Modal transformation applied to double three-phase transmission lines

Double three-phase transmission lines are analyzed in this paper using a modal transformation model. The main attribute of this model is the use of a single real transformation matrix based on line geometrical characteristics and the Clarke matrix. Because of this, for any line point, the electrical values can be accessed for phase domain or mode domain using the considered transformation matrix and without convolution methods. For non-transposed symmetrical lines the errors between the model results and the exact modes are insignificant values. The eigenvector and eigenvalue analyses for transposed lines search the similarities among the three analyzed transposition types and the possible simplifications for a non-transposed case.

Modal transformation analyses for double three-phase transmission lines

Eigenvector and eigenvalue analyses are carried out for double three-phase transmission lines, studying the application of a constant and real phase-mode transformation matrix and the errors of this application to mode line models. Employing some line transposition types, exact results are obtained with a single real transformation matrix based on Clarke's matrix and line geometrical characteristics. It is shown that the proposed technique leads to insignificant errors when a nontransposed case is considered. For both cases, transposed and nontransposed, the access to the electrical values (voltage and current, for example) is provided through a simple matrix multiplication without convolution methods. Using this facility, an interesting model for transmission line analysis is obtained even though the nontransposed case errors are not eliminated. The main advantages of the model are related to the transformation matrix: single, real, frequency independent, and identical for voltage and current.

Single real transformation matrices applied to double three-phase transmission lines

Single real transformation matrices are tested as phase-mode transformation matrices of typical symmetrical systems with double three-phase and two parallel double three-phase transmission lines. These single real transformation matrices are achieved from eigenvector matrices of the mentioned systems and they are based on Clarke's matrix. Using linear combinations of the Clarke's matrix elements, the techniques applied to the single three-phase lines are extended to systems with 6 or 12 phase conductors. For transposed double three-phase lines, phase Z and Y matrices are changed into diagonal matrices in mode domain. Considering non-transposed cases of double three-phase lines, the results are not exact and the error analyses are performed using the exact eigenvalues. In case of two parallel double three-phase lines, the exact single real transformation matrix has not been obtained yet. Searching for this exact matrix, the analyses are based on a single homopolar reference. For all analyses in this paper, the homopolar mode is used as the only homopolar reference for all phase conductors of the studied system. (C) 2008 Elsevier B.V. All rights reserved.

Eigenvalue analyses of two parallel lines using a single real transformation matrix

For a typical non-symmetrical system with two parallel three phase transmission lines, modal transformation is applied using some examples of single real transformation matrices. These examples are applied searching an adequate single real transformation matrix to two parallel three phase transmission line systems. The analyses are started with the eigenvector and eigenvalue studies, using Clarke's transformation or linear combinations of Clarke's elements. The Z C and parameters are analyzed for the case that presents the smallest errors between the exact eigenvalues and the single real transformation matrix application results. The single real transformation determined for this case is based on Clarke's matrix and its main characteristic is the use of a unique homopolar reference. So, the homopolar mode becomes a connector mode between the two three-phase circuits of the analyzed system. ©2005 IEEE.

Non-transposed three-phase line analyses with a single real transformation matrix

In transmission line transient analyses, a single real transformation matrix can obtain exact modes when the analyzed line is transposed. For non-transposed lines, the results are not exact. In this paper, non-symmetrical and non transposed three-phase line samples are analyzed with a single real transformation matrix application (Clarke's matrix). Some interesting characteristics of this matrix application are: single, real, frequency independent, line parameter independent, identical for voltage and current determination. With Clarke's matrix use, mathematical simplifications are obtained and the developed model can be applied directly in programs based on time domain. This model works without convolution procedures to deal with phase-mode transformation. In EMTP programs, Clarke's matrix can be represented by ideal transformers and the frequency dependent line parameters can be represented by modified-circuits. With these representations, the electrical values at any line point can be accessed for phase domain or mode domain using the Clarke matrix or its inverse matrix. For symmetrical and non-transposed lines, the model originates quite small errors. In addition, the application of the proposed model to the non-symmetrical and non-transposed three phase transmission lines is investigated. ©2005 IEEE.

Correction procedure applied to a single real transformation matrix -Untransposed three-phase transmission line cases

Clarke's matrix has been used as an eigenvector matrix for transposed three-phase transmission lines and it can be applied as a phase-mode transformation matrix for transposed cases. Considering untransposed three-phase transmission lines, Clarke's matrix is not an exact eigenvector matrix. In this case, the errors related to the diagonal elements of the Z and Y matrices can be considered negligible, if these diagonal elements are compared to the exact elements in domain mode. The mentioned comparisons are performed based on the error and frequency scan analyses. From these analyses and considering untransposed asymmetrical three-phase transmission lines, a correction procedure is determined searching for better results from the Clarke's matrix use as a phase-mode transformation matrix. Using the Clarke's matrix, the relative errors of the eigenvalue matrix elements can be considered negligible and the relative values of the off-diagonal elements are significant. Applying the corrected transformation matrices, the relative values of the off-diagonal elements are decreased. The comparisons among the results of these analyses show that the homopolar mode is more sensitive to the frequency influence than the two other modes related to three-phase lines. © 2006 IEEE.

Modal transformation obtained from Clarke's matrix - Asymmetrical three-phase case

Some constant matrices can be used as phase-mode transformation matrices for transposed three-phase transmission lines. Clarke's matrix is one of these options. Its application as a phase-mode transformation matrix for untransposed three-phase transmission lines has been analyzed through error and frequency scan comparisons. Based on an actual untransposed asymmetrical three-phase transmission line example, a correction procedure is applied searching for better results from the Clarke's matrix applicaton as a phase-mode transformation matrix. The error analyses are carried out using Clarke's matrix and the new transformation matrices obtained from the correction procedure. Applying Clarke's matrix, the relative errors of the eigenvalue matrix elements can be considered negligible and the relative values of the off-diagonal elements are significant. If the the corrected transformation matrices are used, the relative values of the off-diagonal elements are decreased. Based on the results of these analyses, the homopolar mode is more sensitive to the frequency influence than the two other modes related to three-phase lines. © 2007 IEEE.

