Evaluation of Modal Transformation Matrices for Overhead Transmission Lines and Underground Cables by Optimization Methods

This article develops a numerical procedure based on constrained optimization for evaluating the modal transformation matrices of overhead transmission lines and underground cables. In the method, the eigenvalue-eigenvector equation is transformed into an objective function, which is then minimized for finding eigenvalues and eigenvectors. The developed method completely avoids the problem of eigenvalue switching, which causes the discontinuities of eigenvectors encountered in the QR method. Initial values for starting the optimization at each frequency are the eigenvalues and eigenvectors evaluated at the preceding frequency. This technique leads to the set of smooth eigenvectors that define the modal transformation matrices. The form of the objective function derived in the paper for minimization, together with the sequential quadratic programming technique, guarantees that convergence to valid eigenvalues and eigenvectors is achieved. Using the method, transformation matrices for a wide range of frequency for representative double-circuit line and underground cable are evaluated.