## 3 Proposed V-I Characteristic Model of Electrical Arc
### Grey Wolf Optimization (GWO) Algorithm
The mathematical representation of the S-curves as a function of the \(V_{ex}\) voltage is as follows:

* In Region 2, where \(di/dt<0,\ \ 0<i<30\) kA, and \(V_{arc}>0\), the mathematical correlation between arc voltage and arc current is expressed by Equation (17): \[\overset{\rightarrow}{D}=\left|\overset{\rightarrow}{C}\cdot\overset{ \rightarrow}{X}_{p}(t)-\overset{\rightarrow}{X}(t)\right| \tag{8}\]0 (17)
* For Region 3, under the constraints \(di/dt<0,\ 30<i\), and \(V_{arc}>0\), the mathematical relationship between arc voltage and arc current is articulated by Equation (18): \[\overset{\rightarrow}{D}=\left|\overset{\rightarrow}{C}\cdot\overset{ \rightarrow}{X}_{p}(t)-\overset{\rightarrow}{X}(t)\right| \tag{8}\]1 (18) In Equations (17) and (18);

\[\overset{\rightarrow}{D}=\left|\overset{\rightarrow}{C}\cdot\overset{ \rightarrow}{X}_{p}(t)-\overset{\rightarrow}{X}(t)\right| \tag{8}\]2 (a) In \(di/dt<0\) case; third-degree polynomial expression. (b) Double S-curve. (c) S-curve approach. \begin{tabular}{l l} \(V_{1}\) & is the maximum voltage value of the S-curve for Region 2. \\ \(V_{2}\) & is the maximum voltage value of the S-curve for Region 3. \\ \(V_{ex}\) & is the maximum value of the polynomial expressions obtained through curve fitting. (corresponding to the maximum value of the eighth-degree polynomial defined in Region 1) \\ \(i_{3}\), \(i_{5}\) & is the arc current value of S-curves in the minimum hyper grow state. \\ \(i_{4}\), \(i_{6}\) & is the arc current values of S-curves at the highest hypergrowth condition. \\ \(\alpha\), \(\beta\), \(\lambda\) & is the coefficient values used to define the optimal solution of the polynomial expression. \\ \end{tabular} 

The GWO approach has been employed to determine the optimal parameters of the S-curves based on the third-degree polynomial equation. In the GWO algorithm, Equations (17) and (18) are defined as the goal function. The optimal solution for S-curves has been obtained by setting upper and lower boundary values, based on the limitations outlined below. \(0<V<V_{1}\) (range of arc voltage values derived from experimental data for Region 2) \(0<i<30\) kA (range of arc current values derived from experimental results for Region 2) \(V_{1}<V<V_{2}\) (range of arc voltage values derived from experimental data for Region 3) \(30\) kA \(\leq i\) (range of arc current values derived from experimental results for Region 3)

According to the characteristic curve obtained using the curve-fitting method, as illustrated in Figure 5, the relationships (\(V_{1}\) = 0.255-\(V_{ex}\)), and (\(V_{2}\) = 0.477-\(V_{ex}\)) can be expressed. In Equations (17) and (18), the application of the GWO algorithm, while adhering to the defined constraints, resulted in the calculated coefficients \(\alpha\), \(\beta\), and \(\lambda\) as 1.505, 2.533 and 1.50, respectively. The optimization yields for the following values for the hypergrowth parameters are calculated as follows: \(i_{3}\) = 0 kA, \(i_{4}\) = 20 kA, \(i_{5}\) = 27 kA and \(i_{6}\) = 92 kA. Figure 11 displays the V-I characteristic curves obtained from the third-degree polynomial curve and the GWO technique for utilizing double S-curves. Figure 11: The S-curve derived using a third-degree polynomial curve and the GWO algorithm for the V–I characteristic during arc current reduction. ### Dynamic Characteristic of Electric Arc for Melting Process

During the melting process in the arc furnace, the distance between the electrodes and the molten material continuously fluctuates. The dynamic characteristic of the EAF can be represented by incorporating the flicker effect into the static characteristic model. In this study, the random flicker effect is modeled in Equation (20), which mathematically defines these fluctuations. The frequency of this equation is set at 10 Hz to accurately simulate the flicker behavior observed in real-world EAF operations.