with the same overall length of the conductors. A four-point star grounding has higher stationary resistance, but lower initial transient resistance, due to a larger amount of conductors involved in diversion of current within the first microseconds. Hereto, it is not clear which grounding design is more efficient, especially if taking into account that the actual insulation voltage is very different from grounding voltage with no tower modeling. Of course, there is also a special case for grounding with capacitive behavior, but in most cases tower grounding have a rather inductive behavior.

To avoid the aforementioned problems, a decision was made to select the grounding based on the calculations involving the tower. In the calculations, a lightning stroke into the top of the tower is modeled, and the difference of potentials between the tower cross arms and phase wires is calculated. It is evident that such calculations are more realistic. There exist studies in which the calculations involve the tower. However, these studies only review electric characteristics of the tower, but not the choice of the grounding design [ 7, 12].

Main parameters of model used

Parameters of FDTD method

The size of the cell for the computation mesh is selected as 0.25 m to provide sufficient model specification.

As absorbing boundary conditions, this paper uses 15-cell convolutional perfectly matched layer (CPML) [ 13]. The PML conductivity s_max equals s_opt, and the PML k_max is 1 [ 13]. Polynomial grading of s and k is used with scaling order 3 (Therefore, s_opt is approximately 0.034.) [ 13]. The CPML a_max equals 10^–5 with scaling order 1 [ 13].

Tower conductor modeling is conducted with the help of staircase approximation. As the tower is narrow, and the main conductors installed on its structure are tilted at a small angle, the error of propagation speed along it is only few percents [ 14]. Grounding conductors are modeled by means of thin wire modeling method [ 15, 16]. However, there has appeared a relatively new method enabling the modeling of oblique arbitrary radius conductors [ 17] thanks to which, if necessary, tilted grounding conductors and guyed towers can be modeled more accurately.

Lightning stroke modeling

A lightning channel is modeled as a vertical conductor with a current power source at its base. The other end of the conductor penetrates into the CPML area, at the expense of which an infinitely long conductor is modeled. A lumped source is selected to consider a wide range of lightning currents, calculating insulation voltage with random current via the integral [ 18]:

(2) u(t)={\displaystyle {\int }_{\mathrm{0}}^{t}r(t-x)i^{\prime }(x)dx},

where r(t) is the response to step current, i'(t) is the derivative of arbitrary current (it can also be a voltage derivative), and u(t) is the voltage (it can be a current) at arbitrary current. This integral is also called Duhamel’s integral.

As the paper reviews only direct lightning stroke, in this case, a lumped source can be applied [ 19]. An ideal current source is accepted for the calculations. However, for electrically long objects, the wave impedance of the lightning channel should be considered [ 19]. The influence of wave impedance on calculation results for the considered model is estimated by means of a non-ideal current source together with correction of current through tower top. However, due to the peculiarities of modeling lightning channel by wire, the accuracy of proposed approach is questionable.

Strictly speaking, the wave impedance of the lightning channel varies, depending on the lightning current and makes, by different estimates, from 0.5 to 2.5 kW [ 19]. In these calculations, the internal resistance of the current source is considered as equal to 1 kW. However, the resistance of the lightning channel being modeled is increased by several hundreds of ohms as the conductor, modeling the lightning channel, has wave impedance. Therefore, the wave which has come from the tower base to the current source will reflect from the load, equal to the sum of internal resistance of the current source and wave impedance of the vertical conductor.

The currents of both first and subsequent return strokes are considered.

The form for the current of first lightning strokes is set using three different functions given by Eqs. (3)–(7).

Ramp function [ 20]:

(3) i(t)=\{\begin{array}{ll}{i}_{\max }{A}_{\mathrm{1}}t& \mathrm{if}t\le t_f,\\ i_{\max }[A_{\mathrm{1}}t+A_{\mathrm{2}}(t-t_f)]& \mathrm{if}t>t_f,\end{array}

where i_max is amplitude, t_f is front time, A₁=1/t_f, A₂=(0.5–A₁t_h)/(t_h–t_f), and t_h is time-to-half.

