Arc Quality Index Based on Three-Phase Cassie–Mayr Electric Arc Model of Electric Arc Furnace

Abstract

In steel recycling, the optimization of Electric Arc Furnaces (EAFs) is of central importance in order to increase efficiency and reduce costs. This study focuses on the optimization of electric arcs, which make a significant contribution to the energy consumption of EAFs. A three-phase equivalent circuit integrated with the Cassie–Mayr arc model is used to capture the nonlinear and dynamic characteristics of arcs, including arc breakage and ignition process. A particle swarm optimization technique is applied to real EAF data containing current and voltage measurements to estimate the parameters of the Cassie–Mayr model. Based on the Cassie–Mayr arc parameters, a novel Arc Quality Index (AQI) is introduced in the study, which can be used to evaluate arc quality based on deviations from optimal conditions. The AQI provides a qualitative assessment of arc quality, analogous to indices such as arc coverage and arc stability. The study concludes that the AQI serves as an effective operational tool for EAF operators to optimize production and increase the efficiency and sustainability of steel production. The results underline the importance of understanding electric arc dynamics for the development of EAF technology.

1. Introduction

Steel, a fundamental material in numerous industries such as construction, manufacturing and energy, was produced in the order of 1.9 billion tons worldwide in 2022 [1]. Despite its widespread benefits, steel production is associated with significant environmental problems, in particular high energy consumption and significant ${\mathrm{CO}}_2$ emissions. With steel demand expected to increase by 30% over the next three decades, there is an urgent need for more sustainable production methods [2]. Electric Arc Furnaces (EAFs) offer a promising solution as they enable steel production from scrap steel and direct reduced iron (DRI). In addition, recent advances in EAF technology have led to improvements such as shorter tap-to-tap times, lower energy consumption through the introduction of carbon monoxide post-combustion and melt stirring techniques. Furthermore, digitalization and the application of advanced technologies, e.g., machine learning, have shown great potential for further optimization of the steel production process, i.e., temperature soft sensors [3] and slag foaming estimation [4]. These technologies enable the analysis and utilization of large datasets and provide valuable insights for optimizing production parameters and reducing environmental impact. In the metallurgical industry, especially in the EAF operation, accurate real-time measurements of key performance indicators (KPIs) are essential for high productivity. However, challenges such as cost and data acquisition limitations often hinder the direct measurement of critical variables. Soft sensors, which use indirect measurements to estimate hard-to-measure process variables, have proven to be a cost-effective and adaptable solution. These models, ranging from fuzzy [3] and neural network models [4] to comprehensive theoretical models [5,6,7,8], are central to improving process control and plant efficiency.

One of the critical aspects of three-phase alternating current (AC) EAFs is the architecture of the power supply system, depicted in Figure 1. This system consists of a series of components, starting with the intake of electric power from the utility grid, referred to as $u_{\mathrm{u}}$. At the point of common coupling (PCC), which is characterized by short-circuit resistance $R_{\mathrm{sc},\mathrm{u}}$ and inductance $L_{\mathrm{sc},\mathrm{u}}$, the grid is connected to a high/medium voltage step-down transformer (HV/MV T1). The secondary side of HV/MV T1 then feeds into the arc furnace transformer, a medium/low voltage transformer (MV/LV T2), via series reactor $X_R$ which serves to increase the overall reactance of the system [9]. The MV/LV T2 is equipped with a variable tapping mechanism that makes it possible to adjust the turns ratio under load and thus regulate the voltage supplied to the EAF. The measuring point (MP) is located on the secondary side of the arc furnace transformer and enables the measurement of line-to-ground voltages and line currents. The line current flows through conductors consisting of flexible cables and tubular conductors leading to graphite electrodes, which are modeled as resistance R and inductance L. At the point where these electrodes meet the conductive metal scrap, an electric arc is formed, which closes the circuit that is grounded via the furnace hearth. The dynamic control of arc intensity is ensured by the precise vertical adjustment of the electrodes. This adjustment is performed by an electrode control system designed to maintain specific current or impedance parameters for each line [10].

In an effort to increase efficiency and reduce the operating costs of the EAFs, great attention is being paid to optimizing the electric arcs which are responsible for a significant portion of the energy consumption. It is estimated that around 80% of the energy emitted by an electric arc occurs as radiation [6]. This underlines the importance of directing this radiation to the heating of scrap and molten steel rather than allowing it dissipate on the furnace roof and cooling panels. The intensity of arc radiation is directly proportional to arc length and voltage, which means that longer arcs generally emit more radiation. However, longer arcs can affect stability and lead to increased radiation dispersion away from the melt. The use of foaming slag is therefore crucial for the complete coverage of arcs to minimize radiation losses and improve energy transfer to the molten metal [11], as well as to stabilize the arc’s burning behavior and increase the efficiency of the power input [12]. A major challenge is to accurately measure the coverage of the arc by slag, as direct visual access is not possible and the installation of sensors inside the furnace is impractical [13]. Various methodologies, such as laser vibrometry [4,13], imaging through the slag gate [4] and analyzing the secondary side transformer voltage [14] and current [12] have been explored. One notable approach is the use of a laser vibrometer to measure real-time vibrations of the furnace shell in combination with the operator’s estimation of arc coverage. These data are then used to train an artificial neural network that estimates the arc coverage [13]. However, it is important to note that these estimates depend on the experience of the operator. In addition, Martell et al. [14] proposed an arc coverage index based on the third and ninth harmonic values of the voltage between the virtual neutral and ground voltages. While this index provides a general assessment of arc coverage, it does not provide insight into individual arc coverage. Furthermore, Sedivy et al. [12] proposed an index for foaming slag that suggests an inverse relationship to the Total Harmonic Distortion (THD) of the current. Finally, Son et al. [4] developed a long short-term memory (LSTM) neural network model for estimating slag foaming height in EAF steelmaking by integrating preprocessed EAF heat data with reference data from measured furnace shell vibration and digital imaging [4]. However, a critical challenge is estimating the height of the foaming slag from images taken through the slag gate. This process is inherently complex as the initial level of the molten metal surface is indeterminate, making the accurate determination of the height of the slag foam much more difficult.

Another crucial aspect of EAF operation is arc stability, which significantly affects both electrical phenomena and operational power level. Arc instability can affect the system’s ability to deliver a high-power current, leading to operational inefficiencies. Conversely, stable arcs enable higher power output, resulting in shorter tapping times, higher productivity and steel quality [15]. Various methodologies have been used to evaluate arc stability, including analysis of secondary-side transformer voltages or line currents [14,15,16] and the evaluation of acoustic signals [17]. An important stability indicator in the frequency domain is the THD of the voltage or current signal [16]. THD values vary in the different EAF heating stages and generally show increased values during the initial bore-down and early melting stages, which then decrease during the refining stages. Martell et al. [15] introduced a stability index based on the RMS voltage between virtual neutral voltage of the EAF delta transformer and ground voltage. In addition, arc stability can be evaluated by analyzing acoustic signals, where a stability index is derived by examining both the time and frequency domain of these signals [17]. Arc stability is not only a reflection of the electrical behavior in the furnace, but also a crucial process variable that, together with the specific heat stage, determines the EAF’s operational power level. Understanding and controlling arc stability is critical to operational efficiency as it allows for the power profile to be adjusted to the stability conditions of the arc. Effective control strategies in this area require comprehensive arc models. A variety of approaches to arc modeling can be found in the literature, including linear and nonlinear, static and dynamic methods that include both time-variant and invariant aspects as well as black-box and white-box approaches [18]. The following sections provide an overview of these arc models and explain their application and effectiveness in the operation of EAF.

