Primal-dual Logarithmic Barrier Research Articles

State Estimation of the Tanzanian Power System Network Using Non-Quadratic Criterion and MATLAB Environment

Power system state estimation is an effective online tool for monitoring, control and for providing consistent database in energy management systems. This paper presents an algorithm for state estimation of the Tanzanian power system network using a non-quadratic state criterion. Equality and inequality constraints existing in a power system are included in formulating the estimation problem. Equality constraints are target values used in load flow analysis and are included in power system state estimation in order to restore observability to those parts of the power system network which are permanently or temporarily unobservable. Inequality constraints are limits such as minimum and maximum reactive power generation, transformer tap and phase-shift. The solution techniques used is primal-dual interior point logarithmic barrier functions to treat the inequality constraints. An algorithm is developed using the method and a program coded in MATLAB is applied in implementing the simulation. Computational issues arising in the implementation of the algorithm are presented. The simulation results demonstrate that the primal-dual logarithmic barrier interior point algorithm is a useful numerical tool to compute the state of an electrical power system network. The inequality constraints play essential role in enhancing the reliability of the estimation results. Also, it is expected that significant benefit could be gained from application of the constrained state estimation algorithm to the Tanzanian power system network.

A resolução do problema de despacho ótimo de reativos pelo método da função lagrangiana-barreira relaxada

Este artigo apresenta a aplicação do método da função Lagrangiana-barreira relaxada ao problema de despacho ótimo de reativos, o qual é um problema não-linear não convexo e de grande porte. Nesta abordagem as restrições de desigualdade são tratadas pela associação dos métodos de barreira modificada e primal-dual barreira logarítmica. Essas restrições são transformadas em igualdades através de variáveis auxiliares positivas e são relaxadas pelo parâmetro de barreira. Uma função Lagrangiana é associada ao problema modificado. As condições necessárias de primeira-ordem são aplicadas gerando um sistema não-linear o qual é resolvido pelo método de Newton. A relaxação das variáveis auxiliares resulta na expansão da região factível do problema original, permitindo atingir o limite das restrições de desigualdade. Testes numéricos nos sistemas CESP 53 barras, o equivalente brasileiro sul-sudeste e o teste comparativo com o método primal-dual barreira logarítmica indicam que o método apresentado é eficiente na resolução do problema de despacho ótimo de reativos.

Fluxo de potência ótimo reativo via método da função lagrangiana barreira modificada

Este artigo apresenta uma nova abordagem para a resolução do problema de Fluxo de Potência Ótimo Reativo. Nesta abordagem, as restrições de desigualdade são tratadas pela associação dos métodos de Barreira Modificada e Primal-Dual Barreira Logarítmica (PDBL). As restrições de desigualdade são transformadas em igualdades introduzindo variáveis de folga positivas e são relaxadas através do parâmetro de barreira. Uma função Lagrangiana é associada ao problema modificado. As condições necessárias de primeira ordem são aplicadas à função Lagrangiana, gerando um sistema de equações não-lineares, o qual é linearizado pelo método de Newton. A relaxação das variáveis de folga resulta na expansão da região factível do problema original, permitindo que os limites das restrições de desigualdade sejam atingidos. Testes numéricos utilizando os sistemas CESP-440kV e SUL-SUDESTE BRASILEIRO e um teste comparativo com o método PDBL indicam que a nova abordagem é eficiente na resolução do problema Fluxo de Potência Ótimo Reativo.

Optimal Reactive Dispatch Problem via Improved Newton Approach

This article proposes a new approach to solve the optimal reactive dispatch problem, based on Newton approach and the primal-dual logarithmic barrier method. A Lagrangian function is associated with the modified problem. The first-order necessary conditions for optimality are fulfilled by Newton's method and by updating the penalty and barrier terms. The proposed approach does not require the set of binding constraints to be identified and can be utilized from an infeasible starting point. The effectiveness of the proposed approach has been examined by solving the Brazilian 53-bus and 662-bus systems.

Primal-Dual Logarithmic Barrier and Augmented Lagrangian Function to the Loss Minimization in Power Systems

This article presents a new approach to minimize the losses in electrical power systems. This approach considers the application of the primal-dual logarithmic barrier method to voltage magnitude and tap-changing transformer variables, and the other inequality constraints are treated by augmented Lagrangian method. The Lagrangian function aggregates all the constraints. The first-order necessary conditions are reached by Newton's method, and by updating the dual variables and penalty factors. Test results are presented to show the good performance of this approach.

On a Nonlinear Multiple-Centralitycorrections Interior-Point Method for Optimal Power Flow

Large-scale nonlinear optimal power flow (OPF) problems have been efficiently solved lately by extensions from linear programming to nonlinear programming of the primal-dual logarithmic barrier interior-point method and its predictor-corrector variant. Motivated by the impressive performance of the nonlinear predictor-corrector extension, in this paper we extend from linear programming to nonlinear OPF the efficient multiple centrality corrections (MCC) technique that was developed by Gondzio. The numerical performance of the proposed MCC algorithm is evaluated on a set of power networks ranging in size from 118 buses to 2098 buses. Extensive computational results demonstrate that the MCC technique is fast and robust, and outperforms the successful predictor-corrector technique.

