Alternative proposal for Modal Representation of a Non-transposed Three-Phase Transmission Line 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 a, b 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.

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.

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.

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. © 2003-2012 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.

Um modelo de linha de transmissão bifásica desenvolvido diretamente no domínio das fases

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

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.

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.

A procedure to estimate parameters of a line segment taking into account its representation through pi circuits: Theoretical development

The objective of this paper is to show an alternative methodology to estimate per unit length parameters of a line segment of a transmission line. With this methodology the line segment parameters can be obtained starting from the phase currents and -voltages in receiving and sending end of the line segment. If the line segment is represented as being one or more pi circuits whose frequency dependent parameters are considered lumped, its impedance and admittance can be easily expressed as functions of the currents and voltages at the sending and receiving end. Because we are supposing that voltages and currents at the sending and receiving end of the tine segment (in frequency domain) are known, it is possible to obtains its impedance and admittance and consequently its per unit length longitudinal and transversal parameters. The procedure will be applied to estimate the longitudinal and transversal parameters of a small segment of a single-phase line that is already built.

A procedure to estimate parameters of a line segment taking into account its representation through π circuits: Theoretical development

The objective of this paper is to show an alternative methodology to estimate per unit length parameters of a line segment of a transmission line. With this methodology the line segment parameters can be obtained starting from the phase currents and voltages in receiving and sending end of the line segment. If the line segment is represented as being one or more π circuits whose frequency dependent parameters are considered lumped, its impedance and admittance can be easily expressed as functions of the currents and voltages at the sending and receiving end. Because we are supposing that voltages and currents at the sending and receiving end of the line segment (in frequency domain) are known, it is possible to obtains its impedance and admittance and consequently its per unit length longitudinal and transversal parameters. The procedure will be applied to estimate the longitudinal and transversal parameters of a small segment of a single-phase line that is already built. © 2006 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.

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. © 2012 IEEE.

Estimação de parâmetros de linhas de transmissão utilizando técnicas de decomposição modal: Aplicação em linhas com plano de simetria vertical

Pós-graduação em Engenharia Elétrica - FEIS

An efficient Hopfield network to solve economic dispatch problems with transmission system representation

Economic dispatch (ED) problems have recently been solved by artificial neural network approaches. Systems based on artificial neural networks have high computational rates due to the use of a massive number of simple processing elements and the high degree of connectivity between these elements. The ability of neural networks to realize some complex non-linear function makes them attractive for system optimization. All ED models solved by neural approaches described in the literature fail to represent the transmission system. Therefore, such procedures may calculate dispatch policies, which do not take into account important active power constraints. Another drawback pointed out in the literature is that some of the neural approaches fail to converge efficiently toward feasible equilibrium points. A modified Hopfield approach designed to solve ED problems with transmission system representation is presented in this paper. The transmission system is represented through linear load flow equations and constraints on active power flows. The internal parameters of such modified Hopfield networks are computed using the valid-subspace technique. These parameters guarantee the network convergence to feasible equilibrium points, which represent the solution for the ED problem. Simulation results and a sensitivity analysis involving IEEE 14-bus test system are presented to illustrate efficiency of the proposed approach. (C) 2004 Elsevier Ltd. All rights reserved.