The origins of arc models can be traced back almost a century, with Cassie’s model being introduced in 1939 [19] and Mayr’s model in 1943 [20]. The Mayr model was developed specifically for low-current applications and is particularly effective in the zero-crossing region, whereas the Cassie model is more suitable for high-current situations. The relatively simple Cassie and Mayr differential equation models can effectively model the key characteristics of the arc as a circuit element, with arc resistance defined as a dynamic variable. Based on the simplifications of principal power-loss mechanisms and energy storage in the arc column, they serve as valuable tools to understand arcing phenomena. In an effort to develop more comprehensive models, several approaches have been proposed to merge the arc models of Cassie and Mayr. One approach is to use a current-based transition function, where a Mayr model is active near the zero current while the Cassie arc model is active at higher currents [21,22,23]. Another possibility is to generalize the Cassie and Mayr arc model by introducing an additional parameter denoted as $\alpha$ [5,18,24,25]. Setting $\alpha$ to zero yields the Mayr model, while setting it to one produces the Cassie model. The theoretical range for $\alpha$ is between 0.5 and 1, as stated by Lee et al. [24], although it is worth noting that experimental data suggest that $\alpha$ values greater than one are possible. Khakpour et al. [26] proposed an improved Mayr arc model in which the dissipated power at current passage through the zero point was defined in terms of the arc current and the estimated arc diameter. The hyperparameters of the arc model were then estimated using measurements from a dedicated experimental setup. In addition to the EAFs, the circuit breaker models [27,28,29,30,31,32] also utilized the arc models with more advanced adaptations of Mayr’s and Cassie’s models, i.e., the Schwarz arc model, the Habedank arc model, the KEMA arc model [31,32] and the Schavemaker arc model [28]. However, these models introduce additional parameters, which makes their identification even more difficult. Nonetheless, some efforts have been made in EAF modeling to incorporate the Schavemaker model in combination with auto-regressive moving average (ARMA) models to attain the time variation characteristic of the arc parameters [33].

In 1990, Acha et al. [34] introduced a novel arc modeling methodology known as the power balance equation. This dynamic arc model is based on a system of differential equations, collectively referred to as the power balance equation. The basic premise of this approach revolves around establishing an equilibrium in which the total power generated in the arc equals the power dissipated to the surroundings and the power contributing to the arc’s internal energy. The radius of the arc is chosen as the state variable, and it forms the core of the differential equation. In recent years, increasing attention has been paid to analyzing and determining the parameters inherent in the power balance equation. Various methodologies have been explored, including analytical solutions [35,36], as well as optimization techniques such as Monte Carlo [37], genetic algorithms [37,38,39] and particle swarm optimization (PSO) [38]. In addition, efforts have been made to understand the stochastic properties of these parameters using approaches such as the ARIMA model [37] and the LSTM neural network [39]. In paper [35], it was found that the parameters significantly change during the EAF stages, i.e., the melting stage and the refining stage. However, it is worth noting that these methods often restrict the power balance equation to a particular scenario characterized by values $n = 0$ and $m = 2$.

MagnetoHydroDynamic (MHD) models represent the most complete type of plasma modeling and provide comprehensive insights into the behavior of electric arcs. These models require the solution of a complex set of differential equations that include Maxwell’s equations, the Navier–Stokes equations modified by the Lorentz force, and the energy equations that account for the Joule heating effect [40]. The MHD theory was also applied to understand the parameter values of the Cassie–Mayr hybrid arc model [23]. However, despite their thoroughness, a major challenge in MHD modeling is the scarcity of precise data on the physical properties of the plasma gas. Furthermore, the need for fine discretization in these models, which is essential for accurate system estimations, leads to significant computational requirements, making industrial real-time applications a challenge. In response, Channel Arc Models (CAMs) were developed as a more computationally tractable alternative. These models introduce several simplifications to reduce complexity. Key assumptions include the establishment of a local thermodynamic equilibrium, the conception of the arc as an ionized gas channel with axial symmetry, and the homogeneity of current and temperature distributions across the channel diameter [41,42,43]. Although CAM are less complex than MHD models, they still pose the task of estimating numerous model parameters. While this parameter estimation requirement is less daunting than for MHD models, it remains a non-trivial aspect of channel arc modeling that affects both the accuracy of the model and its applicability in various industrial scenarios.

This article presents a holistic approach for the electrical analysis of three-phase AC EAF and introduces a novel Arc Quality Index (AQI) based on the Cassie–Mayr arc model. The approach is based on a three-phase equivalent circuit that integrates the Cassie–Mayr arc model to capture the nonlinear and time-varying load characteristics of electric arcs. The model also takes into account the processes of arc breakage and ignition, thus improving the accuracy in representing the dynamic behavior of arcs, especially during the chaotic melting stage. The parameters of the Cassie–Mayr arc model, in particular the arc time constant and the Cassie–Mayr $\alpha$ constant for each electrode, are determined using a PSO technique. This estimation is performed using real furnace data collected during a single heat that included both the melting and refining stages. The optimization objective or fitness function is defined by minimizing the discrepancy between the harmonic content of the measured and simulated voltage and current signals. The AQI is formulated based on the deviations from the optimum conditions utilizing the Cassie–Mayr arc parameters and the RMS value of arc resistance. This index provides a thorough qualitative assessment of arc quality, analogous to indices such as the arc coverage index and the arc stability index. In addition, the AQI is based solely on the measurements of current and voltage, which can be measured on-site to assess arc quality. This research demonstrates the importance of understanding arc dynamics in EAFs, as it provides an insight into the process by which more efficient and sustainable steel production techniques can be achieved. Effective monitoring and control of the AQI enables operators to improve the operation of the EAF and thus increase the efficiency of steel production. Achieving high AQI values is desirable, at least in the later stages of EAF operation, in order to minimize energy losses caused by radiation. At this point, the extent of arc coverage by slag proves to be a decisive factor for the AQI value and the overall EAF. Consequently, operators can take targeted measures to improve AQI, including adjustments to power level or arc length and initiatives to increase slag height, thereby optimizing operating conditions and energy utilization.

Contributions of the work include: the development of a holistic electrical circuit for EAF based on the Cassie–Mayr arc model, an optimization method for parameter estimation of the Cassie–Mayr model across all three lines, and the formulation of a novel arc quality index that integrates aspects of arc stability and coverage.

2. Materials and Methods

2.1. Electric Arc Furnace (EAF)

In EAF methodology, scrap steel is melted and then overheated to a target temperature, which is mainly achieved by electric arcs. These arcs form between the tips of the graphite electrodes and the scrap material. The EAF process utilizes the intense heat generated by the conductive channels, supplemented by additional heat from exothermic reactions that occur in various chemical processes. The EAF process can be divided into four main stages: charging, melting, refining and the final tapping of the molten steel. This paper focuses predominantly on the intricacies of the melting and refining phases, examining the critical factors that influence efficiency and quality in these stages.

