## 3 Proposed V-I Characteristic Model of Electrical Arc
### Grey Wolf Optimization (GWO) Algorithm
\[V_{ex}(t)=V_{at}\cdot(1+m\cdot N(t)) \tag{20}\]

In Equation (20), \(V_{at}\) represents the threshold voltage which depends on the arc length, \(m\) denotes the modulation index, and \(N(t)\) signifies the band-limited white noise signal. The value of \(V_{at}\) at a constant arc length is the threshold voltage determined from the static characteristic curves. Equation (20) is integrated into the V-I static characteristic model that defines the structural characteristics of the arc, hence determining the dynamic characteristic structure of the arc during the melting process. The flicker effect of the arc melting process as modeled for the exponential model (Figure 17), the hyperbolic model

Figure 16: The proposed model with constant arc distance case (V\({}_{at}\) = 240 V): (**a**) static V–I characteristic curve; (**b**) voltage and current of arc waveform. Figure 15: The nonlinear resistance model with constant arc distance case (V\({}_{at}\) = 240 V): (**a**) static V–I characteristic curve; (**b**) voltage and current of arc waveform. The exponential model (Figure 17) provides a basic representation but lacks accuracy in capturing rapid arc fluctuations. The hyperbolic model (Figure 18) improves performance at high currents but deviates in low-current conditions. The exponential-hyperbolic model (Figure 19) refines this approach, offering better transient behavior but still has limitations in dynamic conditions. The nonlinear resistance model (Figure 20) accounts for nonlinear effects, improving accuracy across a wider range, yet struggles with high-frequency flicker variations. By contrast, the proposed model (Figure 21), incorporating the modified Heidler function and double S-curves, optimized via the GWO algorithm, demonstrates more accurate performance in replicating experimental results. It effectively adapts to arc length variations and transient behaviors, making it a more reliable model for characterizing dynamic EAF operations. ### Harmonic Contents of Arc Models

This study analyzes the harmonic content of different arc models, including the exponential model (Figure 22), the hyperbolic model (Figure 23), the exponential-hyperbolic model (Figure 24), the nonlinear resistance model (Figure 25), the proposed model (Figure 26), and real-time measurements (Figure 27). This analysis was conducted at

Figure 21: The proposed model during the melting process with flicker effect: (**a**) dynamic V–I characteristic curve; (**b**) dynamic voltage and current of arc waveform. the point of common coupling (PCC) using a fast Fourier transform (FFT) to evaluate current and voltage harmonics. Figure 23: FFT in the melting process for the hyperbolic model: (**a**) harmonic voltage spectrum; (**b**) harmonic current spectrum. Figure 22: FFT in the melting process for the exponential model: (**a**) harmonic voltage spectrum; (**b**) harmonic current spectrum. Figures 22b, 23b, 24b, 25b, 26b and 27b. The harmonic values derived from the FFT spectra are summarized in Table 2 for voltage harmonics and Table 3 for current harmonics. Upon examining the voltage harmonic results in Table 2, it is evident that the hyperbolic model fails to accurately determine the total voltage harmonic distortion (THD). In all models, excluding the proposed model, the third voltage harmonic significantly exceeds the allowable limit. The proposed model provides a more accurate characterization of voltage harmonics, demonstrating a closer match to real-time measurement data across multiple harmonic orders, including the second, third, fifth, and seventh harmonics, rather than focusing solely on the THD. While Model 3 produces a THD value that is slightly closer to the actual measurement, its harmonic distribution deviates significantly from the real-time data, limiting its practical applicability in a power quality analysis. Similarly, from an analysis of the current FFT spectrum and the data in Table 3, it is evident that the current harmonics derived from the nonlinear resistance model (Model 4) significantly exceed the actual values.