Wide-band modelling and transient analysis of the multi-conductor transmission lines system considering the frequency-dependent parameters based on the fractional calculus theory

The transient analysis of multi-conductor transmission lines should consider the frequency-dependent characteristics due to the skin effect. Fully considering the inherent fractional order characteristics of the frequency-dependent effect, a general wide-band modelling method is proposed. The fractional order vector fitting method is adopted to approximate the frequency-dependent parameters and the corresponding fractional differential equations can be obtained by the inverse Laplace transformation. The backward difference is a practical method to solve the fractional differential equations; however, a linear convolution must be calculated, which will lead to a heavy computation complexity. To address this issue, a new recursive convolution method for the fractional differential equations is proposed and an efficient solution is achieved. Furthermore, considering the indispensability of the passivity verification of a system for the transient simulation, the passivity verification by extending the Hamiltonian matrix for the fractional order systems is studied and a practical criterion is proposed. Three examples are considered to validate the proposed method: (i) a single underground cable, (ii) three-phase underground cable, and (iii) an experimental transformer under very fast transient voltage. The simulation results are compared with the results obtained by power systems computer-aided design or measurements and good agreements are achieved.

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