A frequency dependent mode domain model for double three-phase transmission lines

This article presents a model to represent double three-phase transmission lines, including the frequency dependence of longitudinal parameters. For this, frequency dependence modeling is carried out on the mode domain. A single real transformation matrix is used for the entire frequency range (up to 10 kHz). This transformation matrix is obtained from the geometrical properties of the line and is based on Clarke transformation. One of the interesting aspects of this paper is the analysis of the natural modes of the double circuit line. The transposition of the line was supposed ideal, which resulted in five equal modes and one different mode. Afterwards the line was assumed transposed as if it were a six-phase transmission line. Following this, each circuit was considered ideally transposed and there was a coupling term between the circuits. Finally the line was assumed to be nontransposed. The analysis of all this mode behavior produces a very interesting conclusion, such as: the similitude of some modes, and the possibility of working with a real transformation matrix obtained from the geometrical properties of the line as is carried out in our quasi-mode model. Finally the model is employed for an application of double three-phase transmission line using a Microtran program.