Modeling of lightning surges on a grounding system of a tower cascade

Abstract

Propagation of electromagnetic lightning transients modeling in the electrical network is not an easy task because of the large dimensions of its various equipment (length of power lines and ground wires, height of towers…). The full wave numerical methods well known in modeling, generally based on the mesh of the electromagnetic device to construct a system of equations, can prove to be cumbersome or even inapplicable to treat this problem. To study the lightning direct impact on a towers cascade equipped with its lines and earth connections, this article presents a realistic approach based on the transmission line equations. This new modeling, developed directly in the frequency domain, allows taking into account the frequency variations of the ground electrical characteristics in order to analyze properly the grounding systems transient behavior, particularly the Ground Power Rise during a back-flashover.

Appendices

Appendix A: The impedance [Z] and admittance [Y] expressions for grounding conductors, for towers and the ground's impedance expression of an overhead line above a finite conductivity soil

1.1 A.1. Expressions of per-unit length impedance [Z] and admittance [Y] of grounding conductors:

The filiform nature of the conductors constituting the grounding system and the significant spectral content of the fault currents not exceeding 10 MHz, are factors favoring the modeling of a grounding by the transmission lines theory.

The representation of a simple electrode buried horizontally or vertically is introduced for the first time by E. D. Sunde [9]. In his book, E. D. Sunde offers a large work devoted to the calculation of the per-unit length parameters of earth electrodes mainly based on images theory [9]. (Fig. 17).

Fig. 17

Electrodes buried horizontally or vertically

• For an electrode buried horizontally [9]: For an electrode of “l” length buried horizontally at a depth h, in a soil of finite conductivity, the per-unit length parameters are as follows:

  $$ G = \frac{\pi }{{\rho_{{{\text{sol}}}} .\left[ {ln\left( {\frac{2.l}{{\sqrt {2.h.r} }}} \right) - 1} \right]}} $$

  (A.1)
  $$ C = { }\frac{{\pi .\varepsilon_{{{\text{sol}}}} }}{{\left[ {\ln \left( {\frac{2l}{{\sqrt {2.h.r} }}} \right) - 1} \right]}} $$

  (A.2)
  $$ L = \frac{\mu }{\pi }.\left[ {\ln \left( {\frac{2.l}{{\sqrt {2.h.r} }}} \right) - 1} \right] $$

  (A.3)
  $$ R = \frac{{\rho_{C} }}{S} $$

  (A.4)

  $\rho_{{{\text{soil}}}}$ and $\varepsilon_{{{\text{soil}}}}$ are the electrical resistivity and permittivity of the soil, $r$ the radius of the electrode, $\rho_{C}$ its resistivity and $S = \pi .r^{2}$ its section.

• For a vertical rod [9]: By the theory of images, E. D. Sunde subtracts the expressions of the per-unit length parameters of a vertical rod, of “l” length, which are [9]:

  $$ G = { }\frac{2\pi }{{\rho_{s} .\left[ {ln \left( {\frac{4.l}{r}} \right) - 1} \right]}} $$

  (A.5)
  $$ C = \frac{{2\pi .\varepsilon_{s} }}{{\left[ {\ln \left( {\frac{4.l}{r}} \right) - 1} \right]}} $$

  (A.6)
  $$ L = \frac{\mu }{2\pi }\left[ {\ln \left( {\frac{4.l}{r}} \right) - 1} \right] $$

  (A.7)

The internal per-unit length resistance R keeps the same expression as a horizontal electrode, (the equation (A.4)).

1.2 A.2. Expressions of impedance [Z] and admittance [Y] of towers:

In our study, we opt for the modeling of the "tower – grounding" system by the transmission lines theory, taking a more complete modeling and in order to be as close as possible to reality (study of complex structures) and to have a convenient implementation with a tolerable calculation time.

