Predicting the Liquid Steel End-Point Temperature during the Vacuum Tank Degassing Process Using Machine Learning Modeling

Abstract

The present work focuses on predicting the steel melt temperature following the vacuum treatment step in a vacuum tank degasser (VTD). The primary objective is to establish a comprehensive methodology for developing and validating machine learning (ML) models within this context. Another objective is to evaluate the model by analyzing the alignment of the SHAP values with metallurgical domain expectations, thereby validating the model’s predictions from a metallurgical perspective. The proposed methodology employs a Random Forest model, incorporating a grid search with domain-informed variables grouped into batches, and a robust model-selection criterion that ensures optimal predictive performance, while keeping the model as simple and stable as possible. Furthermore, the Shapley Additive Explanations (SHAP) algorithm is employed to interpret the model’s predictions. The selected model achieved a mean adjusted $R^2$ of 0.631 and a hit ratio of 75.3% for a prediction error within ±5 °C. Despite the moderate predictive performance, SHAP highlighted several aspects consistent with metallurgical domain expertise, emphasizing the importance of domain knowledge in interpreting ML models. Improving data quality and refining the model framework could enhance predictive performance.

1. Introduction

The temperature control in secondary metallurgy is a critical aspect since it directly affects the quality and the properties of the final product. By controlling the temperature during the production steps and the chemical composition of the melt, it is possible to ensure that the steel will achieve the target microstructure [1]. The microstructure, indeed, is directly connected to the properties and performance that the steel will exhibit in use [2]. In addition, controlling the temperature after the vacuum treatment step in the vacuum tank degasser (VTD) before reaching the Continuous Casting machine (CC) results in improving the efficiency and the productivity of the steelmaking processes. It is also important that the steel melt temperature resides within a desired range of values in order to avoid nozzle clogging problems in the tundish at the CC [3]. Hence, after the vacuum treatment prior to reaching the CC, the steel melt could undergo final temperature adjustments, e.g., heating procedure if the temperature is too low [3]. Furthermore, monitoring the temperature before the CC contributes to planning the production steps and scheduled time. It is important that the secondary metallurgy processes are carried out in a specific tight range of time that matches the scheduled CC starting time. A mismatch at this stage would cause a loss of productivity, since it implies that the CC would have to wait and ultimately break the sequence of heat to be cast. This restarting procedure requires a considerable amount of time and would directly impact the productivity of the steel plant.

A powerful tool for process optimization is the development and use of numerical models representative of the processes involved. One benefit of numerical modeling is that it does not require an actual intervention to the production line, which might imply a temporary halt of the process. Therefore, it can be used to explore the process design without causing negative economic impact to the steel plant [4,5]. Common numerical modeling frameworks for the steel processes are Computational Fluid Dynamics (CFD) and thermodynamic modeling. Both use physico-chemical equations [6,7]. In recent years, partially due to the Industry 4.0 developments [8], the application of machine learning (ML) modeling approaches has become of interest within the steel industry.

Several studies have shown the usefulness of applying ML modeling in the steel industry, e.g., in the prediction of the temperature of the liquid steel [9,10,11]. However, ML models within the steel industry are surrounded by a general skepticism mainly due to a lack of transparency and interpretability of models that can accurately predict the steelmaking processes [12]. The “black-box” approach of non-linear statistical model frameworks, such as the Artificial Neural Network (ANN) and Random Forest (RF), brings with it a lack of trust from the process engineer due to the difficulty in explaining the impact of the input variables on the model’s predictions [4]. In addition, general skepticism can be found in the difficulty of creating models of practical usefulness for the steel plants due to low data quality [4]. Providing a structured methodological approach for both building and evaluating the ML model in the VTD station will contribute to increasing the trust of process engineers for the application and development of ML models that will be used in secondary metallurgy processes.

The reasons behind this study are related both to the importance of the steel melt temperature reached after the vacuum treatment and the necessity of providing a methodological approach when creating an ML model in this context. Steel processes such as the VTD are characterized by non-linear relations among the several parameters of influence [13]. An RF regressor will be employed to forecast the aforementioned temperature. The choice is dependent on the fact that it has been demonstrated in the literature that RF models are not prone to problems such as overfitting and have the ability to handle non-linear relations between the input variables and the output variable [12,13,14]. Additionally, the goal of the proposed ML model is to achieve a 90% hit ratio for a prediction error within ±5 °C to be deemed practically useful, as per the specifications outlined by the process engineers of the steel plant under study. The choice of the input variables of the models will be motivated by using metallurgical reasoning. To validate the model and to explain the output of the model, addressing the “black-box” behavior of RF, the Shapley Additive Explanations (SHAP) algorithm will be used. This will aid in interpreting the model to ensure its agreement with the expected outcomes of VTD process metallurgy.

2. Background

2.1. Steelmaking Overview

Steelmaking is the process of producing steel from raw materials such as hot metal, pig iron, steel scrap, and direct reduced iron (DRI) [15]. The steelmaking process consists of several steps performed to produce steel with the desired target properties. There are two main process routes for primary steel production:

• Blast Furnace (BF)-Basic Oxygen Furnace (BOF);
• Electric Arc Furnace (EAF).

In both routes, once the targeted temperature and composition are met, the molten steel is tapped into the ladle for further refining. At this stage, the steel is processed to reach a specific composition and temperature (thus, specific properties). To accomplish this, the molten steel is transferred to a ladle furnace (LF) for further refining. Concurrently, both the EAF and the BOF undergo preparation phases to accommodate the next cycle.

In the LF, for both the EAF route and BF-BOF route, additional raw materials are added such as silicon, lime, and carbon. The LF processes promote desulfurization and inclusion removal. In addition, vacuum degassing is generally performed to remove dissolved gasses such as nitrogen and hydrogen gasses. Lastly, temperature adjustments are performed to the liquid metal in order to have an optimal temperature control in the casting stage. The refining procedures are generally referred to as secondary metallurgy treatments. Finally, the steel is cast at the CC machine.

The steelmaking process in the steel plant of study starts with the BF. Thereafter, the hot liquid metal is transferred to the BOF. Here, cooling scrap, hot metal, and slag formers such as limestone and dolomite are added to the liquid metal [15]. Then, oxygen is blown into the melt to reduce the carbon content and other impurities such as silicon and phosphorus to the desired target. After the BOF, the steel is tapped into a ladle and then transferred to the secondary metallurgy station (SMS). Here, slag skimming of the converter slag is performed as a first step to avoid unwanted reaction with the liquid steel. The SMS consists of two units:

• Ladle Furnace (LF);
• Vacuum tank degasser (VTD).

In the LF, alloys are added to the liquid metal. This is performed to remove the impurities such as sulfur, to adjust the alloy content, and to reach the desired chemical composition of the steel, which determines the properties of the final product. Moreover, at this stage, the temperature of the steel is also adjusted to meet the desired casting temperature to avoid possible freezing problems at the CC machine. The adjustment of the temperature is carried out by heating the liquid steel with electric arcs. Once the alloys are dissolved, the vacuum treatment starts in the VTD. After the vacuum treatment, final adjustments are performed by yet again passing through the LF [16]. Once having completed all the steps of the secondary metallurgy station, the ladle is transferred to the CC. The secondary metallurgy processes are carried out under time constraints to meet the CC scheduled starting time. Indeed, if the ladle were to not match the CC scheduled time, it would cause a loss of productivity in the whole steelmaking operation, since the CC machine would have to be restarted, causing a halt in its operations. The restarting procedure typically requires up to 60 min. After reaching the CC, the steel in the ladle is discharged and the ladle is kept empty until the cycle starts once again. During the time the ladle is kept empty after the casting of the liquid steel, the energy accumulated in the refractory wall decreases. As a consequence, the refractories’ temperature is reduced. Therefore, to reduce the heat losses of the steel in the ladle, it is necessary to heat up the ladle refractories, i.e., the whole ladle, prior to restarting the production cycle. The preheating of the ladle refractories prior to tapping is accomplished by burners using oxygen and fuel. Figure 1 presents a schematic representation of the production line at the steel plant of study.

2.2. Ladle Furnace (LF)

Prior to the secondary metallurgy station, the steel melt is tapped from the Basic Oxygen Furnace (BOF) into the ladle. Thereafter, the steel melt is further processed in the secondary metallurgy station (SMS). The first step in the SMS is deslagging. The deslagging process ensures chemical composition control of the liquid steel by avoiding unwanted reactions with the carry over slag from the BOF. If the slag layer were not effectively removed, it would cause unwanted chemical reactions with the liquid steel that can impact the properties of the final product. After completing deslagging, the steel melt is transferred to the Ladle Furnace (LF). A schematic representation of the SMS production steps is shown in Figure 2.

Several adjustments are made in the LF, such as additions of alloying elements and temperature adjustments to ensure the target properties for the final steel product. The steel melt is usually stirred by argon bubbling, while deoxidation agents such as aluminum and/or ferrosilicon are added to further reduce the oxygen content of the steel melt. In addition, metal scrap and other alloying elements can be added to reach a specific target composition. It is important to control the melt composition to control the properties of the solidified steel product. The alloying materials added also change the temperature of the molten steel [17]. Furthermore, the melt can be heated up in order to reach a desired temperature before the degassing procedure. The heating is performed by using electrical energy with the aim to control the steel melt temperature that will be reached after the vacuum treatment in the VTD. Controlling the temperature is fundamental since it ultimately affects the castability of the melt at the CC machine [3].

2.3. VTD Process

After the initial treatments in the LF, the steel melt is transferred to the VTD. During the VTD process, dissolved gasses such as hydrogen ($H_2$) and nitrogen ($N_2$) and dissolved carbon (C) are reduced from the molten bath. A schematic representation of the process can be found in Figure 3.

