Lightning surges in hybrid cable-overhead lines: Part I—voltage estimation for shielding failure

Abstract

The increasing usage of hybrid cable-overhead lines raises concerns over the protection of short cable sections against lightning surges, because of voltage build-up in the cable section. To set a simulation model of the phenomenon is time consuming, with numerous parameters impacting the overvoltage. This paper (Part I) presents formulas able to perform a fast estimation of the maximum overvoltage at a cable-overhead line transition point in the event of shielding failure. The formulas estimate the overvoltage with and without surge arrester, as well as the energy absorbed by the surge arrester. These formulas do not require an electromagnetic transients software and they can be implemented as a script, requiring solely the geometric data of the cable and overhead line, information available in the respective datasheets. The main usefulness is a tool for a fast screening of the overvoltages and a fast evaluation of the impact of different parameters such as cable length, lightning waveform and grounding impedance.

Appendices

Appendix 1

All simulations were done in PSCAD/EMTDC. The cables and OHLs were modelled using geometric frequency-dependent models, with the data from Tables 

Table 2 Cable data

2 and

Table 3 OHL data

3, respectively. Figure 

Fig. 17

Geometric data on OHL (left) and cable (right)

17 shows the position of the phase conductors and earth wires for the OHL, as well as the thickness and position of the core, insulation, sheath and outer insulation for the cable. Figure 

Fig. 18

Schematic for the performed simulation: C₁ corresponds to a shielding failure. Surge arrester is not considered for the cases of chapter IV

18 shows the schematic for shielding failure simulations shown in chapters IV and V.

Appendix 2

2.1 Derivation of Eq. (14)—W _Front

The deduction of (14) is done in (24) for the 2nd reflection, where the voltage is reflected at the cable junction point, after being reflected once at the sending end and twice at the receiving end. In (24), this is represented by the first three terms and the last four terms of the polynomial between brackets, respectively.

$$ W_{{{\text{Front}}}} \left( 2 \right) = \frac{{V_{H\_C} }}{{e^{{n\left( {2A \cdot l_{C} } \right)}} }} \cdot \left( {k_{{{\text{Rec}}}} \cdot k_{{{\text{Rec}}}} \cdot k_{{{\text{Send}}}} + k_{{{\text{Rec}}}} \cdot k_{{{\text{Rec}}}} \cdot k_{{{\text{Send}}}} \cdot k_{{{\text{Send}}}} } \right) $$

(24)

Equation (24) can simplify into (25). For the following reflections, the power of 2 in (25) is replaced by the number of the reflection, resulting in (14).

$$ \begin{gathered} W_{{{\text{Front}}}} \left( 2 \right) = \frac{{V_{H\_C} }}{{e^{{n\left( {2A \cdot l_{C} } \right)}} }} \cdot k_{{{\text{Rec}}}}^{2} \cdot k_{{{\text{Send}}}} \left( {1 + k_{{{\text{Send}}}} } \right) \hfill \\ \Leftrightarrow W_{{{\text{Front}}}} \left( 2 \right) = \frac{{V_{H\_C} }}{{e^{{n\left( {2A \cdot l_{C} } \right)}} }} \cdot \left( {k_{{{\text{Rec}}}} \cdot k_{{{\text{Send}}}} } \right)^{2} \left( {1 + \frac{1}{{k_{{{\text{Send}}}} }}} \right) \hfill \\ \end{gathered} $$

(25)

2.2 Derivation of Eq. (22)—W _{Front_Arr}

Contrary to the case without surge arrester, the value of k_send is not constant, but a function of the current flowing into the surge arrester that changes for every reflection. The value k_{send_Arr} for reflection n is then the multiplication of the previous k_send. Example, k_{send_Arr} (3) = k_send (2)·k_send (1) and k_send (2) ≠ k₀ (1), because of the surge arrester.

$$ \begin{aligned} W_{{{\text{Front}}\_{\text{Arr}}}} (3)&= \frac{{V_{{H\_C}}}}{{e^{{n( {2A \cdot l_{C} } )}} }} \cdot (k_{{{\text{Rec}}}} \cdot k_{{{\text{Rec}}}} \cdot k_{{{\text{Rec}}}} \cdot k_{{{\text{Send}}}} (1)\\ &\quad \cdot k_{{{\text{Send}}}} ( 2 ) + k_{{{\text{Rec}}}}\cdot k_{{{\text{Rec}}}} \cdot k_{{{\text{Rec}}}} \cdot k_{{{\text{Send}}}} ( 1 )\\ &\quad \cdot k_{{{\text{Send}}}} (2) \cdot k_{{{\text{Send}}}} (3)) \\ \end{aligned} $$

(26)

$$ \begin{aligned} W_{{{\text{Front\_Arr}}}} \left( 3 \right) &= \frac{{V_{H\_C} }}{{e^{{n\left( {2A \cdot l_{C} } \right)}} }} \cdot k_{{{\text{Rec}}}}^{3} \left( k_{{{\text{Send}}}} \left( 1 \right) \cdot k_{{{\text{Send}}}} \left( 2 \right) \right. \\ &\quad \left. + k_{{{\text{Send}}}} \left( 1 \right) \cdot k_{{{\text{Send}}}} \left( 2 \right) \cdot k_{{{\text{Send}}}} \left( 3 \right) \right) \hfill \\ &\quad \Leftrightarrow W_{{{\text{Front\_Arr}}}} \left( 3 \right) = \frac{{V_{H\_C} }}{{e^{{n\left( {2A \cdot l_{C} } \right)}} }} \\ &\quad \cdot k_{{{\text{Rec}}}}^{3} \cdot \left( {k_{{{\text{Send}}}} \left( 1 \right) \cdot k_{{{\text{Send}}}} \left( 2 \right)} \right) \cdot \left( {1 + k_{{{\text{Send}}}} \left( 3 \right)} \right) \hfill \\ &\quad \Leftrightarrow W_{{{\text{Front\_Arr}}}} \left( 3 \right) = \frac{{V_{H\_C} }}{{e^{{n\left( {2A \cdot l_{C} } \right)}} }} \\ &\quad \cdot k_{{{\text{Rec}}}}^{3} \cdot k_{{{\text{Send\_Arr}}}} \left( 3 \right) \cdot \left( {1 + k_{{{\text{Send}}}} \left( 3 \right)} \right) \hfill \\ \end{aligned} $$

(27)

For a reflection n, the multiple instance of the number 3 in (27) are replaced by n (22).