2.2. Three-Phase Equivalent Circuit

In AC EAFs, the power supply is provided by a three-phase arc furnace transformer (T2). The primary winding of the T2 is normally delta-connected, while the secondary winding can be either star- or delta-connected. The delta connection is often preferred as it reacts better to unbalanced loads and significantly reduces the phase currents. More precisely, the phase currents (coil currents) are reduced by a factor of $\sqrt{3}$ compared to the line currents. With this configuration, higher line currents can be absorbed without compromising the integrity of the system. This study focuses on an equivalent circuit diagram representing the system from the perspective of the secondary side of the arc furnace transformer. Figure 2 shows a simplified equivalent circuit derived from a more complex setup shown in Figure 1. The circuit receives power from the grid, first via a step-down transformer and then via the arc furnace transformer. In this circuit configuration, the secondary phase voltages in the delta connection are labeled $u_{ij}$ and are subject to variation depending on the transformer tap position. To determine equivalent transformer resistances $R_{ij}$ and inductances $L_{ij}$, all circuit elements form the primary side of T2 are viewed from the perspective of the T2 secondary side. The line currents, shown as $i_l$, flow through the electrodes into the melt, which is grounded by the furnace hearth. The resistances and inductances of the flexible cables and tubular conductors leading to the electrode tips are shown as $R_l$ and $L_l$, respectively. The electric arcs between the electrodes and the scrap or the steel bath are characterized by variable arc resistances symbolized by $R_{al}$.

In the equivalent circuit model, the notation is standardized as follows: Voltages and currents are denoted by u and i, respectively. Capital letters stand for phasors, which represent complex variables. The real part of a phasor is extracted as $u=\Re \left(U\right)$. Vectors and matrices are marked in bold. Resistances and inductances are symbolized by R and L, respectively. Complex impedances are expressed as $Z=R+jX$, where X is the reactance defined by equation $X=\omega L$. Here, $\omega$ is the angular fundamental frequency, which is related to fundamental frequency $f_f$ of the network as follows: $\omega =2\pi f_f$. The secondary transformer voltages are assumed to be symmetrical and are defined as follows:

$$\begin{array}{cc}\hfill U_{12}\left(t\right)& =\sqrt{2}{U}_{\mathrm{LV}}\left(T\right)e^{j(\omega t+\varphi )},\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill U_{23}\left(t\right)& =\sqrt{2}{U}_{\mathrm{LV}}\left(T\right)e^{j(\omega t+\varphi -\frac{2}{3}\pi )},\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill U_{31}\left(t\right)& =\sqrt{2}{U}_{\mathrm{LV}}\left(T\right)e^{j(\omega t+\varphi -\frac{4}{3}\pi )},\hfill \end{array}$$

(3)

where the $U_{\mathrm{LV}}\left(T\right)$ refers to the effective low voltage, which depends on tap position T, and $\varphi$ stands for the initial phase angle. Note that the sum of transformer voltages is zero: $U_{12}+U_{23}+U_{31}=0$. To determine equivalent resistances $R_{ij}$ and inductances $L_{ij}$, the analysis is carried out from the secondary side of transformer T2. This means that all elements from the primary side of T2 must be reflected to the secondary side. The short-circuit power at the PCC, referred to as $S_u$, is strongly dependent on the state of the grid and is treated as a simulation parameter. Utility impedance is determined as follows:

$$|Z_u|=\frac{U_{\mathrm{HV}}^2}{S_u},$$

(4)

where $U_{\mathrm{HV}}$ stands for the effective high voltage at the PCC. The values of short-circuit resistance and reactance values are derived from the reactance/resistance ratio $\frac{X_{\mathrm{sc},\mathrm{u}}}{R_{\mathrm{sc},\mathrm{u}}}$ as follows:

$$\begin{array}{cc}\hfill R_{sc,u}& = |Z_u|cos\left(arctan\left(\frac{X_{\mathrm{sc},\mathrm{u}}}{R_{\mathrm{sc},\mathrm{u}}}\right)\right),\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill X_{sc,u}& = |Z_u|sin\left(arctan\left(\frac{X_{\mathrm{sc},\mathrm{u}}}{R_{\mathrm{sc},\mathrm{u}}}\right)\right).\hfill \end{array}$$

(6)

The impedance of the step down transformer (T1) is calculated as follows:

$$|Z_{\mathrm{T}1}|=\%Z_{\mathrm{T}1}\frac{U_{\mathrm{HV}}^2}{S_{\mathrm{T}1}},$$

(7)

where $\%Z_{\mathrm{T}1}$ represents the percentage impedance, a standardized measure of the impedance of the transformer in relation to its rated power. $S_{\mathrm{T}1}$ is the rated apparent power of T1. Equations (5) and (6) are used to determine short-circuit resistance $R_{\mathrm{sc},\mathrm{T}1}$ and reactance $X_{\mathrm{sc},\mathrm{T}1}$ of T1 using a reactance/resistance ratio of $\frac{X_{\mathrm{sc},\mathrm{T}1}}{R_{\mathrm{sc},\mathrm{T}1}}$. The turns ration of T1 is defined as $a_{\mathrm{T}1}=U_{\mathrm{HV}}/U_{\mathrm{MV}}$, where $U_{\mathrm{MV}}$ stands for middle voltage. The impedance of the arc furnace transformer (T2) is calculated as follows:

$$|Z_{\mathrm{T}2}|=\%Z_{\mathrm{T}2}\frac{U_{\mathrm{MV}}^2}{S_{\mathrm{T}2}},$$

(8)

where $\%Z_{\mathrm{T}2}$ represents the percentage impedance, dependent on the tapping position. $S_{\mathrm{T}2}$ is the rated apparent power of T2. Equations (5) and (6) are used to determine the short-circuit resistance $R_{\mathrm{sc},\mathrm{T}2}$ and reactance $X_{\mathrm{sc},\mathrm{T}2}$ of T2, using a reactance/resistance ratio of $\frac{X_{\mathrm{sc},\mathrm{T}2}}{R_{\mathrm{sc},\mathrm{T}2}}$. The turns ration of T2 is defined as $a_{\mathrm{T}2}=U_{\mathrm{MV}}/U_{\mathrm{LV}}\left(T\right)$. The equivalent resistances and equivalent inductances in the circuit, which are denoted by $R_{ij}$ and $L_{ij}$ respectively, are determined as follows:

$$\begin{array}{cc}\hfill R_{ij}& =3\left(\frac{R_{\mathrm{sc},\mathrm{u}}}{a_{\mathrm{T}1}^2{a}_{\mathrm{T}2}^2}+\frac{R_{\mathrm{sc},\mathrm{T}1}}{a_{\mathrm{T}1}^2{a}_{\mathrm{T}2}^2}+\frac{R_{\mathrm{sc},\mathrm{T}2}}{a_{\mathrm{T}2}^2}\right),\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill X_{ij}& =3\left(\frac{X_{\mathrm{sc},\mathrm{u}}}{a_{\mathrm{T}1}^2{a}_{\mathrm{T}2}^2}+\frac{X_{\mathrm{sc},\mathrm{T}1}}{a_{\mathrm{T}1}^2{a}_{\mathrm{T}2}^2}+\frac{X_{\mathrm{R}}}{a_{\mathrm{T}2}^2}+\frac{X_{\mathrm{sc},\mathrm{T}2}}{a_{\mathrm{T}2}^2}\right),\hfill \end{array}$$

(10)

where $ij=\{12,23,31\}$ stands for the indices corresponding to all phases of the circuit. The arc transformer in this study is configured in vector group Dd0. Consequently, the impedance values are scaled by a factor of three due to wye-delta transform.