Heidler function [ 21]:

(4) i(t)={\displaystyle \frac{i_{\max }}{\eta }}{\displaystyle \frac{a{(t/T)}^k+{(t/T)}^n}{\mathrm{1}+{(t/T)}^n}}\exp \left(-{\displaystyle \frac{t}{\tau }}\right),

where a=1, k=2.5, n=60, T sets the duration of a pulse, t sets the front time, and h serves to normalize the function.

CIGRE function [ 22]:

(5)

where

(6) \begin{array}{l}A={\displaystyle \frac{\mathrm{1}}{n-\mathrm{1}}}(\mathrm{0.9}\times {\displaystyle \frac{i_{\max }n}{t_n}}-S_m),\\ B={\displaystyle \frac{\mathrm{1}}{t_n^n(n-\mathrm{1})}}(S_m{t}_n-\mathrm{0.9}{i}_{\max }),\end{array}

with n=1+2(S_N–1)(2+1/S_N), t_n=0.6t_f[3S²_N/(1+S²_N)] and S_N=S_mt_f/i_max, where S_m is maximum rate of rise;

(7) \begin{array}{l}{I}_{\mathrm{1}}={\displaystyle \frac{t_{\mathrm{1}}{t}_{\mathrm{2}}}{t_{\mathrm{1}}-t_{\mathrm{2}}}}(S_m+\mathrm{0.9}\times {\displaystyle \frac{i_{\max }}{t_{\mathrm{2}}}}),\\ I_{\mathrm{2}}={\displaystyle \frac{t_{\mathrm{1}}{t}_{\mathrm{2}}}{t_{\mathrm{1}}-t_{\mathrm{2}}}}(S_m+\mathrm{0.9}\times {\displaystyle \frac{i_{\max }}{t_{\mathrm{1}}}}),\end{array}

with t₁=(t_h–t_n)/ln2, t₂=0.1i_max/S_m.

For current of subsequent strokes of the lightning, a single Heidler function is used [ 21]:

(8) i(t)={\displaystyle \frac{i_{\max }}{\eta }}{\displaystyle \frac{{(t/T)}^n}{\mathrm{1}+{(t/T)}^n}}\exp \left(-{\displaystyle \frac{t}{\tau }}\right)+\mathrm{0.5}\tau ,

where n=10. The term 0.5t means that the function is shifted by 0.5t, as it increases too slowly at the beginning.

Problem of grounding comparison by examples

As it has already been noted above, the FDTD method can take into account soil inhomogeneities. The soil inhomogeneities reviewed below can be different from those which can be met fairly often; however, the task of the given paper does not include accurate consideration of various inhomogeneity types. If necessary, it is possible to make the calculations including many variants of soil inhomogeneities, as the FDTD method of modeling inhomogeneities is mostly limited by the size of the calculation mesh, which is, as a rule, not bigger than 1 m for calculations with groundings.

Let us review the case with two-layer soil, the upper layer being 2.25 m thick. Specific resistance of the upper layer is 1000 W·m, and the lower layer is 250 W·m. The relative dielectric permittivity is 12. The overall length of the grounding conductors is accepted as 200 m. The diameter of the conductors is 12 mm. The phase conductors have a diameter of 20 mm, and the distance between the cross arms and the phase conductors is 1.25 m. As the tower, a suspension tower for 110 kV lines is used. The tower representation in Yee grid, the tower configuration and dimensions of one of the four foundations are shown in Fig. 1. The tower is not equipped with shield wire.

As the foundations are made of reinforced concrete, there are highly conductive steel reinforcing bars inside them. Moisture decreases concrete resistivity, which should be taken into account in calculations. As long as concrete moisture depends on soil moisture, in Ref. [ 23], it is concluded that the upper layer of concrete has a resistivity close to that of surrounding soil if the soil is moist enough. In this case, foundations can be modeled as made of metal [ 24]. Possible estimation of moisture content in soil can be made by its resistivity. According to Ref. [ 25], the moisture penetrates into foundations for soils up to 500–1000 W·m. However, in Ref. [ 24], foundations were used for grounding and modeled as made of metal for soils with a resistivity of up to 300 W·m only. In the present paper, the foundations are modeled as all made of metal (i.e., upper layer of concrete is neglected). However, since resistivity 1000 W·m is relatively high, more exact calculations could model reinforcing bars and moist concrete or some equivalent foundation model, therefore works dedicated to modeling of foundations in soils with high resistivity are needed. In addition, calculations are made for the case without foundations to estimate their influence on calculation results.

