Electromagnetic interference from variable speed motor drives

A methodology is presented which can be used to produce the level of electromagnetic interference, in the form of conducted and radiated emissions, from variable speed drives, the drive that was modelled being a Eurotherm 583 drive. The conducted emissions are predicted using an accurate circuit model of the drive and its associated equipment. The circuit model was constructed from a number of different areas, these being: the power electronics of the drive, the line impedance stabilising network used during the experimental work to measure the conducted emissions, a model of an induction motor assuming near zero load, an accurate model of the shielded cable which connected the drive to the motor, and finally the parasitic capacitances that were present in the drive modelled. The conducted emissions were predicted with an error of +/-6dB over the frequency range 150kHz to 16MHz, which compares well with the limits set in the standards which specify a frequency range of 150kHz to 30MHz. The conducted emissions model was also used to predict the current and voltage sources which were used to predict the radiated emissions from the drive. Two methods for the prediction of the radiated emissions from the drive were investigated, the first being two-dimensional finite element analysis and the second three-dimensional transmission line matrix modelling. The finite element model took account of the features of the drive that were considered to produce the majority of the radiation, these features being the switching of the IGBT's in the inverter, the shielded cable which connected the drive to the motor as well as some of the cables that were present in the drive.The model also took account of the structure of the test rig used to measure the radiated emissions. It was found that the majority of the radiation produced came from the shielded cable and the common mode currents that were flowing in the shield, and that it was feasible to model the radiation from the drive by only modelling the shielded cable. The radiated emissions were correctly predicted in the frequency range 30MHz to 200MHz with an error of +10dB/-6dB. The transmission line matrix method modelled the shielded cable which connected the drive to the motor and also took account of the architecture of the test rig. Only limited simulations were performed using the transmission line matrix model as it was found to be a very slow method and not an ideal solution to the problem. However the limited results obtained were comparable, to within 5%, to the results obtained using the finite element model.

Characterization and modeling of multi-conductor transmission line using Finite-difference Time-Domain method

A two-dimensional, 2D, finite-difference time-domain (FDTD) method is used to analyze two different models of multi-conductor transmission lines (MTL). The first model is a two-conductor MTL and the second is a threeconductor MTL. Apart from the MTL's, a three-dimensional, 3D, FDTD method is used to analyze a three-patch microstrip parasitic array. While the MTL analysis is entirely in time-domain, the microstrip parasitic array is a study of scattering parameter Sn in the frequency-domain. The results clearly indicate that FDTD is an efficient and accurate tool to model and analyze multiconductor transmission line as well as microstrip antennas and arrays.

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 Clarke's 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 constructive heuristic algorithm to Short Term Transmission Network Expansion Planning

In this paper a method for solving the Short Term Transmission Network Expansion Planning (STTNEP) problem is presented. The STTNEP is a very complex mixed integer nonlinear programming problem that presents a combinatorial explosion in the search space. In this work we present a constructive heuristic algorithm to find a solution of the STTNEP of excellent quality. In each step of the algorithm a sensitivity index is used to add a circuit (transmission line or transformer) to the system. This sensitivity index is obtained solving the STTNEP problem considering as a continuous variable the number of circuits to be added (relaxed problem). The relaxed problem is a large and complex nonlinear programming and was solved through an interior points method that uses a combination of the multiple predictor corrector and multiple centrality corrections methods, both belonging to the family of higher order interior points method (HOIPM). Tests were carried out using a modified Carver system and the results presented show the good performance of both the constructive heuristic algorithm to solve the STTNEP problem and the HOIPM used in each step.

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.

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.

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.

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.

