Deterministic model of electric arc furnace – a closed form solution

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 equation. The goal of this paper 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.

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.

The paper provides a set of parameters for which the solution to the nonlinear differential equation describing electric arc can be obtained in a closed form.

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.

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.

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.