The removal of the dissolved gasses is performed by decreasing the pressure surrounding the molten steel, causing a reduction of their solubility in the melt. Furthermore, argon is injected into the liquid steel to promote the gasses’ removal [3]. The efficacy of the process is dependent on other dissolved elements present in the liquid steel bath such as manganese ($Mn$), silicon ($Si$), phosphorous (P), and sulfur (S) [18]. Depending on the concentration of the dissolved elements, it is possible to influence the removal of the dissolved gasses present in the molten bath. Depending on the specific steel grades, the vacuum treatment duration might vary. Typically, the vacuum treatment is performed for about 30 min. This is due to the fact that different steel grades have different minimum accepted concentration values of dissolved gasses in the steel melt.

2.4. Ladle Heat Loss and Melt Temperature Changes

As mentioned earlier, the temperature changes in the liquid steel are partially related to the ladle heat losses. Being aware of the ladle heat losses during the process step occurring prior to reaching the CC machine is essential to the temperature control of the liquid steel in industrial ladles [19,20,21]. Several studies have highlighted the impact of the heat losses on temperature predictions of the liquid steel [5,17,20,22,23,24,25,26,27,28,29]. There are several phenomena that influence the heat losses of the steel. Radiation, conduction, and convection losses must be considered in order to determine the heat losses in the ladle [21,29,30]. The process steps and observed phenomena influencing the heat losses and temperature changes the most in the secondary metallurgy station are listed in detail below [20,21,23,24,25,26,27,28,29,30,31]:

• Ladle preheating:
  • The ladle refractories have a lower temperature than the steel melt. Therefore, when the liquid steel comes into contact with the ladle, it experiences a decrease in temperature due to phenomena such as conduction, convection, and radiation. The temperature drop depends on the temperature gradient present between the liquid steel and the ladle refractories. Therefore, the preheating of the ladle contributes to reducing the temperature gradient and, thus, the temperature losses. The preheating process is carried out prior to the tapping to increase the accumulated energy of the refractory wall and minimize the heat loss via conduction in the process [30]. An increase in the temperature of the ladle refractories is responsible for a reduction in the thermal gradient existing between the molten steel and the ladle inner walls, which can cause heat losses.
• Tapping time from the BOF:
  • This is a step in which the ladle receives the molten steel from the Basic Oxygen Furnace (BOF). The liquid steel temperature during the tapping from the BOF is reduced by pouring the melt from the BOF into the ladle. This is due to the fact that, during the tapping, the steel is exposed to an atmosphere and surroundings that have a much lower temperature than the steel melt causing heat losses due to radiation and convection [29]. Moreover, the ladle has a lower temperature than the liquid steel. This generates a temperature gradient in the melt and heat losses due to conduction phenomena [29,30].
• Alloying additions and stirring:
  • This is the procedure in which attempts are made to homogenize the steel temperature and chemical composition. By stirring the steel melt, argon is injected, causing a change in the temperature of the metal bath due to the mixing promoted by the gas flow rate. In addition, stirring contributes also to chemical mixing. The efficacy of the process depends on the radial position of the porous plug used [23]. The stirring procedure promotes the heat exchange between hot steel regions and cold ones. By performing alloying additions, the new materials are added into the liquid steel, affecting its temperature due to the chemical reaction occurring, melting, and dissolution, causing thermal gradients in the melt, leading to heat losses.
• Vacuum treatment:
  • The vacuum treatment itself reduces the temperature of the steel melt. By creating the vacuum, the pressure is lowered due to the argon gas injection. This results in a reduction of the steel melt temperature. Furthermore, the injection of argon gas can cause mixing and turbulence in the steel melt, increasing the heat transfer rate. In addition, the argon bubbling is responsible for increasing the melt surface area due to the formation of the bubbles, which contributes to heat lost through radiation and convection.
• Holding time:
  • This is the time between the alloying additions and the teeming, in which the ladle is kept on hold. Generally, the ladle is covered with a removable insulated lid during this stage, and stirring is performed to promote thermal homogenization. However, it can occur that the ladle is exposed to the open surroundings’ atmosphere. This causes a drop in the temperature of the melt due to phenomena such as radiation and convection. Moreover, heat losses due to conduction flow can occur due to heat exchange between the molten metal and the refractory wall in the presence of thermal gradients. It is generally found that the holding time is linear proportional to the thermal stratification, which is also linear proportional to the bulk cooling rate [30]. Therefore, heat losses can be found during this step in the production.
• Slag thickness:
  • Its effect on the heat losses is relatively minor. A thicker slag layer reduces the heat losses up to a critical thickness of around 80 mm [29,30]. The effect of thicker slag layers than the ones with critical thickness has been found to be neglectable [29,30]. Slag layers below 30 mm increase the thermal stratification in the liquid steel. A thick layer generally corresponds to a better thermal insulation of the liquid steel and, thus, reduces the thermal gradients in the melt.
• Teeming:
  • This is the time when the molten steel is poured from the ladle to the tundish in the CC machine. In this step, there is a drop in the liquid steel temperature. Heat losses occur during teeming due to the contact between the liquid steel and the outlet in the bottom wall of the ladle [30]. Furthermore, the exposure of the liquid steel to the tundish is responsible for heat losses.
• Turn around time:
  • This is the time the ladle spends to complete a full cycle in the steelmaking process chain. This influences the hot face wall temperature. A longer time generally implies a lower temperature at the hot face and greater heat losses [29].
• Ladle wear:
  • The ladle refractory’s wearing conditions and its heat status influence the temperature of the melt. This is due to the fact that, depending on the refractory’s wear and the ladle heat status, the efficiency of the steelmaking process varies. Indeed, having non-optimal refractory conditions increases the heat losses in the ladle [20,29,30]. In addition, depending on the heat status of the ladle, the thermal dissipation with the steel melt varies. As previously mentioned, if the ladle is cold, it contributes to generating thermal stratification in the liquid steel, while in contact with the refractories, causing thermal dissipation, i.e., heat losses in the ladle through conduction, convection, and radiation [19,20,29,30]. Generally, thinner refractories cause a higher thermal gradient and heat losses in the steel melt. This is due to a lower ability to keep the hot face wall temperature of the ladle uniform and to provide mechanical stability. This leads to heat losses and thermal stratification in the melt.

All the mentioned processes and phenomena contribute to variations of the liquid steel temperature. Therefore, when modeling the steel melt temperature after the vacuum step in the VTD, variables related to the events, phenomena, and temperature changes discussed above must be considered.

2.5. Model Conceptualization

The proposed ML model will be conceptualized around the secondary metallurgy process. As illustrated in Figure 1, the secondary metallurgy station includes various processes such as deslagging, ladle furnace treatment, and vacuum tank degassing, all of which play crucial roles in refining the steel before it reaches the CC machine.

The conceptualization will treat the secondary metallurgy process as a sequence of repeated “batches” or “heats”, each influenced by multiple internal and upstream effects. This approach allows the model to capture the complexities and variations within each batch. The input variables for the ML model will be selected based on this concept of repeated “batches”, focusing on the “type of effect” they have on the steel melt temperature rather than their specific sequence in the process. This method enables the study to address both causation and correlation considerations, providing the ML algorithm with information on how different effects interact and influence the steel melt temperature.

By organizing and selecting variables based on their “type of effect”, the approach allows for analyzing both individual and combined effects of various process steps, offering deeper insights into their interactions and impact on the final steel temperature. This strategy aims to enhance the understanding of the secondary metallurgy process and improve the model’s predictive performance.

2.6. Secondary Metallurgy Energy Balance Equation

As previously mentioned, the molten steel temperature is affected by various production steps and phenomena causing heat losses that lead to temperature changes. Consequently, estimating the melt temperature involves several assumptions due to the problem’s complexity. By considering an energy balance approach, it is possible to identify the process variables that influence the temperature of the liquid steel in the ladle after the vacuum treatment in the VTD. This would support the variable selection step in the ML model construction. Based on the factors listed in the previous section, the energy balance is described in Equations (1)–(3):

$$E_{in}=E_{tapp}+E_{chem}+E_{electrodes}+E_{burners}$$

(1)

$$E_{out}=E_{steel}+E_{conv}+E_{rad}+E_{cond}+E_{gas}+E_{slag}$$

(2)

$$E_{out}+E_{in}=0$$

(3)

Each energy term is related to a physical–chemical entity (see

Table 1. Energy balance terms governing the secondary metallurgy processes.

