Convergence analysis of the triangular-based power flow method for AC distribution grids

<p>This paper addresses the convergence analysis of the triangular-based power flow (PF) method in alternating current radial distribution networks. The PF formulation is made via upper-triangular matrices, which enables finding a general iterative PF formula that does not require admittance matrix calculations. The convergence analysis of this iterative formula is carried out by applying the Banach fixed-point theorem (BFPT), which allows demonstrating that under an adequate voltage profile the triangular-based PF always converges. Numerical validations are made, on the well-known 33 and 69 distribution networks test systems. Gauss-seidel, newton-raphson, and backward/forward PF methods are considered for the sake of comparison. All the simulations are carried out in MATLAB software.</p>

An improved Newton-Raphson based linear power flow method for DC grids with dispatchable DGs and ZIP loads

Purpose This study aims to provide a solution of the power flow calculation for the low-voltage ditrect current power grid. The direct current (DC) power grid is becoming a reliable and economic alternative to millions of residential loads. The power flow (PF) in the DC network has some similarities with the alternative current case, but there are important differences that deserve to be further concerned. Moreover, the dispatchable distributed generators (DGs) in DC network can realize the flexible voltage control based on droop-control or virtual impedance-based methods. Thus, DC PF problems are still required to further study, such as hosting all load types and different DGs. Design/methodology/approach The DC power analysis was explored in this paper, and an improved Newton–Raphson based linear PF method has been proposed. Considering that constant impedance (CR), constant current (CI) and constant power (CP) (ZIP) loads can get close to the practical load level, ZIP load has been merged into the linear PF method. Moreover, DGs are much common and can be easily connected to the DC grid, so V nodes and the dispatchable DG units with droop control have been further taken into account in the proposed method. Findings The performance and advantages of the proposed method are investigated based on the results of the various test systems. The two existing linear models were used to compare with the proposed linear method. The numerical results demonstrate enough accuracy, strong robustness and high computational efficiency of the proposed linear method even in the heavily-loaded conditions and with 10 times the line resistances. Originality/value The conductance corresponding to each constant resistance load and the equivalent conductance for the dispatchable unit can be directly merged into the self-conductance (diagonal component) of the conductance matrix. The constant current loads and the injection powers from dispatchable DG units can be treated as the current sources in the proposed method. All of those make the PF model much clear and simple. It is capable of offering enough accuracy level, and it is suitable for applications in DC networks that require a large number of repeated PF calculations to optimize the energy flows under different scenarios.

Performance of textured foil journal bearing considering the influence of relative texture depth

Both foil structure and surface texturing have been widely used to improve bearing performance. However, there is little research on their combination, namely, textured gas foil bearing. This paper adopts the Reynolds equation as the pressure governing equation of bump-type foil journal bearing to study the influence of textures located on the top foil. The Newton-Raphson iterative method and the perturbation method are employed to obtain static and dynamic characteristics, respectively. Thereafter, based on three texture distribution types, further analysis about the effect of the relative texture depth and the textured portion is carried out. The results indicate that an appropriate arrangement of textures could improve the performance of gas foil bearing. For #1 texture distribution, the maximum increment of load capacity could exceed 10% when ω = 1.4 × 105 r/min, ε = 0.2.

A Finite Difference Method for the Variational p-Laplacian

We propose a new monotone finite difference discretization for the variational p-Laplace operator, $$\Delta _pu=\text{ div }(|\nabla u|^{p-2}\nabla u),$$ Δ p u = div ( | ∇ u | p - 2 ∇ u ) , and present a convergent numerical scheme for related Dirichlet problems. The resulting nonlinear system is solved using two different methods: one based on Newton-Raphson and one explicit method. Finally, we exhibit some numerical simulations supporting our theoretical results. To the best of our knowledge, this is the first monotone finite difference discretization of the variational p-Laplacian and also the first time that nonhomogeneous problems for this operator can be treated numerically with a finite difference scheme.

Comparative Study of Various Iterative Numerical Methods for Computation of Approximate Root of the Polynomials

The aim of this article is to present a brief review and a numerical comparison of iterative methods applied to solve the polynomial equations with real coefficients. In this paper, four numerical methods are compared, namely: Horner’s method, Synthetic division with Chebyshev method (Proposed Method), Synthetic division with Modified Newton Raphson method and Birge-Vieta method which will helpful to the readers to understand the importance and usefulness of these methods.

Aplicación del método Newton-Raphson por polinomio cúbicos usando el método Tartaglia-Cardano

Algunos de los métodos usados para encontrar raíces de funciones en las matemáticas aplicadas son los algoritmos interactivos, que constituyen un campo de interés muy importante dentro de la disciplina debido a que permiten encontrar soluciones para usar la menor capacidad de recursos dentro de los sistemas computacionales. La propuesta de estos algoritmos iterativos implica la relación de diferentes áreas de las matemáticas: álgebra, cálculo diferencial, cálculo vectorial y análisis numérico. Esta relación conjuga la teoría de cálculo diferencial y el análisis numérico realizado por matemáticos ingleses, como Newton y Raphson. Estos propusieron la primera forma de buscar una raíz de una función usando iteraciones, la que a su vez se sustentó en teorías algebraicas desarrolladas por algunos matemáticos italianos, como Tartaglia y Cardano, quienes anteriormente habían encontrado las soluciones a las raíces de un polinomio de orden cúbico. Dentro del proceso deductivo que se siguió en el proceso, se utilizaron las aproximaciones recomendadas que se generaron al aplicar las Series de Taylor. Por otro lado, durante el proceso de experimentación se han identificado determinados patrones repetitivos, especialmente, cuando el punto de inicio se aleja de la raíz. Esto permitió mejorar el algoritmo iterativo al identificar el lugar de la curva de convergencia. Finalmente, se generalizó la ecuación iterativa para muchas variables, a partir de las deducciones utilizadas de 1 y 2 variables.