# Modeling of Voltage Fluctuations Generated by Arc Furnaces
## 1 Introduction  
The issues of voltage fluctuations in power systems and their mathematical description are presented, among others, in the following publications [1; 2; 3; 4; 5; 6; 7]. The voltage fluctuations generated mainly by arc furnaces have been published in [8; 9; 10; 11; 12; 13; 14; 15; 16]. The significant contribution of the UIE (International Union for Electricity) to the research on voltage fluctuations as a source of the flicker phenomenon should also be emphasized. The effect of many years of research was the development of the concept and implementation for production of a flickermeter [17]. Voltage fluctuations are defined according to the standard [18] as cyclic changes of the voltage envelope or a series of random changes of the root mean square (RMS) voltage around the nominal value. Voltage changes from U\({}_{\rm A}\) to U\({}_{\rm B}\) determine fluctuations:  
\[\Delta\rm U=U_{A}-U_{B} \tag{1}\]  
\[\Delta\rm U=\frac{U_{A}-U_{B}}{U_{N}}100[\%] \tag{2}\]  
or are expressed as a percentage of the supply voltage:  
Arc furnaces are devices that use an electric arc between electrodes and the charge to melt scrap. Arc devices are among the restless receivers of electricity. Due to the unstable operation of the arc furnaces, rapid changes in power consumption when melting scrap are experienced. Very fast changes in the value of the furnace currents cause equally rapid voltage fluctuations in the supply network. In the case of arc furnaces, voltage changes (fluctuations) range from a few to several changes per second. Figures 1-3 show the voltage waveforms recorded by the author in the networks supplying arc furnaces. Measurements were made at different short-circuit powers of the mains supplying arc furnaces (steel mills), during the various stages of smelting in the arc furnace, and at a different number of working parallel arc furnaces. Figure 1a shows the voltage oscillogram recorded in the supply network of arc furnaces. Figure 1b in magnification shows the maximum values of the oscillogram. You can see the change of the voltage oscillogram envelope. Figure 1c presents changes Urms voltage. Urms voltage was calculated in 10 ms intervals. Figure 1: Voltage changes in the supply line to arc furnaces: voltage oscillogram (**a**), maximum values of the oscillogram (**b**), Urms voltage (**c**). Both the value of the voltage curve (oscillogram) and the RMS voltage U\({}_{\text{rms}}\) are presented in relation to the rated voltage of Equation (3). The rated voltage of the network was adopted as U\({}_{\text{n}}\) = 100%. For example, the RMS value was converted to the form expressed as a percentage:  
\[\text{U}_{\text{rms}}[\%]=\frac{\text{U}_{\text{rms}}[\text{V}]}{\text{U}_{ \text{n}}[\text{V}]}100\% \tag{3}\]  
The advantage of such a solution is the unification of the analysis of measurement results and model tests at various voltage levels. In publication [19], standard deviation of the voltage SD(U) was used as a reliable source for the evaluation of randomly changing voltage fluctuations:  
\[\text{SD}(\text{U})=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(\text{U}_{i}-\text{U}_{ \text{mean}})^{2}} \tag{4}\]  
Figure 3: Voltage changes as a function of furnace current changes. Figure 2: Voltage fluctuation changes in the arc furnace supply line in various phases of the melting of scrap. where:  
SD(U)--standard deviation of voltage  
Ui--value for the i-th rms voltage sample  
Umean--RMS voltage mean for n samples  
n--number of voltage samples  
In the case of arc furnaces, the range of changes in voltage fluctuations changes in individual phases of the smelting. The greatest fluctuations occur during the initial scrap melting phase. The arc furnace transformer works from an idle state with the electrodes raised to a short circuit when the electrodes are in contact with the charge. With the melting of the scrap under the electrodes, the arc stabilizes. This reduces the amplitudes of voltage fluctuations.