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Lightning-induced voltages on overhead lines over irregular terrains
Electric Power Systems Research 176 (2019) 105941
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Lightning-induced voltages on overhead lines over irregular terrains a,⁎
Edison Soto , Ernesto Perez a b
b
T
Escuela de Ingeniería Eléctrica, Electrónica y de Telecomunicaciones, Universidad Industrial de Santander, Bucaramanga, Colombia Departamento de Energía Eléctrica y Automática. Universidad Nacional de Colombia, Medellín, Colombia
ARTICLE INFO
ABSTRACT
Keywords: Finite-Difference Time-Domain (FDTD) method Electromagnetic fields Lightning-induced voltages Mountains
This paper analyzes lightning-induced voltages on distribution lines located on irregular terrains since it is common, in some places in the world, that real distribution systems are placed on those terrains. Simplified topographies were used in this analysis to approximate to those found on real distribution systems. The lightning-induced voltages were calculated with a full-wave analysis approach based on the 3D Finite-Difference Time-Domain (FDTD) method. It was found that lightning-induced voltages on overhead lines above irregular topographies, in the majority of cases, are greater than those found for flat terrain. The enhancement factor can reach values up to 5.2 times. In a few cases, factors below unity were found. These results may give a new insight on the design of overhead distribution lines for improving lightning performance on real distribution networks.
1. Introduction Lightning-induced voltages are one of the major issues on overhead distribution lines. These kind of lines, due to its reduced insulation level compared with transmission lines, experiment overvoltages produced by indirect lightning that cause insulation flashover and frequently outages. Modeling the mechanism of induction of the electromagnetic field produced by lightning has been an important task in the past [1–4]. The calculation of induced voltages is, commonly, based on the computation of three steps, which include, the representation of the current along the lightning channel, the electromagnetic field calculation and the coupling mechanism to the conductors [5,3,6,7]. The current distribution along the lightning channel has been represented with different types of models which aim to reproduce the return stroke behavior, being more used for induced voltage calculation, the so-called “engineering models” [7]. The electromagnetic fields equations have, traditionally, been derived by analytical representation of antenna theory using dipoles or monopoles above a flat terrain with finite [8] or infinite ground conductivity [9,10]. Since the horizontal component of the electric field is highly affected by the ground conductivity, several authors have contributed equations that represent this effect [11–16]. These equations have been widely used to calculate the lightning electromagnetic pulse (LEMP) [7,17]. In order to represent more accurately the complexity of the lightning phenomenon, several numerical methods have been used to compute the LEMP in more complex scenarios: the FDTD method [18–22], the method of moments (MoM), applied to the calculation of
⁎
electromagnetic fields produced by inclined lightning channels [23] and lightning channel tortuosity [24] and, the hybrid electromagnetic/ circuit model (HEM) [25,26]. Recent studies have conducted investigations to analyze the effect of the LEMP due to different considerations, such as the characteristics of the lightning channel [23,27–29], the presence of nearby towers [30,31] or buildings [32], among others. Some other works have analyzed the effect of irregular terrains on the electric fields produced by lightning, showing an enhancement of the electric field [33,34]. The interaction of the LEMP with overhead distribution lines can be analyzed by two methodologies: (i) a transmission line approach or (ii) full-wave analysis. In the first methodology, the electromagnetic field is calculated without considering the distribution line, and then, a coupling model is used to calculate the induced voltage along the line [3,35]. The majority of works that use this methodology, have assumed lines over flat terrain [6,36–43], though recently, it has been considered the effect of irregular terrains [44,45,16]. On the other hand, the methodology of full-wave analysis incorporates the conductors in the electromagnetic field calculation and directly calculates the induced voltage [46,47]. It has been used to analyze different aspects that were difficult to consider with the other approach: modeling corona in the conductors, take into account the effect of nearby buildings or tall structures, among others. Nevertheless, the developed works have only considered overhead lines above flat ground planes [48,49,38,50,51]. In many places around the world, the distribution lines are built over mountains. Since, electromagnetic field propagation depends on topography [33,34] and hence lightning-induced voltages, it is
Corresponding author. E-mail addresses: [email protected] (E. Soto), [email protected] (E. Perez).
https://doi.org/10.1016/j.epsr.2019.105941 Received 14 November 2018; Received in revised form 23 March 2019; Accepted 9 July 2019 Available online 05 August 2019 0378-7796/ © 2019 Elsevier B.V. All rights reserved.
Electric Power Systems Research 176 (2019) 105941
E. Soto and E. Perez
Fig. 1. Simulated circuit.
Fig. 3. Case A. Line that crosses a V-shaped valley.
Fig. 2. Example of a irregular terrain.
discussion are presented in Section 4.