Energy Term	Description	Equation	Proportionality
$E_{tapp}$	Energy of the steel during the tapping process	$E_{tapp}=H_{melt,steel}+w_{steel}{\int }_{T_{tapp}}^{T_{steel}}{c}_{p,steel}dT$	$E_{tapp}\propto T_{tapp},T_{steel}$
$E_{chem}$	Total energy from chemical reactions in steel and slag due to the alloying	$E_{chem}={\sum }_i^n{w}_i({\int }_{298}^{T_{m,i}}{c}_{p_i}dT+H_{melt,i}+H_{diss,i}+{\int }_{T_{m,i}}^{T_{steel}}{c}_{p_i}dT)$	$E_{chem}\propto w_i,T_m,T_{steel}$
$E_{electrodes}$	Total energy from electrodes’ heating	$E_{electrodes}={\eta }_{trans}{P}_{avg}{t}_{pon}$	$E_{electrodes}\propto {\eta }_{trans},t_{pon}$
$E_{burners}$	Total energy from burners’ heating	$E_{burners}={\eta }_{fuel}{h}_{fuel}{V}_{fuel}$	$E_{burners}\propto V_{fuel},{\eta }_{fuel}$
$E_{steel}$	Energy of the steel after the secondary metallurgy station	$E_{steel}=H_{melt,steel}+w_{steel}{\int }_{T_{steel}}^{T_{cast}}{c}_{p,steel}dT$	$E_{steel}\propto T_{steel},T_{cast}$
$E_{conv}$	Total energy lost through convection	$E_{conv}=ϵh{A}_{conv}(T_{conv}-T_{env})t_{holding}$	$E_{conv}\propto T_{conv},T_{env},t_{holding}$
$E_{rad}$	Total energy lost through radiation	$E_{rad}=ϵ\sigma A_{rad}{T}_A^4{t}_{holding}$	$E_{rad}\propto T_A^4,t_{holding}$
$E_{gas}$	Total energy lost during the gas treatments	$E_{gas}=\frac{w_{gas}P{V}_{gas}}{R}{t}_{gas}{\int }_{T_{steel}}^{T_{offgas}}{c}_{p_{gas}}\frac{dT}{T}$	$E_{gas}\propto V_{gas},t_{gas},ln\left(T_{steel}\right),ln\left(T_{offgas}\right)$
$E_{slag}$	Total energy lost in slag	$E_{slag}=H_{melt,slag}+w_{slag}{\int }_{T_{steel}}^{T_{Tapp}}{c}_{p,slag}dT$	$E_{slag}\propto T_{steel},T_{tapp}$
$E_{cond}$	Total energy lost through conduction	$E_{cond}=-k{\int }_A(T\left(r\right)-T_{amb})dA$	$E_{cond}\propto T\left(r\right),T_{amb}$

). Several assumptions have been made in the description of the energy terms reported in Table 1):

• The weight of the steel melt and the enthalpy of melting of the steel are assumed to be constant.
• The ladle walls are at ambient temperature.
• The gasses have been assumed to adhere to the ideal gas law.
• The enthalpy of melting of the slag and the weight of the slag have been considered as constants.

Therefore, while constructing a temperature model to predict the temperature after the vacuum step in the VTD, the variables chosen must consider all the factors mentioned above. Thus, as described in Equations (1)–(3) and in Table 1, the model proposed will include variables that represent and characterize the different processes occurring and the temperature sampling in the critical stages of the processes, i.e., the variable belonging the “Proportionality” column in Table 1.

2.7. Statistical Modeling

Statistical models are mathematical models that are not deterministic. Physico-chemical models, on the other hand, rely on a set of phenomena and determined relations and directly predict the output variable from the input variables’ values according to an established relation. As a contrast, statistical models predict the output variable as the most probable one in the context to which the model has been adapted. This means that statistical models may not follow physico-chemical relations between the variables [4]. Statistical models allow the possibility of considering the context under description with all its influencing factors, thus reducing the assumptions and simplifications needed when modeling the industrial context [32]. This is connected to the fact that statistical models make their predictions based on a set of probability distributions and not on established relations. Statistical modeling can be divided into two main approaches [12,33]. One is referred to as data modeling and relies on solid mathematical theories. It produces models that are easy to interpret. An example of a data-modeling framework is linear regression. These models are not optimal when the data are characterized by non-linear relations. The other approach is referred to as algorithmic modeling and has the ability to achieve high predictive performance on data that have a complex relationship with the output variable [33]. Depending on the model framework, algorithmic modeling can also handle non-linear relations between input variables and output variables. However, this could lead to complex models that have low or limited interpretability and lack transparency in explaining the underlying relationship between the models’ variables [34]. Most data collected in the steel industry appear to be characterized by high-dimensional relationships, of which most are not linear [13]. Therefore, the algorithmic modeling approach is appropriate to model steel process systems.

2.7.1. Data Quality

To represent reality, which is the aim of all models, a fundamental prerequisite for an ML model is high data quality [4]. High-quality data accurately mirror the parameters and consistently align with their counterparts in the reality attempted to be modeled. On the other hand, data with low quality will negatively impact the predictive performance of the model. There are various possible sources of errors in the data-collection procedures. Generally, these are due to human factors, hardware limitations, calibration systems, and failure in the used equipment [35,36]. An example in the steelmaking processes could be a malfunctioning of the temperature measurement system while recording the temperature of the steel melt. In this case, there will be missing values or the table will have erroneous entries, which will not depict the actual steel melt temperature.

Identifying the sources of errors in the data-collection procedures can be challenging. Therefore, before building ML models, it is important to be aware of the data quality to evaluate the worth of investing in this modeling methodology [12,36,37,38]. Several methods can be used to assess the data quality [37,38]. In this paper, we primarily focused on data cleaning, which indicates the procedure of correcting or removing errors and missing values in the data, and removing duplicate or corrupted data points from the dataset. By utilizing this method, it is possible to improve the data quality before it affects the performance of an ML model.

2.7.2. Correlation and Causation

Another important factor to consider when building an ML model is correlation. ML models, which are statistical models, are not able to distinguish between correlation and causation [39]. A correlation between two variables does not indicate a causal relationship between them. However, correlation always indicates a possibility of causal relations. Input variables that are strongly correlated amongst one another may be redundant in ML models. The degree of redundancy depends on the inter-relations between the input variables. Therefore, upon selecting the input variables, it is crucial to have domain knowledge about the entity that is to be modeled [4].

An example of correlation can be given by considering the vacuum treatment process. The total amount of injected argon gas into the steel melt is correlated with the duration of the vacuum treatment step. Another example is the electrical energy consumption by the electrode transformer system and the temperature of the molten steel. The importance of correlation is that it indicates and quantifies the relation between the input variables to the output variable. An example of this can be expressed as follows:

$$F1\stackrel{corr}{\to }F2\stackrel{corr}{\to }{F}_{output}$$

(4)

where $F1$ and $F2$ are two different input variables and $F_{output}$ is the output variable. If $F2$ is known to be correlated with the output variable $F_{output}$ and $F1$ is correlated with $F2$, then $F1$ is also correlated with $F_{output}$. Figure 4 illustrates the expected correlations between the input variables and the output variable in the proposed statistical model predicting the steel melt temperature reached after the vacuum treatment. The arrows illustrate the correlative relation between the variables.

2.7.3. Non-Linearity

While building an ML model, it is important to be aware of the type of relationship that exists between the selected input variables and the output variable. The temperature of the molten steel after the vacuum treatment step is influenced by several phenomena that do not follow linear relations. As previously mentioned, the steel melt is subjected to convection, conduction, and radiation phenomena, which cause heat losses in the melt, which affect the steel melt temperature. Such phenomena (Equations (1)–(3) and Table 1) are not linear. Therefore, linear statistical models, such as Multivariate Linear Regression (MLR), and linear statistical metrics, such as the Pearson correlation, are not optimal tools in the context of creating models that aim to predict the temperature of the molten steel after the vacuum treatment step.

2.7.4. Parsimony

Upon building an ML model, adhering to the principle of parsimony is important. A model, in fact, should not use variables that increase its complexity, but do not significantly improve its predictive performance. The complexity of a model is related to the type of statistical model chosen, as well as the number of input variables used. The choice of the final model should, thus, rely on the simplest one in a group of models with similar predictive performance.

Non-linear model frameworks have the ability to create models that can learn more complex relations between the input variables and the output variable [40]. However, this might result in the ML algorithm learning the noise within the data, which is the irreproducible part [41,42]. If this occurs, the ML model will struggle on new data that were not used to adjust its parameters [41,42]. This is also known as overfitting [42]. On the other hand, it is essential to avoid over-simplistic models, as they will fail to capture all the relevant patterns in the data.

2.7.5. Model Stability and Predictive Performance

When developing an ML model, it is crucial to distinguish between model type and model instance. A model type refers to a specific configuration of parameters from the model framework, along with variables chosen based on domain expertise. Conversely, a model instance represents a particular instance of a model type. Ensuring stability in the model type guarantees that the predictive performance of the ML model is not reliant on a random outcome produced by the ML algorithm. Randomization is commonly used in many ML model algorithms such as the ANN and RF, and it has the aim of reducing overfitting and improving the generalizability of the model [43]. By training several instances of the same model type, it is possible to assess the stability of the model type considered. This is achieved by examining the distribution of predictive performance across various model instances. There is no singular method for assessing a model’s stability; thus, it must be defined by the modeler according to the particular use case.

Thus, the aim of building an ML model is to find a model type that achieves high predictive performance, high stability, and low model complexity. This model type is referred to as the most optimal model type [4]. Therefore, a trade-off between predicting performance, stability, and complexity is expected. To achieve this balance, a grid search procedure is commonly employed. The grid search procedure refers to a method used to systematically test through multiple combinations of parameter values, cross-validating for each parameter combination as it iterates in the calculation, to determine the optimal parameters for a given model.

2.7.6. Interpretability

One problem of ML models is the fact that their predictions are difficult to interpret when using complex model frameworks. This often results in a general lack of trust among experts in the application domain. This lack of trust arises from two main factors: a lack of understanding of how the ML model generates its predictions and concerns about its ability to capture the nuances and intricacies of the real-world phenomena it is meant to model. The reasons behind the difficulty in interpreting the results produced by an ML model are related to the “black-box” behavior of ML models [9]. The “black-box” behavior refers to the way the ML model acts while achieving its predictions, which does not adhere to established relations within the field of study, since the model cannot be expressed as analytical equations. However, recent developments in the field of interpretable machine learning have generated algorithms that can be utilized to detect domain-specific validity influencing the predictions of ML models [44].

2.8. Applied ML Modeling in Previous Work on Predicting End-Point Temperatures in Secondary Metallurgy Processes

Predicting the end-point temperature after the secondary metallurgy station is a critical aspect for temperature control in steelmaking. As reported in the literature, several methods have been adopted to predict the end-point temperature of the molten steel after the refining units, including theoretical models based on heat transfer studies, data-driven models based on ML techniques, and integrated models combining the two previously named methodologies [7,9,19,22,24,26,45,46,47].

