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TIME DOMAIN SOLUTIONS FOR TRANSIENT RESPONSE OF TRANSMISSION TOWER STRUCTURES AND ASSOCIATED GROUNDING SYSTEMS Boris Caric*, Selva Moorthy*, Srete Nikolovski** and Damir Sljivac** *

Department of Electrical Engineering Royal Melbourne Institute of Technology GPO Box 2476V, Melbourne, Vic. 3001, Australia **

Faculty of Electrical Engineering University of Osijek Kneza Trpimira 2B, 31 000 Osijek, Croatia Abstract Transient behaviour of transmission towers and associated grounding systems exposed to a lightning surge current having a double exponential pulse waveshape is investigated and analyzed. Frequency domain solutions for electromagnetic fields generated by the network of overhead and buried cylindrical conductors subject to the thin – wire approximation are derived based on the field theory approach. Once the scalar potentials and electromagnetic fields in the vicinity of the tower have been computed at selected frequencies, the time domain response is obtained using the inverse Fast Fourier Transform. Simulation models are obtained by the use of HIFREQ and FFTSES modules of the CDEGS (Current Distribution, Electromagnetic Fields, Grounding and Soil Structure Analysis) software package. 1.

INTRODUCTION

Elevated structures such as transmission line towers and telecommunication centers are frequently exposed to lightning strikes, resulting in ultra fast transient electromagnetic fields near the strike site. The design of lightning protection systems, using modern electronic equipment inherently susceptible to voltage and current surges, has become rigorous lately, making the analysis of transient electromagnetic response of tower structures exposed to a direct lightning strike increasingly important [6]. This paper presents solutions for electromagnetic fields radiated by an energized network modeled by an ensemble of cylindrical metallic conductors having arbitrary orientation. The conductors are assumed to be subject to the thin – wire approximation, which reduces the original two – dimensional analysis to one – dimensional line integration. This approximation makes it permissible to assume that the longitudinal current has negligible circumferential variation so that conductor current and charge distribution can be replaced by filaments of current and charge along the wire axis, while the boundary condition is matched at the wire surface [4]. It, however, does not imply that the finite conductor thickness and the radial leakage currents along buried conductors are neglected [7]. The simulation models are developed for the case of a 400 kV, 39 m tall transmission tower connected to two different grounding grids and subjected to a 20 kA current surge.

The lightning surge current considered in the simulation is given by the following double – exponential type function:

[

i (t ) = I m e − α ⋅ t − e − β ⋅ t

]

(1)

where: I m = 20 kA , α = 1.386 × 104 s −1 and β = 6 × 106 s −1 The waveform is characterized by the rise time of 1.2 µs and a half value time of 50 µs . The current distribution in the tower and the associated grounding network, the scalar potential and the electromagnetic fields are computed at the selected frequencies using the HIFREQ module of the CDEGS software package [5]. The time domain response is recovered using the inverse Fast Fourier Transform in the FFTSES module. The computer program procedure is only briefly discussed as it has been comprehensively treated in [10]. 2.

INTEGRAL EQUATIONS

Figure 1 illustrates two conductor segments of the given network in free space. The position on the axis of segment m with respect to the origin of the general ρ coordinate system is given by r ′ and that of a point ρ on the surface of segment n (observation point) by r . ρ ρ Unit vectors t ′ and t denote the unit vector along

the axis of segment m and the unit vector tangential to the surface of segment n, respectively. l

ρ ρ ρ

Segment m

ρ

Z s (ω ) =

t

ρ

r

Segment n

The total electric field E at the observation point can ρ be expressed as a sum of an incident field E I and a ρ scattered field E S . Assuming a perfectly conducting body, the field component tangential to the conductor surface is expected to vanish: ρ

[ρ

ρ

ρ

]

ρ

(2)