In the analysis of the circuit presented in Figure 2, state variables $i_1$, $i_2$ and $i_{12}$ are chosen, represented as $\mathit{i} = {[i_1,i_2,i_{12}]}^{\top }$. Using Kirchhoff’s current law, the remaining currents are determined by the following equations:

$$\begin{array}{cc}\hfill i_3& =-i_1-i_2,\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill i_{23}& =-i_2+i_{12},\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill i_{31}& =i_1+i_{12}.\hfill \end{array}$$

(13)

Applying Kirchhoff’s voltage law, we derive the relationship between inductances, resistances and voltage sources as follows:

$$\mathit{L}\frac{d\mathit{i}\left(t\right)}{dt}+\mathit{R}\left(t\right)\mathit{i}\left(t\right)=\mathit{B}\mathit{u}\left(t\right),$$

(14)

where the input vector is defined as $\mathit{u}={[u_{12},u_{23},u_{31}]}^{\top }$ and $\mathit{B}$ is defined as follows:

$$\mathit{B}=\left[\begin{array}{ccc}-1& 0& 0\\ 0& -1& 0\\ 0& 0& 1\end{array}\right],$$

(15)

while $\mathit{L}$ and $\mathit{R}$ are defined as follows:

$$\mathit{L}=\left[\begin{array}{ccc}{L}_1& -L_2& -L_{12}\\ L_3& L_2+L_3+L_{23}& -L_{23}\\ L_1+L_3+L_{31}& L_3& L_{31}\end{array}\right],$$

(16)

$$\mathit{R}\left(t\right)=\left[\begin{array}{ccc}{R}_1+R_{a1}\left(t\right)& -R_2-R_{a2}\left(t\right)& -R_{12}\\ R_3+R_{a3}\left(t\right)& R_2+R_{a2}\left(t\right)+R_3+R_{a3}\left(t\right)+R_{23}& -R_{23}\\ R_1+R_{a1}\left(t\right)+R_3+R_{a3}\left(t\right)+R_{31}& R_3+R_{a3}\left(t\right)& R_{31}\end{array}\right].$$

(17)

Finally, the derivatives of currents can be expressed as follows:

$$\frac{d\mathit{i}\left(t\right)}{dt}={\mathit{L}}^{-1}\left(-\mathit{R}\left(t\right)\mathit{i}\left(t\right)+\mathit{B}\mathit{u}\left(t\right)\right).$$

(18)

The electric arcs across all three lines exhibit nonlinear and time-varying load characteristics. To accurately model this behavior, the Cassie–Mayr arc model is used, which provides a comprehensive approach to capture the dynamic nature of the arcs. A detailed description of this model can be found in the following section.

2.3. Cassie–Mayr Arc Model

Extensive theoretical and experimental investigations have revealed the complexity of dynamic arc-plasma phenomena, which precludes the development of complete mathematical models that match the simplicity and precision of those for power electronic circuits. Nonetheless, despite their simplified approach to the mechanisms of power loss and energy storage in the arc column, Cassie and Mayr’s differential equation models provide a valuable qualitative understanding of arc dynamics [21].

The Cassie model assumes an arc with constant current density and stored energy per unit volume, which leads to the following formulation for arc resistance $R_a$:

$$\frac{1}{R_a}\frac{d{R}_a}{dt}=\frac{1}{\tau }\left(1-\frac{u_a^2}{E_0^2}\right),$$

(19)

where $u_a$ is the arc voltage, $\tau$ is the arc time constant and $E_0$ is the constant steady-state arc voltage. However, this model does not take arc interruption into account, as it primarily describes the behavior of the arc at high currents.

Mayr’s arc model, on the other hand, assumes that the heat loss comes mainly from the periphery of the arc, with the arc conductance varying with the stored energy [20]. It is expressed as

$$\frac{1}{R_a}\frac{d{R}_a}{dt}=\frac{1}{\tau }\left(1-\frac{u_a{i}_a}{P_0}\right),$$

(20)

where $i_a$ is the arc current and $P_0$ is the power loss factor of the arc. The physical idea behind this model is that the arc is heated by the current until it reaches the steady state, $v_a{i}_a=P_0$. This model allows the arc to be interrupted and has hyperbolic steady-state properties, which are characteristic of low current ranges.

Lee et al. [24] proposed a generalized Cassie–Mayr model that integrates the properties of both models:

$$\frac{1}{R_a}\frac{d{R}_a}{dt}=\frac{1}{\tau }\left(1-\frac{u_a{i}_a}{P_0{R}_a^{-\alpha }}\right),$$

(21)

where the power loss of the arc is defined as $P_0{R}_a^{-\alpha }$. The model contains a new parameter $\alpha$ that covers the range between the models of Mayr ($\alpha =0$) and Cassie ($\alpha =1$). Theoretical values of $\alpha$ in EAFs range from 0.5 to 1, although experimental evidence suggests that values greater than 1 are possible [24]. The power loss factor of the arc is defined as follows:

$$P_0=\sqrt{\frac{4\pi l_a^3}{{\sigma }_a}}{\sigma }_{\mathrm{SB}}{T}_a^4,$$

(22)

where $l_a$ stands for the length of the arc and $T_a$ for its temperature. Expression ${\sigma }_a$ stands for the conductivity of the arc, while ${\sigma }_{\mathrm{SB}}$ denotes the Stefan–Boltzmann constant. To improve the numerical stability, the Cassie–Mayr model is adjusted for each of the three arcs as follows:

$$\frac{d{s}_l}{dt}=\frac{1}{{\tau }_l}\left(1-\frac{1}{P_{l,0}}{i}_l^2{e}^{({\alpha }_l+1)s_l}\right),\mathrm{for}l\in \{1,2,3\},$$

(23)

where arc resistance $R_{al}$ is expressed in logarithmic form $s_l$ as follows:

$$s_l=ln{R}_{al}.$$

(24)

The comprehensive three-phase electrical circuit model of the EAF, which integrates the properties of arc resistance according to Cassie–Mayr, is presented concisely using a state-space approach. This mathematical model is of central importance for the exact representation of the dynamic behavior of the EAF system. The state-space representation is formalized as

$$\dot{\mathit{x}}\left(t\right)=f(\mathit{x}\left(t\right),\mathit{u}\left(t\right)),$$

(25)

where $\mathit{x}$ denotes the state vector and $\mathit{u}$ represents the input vector. State vector $\mathit{x}\left(t\right)$ comprises two line currents, a phase current and logarithmically transformed arc resistances, configured as $\mathit{x}={[i_1,i_2,i_{12},s_1,s_2,s_3]}^{\top }$. Function f summarizes differential Equations (18) and (23) in a uniform formulation:

$$f(\mathit{x}\left(t\right),\mathit{u}\left(t\right))=\left[\begin{array}{c}{\mathit{L}}^{-1}\left(-\mathit{R}\left(t\right)\mathit{i}\left(t\right)+\mathit{B}\mathit{u}\left(t\right)\right)\\ \frac{1}{{\tau }_1}\left(1-\frac{1}{P_{1,0}}{i}_1^2\left(t\right)e^{({\alpha }_1+1)s_1\left(t\right)}\right)\\ \frac{1}{{\tau }_2}\left(1-\frac{1}{P_{2,0}}{i}_2^2\left(t\right)e^{({\alpha }_2+1)s_2\left(t\right)}\right)\\ \frac{1}{{\tau }_3}\left(1-\frac{1}{P_{3,0}}{i}_3^2\left(t\right)e^{({\alpha }_3+1)s_3\left(t\right)}\right)\end{array}\right].$$

(26)

To simulate differential Equation (25), the Euler method is used. It is crucial to choose the sampling time in accordance with parameter $\tau$ to ensure the stability of the simulation.