The current front time is 1 µs, the time-to-half value is 200µs, and amplitude is 30 kA. The current is calculated as an integral of magnetic field around wire. The voltage is calculated as an integral of an electric field along the shortest distance between the cross arms and phase conductors (The path of the integral is shown on Fig. 1.):

(9) U={\displaystyle \int Edl}\approx {\displaystyle \sum _{k=\mathrm{1}}^n{E}_k\Delta l},

where n is the number of cells between the phase wire and cross arm, and Dl is FDTD cell size.

The calculations consider substantially voltage at the top phase. However, the influence of phase position and instantaneous AC phase voltage on calculation results is also estimated below. Like the conductor modeling the lightning channel, phase conductors penetrate into the CPML area.

Grounding variants are schematically demonstrated in Figs. 2 and 3 (not all proportions maintained). In the first case, a four-point star grounding is applied (Each conductor is 50 m long.), placed at a depth of 0.5 m. In the second one, the same grounding is located at a depth of 1 m. In the third one, each conductor is shortened to 30 m (The depth is 0.5 m.) at the expense of adding four vertical electrodes of 5 m each, and the distance between electrodes is 7.5 m. In the fourth case, each conductor is also 30 m long, however, at the expense of two vertical electrodes, each of them 10 m long, and the distance between electrodes is 15 m.

The calculation results are displayed in Figs. 4 and 5. It can be seen that the additional deepening of half a meter hardly affects the voltage. Yet, vertical groundings have a bigger effect. It can be also noted that for this case, 10-m electrodes are only a little more efficient than 5-m ones. Therefore, in this case, while selecting the length of electrodes, the convenience of the installation of the grounding should be considered. The inclusion of foundations in the calculation model reduces the difference between insulation voltages for various groundings. The figures with calculation results also show flashover time moment (The insulation strength method used and parameters of the insulation are described below.).

Also, calculations can be done for horizontally inhomogeneous soil, when, for example, the tower is embedded in the soil with high specific resistance, whereas there is an adjoining area with lower specific resistance of the soil. Let us review the case when the tower is in the soil with a specific resistance of 1000 W·m, with an area with a specific resistance of 250 W·m located nearby. The variants of groundings involved in calculations are depicted in Figs. 6 and 7. The diameter of conductors is also set as 12 mm, and the overall length for each variant is 200 m. Conductors are placed at a depth of 0.5 m. The tower parameters are the same.

The results of calculations for the reviewed cases are exhibited in Figs. 8 and 9. With no use of information about insulation flashover when the pulses are of nonstandard waveshape, it is hard from the figures to make a choice among groundings.

It is clear from the calculations that in some cases one can determine preferable grounding by phase voltage, as the voltage for some grounding is lower than the voltage for other groundings during the entire (or almost entire) considered time interval, when such a grounding is applied, will turn lower than the voltage for other groundings. However, in many cases there remains a problem which is similar to the case when transient resistance of the groundings is compared. However, different from resistance values, real insulation voltages are compared in this paper, which can be conducted by means of the methods that evaluate electric strength of insulation for the voltages of nonstandard waveshape [ 26– 28]. In this paper, a method is applied, which has been proposed in Ref. [ 27], as it has been checked for numerous pulses of different forms. Calculations are made for the negative polarity pulse for the rod-rod spacing, 114 cm of length, which corresponds to the insulation for 110 kV lines [ 26].

In the case when a non-ideal current source is used for modeling wave impedance of the lightning channel, part of the current is branched to interior resistance of the current source. As a consequence, the current coming to the tower is essentially smaller than the full current of the source. To increase the current pointing to the tower, it can be normalized by its peak value. However, current normalization only may not be sufficient, as what is also changed is time-to-half value of current pulse (Fig. 10) and, to some extent, front time can be also changed. There yet exists a possibility to select the current source parameters by means of iterations so that the resulting current pointed to the tower should have an amplitude, front time, and time-to-half values which have to be set (Fig. 10). It can be reached by means of Duhamel’s integral, knowing the current pointed to the tower at the step current of the power source. The amplitude of the current pointed to the tower is normalized by the first current peak (Fig. 10). The second peak in this case is caused by reflections from the grounding and model of lightning channel; it was not set by current source (which is used sometimes to model double-peaked waveform).