The time harmonic incident field at the specific frequency ω is given by: ρ ρ

 jωµ 0 ρ  1 t ⋅  2 ∇∇ + I  ⋅ 4π  k0 

ρ

t ⋅ E I (r ,ω ) =

ρ

ρ

⋅ ∫ t ′I l ( r ′, ω ) l

ρ

(3)

ρ

where I is the unit dyad, R = r − r ′ and k 0 = ω c . A brief description of the steps involved in the derivation is given in Appendix A. The corresponding integral equation in time domain may be expressed in the form [9]: ρ

ρ ρI ρ

t µ ∂ ρ ρ t ⋅ E (r , t) = ⋅∫  0 t ′I l ( r ′,τ ) − 4π l  R ∂τ ρ

ρ

ρ

where retarded time is τ = t − R c and where axial ρ

ρ

current t ′I l (r ′) and linear charge density σ l (r ′) at ρ r ′ are related through the continuity equation: ρ

ρ

∇ ⋅ t ′I l ( r ′, t ) +

ρ ∂ σ l ( r ′, t ) = 0 ∂t

ρ ρ ρ

t ⋅ E(r , t) =

t

ρ

∫ z (τ ) I ( r ,τ − t )dτ s

(8)

−∞

where z s (t ) represents the inverse Fourier transform of the surface impedance function Z s (ω ) . The integral equation for current on the lossy conductor is obtained by adding the convolution term in (8) to the right hand side of (4). FREQUENCY DOMAIN ANALYSIS

The scattered electric field appearing in (2) results from the currents and charges induced on the conductor by the incident field. For conductors immersed in an infinite conducting medium it is expressed in the form: ρ

ρ

E S ( r ,ω ) = −

S ρ ρ ρ jωµ ρ ⋅ ∫ t ′ ⋅ I l (r ′, ω )G ∞ ( r , r ′)dl 4π l

(9)

The infinite space dyadic Green’s function in (9) is given by:

σ ( r ′,τ ) R ρ 1 ∂ R  − l ⋅ 3 − σ l ( r ′,τ ) ⋅ 2 dl (4) ε0 ε 0 ∂τ R cR  ρ

In time domain, (6) may be written as [12]:

3. e − jk0 R dl R

(7)

metallic conductors, in which case λ2c ≅ jωµcκ c .

ρ

t ⋅ E ( r ) = t ⋅ E I (r ) + E S (r ) = 0

J (λ a ) λc ⋅ 0 c 2πaκ c J 1 (λc a )

where J0 and J1 denote Bessel functions of the first kind, order zero and one and a is the conductor radius. The approximation κ c ωε c >> 1 is valid for all

Figure 1 – Conductor segments in free space

ρ ρ ρ

(6)

The surface impedance appearing in (6) is obtained in the form [11]:

t′

0

ρ

t ⋅ E ( r , ω ) = I ( r , ω ) ⋅ Z s (ω )

ρ

ρ

r′

For a conductor having finite conductivity κc, the total tangential field at the segment surface equals the internal voltage drop per unit length along the segment:

(5)

ρ ρ ρ ρ G ∞ ( r , r ′) =  ∇∇ k 2 + I  ⋅ g ∞ ( r , r ′)   S

where: ρ ρ

g ∞ ( r , r ′) = e − jkR R I = xˆxˆ + yˆyˆ + zˆzˆ k = ω 2 µε ε = ε + κ jω

(10)

ρ ρ

In the previous expressions, g ∞ ( r , r ′) denotes an unbounded medium Green’s function and ε is the complex conductivity of the lossy medium.

the current conducted to the network at n = 1 ) and z n1 ⋅ I 1 + z n 2 ⋅ I 2 + Κ + z nN ⋅ I N = 0 for 2 ≤ n ≤ N , the impedance matrix elements are expressed as [7]:

3.1 Conductors in Presence of a Lossy Interface

z nm =

For the conductors near a conducting boundary separating two linear and isotropic mediums, say earth and air, the expression (2) needs to be modified to include the field component scattered from the interface: ρ ρ

