Modal Representation Of Three-phase Lines Applying Two Transformation Matrices: Evaluation Of Its Eigenvectors

The objective of this paper is to show an alternative representation in time domain of a non-transposed three-phase transmission line decomposed in its exact modes by using two transformation matrices. The first matrix is Clarke's matrix that is real, frequency independent, easily represented in computational transient programs (EMTP) and separates the line into Quasi-modes α, β and zero. After that, Quasi-modes a and zero are decomposed into their exact modes by using a modal transformation matrix whose elements can be synthesized in time domain through standard curve-fitting techniques. The main advantage of this alternative representation is to reduce the processing time because a frequency dependent modal transformation matrix of a three-phase line has nine elements to be represented in time domain while a modal transformation matrix of a two-phase line has only four elements. This paper shows modal decomposition process and eigenvectors of a non-transposed three-phase line with a vertical symmetry plane whose nominal voltage is 440 kV and line length is 500 km. ©2006 IEEE.Marti, J.R., Accurate modeling of frequency-dependent transmission line in electromagnetic transient simulations (1982) IEEE Trans. Power App. and Systems, PAS-101 (N2), pp. 147-155. , 1, pp, JanNguyen, T.T., Chan, H.Y., Evalution of modal transformation matrices for overhead transmission lines and undergraund cables by optimization method (2002) IEEE Trans. on Power Delivery, 17 (N2), pp. 200-209. , 1, pp, JanWedephol, L.M., Nguyen, H.V., Irwin, G.D., Frequency-dependent transformation matrices for untransposed transmission lines using Newton-Raphson method (1996) IEEE Trans. Power Systems, 11 (N2), pp. 1538-1546. , 3, pp, AugB. Gustavsen and A. Semlyen Simulation of transmission line transients using vector fitting and modal decomposition, IEEE Trans. Power Delivery, 13, n 2 2, pp. 605-612, April 1998Tavares, M.C., Pissolato, J., Portela, C.M., Quasi-modes multiphase transmission line model (1999) Electric Power Systems Research, (49), pp. 159-167Portela, C., Tavares, M.C., Modeling, simulation and optimization of transmission lines. Applicability and limitations of some used procedures (2002) IEEE PES Transmission and Distribution, , São Paulo, BrazilBrandão Faria, J.A., Briceno Mendez, J.H., Modal Analysis of untransposed bilateral three-phase lines - A perturbation approach (1997) IEEE Trans. Power Delivery, 12 (1), pp. 497-504. , JanBudner, A., Introduction of frequency-dependent line parameters into an electromagnetic transients programs (1970) IEEE Trans. Power App. and Systems, PAS-89 (1), pp. 88-97. , JanDommel, H.W., (1986) EMTP theory book, , VancouverMarti, L., Low-order approximation of transmission line parameters for frequency-dependent models (1983) IEEE Trans. Power App. and Systems, PAS-102 (N2), pp. 3582-3589. , 11, pp, NovGustavsen, B., Semlyen, A., Combined phase and modal domain calculation of transmission line transients based on vector fitting (1998) IEEE Trans. Power Delivery, 13 (N2), pp. 596-604. , 2, pp, April, Discussion of the pape