# Modeling of Voltage Fluctuations Generated by Arc Furnaces
## 4 Verification of the Proposed Furnace Power Supply System with the Electric Arc Model
The set of consecutive amplitudes consists of n = 151 values, which have been classified into seven classes; according to the formula we get:  
Figure 4.1  
Figure 4.2 Changes of total harmonic distortion (THD) recorded at the same time in points with different short-circuit powers: (a) \(\mathrm{S_{cc}}/\mathrm{S_{intr}}=150\), (b) \(\mathrm{S_{cc}}/\mathrm{S_{intr}}=20\). hence the assumption of seven class ranges. For seven class intervals, the lengths of the intervals \(\Delta\)U = 0.81 were selected according to the formula:  
Figure 4.3  
The use of class intervals in Table 2 as negative numbers is intended to denote amplitudes lower than the mean, and the class interval marked with zero corresponds to the middle class interval, i.e., its center (arithmetic mean) is close to zero. For the analyzed data group, the basic statistical parameters are as follows:  
- Maximum value: 2.8361  
- Minimum value: \(-\)2.8077  
- Arithmetic mean: 0.012  
- Variance: 1.452  
- Standard deviation: 1.205  
To verify the hypothesis that the voltage fluctuation distribution is a normal distribution, the \(\chi^{2}\) Pearson compatibility test was performed. Table 2 summarizes the basic parameters characterizing the given class ranges, including class number, class range, center class, amount periodicity, cumulative amount, and cumulative periodicity characterizing the course, as shown in Figure 4.4  
The graphical representation of the prepared distribution series, presented in Table 2, by means of appropriate diagrams, allows for the convenient illustration of the obtained measurement results. Figure 21 presents histograms of changes in the periodicity in classes in the middle of class range. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Class Number** & **Class Range** & **Center Class** & **Amount** & **Periodicity** & **Cumulative Amount** & **Cumulative Periodicity** \\ \hline i & - & U & n\({}_{\text{i}}\) & f\({}_{\text{i}}\) & n\({}_{\text{i}}\)* & f\({}_{\text{i}}\)* \\ \hline \(-3\) & \(-2.81:-2.00\) & \(-2.405\) & 7 & 0.0464 & 7 & 0.0464 \\ \(-2\) & \(-1.99:-1.19\) & \(-1.590\) & 20 & 0.1325 & 27 & 0.1788 \\ \(-1\) & \(-1.18:-0.38\) & \(-0.780\) & 32 & 0.2119 & 59 & 0.3907 \\ \(0\) & \(-0.37:0.44\) & 0.035 & 40 & 0.2649 & 99 & 0.6556 \\ \(1\) & \(0.45:1.25\) & 0.850 & 26 & 0.1722 & 125 & 0.8278 \\ \(2\) & \(1.26:2.06\) & 1.660 & 19 & 0.1258 & 144 & 0.9536 \\ \(3\) & \(2.07:2.88\) & 2.475 & 7 & 0.0464 & 151 & 1.0000 \\ \hline \end{tabular} \(\text{n}_{\text{i}}\)* cumulative amount. f\({}_{\text{i}}\)* cumulative periodicity.  
\end{table} 
Table 2: Amount and periodicity of the resolving series and the cumulative resolving series of voltage fluctuations. Figure 20: Changes in voltage fluctuations with arc furnace on and off. Figure 22 shows the periodicity expansion (a) and distribution function (b) based on the function approximation [56]:  
\[\mathrm{f(x)}=\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{1}{2}\bigg{(}\frac{ \mathrm{x}-\mu}{\sigma}\bigg{)}^{2}\right] \tag{44}\]  
where:  
\(\sigma\)- standard deviation--deciding on the degree of flattening of the distribution curve. \(\mu\)- the mean value by locating the symmetrical normal distribution curve with respect to the \(x\) axis, which is a function of the probability density of the normal distribution. As already mentioned, in order to find out whether the analyzed voltage fluctuation distribution is a normal distribution, the \(\chi^{2}\) Pearson compatibility test was performed. On the basis of the sample, the parameters of the normal distribution were estimated--mean value and standard deviation. The results necessary to determine the statistics are presented in tabular form (Table 3). Figure 21: Histograms of changes in the periodicity in the middle of class range. Figure 22: Periodicity expansion (**a**) and distribution function (**b**) as a function of voltage fluctuations.