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.

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 alpha, beta 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 nontransposed three-phase line with a vertical symmetry plane whose nominal voltage is 440 kV and line length is 500 km.

Step by step analyses of Clarke's matrix correction procedure for untransposed three-phase transmission line cases

The correction procedure for Clarke's matrix, considering three-phase transmission line analyzes, is analyzed step by step in this paper, searching to improve the application of this procedure. Changing the eigenvectors as modal transformation matrices, Clarke's matrix has been applied to analyses for transposed and untransposed three-phase transmission line cases. It is based on the fact that Clarke's matrix is an eigenvector matrix for transposed three-phase transmission lines considering symmetrical and asymmetrical cases. Because of this, the application of this matrix has been analyzed considering untransposed three-phase transmission lines. In most of these cases, the errors related to the eigenvalues can be considered negligible. It is not true when it is analyzed the elements that are not in main diagonal of the quasi-mode matrix. This matrix is obtained from the application of Clarke's matrix. The quasi-mode matrix is correspondent to the eigenvalue matrix. Their off-diagonal elements represent couplings among the quasi-modes. So, the off-diagonal quasi-mode element relative values are not negligible when compared to the eigenvalues that correspond to the coupled quasi-modes. Minimizing these relative values, the correction procedure is analyzed in detail, checking some alternatives for the correction procedure application.

Unbalanced Three-Phase Induction Motors Starting and Arc Welding Machines Short-Circuit Modeling

Three-Phase Induction Motors (TIM) and Arc Welding Machines (AWM) are loads of special behavior widely used in industrial and commercial installations, and therefore may contribute significantly to the deterioration of the quality of energy supplied by utilities. This paper proposes a modeling in constant power of the unbalanced TIM starting using Genetic Algorithm (GA) and AWM short-circuit based on their statics characteristics curves. The proposed models are compared with the conventional models in the literature. The results showed the good performance of the proposed models, allowing a more precise analysis of the real requests of these loads.

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.

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.

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.

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.

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.

A true programmable HPF hybrid three-phase rectifier

In this paper is proposed and analyzed a digital hysteresis modulation using a FPGA (Field Programmable Gate Array) device and VHDL (Hardware Description Language), applied at a hybrid three-phase rectifier with almost unitary input power factor, composed by parallel SEPIC controlled single-phase rectifiers connected to each leg of a standard 6-pulses uncontrolled diode rectifier. The digital control allows a programmable THD (Total Harmonic Distortion) at the input currents, and it makes possible that the power rating of the switching-mode converters, connected in parallel, can be a small fraction of the total average output power, in order to obtain a compact converter, reduced input current THD and almost unitary input power factor. The proposed digital control, using a FPGA device and VHDL, offers an important flexibility for the associated control technique, in order to obtain a programmable PFC (Power Factor Correction) hybrid three-phase rectifier, in agreement with the international standards (IEC, and IEEE), which impose limits for the THD of the AC (Alternate Current) line input currents. Finally, the proposed control strategy is verified through experimental results from an implemented prototype. ©2008 IEEE.

Hybrid three-phase HPF rectifier with programmable input current THD, using FPGA device and VHDL language

In this paper is proposed and analyzed a digital hysteresis modulation using a FPGA (Field Programmable Gate Array) device and VHDL (Hardware Description Language), applied at a hybrid three-phase rectifier with almost unitary input power factor, composed by parallel SEPIC controlled single-phase rectifiers connected to each leg of a standard 6-pulses uncontrolled diode rectifier. The digital control allows a programmable THD (Total Harmonic Distortion) at the input currents, and it makes possible that the power rating of the switching-mode converters, connected in parallel, can be a small fraction of the total average output power, in order to obtain a compact converter, reduced input current THD and almost unitary input power factor. Finally, the proposed digital control, using a FPGA device and VHDL, offers an important flexibility for the associated control technique, in order to obtain a programmable PFC (Power Factor Correction) hybrid three-phase rectifier, in agreement with the international standards (IEC, and IEEE), which impose limits for the THD of the AC (Alternate Current) line input currents. The proposed strategy is verified by experiments. © 2008 IEEE.

A comparative analysis of FBD, PQ and CPT current decompositions - Part II: Three-phase four-wire systems

This paper investigates the major similarities and discrepancies among three important current decompositions proposed for the interpretation of unbalanced and/or non linear three-phase four-wire power circuits. The considered approaches were the so-called FBD Theory, the pq-Theory and the CPT. Although the methods are based on different concepts, the results obtained under ideal conditions (sinusoidal and balanced signals) are very similar. The main differences appear in the presence of unbalanced and non linear load conditions. It will be demonstrated and discussed how the choice of the voltage referential and the return conductor impedance can influence in the resulting current components, as well as, the way of interpreting a power circuit with return conductor. Under linear unbalanced conditions, both FBD and pq-Theory suggest that the some current components contain a third-order harmonic. Besides, neither pq-Theory nor FBD method are able to provide accurate information for reactive current under unbalanced and distorted conditions, what can be done by means of the CPT. © 2009 IEEE.

AC-DC three-phase multipulse converter with boost DC-DC stage and constant hysteresis control

This work proposes a new three-phase multipulse rectifier based on the delta autotransformer connection with DC-DC Boost stages and constant hysteresis control which has the objective of providing a reliable DC bus for on-board applications, electric motor drives and similars, always considering power quality issues. Thus, the proposal presents 0.99 power factor, 6% harmonic distortions in the currents from the mains and enhanced magnetic core utilization, which results in low weight and volume for the overall converter. The proposed control technique uses the simple constant hysteresis concept, thus leading to a low-cost but effective and reliable strategy. © 2011 IEEE.

Comparative analysis for reactive energy measurement methodologies, under non-sinusoidal conditions in three-phase four-wire circuits

This paper presents some methodologies for reactive energy measurement, considering three modern power theories that are suitable for three-phase four-wire non-sinusoidal and unbalanced circuits. The theories were applied in some profiles collected in electrical distribution systems which have real characteristics for voltages and currents measured by commercial reactive energy meters. The experimental results are presented in order to analyze the accuracy of the methodologies, considering the standard IEEE 1459-2010 as a reference. Finally, for additional comparisons, the theories will be confronted with the modern Yokogawa WT3000 energy meter and three samples of a commercial energy meter through an experimental setup. © 2011 IEEE.

An analysis of Generalized Symmetrical Components in non sinusoidal three phase systems

This paper presents the study of the so called Generalized Symmetrical Components, proposed by Tenti et. al. to the analysis of unbalanced periodic non sinusoidal three phase systems. As a result, it was possible to establish a proper relationship between such of generalized symmetrical components and Fortescue symmetrical components to the harmonic frequencies that compose a generic periodic non sinusoidal three phase system. © 2011 IEEE.