Clarke's Matrix Correction Procedure For Non Transposed Three-phase Transmission Lines

Clarke's matrix has been applied as a phase-mode transformation matrix to three-phase transmission lines substituting the eigenvector matrices. Considering symmetrical untransposed three-phase lines and the frequency range into 10 kHz, some transient simulations have been made with the application of this single real matrix. An actual symmetrical three-phase line on untransposed conditions is associated with Clarke's matrix for error and frequency scan analyses in this paper. Error analyses are calculated for the eigenvalue diagonal elements obtained from Clarke's matrix. The eigenvalue off-diagonal elements from the Clarke's matrix application are compared to the correspondent exact eigenvalues. Based on the characteristic impedance and propagation function values, the frequency scan analyses show that there are great differences between the Clarke's matrix results and the exact ones, considering frequency values from 10 kHz. A correction procedure is applied obtaining two new transformation matrices. These matrices lead to good approximated results when compared to the exact ones. Based on the frequency scan analyses and neglecting the imaginary part of these transformation matrices, similar results to those from new transformation matrices can be obtained. With the correction procedure applied to Clarke's matrix, the relative values of the eigenvalue matrix off-diagonal element obtained from Clarke's matrix are decreased while the frequency scan results are improved. © 2008 IEEE.Dommel, H.W., (1984) Electromagnetic Transients Program - Rule Book, , Oregon(1992) Microtran Power System Analysis Corporation, Transients Analysis Program Reference Manual, , Vancouver, CanadaBrandão Faria, J.A., Briceño Mendez, J., Modal Analysis of Untransposed Bilateral Three-phase Lines - a Perturbation Approach (1997) IEEE Transactions on Power Delivery, 12 (1). , JanuaryNguyen, T.T., Chan, H.Y., Evaluation of modal transformation matrices for overhead transmission lines and underground cables by optimization method (2002) IEEE Trans. on Power Delivery, 17 (1). , JanuaryMorched, A., Gustavsen, B., Tartibi, M., A Universal Model for Accurate Calculation of Electromagnetic Transients on Overhead Lines and Underground Cables (1999) IEEE Trans. on Power Delivery, 14 (3), pp. 1032-1038. , JulyNobre, D.M., Boaventura, W.C., Neves, W.L.A., Phase-Domain Network Equivalents for Electromagnetic Transient Studies (2005) The 2005 IEEE Power Engineering Society General Meeting, , 12-16 June, CD-ROM, San Francisco, USABudner, A., Introduction of Frequency Dependent Transmission Line Parameters into an Electromagnetic Transients Program (1970) IEEE Trans. on Power Apparatus and Systems, PAS-89, pp. 88-97. , JanuaryCarneiro Jr., S., Martí, J.R., Dommel, H.W., Barros, H.M., An Efficient Procedure for the Implementation of Corona Models in Electromagnetic Transients Programs (1994) IEEE Transactions on Power Delivery, 9 (2). , AprilT. F. R. D. Martins, A. C. S. Lima, S. Carneiro Jr., Effect of Impedance Approximate Formulae on Frequency Dependence Realization, The 2005 IEEE Power Engineering Society General Meeting, 12-16 June 2005, CD-ROM, San Francisco, USAMarti, J.R., Accurate modelling of frequency-dependent transmission lines in electromagnetic transients simulations (1982) IEEE Trans. on PAS, 101, pp. 147-155. , JanuaryClarke, E., (1950) Circuit Analysis of AC Power Systems, 1. , Wiley, New YorkWedepohl, L.M., Application of Matrix Methods to the Solution of Travelling-wave Phenomena in Polyphase Systems (1963) Proceedings IEE, 110, pp. 2200-2212. , DecemberWedepohl, L.M., Wilcox, D.J., Transient analysis of underground power-transmission system - system model and wave propagation characteristics (1973) Proceedings of IEE, 120 (2), pp. 253-260Wedepohl, L.M., Nguyen, H.V., Irwin, G.D., Frequency-dependent transformation matrices for untransposed transmission lines using Newton-Raphson method (1996) IEEE Trans. on Power Systems, 11 (3), pp. 1538-1546. , Augus