Table 2. Application of ML in secondary metallurgy temperature prediction (Part 1), adapted from [45]. “-” indicates that the method is not mentioned.

Author	Year	Process	Algorithm	Predictive Performance	Domain-Informed Variables	Data Treatment	Temporal Data Split	Non-Linear Algorithm	Grid Search	Robust Model Selection	Interpretability Algorithm
He et al. [48]	2012	LF	CBR	Accuracy (±5 °C): 76.38% R: 0.7237 RMSE: 5.38 °C	Yes	Yes	-	Yes	-	Stability–complexity not discussed	-
Tian et al. [49]	2017	LF	AdaBoost	Accuracy (±5 °C): 92.85% RMSE: 5.1506 °C	Yes	Yes	-	Yes	-	Stability–complexity not discussed	-
Yang et al. [50]	2019	LF	FNN	Accuracy (±5 °C): 88% RMSE: 3.8563 °C MAE: 3.066 °C	Yes	Yes	-	Yes	-	Stability–complexity not discussed	-
Yang et al. [51]	2021	LF	DNN	Accuracy (±3 °C): 77.0% Accuracy (±5 °C): 93.6% R2: 0.9286 RMSE: 3.1647 °C	Yes	Yes	-	Yes	-	Stability–complexity not discussed	-
Xin et al. [52]	2022	LF	DNN	R2: 0.897 RMSE: 1.710 °C	Yes	Yes	-	Yes	-	Stability–complexity not discussed	-
Feng et al. [53]	2019	RH	CBR	Accuracy (±5 °C): 78.31% Accuracy (±7 °C): 91.17%	Yes	Yes	-	Yes	-	Partially, limited discussion Stability–complexity	-
Gu et al. [54]	2020	RH	CBR	Accuracy (±7 °C): 83.67%	Yes	Yes	-	Yes	-	Stability-Complexity not discussed	-
Bao et al. [55]	2019	RH	Cloud model	Accuracy (±10 °C): 93.32%	Yes	Yes	-	Yes	-	Partially, limited discussion Stability–complexity	-
Wang et al. [13]	2019	VD	ELM	Accuracy (classification): 83.67% MAE: 4.873 °C	Yes	Yes	-	Yes	-	Stability-Complexity not discussed	-
Chen et al. [56]	2020	VD	ELM	MARE: 0.02801 RMSE: 6.299 °C	Yes	Yes	-	Yes	-	Stability–complexity not discussed	-
Jianwen et al. [57]	2013	VOD	RBFNN	Accuracy (±50 °C): 85.7%	Yes	Yes	-	Yes	-	Stability–complexity not discussed	-
Fang et al. [58]	2023	LF, RH	Improved BPNN	Hit rate of over 90% for deviations within ±20 °C	Yes	Yes	-	Yes	-	Partially, limited discussion Stability–complexity	-
Viana et al. [59]	2018	BOF to RH	Hybrid ANN Thermodynamic model	Standard deviation of error: 5.46 °C Predicted error interval: −10.79 °C to +10.66 °C	Yes	Yes	-	Yes	-	Stability–complexity not discussed	-
Tian et al. [9]	2008	LF	Hybrid ELM Thermodynamic model	Error range within ±5 °C for over 90% of cases	Yes	Yes	-	Yes	-	Stability–complexity not discussed	-
Xin et al. [60]	2023	LF	IF–ZCA–DNN	R2: 0.916 RMSE: 2.827 MAE: 2.048 Hit ratio for ±10 °C: 99.6%	Yes	Yes	-	Yes	-	Stability–complexity not discussed	-

and

Table 3. Application of ML in secondary metallurgy temperature prediction (Part 2), continued from Table 2, adapted from [45]. “-” indicates that the method is not mentioned.

Author	Year	Process	Algorithm	Predictive Performance	Domain-Informed Variables	Data Treatment	Temporal Data Split	Non-Linear Algorithm	Grid-Search	Robust Model Selection	Interpretability Algorithm
Qiao et al. [61]	2021	LF	Dynamic Ensemble	RMSE of 1.31 °C	Yes	Yes	-	Yes	-	Stability–complexity not discussed	-
Mao et al. [62]	2012	LF	Optimally Pruned Bagging with PLELM	Specific performance metrics not detailed.	Yes	Yes	-	Yes	-	Stability–complexity not discussed	-
Wang et al. [63]	2022	LF	Dynamic Selective GPR	RMSE: 3.024 °C	Yes	Yes	-	Yes	-	Stability–complexity not discussed	-
Wang et al. [64]	2022	LF	Dynamic Outlier Ensemble model	PC: 0.985 Accuracy: 90.6%	Yes	Yes	Partially	Yes	-	Partially, limited discussion Stability–complexity	-
Li et al. [65]	2011	LF	BP neural network with Expert System	Hit probability of 85% for deviations within ±5 °C	Yes	Yes	-	Yes	-	Stability–complexity not discussed	-
Singh et al. [66]	2023	BOF, LF, RH, CC	Stacking Regression	Accuracy: 95%-99%	Yes	Yes	-	Yes	-	Partially, limited discussion Stability–complexity	-
Chen et al. [67]	2022	LF	Heterogeneous Ensemble	R2: 0.545 to 0.574 MAE: 5.88 to 5.85 MSE: 78.42 to 73.01 RMSE: 8.82 to 8.51	-	Yes	-	Yes	Yes	Partially, limited discussion Stability–complexity	-
Wang et al. [68]	2016	LF	BFSE-RTs	RMSE < 2.4 °C Max Error < 6.7 °C	Yes	Yes	No	Yes	-	Yes, focuses on performance and stability	-
Wang et al. [11]	2016	LF	TSE-GRNNs	RMSE: less than 0.33 °C Max Error: less than 0.89 °C	Yes	Yes	No	Yes	-	Yes, focuses on performance and stability	-
He et al. [69]	2014	BOF, LF, CC	Hybrid ANN	End LF Hit rate for ±10 °C: 97.7%	Yes	Yes	-	Yes	-	Stability–complexity not discussed	-
Wang et al. [70]	2012	RH-TOP	Multiple Regression	Hit rate above 95% for ±10 °C	Yes	Yes	-	Yes	-	Stability–complexity not discussed	-
Yuan et al. [71]	2021	LF	Improved CBR-HTC	MAE: 3.12 °C RMSE: 3.86 °C	Yes	Yes	-	Yes	-	Stability–complexity not discussed	-
Wang et al. [28]	2017	LF	Random Forest	RMSE: 2.8 °C Max Error: 7.6 °C	Yes	Yes	No	Yes	-	Stability–complexity not discussed	-
Wang et al. [72]	2022	VTD	AdaBoost.RT + ELM	Accuracy of 91.02% within ±10 °C	Yes	Yes	-	Yes	-	Yes	-

present a summary of previous research employing ML modeling predicting the steel melt temperature in secondary metallurgy, which is the focus of the present study. Notably, it highlights their methodological approaches with respect to the present work and the reported predictive performance of the created ML models.

An analysis of these studies points out that, while ML models have been used to predict temperatures with various degrees of predictive performance, several gaps stand out in existing methodologies. Firstly, the methodologies frequently do not utilize a temporal split approach to divide the testing and training dataset. This is paramount for capturing the dynamic nature of the VTD process over time. This type of split permits generalizing predictions for forecasting purposes. Without it, models may fail to accurately reflect process dynamics, impacting the applicability of the model for forecasting. Moreover, previous works often do not address the stability and complexity of the models during the model-selection process. This can lead to the development of models that are either overfitted or overly simplistic, thus failing to generalize well across different operational settings. Additionally, the predictive performance of existing models for the Vacuum Degasser (VD), VTD, and Vacuum Oxygen Decarburization (VOD) ranges widely, as summarized in the table. For VD models, the accuracy generally falls between 83% and 84% with an error margin of around ±5 °C. VTD models demonstrate accuracy between 90% and 91% within a ±10 °C error range. VOD models show accuracy around 85%, but within a much broader ±50 °C range. Previous work, as reported in Table 2 and Table 3, outlines the potential of ML models predicting the molten steel temperature during secondary steelmaking, but also underscore the need for improved methodologies. The approach proposed in the present work will be used to attempt to create a model that meets a hit ratio of 90% within ±5 °C, providing a reliable tool for process developers in the steel plant of study. Finally, interpretability algorithms, such as SHAP, are never employed to elucidate the decision-making mechanisms of the models. This often leaves the ML models as “black-boxes”, where the reasoning behind predictions remains unclear. The present work intends to use interpretability algorithms to make the model’s decisions transparent and understandable to users. This will enhance the trust of process engineers in the application and development of ML models within steelmaking. The present work will also focus on providing a clear methodological approach for creating and evaluating an ML model in the context of predicting the liquid steel temperature after the vacuum tank degasser (VTD) treatment step in the steel plant process chain.

3. Method

This section provides a detailed overview of the methodology employed in the current research. This methodology is derived from a methodological approach published previously [12]. Specifically, the proposed methodology incorporates its approach to establish the variable-selection process on metallurgical considerations. Furthermore, it employs techniques for creating variable batches and groups, as well as model-selection techniques aimed at maximizing predictive performance and stability while minimizing the complexity of the selected model. Additionally, the methodology includes part of the model-validation and -verification procedures outlined in the above-mentioned research, in particular as concerns the elucidation of the influence of the input variables on prediction outcomes, thereby opening the “black-box” of the model by utilizing the SHAP algorithm.

