## 3 Proposed V-I Characteristic Model of Electrical Arc
### Grey Wolf Optimization (GWO) Algorithm
The harmonic components derived from the exponential model (Model 1), the hyperbolic model (Model 2), and the exponential-hyperbolic model (Model 3) yield total THD values that closely approximate the real-time measured THD value. Nonetheless, for basic harmonic components, the values do not yield the resultant harmonic current values. The proposed model demonstrates better compatibility with the measured data, particularly in representing harmonic components. It achieves improved performance in current harmonics, particularly in the third harmonic content of the current spectrum, which is a critical factor in power system stability. Additionally, it exhibits lower deviation in higher-order harmonics, which is critical for accurately evaluating arc furnace behavior. The THD for the voltage is reduced to 2.34%, closely aligning with the real-time measured value of 2.30%, while the THD for the current is lowered to 11.40%, compared to 13.76% in real-time measurements. However, its total current harmonic distortion deviates from the real value due to the SVC system in real-time applications, which amplifies the intermediate harmonics between the fundamental and second harmonic ranges. Overall, the results indicate that the proposed model for the EAF outperforms existing time-domain methods in accurately analyzing the current and voltage harmonics in arc furnaces, making it a more reliable tool for power quality assessment. \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & **Harmonics** & **Actual** & **Model 1** & **Model 2** & **Model 3** & **Model 4** & **Proposed** \\ \hline \multirow{5}{*}{Arc} & 2 nd & 0.093\% & 0.07\% & 0.08\% & 0.08\% & 0.08\% & 0.079\% \\  & 3 rd & 0.801\% & 1.37\% & 1.53\% & 1.51\% & 1.53\% & 0.660\% \\  & 4 th & 0.056\% & 0.03\% & 0.07\% & 0.04\% & 0.05\% & 0.060\% \\ \cline{1-1}  & 5 th & 0.531\% & 0.62\% & 0.78\% & 0.78\% & 0.49\% & 0.435\% \\ \cline{1-1}  & 7 th & 0.332\% & 0.30\% & 0.43\% & 0.45\% & 0.20\% & 0.32\% \\ \cline{1-1}  & THD & 2.300\% & 1.80\% & 4.19\% & 2.13\% & 2.97\% & 2.34\% \\ \hline \end{tabular}
\end{table} 
Table 2: Real-time measurement variables for arc voltage. \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & **Harmonics** & **Actual** & **Model 1** & **Model 2** & **Model 3** & **Model 4** & **Proposed** \\ \hline \multirow{5}{*}{Arc} & 2 nd & 1.20\% & 1.04\% & 1.09\% & 1.16\% & 0.87\% & 0.92\% \\  & 3 rd & 3.12\% & 7.65\% & 9.91\% & 9.76\% & 28.04\% & 5.36\% \\  & 4 th & 0.79\% & 0.55\% & 0.56\% & 0.65\% & 1.33\% & 0.62\% \\ \cline{1-1}  & 5 th & 0.89\% & 2.05\% & 3.16\% & 3.21\% & 7.96\% & 1.34\% \\ \cline{1-1}  & 7 th & 0.31\% & 0.68\% & 1.31\% & 1.42\% & 3.57\% & 0.81\% \\ \cline{1-1}  & THD & 13.76\% & 11.47\% & 14.55\% & 14.50\% & 32.54\% & 11.40\% \\ \hline \end{tabular}
\end{table} 
Table 3: Real-time measurement variables for arc current. The results show that the proposed model significantly reduces voltage and current harmonic distortion compared to the existing models. The proposed mathematical model can be effectively applied to an analysis of the harmonic effects induced by EAFs. The objective of visually comparing the arc voltage and current waveforms generated by the model is illustrated in Figures 28a and 28b, respectively. The model aims to illustrate the alignment between the arc current and voltage variations with the actual measurement data, as depicted in the images. The novel introduced mathematical model can provide a more accurate characterization of the arc current behavior and enables a detailed analysis of the harmonic effects produced by arc furnaces. ## 5 Conclusions

This study presents a novel mathematical methodology based on field data to characterize the EAF behavior during the melting process. In contrast to existing models, the proposed mathematical model integrates polynomial curve fitting, the modified Heidler function, and double S-curves, optimized using the GWO algorithm, to accurately represent the dynamic V-I characteristics of the arc.