Keywords

1 Introduction

Shielding effect of nearby objects on the number of strokes in case of critical sections of a transmission line is an important question. Such a critical section can be the one near a substation or a tower equipped by telecommunication devices, etc. There are different methods applied for the calculation of the number of strokes as it was overviewed in [1]. The method described in that paper was selected to use it in the examination of the following problem.

Until nowadays high number of wind turbines were installed all over the world. Some of them were installed quite close to a transmission line. Regarding that the wind turbines are much higher than the transmission lines they have a significant effect on the expected number of strokes. The question is, how this effect can be calculated and how the results can be taken into consideration.

There is another interesting situation with lot of questions that it is necessary to install lightning protection system for the installed solar farms or not [1]. The protection of the inverters is clear because that is one of the main cost. It is possible to be 1/3 of the whole costs. In this paper there will be shown a calculation for a small solar power system, which could be equivalent as a part to a bigger one.

2 Relationship to Resilient Systems

The lightning protection and the design of a lightning protection system (LPS) is an important part of the transmission systems, too like the LPS of a building (e.g. family house, hospital, skyscapers, etc.).

Nowadays with the dynamic system regulation (like DLR – Dynamic Line Rating), for which is a possible solution for the transmission of the produced energy from renewable sources, it is necessary to consider other aspects during the designing and planning. This method could not work well if there is no high efficiently lightning protection. Therefor it is relevant to deal in this aspect with the protection of transmission lines and wind turbines.

3 About the PMAS Method

The PMAS method (Probability Modulated Attractive Space) is a mode to determinate the expected frequency of lightning strokes attached to a certain object. The basis of this method is similar to the electro-geometric model (EGM), [2].

From the point of the application of PMAS it is very important to determinate the boundary (50% regression surface) of the attractive space. According to the experience, this boundary differs from the one generated according to pure geometric method (points that have the same distance from the object and the ground/air termination). In additional to, the boundary surface is polarity-dependent.

In case of high objects, a certain number of upward lightning strokes appear. (Their number increases with increasing height.) When the upward leader is initiated by the electric field of the space charge in the thundercloud, striking point cannot be defined in such a way as in case of downward lightning. However such examination which results in the “attachment point” of lightning in the previous case is still interesting.

The striking point is that point, where the endpoint of the downward leader is at the moment, when the return stroke is starting. The distance, between the striking point and the point of strike, is the striking distance, which is dependent on the lightning current. There are different methods to describe this dependence [3, 4].

$$ N_{F} = N_{G} \int_{V} {\frac{dP}{dr}dV\,b} $$
(1)
$$ N_{Fa} = N_{G} \int_{{V_{a} }} {\frac{dP}{dr}dV} $$
(2)

In the Eqs. (1) and (2) the P is the probability, the r is the striking distance, V is the volume, NG is the number of the strikes for a km2 and a year.

In reality, the attractive space is an abstract term. Although it can be considered, that from a given striking point the lightning attachment will happen to the closest object, it happens only with a given probability. (Let us denote it by b.) So, in the reality, the attractive space of an object gives that points where the probability is higher that the stroke will be attached there than to other point. Therefore, the interface are determined by the points, where the probability of the stroke is b = 0.5. (Equation 1 and for the attractive space only the Eq. 2.) To each point of the attractive space it is possible to give a dP/dr value:

$$ \frac{dP}{dr} = \frac{kp}{{\sqrt {2\pi \,r} }}\exp \left( { - \frac{1}{2}k^{2} p^{2} \left( {{ \ln }\frac{r}{{r_{m} }}} \right)^{2} } \right) $$
(3)

Where

k:

is the parameter which is dependent to polarity of the lightning,

p:

is a value between 1.2 and 2,

rm:

is the median value of the striking distance

r:

is the striking distance.

The attractive space for positive and negative polarity is different.

For a PMAS simulation it is necessary to know the dependence of these parameters on different influencing factors, like environment, geography, climate and the arrangement of the objects.