What distinguishes the proposed methodology is the incorporation of approaches to imputing missing data, involving the testing of various imputation methods. In the model selection, the methodology employs a filtering approach to reduce the complexity of the selected model by focusing on models with similar predictive performance. Additionally, it incorporates a temporal cross-validation method enhancing the models’ generalizability, to ultimately select the model with the highest predictive performance and stability and lowest complexity. Furthermore, to enhance the reproducibility of the results and mitigate the effect of randomness by the RF algorithm, a fixed random state is set for each model instance of a model type produced. Through this comprehensive methodology, the study aims to provide robust and reliable predictions in the context of temperature prediction post vacuum treatment in the VTD process. Figure 5 shows a flowchart for the methodology flow of the present work. Notably, the dashed boxes highlight the proposed steps in the methodology, while the other boxes outline the steps derived from previous research.

3.1. Variable Selection

The variable-selection process has been based on the discussion around the heat losses in the ladle, the specification of the VTD process itself, and the process steps between the BOF and the VTD stations together with their influence on the steel melt temperature variations (Section 2.1, Section 2.2, Section 2.3, Section 2.4, Section 2.5, Section 2.6 and Section 2.8). The selected variables are presented in

Table 4. Variables used in the model and their corresponding energy terms, as shown in Table 1.

Variable	Unit	Definition	Energy Term
BOF-to-ladle addition	kg	The addition of scrap in the ladle during the tapping from the BOF	$E_{tapp}$
Energy transformer	MWh	The energy consumed by the electrodes to heat up the steel melt between the temperature sample before the vacuum step and after one vacuum step	$E_{electrodes}$
Heat status ladle	-	The heat status of the ladle	$E_{rad},E_{conv}$$E_{burners}$
Number of boosts	-	The total number of boosts that has been performed for each heat	$E_{gas}$
Number of heats ladle	-	The number of heats the ladle has been reused after the last relining	$E_{conv},E_{rad}$
Porous plug gas volume	m3	The volume of the injected argon gas during stirring between the temperature sample before the vacuum step and after one vacuum step	$E_{gas}$
Steel code	-	The steel grade code for each heat	$E_{steel},E_{slag}$$E_{chem}$
Steel weight	kg	The steel start weight at the beginning of the VTD station plus the addition weight before the last temperature sample before the vacuum	$E_{tapp},E_{chem}$
Stirring time	min	The total stirring time between temperature before and temperature final	$E_{rad},E_{conv}$$E_{gas}$
Tapping temperature	°C	The temperature of the melt in the tapping step	$E_{tapp}$
Temperature before	°C	The last temperature of the melt before the vacuum treatment	$E_{tapp}$
Temperature final	°C	The first temperature of the melt after the vacuum treatment; it is the target variable of the ML models in the present work	$E_{steel}$
Temperature predicted before	°C	The last temperature of the melt before the vacuum treatment predicted by the current online model; the prediction is taken at the same time as the samples of temperature before	$E_{tapp}$
Temperature predicted final	°C	The first temperature of the melt after the vacuum treatment predicted by the current online model; the prediction is taken at the same time as the samples of temperature final	$E_{steel}$
Temperature variation additions	°C	The variation of the melt temperature between temperature before and temperature final due to the alloying additions	$E_{steel},E_{chem}$
Time between temperature samples	min	The total time between the temperature samples, respectively before and after the vacuum treatment	$E_{rad},E_{conv},E_{cond}$
Time burners	min	The total time in which the ladle is under the burners	$E_{burners}$
Time burners-to-tapping	min	The total time between the end of the burners and the tapping for each ladle	$E_{burners}$
Time from BOF	min	The time between the tapping and the starting of the vacuum	$E_{rad},E_{conv}$
Time of empty ladle	min	The time in which the ladle has been empty between different campaigns	$E_{rad},E_{conv}$
Total addition weight	kg	The total amount of scrap and alloying elements added between temperature before and Temperature final	$E_{chem}$
Total boost time	min	The total time of all the boosts performed for each heat	$E_{rad},E_{conv}$, $E_{gas}$
Total heat ladle	-	The number of times the ladle has been used	$E_{rad},E_{conv}$
Vacuum process time	min	The time of the vacuum treatment	$E_{rad},E_{conv}{E}_{gas}$

with their corresponding energy terms presented in Section 2.6.

3.2. Variable Combinations

The key objective is to create a model type that achieves a high predictive performance, is stable, and is as simple as possible. However, achieving this goal collides with the necessity of having high computational power. In the current study, the number of input variables is 23. Therefore, determining the number of possible combinations is straightforward, involving only 23 multiplication procedures. However, training $2^{23}$ models incurs substantial computational costs, amounting to 8,388,608 model types. In light of this computational burden, it has been decided to create variable groups informed by domain knowledge. Subsequently, 16 distinct possible combinations of those groups referred to as variable batches have been appointed. To address this, the approach of combining variable groups and variable batches is proposed as a solution. The next paragraphs explain, in detail, the procedures behind the construction of the variable groups and variable batches.

3.2.1. Variable Groups

The creation of variable groups poses advantages in selecting a model type characterized by optimal predictive performance while minimizing its complexity. This is due to the fact that creating variable groups and training the models across these group combinations facilitate the possibility of reducing the number of input variables. The model’s complexity is proportional to the number of input variables, with fewer input variables corresponding to a less complex model. These groups, derived from domain knowledge, serve as building blocks for several variable batches. The formation of each group is grounded in domain-specific insights.

To clarify the concept of the “type of effect” mentioned in Section 2.5, it was chosen to consider different categories of variables based on their impact, such as temperature-related effects, chemical composition effects, and operational parameters. This approach ensures that the model captures the essential influences on the steel melt temperature without being overwhelmed by the sheer number of variables.

Additionally, it was chosen to introduce a “Mix” group containing the expected most influential variables from each production step on the temperature of the molten steel. This aggregation seeks to capture the overall impact of the production steps, from the BOF to the VTD, affecting the temperature of the molten steel at the end of the vacuum tank degassing process in the VTD. This strategy allows for the study of both causation and correlation, considering how different types of effects interact and influence the steel melt temperature. The variable groups are shown in

Table 5. Variable groups and the number of input variables in each group.

Group	Number of Variables
Temperature Group
Tapping Temperature Temperature before Temperature predicted before Temperature predicted final	4
Mix Group
BOF-to-Ladle addition Energy transformer Heat status ladle Number of boosts Porous plug gas volume Steel weight Time between temperature samples Time of empty ladle Vacuum process time	9
Gas Group
Stirring time Total boost time	2
Ladle Group
Number of heats ladle Time burners Time burners-to-tapping Time from BOF Total heats ladle	5
Addition Group
Steel code Total addition weight Temperature variation additions	3
Number of Variables	23

.

3.2.2. Variable Batches

Table 6. The variable batches created for the numerical experiments.

Group\Batch	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Temperature group	X	X	X	X	X	X	X	X	X	X	X	X	X	X	X	X
Mix group	X	X	X	X	X	X	X	X
Gas group	X	X	X	X					X	X	X	X
Ladle group	X	X			X	X			X	X			X	X
Addition group	X		X		X		X		X		X		X		X

provides an overview of the 16 variable batches considered. As previously mentioned, the choice of creating variable batches relies on the purpose of identifying the model type that optimally balances a high predictive performance and stability and minimal complexity.

Notably, the temperature group is present in every variable batch, aligning with the core purpose of the model, which is to predict the final temperature of the steel melt after the vacuum treatment step. The logic behind creating these batches is twofold: firstly, to systematically test the model’s predictive performance by evaluating the impact of the different variable groups presented in Table 5 and, secondly, to account for metallurgical considerations. Specifically, the selection of variable groups was influenced by their known relevance to the steelmaking process under description and their potential impact on the melt temperature.

3.3. Data Treatment

The data used encompassed approximately one year of production. The aim of the data treatment was twofold: to identify and remove erroneous data instances and to manage data points that could lead to a distorted representation of regular production. The data treatment step unfolds across three sub-steps: data cleaning, splitting into training data and test data, and the treatment of missing data. The subsequent subsections elaborate on each of the procedures. A schematic representation of the strategy adopted in the data treatment steps is shown in Figure 6.

3.3.1. Data Cleaning

The primary objective of the data cleaning step is to systematically eliminate potential errors inherent in the data, particularly focusing on rarely occurring events. Such events, if not addressed, could compromise the predictive performance of the model [4].

The intricacy of data cleaning lies in the delicate balance between two conflicting aspects. On the one hand, the removal of sporadic events and errors leads to improved predictive performance. On the other hand, the elimination of sporadic events risks rendering the model less representative of reality. Hence, the application of solely statistical methods for outlier detection becomes insufficient in navigating this nuanced terrain. This stems from the limitation of statistical methods for outlier detection, which primarily focus on the probability distributions of the data and lack consideration for the demands specific to the application domain. In addition, statistical methods for outlier detection generally prove inadequate for addressing multidimensional data problems [73]. In dynamic environments, characterized by continuous changes influenced by specific production demands, as observed in the steelmaking industry, the domain knowledge of the modeler plays a crucial role in guiding the data cleaning step. This enables a more comprehensive identification of sporadic events and low data quality.

In this study, the chosen approach involves replacing all the errors present in the database, including inaccurate entries and negative values for physical entities, such times and temperatures, with missing values (NaN). The treatment of missing values will be extensively covered in Section 3.3.3. In addition, it was chosen to exclude all the heats characterized by logged data with a process time of zero for the recorded events. For instance, where the stirring and its injected gas volume was registered as an occurring process, if the stirring process time was logged as zero, the heats were removed.

