# Modeling of Voltage Fluctuations Generated by Arc Furnaces
## 2 Modeling of Voltage Fluctuations in the Operation of a Single Arc Furnace  
There are many methods of electric arc modeling. The most important methods include dynamic [26; 27; 28], classic based on nonlinear differential equations [29], based on resistance and inductance [30], using a voltage source variable in time, and defined as a nonlinear function depending on the length of the electric arc [31; 32; 33]. For modeling arc devices, computer simulation programs are widely used [34; 35; 36; 37; 38; 39; 40; 41; 42]. The method proposed in the article can be classified as quasi-dynamic. The proposed model of the arc device allows for the estimation of the influence of the steel plant supply conditions on the amount of voltage fluctuations generated by arc furnaces (by changing the parameters of the \(\mathrm{R_{L}}\) and \(\mathrm{X_{L}}\) supply networks). The supply voltage \(\mathrm{U_{L}}\) represents an ideal constant voltage sinusoidal source, which has been adopted as \(\mathrm{U_{L}}=100\%\). The electric arc of the furnace is represented by the voltage \(\mathrm{U_{Arc}}\). The arc voltage \(\mathrm{U_{Arc}}\) varies depending on the arc length. In the research, only the first voltage harmonic was assumed, which means that the shape of the arc voltage was ignored. The high-current path with the furnace transformer is represented by resistance \(\mathrm{R_{T}}\) and reactance \(\mathrm{X_{T}}\). The performed measurements and statistical analysis of the actual voltage fluctuations recorded in the networks supplying the arc furnaces allowed for the development of a model of the furnace installation. The arc furnace, due to the nature of the electric arc, is a non-linear receiver. After ignition, an electric arc occurs between the electrodes and the molten scrap. The electric arc is represented by the voltage \(\mathrm{U_{Arc}}\). As a result of the movement of the electrodes, the distance of the electric arc changes. The effect of changing the arc length is the change of the voltage \(\mathrm{U_{Arc}}\). The arc voltage \(\mathrm{U_{Arc}}\) was simulated using a sinusoidal voltage source with an amplitude depending on the arc length, which is the basic harmonic of the arc voltage. Based on the analysis of measurement data, a power supply scheme for the arc furnace has been proposed. The arc furnace power supply system is shown in Figure 4.  
The range of current changes directly affects the amount of voltage fluctuations in the supply line of the arc furnace. After ignition of the arc between the electrodes and the charge, as a result of the changes in the arc length, the current \(\mathrm{I}\) changes its value dynamically. For arc furnaces, the scrap (charge) melting current typically varies around 120% of the rated (\(\mathrm{I_{N}}\) nominal current) current. In the initial stage of smelting, the current changes are the highest and it is assumed that they take place between the values specified by \(\mathrm{I_{A}}\) and \(\mathrm{I_{B}}\), where \(\mathrm{I_{A}}\) is 90% \(\mathrm{I_{N}}\) and \(\mathrm{I_{B}}\) is 150% \(\mathrm{I_{N}}\). The changes in current between \(\mathrm{I_{A}}\) and \(\mathrm{I_{B}}\), as defined \(\mathrm{\Delta I_{AB}}\), cause voltage changes between \(\mathrm{U_{A}}\) and \(\mathrm{U_{B}}\), defined as voltage fluctuations and marked \(\mathrm{\Delta U_{AB}}\)  
Figure 4: Single-phase power supply scheme for an arc furnace.