2.4. Arc Extinction and Ignition

The state-space model described in Equation (25) does not currently account for the critical phenomenon of arc extinction, which is characterized by the interruption of the plasma column. This phenomenon, which mainly occurs during the melting stage of the EAF, has a significant effect on the system behavior. To ensure a holistic representation of EAF operation, it is essential to include both arc extinction and ignition in the model. For this purpose, we introduce a binary variable $B_l$, defined as

$$B_l=\left\{\begin{array}{cc}0,\hfill & R_{al}\ge R_a^{\mathrm{MAX}},\hfill \\ 1,\hfill & R_{al}<R_a^{\mathrm{MAX}},\hfill \end{array}\right.$$

(27)

where $B_l$ indicates the operating state of the arc: 0 for an extinguished arc and 1 for an active/burning arc. Threshold resistance $R_a^{\mathrm{MAX}}$ defines the point at which the arc is extinguished. If $B_l=0$, which indicates that the arc is extinguished, arc resistance $R_{al}$ is set to $R_a^{\mathrm{MAX}}$ and remains at this value until ignition occurs.

During the melting stage of EAFs, the current occasionally drops to zero, which is a sign of arc extinction, followed by a rapid rise, indicating arc re-ignition. We assume that ignition voltage depends on arc length. Due to the dynamic nature of arcs and the harsh environmental conditions, direct measurement of arc length is challenging. Therefore, researchers estimate arc length based on the relationship between arc voltage and arc length, which has been extensively studied and often modeled as an affine linear function, as shown in works such as Garcia-Segura et al. [10], Nikolaev et al. [44] and Pauna et al. [11]. This relationship is represented as follows:

$$U_a^{\mathrm{rms}}=E{l}_a+U_{\mathrm{AK}},$$

(28)

where $U_a^{\mathrm{rms}}$ denotes the RMS value of the arc voltage, the term $U_{\mathrm{AK}}$ stands for the combined voltage drop at the anode and the cathode and E for the intensity of the electric field within the plasma column. The voltage required to reignite the arc is set according to Equation (28) multiplied by the square root of two in order to obtain the peak value:

$$U_l^{\mathrm{ig}}=\sqrt{2}{U}_{al}^{\mathrm{rms}}.$$

(29)

The condition for ignition of the arc in line l is determined by comparing arc voltage $u_{al}$ with threshold voltage for ignition $U_l^{\mathrm{ig}}$:

$$B_l=\left\{\begin{array}{cc}1,\hfill & u_{al}\ge U_l^{\mathrm{ig}},\hfill \\ B_l,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

(30)

where arc voltage $u_{al}$ in line l is given by

$$u_{al}\left(t\right)=R_{al}\left(t\right)i_l\left(t\right).$$

(31)

It should be noted that if the current $i_l$ is zero, the voltage must be calculated using an alternative approach, as shown in the circuit diagram in Figure 2 and the application of Kirchhoff’s voltage law.

3. Parameter Optimization

This section describes the methodology used to estimate the arc parameters in an EAF during the melting and refining stages. The dataset utilized for the analysis consists of line-to-ground voltage measurements, denoted by $u_{lg}^m$ (where subscript l denotes line $l\in 1,2,3$ and m denotes the measurement), and line current measurements, represented by $i_l^m$. These measurements are then formulated as vectors, e.g., ${\mathit{u}}_{lg}^m={[u_{lg}^m\left(1\right),\dots ,u_{lg}^m\left(N\right)]}^{\top }$, where N is the total number of measurements in the observed time window. To enable effective parameter estimation, the analysis focuses on a single signal period and assumes that the arc parameters remain constant within this time frame. This hypothesis simplifies the analysis while maintaining its relevance within the dynamic environment of the EAF. This approach is in line with similar practices in the field, where the authors have opted for a half-period for optimization [18,38].

The main objective is to estimate two critical Cassie–Mayr parameters for each line l: arc parameter ${\alpha }_l$ and time constant ${\tau }_l$. To achieve this, Equation (25) is used to simulate the EAF system over a complete signal period. The simulated line-to-ground voltage ${\mathit{u}}_{lg}^s$ and line current ${\mathit{i}}_l^s$ are derived from the state space vector and its derivative, where s denotes the simulated values. The line-to-ground voltage $u_{lg}$ is calculated as follows:

$$u_{lg}\left(t\right)=L_l\frac{d{i}_l\left(t\right)}{dt}+R_l{i}_l\left(t\right)+u_{al}\left(t\right).$$

(32)

The analytical process includes a Fourier series analysis to determine the magnitude of the measured and simulated signals for harmonics up to the $N_H$th order. This harmonic analysis is crucial for capturing the complex electrical behavior of the EAF. The term h in the analysis stands for the hth harmonic and enables a detailed investigation of the electrical properties at different harmonic levels. For example, $M_{l,h}^{U,m}$ stands for the magnitude of the lth line-to-ground voltage measurement for the hth harmonic, which provides insight into the harmonic distribution and its impact on the operation of the EAF. To optimize arc parameters, Particle Swarm Optimization (PSO) is employed, leveraging its capability for nonlinear function optimization [45,46]. This technique involves multi-candidate solutions, known as ’particles’, concurrently exploring the search space to minimize a predefined fitness function. Particles dynamically adjust their positions based on their own experiences and those of their neighbors. The fitness function is defined as follows:

$$J=\lambda J^U+(1-\lambda )J^I,J^U=\frac{1}{3}\sum _{l=1}^3{J}_l^U,J^I=\frac{1}{3}\sum _{l=1}^3{J}_l^I,$$

(33)

where $J_l^U$ and $J_l^I$ represent the voltage and current fitness functions for the lth line, respectively. Hyperparameter $\lambda$, ranging from zero to one, balances the focus between minimizing voltage $J^U$ and current $J^I$ components. Higher $\lambda$ values prioritize voltage minimization, while lower values focus on current minimization. The fitness functions for voltage and current part are thus defined as follows:

$$J_l^U=\frac{1}{\underset{l\in \{1,2,3\}}{max}{M}_{l,1}^{U,m}}\sqrt{\frac{1}{N_H}\sum _{h=1}^{N_H}{(M_{l,h}^{U,s}-M_{l,h}^{U,m})}^2},$$

(34)

$$J_l^I=\frac{1}{\underset{l\in \{1,2,3\}}{max}{M}_{l,1}^{I,m}}\sqrt{\frac{1}{N_H}\sum _{h=1}^{N_H}{(M_{l,h}^{I,s}-M_{l,h}^{I,m})}^2},$$

(35)

where the aim is to minimize the root mean square of the magnitude differences of measured and simulated signals for harmonics up to $N_H$. In addition, the fitness functions are normalized by the maximum fundamental harmonic of the measured signal in all the lines. This ensures that the values of $J_l^U$ and $J_l^I$ are comparable in scale, which enables parameter $\lambda$ to be more easily tuned.