important to analyze the effect of terrain shape on the calculation of lightning-induced voltages on distribution lines that may give new insight about the design of distribution networks located irregular terrains. In this paper, lightning-induced voltages calculated on overhead distribution lines located over three arrangements of irregular terrains similar to those found in rural zones of Colombia are presented, using a developed code of full-wave analysis based on the 3D-FDTD method. Conclusions about the differences in overvoltages found on lines placed over flat terrains are given. This paper is organized as follows: Section 2 provides the methodology used for lightning-induced voltage calculation. The test cases are detailed in Section 3. Finally, conclusions and
2. Methodology In order to calculate lightning-induced voltages [7] on overhead lines located over non-flat ground is important to take into consideration two issues: (a) the analytical formulation of electromagnetic field produced by lightning developed by Master and Uman [9] are only valid for flat terrain and (b) the uncertainty of the performance of the Transmission Line Model to calculate lightning-induced voltages on lines placed over irregular terrains. For these reasons a numerical methodology based on full-wave analysis performed with the 3D-FDTD
2
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Fig. 4. Lightning-induced voltages at both line extremities for Case A (Fig. 4) for an inclination angle α = 20°, a distance d = 400 m and r = 100 m. Table 1 Summary of maximum lightning-induced voltages in kV obtained for Case A for flat terrain and the enhancement factor (ρ) varying d and α in Fig. 3, keeping r = 100 m and considering lossy ground. σ (S/m)
α
d=0m
200 m
400 m
500 m
0° 10° 20° 30°
23.2 kV 2.0 2.6 2.9
42.1 kV 3.5 4.6 5.2
46.3 kV 4.5 5.2 5.2
46.3 kV 4.7 5.2 5.2
0.01
0° 10° 20° 30°
27.9 kV 1.8 2.2 2.5
56.4 kV 2.7 3.5 3.8
56.4 kV 3.8 4.3 4.4
56.3 kV 3.9 4.3 4.4
0.001
0° 10° 20° 30°
40.0 kV 1.5 1.8 2.0
79.7 kV 2.1 2.7 2.9
80.5 kV 2.8 3.1 3.2
80.5 kV 2.9 3.2 3.2
∞
n+ 1 2
Hy
2 2 +
z z
Hy
n+ 1 2 t (Hx 2 (i , j, k ) (2 z + t z ) y
Hx
(2
z
n+ 1 2 n+ 1 2
(i (i , j
t
y
2µ y + t
y
n 1 2
Hy
(i , j , k )
2 t (Ezn (i + 1, j, k ) (2µ y + t y ) y
Ezn (i , j, k ))
2 t (Exn (i , j, k + 1) (2µ y + t y ) z
Exn (i , j, k ))
(2)
where: εu, σu, μu are the permittivity, conductivity and permeability in each direction respectively, at the location of the associated calculated field. Δx, Δy and Δz are the space step in x, y and z respectively of the FDTD method. Δt is the time step used in the FDTD method. Is represents the return stroke current. The overhead distribution line is represented as a Thin Wire according to [48,49]. The electric fields along the line are forced to be zero, and the parameters μm and εm around the wire are updated differently than the rest of the working space according to:
µm = µ 0 / m ,
m
=m
0
(3)
where
m=
1.471 s
ln( r )
(4)
and Δs is the spatial step and r is the radius of the conductor. The intrinsic radius r0 for this method is 0.2298Δs. It is defined as the radius for which the real distribution of electric and magnetic fields around the wire is the same as the one obtained by the FDTD method by simply forcing electric fields along a line to be zero [48]. The overhead line was modeled as a thin wire spanning across the x direction. For each arrangement of non-flat terrain, a different working volume was chosen. According to the available computational capability, a spatial resolution of 5 m was used. Nonetheless, in order to check the accuracy of this resolution, several simulations were done considering a spatial resolution of 2.5 m and getting relative errors on the peak value lower than 2%. In order to limit the computational space, all the cubic surfaces were treated as Liao's second-order
t z n Ez (i, j, k ) + t z n+ 1 2 t (H y 2 (i , j , k ) + t z) x
2µ y +
method is employed in this paper. The Maxwell equations could be solved in Cartesian Coordinates in three dimensions, by means of the Finite-Difference Time-Domain Method (FDTD) [52,53]. In total, six equations are updated: three for the electric fields (Ex, Ey, Ez) and three for the magnetic fields (Hx, Hy, Hz). The equations for calculating Ez and Hy in a non-homogeneous and anisotropic medium are as follows:
Ezn + 1 (i, j, k )=
(i , j , k )=
1, j, k )) 1, k ))
n+ 1 2 t Is 2 (i , j, k ) (2 z + t z ) x y
(1)
3
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cases were selected based on a real distribution line which feeds a rural zone in Colombia across a chain of mountains. Fig. 1 shows the online diagram of an 11.4 kV distribution circuit located in a mountainous rural zone of Colombia. Several segments of this line were selected to analyze its behavior due to the surrounding topography. One example is shown in Fig. 2 where the distribution line connects the tip of two mountains. The relief under each segment was approximated by a mathematical function and modeled in the FDTD code. The simulated line is a single overhead conductor without ground wires, with a length of 1 km, a radius of 5 mm and at a height of 10 m connected to its surge impedance at both ends. The DC resistance of the line was assumed as 0.5 Ω/km. It is an approximation to a typical distribution line of 11.4 kV. The line voltage and the insulator parameters were not included in the current simulation since the purpose of the study is observed the maximum induced overvoltage due to a nearby lightning. Additionally, the line was set straight on its route for each arrangement. Although, it is not a real condition, because the presence of the sag, it could give some insights of the behavior of lightning-induced overvoltages on overhead lines that cross several heights. The lightning-induced voltages were observed at both ends of the overhead line and compared with the results for a flat terrain configuration. The return stroke channel was assumed vertical and placed at several points toward the line at the highest point of the relief, this being the most likely point for a strike. It was represented by the MTLE model [55] with an attenuation constant λ = 2000 m and a return stroke velocity of 130 m/μs. The return stroke current at the channel base has a peak value of 12 kA and a maximum time-derivative of 40 kA/μs [56] which are the median values of the peak current and current derivative, respectively, of subsequent currents measured by Berger [57], the main reference worldwide on lightning parameters. 3.1. Case A The overhead line crosses a relief formed by a V-shaped configuration representing a river valley, as shown in Fig. 3(a). Fig. 3(b) and (c) present the plan view and the front view of the line, respectively. Several strike locations were simulated at a distance d from the left extremity of the line and at a distance r in front of the line. The inclination angle α was varied between 10° and 30° (the highest practical value of the slope). Additionally, conductivity values of 0.01 and 0.001 S/m were simulated which are typical values found in literature [17]. The lightning-induced voltages are obtained at four points of the line in Fig. 3(b) (the most relevant): the beginning of the line (B), the end of the line (E), the nearest to the strike location and at the middle of the line. The result considering α = 20°, d = 400 m and r = 100 m is shown in Fig. 4(a). For flat terrain is shown in Fig. 4(b). It is possible to see a high increase of the peak voltage in the waveforms for non-flat ground compared with the flat ground ones, in all the calculation points. As expected the high values of the induced voltage are seen at the points (N) and (M), for being closest to the strike. In this simulation, the maximum induced voltage on the irregular terrain is between 2.8 and 5.2 times higher than the calculated for flat terrain. The effect of the ground conductivity is to reduce the induced voltage at the beginning (B) and at the end of the line (E) and to increase the peak voltage at the points (N) and (M), especially for flat terrain; for non-flat ground, the effect is less pronounced. It is important to note, the high value of the induced voltage at the middle of the line (M) which is explained by the effect of the risers (the integral of the vertical electric field Ez below that point, which coincide with the deeper point of the terrain). Table 1 summarizes the maximum induced voltage observed along the distribution line, when the angle of the slope α in Fig. 3(c) varies between 0° (flat terrain) and 30° degrees, and the distance from the beginning of the line d changes between 0 m and 500 m, keeping r = 100 m. The voltage for flat terrain (α = 0°) is presented in kV and
Fig. 5. Case B. A line on the top of a V shape inverted terrain.