An alternative procedure to decrease the dimension of the frequency dependent modal transformation matrices: Application in three-phase transmission lines with a vertical symmetry plane

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 α 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.

Eigenvalue analyses for non-transposed three-phase transmission line considering non-implicit ground wires

This paper presents a method for analyzing electromagnetic transients using real transformation matrices in three-phase systems considering the presence of ground wires. So, for the Z and Y matrices that represent the transmission line, the characteristics of ground wires are not implied in the values related to the phases. A first approach uses a real transformation matrix for the entire frequency range considered in this case. This transformation matrix is an approximation to the exact transformation matrix. For those elements related to the phases of the considered system, the transformation matrix is composed of the elements of Clarke's matrix. In part related to the ground wires, the elements of the transformation matrix must establish a relationship with the elements of the phases considering the establishment of a single homopolar reference in the mode domain. In the case of three-phase lines with the presence of two ground wires, it is unable to get the full diagonalization of the matrices Z and Y in the mode domain. This leads to the second proposal for the composition of real transformation matrix: obtain such transformation matrix from the multiplication of two real and constant matrices. In this case, the inclusion of a second matrix had the objective to minimize errors from the first proposal for the composition of the transformation matrix mentioned. © 2012 IEEE.

Análise de transitórios eletromagnéticos utilizando matrizes de transformação reais-sistemas trifásicos com a presença de cabos pára-raios

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Time Domain Analyses for Three-phase Lines with Corrected Modal Transformation Matrix

The results presented in this paper are based on a research about the application of approximated transformation matrices for electromagnetic transient analyses and simulations in transmission lines. Initially, it has developed the application of a single real transformation matrix for a double three-phase transmission lines, because the symmetry of the distribution of the phase conductors and the ground wires. After this, the same type of transformation matrix has applied for symmetrical single three-phase transmission lines. Analyzing asymmetrical single three-phase lines, it has used three different line configurations. For these transmission line types, the errors between the eigenvalues and the approximated results, called quasi modes, have been considered negligible. on the other hand, the quasi mode eigenvalue matrix for each case was not a diagonal one. and the relative values of the off-diagonal elements of the approximated quasi mode matrix are not negligible, mainly for the low frequencies. Based on this problem, a correction procedure has been applied for minimizing the mentioned relative values. For the correction procedure application, symmetrical and asymmetrical single three-phase transmission line samples have been used. Checking the correction procedure results, analyses and simulations have been carried out in mode and time domain. In this paper, the last results of mentioned research are presented and they related to the time domain simulations.

Representation of transmission lines in modal domain using analytical transformation matrices

The phases of a transmission line are tightly coupled due to mutual impedances and admittances of the line. One way to accomplish the calculations of currents and voltages in multi-phase lines consists in representing them in modal domain, where its n coupled phases are represented by their n propagation modes. The separation line in their modes of propagation is through the use of a modal transformation matrix whose columns are eigenvectors associated with the parameters of the line. Usually, this matrix is achieved through numerical methods which does not allow the achievement of an analytical model for line developed directly in the phases domain. This work will show the modal transformation matrix of a hypothetical two-phase obtained with numerical and analytical procedures. It will be shown currents and voltage s at terminals of the line taking into account the use of modal transformation matrices obtained by using numerical and analytical procedures. © 2011 IEEE.

Mutual coupling modeling in transmission lines directly in the phase domain

This paper describes a computational model based on lumped elements for the mutual coupling between phases in three-phase transmission lines without the explicit use of modal transformation matrices. The self and mutual parameters and the coupling between phases are modeled using modal transformation techniques. The modal representation is developed from the intrinsic consideration of the modal transformation matrix and the resulting system of time-domain differential equations is described as state equations. Thus, a detailed profile of the currents and the voltages through the line can be easily calculated using numerical or analytical integration methods. However, the original contribution of the article is the proposal of a time-domain model without the successive phase/mode transformations and a practical implementation based on conventional electrical circuits, without the use of electromagnetic theory to model the coupling between phases. © 2011 IEEE.

A transmission line model developed directly in phase domain

The phases of a transmission line are tightly coupled due to mutual impedances and admittances of the line. One way to accomplish the calculations of currents and voltages in multi-phase lines consists in representing them in modal domain, where its n coupled phases are represented by their n propagation modes. The separation line in their modes of propagation is through the use of a modal transformation matrix whose columns are eigenvectors associated with the parameters of the line. Usually, this matrix is achieved through numerical methods which do not allow the achievement of an analytical model for line developed directly in the phases domain. This work will show an analytical model for phase currents and voltages of the line and results it will be applied to a hypothetical two-phase. It will be shown results obtained with that will be compared to results obtained using a classical model. © 2012 IEEE.

A transmission line model developed directly in phase domain

The phases of a transmission line are tightly coupled due to mutual impedances and admittances of the line. One way to accomplish the calculations of currents and voltages in multi phase lines consists in representing them in modal domain, where its n coupled phases are represented by their n propagation modes. The separation line in their modes of propagation is through the use of a modal transformation matrix whose columns are eigenvectors associated with the parameters of the line. Usually, this matrix is achieved through numerical methods which do not allow the achievement of an analytical model for line developed directly in the phases domain. This work will show an analytical model for phase currents and voltages of the line and results it will be applied to a hypothetical two-phase. It will be shown results obtained with that will be compared to results obtained using a classical model © 2003-2012 IEEE.