# Modeling of Voltage Fluctuations Generated by Arc Furnaces
## 4 Verification of the Proposed Furnace Power Supply System with the Electric Arc Model
In similar cases, Electricite de France [51] recommends taking into account the measurement results in the phase with the largest voltage fluctuations. The electric arc is represented by the arc voltage \(\mathrm{U_{Arc}}\). The arc has been mapped using an ideal source of sinusoidal voltage with an amplitude dependent on the arc length, which is the fundamental harmonic of the arc voltage. Figure 18a shows the changes in the RMS voltage \(\mathrm{U_{RMS}}\) and the first harmonic of the voltage \(\mathrm{U_{HI}}\). \(\mathrm{U_{HI}}\) is the RMS (root mean square) value of the first voltage harmonic determined in a five-second interval. The correlation coefficient is respectively--Figure 18b:  
Figure 16: Changes in the short-term flicker severity \(\mathrm{P_{st}}\) recorded during one melt in the arc furnace. Figure 17: Relationships between the short-term flicker severity \(\mathrm{P_{st}}\) recorded in individual phases. The amount of voltage distortion in the power line of the steel plant depends mainly on the short-circuit power of the supply network, the melting phase in the arc furnace, and the daily load on the power system. Figure 19 shows changes of total harmonic distortion (THD) recorded at the same time in points with different short-circuit powers in the supply network electric arc furnace. Figure 19a shows the THD changes for S\({}_{\rm{cc}}\)/S\({}_{\rm{ntr}}\) = 150 and Figure 19b for S\({}_{\rm{cc}}\)/S\({}_{\rm{ntr}}\) = 25. The total harmonic distortion (THD) waveforms presented in Figure 19a,b show the influence of the power system on the voltage distortions in the mains supplying arc furnaces. In both cases, the THD limits according to [52; 53] were not exceeded. In the case of significant short-circuit power of the network (S\({}_{\rm{cc}}\)/S\({}_{\rm{ntr}}\) = 150--Figure 19a), it can be assumed that the influence of the furnace on voltage distortion is negligible. Therefore, it seems to be a correct assumption for the adoption of the first harmonic of the arc voltage. One of the assumptions in the model tests (especially for the initial phase of scrap melting) was the adoption of two-stage changes in the arc furnace current from I\({}_{\rm{A}}\) to I\({}_{\rm{B}}\) (symmetrically with respect to the scrap melting current of I\({}_{\rm{M}}\) = 1.2 I\({}_{\rm{N}}\)). The confirmation of the correctness of the adopted assumptions required a statistical analysis of voltage fluctuations. Figure 18: Changes in the root mean square (RMS) voltage U\({}_{\rm{RMS}}\) and the first harmonic of the voltage U\({}_{\rm{H1}}\) (**a**) and relationships between U\({}_{\rm{RMS}}\) and U\({}_{\rm{H1}}\) (**b**). The results of the statistical analysis of voltage fluctuations are presented below. The amplitude of voltage fluctuations \(\Delta\)U is presented in percentages. According to the recommendations of the statistical analysis of measurement data, when the number of samples exceeds 20, grouping them into a distributive series should be used [54]. The grouping is aimed at systematizing the obtained results and reducing the impact of individual deviations of voltage fluctuations registered during the steel smelting process. The resulting groups are called class intervals or otherwise classes, and the value representing each class is their center. The classes and the corresponding samples form a distributive series, which is obtained by determining the area of variability of a given feature, dividing it into classes and determining the number of samples in a given class [55]. The number of class intervals should not be less than 7 and greater than 15 and should satisfy the inequality [54]:  
Figure 4.0  
where:  
n--sample size  
m--number of classes  
and the size of any of the class ranges should not be less than 5 [54]. When the multiplicity in a given class is lower than 5, adjacent classes can be joined, which, however, leads to a reduction in the degrees of freedom for the tested sample [55].