Absorbing Boundaries [54]. In the case of the terrain, it was modeled as staircases with defined properties of conductivity, permeability, and permittivity. 3. Simulated cases Three irregular terrains were chosen to evaluate the lightning-induced voltages on distribution lines placed on irregular terrains. These 4
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Fig. 6. Induced voltages at both line extremities of line in Case B (Fig. 5) with an inclination angle α = 30°, d = 200 m and r = 50 m.
induced voltages in a mountainous configuration are higher than for flat terrain. Previous works had found the enhancement of the lightning electromagnetic field in zones of non-flat grounds by means of numerical methods [33,34] and also experimentally [34]. Additionally, in [44] the calculation of lightning-induced voltages in non-flat terrains with rotational symmetry was done founding the dependence of the voltages upon the angle. The explanation in our case, according to the analyzed electric fields (some of them presented in Appendix B), is as follows: the vertical electric field Ez increases its value as far as the angle α rises. The same happens with the horizontal electric field Ex. Both fields Ez and Ex contribute to the total voltage along the line, depending especially on the location of the strike respect the extremities of the line (the effect of the risers are more important for measurements points near the strike points [7]). In the current case, the enhancement of the electric and magnetic fields plays an important role in the coupling mechanism increasing the induced voltage
Table 2 Summary of maximum lightning-induced voltages in kV obtained for Case B for flat terrain and the enhancement factor (ρ) varying d in Fig. 5b, keeping r = 100 m and considering lossy ground. σ (S/m)
α
d=0m
d = 200 m
d = 500 m
∞
0° 10° 20° 30°
23.2 kV 1.1 1.1 1.1
42.1 kV 1.2 1.2 1.2
46.3 kV 1.1 1.1 1.1
0.01
0° 10° 20° 30°
27.9 kV 1.0 1.0 1.1
56.4 kV 1.0 1.0 1.1
56.3 kV 1.0 1.0 1.1
0.001
0° 10° 20° 30°
40.0 kV 0.9 0.9 0.9
79.7 kV 1.0 1.1 1.1
80.5 kV 1.0 1.1 1.1
3.2. Case B
for the rest of the angles, the enhancement factor (ρ) is presented, which is the relation between maximum induced voltage on the line for non-flat terrain and for flat terrain:
=
Vmax Non flat terrain Vmax Flat terrain
In this case, the line is located on the top of a mountain with an inverted V shape as illustrated in Fig. 5(a). As depicted in Fig. 5(b), the return stroke channel is located at a distance d of the left extremity and at a distance r in front of the line. Fig. 6(a) indicates the induced voltage along the line for different ground conductivities when d = 200 m, r = 50 m and α = 30°, while the induced voltage for flat terrain is displayed in Fig. 6(b). In this case, the peak values are similar for nonflat terrain and for flat terrain. The ground conductivity as in case A, tends to reduce the peak voltage in all the observations points, except at (N) where the tendency is the opposite. Also observed is a change of the polarity as the ground conductivity is reduced. In order to analyze other strike locations in configuration B, Table 2 summarizes the induced voltages for flat terrain and the enhancement factor (ρ) for several values of distance d in Fig. 5(b), varying the ground conductivity, and taking r = 100 m. In general, factor ρ is toward the unity, reaching a maximum of 1.2 for the lossless ground case and 0.9 for the case of σ = 0.001 (S/m), which implies that the induced voltage is less than for flat terrain. In general, the extreme values are
(5)
This factor helps to see the effect of the shape of the mountain in the increase of the lightning-induced voltage if the ground conductivity is kept constant. Additionally, helps to understand what is the effort of the insulation if the line is located in a mountainous zone. For lossless ground (σ =∞) the factor (ρ) is higher, with maximum values of 5.2. For poor conductivities, it tends to decrease, reaching maximum values of 3.2 for σ = 0.001 S/m. Nevertheless, the peak voltages are maximum for this conductivity, values of 260 kV. According to Table 1, distance d has an important effect on the enhancement factor ρ, being maximum at 400 m and 500 m. This result could be explained by the fact that at a distance d of 400 m and 500 m, the lightning is almost at the middle of the line, where the highest magnitude of radiated electromagnetic field will be in the largest cross-section area of the line. In general, the 5
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Fig. 8. Induced voltages at both line extremities of line on Case C (Fig. 7) with an inclination angle α = 30°, d = 50 m and r = 100 m.
3.3. Case C In this case, the line crosses two mountains, each one with a slope of α. (see Fig. 7(a)). Some strikes were located at distances r and d of the beginning (B) of the line (see Fig. 7(b)). Taking r = 100 m, d = 50],m and α = 30°, the induced voltages at both line extremities ((B) and (E)) and at the middle of the line (M) for this configuration are shown in Fig. 8(a). (The nearest point (N) is not shown because the maximum voltage is at the point (B). Compared with the flat-ground case (Fig. 8(b)), the induced voltages are approximately 3.8 times greater than for perfect ground conductivity, For the other conductivities, this factor is 3.1 and 2.0. A change of polarity is noted at the point (B) and (E), but less pronounced than for flat terrain. The enhancement factor ρ for distances r of 0 and 100 and d = 50 and 100 m is presented in Table 3. In general, there is a tendency of decrease of (ρ) for poor conductivities, with maximum values for perfect ground conductivities. Again as in case A, the maximum voltage is dependent on the angle of the terrain α. The distance d and r influence also the maximum induced voltage along the line, according to the projected incidence angle [58]. The explanation of the results is as follows: the vertical electric field Ez reduces its value as far as the angle of the terrain α increases. (the tendency at the end of the line is the
Fig. 7. Case C. Line between two mountains.
found when a polarity change is present. Again, the explanation of the results is given analyzing the strong influence of the electric fields in the lightning-induced voltage. In Appendix B, is shown that the vertical electric field Ez increases its value, while the horizontal electric field Ex decreases slightly in function of the angle α. For this reason, the induced voltage tends to reduce slightly its value too. In this case, the shape of the terrain does not influence significantly, the wave-shape and peak value of the lightninginduced voltage. 6
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E. Soto and E. Perez
and Ex found for some of the simulations done for Case C are presented.
