E. C. M. Costa

Proposal of a Transmission Line Model Based on Lumped Elements: An Analytic Solution

Abstract This article proposes a frequency-dependent transmission line model based on state-space techniques. To include the frequency effect associated with distributed parameters in the state matrices, the line frequency-dependent longitudinal parameters are fitted by a rational function and then introduced in lumped elements representation by an equivalent RL circuit; this procedure is known as vector fitting. Subsequently, the system of ordinary differential equations that represents the transients of currents and voltages on a three-phase transmission line is represented in state space. After the state equations are solved by numerical and analytic methods, the results are compared to a frequency-dependent lumped model implemented by the commercial software MICROTRAN (an electromagnetic transient program; Microtran Power Systems Analysis Corporation, Vancouver, Canada). Following this, the results obtained from the proposed lumped model are compared to results from a distributed-parameters model based on Fourier transform. Some of the principal advantages for using the proposed state-space model instead of a distributed-parameters modeling are the easy modeling of several electromagnetic phenomena over transmission lines directly in the time domain without the use of inverse transforms and that a detailed description of the voltages and currents along the line can be described with good approximation. These attributes are not observed for models based on distributed parameters. All simulations are performed considering the switching operation of a 440-kV three-phase line, and by this procedure, it is possible to analyze the accuracy of the proposed line modeling.

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.

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.

Analysis of the Electrical Characteristics of an Alternative Solution for the Brazilian-Amazon Transmission System

Abstract This article analyzes the electrical parameters of a 3-phase transmission line using a 280-m-high steel tower that has been proposed for the Amazon transmission system in Brazil. The height of the line conductors and the distance between them are intrinsically related to the longitudinal and transverse parameters of the line. Hence, an accurate study is carried out in order to show the electrical variations between a transmission line using the new technology and a conventional 3-phase 440-kV line, considering a wide range of frequencies and variable soil resistivity. First, a brief review of the fundamental theory of line parameters is presented. In addition, by using a digital line model, simulations are carried out in the time domain to analyze possible and critical over-voltage transients on the proposed line representation.

Equivalent Representation of Mutual Coupling in Transmission Lines by Discrete Element

This paper describes a computational model based on lumped elements for the mutual coupling between phases in 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 ofthe 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.

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.

Proposal of a simplified process to correct the phase decoupling using modal analysis

Modal analysis is widely approached in the classic theory of transmission line modeling. This technique is applied to model the three-phase representation of conventional electric systems taking into account their self and mutual electrical parameters. However the methodology has some particularities and inaccuracies for specific applications which are not clearly described in the basic references of this topic. This paper provides a thorough review of modal analysis theory applied to line models followed by an original and simple procedure to overcome the possible errors embedded in the modal decoupling through the three-phase system modeling.

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.

Corona Discharge Model for Transmission Lines by Lumped Elements

This paper shows the insertion of corona effect in a transmission line model based on lumped elements. The development is performed considering a frequency-dependent line representation by cascade of π sections and state equations. Hence, the detailed profile of currents and voltages along the line, described from a non-homogeneous system of differential equations, can be obtained directly in time domain applying numerical or analytic solution integration methods. The corona discharge model is also based on lumped elements and is implemented from the well-know Skilling-Umoto Model.

Assessment of expert panels

Expert panels are groups of people who work together to perform a given task, make diagnostics, or reach a decision. A good example of an expert panel is a software development team. The Condorcet-List theorem states that under certain conditions the proposals of a group may be more reliable than those taken by a single expert. In order to estimate the probability of an individual arriving at the correct choice, the authors of the present work use the Rasch model, as is usual in assessments. Besides evaluating individuals, the present paper uses the Rasch model to assess software development teams. To evaluate a team, it is proposed that each member of the group is able to give educated advice in each subtask of the problem. This is necessary, since the Condorcet-List theorem requires that decisions be taken by a majority vote, and that each voter has a probability greater than 0.5 of giving a correct feedback to his/her co-workers.