Using Universal Line Model (ULM) for Representing Electromagnetic Transients in Three-Phase Transmission Lines

The second-order differential equations that describe the polyphase transmission line are difficult to solve due to the mutual coupling among them and the fact that the parameters are distributed along their length. A method for the analysis of polyphase systems is the technique that decouples their phases. Thus, a system that has n phases coupled can be represented by n decoupled single-phase systems which are mathematically identical to the original system. Once obtained the n-phase circuit, it's possible to calculate the voltages and currents at any point on the line using computational methods. The Universal Line Model (ULM) transforms the differential equations in the time domain to algebraic equations in the frequency domain, solve them and obtain the solution in the frequency domain using the inverse Laplace transform. This work will analyze the method of modal decomposition in a three-phase transmission line for the evaluation of voltages and currents of the line during the energizing process.

Formulas for algorithmic computation of impedance matrix using general circuit constants

A set of formulas is derived from general circuit constants which facilitates formation of the impedance matrix of a power system by the bus-impedance method. The errors associated with the lumpedparameter representation of a transmission line are thereby eliminated. The formulas are valid for short lines also, if the relevant general circuit constants are employed. The mutual impedance between the added line and the existing system is not considered, but the approach suggested can well be extended to it.

Optimizing Product Line Designs: Efficient Methods and Comparisons

We compare a broad range of optimal product line design methods. The comparisons take advantage of recent advances that make it possible to identify the optimal solution to problems that are too large for complete enumeration. Several of the methods perform surprisingly well, including Simulated Annealing, Product-Swapping and Genetic Algorithms. The Product-Swapping heuristic is remarkable for its simplicity. The performance of this heuristic suggests that the optimal product line design problem may be far easier to solve in practice than indicated by complexity theory.

Esquemas alternativos para o passo de parametrização do método da continuação baseados em parâmetros físicos

O método de fluxo de carga convencional é considerado inadequado para se obter o ponto de máximo carregamento (PMC) de sistemas de potência, devido à singularidade da matriz Jacobiana neste ponto. Os métodos da continuação são ferramentas eficientes para a solução deste tipo de problema, visto que técnicas de parametrização podem ser utilizadas para evitar a singularidade da matriz Jacobiana. Neste trabalho, novas opções para a etapa de parametrização do método da continuação são apresentadas. Mostra-se que variáveis com claro significado físico podem ser utilizadas na etapa de parametrização. As seguintes variáveis foram testadas: perda total de potência ativa e reativa, potência ativa e reativa na barra de referência, potência reativa das barras de geração, e as perdas de potência ativa e reativa nas linhas de transmissão (LT). Além de facilitar a implementação computacional do método de continuação, as técnicas de parametrização apresentadas simplificam a definição matemática e o entendimento do método por parte de engenheiros de potência, visto que os métodos de continuação existentes na literatura sempre utilizam técnicas de parametrização complexas, e de interpretação puramente geométrica. Resultados obtidos com a nova metodologia para os sistemas testes do IEEE (14, 30, 57 e 118 barras) mostram que as características de convergência do método de fluxo de carga convencional são melhoradas na região do PMC. Além disso, durante o traçado das curvas PV, as diversas técnicas de parametrização podem ser comutadas entre si possibilitando o cálculo de todos os pontos da curva com um número reduzido de iterações. Diversos testes são realizados para proporcionar a comparação do desempenho dos esquemas de parametrização propostos.

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 pi 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.

Study of alternative schemes for the parameterization step of the continuation power flow method based on physical parameters, part I: Mathematical modeling

The conventional power flow method is considered to be inadequate to obtain the maximum loading point because of the singularity of Jacobian matrix. Continuation methods are efficient tools for solving this kind of problem since different parameterization schemes can be used to avoid such ill-conditioning problems. This paper presents the details of new schemes for the parameterization step of the continuation power flow method. The new parameterization options are based on physical parameters, namely, the total power losses (real and reactive), the power at the slack bus (real or reactive), the reactive power at generation buses, and transmission line power losses (real and reactive). The simulation results obtained with the new approach for the IEEE test systems (14, 30, 57, and 118 buses) are presented and discussed in the companion paper. The results show that the characteristics of the conventional method are not only preserved but also improved.

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.