Table 3 Summary of maximum lightning-induced voltages in kV obtained for Case C for flat terrain and the enhancement factor (ρ) varying r and d in Fig. 7(b) and considering lossy ground. r=0m
4. Conclusions Using a 3D FDTD method, the lightning-induced voltages on a distribution line placed on a non-flat terrain was obtained. Three arrangements of irregular terrains were used to observe the effect of this topography. The results showed that the topography has a significant influence on the induced voltage amplitude, where two out of the three configurations used present an enhancement which could be up to 5.2 times the maximum induced voltage calculated for flat terrain. The enhancement factor depends considerably on the stroke location, the slope angle and the ground conductivity. In general, the effect of finite conductivity is to reduce the (ρ) value. In case A, the lightning-induced voltages achieve values between 1.5 and 5.2 higher than for flat terrain. In case B, a slightly change of the lightning-induced voltage for the nonflat terrain compared with the flat terrain case is observed. In case C, the induced voltages for irregular terrain are greater than those found for flat terrain, with enhancement factors (ρ) between 1.1 and 4.2. The reduction of (ρ) for finite conductivity is more important in this case (approximately half in some cases). According to the results of this paper, it is expected that the insulation requirements of lines located in mountainous zones are higher than the located in flat terrain. Nevertheless, it is important to make a complete Monte Carlo simulation in the presented configurations that enables to conclude about required insulation levels of lines over irregular terrains.
r = 100 m
σ (S/m) ∞
α 0° 10° 20° 30°
d = 50 m 36.1 kV 1.4 1.8 2.0
100 m 17.9 kV 1.9 2.7 3.2
50 m 17.8 kV 2.3 3.2 3.8
100 m 13.2 kV 2.4 3.5 4.2
0.01
0° 10° 20° 30°
44.7 kV 1.3 1.5 1.7
23.0 kV 1.7 2.2 2.6
22.5 kV 2.0 2.7 3.1
17.4 kV 2.0 2.8 3.3
0.001
0° 10° 20° 30°
64.1 kV 1.1 1.3 1.4
44.7 kV 1.1 1.3 1.4
36.2 kV 1.4 1.8 2.0
35.1 kV 1.2 1.5 1.7
opposite). The horizontal electric field Ex tends to increases in function of the angle of the terrain (at point B, the field changes its polarity). Taking into account the important influence of the field Ex on the coupling mechanism, the lightning-induced voltage follows the same tendency of this field. In the case of finite conductivity, the fields Ez and Ex decrease its values for lower conductivities, which also diminish the induced voltage value. In Appendix B, the calculated electric fields Ez
Appendix A. Validation of lightning-induced voltage calculation for flat ground With the aim of validating the developed FDTD code to calculate lightning-induced voltages, it is simulated a distribution line 1000 m long (see Fig. 9), 10 m high connected to a resistance of 500 Ω (approaching to the surge impedance of the line) at both extremities and above a perfectly conducting flat ground plane. The return stroke channel is located 50 m in front of the middle of the line. The current at the channel base and the return stroke model are the same as the described in Section 3. Fig. 10(a) and (b) shows the induced voltage at both line extremities and at the middle of line, respectively, calculated by two methodologies: (1) using Agrawal coupling model and calculating electric fields by means of Master and Uman equations [9] (Agrawal) and (2) 3D FDTD method to calculate both electric fields and induced voltages (3D-FDTD). It is seen a good agreement between the two methodologies, with a small difference in the wave tail, explained by the non-perfect absorption of Liao's boundaries.
Fig. 9. Arrangement of flat terrain to validate FDTD code.
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Fig. 10. Induced Voltages at the line in Fig. 9.
Appendix B. Calculation of the electric fields for each case In this section, it is presented the vertical Ez and horizontal Ex electric fields calculated in some of the presented arrangements.
Fig. 11. Vertical electric field Ez at the beginning of the line in Fig. 3(b), taking d = 200 m, r = 100 m and σ =∞. 8
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E. Soto and E. Perez
B.1 Case A The calculated electric fields Ez and Ex for case A at the beginning of the line taking d = 200 m, r = 100 m and σ =∞ in Fig. 3(b) are presented in Figs. 11 and 12 , respectively.
Fig. 12. Horizontal electric field Ex at the beginning of the line in Fig. 3(b), taking d = 200 m, r = 100 m and σ =∞.
B.2 Case B The calculated electric fields Ez and Ex for case B at the beginning of the line in Fig. 5(b), taking d = 200 m, r = 100 m and σ =∞, are presented in Figs. 13 and 14 , respectively.
Fig. 13. Vertical electric field Ez at the beginning of the line in Fig. 5(b), taking d = 200 m, r = 100 m and σ =∞.
Fig. 14. Horizontal electric field Ex at the beginning of the line in Fig. 5(b), taking d = 200 m, r = 100 m and σ =∞. 9
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B.3 Case C The calculated electric fields Ez and Ex for case C at the beginning of the line in Fig. 7(b), taking d = 100 m, r = 50 m and σ =∞, are presented in Figs. 15 and 16 , respectively.
Fig. 15. Vertical electric field Ez at the beginning of the line in Fig. 7(b), taking d = 100 m, r = 50 m and σ =∞.
Fig. 16. Horizontal electric field Ex at the beginning of the line in Fig. 7(b), taking d = 100 m, r = 50 m and σ =∞.
