José Pissolato Filho

Quasi-modes multiphase transmission line model

Abstract This article presents a new model to represent transmission lines including the frequency dependence of longitudinal parameters. The model uses the natural modes, for ideally transposed lines, and ‘quasi-modes’ for non-transposed lines, and is applied to lines that have a vertical symmetry plane. The line is represented through π -circuits, with one π -circuit for each mode. The transformation matrix is modeled using ideal transformers. The model is described for three-phase lines, dc lines, double three-phase lines and six phase lines. A 440 kV three-phase transmission line illustrates it and is compared with a frequency dependent EMTP line model, the Semlyen one.

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 specific 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.

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 Clarkes 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.

A new formulation for skin-effect resistance and internal inductance frequency-dependent of a solid cylindrical conductor

This paper presents a new and quite simple formulation to calculate the skin-effect resistance and internal inductance of a solid cylindrical conductor. The formulation is obtained from the exact solution of the Maxwells wave equation of an electrical field in the direction of propagation. The Fourier transform method is used to obtain the frequency-domain solution. The results are presented in graphical form showing the conductor resistance and internal inductance as a function of frequency.

Correction Procedure Applied to a Single Real Transformation Matrix -Untransposed Three-phase Transmission Line Cases

Clarkes 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, Clarkes 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 Clarkes matrix use as a phase-mode transformation matrix. Using the Clarkes 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

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 Clarkes 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.

Influence of earth conductivity and permittivity frequency dependence in electromagnetic transient phenomena - more measurements results in new sites

In this work some procedures to measure and model soil electromagnetic behavior in frequency domain are presented. The proposed model takes into account the earth conductivity and permittivity frequency dependence, which normally are not considered. Some procedures to reduce noise signals in the field measurements values are presented. For an actual 440 kV single three-phase transmission line, the importance to consider the soil behavior is represented through a unique real value of conductance (the normal approach) and through the proposed model.

Influences of Damping Resistances on Transient Simulations in Transmission Lines

Simulations of electromagnetic transients in transmission lines can be carried out using simple circuit model. In the case of applications of simple circuit models based on π circuits, there are problems mainly caused by numeric oscillations. The lumped-parameter π equivalent model can be used with some advantages. The numeric integration method applied to the simulations of electromagnetic transients is the trapezoidal rule. If this numeric method is associated with the π equivalent model, results obtained from the simulations are distorted by Gibbs’ oscillations or numeric ones. The introduction of damping resistance parallel to the series RL branch of the π equivalent model can mitigate Gibbs’ oscillations in obtained results. Voltage peaks caused by these oscillations can also be decreased. So, in this paper, the combined influences of the introduction of damping resistance, the number of π circuits and the time step are investigated searching for minimizing Gibbs’ oscillations and the voltage peaks in electromagnetic transient simulations. For this, transient simulations are exhaustively carried out for determining the highest voltage peaks, ranges of damping resistances and other parameters of the model, which minimize these voltage peaks caused by Gibbs’ oscillations.

Transmission Lines Model with Different Basic Structures Applied to Transient Electromagnetic Simulations

Models based on lumped elements are simple ones to represent systems composed by distributed parameters when models with lumped elements use a lot of these elements. However, such mentioned models usually present numerical problems because it is subject to the influence of numerical oscillations. When introducing new elements, such as damping resistors, in the mentioned models, the numerical oscillations can be minimized. The introduction of new elements increases the computational time for the analysis and simulations with the modified model. Thus, by applying π circuit cascades, simulations with abrupt changes in voltage or current are carried out using two different structures of π circuits with the purpose to achieve a similar accuracy and less computational time than those obtained with application damping resistor in every π circuit units.

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.