It is seen from Fig. 10 that the second peak is not much bigger than the first peak. Besides, it can be noticed that the second peak appears only for the short front times, i.e., the tower is relatively small electrically. Therefore, using non-ideal current source with proposed current correction has negligible effect on calculated back flashover probabilities in this case.

Calculation of back flashover probability to compare grounding efficiency

Above, there were calculations only for a certain form and amplitude of lightning current pulse. However, if any of the groundings is better than another one for one form of current pulse, it can be vice versa for another form of current pulse. Therefore, for objective estimation of grounding design efficiency, (ideally) as many as possible lightning current parameters and their probabilistic nature should be taken into account.

It is not possible to take into account all probable currents. Therefore, the most important parameters of a lightning current shall be chosen (such as, for example, amplitude, front time) and calculations shall be made with their different combinations. Further, knowing what combination of parameters results in insulation flashover and the probabilistic data for those parameters, the probability of back flashover can be calculated at a lightning stroke. The problem is that it is not simple to say in advance which parameters, besides the amplitude, are more important, and which are less important. Another problem is that the results of calculations pertaining to parameter correlation coefficients are different by available lightning current measurement results [ 29, 30]. This paper reviews two possible approaches to apply probability data of lightning currents.

Not to make lengthy calculations for different forms of currents, it is sufficient to make one calculation for step current, and then to apply Duhamel’s integral to calculate voltages on insulation at arbitrary current. Further, depending on approach, the probability of back flashover is calculated for the lightning to strike the tower. If ground wire is applied, it is also necessary to review the lightning stroke in the ground wire. So, when groundings are compared, the probabilities of back flashover are just compared. Using the flashover probability, a more efficient grounding can be determined and estimation of its effectiveness comparing to other groundings can be made.

First approach for flashover probability calculation

In the first approach, the distributions of current amplitudes and rates of rise are taken into account. With such an approach, the so-called curve for dangerous parameters is calculated, which shows the critical values of the amplitude and rate of rise. Exceeding these values results in insulation flashover [ 31]. An example of the curve for dangerous parameters corresponding to grounding No. 1 in Fig. 6 for the first strokes of a lightning discharge is presented in Fig. 11 (Heidler function (4) is used for the calculations.). The curve is calculated by searching rates of rise and amplitudes which lead to back flashover. There is no need to select very high maximum values for steepness and amplitude up to which the curve is calculated, as the curve of dangerous parameters can be prolonged by means of straight lines, parallel to axes (Fig. 11). This will not incur a gross error, as the probability density functions reduce very fast with the growth of the rate of rise and amplitude [ 31].

The probability of insulation flashover is determined by the probability of the fact that a certain combination of the steepness and amplitude will occur in the region D lying to the right of the curve for dangerous parameters. So, the probability of insulation flashover is calculated as a double integral:

(10) P={\displaystyle \underset{D}{\iint }p(I,S)dIdS},

where p(I,S) is a joint probability density for the combination of values of amplitude I and steepness S.

Integral (10) can be approximated as

(11) P={\displaystyle \sum _{k=\mathrm{1}}^n\Delta P_k},

where \Delta P_k is integral for a separate element:

(12) \Delta P_k={\displaystyle \underset{I_i}{\overset{\infty }{\int }}{\displaystyle \underset{S_j}{\overset{S_{j+\mathrm{1}}}{\int }}p(I,S)dIdS}},

which can be also represented via product of probabilities:

(13) \Delta P_k=P(I>I_i)[P(S>S_j)-P(S>S_{j+\mathrm{1}})].

For the top element of area, S_j=\infty .