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Lightning-induced voltages on overhead lines over irregular terrains a,⁎
Edison Soto , Ernesto Perez a b
b
T
Escuela de Ingeniería Eléctrica, Electrónica y de Telecomunicaciones, Universidad Industrial de Santander, Bucaramanga, Colombia Departamento de Energía Eléctrica y Automática. Universidad Nacional de Colombia, Medellín, Colombia
ARTICLE INFO
ABSTRACT
Keywords: Finite-Difference Time-Domain (FDTD) method Electromagnetic fields Lightning-induced voltages Mountains
This paper analyzes lightning-induced voltages on distribution lines located on irregular terrains since it is common, in some places in the world, that real distribution systems are placed on those terrains. Simplified topographies were used in this analysis to approximate to those found on real distribution systems. The lightning-induced voltages were calculated with a full-wave analysis approach based on the 3D Finite-Difference Time-Domain (FDTD) method. It was found that lightning-induced voltages on overhead lines above irregular topographies, in the majority of cases, are greater than those found for flat terrain. The enhancement factor can reach values up to 5.2 times. In a few cases, factors below unity were found. These results may give a new insight on the design of overhead distribution lines for improving lightning performance on real distribution networks.
1. Introduction Lightning-induced voltages are one of the major issues on overhead distribution lines. These kind of lines, due to its reduced insulation level compared with transmission lines, experiment overvoltages produced by indirect lightning that cause insulation flashover and frequently outages. Modeling the mechanism of induction of the electromagnetic field produced by lightning has been an important task in the past [1–4]. The calculation of induced voltages is, commonly, based on the computation of three steps, which include, the representation of the current along the lightning channel, the electromagnetic field calculation and the coupling mechanism to the conductors [5,3,6,7]. The current distribution along the lightning channel has been represented with different types of models which aim to reproduce the return stroke behavior, being more used for induced voltage calculation, the so-called “engineering models” [7]. The electromagnetic fields equations have, traditionally, been derived by analytical representation of antenna theory using dipoles or monopoles above a flat terrain with finite [8] or infinite ground conductivity [9,10]. Since the horizontal component of the electric field is highly affected by the ground conductivity, several authors have contributed equations that represent this effect [11–16]. These equations have been widely used to calculate the lightning electromagnetic pulse (LEMP) [7,17]. In order to represent more accurately the complexity of the lightning phenomenon, several numerical methods have been used to compute the LEMP in more complex scenarios: the FDTD method [18–22], the method of moments (MoM), applied to the calculation of
⁎
electromagnetic fields produced by inclined lightning channels [23] and lightning channel tortuosity [24] and, the hybrid electromagnetic/ circuit model (HEM) [25,26]. Recent studies have conducted investigations to analyze the effect of the LEMP due to different considerations, such as the characteristics of the lightning channel [23,27–29], the presence of nearby towers [30,31] or buildings [32], among others. Some other works have analyzed the effect of irregular terrains on the electric fields produced by lightning, showing an enhancement of the electric field [33,34]. The interaction of the LEMP with overhead distribution lines can be analyzed by two methodologies: (i) a transmission line approach or (ii) full-wave analysis. In the first methodology, the electromagnetic field is calculated without considering the distribution line, and then, a coupling model is used to calculate the induced voltage along the line [3,35]. The majority of works that use this methodology, have assumed lines over flat terrain [6,36–43], though recently, it has been considered the effect of irregular terrains [44,45,16]. On the other hand, the methodology of full-wave analysis incorporates the conductors in the electromagnetic field calculation and directly calculates the induced voltage [46,47]. It has been used to analyze different aspects that were difficult to consider with the other approach: modeling corona in the conductors, take into account the effect of nearby buildings or tall structures, among others. Nevertheless, the developed works have only considered overhead lines above flat ground planes [48,49,38,50,51]. In many places around the world, the distribution lines are built over mountains. Since, electromagnetic field propagation depends on topography [33,34] and hence lightning-induced voltages, it is
Corresponding author. E-mail addresses: [email protected] (E. Soto), [email protected] (E. Perez).
https://doi.org/10.1016/j.epsr.2019.105941 Received 14 November 2018; Received in revised form 23 March 2019; Accepted 9 July 2019 Available online 05 August 2019 0378-7796/ © 2019 Elsevier B.V. All rights reserved.
Electric Power Systems Research 176 (2019) 105941
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Fig. 1. Simulated circuit.
Fig. 3. Case A. Line that crosses a V-shaped valley.
Fig. 2. Example of a irregular terrain.
discussion are presented in Section 4.