On a nonlinear multiple-centrality-corrections interior-point method for optimal power flow

Large scale nonlinear optimal power flow (OPF) problems have been efficiently solved by extensions from linear programming to nonlinear programming of the primal-dual logarithmic barrier interior-point method and its predictor-corrector variant. Motivated by the impressive performance of the nonlinear predictor-corrector extension, in this paper we extend from linear programming to nonlinear OPF the efficient multiple centrality corrections (MCC) technique that was developed by Gondzio. The numerical performance of the proposed MCC algorithm is evaluated on a set of power networks ranging in size from 118 buses to 2098 buses. Extensive computational results demonstrate that the MCC technique is fast and robust, and outperforms the successful predictor-corrector technique.

Atomic Decomposition by Basis Pursuit

The time-frequency and time-scale communities have recently developed a large number of overcomplete waveform dictionaries---stationary wavelets, wavelet packets, cosine packets, chirplets, and warplets, to name a few. Decomposition into overcomplete systems is not unique, and several methods for decomposition have been proposed, including the method of frames (MOF), matching pursuit (MP), and, for special dictionaries, the best orthogonal basis (BOB). Basis pursuit (BP) is a principle for decomposing a signal into an "optimal"' superposition of dictionary elements, where optimal means having the smallest l1 norm of coefficients among all such decompositions. We give examples exhibiting several advantages over MOF, MP, and BOB, including better sparsity and superresolution. BP has interesting relations to ideas in areas as diverse as ill-posed problems, abstract harmonic analysis, total variation denoising, and multiscale edge denoising. BP in highly overcomplete dictionaries leads to large-scale optimization problems. With signals of length 8192 and a wavelet packet dictionary, one gets an equivalent linear program of size 8192 by 212,992. Such problems can be attacked successfully only because of recent advances in linear and quadratic programming by interior-point methods. We obtain reasonable success with a primal-dual logarithmic barrier method and conjugate-gradient solver.

Comparative studies of optimization methods for the optimal power flow problem

Abstract In this paper we have investigated three optimization approaches for solving the optimal power flow (OPF) problem: active set and penalty (ASP), primal–dual (PD), and primal–dual logarithmic-barrier (PDLB). All three approaches are Newton-based methods. This paper compares the main features of the above and reviews their methodologies. Results obtained in performance tests with traditional networks are summarized.

Atomic Decomposition by Basis Pursuit

The time-frequency and time-scale communities have recently developed a large number of overcomplete waveform dictionaries --- stationary wavelets, wavelet packets, cosine packets, chirplets, and warplets, to name a few. Decomposition into overcomplete systems is not unique, and several methods for decomposition have been proposed, including the method of frames (MOF), Matching pursuit (MP), and, for special dictionaries, the best orthogonal basis (BOB). Basis Pursuit (BP) is a principle for decomposing a signal into an "optimal" superposition of dictionary elements, where optimal means having the smallest l1 norm of coefficients among all such decompositions. We give examples exhibiting several advantages over MOF, MP, and BOB, including better sparsity and superresolution. BP has interesting relations to ideas in areas as diverse as ill-posed problems, in abstract harmonic analysis, total variation denoising, and multiscale edge denoising. BP in highly overcomplete dictionaries leads to large-scale optimization problems. With signals of length 8192 and a wavelet packet dictionary, one gets an equivalent linear program of size 8192 by 212,992. Such problems can be attacked successfully only because of recent advances in linear programming by interior-point methods. We obtain reasonable success with a primal-dual logarithmic barrier method and conjugate-gradient solver.

Solving a class of LP problems with a primal-dual logarithmic barrier method

Applying a higher order primal-dual logarithmic barrier method for solving large real-life linear programming problems is addressed in this paper. The efficiency of an interior point algorithm on these problems is compared with the one of the state-of-the-art simplex code MINOS version 5.3. Based on such experience, a wide class of LP problems is identified for which logarithmic barrier approach seems advantageous over the simplex one. Additionally, some practical rules for model builders are derived that should allow them to create problems that can easily be solved with logarithmic barrier algorithms.

Primal-dual algorithms for linear programming based on the logarithmic barrier method

In this paper, we deal with primal-dual interior point methods for solving the linear programming problem. We present a short-step and a long-step path-following primal-dual method and derive polynomial-time bounds for both methods. The iteration bounds are as usual in the existing literature, namely $$O(\sqrt n L)$$ iterations for the short-step variant andO(nL) for the long-step variant. In the analysis of both variants, we use a new proximity measure, which is closely related to the Euclidean norm of the scaled search direction vectors. The analysis of the long-step method depends strongly on the fact that the usual search directions form a descent direction for the so-called primal-dual logarithmic barrier function.