Frequency-dependent impedance modelling of HVDC cable

Understanding such phenomena becomes important when assessing the system response during transient events or when analysing the effects of DC converter harmonics on the currentcarrying capability of the cable. In order to understand such effects, one must derive the analytical frequency-dependent formulae corresponding to HVDC cables. CIGRE Technical

Brochure (TB) 531 [1] offers a method to calculate the sequence and surge impedances of

Alternating Current (AC) cable systems at 45 kV and above. Subsequently, CIGRE TB 908 [2]

(which is based on works presented in [3][4][5]) and the CIGRE session paper B1-10693 [6] developed calculations to better describe armoured three-core submarine cables, partially incorporating magnetic effects of the armour. However, there is a lack of consensus regarding similar methods for HVDC cable pairs. Even though schematic approaches have been outlined in the literature, no standardised analytical methodology has been presented to the authors’ knowledge.

This paper presents frequency-dependent impedance formulae for a single HVDC submarine cable, a pair of cables as well as two pairs of cables (including both polarity arrangements).

These formulae are fully analytical and do not rely on any tuning parameters. The analytical model includes key contributions, such as from the skin effect, screening effects, magnetic effects of the armour and ground effects. Additionally, low- and high-frequency approximations of the impedance are provided for all cases. Our analytical results for the single-cable case considered here reproduce those obtained by Schelkunoff [7], Wedepohl & Wilcox [8] and

Ametani [9][10] to within ~10%. Moreover, the low- and high-frequency approximations of our results match the behaviour of Schelkunoff’s model exactly. Furthermore, the analytical results are compared with two-dimensional (2D) Finite Element

Method (FEM) modelling. We compare the results of both methods over the range of frequencies 0.01 Hz to 1 MHz for values of the armour magnetic permeability corresponding to Grade 34 and Grade 65 galvanized steel as well as for non-magnetic armour. The analytical and FEM results agree within ~10% for all considered cases. In addition, the higher the magnetic permeability, the smaller the discrepancy.