An Invertible Extended Sequence Transform for Untransposed Three-Phase Overhead Lines

The classical Fortescue symmetrical-component transform remains fully invertible as a similarity transform when the complete sequence-domain matrix is retained. In practice, however, untransposed three-phase overhead lines are often summarized by the diagonal sequence quantities Z0 and Z1, while the asymmetry appears through coupled off-diagonal terms that are less convenient for compact parameterization and measurement-based interpretation. This paper presents an Extended Sequence Transform that reorganizes the six independent entries of the phase-domain impedance matrix into six structured parameters by means of a lossless, invertible 6×6 linear mapping. The first two parameters are identical to the classical zero- and positive-sequence impedances, which preserves backward compatibility. The remaining four parameters isolate asymmetry information in a form from which all entries of the classical sequence matrix can be recovered exactly, and from which the full phase-domain matrix is reconstructed to machine precision. The proposed representation does not diagonalize an untransposed line; rather, it provides a compact and explicit six-scalar parameterization that separates the classical sequence pair from four asymmetry descriptors. Numerical validation on a 50 km untransposed overhead line confirms exact round-trip reconstruction and exact agreement of the unbalance factors obtained from the classical and extended representations. A stochastic perturbation study further shows that round-trip reconstruction remains at numerical precision and that, within the tested perturbation grid, the M2 factor is more sensitive than M0. Line-constant calculations performed in OpenDSS for two typical 400 kV tower geometries link the four asymmetry parameters to specific geometric features, a worked offline measurement example recovers them from simulated three-phase terminal tests, and a distributed-model study confirms that the lumped-parameter description of the asymmetry remains accurate to within about 0.6% up to 150 km.