• The impedance and admittance of the vertical tower segments: Ametani et al Approach [10]

Consider a multi-conductor system of vertical columns (Fig. 18). For this system A. Ametani et al. [10] expresses the mutual impedance between two cylindrical vertical conductors, as follows:

$$ Z_{ij} = j{\upomega }\left( {{\upmu }_{0} /2{\uppi }} \right).P_{ij} $$

(A.8)

Fig. 18

A multi-conductor system of vertical columns

With:

$$ P_{ij} = \left( {M + M^{\prime}} \right)/2.X_{ij} $$

(A.9)

$$ M = \mathop \int \limits_{{h_{m} }}^{{h_{n} }} \mathop \int \limits_{{h_{r} }}^{{h_{s} }} \frac{{{\text{d}}x_{i} {\text{d}}x_{j} }}{S}\;{\text{and}}\;M^{\prime} = \mathop \int \limits_{{h_{m} }}^{{h_{n} }} \mathop \int \limits_{{ - h_{r}^{^{\prime}} }}^{{h_{s}^{^{\prime}} }} \frac{{{\text{d}}x_{i} {\text{d}}x_{j}^{^{\prime}} }}{{S^{\prime}}} $$

(A.10)

Whose different geometric dimensions, see (Fig. 18), are calculated as follows:

$$ S = \sqrt {\left( {y_{i} - y_{j} } \right)^{2} + \left( {x_{i} - x_{j} } \right)^{2} } = \sqrt {\left( {d^{2} + \left( {x_{i} - x_{j} } \right)^{2} } \right)} $$

(A.11)

$$ S^{\prime} = \sqrt {\left( {d^{2} + \left( {x_{i} - x^{\prime}_{j} } \right)^{2} } \right)} $$

(A.12)

$$ X_{ii} = h_{n} - h_{m} ; X_{jj} = h_{S} - h_{r} ; X_{ij} = \left( {X_{ii} + X_{jj} } \right)/2 $$

(A.13)

$$ h\prime_{r} = h_{r} + 2h_{e} ; h\prime_{S} = h_{S} + 2h_{e} ; h_{e} = \sqrt {\rho_{{{\text{soil}}}} /j\omega \mu_{{{\text{soil}}}} } $$

(A.14)

A. Ametani [10] formulates the following expression for the coefficients ${P}_{ij}$:

$$ P_{ij} = \left( {\frac{1}{{2X_{ij} }}} \right)\left[ \begin{gathered} - h_{1} lnA_{1} + h_{2} lnA_{2} + h_{3} lnA_{3} - h_{4} lnA_{4} + a_{1} - a_{2} \hfill\\ - a_{3} + a_{4} - \left( {h_{5} lnA_{5} - h_{6} lnA_{6} - h_{7} lnA_{7} } \right. \hfill \\ \qquad \left.{ + h_{8} lnA_{8} - a_{5} + a_{6} + a_{7} - a_{8}} \right) \hfill \\ \end{gathered} \right] $$

(A.15)

In which:

$h_{1} = h_{n} - h_{S}$,$h_{2} = h_{m} - h_{S}$,$h_{3} = h_{n} - h_{r}$,

$h_{4} = h_{m} - h_{r}$,$h_{5} = h_{n} + h_{S} + 2h_{e}$,

$h_{6} = h_{m} + h_{S} + 2h_{e}$,$h_{7} = h_{n} + h_{r} + 2h_{e}$,

$$ h_{8} = h_{m} + h_{r} + 2h_{e} $$

(A.16)

$$ a_{k} = \sqrt {\left( {h_{k}^{2} + d^{2} } \right)} ;A_{k} = a_{k} + h_{k } , k = \left( {1,2 \ldots .8 } \right) $$

(A.17)

The self-impedance of a conductor i is calculated from the self-coefficients $P_{ii}$, which are obtained by operating, in the expression of the $P_{ij}$, the following changes:$d = r_{i} , h_{S} = h_{n}$ and $h_{r} = h_{m}$. This gives:

$$ Z_{ii} = j{\upomega }\left( {{\upmu }_{0} /2{\uppi }} \right).P_{ii} $$

(A.18)

Finally, in order to calculate the self per-unit length capacity, the soil is considered perfectly conductive, by laying the penetration depth $h_{e} = 0$ in the expression of the coefficients $P_{ii}$ to have the coefficients $P_{0ii}$ characterizing a perfect soil [10]. Which gives:

$$ C = 2.\pi \varepsilon_{0} .\;P_{0ii}^{ - 1} $$

(A.19)

The per-unit length impedance and admittance are $\underline {Z} = Z_{ii}$ and $\underline {Y} = j\omega C$.

