The calculation of lightning induced voltages on distribution lines is of crucial importance because the indirect impacts are a major cause of outages in this kind of lines [1]. This is because the high frequency of indirect strikes compared to direct ones and the reduced insulation level of the distribution lines, which makes them vulnerable to indirect lightning-induced voltages [2]. For the calculation of lightning-induced voltages, a very well-known methodologies adopted, which considered the following steps [3]:

1. To adopt a spatial-temporal distribution of the current along the channel i (z´ , t).

2. On the basis of the return stroke current, the electromagnetic field is calculated along the distribution line.

3. The electromagnetic field along the line is used in the coupling model to calculate the voltage on the line.

From the previous steps, the calculation of the electromagnetic field has the highest computational effort, because Equations (1), (2) and (3) are solved with numerical methods [4][5]. Some methods could be used to decrease the computational time required: Thottappillil et al [6] derived a formula that allows the calculation of electric and magnetic fields for a return stroke with a velocity equal to the speed of light v = c. Rubinstein et al [7] developed expressions of the electromagnetic field for a spatial-temporal distribution of the given current for a step function calculated at ground level. Barbosa et al [8] proposed a formula for the calculation of the horizontal electric field considering finitely conducting earth in time domain. Napolitano [9] developed an analytical formula to calculate the electromagnetic field generated by lightning assuming a current wave shape linearly rising with flat top that allows the reduction of the computation time of the lightning-induced voltage calculation, thus avoiding the use of slower numerical methods. Nevertheless, in the analytical solution derived by Napolitano in [9] only the source terms are presented, which are not enough to compute the electromagnetic field. In order to compute lightning-induced voltage on an overhead distribution line adopting the traditional methodologies [3] the vertical electric field E_z and the azimuthal magnetic field are calculated at ground level, meanwhile the horizontal electric field E_p is calculated at the height of the line. Each field is obtained by means of the Master and Uman equations [10][11] deduced for perfectly conducting ground. In the case of the vertical electric field E_z and the azimuthal magnetic field components at ground level, it is just necessary to derive the equations for either the source or the dipole contribution and then obtain the total field multiplying it by two, because of the equal contribution from the source or the image dipoles at ground level. In the case of the horizontal electric field E_p , it must be calculated at height h, where the contribution of the source and dipole terms are required.

(1)

(2)

(3)

In this paper, the image terms to calculate the horizontal electric field are derived and implemented in Yaluk Draw software [12] to calculate lightning-induced voltages and the lightning performance of a typical distribution line. The implemented analytical formulation with traditional numerical methodologies is presented.

This paper is organized as follows: Section 2 presents the derived terms of the horizontal electric field corresponding to image dipoles. Section 3 presents the calculation of lightning-induced voltages using the analytical expression for calculating the electromagnetic field. Section 4 presents the lightning performance of a distribution line in comparison with the results found in the standard IEEE 1410 – 2010. Finally, the conclusions of the work are presented.

Master and Uman [10] derived analytical expressions for computing the electromagnetic field generated by lightning. These equations in cylindrical coordinates are [11].

Equations have been traditionally solved by means of numerical methods [3] [5]. Napolitano et al [9] solved Equations (1), (2) and (3) assuming a linear-front and flat-top lightning current wave shape and a TL model. The calculation of the electromagnetic field requires the sum of the contribution of the sources and image dipoles.

where is the current along the return-stroke channel at height ; , the variable of the height along the channel (see Fig. 1); c, the speed of light in free space; and r, the distance between point P and the infinitesimal current, given by (4).

Fig. 1. Fig. 1. Geometry for electromagnetic field calculation in cylindrical coordinates. Source:[9]

(4)

The expressions to calculate the image terms of the horizontal electric field are presented below:

The wave shape of the lightning current is linear-front and flat-top, according to Equations (5) and (6):

(5)

(6)

where I_{p is the peak value of the current and tf is the front time.}

The solutions of (1), (2) and (3) need to be divided into two integration limits: the first one from to , and the second between and zero, as seen in Fig.2

Fig. 2. Fig. 2. Integration limits of the dipole image. Source: Authors.

represents the height of the last dipole turned on when the front-wave travels from the ground to height , which is obtained from equation (8) derived from condition (7):

(7)

(8)

On the other hand, is the height which below that point the current has a constant value . is obtained from condition (9) and calculated from (10).