ρ ρ

ρ

ρ ρ

ρ

ρ

t ⋅ E±I (r ,ω ) = −t ⋅ E±S ( r ,ω ) − t ⋅ E ±R ( r ,ω )

(11)

Following the approach of [4], this component is given in the form: ρ

R ρ ρ ρ jωµ ρ E ( r ,ω ) = − ⋅ ∫ t ′ ⋅ I l ± ( r ′,ω )G ± ( r , r ′)dl 4π l R ±

ρ

R

ρ ρ

I

ρ ρ

(12)

ρ

ρ

G ± ( r , r ′) = − I R ⋅ G ± ( r , I R ⋅ r ′)

where subscripts (+) and (-) denote the location of the source and the observation point in the air and in the earth respectively, so that z ⋅ z ′ ≥ 0 . The vector components used to construct the correction term ρ ρ

R ± ( r , r ′) are given in Appendix B. 3.2 Current Distribution The method of moments permits the integral equation of the form given by (3) to be reduced to a system of linear equations that can be expressed in the matrix form [8]. This is done by expanding the axial current distribution in the network in a sequence of subsectional expansion functions, each being non zero only over a small portion of the wire [1]. For a total of N expansion functions, the matrix equation is given in the form:  z11 z  21  Μ  zN 1

z12 z22

Λ

Μ

Ο

Λ

zN 2 Λ

z1 N   I1  z2 N   I 2  ⋅ Μ   Μ    z NN   I N 

(rn , rm ′ )dl ρ ρ

(14)

∆l m

In (14), ∆l n and ∆l m denote the lengths of the nth and ρ ρ mth segment respectively, while r and r ′ indicate n

m

the position of the mid – points on the surface of the nth and the axis of the mth segment. Taking the lossy nature of the conductors into account, the main diagonal elements in (13) are expressed in terms of the surface impedance given by (7), as follows:

4.

(15)

TIME DOMAIN ANALYSIS

x (t ) = F −1 {W ( jω ) ⋅ F [i (t )]}

I R = xˆxˆ + yˆ yˆ − zˆzˆ

 V1  V   2 =  Μ   VN 

±

Once the transient problem has been formulated in frequency domain, the time domain solutions are obtained by application of the suitable Fourier transformation. If i (t ) is the current injected in the network, the observed response may be written as [7]:

I ρ ρ ρ ρ k µ2 − k ±2 ⋅ G ± (r , r ′) + R ± ( r , r ′) 2 2 k+ + k− S

∫G

′ = z nn + Z s ⋅ ∆l n z nn

In the previous expression: G ± ( r , r ′) =

∆l n jωµ 4π

(13)

Assuming the network to have been constructed of lossless conductors, in which case I1 = I g (Ig being

(16)

where F and F −1 denote the Fourier and inverse Fourier transforms [3] and W ( jω ) is the transfer function. 5.

COMPUTATION RESULTS

Simulation has been carried out on a 400 kV, 39 m tall transmission tower subjected to a 20 kA lightning current surge. In order to obtain the time domain electromagnetic response of the tower structure and the associated grounding system, the lightning current energizing the network is decomposed into its frequency spectrum by the forward Fast Fourier Transform in the FFTSES module of the CDEGS software package. The spatial distributions of the scalar potential and the electromagnetic fields in the vicinity of the tower at selected frequencies are obtained using the HIFREQ module. Upon computing the frequency domain response of the network energized by injecting ( 20.0 + j 0.0) kA at the top of the tower, the time domain response is recovered by application of the inverse Fast Fourier Transform. To select an appropriate number of representative frequencies, the unmodulated field response is computed at an observation point near the tower structure by energizing the network using a unit current (1.0 + j 0.0) A .

5.1 Frequency Domain Results The simulated system consisting of the tower structure connected to a grid in the shape of a 8.2 m × 6 m rectangular contour buried at a depth 0.7 m is shown in Figure 2.