3.1. Initial Parameter Selection

Before optimization, initialization of the three-phase Cassie–Mayr model of the EAF is essential. This step requires setting the initial conditions for state vector $\mathit{x}\left(1\right)$. In addition, the initial phase angle $\varphi$ of the secondary transformer voltages is determined at this stage. The initial states of the first two line currents, $i_1^s\left(1\right)$ and $i_2^s\left(1\right)$, are determined directly on the basis of the measured line currents $i_1^m\left(1\right)$ and $i_2^m\left(1\right)$. However, as phase current $i_{12}$ is not measured directly, its initial state is estimated using the phasor analysis as follows:

$${\mathit{I}}_{12}^m=\frac{e^{-j\frac{2\pi }{3}}({\mathit{U}}_{1g}^m-{\mathit{U}}_{2g}^m)-({\mathit{U}}_{2g}^m-{\mathit{U}}_{3g}^m)-Z_{23}{\mathit{I}}_2^m}{e^{-j\frac{2\pi }{3}}{Z}_{12}-Z_{23}},$$

(36)

where initial state $i_{12}^s\left(1\right)$ is defined as the real part of $I_{12}^m\left(1\right)$, which is the the first complex number in vector ${\mathit{I}}_{12}^m$. In addition, the initial phase angle of the secondary transformer voltages is initialized using the following expression:

$$\varphi =arg\left(U_{2g}^m\left(1\right)-U_{1g}^m\left(1\right)+Z_{12}{I}_{12}^m\left(1\right)\right),$$

(37)

where function arg calculates the argument of a complex number. Initial arc resistances $R_{al}^s\left(1\right)$ in the model are calculated according to Equation (31) as follows:

$$R_{al}^s\left(1\right)=\frac{u_{al}^m\left(1\right)}{i_l^m\left(1\right)}.$$

(38)

Note that the initial arc resistance tends to infinity as current $i_l\left(1\right)$ approaches zero, which is a mathematical consequence of their inverse relationship. To take this into account and maintain numerical stability, the resistance is saturated at threshold $R_a^{\mathrm{MAX}}$. In addition, arc voltage $u_{al}$ cannot be measured directly and is therefore estimated using the basic relationship equation from (32) as follows:

$$u_{al}^m\left(k\right)=u_{pg}^m\left(k\right)-L_l\frac{d{\widehat{i}}_l^m\left(k\right)}{dt}-R_l{i}_l^m\left(k\right),$$

(39)

where k represents the kth measurement and $d{\widehat{i}}_l^m/dt$ denotes the estimated derivative of the current. Since phase $\varphi$ and initial phase current $i_{12}$ are estimated using phasors, the nonlinear nature of the system leads to some estimation errors. To improve model initialization, the system is simulated over a complete time period, with the final simulation value serving as initial conditions $\mathit{x}\left(1\right)$. This procedure assumes that the arc reaches a stable state at the end of the simulation period. Given the periodicity of the system, a consistent value of the state vector is expected over successive periods. However, the applicability of this method depends on the stability of all arcs within the system, a condition that is not always fulfilled during the melting phase. Therefore, this reinitialization strategy is only applied if no interruptions of the arcs occur during the entire simulation period.

3.2. Equivalent Circuit Parameter Selection

The grid in this study was operated at a fundamental frequency of 50 Hz and the short-circuit power at the PCC rated at 3000 MVA. However, it is important to note that this value can drop significantly under worst-case conditions. In accordance with the observations of Ciotti et al. [9], the reactance/resistance ratio of the grid, expressed as $\frac{X_{\mathrm{sc},\mathrm{u}}}{R_{\mathrm{sc},\mathrm{u}}}$, was set to a value of eight. Similarly, the reactance/resistance ratio for the step-down transformer $\frac{X_{\mathrm{sc},\mathrm{T}1}}{R_{\mathrm{sc},\mathrm{T}1}}$ was set to 10. The arc furnace transformer was rated at 80 MVA. The reactance/resistance ratio and the percentage impedance of the arc furnace transformer depend on the transformer tap position, which changes during the heat. The values for short-circuit resistance $R_{\mathrm{sc},\mathrm{T}2}$ and reactance $X_{\mathrm{sc},\mathrm{T}2}$ were derived from the transformer data sheet. However, no values were given for individual phases in this data sheet, so uniform values were assumed for all phases for the purposes of this study.

In the field of arc voltage and arc length modeling, the determination of the parameters is crucial, as described in Equation (28). The anode and cathode voltage drop $U_{\mathrm{AK}}$ and the electric field intensity E are of particular interest. Recent studies have suggested different ranges for these parameters. Pauna et al. [11] suggest $U_{\mathrm{AK}}$ values between 10 V and 80 V and E values between 500 V/m and 3000 V/m, indicating a large variability in arcing conditions. In contrast, Nikolaev et al. [44] recommend a narrower range for $U_{\mathrm{AK}}$ (20 V to 40 V) and E (600 V/m to 1000 V/m). Garcia-Segura et al. [10] provide more specific values and argue for $U_{\mathrm{AK}}$ at 40 V and E at 1150 V/m. In the absence of precise measurements of arc length in our operational context, we took an empirical approach to parameter selection and adjusted our values to these recommendations. We set $U_{\mathrm{AK}}$ to 40 V in agreement with Garcia-Segura et al. [10] and chose E at 1000 V/m. For the power loss factor in Equation (22), determining the length of the arc, temperature and conductivity is challenging due to measurement issues. We estimated the arc length using Equation (28) and the effective arc voltage. According to Pauna et al. [11], arc conductivity ${\sigma }_a$ is influenced by both the temperature and the length. Following the suggestion of Logar et al. [5], we set the specific conductivity to a constant value of $2 \times {10}^3$ S/m. While Logar suggests an arc temperature of 4500 K, our analysis showed that reducing this parameter to 3000 K minimizes the mean error in the fitness function (33). Resistance $R_a^{\mathrm{MAX}}$ that defines the threshold at which the arc extinguishes is set to the value of 250 m$\Omega$.

In the process of optimizing parameters using PSO, it is imperative to initially establish the boundaries within which these parameters will vary. Lee et al. [24] postulate a theoretical range for $\alpha$ from 0.5 to 1. In addition, they hypothesize that $\alpha$ can exceed this range based on empirical observations. To capture a broader range of possible values and increase the robustness of our search, we delimited the range for $\alpha$ as ${\alpha }^{\mathrm{MIN}}=0$ and ${\alpha }^{\mathrm{MAX}}=2$. In addition, the arc time constant, a critical parameter in our model, was determined by synthesizing findings from previous studies and empirical experiments. Nikolaev et al. [44] determine that the arc time constant is between 0.2 ms and 5 ms. At the same time, the analysis of the data of Golestani et al. [18] shows that their arc time constant is between 0.4 ms and 2 ms. Considering these results, we set the bounds for the arc time constant in our study as ${\tau }^{\mathrm{MIN}}=0.1$ ms and ${\tau }^{\mathrm{MAX}}=5$ ms. This range was intended to reflect the different results observed in previous research while allowing a comprehensive exploration of the possible values in the context of the PSO framework.

3.3. Hyperparameter Selection

In formulating our optimization strategy, particular emphasis was placed on balancing the importance of voltage and current within the objective function, as shown in Equation (33). This balance was achieved by including parameter $\lambda$. We assigned a value of 0.5 to $\lambda$ to ensure equal weighting of voltage and current in the optimization process. Furthermore, the fitness functions for voltage and current defined in Equations (34) and (35) are based on Fourier series analysis. This analytical approach requires the specification of the number of harmonics, $N_H$. After careful consideration and experimental verification, we decided to set $N_H$ to 10. This decision was supported by the realization that the inclusion of a much higher number of harmonics does not significantly increase the accuracy of the analysis. For the optimization procedure, the number of particles $N_{part}$ was set to 15 and the maximum number of PSO iterations $N_{\mathrm{iter}}^{\mathrm{MAX}}$ was set to 200.

