# Modeling of Voltage Fluctuations Generated by Arc Furnaces
## 2 Modeling of Voltage Fluctuations in the Operation of a Single Arc Furnace
For the single-phase power supply scheme for the arc furnace shown in Figure 4 and the current changes with the assumed values of \(I_{A}\) and \(I_{B}\), a vector diagram can be drawn--Figure 6.  
Based on the vector diagram of the arc furnace currents and voltages shown in Figure 4, \(U_{ArcA}\) and \(U_{ArcB}\) arc voltages can be calculated as:  
\[U_{Arc_{A}}=\sqrt{U_{L}^{2}-I_{A}^{2}\left(X_{L}+X_{T}\right)^{2}}-\left(R_{L} +R_{T}\right)I_{A} \tag{5}\]  
\[U_{Arc_{B}}=\sqrt{U_{L}^{2}-I_{B}^{2}\left(X_{L}+X_{T}\right)^{2}}-\left(R_{L} +R_{T}\right)I\ \ \text{B} \tag{6}\]  
and voltages \(US_{A}\) and \(US_{B}\):  
\[U_{S_{A}}=\sqrt{\left(U_{Arc_{A}}+R_{T}I_{A}\right)^{2}+I_{A}^{2}X_{T}^{2}} \tag{7}\]  
Figure 5: Voltage changes as a function of furnace current changes. Figure 6: Vector diagram of the currents and voltages of the arc furnace. \[\mathrm{U_{S_{B}}}=\sqrt{\left(\mathrm{U_{Arc_{B}}+R_{T}I_{B}}\right)^{2}+I_{B}^{2 }\chi_{T}^{2}} \tag{8}\]  
Voltage changes from \(\mathrm{US_{A}}\) to \(\mathrm{US_{B}}\) determine fluctuations in point S--Figure 4:  
Figure 4:0  
or expressed as a percentage of the \(\mathrm{U_{L}}\) reference voltage:  
Figure 4:1  
or as a standard deviation voltage:  
Figure 4:2  
The assumed changes in the current from \(\mathrm{I_{A}}\) to \(\mathrm{I_{B}}\), around the melting current \(\mathrm{I_{M}}\), represent the greatest disturbances occurring in the initial phase of melting the scrap--Figure 4:3  
As the amount of molten scrap increases, the changes in the current consumed by the arc furnace decrease. It is influenced by the stabilization of the electric arc between the electrodes and the charge (molten scrap). The reduction of the range of current changes is represented by different histograms corresponding to the various phases of melting in an arc furnace. Figure 7 shows sample histograms corresponding to changes in the furnace current in the various stages of the smelting. In the initial phase of scrap melting, the greatest changes in the furnace current occur between the \(\mathrm{I_{A}}\) and \(\mathrm{I_{B}}\) values. These changes are characterized by a histogram (red), for which the standard deviation is \(\mathrm{SD(I)}=3\). After melting a part of the scrap, the current assumes the values \(\mathrm{I_{A}}\), \(\mathrm{I_{B}}\), \(\mathrm{I_{C}}\), \(\mathrm{I_{D}}\). The probability of the occurrence of currents is characterized by the histogram (blue), where \(\mathrm{SD(I)}=1.732\). In the further part of the smelting in the arc furnace, the electric arc between the electrodes and the charge (molten scrap) is stabilized. The current fluctuates between \(\mathrm{I_{C}}\) and \(\mathrm{I_{D}}\), which corresponds to the histogram (yellow), for which \(\mathrm{SD(I)}=1\). Figure 7 also shows the standard deviation of the current arc furnace corresponding to the various histograms. Figure 8 shows examples of voltage fluctuations for various  
Figure 4:4 Histograms of arc furnace current changes in various stages of melting scrap. furnace current histograms and the standard deviations of the currents from the mean value of \(\text{I}_{\text{M}}=1.2\text{I}_{\text{N}}\). The presented histograms are only selected examples of the ranges of changes in the arc furnace current adopted for the research. Analyzing the impact of arc furnaces on the power system, it was found that the greatest disturbances arise in the initial phase of steel smelting. The presented current change histograms are representative of the beginning of the melting of the scrap and subsequent addition of the scrap to the furnace ladle. One of the most important parameters affecting the voltage fluctuations generated by the arc furnace is the ratio of the short-circuit power of the supply network to the power of the furnace transformer. The short-circuit power \(\text{S}_{\text{CC}}\) characterizes the supply network in terms of resistance to disturbances caused by loads with rapidly changing load. Arc furnaces are such receivers.