• The impedance and admittance of the horizontal arms: Ametani et al Approach [11]

Either a multi-conductor system of finite lengths, whose return is through the ground (Fig. 19).

Fig. 19

Multi-conductor system of finite lengths

The self and mutual per-unit length impedances of this system can be determined as follows [11]:

$$ Z_{ij} = j{\upomega }\left( {{\upmu }_{0} /2{\uppi }} \right).P_{ij}^{^{\prime}} /l $$

(A.20)

With:

$$ P_{ij}^{^{\prime}} = \left( {M_{d} + M_{i} } \right)/2 $$

(A.21)

$M_{d}$: Direct mutual inductance between conductors i and j [11];

$M_{i}$: The mutual inductance between conductor i and image j' of conductor j relative to the ground plane.

If the terminal of conductor j,$x_{j1} = 0$, and the terminals $x_{i} = x_{j2} = l,$ the coefficients $P_{ij}^{^{\prime}}$ are expressed:

$$ P_{ij}^{^{\prime}} = \left[ {l\ln \frac{{1 + \sqrt {1 + \left( {d_{ij} /l} \right)^{2} } }}{{1 + \sqrt {1 + \left( {S_{ij} /l} \right)^{2} } }} + l ln\frac{{S_{ij} }}{{d_{ij} }} - \sqrt {l^{2} + d_{ij}^{2} } + \sqrt {l^{2} + S_{ij}^{2} } + d_{ij} - S_{ij} } \right] $$

(A.22)

Hence:

$$ d_{ij} = \sqrt {\left( {h_{i} - h_{j} } \right)^{2} + y^{2} } $$

(A.23)

$$ S_{ij} = \sqrt {\left( {h_{i} + h_{j} + 2h_{S} } \right)^{2} + y^{2} } , \left( {y = y_{ij} } \right) $$

(A.24)

$$ h_{S} = \sqrt {\rho_{{{\text{soil}}}} /\left( {j\omega \mu_{{{\text{soil}}}} } \right)} $$

(A.25)

In order to have the self per-unit length impedance $Z_{ii} = j{\upomega }\left( {{\upmu }_{0} /2{\uppi }} \right).P_{ii}^{^{\prime}} /l$, we put $j = i , d_{ij} = r_{i} , S_{ij} = 2\left( {h_{i} + h_{S} } \right)$.

For a perfectly conductive ground $\left( {\rho_{soil} = 0} \right)$, skin depth $h_{s} = 0$, this gives for the coefficients $P_{ij}^{^{\prime}}$[11]:

$$ P_{0ij}^{^{\prime}} = \bigg[ {l\ln \frac{{1 + \sqrt {1 + \left( {d_{ij} /l} \right)^{2} } }}{{1 + \sqrt {1 + \left( {D_{ij} /l} \right)^{2} } }} + l \ln\frac{{D_{ij} }}{{d_{ij} }} - \sqrt {l^{2} + d_{ij}^{2} }+ \sqrt {l^{2} + D_{ij}^{2} } + d_{ij} - D_{ij} } \bigg] $$

(A.26)

For which:

$$ D_{ij} = \sqrt {\left( {h_{i} + h_{j} } \right)^{2} + y_{ij}^{2} } $$

(A.27)

If we put $j = i , d_{ij} = r_{i} , D_{ij} = 2h_{i}$, in the expression of $P_{0ij}^{^{\prime}}$, we come to the self-coefficients $P_{0ii}^{^{\prime}}$ and we then deduce the self per-unit length capacity of a finite length horizontal conductor:

$$ C = 2\pi \varepsilon_{0} .\left( {P_{0ii}^{^{\prime}} /l} \right)^{ - 1} $$

(A.28)

Idem, the per-unit length impedance and admittance are therefore: $\underline {Z} = Z_{ii}$ and $\underline {Y} = j\omega C$.

1.3 A.3. Expression of the ground impedance of an overhead line above a finite conductivity soil:

For a multi-conductor overhead line (Fig. 20)whose return is via the ground, E. D. Sunde [9], in the form of an integral, gives the correction terms (due to the ground’s finite conductivity) of the per-unit length impedance.