The probability of exceeding a certain value of current amplitude I_i is defined from the formula [ 19]:

(14) P(I>I_i)={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\mathrm{erfc}\left({\displaystyle \frac{z}{\sqrt{\mathrm{2}}}}\right),

where erfc is the complementary error function,

(15) z={\displaystyle \frac{\ln I_i-\ln M_I}{{\beta }_I}},

where M_I is the median value of current and b_I is the standard deviation of ln I.

The probability of exceeding a certain value of the current steepness is defined in a similar way.

To calculate the back flashover probability when correlation coefficient between steepness and amplitude is not equal to zero, the probability density function of the bivariate lognormal distribution is used

(16) \begin{array}{l}p(I,S)={\displaystyle \frac{\mathrm{1}}{\mathrm{2}\pi IS\sqrt{\mathrm{1}-P}}}{e}^{-q/\mathrm{2}},\\ q={\displaystyle \frac{\mathrm{1}}{\mathrm{1}-p^{\mathrm{2}}}}({{\displaystyle z}}_I^{\mathrm{2}}-\mathrm{2}p{{\displaystyle z}}_I{{\displaystyle z}}_S+{{{\displaystyle z}}_S}^{\mathrm{2}}),\\ {{\displaystyle z}}_I={\displaystyle \frac{\ln I-\ln {{\displaystyle M}}_I}{{{\displaystyle \beta }}_I}},\\ {{\displaystyle z}}_S={\displaystyle \frac{\ln S-\ln {{\displaystyle M}}_S}{{{\displaystyle \beta }}_S}},\end{array}

where r is correlation coefficient, M_S is the median value of steepness and β_S is the standard deviation of InS.

Again, the probability of insulation flashover is calculated by using Eq. (10). However, in this case, it should be approximated with smaller elements to perform numerical integration:

(17) \Delta P_k={\displaystyle \underset{I_i}{\overset{I_{i+\mathrm{1}}}{\int }}{\displaystyle \underset{S_j}{\overset{S_{j+\mathrm{1}}}{\int }}p(I,S)dIdS}}.

Eqs. (4) and (8) which define the form of the lightning current have a feature in accordance to which it is not possible to directly set the value for the front time and time-to-half value. Therefore, an additional procedure is required to calculate the indices which establish the necessary front times and time-to-half values.

A complete algorithm to calculate insulation flashover probability is given in Fig. 12.

The maximum current steepness S_m is used in the calculations. For the calculation of the curve for dangerous parameters for the first strokes of lightning discharge, steepnesses of 6–100 kA/ms and amplitudes 5–80 kA are used. The time-to-half value is accepted as 77.5 ms. For subsequent strokes, steepnesses of 4–120 kA/ms, amplitudes 3–50 kA are used, and the time-to-half value is 30.2 ms.

Distribution of the first stroke amplitudes used in calculations is set by M_I – 27.7 kA and b_I – 0.461. The distribution of first stroke rates of rise (S_m) is set by M_S – 24.3 kA/us and b_S – 0.599 [ 19]. The corresponding values for subsequent strokes are M_I – 11.8 kA and b_I – 0.530 (amplitudes), and M_S – 39.9 kA/us and b_S – 0.852 (rates of rise S_m) [ 19].

Due to lengthy time of calculation, the calculations are conducted only for the first 40 ms. A more accurate calculation shall probably include a whole current pulse.

Second approach for flashover probability calculation

In another approach to the calculation of insulation flashover probability, an assumption is accepted that the front time can be identified via the current amplitude. Equation (18) is used for that purpose [ 29, 32]:

(18) t_f=\mathrm{1.31}\exp (I/\mathrm{230}),

where t_f is the front time (ms), I is the current amplitude (kA). The front time t_f is calculated from maximum steepness [ 29]. The equation is used as a function in this paper. However, of course, it actually deals with the averaged value of front time, whereas the real front time can have other values, too, which shall be understood if more accurate estimation of flashover probability is needed.

In this approach, the calculation is focused on the determination of the critical current value, exceeding which will result in insulation flashover. Critical current can be found by gradual increasing of lightning current and calculating insulation voltage with Duhamel’s integral until occurrence of flashover. After that, what is determined is the probability for the excess of this critical current value (see Fig. 13).