important to analyze the effect of terrain shape on the calculation of lightning-induced voltages on distribution lines that may give new insight about the design of distribution networks located irregular terrains. In this paper, lightning-induced voltages calculated on overhead distribution lines located over three arrangements of irregular terrains similar to those found in rural zones of Colombia are presented, using a developed code of full-wave analysis based on the 3D-FDTD method. Conclusions about the differences in overvoltages found on lines placed over flat terrains are given. This paper is organized as follows: Section 2 provides the methodology used for lightning-induced voltage calculation. The test cases are detailed in Section 3. Finally, conclusions and
2. Methodology In order to calculate lightning-induced voltages [7] on overhead lines located over non-flat ground is important to take into consideration two issues: (a) the analytical formulation of electromagnetic field produced by lightning developed by Master and Uman [9] are only valid for flat terrain and (b) the uncertainty of the performance of the Transmission Line Model to calculate lightning-induced voltages on lines placed over irregular terrains. For these reasons a numerical methodology based on full-wave analysis performed with the 3D-FDTD
2
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Fig. 4. Lightning-induced voltages at both line extremities for Case A (Fig. 4) for an inclination angle α = 20°, a distance d = 400 m and r = 100 m. Table 1 Summary of maximum lightning-induced voltages in kV obtained for Case A for flat terrain and the enhancement factor (ρ) varying d and α in Fig. 3, keeping r = 100 m and considering lossy ground. σ (S/m)
α
d=0m
200 m
400 m
500 m
0° 10° 20° 30°
23.2 kV 2.0 2.6 2.9
42.1 kV 3.5 4.6 5.2
46.3 kV 4.5 5.2 5.2
46.3 kV 4.7 5.2 5.2
0.01
0° 10° 20° 30°
27.9 kV 1.8 2.2 2.5
56.4 kV 2.7 3.5 3.8
56.4 kV 3.8 4.3 4.4
56.3 kV 3.9 4.3 4.4
0.001
0° 10° 20° 30°
40.0 kV 1.5 1.8 2.0
79.7 kV 2.1 2.7 2.9
80.5 kV 2.8 3.1 3.2
80.5 kV 2.9 3.2 3.2
∞
n+ 1 2
Hy
2 2 +
z z
Hy
n+ 1 2 t (Hx 2 (i , j, k ) (2 z + t z ) y
Hx
(2
z
n+ 1 2 n+ 1 2
(i (i , j
t
y
2µ y + t
y
n 1 2
Hy
(i , j , k )
2 t (Ezn (i + 1, j, k ) (2µ y + t y ) y
Ezn (i , j, k ))
2 t (Exn (i , j, k + 1) (2µ y + t y ) z
Exn (i , j, k ))
(2)
where: εu, σu, μu are the permittivity, conductivity and permeability in each direction respectively, at the location of the associated calculated field. Δx, Δy and Δz are the space step in x, y and z respectively of the FDTD method. Δt is the time step used in the FDTD method. Is represents the return stroke current. The overhead distribution line is represented as a Thin Wire according to [48,49]. The electric fields along the line are forced to be zero, and the parameters μm and εm around the wire are updated differently than the rest of the working space according to:
µm = µ 0 / m ,
m
=m
0
(3)
where
m=
1.471 s
ln( r )
(4)
and Δs is the spatial step and r is the radius of the conductor. The intrinsic radius r0 for this method is 0.2298Δs. It is defined as the radius for which the real distribution of electric and magnetic fields around the wire is the same as the one obtained by the FDTD method by simply forcing electric fields along a line to be zero [48]. The overhead line was modeled as a thin wire spanning across the x direction. For each arrangement of non-flat terrain, a different working volume was chosen. According to the available computational capability, a spatial resolution of 5 m was used. Nonetheless, in order to check the accuracy of this resolution, several simulations were done considering a spatial resolution of 2.5 m and getting relative errors on the peak value lower than 2%. In order to limit the computational space, all the cubic surfaces were treated as Liao's second-order
t z n Ez (i, j, k ) + t z n+ 1 2 t (H y 2 (i , j , k ) + t z) x
2µ y +
method is employed in this paper. The Maxwell equations could be solved in Cartesian Coordinates in three dimensions, by means of the Finite-Difference Time-Domain Method (FDTD) [52,53]. In total, six equations are updated: three for the electric fields (Ex, Ey, Ez) and three for the magnetic fields (Hx, Hy, Hz). The equations for calculating Ez and Hy in a non-homogeneous and anisotropic medium are as follows:
Ezn + 1 (i, j, k )=
(i , j , k )=
1, j, k )) 1, k ))
n+ 1 2 t Is 2 (i , j, k ) (2 z + t z ) x y
(1)
3
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cases were selected based on a real distribution line which feeds a rural zone in Colombia across a chain of mountains. Fig. 1 shows the online diagram of an 11.4 kV distribution circuit located in a mountainous rural zone of Colombia. Several segments of this line were selected to analyze its behavior due to the surrounding topography. One example is shown in Fig. 2 where the distribution line connects the tip of two mountains. The relief under each segment was approximated by a mathematical function and modeled in the FDTD code. The simulated line is a single overhead conductor without ground wires, with a length of 1 km, a radius of 5 mm and at a height of 10 m connected to its surge impedance at both ends. The DC resistance of the line was assumed as 0.5 Ω/km. It is an approximation to a typical distribution line of 11.4 kV. The line voltage and the insulator parameters were not included in the current simulation since the purpose of the study is observed the maximum induced overvoltage due to a nearby lightning. Additionally, the line was set straight on its route for each arrangement. Although, it is not a real condition, because the presence of the sag, it could give some insights of the behavior of lightning-induced overvoltages on overhead lines that cross several heights. The lightning-induced voltages were observed at both ends of the overhead line and compared with the results for a flat terrain configuration. The return stroke channel was assumed vertical and placed at several points toward the line at the highest point of the relief, this being the most likely point for a strike. It was represented by the MTLE model [55] with an attenuation constant λ = 2000 m and a return stroke velocity of 130 m/μs. The return stroke current at the channel base has a peak value of 12 kA and a maximum time-derivative of 40 kA/μs [56] which are the median values of the peak current and current derivative, respectively, of subsequent currents measured by Berger [57], the main reference worldwide on lightning parameters. 3.1. Case A The overhead line crosses a relief formed by a V-shaped configuration representing a river valley, as shown in Fig. 3(a). Fig. 3(b) and (c) present the plan view and the front view of the line, respectively. Several strike locations were simulated at a distance d from the left extremity of the line and at a distance r in front of the line. The inclination angle α was varied between 10° and 30° (the highest practical value of the slope). Additionally, conductivity values of 0.01 and 0.001 S/m were simulated which are typical values found in literature [17]. The lightning-induced voltages are obtained at four points of the line in Fig. 3(b) (the most relevant): the beginning of the line (B), the end of the line (E), the nearest to the strike location and at the middle of the line. The result considering α = 20°, d = 400 m and r = 100 m is shown in Fig. 4(a). For flat terrain is shown in Fig. 4(b). It is possible to see a high increase of the peak voltage in the waveforms for non-flat ground compared with the flat ground ones, in all the calculation points. As expected the high values of the induced voltage are seen at the points (N) and (M), for being closest to the strike. In this simulation, the maximum induced voltage on the irregular terrain is between 2.8 and 5.2 times higher than the calculated for flat terrain. The effect of the ground conductivity is to reduce the induced voltage at the beginning (B) and at the end of the line (E) and to increase the peak voltage at the points (N) and (M), especially for flat terrain; for non-flat ground, the effect is less pronounced. It is important to note, the high value of the induced voltage at the middle of the line (M) which is explained by the effect of the risers (the integral of the vertical electric field Ez below that point, which coincide with the deeper point of the terrain). Table 1 summarizes the maximum induced voltage observed along the distribution line, when the angle of the slope α in Fig. 3(c) varies between 0° (flat terrain) and 30° degrees, and the distance from the beginning of the line d changes between 0 m and 500 m, keeping r = 100 m. The voltage for flat terrain (α = 0°) is presented in kV and
Fig. 5. Case B. A line on the top of a V shape inverted terrain.
