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Electric arc furnaces are very often modeled using combined models which cover separately deterministic and stochastic phenomena taking place in the furnace. The deterministic part is expressed by nonlinear differential equations. A closed form of the solution to one of the most popular nonlinear differential equations used for the AC electric arc modeling does not exist for some values of the parameters. The purpose of this paper is to convert electric arc furnace equation for these parameters to the quadratic polynomial form which significantly simplifies solution.

The solution has been obtained in the time domain by a sequence of transformations of the original nonlinear equation which lead to a system of quadratic equations, for which a periodic solution can be found easily using harmonic balance method (HBM).

Quadratic polynomial form of electric arc furnace nonlinear equation in the case for which the solution to the nonlinear differential equation describing electric arc cannot be obtained in a closed form.

The complete model of the arc requires extension of the deterministic solution obtained for the quadratic polynomial form using stochastic or chaotic component.

The obtained results simplify determination of the arc voltage or radius time waveforms if a closed form solution does not exist. 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.

In order to avoid problems occurring when a large number of harmonics is required or the system contains strong nonlinearities, a transformation of the original equation has been proposed. Nonlinearities present in the equation have been transformed into purely quadratic polynomial terms. It facilitates application of the classical HBM and allows to follow periodic solutions of the arc equation when its parameters are varied. It also enables better understanding of the phenomenon described by the equation and makes easier the extension of the arc model in order to cover the time-varying character of the arc waveforms.

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.

The change in the way of active power filters (APF) location can lead to overall cost reduction due to less number or less power of APFs required. The goal of this paper was to minimize the APF currents what is equivalent to solution with less apparent power of installed devices. The next step consists in development of new methods of APF optimal location.

Some scripts integrating optimization and harmonic analysis methods in Matlab and PCFLO software environments have been developed in order to achieve the goal.

Solution to the minimization problem determines the current spectrum of an APF connected to a selected system bus in accordance with some optimization strategies which among others enable minimization of THDV coefficients.

The APF control algorithm defined in the frequency domain and based on given current spectrum could lead to some problems with synchronization between APF instantaneous current and compensated current waveforms.

There are many papers on APFs but usually systems in which an APF is connected near a nonlinear load are analyzed. Some attempts to solve the more complex problems of synchronized multipoint compensation have been already made but there is still no generally accepted and commonly used solution.