Fig. 20

Multi-conductor overhead line above finite conductivity ground

On a frequency band of a few MHz, Sunde [9] offers an excellent logarithmic approximation for these self and mutual correction terms, such as:

$$ Z_{sii} \cong \frac{{j\omega \mu_{0} }}{2\pi }.ln\left( {\frac{{1 + \gamma_{s} .h_{i} }}{{\gamma_{s} .h_{i} }}} \right) $$

(A.29)

$$ Z_{sij} \cong \frac{{j\omega \mu_{0} }}{4\pi }.ln\left( {\frac{{\left( {1 + \gamma_{s} .\left( {\frac{{h_{i} + h_{j} }}{2}} \right)} \right)^{2} + \left( {\gamma_{s} .\frac{{d_{ij}^{^{\prime}} }}{2}} \right)^{2} }}{{\left( {\gamma_{s} .\left( {\frac{{h_{i} + h_{j} }}{2}} \right)} \right)^{2} + \left( {\gamma_{s} .\frac{{d_{ij}^{^{\prime}} }}{2}} \right)^{2} }}} \right) $$

(A.30)

With:

$$ \gamma_{s}^{2} = j\omega \mu_{0} .\left( {\sigma_{{{\text{soil}}}} + j\omega \varepsilon_{0} \varepsilon_{{{\text{rsoil}}}} } \right) $$

(A.31)

$\sigma_{{{\text{soil}}}}$,$\varepsilon_{{{\text{rsoil}}}}$ respectively, conductivity, relative permittivity of the soil and $\gamma_{s}$ is the propagation constant in the soil..

Appendix B: The expressions of the soil electrical parameters variation with frequency

The expressions giving the variation of the soil’s conductivity and relative permittivity as a function of the frequency are as follows:

• For the Visacro and Alipio formulation [22] :

  $$ \sigma_{{{\text{soil}}}} \left( f \right) = \sigma_{{\text{soil 100Hz}}} \times \left\{ {1 + \left[ {1.2.10^{ - 6} \left( {\frac{1}{{\sigma_{{\text{soil 100Hz}}} }}} \right)^{0.73} } \right]\left( {f - 100} \right)^{0.65} } \right\} $$

  (B.1)
  $$ \varepsilon_{rsoil} \left( f \right) = 7.6.10^{3} .f^{ - 0.4} + 1.3\, for\;f \ge 10 kHz $$

  (B.2)

where $\sigma_{soil 100Hz} { }$ is the ground conductivity at low frequencies ($f$ = 100 Hz).

• For the Longmire and Smith formulation [23] :

  $$ \sigma_{{{\text{soil}}}} \left( f \right) = \sigma_{{\text{soil 0}}} + 2\pi .\varepsilon_{0} \mathop \sum \limits_{n = 1}^{K} \frac{{a_{n} . \left( {\left( \frac{p}{10} \right)^{1.28} . 10^{n - 1} } \right). \left( {\frac{f}{{\left( \frac{p}{10} \right)^{1.28} . 10^{n - 1} }}} \right)^{2} }}{{1 + \left( {\frac{f}{{\left( \frac{p}{10} \right)^{1.28} . 10^{n - 1} }}} \right)^{2} }} $$

  (B.3)
  $$ \varepsilon_{{{\text{rsoil}}}} \left( f \right) = \varepsilon_{\infty } + \mathop \sum \limits_{n = 1}^{K} \frac{{a_{n} }}{{1 + \left( {\frac{f}{{\left( \frac{p}{10} \right)^{1.28} . 10^{n - 1} }}} \right)^{2} }} $$

  (B.4)

With:

$$ \sigma_{{\text{soil 0}}} = \sigma_{{\text{soil 100Hz}}} = 8. 10^{ - 3} .\left( \frac{p}{10} \right)^{1.54} $$

(B.5)

$p$ is the water percentage in the soil, $a_{n} { }$ are coefficients shown in the reference [23]. According to Longmire and Smith [23], the soil’s conductivity and relative permittivity, at each frequency, are a function of its water content.