The frequency dependence of the electric field magnitude near the base of the structure is given in Figures 5 and 6.

Figure 5 - Electric field magnitude at 50 Hz Figure 2 - Simulated System, Scenario 1 The 3D perspective plots of the electric scalar potential at the surface of the earth near the tower base are shown in Figures 3 and 4. The computations have been carried out at 50 Hz and 1 MHz by injecting ( 20.0 + j 0.0) kA at the top of the tower.

Figure 6 - Electric field magnitude at 1 MHz The surface plot of the magnetic field magnitude is shown in Figure 7.

Figure 3 - Electric scalar potential at 50 Hz

Figure 7 - Magnetic field magnitude at 50 Hz and 1 MHz

Figure 4 - Electric scalar potential at 1 MHz

As the results of the simulation indicate, the electric scalar potential and the electric field intensity are significantly dependent on the frequency, while the magnetic field in the vicinity of the tower structure shows rather weak frequency dependence.

5.2 Time Domain Results V/m

6500

The time domain response has been computed for the case of tower structure connected to three 5 m long horizontal grounding conductors buried at a depth 0.7 m, shown in Figure 8.

5200 3900 2600 1300

µs 0

20

40

60

80

100

Figure 11 - Transient total electric field at x = 7.5 m, y = - 2.5 m, z = 0.0 m Figure 8 - Simulated System, Scenario 2 6500

The frequency spectrum of unmodulated scalar potential and the transient electromagnetic quantities near the tower are shown in Figures 9 - 12.

A/m

5200

3900 V

1000

2600

1300 500

µs 0

20

40

60

80

100

0

Figure 12 - Transient total magnetic field at x = 7.5 m, y = - 2.5 m, z = 0.0 m

MHz -500

.0

1.5

3.0

4.5

6.

6.0

Figure 9 - Frequency spectrum of unmodulated scalar potential at x = 7.5 m, y = - 2.5 m, z = 0.0 m 150

kV

100 50 0

-50 -100

µs 0

20

40

60

80

100

Figure 10 - Transient scalar potential at x = 7.5 m, y = - 2.5 m, z = 0.0 m

CONCLUDING REMARKS

The analysis of the transient response of transmission tower structures and associated grounding systems exposed to a direct lightning strike has been carried out. The mathematical model for electromagnetic fields radiated by thin - wire conductor networks is developed based on the field theory approach. The spatial distributions of the electric scalar potential, the electric and magnetic fields are computed at selected frequencies at a number of observation points on the earth's surface over the grounding grid using the HIFREQ module of the CDEGS (Current Distribution, Electromagnetic Fields Grounding and Soil Structure Analysis) software package. The temporal response is obtained by application of a suitable Fourier inversion technique in the FFTSES/CDEGS module. The time domain simulation obtained is as observed in practical systems.

Appendix A

where:

The scattered field appearing in (2) may be expressed in terms of retarded potential integrals as follows:

k +2 = ω 2 µ 0 ε 0

ρ

ρ

E S ( r ,ω ) = −

jωµ 0 ρ ρ ρ ρ t ′I l ( r ′, ω ) g ∞ ( r , r ′)dl − ∫ 4π l −

ρ ρ ρ 1 ∇ ⋅ ∫ σ l (r ′, ω ) g ∞ (r , r ′)dl 4πε 0 l

γ ± = λ2 − k ±2 (17)

with the continuity relation in frequency domain given by: ρ

σ l ( r ′, ω ) = −

ρ ρ 1 ∇ ⋅ t ′I l (r ′, ω ) jω

(18)

Substitution of (17) and (18) into (2) leads to (3). The expression (9) is obtained by assuming that the order of integration and differentiation in (3) is irrelevant, ρ ρ ρ which is possible when r ≠ r ′ and r ′ ∈ l [7, 9]. Appendix B Vector components used to construct the correction ρ ρ

term R ± ( r , r ′) are obtained from the expressions for electric fields of vertical and horizontal dipoles immersed in a semi - infinite medium [2], in the form: R±Vρ =

