Step by step analyses of Clarke's matrix correction procedure for untransposed three-phase transmission line cases

Abstract

The correction procedure for Clarke's matrix, considering three-phase transmission line analyzes, is analyzed step by step in this paper, searching to improve the application of this procedure. Changing the eigenvectors as modal transformation matrices, Clarke's matrix has been applied to analyses for transposed and untransposed three-phase transmission line cases. It is based on the fact that Clarke's matrix is an eigenvector matrix for transposed three-phase transmission lines considering symmetrical and asymmetrical cases. Because of this, the application of this matrix has been analyzed considering untransposed three-phase transmission lines. In most of these cases, the errors related to the eigenvalues can be considered negligible. It is not true when it is analyzed the elements that are not in main diagonal of the quasi-mode matrix. This matrix is obtained from the application of Clarke's matrix. The quasi-mode matrix is correspondent to the eigenvalue matrix. Their off-diagonal elements represent couplings among the quasi-modes. So, the off-diagonal quasi-mode element relative values are not negligible when compared to the eigenvalues that correspond to the coupled quasi-modes. Minimizing these relative values, the correction procedure is analyzed in detail, checking some alternatives for the correction procedure application.Index Terms--Clarke's matrix, eigenvector, eigenvalue, phase mode transformation, error analysis, non symmetrical lines, frequency dependent parameters. M

Simplified computational routine to correct the modal decoupling in transmission lines and power systems modelling

“…These variations reflect the characteristics previously verified in 5 Simplified routine to correct the errors associated with the modal decoupling As discussed in the introduction, there are several well-established transmission line models based on modal analysis techniques using the approach by the Clarke's matrix. Although this subject-matter has been studied over a few decades, alternative modelling techniques have been even more discussed over the last years to improve the current computational models or to propose new equivalent representations [7,13,15]. There are efficient methods, however, they are sometimes very complex, unstable and have a hard computational processing, for example, the procedure given in [13].…”

“…Although this subject-matter has been studied over a few decades, alternative modelling techniques have been even more discussed over the last years to improve the current computational models or to propose new equivalent representations [7,13,15]. There are efficient methods, however, they are sometimes very complex, unstable and have a hard computational processing, for example, the procedure given in [13]. The mentioned method is based on the correction of the eigenvectors for each discrete frequency, which are equivalently approximated by the constant and real columns of the Clarke's matrix.…”

“…Thereafter, a simple variation is proposed for the use of the transformation matrices in the modelling/simulation algorithm to correct the errors intrinsic to the modal decoupling of the line, without the handling of matrix elements and complex correction processes [13].…”

“…Concerning a transposed transmission line or then an untransposed system, but with vertical symmetry plane, a real and constant matrix can be used as a transformation matrix. This approach is very useful for models developed directly in the time domain [11][12][13]. A real and constant matrix conventionally used to model transmission systems is the well-known Clarke's matrix…”

“…First, from the qualitative descriptions given above, this paper presents a brief review of modal analysis theory applied for multiconductor transmission line modelling. Thereafter, a simple variation is proposed for the use of the transformation matrices in the modelling/simulation algorithm to correct the errors intrinsic to the modal decoupling of the line, without the handling of matrix elements and complex correction processes [13].…”