Of course, in this case, only minimum current amplitudes resulting in flashover can be compared. However, the probability of insulation flashover is more indicative for selecting the grounding, as in some cases small variations of critical current values can result in considerable variations of flashover probability (In such a case, the grounding design has a great effect.), and in some cases—on the opposite.

The time-to-half value and the probabilistic parameters of lightning currents used in the calculations, are the same as earlier.

Total flashover probability

The exact calculation of total back flashover probability needs the distribution of the number of lightning discharge strokes in one point. Existing distributions are based on video or electric field measurements. These distributions give information about all strokes in lightning discharge (neglecting spatial separation). However, it is known that the average amount of strokes in one point is about three [ 19, 33]. Taking into account that the correlation between the first and the subsequent stroke current amplitudes is not observed [ 34], as an approximation, it can be accepted that the lightning discharge to tower top contains three strokes (first stroke and two subsequent ones) and parameters of these strokes are not correlated. In this case, the total probability can be calculated by using

(19) P_t=\mathrm{1}-(\mathrm{1}-P_f)(\mathrm{1}-P_s)(\mathrm{1}-P_s),

where P_f is the probability of back flashover caused by the first stroke, and P_s is the probability for subsequent stroke.

Calculation results

First, two different approaches using two different functions for the currents of the first strokes are compared. Table 1 presents the results of calculating insulation flashover probability for the case with groundings in a two-layer soil taking into account foundations. It is seen clearly from Table 1 that the values of insulation flashover probability strongly depend on assumptions made when the probabilities are calculated. However, as different grounding designs are compared, relative values of insulation flashover probability are different to approximately the same extent (little in this case) as insulation voltages, and this makes it possible to estimate the efficiency of the grounding design.

What seems more interesting is the case with reviewed groundings in horizontally inhomogeneous soil, as in this case it can be hard to make a conclusion about the most preferable grounding using only insulation voltages. Also, it is important to find out why probability calculation approach and current function influence calculation results greatly.

Three different first stroke current functions are used to estimate their influence on calculated back flashover probability. Back flashover probabilities for subsequent strokes are also calculated. Table 2 lists the calculation results. It is seen that the calculated probability strongly depends on current function. The main difference between these functions is the ratio of maximum steepness to average steepness (or concavity). The concavity of CIGRE function is set in accordance to measurements (in the calculations, the ratio of S_m to S_30/90 is set to 3.375 as the S_m median value is 24.3 kA/us and the S_30/90 median value is 7.2 kA/us.). Heidler function (Eq. (4)) has a smaller maximum to average steepness ratio compared to measurements, i.e., it is not concave enough. Ramp function (Eq. (3)) is not concave. It can be expected, therefore, that calculation results with CIGRE function (Eq. (5)) are more accurate than others. The total back flashover probability is calculated using the probability values corresponding to Eqs. (5) and (8).

Calculations with different correlation coefficients are conducted to estimate the influence of correlation between maximum steepness and amplitude on calculated back flashover probability using the first approach with correlation coefficients of 0, 0.16 [ 30], 0.5 and 0.846 [ 29] and the second approach (In this case, the coefficient equals 1 because of functional dependence.). For the calculations, the CIGRE current function is used. In order to make a more reasonable comparison of the first approach with Eq. (18), the probabilistic data from Ref. [ 29] is used. The distribution of the first stroke amplitudes used in the calculations is set to M_I – 27.7 kA and b_I – 0.28ln(10). The distribution of the first stroke rates of rise (S_m) is set to M_S – 18.9 kA/µs and b_S – 0.26ln(10). The S_m to S_30/90 ratio is 2.148 (as M_S for S_30/90 is 8.8 kA/us). Table 3 lists the calculation results. It can be seen that the correlation between the amplitude and the steepness (or probability calculation approach) has a smaller significance than the shape of current function. It can also be seen that the probability grows fast with the increase in correlation coefficient.