1 ∂2 2 k µ V± k ±2 ∂z∂ρ

R±Vz =

1 k ±2

(19)

 ∂2 2 2  2 + k ±  ⋅ k µ V± ∂ z  

(20)

  ∂2 2 1 2 cos ϕ  2 k ± V± + k ± U ±  2 k±   ∂ρ 1 ∂ 2  1 = − 2 sinϕ  k ± V± + k ±2U ±  k±  ρ ∂ρ 

R±Hρ =

(21)

R±Hϕ

(22)

R±Hz = −

1 ∂2 2 cosϕ k µ V± 2 ∂z∂ρ k±

k −2 = ω 2 µε

(23) ρ

where the elementary source is at r ′ = z ′zˆ and the ρ field is observed at r = ρ (cosϕ ⋅ xˆ + sinϕ ⋅ yˆ ) + zzˆ . The superscripts V and H in (19 - 23) stand for vertical and horizontal dipoles respectively, and the Sommerfeld integral terms are given by [4]: ∞  2  −γ ± z + z ′ 2k ±2 U± = ∫  − ⋅e J 0 ( λρ )λdλ 2 2  γ + γ − γ ± (k + + k − )  0  + ∞   −γ ± z + z ′ 2 2 V± = ∫  2 − ⋅e × 2 2 2  k γ + k + γ − γ ± (k+ + k − )  0  − + × J 0 (λρ )λdλ

7.

REFERENCES

[1] Adams, A. T., Strait, B. J., Warren, D. E., Kuo, D., Baldwin, T. E., “Near Fields of Wire Antennas by Matrix Method”, IEEE Transactions on Antennas and Propagation, Vol. AP – 21, No. 5, September 1973, pp. 602. – 610. [2] Banos, A., “Dipole Radiation in the Presence of a Conducting Half – Space”, Pergamon Press, Oxford, 1966. [3] Brigham, E. O., “The Fast Fourier Transform”, Prentice - Hall, New Jersey 1974. [4] Burke, G. J., Miller, E. K., “Modeling Antennas Near to and Penetrating a Lossy Interface”, IEEE Transactions on Antennas and Propagation, Vol. AP – 32, No. 10, October 1984, pp. 1040. – 1049. [5] CDEGS User’s Manual, “HIFREQ and FFTSES User Manuals”, Safe Engineering Services and Technologies Ltd., 1997. [6] Dawalibi, F. P., Ruan, W., Fortin, S., “Lightning Transient Response of Communication Towers and Associated Grounding Networks”, Proceedings of the International Conference on Electromagnetic Compatibility, ICEMC ’95, Kualalumpur, Malaysia 1995. [7] Grcev, L., Dawalibi, F. P., “An Electromagnetic Model for Transients in Grounding Systems”, IEEE Transactions on Power Delivery, Vol. 5, No. 4, November 1990, pp. 1773. – 1781. [8] Harrington, R. F., “Field Computation by Moment Methods”, The Macmillan Co., New York 1968. [9] Mittra, R., “Integral Equation Methods for Transient Scattering”, in “Transient Electromagnetic Fields”, Edited by Felsen, L. B., Springer – Verlag, Berlin 1976, Ch. 2, pp. 73. – 128. [10] Nikolovski, S., Sljivac, D., Fortin, S., “Lightning Transient Response of a 400 kV Transmission Tower With Associated Grounding System”, Proceedings of the International Symposium on Electromagnetic Compatibility, EMC ’98, Rome, Italy 1998. [11] Stratton, J. A., “Electromagnetic Theory”, McGraw – Hill, New York 1941. [12] Tesche, F. M., “On the Inclusion of Loss in Time – Domain Solutions of Electromagnetic Interaction Problems”, IEEE Transactions on Electromagnetic Compatibility, Vol. 32, No. 1, February 1990, pp. 1. – 4.