Absorbing Boundaries [54]. In the case of the terrain, it was modeled as staircases with defined properties of conductivity, permeability, and permittivity. 3. Simulated cases Three irregular terrains were chosen to evaluate the lightning-induced voltages on distribution lines placed on irregular terrains. These 4
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Fig. 6. Induced voltages at both line extremities of line in Case B (Fig. 5) with an inclination angle α = 30°, d = 200 m and r = 50 m.
induced voltages in a mountainous configuration are higher than for flat terrain. Previous works had found the enhancement of the lightning electromagnetic field in zones of non-flat grounds by means of numerical methods [33,34] and also experimentally [34]. Additionally, in [44] the calculation of lightning-induced voltages in non-flat terrains with rotational symmetry was done founding the dependence of the voltages upon the angle. The explanation in our case, according to the analyzed electric fields (some of them presented in Appendix B), is as follows: the vertical electric field Ez increases its value as far as the angle α rises. The same happens with the horizontal electric field Ex. Both fields Ez and Ex contribute to the total voltage along the line, depending especially on the location of the strike respect the extremities of the line (the effect of the risers are more important for measurements points near the strike points [7]). In the current case, the enhancement of the electric and magnetic fields plays an important role in the coupling mechanism increasing the induced voltage
Table 2 Summary of maximum lightning-induced voltages in kV obtained for Case B for flat terrain and the enhancement factor (ρ) varying d in Fig. 5b, keeping r = 100 m and considering lossy ground. σ (S/m)
α
d=0m
d = 200 m
d = 500 m
∞
0° 10° 20° 30°
23.2 kV 1.1 1.1 1.1
42.1 kV 1.2 1.2 1.2
46.3 kV 1.1 1.1 1.1
0.01
0° 10° 20° 30°
27.9 kV 1.0 1.0 1.1
56.4 kV 1.0 1.0 1.1
56.3 kV 1.0 1.0 1.1
0.001
0° 10° 20° 30°
40.0 kV 0.9 0.9 0.9
79.7 kV 1.0 1.1 1.1
80.5 kV 1.0 1.1 1.1
3.2. Case B
for the rest of the angles, the enhancement factor (ρ) is presented, which is the relation between maximum induced voltage on the line for non-flat terrain and for flat terrain:
=
Vmax Non flat terrain Vmax Flat terrain
In this case, the line is located on the top of a mountain with an inverted V shape as illustrated in Fig. 5(a). As depicted in Fig. 5(b), the return stroke channel is located at a distance d of the left extremity and at a distance r in front of the line. Fig. 6(a) indicates the induced voltage along the line for different ground conductivities when d = 200 m, r = 50 m and α = 30°, while the induced voltage for flat terrain is displayed in Fig. 6(b). In this case, the peak values are similar for nonflat terrain and for flat terrain. The ground conductivity as in case A, tends to reduce the peak voltage in all the observations points, except at (N) where the tendency is the opposite. Also observed is a change of the polarity as the ground conductivity is reduced. In order to analyze other strike locations in configuration B, Table 2 summarizes the induced voltages for flat terrain and the enhancement factor (ρ) for several values of distance d in Fig. 5(b), varying the ground conductivity, and taking r = 100 m. In general, factor ρ is toward the unity, reaching a maximum of 1.2 for the lossless ground case and 0.9 for the case of σ = 0.001 (S/m), which implies that the induced voltage is less than for flat terrain. In general, the extreme values are
(5)
This factor helps to see the effect of the shape of the mountain in the increase of the lightning-induced voltage if the ground conductivity is kept constant. Additionally, helps to understand what is the effort of the insulation if the line is located in a mountainous zone. For lossless ground (σ =∞) the factor (ρ) is higher, with maximum values of 5.2. For poor conductivities, it tends to decrease, reaching maximum values of 3.2 for σ = 0.001 S/m. Nevertheless, the peak voltages are maximum for this conductivity, values of 260 kV. According to Table 1, distance d has an important effect on the enhancement factor ρ, being maximum at 400 m and 500 m. This result could be explained by the fact that at a distance d of 400 m and 500 m, the lightning is almost at the middle of the line, where the highest magnitude of radiated electromagnetic field will be in the largest cross-section area of the line. In general, the 5
Electric Power Systems Research 176 (2019) 105941
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Fig. 8. Induced voltages at both line extremities of line on Case C (Fig. 7) with an inclination angle α = 30°, d = 50 m and r = 100 m.
3.3. Case C In this case, the line crosses two mountains, each one with a slope of α. (see Fig. 7(a)). Some strikes were located at distances r and d of the beginning (B) of the line (see Fig. 7(b)). Taking r = 100 m, d = 50],m and α = 30°, the induced voltages at both line extremities ((B) and (E)) and at the middle of the line (M) for this configuration are shown in Fig. 8(a). (The nearest point (N) is not shown because the maximum voltage is at the point (B). Compared with the flat-ground case (Fig. 8(b)), the induced voltages are approximately 3.8 times greater than for perfect ground conductivity, For the other conductivities, this factor is 3.1 and 2.0. A change of polarity is noted at the point (B) and (E), but less pronounced than for flat terrain. The enhancement factor ρ for distances r of 0 and 100 and d = 50 and 100 m is presented in Table 3. In general, there is a tendency of decrease of (ρ) for poor conductivities, with maximum values for perfect ground conductivities. Again as in case A, the maximum voltage is dependent on the angle of the terrain α. The distance d and r influence also the maximum induced voltage along the line, according to the projected incidence angle [58]. The explanation of the results is as follows: the vertical electric field Ez reduces its value as far as the angle of the terrain α increases. (the tendency at the end of the line is the
Fig. 7. Case C. Line between two mountains.
found when a polarity change is present. Again, the explanation of the results is given analyzing the strong influence of the electric fields in the lightning-induced voltage. In Appendix B, is shown that the vertical electric field Ez increases its value, while the horizontal electric field Ex decreases slightly in function of the angle α. For this reason, the induced voltage tends to reduce slightly its value too. In this case, the shape of the terrain does not influence significantly, the wave-shape and peak value of the lightninginduced voltage. 6
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E. Soto and E. Perez
and Ex found for some of the simulations done for Case C are presented.
