Alternative Model of Three-Phase Transmission Line Theory-Based Modal Decomposition

In this paper is shown the development of a transmission line, based on discrete circuit elements that provide responses directly in the time domain and phase. This model is valid for ideally transposed rows represent the phases of each of the small line segments are separated in their modes of propagation and the voltage and current are calculated at the modal field. However, the conversion phase-mode-phase is inserted in the state equations which describe the currents and voltages along the line of which there is no need to know the user of the model representation of the theory in the field lines modal.

Real-time estimation of transmission line impedance based on modal analysis theory

The objective of this paper is to show a methodology to estimate the longitudinal parameters of transmission lines. The method is based on the modal analysis theory and developed from the currents and voltages measured at the sending and receiving ends of the line. Another proposal is to estimate the line impedance in function of the real-time load apparent power and power factor. The procedure is applied for a non-transposed 440 kV three-phase line. © 2011 IEEE.

Behavior of overhead transmission line parameters on the presence of ground-wires

Initially this paper shows the ground wire reduction process for generic multiphase transmission lines and after, the ground wire reduction process for a specilic 440-kV three-phase overhead transmission line. Following this, the influence of the ground wire reduction process considering two situations is shown: first, considering frequency independence and second, when these parameters are considered as frequency dependent. This paper presents analytical results for generic multiphase transmission lines. For a specific 440-kV three-phase overhead transmission line, analytical and graphic results are shown considering real data for every frequency between 10 Hz and 1 MHz.

An alternative modal representation of a symmetrical nontransposed three-phase transmission line

The objective of this letter is to propose an alternative modal representation of a nontransposed three-phase transmission line with a vertical symmetry plane by using two transformation matrices. Initially, Clarke's matrix is used to separate the line into components a, 0, and zero. Because a and zero components are not exact modes, they can be considered as being a two-phase line that will be decomposed in its exact modes by using a 2 x 2 modal transformation matrix. This letter will describe the characteristics of the two-phase line before mentioned. This modal representation is applied to decouple a nontransposed three-phase transmission line with a vertical symmetry plane whose nominal voltage is 440 kV.

Some considerations about a new procedure to derive transmission line parameters concerning three-phase lines

The paper shows an alternative methodology to calculate transmission line parameters per unit length and to apply it in a three-phase line with a vertical symmetry plane. This procedure is derived from a general procedure where the modal transformation matrix of the line is required. In this paper, the unknown modal transformation matrix requested by general procedure is substituted by Clarke's matrix. With the substitution that is shown in the paper, the transmission line parameters can be obtained starting from impedances measured in one terminal of the line. First, the article shows the classical methodology to calculate frequency dependent transmission line parameters by using Carson and Pollaczeck's equations for representing the ground effect and Bessel's functions to represent the skin effect. After that, a new procedure is shown to calculate frequency dependent transmission line parameters directly from currents and voltages of an existing line. Then, this procedure is applied in a non-transposed three-phase transmission line whose parameters have been previously calculated by using the classical methodology. Finally, the results obtained by using the new procedure and by using the classical methodology are compared. The article shows simulation results for typical frequency spectra of switching transients (10 Hz to 10 kHz). Results have shown that procedure has © 2006 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.

Corona effect in frequency dependent transmission line models

The inclusion of the corona effect in a frequency dependent transmission line model is proposed in this paper. The transmission line is represented through a cascade of π circuits and the frequency dependence of the longitudinal parameters is synthesized with series and parallel resistors and inductors. The corona effect will be represented using the Gary and Skilling-Umoto models. The currents and voltages along the line are calculated by using state-space technique. To demonstrate the accuracy and validity of the proposed frequency dependent line model, time domain simulations of a 10 km length single-phase line response under energization procedure will be presented. ©2008 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.

