Deterministic model of electric arc furnace – a closed form solution

Abstract

Purpose -Electric arc furnaces are usually modelled using combined models which divide the phenomenon taking place in real objects into a deterministic and a stochastic or chaotic parts. The former is expressed by a nonlinear differential equations. The paper goal was to obtain a closed form of the solution to one of the most popular nonlinear differential equations used for the AC electric arc modelling. Design/methodology/approach -The solution has been obtained in the time domain by a sequence of transformations of the original nonlinear equation which lead to a linear equation, for which a closed form solution is known. Findings -A set of parameters for which the solution to the nonlinear differential equation describing electric arc can be obtained in a closed form. Research limitations/implications -There are still some parameter values for which the solution can be found only numerically. Moreover, due to the nature of the phenomena occurring in electric arc furnaces in order to build a complete model of the arc the deterministic model must be extended using for example stochastic approach. Practical implications -The obtained results enable determination of exact waveforms of the arc voltage or radius without application of numerical algorithms for ODE solving. The arc model can be used to evaluate the impact of arc furnaces on power quality during the planning stage of new plants. The proposed approach facilitates calculation of the arc characteristic. Originality/value -The importance of having a closed form of the solution instead of the numerical ones comes from new possible ways of extension of the arc model in order to cover the time-varying nature of the arc waveforms. So far the equation has been solved only using numerical algorithms.

Electric Arc Furnace Power Quality Analysis Based on a Stochastic Arc Model

“…(2) The arc equation 1for n = 2 and m = 0, 1 or 2 can be brought to a linear equation [17], [18]. Its solution is expressed by [19]: β e τ e dτ ,…”

Quadratic polynomial form of electric arc furnace equation

“…Application of a substitution given in Grabowski and Walczak (2013) allows to convert (1) into the following first-order differential equation:…”

“…It is made with the aid of chaotic, stochastic or modulated components which are added to the deterministic solution [2], [3], [4]. This paper is a continuation of works [5] and [6] in which a closed form solution to one of the most popular deterministic nonlinear differential equations used for the arc modeling [7] has been developed for some arc parameters. It seems that a closed form solution for the other arc parameters does not exists [8].…”