Table 3 Summary of maximum lightning-induced voltages in kV obtained for Case C for flat terrain and the enhancement factor (ρ) varying r and d in Fig. 7(b) and considering lossy ground. r=0m
4. Conclusions Using a 3D FDTD method, the lightning-induced voltages on a distribution line placed on a non-flat terrain was obtained. Three arrangements of irregular terrains were used to observe the effect of this topography. The results showed that the topography has a significant influence on the induced voltage amplitude, where two out of the three configurations used present an enhancement which could be up to 5.2 times the maximum induced voltage calculated for flat terrain. The enhancement factor depends considerably on the stroke location, the slope angle and the ground conductivity. In general, the effect of finite conductivity is to reduce the (ρ) value. In case A, the lightning-induced voltages achieve values between 1.5 and 5.2 higher than for flat terrain. In case B, a slightly change of the lightning-induced voltage for the nonflat terrain compared with the flat terrain case is observed. In case C, the induced voltages for irregular terrain are greater than those found for flat terrain, with enhancement factors (ρ) between 1.1 and 4.2. The reduction of (ρ) for finite conductivity is more important in this case (approximately half in some cases). According to the results of this paper, it is expected that the insulation requirements of lines located in mountainous zones are higher than the located in flat terrain. Nevertheless, it is important to make a complete Monte Carlo simulation in the presented configurations that enables to conclude about required insulation levels of lines over irregular terrains.
r = 100 m
σ (S/m) ∞
α 0° 10° 20° 30°
d = 50 m 36.1 kV 1.4 1.8 2.0
100 m 17.9 kV 1.9 2.7 3.2
50 m 17.8 kV 2.3 3.2 3.8
100 m 13.2 kV 2.4 3.5 4.2
0.01
0° 10° 20° 30°
44.7 kV 1.3 1.5 1.7
23.0 kV 1.7 2.2 2.6
22.5 kV 2.0 2.7 3.1
17.4 kV 2.0 2.8 3.3
0.001
0° 10° 20° 30°
64.1 kV 1.1 1.3 1.4
44.7 kV 1.1 1.3 1.4
36.2 kV 1.4 1.8 2.0
35.1 kV 1.2 1.5 1.7
opposite). The horizontal electric field Ex tends to increases in function of the angle of the terrain (at point B, the field changes its polarity). Taking into account the important influence of the field Ex on the coupling mechanism, the lightning-induced voltage follows the same tendency of this field. In the case of finite conductivity, the fields Ez and Ex decrease its values for lower conductivities, which also diminish the induced voltage value. In Appendix B, the calculated electric fields Ez
Appendix A. Validation of lightning-induced voltage calculation for flat ground With the aim of validating the developed FDTD code to calculate lightning-induced voltages, it is simulated a distribution line 1000 m long (see Fig. 9), 10 m high connected to a resistance of 500 Ω (approaching to the surge impedance of the line) at both extremities and above a perfectly conducting flat ground plane. The return stroke channel is located 50 m in front of the middle of the line. The current at the channel base and the return stroke model are the same as the described in Section 3. Fig. 10(a) and (b) shows the induced voltage at both line extremities and at the middle of line, respectively, calculated by two methodologies: (1) using Agrawal coupling model and calculating electric fields by means of Master and Uman equations [9] (Agrawal) and (2) 3D FDTD method to calculate both electric fields and induced voltages (3D-FDTD). It is seen a good agreement between the two methodologies, with a small difference in the wave tail, explained by the non-perfect absorption of Liao's boundaries.
Fig. 9. Arrangement of flat terrain to validate FDTD code.
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Fig. 10. Induced Voltages at the line in Fig. 9.
Appendix B. Calculation of the electric fields for each case In this section, it is presented the vertical Ez and horizontal Ex electric fields calculated in some of the presented arrangements.
Fig. 11. Vertical electric field Ez at the beginning of the line in Fig. 3(b), taking d = 200 m, r = 100 m and σ =∞. 8
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B.1 Case A The calculated electric fields Ez and Ex for case A at the beginning of the line taking d = 200 m, r = 100 m and σ =∞ in Fig. 3(b) are presented in Figs. 11 and 12 , respectively.
Fig. 12. Horizontal electric field Ex at the beginning of the line in Fig. 3(b), taking d = 200 m, r = 100 m and σ =∞.
B.2 Case B The calculated electric fields Ez and Ex for case B at the beginning of the line in Fig. 5(b), taking d = 200 m, r = 100 m and σ =∞, are presented in Figs. 13 and 14 , respectively.
Fig. 13. Vertical electric field Ez at the beginning of the line in Fig. 5(b), taking d = 200 m, r = 100 m and σ =∞.
Fig. 14. Horizontal electric field Ex at the beginning of the line in Fig. 5(b), taking d = 200 m, r = 100 m and σ =∞. 9
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B.3 Case C The calculated electric fields Ez and Ex for case C at the beginning of the line in Fig. 7(b), taking d = 100 m, r = 50 m and σ =∞, are presented in Figs. 15 and 16 , respectively.
Fig. 15. Vertical electric field Ez at the beginning of the line in Fig. 7(b), taking d = 100 m, r = 50 m and σ =∞.
Fig. 16. Horizontal electric field Ex at the beginning of the line in Fig. 7(b), taking d = 100 m, r = 50 m and σ =∞.
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