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This paper proposes new iterative methods for the efficient computation of the smallest eigenvalue of the symmetric nonlinear matrix eigenvalue problems of large order with a monotone dependence on the spectral parameter. Monotone nonlinear eigenvalue problems for differential equations have important applications in mechanics and physics. The discretization of these eigenvalue problems leads to ill-conditioned nonlinear eigenvalue problems with very large sparse matrices monotone depending on the spectral parameter. To compute the smallest eigenvalue of large matrix nonlinear eigenvalue problem, we suggest preconditioned iterative methods: preconditioned simple iteration method, preconditioned steepest descent method, and preconditioned conjugate gradient method. These methods use only matrix-vector multiplications, preconditioner-vector multiplications, linear operations with vectors and inner products of vectors. We investigate the convergence and derive grid-independent error estimates of these methods for computing eigenvalues. Numerical experiments demonstrate practical effectiveness of the proposed methods for a class of mechanical problems.

This paper proposes new iterative methods for the efficient computation of the smallest eigenvalue of the symmetric nonlinear matrix eigenvalue problems of large order with a monotone dependence on the spectral parameter. Monotone nonlinear eigenvalue problems for differential equations have important applications in mechanics and physics. The discretization of these eigenvalue problems leads to ill-conditioned nonlinear eigenvalue problems with very large sparse matrices monotone depending on the spectral parameter. To compute the smallest eigenvalue of large matrix nonlinear eigenvalue problem, we suggest preconditioned iterative methods: preconditioned simple iteration method, preconditioned steepest descent method, and preconditioned conjugate gradient method. These methods use only matrix-vector multiplications, preconditioner-vector multiplications, linear operations with vectors and inner products of vectors. We investigate the convergence and derive grid-independent error estimates of these methods for computing eigenvalues. Numerical experiments demonstrate practical effectiveness of the proposed methods for a class of mechanical problems.

Submitted by Geandra Rodrigues (geandrar@gmail.com) on 2018-03-27T14:05:45Z No. of bitstreams: 1 franciscocarlosrodriguescoelho.pdf: 4772391 bytes, checksum: e11633134429c05832808dad96be9940 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2018-03-27T14:28:31Z (GMT) No. of bitstreams: 1 franciscocarlosrodriguescoelho.pdf: 4772391 bytes, checksum: e11633134429c05832808dad96be9940 (MD5) / Made available in DSpace on 2018-03-27T14:28:31Z (GMT). No. of bitstreams: 1 franciscocarlosrodriguescoelho.pdf: 4772391 bytes, checksum: e11633134429c05832808dad96be9940 (MD5) Previous issue date: 2018-02-22 / A alocação de Geração Distribuída (GD) em sistemas de distribuição de energia elétrica consiste em determinar os barramentos para conexão destas unidades geradoras, e o montante de potência a ser injetado, visando um ou mais objetivos, que podem ser: redução das perdas de potência ativa, melhorias no perfil de tensão, minimização dos custos operacionais, maximização da geração de energia, ganhos ambientais, dentre outros. O principal objetivo considerado neste trabalho é a minimização das perdas de potência ativa, mantendo as tensões dos barramentos dentro de limites recomendados. Para alcançar este objetivo, uma metodologia de otimização é proposta, tratando separadamente os problemas de localização das unidades geradoras no sistema, e o dimensionamento destas unidades. A determinação das barras com conexão de GD é realizada através de uma nova técnica de otimização metaheurística, implementada no MATLAB, denominada Metaheurística Empírica Discreta (MED). Já o dimensionamento das unidades de GD é realizado de duas formas distintas, a depender do tipo de sistema de distribuição analisado. No caso dos sistemas cujos dados são equivalentes monofásicos, o montante de potencia é determinado por um Fluxo de Potência Ótimo implementado no software comercial LINGO. A segunda estratégia de determinação da potência despachada é empregada no caso dos testes realizados com sistemas trifásicos desbalanceados, cujo dimensionamento é feito pelo método do gradiente descendente e o cálculo do fluxo de potência é realizado pelo software OpenDSS. Os três sistemas equivalentes monofásicos utilizados são compostos por 33, 69 e 476 barras, enquanto os dois trifásicos desequilibrados possuem 34 e 123 barras. A qualidade da metodologia proposta na resolução do problema de alocação de geração distribuída é avaliada através de comparações com a literatura especializada, comparações com outras metaheurísticas e testes de robustez. Os resultados provenientes de simulações com alocação de três e quatro unidades de GD em sistemas de distribuição de energia elétrica mostram que a metodologia proposta é eficiente, sendo capaz de produzir resultados com significativas reduções nas perdas de potência ativa e perfis de tensão adequados. / The optimal Distributed Generation (DG) allocation problem consists in choosing the best locations of those distributed power plants at the distribution system, and to define its amount of power injection. The approach can be either single or multiobjective. The main objectives are: minimization of total power loss, voltage profile improvement, operational cost minimization, maximization of distributed generation capacity, environmental gains, among others. In this work, the main goal pursued is the total power loss minimization of the distribution system, keeping the buses voltages within the predetermined limits. To achieve this goal, an optimization methodology is proposed. This approach treats separately the location problem and the power dispatched by the generation units. The busbars connected to distributed generation are determined through a new metaheuristic algorithm, implemented in MATLAB, named Empirical Discrete Metaheuristic (EDM). The amount of power injection is solved by an Optimum Power Flow implemented in the commercial software LINGO, or by the Steepest Descent Method in the MATLAB environment. The first strategy to determine the DG dispatch is used on simulations with single phase equivalents systems. The second one is employed in the amount of power determination in unbalanced three phase systems, which the power flow is carried out by the open source software OpenDSS. The three single phase equivalent test systems analyzed are composed by 33, 69 and 476 buses, while the two systems with three phases have 34 and 123 buses, each. To evaluate the proposed methodology quality, comparisons to published works in the specialized literature are made. Also, robustness tests and comparisons to other well succeed metaheuristics are carried out. The results were obtained from simulations with three and four DG units in electric power distribution systems. These results consistently show that the proposed methodology is efficient, providing DGs configurations that significantly reduces the active power losses and keep the voltages at adequate levels.

O objetivo principal deste trabalho é o estudo da regularidade anallítica global de certos operadores diferenciais definidos no toro. Uma ferramenta fundamental utilizada neste estudo são as séries parciais de Fourier, que nos permitem caracterizar tanto as distribuições periódicas quanto as funções anallíticas reais periódicas através do comportamento assintótico de seus coeficientes parciais de Fourier. Neste sentido, apresentamos também um estudo detalhado das relações destes objetos com seus coeficientes parciais de Fourier / The main goal of this work is to study global analytic regularity properties of certain differential operators acting in the torus. A main tool that will be used to achieve our goals are the partial Fourier series, which allow us to characterize objects such as periodic distributions or periodic real analytic functions in terms of the growth of their partial Fourier coefficients

O objetivo principal deste trabalho é o estudo da regularidade anallítica global de certos operadores diferenciais definidos no toro. Uma ferramenta fundamental utilizada neste estudo são as séries parciais de Fourier, que nos permitem caracterizar tanto as distribuições periódicas quanto as funções anallíticas reais periódicas através do comportamento assintótico de seus coeficientes parciais de Fourier. Neste sentido, apresentamos também um estudo detalhado das relações destes objetos com seus coeficientes parciais de Fourier / The main goal of this work is to study global analytic regularity properties of certain differential operators acting in the torus. A main tool that will be used to achieve our goals are the partial Fourier series, which allow us to characterize objects such as periodic distributions or periodic real analytic functions in terms of the growth of their partial Fourier coefficients