In addition, the influence of phase voltage on calculated back flashover probability is estimated. In the calculations, the amplitude voltage of V_a= \mathrm{110}\sqrt{\mathrm{2}/\mathrm{3}} kV is added to or subtracted from the voltage caused by lightning discharge. Again, CIGRE current function (5) is used. The distributions of the first stroke amplitudes and rates of rise are the same as those in section 5.1. The calculation results are presented in Table 4. It is seen that the phase voltage significantly influences the calculation results. However, the difference between back flashover probability values for different groundings (but same phase and voltage) remains approximately the same.

For the case with horizontally inhomogeneous soil, the least probability of flashover among the reviewed groundings is provided by a four-point star grounding, independently of a calculation method. It can be seen from Figs. 8 and 9 that the overvoltage for grounding No. 4 tends to reach a lower value than overvoltages for other groundings (i.e., it very likely has smallest stationary resistance), but the higher overvoltage until about 2 ms (Fig. 8), together with the results in Tables 2–4 indicate that a high grounding inductance can increase the back flashover probability noticeably.

If the only aim is to choose the grounding, at least for the cases considered in this paper, the second approach is more preferable, as it is simpler. However, if there is a need to calculate the back flashover probability more exactly for accurate transmission line grounding design, the approach which uses more probabilistic information is a better choice. The first approach uses more information, but for better accuracy the parameters should be chosen more carefully (based on their influence on the calculated probability).

It should be noted that for the towers, which are essentially different by their design, various constructions of the grounding can be optimal. For transmission lines with shield wire, the insulation voltages are sufficiently different from those presented in this paper due to many reflections from adjacent towers and their groundings. Therefore, preferable grounding constructions may be also different.

Conclusions

A calculation strategy to choose grounding configurations for transmission line towers has been suggested. The approach makes it possible to compare grounding configurations relatively properly, as it takes into account numerous factors: accurate calculation of wave processes by FDTD method, insulation flashover in the event of nonstandard voltages by the method suggested in Ref. [ 27], and probability parameters of lightning currents. Based on the approach, a comparison of several grounding configurations has been made, which helps to estimate the significance of grounding transient characteristics and the influences of these characteristics.

It can be seen from the calculation results that lowering of wires by 0.5 m and using vertical electrodes in two-layer soil do not decrease the back flashover probability significantly. Also, the considered case with shifting of grounding in horizontally inhomogeneous soil toward the region with a lower resistivity turns out to be not efficient. It can be concluded that it is important for the grounding to have small inductance as it increases insulation voltage when it is already high because of tower inductance (or traveling time if the tower is presented as transmission line). Therefore, highly inductive grounding constructions should be avoided even if their stationary resistance is low.

It has been found that it is very important to represent the front part of current waveform correctly, as it strongly affects the accuracy of back flashover probability calculations.

It should be noted, however, that the suggested approach does not consider such phenomena as soil ionization and corona. Although it is possible to consider those phenomena [ 9, 10], it will result in impossibility to apply Duhamel’s integral, which will not allow the calculations for a wide range of parameters of lightning currents (as the calculation time is yet too long for modern personal computers), and, consequently, the calculation of insulation flashover probability.

It is probable that the insulation flashover probability can be approximately calculated taking into account non-linear effects by means of only limited number of currents. To calculate it, the calculation grid size can be increased, and modern video cards can be used, as they are capable of providing a calculation speed increase of up to 10 and more times [ 35].

Also, the following important factors can be noted:

1) Methods for the calculation of insulation flashover with nonstandard pulse forms give different results [ 36]. Their accuracy is low in some cases (e.g., for line insulation, in case of short duration pulses, the error can make tens of percents [ 27]).

2) Correct modeling of lightning channel is not so obvious. Besides, better representation of the first return stroke current waveform may need application of more sophisticated functions [ 37]. In addition, the calculated insulation flashover probability depends on probability characteristics of lightning currents. Therefore, it is necessary to know their accurate values and to treat them correctly.

3) In this paper, such important parameter as grounding installation cost is not taken into account. However, knowing the back flashover probability value, it is possible to use the methodology similar to the one suggested in Ref. [ 38] for optimal grounding design.

4) For accurate calculations, the frequency dependence of soil parameters should be taken into account, as it influences grounding potential rise greatly [ 39].

5) During grounding design, attention should be paid to the hazards associated with transmission tower voltages caused by ground faults.

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