3.3.2. Splitting into Training Data and Test Data

An important preliminary step preceding the execution of the ML experiments involves the division of the data into training and test datasets. This partitioning procedure needs to be performed before addressing the treatment of missing data to prevent any information leakage from the test dataset into the training dataset. The occurrence of such leakage would introduce bias to the prediction generated by the ML models, as the ML model would inadvertently be exposed to information from the test dataset, which is intended solely for evaluating the predictive performance of the ML model.

A common practice involves random sampling the dataset as a means to select the test data. However, in the present study, this approach is not applicable since it would result in a lack of chronological intertwining between the training data and the test data. The purpose of the model, indeed, is to predict the temperature after the vacuum for heats from a future perspective. Therefore, it is imperative that the split adheres to a temporal sequence, enabling the testing and evaluation of the model in a more realistic scenario. Thus, it was chosen to perform a time series cross-validation. The time series cross-validation involves a sequential augmentation of the number of data points in the training dataset. This method ensures that the model is trained on older data and tested on more recent data, capturing any temporal changes in the secondary metallurgical process over long periods of time. The choice of 9-fold cross-validation guarantees that the size of the test dataset always corresponds to 10% of the total number of data points. As a result, cross-validation allows for the evaluation of heats from a forward-looking perspective, enhancing the generalizability of the model. In the final fold, the total number of heats considered is 4162. The training data represent 3745 heats, while the test data encompass 417 heats. A schematic representation of the time series cross-validation is shown in Figure 7.

This approach ensures that the model’s performance is evaluated in a manner that closely mimics its intended industrial scenario, providing insights into how well the model can adapt to changes in the process over time. By training with older data and validating with more recent data, the study aims to capture the effects of long-term changes in production, thereby making the model more robust and reliable for future predictions.

3.3.3. Treatment of Missing Data

The VTD process is characterized by a significant variability, leading to an inconsistent sequence of events for each heat. This variability translates into missing values in the database, often indicating processes that were not performed. To address this, missing values were imputed with zeros to signify the non-occurrence of the process. Furthermore, errors logged in the datasets were replaced with NaN, as discussed in Section 3.3.1. Imputing these values is crucial before training the ML model, as the RF algorithm used in the present work cannot handle missing data.

Handling missing values is a common challenge in ML modeling. While one approach is to remove instances with missing values, this can lead to significant data loss [74], reducing dataset diversity and potentially losing valuable insights. Such loss may compromise the robustness and generalizability of the ML model. Hence, careful considerations are necessary when deciding on strategies for handling missing values. Balancing dataset integrity with model effectiveness is key to maintaining the applicability of the ML model.

Another common approach is imputing missing values with either the mean or median of the respective variable [75]. However, this simplistic strategy may not adequately represent reality, as it operates solely on individual variables without accounting for their relationships [75]. To address this limitation, various multivariate imputation algorithms have been developed. These algorithms predict missing values by considering the interactions between variables in the dataset [74,75]. Despite their complexity and computational demands, these techniques are often considered optimal as they are able to capture the relationships between all variables in the dataset [74].

In this study, it was chosen to explore different imputation strategies to compare and select the optimal option based on predictive performance, stability, and model complexity. The preferred choice will exhibit high predictive performance, stability, and low complexity. To mitigate bias, the imputation approach was adapted using the training data. Subsequently, the imputed values derived from the training data were applied to both the training and test datasets. The imputed training data were used for the model training, and the resulting predictive performance on the imputed test data was recorded. The imputation strategies included were the following:

• Imputation of the mean;
• Imputation of the median;
• K-nearest neighbor algorithm (KNN);
• Random Forest regressor (RF).

Each multivariate imputation technique was evaluated with 3 different sets of algorithm parameters, resulting in 3 trials for each algorithm. Specifically, the number of nearest neighbors for the KNN was varied, considering values of 3, 5, and 7, respectively. Additionally, the number of estimators for the RF was varied, considering values of 100, 250, and 500, respectively. The imputation techniques were enclosed into the grid search procedure (see Section 3.6), allowing selecting the most optimal technique to use.

3.4. Random Forest

While selecting a model framework, it is imperative to ensure that it has the ability to effectively capture both the intra-relationship among the designed input variables and the inter-relationship between these inputs and the output variable. The temperature of the liquid steel in the VTD process, as previously mentioned, is affected by non-linear phenomena. Consequently, the choice of the ML model framework should exhibit non-linear capabilities. One such algorithm is RF.

The RF model framework utilizes multiple decision trees for making predictions. Notably, RF can handle both continuous data and discrete data due to the property of the decision trees. By incorporating a bootstrapping technique while training, which involves sampling with replacement from the entire training dataset to fit each tree, the RF models mitigate the risk of overfitting. Additionally, the reduction of the overfitting risk is also obtained by random sampling of the input variables coupled with the utilization of a loss function to attain the optimal split at each node. This ensures a high likelihood of each tree being unique, contributing to diversity within the model [76]. Finally, each prediction results from an average of the predictions made by each decision tree. A schematic representation of the RF algorithm is presented in Figure 8.

Moreover, the choice of the RF model is influenced by its compatibility with the Tree Shapley Additive Explanations (SHAP) algorithm. The SHAP and Tree SHAP algorithms are explained in detail in Section 3.8. The RF algorithm encompasses various parameters that can be adjusted to optimize the model’s predictive performance. As concerns the current work, it has been chosen to adjust the following three hyperparameters [77]:

• Number of decision trees;
• Maximum number of features for splitting a node;
• Maximum depth of the tree.

The number of decision trees serves as an indicator of the model complexity, reflecting the total number of decision trees generated in the RF model. In this study, it was chosen to vary the number of trees from 100 to 800, encompassing a selection of 10 specific values. The decision to extend the exploration up to 800 was motivated by the purpose of gaining a comprehensive understanding of how the model complexity unfolds across a broad spectrum value and how it influences the model’s predictive performance. During each split, the number of input variables was determined by either the square root of the total number of input variables or the total number of input variables. This parameter, influencing the model complexity, operates as follows: in the former case, the RF model employs a random selection of variables, while in the latter, all the designed input variables are taken into account. This parameter’s influence on the model complexity arises from the trade-off between randomness and determinism introduced. A lower value of this parameter introduces more randomness to the model, leading to higher diversity of the trees, while a higher value leans more towards a deterministic approach, resulting in trees more closely aligned. The maximum depth, representing the utmost number of permitted splits for any tree within the RF model, is another hyperparameter directly impacting the model complexity. Throughout the experiments, the maximum depth was varied from 10 to 200 considering only 10 specific values within this range. The choice of extending the exploration up to 200 aligns with the reasoning regarding the hyperparameter that determines the number of decision trees. Other hyperparameters within the RF model were maintained at their default settings [78].

3.5. Predictive Performance Metrics

Equation (5) will be used to calculate the prediction error of the models:

$$E_i=y_i-{\widehat{y}}_i$$

(5)

where $y_i$ is the true value, ${\widehat{y}}_i$ is the predicted value, and $i\in 1,2,\dots ,n$. n is the number of data points. Consequently, the mean error and the standard deviation of the error are calculated as shown in Equation (6) and Equation (7), respectively:

$$\widehat{E}=\frac{1}{n}{\displaystyle \sum _{i=1}^n}{y}_i-{\widehat{y}}_i$$

(6)

$$E_{\sigma }=\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^n}{\left(E_i-\widehat{E}\right)}^2}$$

(7)

The $E_{min}$ and $E_{max}$ errors indicate, respectively, the lowest and highest predicted error from the dataset. A common predictive performance metric used to test the quality of the fitness in an ML regressor model is the $R^2$-metric, which is calculated according to:

$$R^2=1-\frac{{\sum }_{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}{{\sum }_{i=1}^n{\left(y_i-\overline{y}\right)}^2}$$

(8)

${\overline{y}}_i$ represents the mean of the true values, $y_i$, and ${\widehat{y}}_i$ is the predicted value. An $R^2$ value of 1 signifies a perfect model, where predictions exactly match the true values. Conversely, a negative $R^2$ value suggests that the model’s predictions are worse than simply using the average of the actual values.

Adding more input variables generally results in a slight increase in the $R^2$ value. Consequently, when comparing models with varying numbers of input variables, it is necessary to adjust the $R^2$ value [79]. For this reason, the adjusted $R^2$ value will be utilized in this study. The adjusted $R^2$ value is computed as follows:

$$R_{adj}^2-=1-\left(1-R^2\right)\frac{n-1}{n-p-1}$$

(9)

where $R^2$ represents the standard $R^2$-value, p indicates the number of input variables, and n represents the number of data points.

3.6. Grid Search

The grid search methodology enables the identification of an optimal set of parameters that balances the considerations of stability, predictive performance, and complexity. Acknowledging the heavy computational nature of grid searches, it was chosen to limit the exploration to a defined number of parameter combinations. The outcomes of the grid search inform the selection of the most optimal model. The designed grid search encompasses both modeling framework parameters and domain-specific parameters, as depicted in

Table 7. Grid search parameters and combinations. The specifications of the RF algorithm used can be found in [78].

Parameter	Variations	Combinations
Number of trees	100, 177, 255, 333, 411, 488, 566, 644, 722, 800	10
Max depth of the tree	10, 31, 52, 73, 94, 115, 136, 157, 178, 200	10
Max features	auto, square root	2
Time series cross-validation	9-fold	9
Variable batches	see Table 4	16
Imputation	Mean, Median, 3 KNN, 3 RF	8
Total number of combinations		230,400

. Notably, the parameters for the number of neighbors in the KNN and the number of trees in RF refers to the adjustment parameters selected for the imputation algorithms KNN and RF, respectively (see Section 3.3.3). Including the imputation procedures in the grid search facilitates the identification of the most optimal imputation technique to implement.