3.4. Parameter Estimation Algorithm

Algorithm 1 describes the procedure for optimizing the parameters of the Cassie–Mayr model in a three-phase AC system. This algorithm uses PSO to effectively tune the model parameters to ensure an accurate representation of the arc phenomena in EAFs. This algorithm is designed to systematically process the available EAF data and adjust and refine the parameters of the Cassie–Mayr model for each line. The process is iterative and utilizes the strengths of PSO to converge to the most accurate parameter set that reflects the complex dynamics of the EAF system. The fitness of each particle in the swarm is evaluated using predefined criteria, which facilitates the identification of the optimal parameter set for the model.

Algorithm 1: Three-phase Cassie–Mayr model optimization

1: Define PSO hyperparameters: $\lambda$, $N_H$, $N_{\mathrm{iter}}^{\mathrm{MAX}}$, $N_{part}$   2: Define power supply parameters   3: Define arc parameters: $T_a$, ${\sigma }_a$, E, $U_{\mathrm{AK}}$, $R_a^{\mathrm{MAX}}$   4: Define the range of arc parameters: $\alpha$, $\tau$   5: repeat Through all the data   6:     Acquire one period of measurements: ${\mathit{u}}_{pg}^m$, ${\mathit{i}}_l^m$   7:     Define $U_{\mathrm{LV}}$ based on T and calculate $R_{ij}$, $L_{ij}$   8:     Based on ${\mathit{u}}_{al}^m$, estimate $l_{al}$, $U_l^{\mathrm{ig}}$   9:     Initialize $\mathit{x}\left(1\right)$, $\varphi$ 10:     repeat Through all PSO iterations 11:         Calculate arc parameters ${\alpha }_l$, ${\tau }_l$ according to PSO rules 12:         Simulate one period of operation 13:         Calculate fitness for all particles 14:     until $N_{\mathrm{iter}}\le N_{\mathrm{iter}}^{\mathrm{MAX}}$ 15:     Save best particle: ${\alpha }_l$, ${\tau }_l$ 16: until No data are left

4. Arc Quality Index (AQI)

In our study, we introduce the Arc Quality Index (AQI) as a novel metric to evaluate the efficiency of the electric arc process. This index is a composite function of three key arc parameters: arc time constant $\tau$, arc parameter $\alpha$ and the RMS value of arc resistance $R_a^{\mathrm{RMS}}$. The AQI is mathematically formulated as follows:

$${\mathrm{AQI}}_l=k_0-k_{\tau }\frac{|{\tau }_l-{\tau }_{\mathrm{opt}}|}{\Delta \tau }-k_{\alpha }\frac{|{\alpha }_l-{\alpha }_{\mathrm{opt}}|}{\Delta \alpha }-k_{R_a}\frac{|R_{al}^{\mathrm{RMS}}-R_{\mathrm{opt}}^{\mathrm{RMS}}|}{\Delta R_a},$$

(40)

where $\Delta$ represents the range of each variable. The underlying premise of the AQI is that deviations from the optimum arc parameters are associated with a decrease in electric arc quality. Therefore, determining the optimal parameters is a crucial aspect of this model. With regard to the arc time constant, we assume that a higher time constant correlates with better arc quality. Consequently, we label the optimal time constant ${\tau }_{\mathrm{opt}}$ as the upper limit of its range ${\tau }^{\mathrm{MAX}}$. We also assume that the ideal RMS value of the arc resistance $R_{\mathrm{opt}}^{\mathrm{RMS}}$ is the lower limit of the assumed range, which is zero.

The calibration of AQI coefficients is crucial for the accuracy and relevance of the index. We weigh all three factors equally and determine that $k=k_{\tau }=k_{\alpha }=k_{R_a}$. The AQI is scaled so that it ranges from 0 to 100 percent and reflects the relative quality of the electric arc. This normalization process is facilitated by a linear transformation in which the minimum and maximum values of the unscaled AQI are mapped to zero and one, respectively, defining coefficients $k_0$ and k in a unique way. This approach ensures that the AQI is a quantifiable measure of arc quality that EAF operators can incorporate into their operations.

5. Results

This section presents the results obtained with the proposed procedure to optimize the parameters of the Cassie–Mayr model followed by the AQI. This includes data collected during a single heat spanning from the last bucket loading through the refining stage to the final heat tapping. The voltage and current signals were sampled at a frequency of 5 kHz, resulting in a dataset covering slightly less than 1000 s of EAF operation. The optimization algorithm described in Algorithm 1 uses one period of data to determine the arc parameters. It was determined that not every period should be used in this procedure, and a sampling frequency of 8 Hz was chosen as appropriate to capture the rapidly changing arc parameters while ensuring detailed and granular analysis.

The comprehensive dataset facilitated the calculation of estimated arc length values for all three lines using the Equation (28), as shown in Figure 3. It is important to note that the dataset does not include the entire heat, but covers the measurements from the last bucket charge through the melting phase to the refining phase. The method for estimating the arc length is based on measuring the effective arc voltage over a single signal period. As shown in Figure 3, a significant variability in arc length was observed during the melting phase, which then stabilizes during the refining phase. This pattern is reflected in the analysis of the moving average arc length, which also shows fluctuations during the melting phase before stabilizing in the refining phase. These observations highlight the dynamic nature of the melting process and underline the crucial role of arc length monitoring in the efficient control and optimization of EAF operation.

The optimization process described in Algorithm 1 was used to estimate the parameters of the Cassie–Mayr model in a three-phase AC system over the entire dataset. The performance of this algorithm was quantitatively evaluated using the values of the fitness function shown in Figure 4. The stochastic properties of the arc length are reflected in the values of the fitness function and illustrate the sensitivity of the algorithm to the inherent variability of the arc conditions.

The analysis of Figure 4 shows that the fitness function has higher values during the melting stage, indicating less stable arc conditions. This instability is further evidence of the challenge of accurately modeling the instantaneous voltage and current values due to the stochastic dynamics of the melting process. These dynamics are characterized by frequent arc interruptions and abrupt voltage fluctuations, which can be seen in Figure 5. During the transition to the refining phase, a clear stabilization of the arc behavior can be seen, which is reflected in reduced values of the fitness function. This transition means a relative relaxation of the stochastic conditions and enables a more predictable and stable arc operation.

Table 1. Mean values of the fitness function components for the entire dataset, the melting stage and the refining stage.

Fitness [%]	All	Melting	Refining
$\overline{J}$	1.40	2.39	0.77
${\overline{J}}^U$	1.97	3.19	1.19
${\overline{J}}^I$	0.83	1.59	0.35

provides an analysis of the mean values of the fitness function across different operating phases: covering the entire dataset (“All”), the melting phase (“Melting”) and the refining phase (“Refining”). In addition, the mean values for the voltage and current components of the fitness function are given in this table. A notable observation is that the mean values of the current fitness function, ${\overline{J}}^I$, are consistently lower compared to the voltage fitness function, ${\overline{J}}^U$, across all operating stages. The discrepancy between the mean values of ${\overline{J}}^I$ and ${\overline{J}}^U$ can be attributed to the fact that the arc model does not consider abrupt voltage fluctuations, a characteristic feature of the arc’s behavior that is particularly evident during the melting phase. These fluctuations lead to a considerable variability in the voltage component and thus increase the mean values of the voltage fitness function.