(9)

(10)

The expression for the contribution of the image dipoles to the horizontal electric field is given by (11), (12) and (13).

(11)

(12)

(13)

The expression for the contribution of the source dipoles to the horizontal electric field was presented in [9] in terms of and , as in Fig.2 The total horizontal field is the sum of the image and source terms (14)

(14)

In order to calculate the vertical electric field and the azimuthal magnetic field at ground level, it is possible to use the equations developed by Napolitano [9] derived only for the source terms and multiplied by (2). This is explained by the fact that fields and calculated at ground level receive equal contribution from the source and image dipoles; therefore, only one of them needs to be calculated.

In order to test the validity of the equations presented in the previous section, a comparison with some approximations in the literature is presented.

The analytical formula presented in the previous section could be employed to calculate the line flashover rate of overhead distribution lines. For this purpose, the methodology in [2] and adopted for the IEEE 1410 – 2010 standard [1] was followed. This methodology was implemented in Yaluk Draw software [12] and it consists of the following steps:

Step 1. Randomly generate a large number of lightning strike events , including parameters that allow to characterize its waveform and location, peak current , front time t_f , and spatial location based on Cartesian coordinates and . Parameters and t_f are represented by a lognormal distribution, according to CIGRÉ [15]. The front time t_{f is obtained from the peak current distribution assuming a 0.47 correlation factor. The coordinates of the strike ( and ) are assumed to be uniformly distributed over a rectangular area with dimensions and . Those dimensions are chosen because induced voltages associated with strokes beyond this distance are generally lower than the CFOs used in this paper.}

Step 2. Lightning strikes located at a distance equal to or less than from the line are identified as direct strikes and are not taken into account in the lightning-induced flashover analysis. The distance (15) is obtained from the electrogeometric model in [16][17][18], where h is the line height, and .

(15)

Step 3. The remaining events are considered indirect lightning strikes and induced voltages are calculated using Yaluk Draw software [12], which included the analytical formulation for the electromagnetic field computation. The normalization of the annual number of flashovers due to indirect lightning is given by (16):

(16)

where n is the number of events that generate induced voltages that exceed 1.5 CFO and

N_g is the annual ground flash density (in flashes/km/yr).

is the distance beyond added none of the events causes flashovers on overhead lines.

This section details the application of the last methodology to the simulation of an overhead line, such as the one in Fig. 4, which matches the simulation in [2] and meets IEEE 1410-2010 standard [1]. The total number of simulated strikes n_{tot was 120,000, and the ground flash density was 1 flash/km²-year. A modern computer was used: 3.4 GHz processor and 8 GB RAM.}

Fig. 10 and Fig. 11 compare the flashover rates obtained by applying the methodology described in this paper (called Yaluk-MC) and the rate in IEEE 1410, considering infinite conductivity and , respectively. A good match between the results of the two methodologies under analysis can be observed, with errors lower than 5%.

Fig. 10 . Fig. 10 . Line flashover rate using the methodology in this paper (YALUK-MC) and IEEE Standard in case of a 10-m-high, 2-km-long single conductor above an ideal ground. Source: Authors.

Fig. 11 . Fig. 11 . Line flashover rate using the methodology in this paper (YALUK-MC) and IEEE Standard in case of a 10-m-high, 2-km-long single conductor with soil conductivity of 0.001 S/m. Source: Authors.

On average, a simulation takes 5.2 s when the numerical methodology is used. By contrast, using the analytical formulation, it takes 2.4 s. This is a 50 % reduction, which is also applicable to the calculation of the flashover rate.

The assumption of a current wave shape linearly rising with flat top allows to derive an analytical expression for a lightning electromagnetic field over flat ground. According to the results presented in this work, such electromagnetic fields have good agreement compared with the numerical solution using a much more complex wave shape, such as Heidler function. The errors were found to be lower than 5%, if the Heidler function is assumed to be the most accurate one.

Since to calculate the lightning-induced voltages it is necessary to compute the horizontal electric field above ground, the contribution of the image dipole must be derived, as shown in Section 2.

Since the results of induced voltages’ peak value exhibit good agreement with the numerical simulation, the lightning performance of a distribution line can be assessed with lower computational effort and time. Thanks to the implementation presented in this paper, the analytical formulation was two times faster than using a numerical computation.