###### Abstract  
The control of nonlinear systems cannot be effectively achieved using linear mathematical methods. This paper introduces a novel mathematical model to characterize the voltage-current (V-I) characteristics of the electric arc furnace (EAF) melting process, incorporating experimental field data for validation. The proposed model integrates polynomial curve fitting, the modified Heidler function, and double S-curves, with the grey wolf optimization (GWO) algorithm applied for parameter optimization, enhancing accuracy in predicting arc dynamics. The performance of the model is compared against the exponential, hyperbolic, exponential-hyperbolic, and nonlinear resistance models, as well as real-time measurement data, to assess its effectiveness. The results show that the proposed model significantly reduces voltage and current harmonic distortion compared to existing models. Specifically, the total harmonic distortion (THD) for voltage is reduced to 2.34%, closely matching the real-time measured value of 2.30%. Similarly, in the current spectrum, the proposed model achieves a significant reduction in third harmonic distortion and a THD of 11.40%, compared to 13.76% in real-time measurements. Consequently, a more precise characterization of the EAF behavior enables more effective mitigation of harmonics and vibrations, enhancing the stability and power quality of the electrical networks to which they are connected. electric arc furnace (EAF); S-curve; nonlinear loads; grey wolf optimization (GWO) +
Footnote †: journal: Applied Sciences  
0000-0001-5000-5000-5000]E. M. A. The iron and steel industry has a significant impact on the global economy. The industry supplies primary materials to numerous sectors, including the automotive, construction, and manufacturing sectors [3]. EAFs and induction furnaces (IFs) are used in steel production facilities to perform the melting process of metals. Technological advances in electricity, metallurgy, and mechanics have led to the rapid development of EAFs over the past fifty years. The melting capacity of EAFs has increased due to several key improvements, including increased furnace transformer power [4], the development of cooling systems [5], refractory design to increase its longevity [6], the use of O\({}_{2}\) burners to accelerate melting [4], and the ability to manage all systems with computer assistance [7; 8]. In comparison to other metal melting technologies, EAFs have higher melting capabilities and lower operational costs. Currently, EAFs account for approximately 40% of global steel production, and this ratio is projected to reach approximately 50% by 2050 [9; 10]. The thermal energy generated by the electric arc formed between the electrodes and the molten metal in EAFs is used to initiate the melting process. By employing this principle of operation, metals are melted, and electrical energy is transformed into thermal energy [11]. The heat generated by the electric arc exceeds 1500 \({}^{\circ}\)C when this method is implemented during the melting procedure. The electric arc is distinguished by its high current and low voltage during the melting process. The three primary operating processes are boring, melting, and refining [12]. The melting operation commences with the boring process. The electrodes are charged to initiate the melting process, and they are submerged in the metal reservoir. The electrodes exhibit a short arc length during the charging procedure [13]. The process proceeds to melt once the electrodes have been sufficiently charged. During the melting process, a broad region develops in the center of the molten material. The arc length undergoes a continuous fluctuation due to the dynamic structure of the liquefied metal. The arc length of the electric arc is nearly constant, and the metal basin contains nearly all the metal in a molten state during the refining process. At this point, the arc reaches a steady structure.