Below, we examine the graphical representations of the results in two specific cases: during the melting stage and during the refining stage. But first, it is important to note that the signals are normalized with respect to the peak values of line-to-ground voltage $|U_{lg}|$ and line current $|I_l|$, which are calculated as follows:

$$|U_{lg}|=\sqrt{2}\frac{U_{\mathrm{LV}}\left(T\right)}{\sqrt{3}},$$

(41)

$$|I_l|=\sqrt{2}\frac{S_{T2}}{\sqrt{3}{U}_{\mathrm{LV}}\left(T\right)},$$

(42)

where the square root of two is used to obtain the peak value from the RMS value.

Figure 5 shows a detailed analysis of the line-to-ground voltages and line currents during a specific interval of the melting stage, comparing measured and simulated signals. It is noteworthy that four arc interruptions occur during this period—two in the second line and one each in the first and third lines. This observation highlights the inherent instability of the arc during the melting stage, a critical aspect of the EAF process. The model successfully captures the dynamics of these arc interruptions, proving its effectiveness in replicating arc behavior.

Figure 6 shows the calculated arc resistance for all three lines based on the Cassie–Mayr model, which corresponds to the same time period shown in Figure 5. In addition, the operating state of the arc is represented by a binary variable, $B_l$, that distinguishes between active and extinguished states. This figure clearly illustrates the four cases of arc interruptions mentioned above. In particular, the second line shows the longest interruptions, which exceed a duration of 2 ms. Interestingly, the initial peak of arc resistance in the third line approaches the threshold value, $R_a^{\mathrm{MAX}}$, but it remains slightly below this limit.

Figure 7 shows a descriptive example of a particular interval during the refining stage of an EAF, showing both the observed and simulated line-to-ground voltages and line currents. A notable aspect of this figure is the high degree of correlation between the simulated signals and the actual measured data, indicating the accuracy of the model in capturing the electrical behavior during the refining process.

In addition, Figure 8 shows the arc resistances for all three lines during the refining stage. These resistances are calculated using the Cassie–Mayr model, which is based on data from the same time period as Figure 7. In comparison to the resistance profiles observed during the melting stage (Figure 6), a strong contrast is evident. During the refining stage, the arc resistances are significantly lower and show a periodic pattern, indicating stable arc operation with no signs of instability.

A graphical analysis of the AQI across all three lines of an EAF, from the final part of the melting stage to the refining stage, is shown in Figure 9. The AQI provides a quantitative assessment of arc performance and stability, with higher values indicating more favorable arc conditions. To improve the interpretability of the data and reduce the impact of transient fluctuations, a smoothing process is applied using a moving average filter and a standard deviation for the moving average. The filtering is applied over the past 50 samples, effectively tracing the underlying AQI trends and clarifying the development curve over time. Through a careful Cassie–Mayr parameter $\alpha$ analysis, we set the optimal value for arc parameter ${\alpha }_{\mathrm{opt}} = 1.12$. In order for AQI to provide a qualitative assessment of the arc quality, we set the value between zero and one, resulting in the following parameters: $k_0 = 1.029$ and $k = 0.823$.

Figure 9 serves as an important analytical tool for evaluating EAF performance and provides important insights into the melting and refining processes. The visual representation of the AQI enables the identification of operating patterns and areas for improvement. The initial low AQI values reflect the instability of the arc during the melting stage, which is confirmed by the variable arc length estimates in Figure 3. A significant dip in AQI around 350 s in the first and second lines, which is not reflected in the third line, demonstrates the ability of the index to distinguish between different lines, allowing for EAF operators to optimize individual arcs. At the transition to the refining phase, around 400 s, a significant improvement in arc quality can be observed. AQI values close to 100% denote that the arcs are operating under stable conditions and are completely covered by the slag. A slight decreasing trend in AQI values beyond 500 s is attributed to insufficient arc coverage by the slag.

6. Discussion

The analysis of the parameters of the Cassie–Mayr arc model reveal that the arc time constant $\tau$ is the most informative factor. This observation is confirmed by Nikolaev et al. [44], who found that the arc time constant is closely related to changes in the heat stage. In particular, the arc shows greater instability during the melting stage, characterized by a lower value of $\tau$, in contrast to the more stable refining stage. In our optimization approach, we focus on parameters $\alpha$ and $\tau$ and set the other variables as constants. This decision is made strategically to minimize the complexity of the optimization process, which is particularly prone to local minima, especially in the melting stage and in introduction of other optimization parameters. Although we focus primarily on $\alpha$ and $\tau$, it is important to recognize that the arc model used does not include all possible factors. Certain aspects are not considered in our analysis. However, the optimization results suggest that the estimated parameters and the developed AQI provide valuable insights into the arc dynamics. Some researchers attribute a reactance to the electric arc [5], which is determined using the Köhle equation [47]. In our study, however, it is found that the arc impedance can effectively be represented as purely resistive. The inclusion of the reactance of the arc does not bring any significant improvements in our model and is therefore omitted.

This study represents an initial investigation into the development of the AQI and is based on data from a single, partial heat. This limitation emphasizes the need for improvements by acquiring a more comprehensive dataset that covers a wider range of heats. Expanding the dataset is critical for a more nuanced assessment of arc conditions and will facilitate the refinement of k and $k_0$ parameters, allowing for a more accurate representation of arc quality.

Furthermore, the current methodology assigns equal weight to all three components within the AQI formulation, as indicated in Equation (40). This approach may not optimally reflect the relative importance of individual parameters $k_{\tau }$, $k_{\alpha }$ and $k_{R_a}$ in influencing arc quality. Therefore, further analytical efforts are warranted to evaluate the effects of these parameters more accurately. Such an analysis could lead to a refined weighting scheme that better reflects the empirical influence of each parameter, thereby increasing the predictive accuracy of the AQI and its utility in EAF operations.

The three-phase Cassie–Mayr model offers the possibility to simulate different scenarios, e.g., a stochastic parameter for the arc length could be introduced to account for different operating conditions. This parameter would reflect the high variance of the arc length during the melting stage and the low variance during the refining stage. In addition, the influence of the grid can be investigated by manipulating the short-circuit power at the PCC. Traditional analyses of arcing phenomena often treat the phases individually, overlooking the critical interactions between them. Our study fills this gap, although the mutual inductance between the phases is not considered, which is a possible direction for future research.

7. Conclusions

This study presents the development of a three-phase electrical circuit model for Electric Arc Furnaces (EAFs) utilizing the Cassie–Mayr arc model. The particle swarm optimization method is used to estimate the key parameters of the Cassie–Mayr model in all three lines. Based on these parameters, a novel Arc Quality Index (AQI) is developed that incorporates aspects of arc stability and arc coverage. The AQI is primarily developed to assist EAF operators in making informed decisions. In order to reduce energy losses due to radiation, high AQI values are desirable at least in the later stages of EAF operation, where arc coverage by the slag is the crucial factor affecting AQI as well as EAF performance. In this manner, appropriate actions can be performed by the operators to increase the AQI, such as reducing the arc length in a combination with an effort to increase slag height. The ultimate goal is to integrate the AQI into EAF power control systems to further improve the efficiency of the arc process. Future research directions include a more detailed investigation of the three-phase Cassie–Mayr model and its comparison with power balance equations. It is also planned to investigate the applicability of the model in flicker analysis, its impact on the power grid and other related areas.