4 Formulation of the Problem for Wind Turbines Near Transmission Line

In this paper, simulations was made to determinate the equivalent or collection area, Aeq (see (1)) depending on the relative position of the wind turbine and the transmission line as it is illustrated in Fig. 1. In this arrangement, it is possible to see calculations has been made for the following cases. First, when the bottom of the wind turbine and a tower have the same x-coordinate. In the second case, the wind turbine is in the same distance from two towers.

Fig. 1.
figure 1

The arrangement in two positions (position one: brown; position two: red). (Color figure online)

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The wind turbine has the following parameters:

The parameters of the transmission line are the following (Table 2):

Regarding that most of the calculation methods result in the so-called equivalent area (or collection area), that gives the number of stroke according to (Eq. 4), our goal is to calculate this parameter. In (4) N represents the number of stroke/year to the object, Ng denotes the annual ground flash density, and Aeq is the equivalent area.

$$ N = N_{g} A_{eq} $$
(4)

As a reference, \( A_{eq}^{{\prime }} \) was calculated according to the equation from [4] for the previously mentioned transmission line. According to this \( A_{eq}^{{\prime }} = \, 7 000\, {\text{m}}^{2} \) (by Eq. 5 and Fig. 2.)

Fig. 2.
figure 2

Comparison of the width of equivalent area, obtained from several calculation methods [6].

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$$ A_{eq}^{{\prime }} = 2\,B_{a} \,l $$
(5)

where B a is the half-width of an equivalent area (A eq ) (Table 3).

For the further calculations distance between the wind turbine and the centre line of the transmission line is denoted by “y”.

5 Study Case: The Necessity the Lightning Protection of a Solar Panel String Table

In this study case there will be shown a calculation for the efficiency of a lightning protection system for a smaller solar farm. The parameters of the solar panel string table are the following (Table 4).

After the calculation, the results are in the Tables 5 and 6.

The air-termination reduces the risk of direct lightning stroke significantly. However, the comparison of the cost of selected air-termination system to the expected cost of damage in the unprotected case is very necessary [7, 8]. It is especially useful to see how the resultant risk is changing with the lightning current.

6 Summarizing the Results

The calculated values by the PMAS method and by [5] have different result. The value of \( A_{eq}^{{\prime }} = 7000\, {\text{m}}^{2} \) and the calculated for the overhead lines without turbines give the result which is shown in the Table 1.

Table 1. The parameters of the wind turbine.
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Table 2. The parameters of the transmission line.
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Table 3. Calculated Aeq[m2] for the overhead lines.
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Table 4. The parameters of the solar panel string table
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Table 5. Calculated Aeq[m2] for solar panel string table (without air termination)
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Table 6. Calculated Aeq[m2] for solar panel string table (with air termination)
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The calculated values for our arrangement give the expected values. The wind turbine has a risk reduction effect from the point of the transmission line. Therefore the striking frequency (for the case of power lines) decreases [9]. The measure of this effect depends on the relative position of the wind turbine and the transmission line. From the point of PMAS it can be considered that the attractive space of the transmission line is reduced by the one of the wind turbine. This can be interesting when a critical section of a line is situated near the wind turbine. Although the phase wires are protected by the grounded wires (thus the reduction effect of their equivalent area is not significant), the severity of the back-flashovers and the secondary effects are depending on the number strokes to the grounded overhead wires. In case of a solar power plant with a well-selected and well-designed lightning protection system, that is possible to reduce the probability of the risk of a fault caused by a lightning strike. But it is important to know the high of this risk and be more reliable calculation for the expected stroke.

7 Conclusion

In the paper was shown, the wind turbine has a risk reduction effect from the point of the transmission line. Therefore the striking frequency decreases. The measure of this effect depends on the relative position of the wind turbine and the transmission line. From the point of PMAS it can be considered that the attractive space of the transmission line is reduced by the one of the wind turbine.

The solar power should be in the middle point of the lightning protection, too. With a lightning protection system, that is possible to reduce the probability of the risk of a fault caused by a lightning strike. But it is important to know the high of this risk and be more reliable calculation for the expected stroke. There should be other researches for another important situation when there will take into account the environment, too.