By performing the grid search method, each combination of parameters is represented by one model type. The total number of combinations for the grid search is $10\times 10\times 2\times 9\times 16\times 8=$ 230,400, which corresponds to the 230,400 model types created.

3.7. Model Selection

It is crucial to opt for the model types that guarantee stability. In order to assess and validate the stability of a chosen model type, it is essential to produce multiple instances of the same model type and measure variations in their predictive performance. This method ensures that the predictive performance of the selected model is consistent and not dependent on random outcomes. To address the inherent randomness introduced by the RF algorithm, 10 instances for each model type were generated. This approach permitted an examination of model stability by assessing the variations in predictive performance metrics across different instances. Each of the 10 instances of a model type created was trained with a specific random state, a parameter set prior to training the model instance [78], resulting in the selection of 10 different random states. This was chosen to ensure that each model instance of a specific model type consistently produces the same predictive performance at each execution, provided the same random state is kept. The analysis focused solely on the predictive performance values of the test data to determine the most optimal model. This approach is justified because the test data consist of instances that the model has not been exposed to during training. Equation (10) outlines the 10 different random state considered:

$$Randomstate=901,56,201,1002,281,508,250,71,4,972$$

(10)

To assess the stability of the model types, it was chosen to use the $R_{adj}^2$ metric. The metrics used are explained in

Table 8. Aggregate predictive performance metrics.

Adjusted ${\mathit{R}}^2$ Statistics	Definition
$R_{adjmean}^2$	mean of the mean of the $R_{adj}^2$ values of a model instance of a model type between the 9 folds
$R_{adjstd}^2$	standard deviation of the mean of the $R_{adj}^2$ values of a model instance of a model type between the 9 folds
$R_{adjmin}^2$	minimum of the mean of the $R_{adj}^2$ values of a model instance of a model type between the 9 folds
$R_{adjmax}^2$	maximum value of the mean of the $R_{adj}^2$ values of a model instance of a model type between the 9 folds
$R_{adjmeanmax}^2$	maximum value of the $R_{adjmean}^2$ values within the 10 model instances created between the 9 folds
$R_{adjmeanmin}^2$	minimum value of the $R_{adjmean}^2$ values within the 10 model instances created between the 9 folds
Stability	$R_{adjmeanmax}^2$ − $R_{adjmeanmin}^2$ within the 10 model instances created between the 9 folds

and are all calculated on the test data.

A model type was considered stable over the 9 folds if the following condition was respected:

$$Stability<0.05$$

(11)

The threshold of 0.05 was set by the modeler. Whereas multiple model types fulfilled this constraint, the model type with the highest $R_{adjmean}^2$ was selected. This model is denoted as the Preliminary Model (PM). Finally, a threshold has been established to inspect the predictive performance of other model types ($M_i$) compared to the PM as follows:

$$PM{R}_{adjmean}^2-M_i{R}_{adjmean}^2<0.005$$

(12)

This decision emerges from the possibility that numerous model types exhibit negligible differences in predictive performance. The threshold value of 0.005 was chosen to maintain as high a performance as possible, keeping it close to the PM. Hence, the decision was made to explore the potential of identifying model types with lower complexity than the PM, while maintaining comparable predictive performance.

The final model relied on the minimization of the complexity of the model. Therefore, the model-selection process took into account simultaneously the concepts of stability, predictive performance, and complexity. Figure 9 shows a flowchart for the grid search and model selection procedure.

3.8. Interpretability Algorithms

3.8.1. Shapley Additive Explanations (SHAP)

The Shapley Additive Explanations (SHAP) is an algorithm that is derived from game theory, enabling it to explain the output of any ML model. Introduced by Lundberg in 2017 [80], it facilitates the calculation of each input variable’s contribution to the output variable’s prediction. The SHAP algorithm operates based on the following equation:

$$g\left(z^{\prime }\right)={\varphi }_0+\sum _{i=1}^M{\varphi }_i{z}_i^{\prime }$$

(13)

where M is the number of input variables, $z^{\prime }$ is an M-dimensional binary variable, and ${\varphi }_i$ is the weight of the input variable i on the prediction.

3.8.2. Tree SHAP

The Tree SHAP algorithm permits the determination of the exact contribution of each input variable in tree-based models. This is achievable because the Tree SHAP algorithm does not presume independence among the variables. Consequently, for RF models, the SHAP algorithm provides insights into the impact of each input variable on the prediction without being affected by the other input variables [81].

Given a variable pair ${\varphi }_{i,j}$, the Tree SHAP algorithm treats them as equally split such that ${\varphi }_{i,j}$ = ${\varphi }_{j,i}$ [81]. The relation between the SHAP interaction and the regular SHAP is described by Equation (14):

$${\varphi }_i={\varphi }_{i,i}+\sum _{i\ne j}^M{\varphi }_{i,j}$$

(14)

The model’s prediction is equivalent to the sum of the main SHAP interaction values and all SHAP main interaction values:

$$\sum _{i=0}^M\sum _{j=0}^M{\varphi }_{i,j}(f,x)=f\left(x\right)$$

(15)

As the number of variables and data points increases, the computational expense of SHAP rises significantly. Additionally, it has the limitation of being unable to determine the exact change in the output variable resulting from a specific change in an input variable or a set of input variables. This is due to the fact that the SHAP value represents the mean contribution of multiple combinations of the input variables.

3.8.3. SHAP as Interpretability Tool

The Tree SHAP algorithm enables a comprehensive understanding of the significance of the input variables in prediction. The analysis of the SHAP values’ distribution offers insights into the model behavior upon usage. This approach helps in characterizing the model’s representation of the real industrial process. Specifically, SHAP serves as an insight tool during the exploration of variable contributions to the predictions, elucidating the model’s fidelity to the real world [82,83,84].

In this study, the SHAP algorithm is applied to both the test and training datasets. However, since the model is tailored to the training data, the SHAP values calculated for the test data are dependent on those from the training data. On the one hand, using Tree SHAP on the test data helps demonstrate each input’s contribution to predicting the temperature on new, unseen data points. This approach provides insight into the model’s predictive capability, as the test data come after the training data chronologically. On the other hand, applying Tree SHAP to the training data aids in understanding the model itself, given that the model is a function of the training data. A discrepancy in the SHAP value distributions between the test and training data is anticipated. It is important to emphasize that the results from SHAP should be interpreted with caution if the model exhibits low predictive performance. In this case, the SHAP values should be viewed as indicative rather than exhaustive.

4. Result and Discussion

4.1. Model Selection

In line with what was illustrated in Section 3.6, this study has embraced a grid search approach with the scope of identifying the most suitable models from a diverse array of model types by adjusting the parameters reported in Table 7. The selection of the most optimal model hinges upon its ability to simultaneously optimize predictive performance, stability, and model complexity, echoing the principle of parsimony.

A minimal difference in predictive performance is observed across several variable batches, as detailed in

Table 9. Parameters and predictive performances of models with similar predictive performance as the Preliminary Model (PM).

Model	Variable Batch	Imputer	N° Estimator	Max Feature	Max Depth	${\mathit{R}}_{\mathit{adj}\mathit{mean}}^2$	Stability
Preliminary Model (PM)	7	Simple imputer median	100	auto	10	0.631	0.017
Model type 1	1	Random Forest regressor	100	auto	10	0.626	0.017
Model type 2	2	Random Forest regressor	200	auto	94	0.617	0.016
Model type 3	3	Random Forest regressor	100	auto	31	0.629	0.015
Model type 4	4	K-nearest neighbors	500	sqrt	73	0.617	0.016
Model type 5	5	Simple imputer median	800	sqrt	115	0.626	0.015
Model type 6	6	K-nearest neighbors	800	sqrt	136	0.617	0.013
Model type 8	8	K-nearest neighbors	800	auto	200	0.618	0.017

. The aggregated predictive performance metrics are explained in Table 8. In particular, across Batch 1 to Batch 8, Batch 7 and Batch 8 stand out for having the lowest number of input variables compared to the rest. This observation is based on the analysis of the models’ $R_{adjmean}^2$ across different batches of input variables. Notably, the Preliminary Model (PM) signifies the model type achieving the highest predictive performance, standing out as having the lowest number of input variables compared to the rest. This highlights the fact that higher model complexity does not necessarily equate to improved predictive performance of the model. Furthermore, the RF hyperparameters for the PM result have the lowest value within the available values in the grid search.

By applying Equation (12), only Batch 1, Batch 3, Batch 5, and Batch 7 persist. It is important to highlight that all imputation techniques used (see Section 3.3.3) appear in the model types surviving the above-mentioned threshold (Equation (12)). Therefore, the least complex imputation technique should be chosen. The selected model for further analysis from the surviving model types is based on the least complex option and corresponds to the PM (see Table 9). It is important to note that the PM being the least complex model from the grid search may not always be the case. For reference purposes, the selected model will be referred to as the Model of Choice (MC).

4.2. Predictive Performance Analysis

Figure 10 displays a graph illustrating the predicted temperature alongside the actual temperature measurement for both the test and the training datasets. The selected model type corresponds to the MC. The test data points are marked in red, while the training data points are represented by the green dots. It is important to state that this plot depicts predictions from a singular model instance, rather than the average of all 10 model instances, specifically focusing on the final fold, which allocates 90% of the data to training and 10% of the data to testing.

Notably, the plot exhibits outliers in predictions on both the test data (red dots) and training data (green dots). Furthermore, the MC displays a mean minimum prediction error of −52.7 °C and a mean maximum error of 35.2 °C across the 10 model instances and the nine folds (see

Table 10. Model of Choice (MC) predictive performance metrics.