Development of a simplified transmission line model directly in the phase domain

A transmission line digital model is developed direct in the phase and time domains. The successive modal transformations considered in the three-phase representation are simplified and then the proposed model can be easily applied to several operation condition based only on the previous knowing of the line parameters, without a thorough theoretical knowledge of modal analysis. The proposed model is also developed based on lumped elements, providing a complete current and voltage profile at any point of the transmission system. This model makes possible the modeling of non-linear power devices and electromagnetic phenomena along the transmission line using simple electric circuit components, representing a great advantage when compared to several models based on distributed parameters and inverse transforms. In addition, an efficient integration method is proposed to solve the system of differential equations resulted from the line modeling by lumped elements, thereby making possible simulations of transient and steady state using a wide and constant integration step. © 2012 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.

Identification of transmission line parameters from temporal measurements of currents and voltages in their terminals: Influence oh the length line

The objective of this paper is to show a methodology to estimate transmission line parameters. The method is applied in a single-phase transmission line using the method of least squares. In this method the longitudinal and transversal parameters of the line are obtained as a function of a set of measurements of currents and voltages (as well as their derivatives with respect to time) at the terminals of the line during the occurrence of a short-circuit phase-ground near the load. The method is based on the assumption that a transmission line can be represented by a single circuit π. The results show that the precision of the method depends on the length of the line, where it has a better performance for short lines and medium length. © 2012 IEEE.

Digital filtering of oscillations intrinsic to transmission line modeling based on lumped parameters

A correction procedure based on digital signal processing theory is proposed to smooth the numeric oscillations in electromagnetic transient simulation results from transmission line modeling based on an equivalent representation by lumped parameters. The proposed improvement to this well-known line representation is carried out with an Finite Impulse Response (FIR) digital filter used to exclude the high-frequency components associated with the spurious numeric oscillations. To prove the efficacy of this correction method, a well-established frequency-dependent line representation using state equations is modeled with an FIR filter included in the model. The results obtained from the state-space model with and without the FIR filtering are compared with the results simulated by a line model based on distributed parameters and inverse transforms. Finally, the line model integrated with the FIR filtering is also tested and validated based on simulations that include nonlinear and time-variable elements. © 2012 Elsevier Ltd. All rights reserved.

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.

A distributed-parameters transmission line model developed directly in the phase domain

This article shows a transmission line model developed directly in the phase domain. The proposed model is based on the relationships between the phase currents and voltages at both the sending and receiving ends of a single-phase line. These relationships, established using an ABCD matrix, were extended to multi-phase lines. The proposed model was validated by using it to represent a transmission line during short-and open-circuit tests. The results obtained with the proposed model were compared with results obtained with a classical model based on modal decomposition. These comparisons show that proposed model was correctly developed. © 2013 Taylor and Francis Group, LLC.

Proposal of an alternative transmission line model for symmetrical and asymmetrical configurations

This article shows a transmission line model for simulation of fast and slow transients, applied to symmetrical or asymmetrical configurations. A transmission line model is developed based on lumped elements representation and state-space techniques. The proposed methodology represents a practical procedure to model three-phase transmission lines directly in time domain, without the explicit or implicit use of inverse transforms. In three-phase representation, analysis modal techniques are applied to decouple the phases in their respective propagation modes, using a correction procedure to set a real and constant matrix for untransposed lines with or without vertical symmetry plane. The proposed methodology takes into account the frequency-dependent parameters of the line and in order to include this effect in the state matrices, a fitting procedure is applied. To verify the accuracy of the proposed state-space model in frequency domain, a simple methodology is described based on line distributed parameters and transfer function associated with input/output signals of the lumped parameters representation. In addition, this article proposes the use of a fast and robust integration procedure to solve the state equations, enabling transient and steady-state simulations. The results obtained by the proposed methodology are compared with several established transmission line models in EMTP, taking into account an asymmetrical three-phase transmission line. The principal contribution of the proposed methodology is to handle a steady fundamental signal mixed with fast and slow transients, including impulsive and oscillatory behavior, by a practical procedure applied directly in time domain for symmetrical or asymmetrical representations. (C) 2011 Elsevier Ltd. All rights reserved.