Time-Invariant State-Space model of an AC Cable by $dq$-representation of Frequency-Dependent $\pi$-sections

This paper introduces an approach for frequency-dependent pi-section (FD-$\pi$) modelling of ac cables which enables a time-invariant state-space representation that accounts for the frequency-dependent cable characteristics. The proposed approach relies on vector fitting to obtain parametric values for parallel $RL$-branches representing the frequency-dependency of the series impedance of the cable in the stationary frame. The number of cascaded FD-$\pi$ sections with their corresponding shunt capacitance is then selected in order to represent the internal resonance frequencies of the cable up to the required modelling bandwidth. The resulting stationary frame model is transformed to a synchronously rotating dq reference frame to obtain a state-space formulation compatible with the time-invariant representation of other system components. The presented model allows for accurate time-invariant state-space modelling of power systems with long ac-cables where electromagnetic transients cannot be neglected in small-signal stability studies. The characteristics of the time-invariant cable model is evaluated in the frequency domain, illustrating the impact of the number of $\pi$ sections and parallel $RL$ branches on the accuracy and the associated oscillation modes of the state-space model.