${\mathit{R}}_{\mathit{adj} \mathit{mean}}^2$	${\mathit{R}}_{\mathit{adj} \mathit{min}}^2$	${\mathit{R}}_{\mathit{adj} \mathit{max}}^2$	STD	Max Error	Min Error	Mean Error	Hit Ratio ±5 °C	Hit Ratio ±10 °C
0.631	0.623	0.640	8.42	35.2	−52.7	6.2	75.3%	91.4%

). Given the substantial variability in the data, the RF algorithm currently struggles to effectively capture underlying patterns for higher prediction accuracy, as indicated by the elevated standard deviation of the error. Moreover, the presence of numerous anomaly data points, possibly attributed to special requirements such as delays, hinders the model’s ability to achieve higher predictive performance. Thus, it suggests exploring alternative ML model frameworks to discern whether the observed low predictive performance stems from the chosen ML model framework or potential issues related to the data quality.

Evaluating the adequacy of the model predictive performance for the specific context application is imperative. Table 10 provides the calculated metrics for the MC. The error is calculated in agreement with Equation (5).

When comparing with models reported in the literature, which was compiled in Section 2.8, Li et al. [57] reported an 85.7% accuracy for the VOD within a much broader ±50 °C range, which, although having a higher hit ratio, lacks the precision of the presented model’s ±5 °C error range. Wang et al. [13] achieved an 83.67% accuracy with an MAE of 4.873 °C for the VD, indicating a slightly better predictive performance in terms of accuracy, but within a similar precision range. Chen et al. [56] reported an RMSE of 6.299 °C, which suggests that the MC’s mean error of 6.2 °C indicates a similar level of predictive performance. For the VTD, Wang et al. [72] demonstrated a 91.02% accuracy within a ±10°C error range. In comparison, the MC achieves a higher hit ratio of 91.4% within the same ± 10°C range. Despite these comparisons, as per the process engineers’ requirement in the steel plant of study, predictions should ideally fall within a 90% hit ratio while maintaining a prediction error within ±5°C. However, the MC’s hit ratio of 75.3% does not meet these specifications, indicating the need for further improvements.

4.3. Influence of Input Variables on the Predictions

This section encompasses the evaluation of each input variable’s influence on the predicted temperature post vacuum treatment in the VTD, facilitated by the Model of Choice (MC). The theoretical fundament of the SHAP algorithm has been elaborated in Section 3.8.

Figure 11 presents the SHAP summary plot for both the test and training data. These offer insights into the combination of feature importance and their respective impacts on predictions. The y-axis shows the features, while the x-axis represents their corresponding SHAP values. Each plotted point corresponds to a specific SHAP value associated with a feature and a data instance. The x-axis mirrors the influence of each feature on the model’s predictions. Positive values represent features pushing predictions higher, while negative values suggest features lowering predictions. Thus, the x-axis gives a comprehensive view of how individual features influence the model’s output. Notably, on the y-axis, overlapping values are shown in a way that visually conveys the distribution of the SHAP values for each input variable. Furthermore, the features are ordered according to their significance.

The analysis of the SHAP summary plot obtained from the training data reveals the most significant features guiding the model’s predictions. Mainly, these include the predicted temperature by the current online physico-chemical model, the time interval between temperature samples before and after the vacuum treatment, the ladle’s heat status, the aggregate alloying additions made between the temperature measurements prior and post vacuum, the temperature before the vacuum treatment, and the energy supplied by electrodes to the melt before vacuum treatment. These findings are in agreement with metallurgical considerations, underlining their importance in the model’s predictive efficacy. Moreover, despite the relatively low accuracy of the temperature model currently in use, its predominance within the overall model’s feature importance is noteworthy. This can be associated with its function as a reference point for temperature, supporting the predictions made by the ML model. It is, therefore, not surprising that it plays a significant role in the model’s decision-making process. However, discrepancies between the SHAP values from the training and test data are observed. Specifically, in the test data, the significance of the temperature sample prior to the vacuum treatment is reduced in comparison to the energy provided by the electrodes to the melt. These observations are crucial for model deployment, as they offer insights into the model’s behavior when applied to new datasets. Understanding these intricacies is paramount in ensuring the model’s reliability and effectiveness in real-world scenarios.

To analyze the impact of the individual input variables on predictions, dependence plots were employed. Figure 12 displays the dependence plots for the most influential input variables observed in both the training and test data. This strategy enhances the understanding of the model, revealing how the selected variables influence the model’s predictions. Similar trends seem to appear in both the training and test data as concerns the SHAP values’ distribution. In agreement with domain-specific knowledge, several key insights arise. Firstly, an extended time between temperature samples prior to and after the vacuum treatment exhibits a negative correlation with the melt’s temperature. This phenomenon can be associated with prolonged vacuum treatment and potential holding time during which the ladle is exposed to the surrounding atmosphere. This leads to heat loss through radiation, subsequently resulting in a lower liquid steel temperature. Moreover, a higher initial temperature before the vacuum treatment corresponds to a higher temperature post-treatment. This relationship is intuitive, as a hotter liquid steel before the treatment leads to a higher final temperature. Furthermore, the final predicted temperature by the online model showcases a strong correlation with the actual final temperature. This points out the pivotal role of the online model’s final temperature prediction, as suggested by the SHAP summary plots, where it emerges as a dominant feature. In addition, the variable associated with the ladle heat status indicates that a cold ladle corresponds to higher values of the variable, which inversely correlates with the temperature. Although this inverse correlation is not as evident in the dependence plots, it is clearly suggested by the summary plots. Furthermore, the feature representing alloying additions demonstrates that a higher volume of additions between the two temperature samples before and after the vacuum treatment tends to increase the temperature, suggesting the predominantly exothermic nature of the additions made during this part of the process. Exothermic additions are not predominant in this part of the process in the steel plant of study. Thus, the model misrepresents this variable, which may be connected to the relatively low predictive performance of the MC model.

Indeed, although most of the insights gained from the previous dependence plots seem to align with metallurgical considerations, it must be acknowledged that, due to the low predictive performance achieved by the MC, the SHAP values must be regarded as indicative rather than definitive in this context. Therefore, the SHAP values cannot be used in this case to draw exhaustive conclusions, as they may reflect limitations inherent in the model.

5. Conclusions

This study focused on predicting the temperature after the vacuum treatment step in the VTD station. The primary goal was to establish a comprehensive methodology for creating and evaluating machine learning (ML) models within this context. Additionally, the objective for the developed ML model was to achieve a hit ratio of 90% for a prediction error within ±5 °C to be considered practically useful by the process engineers of the steel plant of study. Moreover, another goal was to analyze the model’s agreement with metallurgical domain expertise. Based on the results and discussion obtained, the corresponding conclusion related to the research goals can be drawn:

Methodology:

• As outlined by the analysis of the models’ predictions and the influence of the variable batches, the predictive performance varied from an $R_{adjmean}^2$ of 0.496 in Batch 16 to an $R_{adjmean}^2$ of 0.631 for the MC (Batch 7). This denotes a significant influence of the number and types of input variables on the predictive performance. Therefore, the selection of input variables should always rely on metallurgical domain knowledge.
• The selected model must showcase satisfactory stability, predictive performance, and simplicity, adhering to the principle of parsimony. Setting a stability threshold as filtering criteria ensures the reliability of the model’s predictive performance. Additionally, achieving high predictive performance does not necessarily entail high model complexity. Models from various variable batches, in particular 1, 3, 5, and 7, exhibited similar predictive performance despite differences in the number of input variables.
• The predictive performance of ML models highly relies on the quality of the data used for training and testing. Improving data quality through carefully designed data-collection strategies and measurement techniques is crucial. This would likely enhance the predictive performance of the Model of Choice (MC).

Requirements of process engineers:

• The evaluation of ML models’ usefulness relied on their predictive performance on test data. The MC achieved a satisfactory $R_{adjmean}^2$ value. However, the hit ratio of 75.3% within a prediction error of ±5 °C does not meet the process engineers’ requirements, suggesting the necessity of data quality improvements and possibly testing other model frameworks. Additionally, it is essential to consider the potential impacts of variable selection and variable grouping on these results. The choice to model with specific variable groups likely influenced the observed predictive performance, highlighting areas for future improvement and refinement. Including more key temperature-related variables throughout the entire process line or incorporating more parameters related to the heat losses of the ladle could potentially enhance the model’s accuracy and robustness.

The selected model’s agreement with metallurgical domain expertise:

• A combination of both metallurgical and machine learning expertise is fundamental in order to produce an ML model that accurately represents the metallurgical context under consideration. The interpretability of the model was outlined to ensure its relevance and trust by the process engineers. The application of SHAP offered insights into how the various input variables impact the predicted temperature within the VTD process. The analysis revealed that the time between the temperature samples, the heat status of the ladle, the steel temperature before the vacuum treatment, and the energy consumed by the electrodes significantly affect the steel’s final temperature, underscoring their critical roles in process control. These findings verify the MC alignment with metallurgical expectations. Furthermore, the dependence plots derived from SHAP values elucidate the intricate relationships between these key variables and the final outcomes, emphasizing the need for domain expertise in interpreting the data and making informed decisions. However, these plots also highlight that the model misinterpreted the impact of alloying additions, which could be associated with the relatively low predictive performance of the MC model. By highlighting specific contributions of the input variables, the SHAP analysis enhances the understanding of their influence on the process, guiding possible improvements in both model accuracy and operational strategies. However, due to the low predictive performance achieved by the MC, the SHAP values cannot be used to draw definitive conclusions and need to be regarded as indicative in this context.

Comparison with previously published models:

• The predictive performance of the presented model, with a mean error of 6.2 °C, a hit ratio of 91.4% within a ±10 °C range, and a hit ratio of 75.3% within a ±5 °C range, demonstrates competitive accuracy compared to previously published models.