Interpretable Machine Learning Models for Estimating Electric Energy Consumption in Steel Industries

Abstract

The substantial energy consumption and associated CO₂ emissions from industrial operations pose significant environmental and economic challenges for factories and surrounding communities. Within the context of industrial energy management, the steel industry represents a major energy consumer. The imperative to optimize energy use in this sector is driven by a combination of environmental concerns, economic incentives, and technological advancements. This study presents a machine learning model that integrates the whale optimization algorithm (WOA) with multivariate adaptive regression splines (MARS) to forecast electric energy consumption. Utilizing a dataset comprising 35,040 real-world energy consumption records from Gwangyang Steelworks in South Korea, the model was benchmarked against other regression techniques (ridge, lasso, and elastic-net), demonstrating that the proposed WOA-MARS approach achieves a significant improvement in the RMSE (vs. elastic-net or lasso regression techniques) while maintaining interpretability through hinge function analysis. The WOA-tuned MARS model achieves a coefficient of determination (R²) of 0.9972, underscoring its effectiveness for energy optimization in steel manufacturing. The key findings reveal that CO₂ emissions and reactive power variables are the strongest predictors.

1. Introduction

As industrialization advances, energy demand correspondingly increases, necessitating the continual adaptation of national policies. Economic growth further intensifies energy consumption, rendering energy both an essential and finite resource. Promoting energy conservation benefits consumers, producers, and overall sustainability, while safeguarding long-term efficiency. The acceleration of economic expansion amplifies the demand for electricity, a fundamental component of daily life and a critical driver of economic development [1]. Environmental challenges stem from the steel industry’s substantial energy consumption and associated CO₂ emissions [2]. Carbon dioxide, the principal greenhouse gas, is predominantly released during blast furnace and coking operations, which together account for approximately 70% of total emissions [3,4]. On a global scale, steel production is responsible for nearly 7% of anthropogenic CO₂ emissions [5]. To mitigate climate change, advanced technologies are being developed to quantify and reduce energy consumption and CO₂ emissions in steel production. Current research focuses on emission-reduction strategies such as carbon capture and the adoption of green hydrogen technologies [6,7,8]. By 2030, these innovative approaches are projected to decrease CO₂ emissions by approximately 14–21% and energy consumption by 7–11% [9]. Consequently, forecasting energy demand has become a critical priority for both industrial sectors and national policymakers [10]. Recent research has increasingly employed data-mining techniques to predict energy consumption, utilizing methods such as gradient-boosted machines [11], neural networks [12], time-series models [13], support vector machines [14], regression analysis [15], and deep learning architectures [16]. One study evaluated predictive models, including generalized linear models (GLMs), support vector regression (SVR), k-nearest neighbors (k-NN), random forest, and M5 model trees within a smart city framework [17], although further enhancements in predictive accuracy remain necessary.

In contrast with time-series forecasting methods, which predict future energy demand using lagged variables, this study focuses on estimating instantaneous energy consumption as a function of concurrent operational parameters. The objective is to model the nonlinear relationships between real-time measurements (e.g., reactive power, CO₂ emissions, and load type) and energy usage, providing interpretable insights for operational optimization rather than temporal prediction.

This study accurately predicts the electric energy consumption of Gwangyang Steelworks (see Figure 1) using a multivariate adaptive regression splines (MARS) model [18,19,20,21,22,23,24,25], optimized through the whale optimization algorithm (WOA) [26,27,28,29]. To the best of our current knowledge, this specific combination has not yet been explored in the existing literature. For comparative analysis, the same dataset is also evaluated using ridge regression (RR), lasso regression (LR), and elastic-net regression (ENR) techniques [20,21,22,30,31,32,33,34,35]. MARS is a statistical learning approach capable of capturing nonlinearities and variable interactions, evolving from linear models to effectively represent complex relationships [18,19,20,21,22,23,24,25,26,27,28,30,31]. It offers several advantages over both traditional and metaheuristic regression techniques [18,19,20,21,22,23,24,25,30,31,36,37]: (i) straightforward implementation compared with linear regression, (ii) consistent basis functions derived from identical datasets, (iii) ease of interpretability, (iv) ability to handle both continuous and categorical variables, and (v) an explicit mathematical formulation of the dependent variable through fundamental basis functions. This latter characteristic distinguishes MARS from black-box machine learning models such as multilayer perceptrons. Previous studies have demonstrated the effectiveness of MARS in predicting diverse phenomena, including antitumor activity [38], soil organic carbon [39], regolith geochemical grades [40], drilling mud viscosity [41], used car prices [42], benzene concentration [43], and dissolved oxygen levels [44]. However, to date, it has not been applied to predicting the electric energy consumption of Gwangyang Steelworks.

The structure of this manuscript is organized as follows: It begins with a detailed description of the experimental design, followed by an outline of the parameters utilized in this investigation. Section 2 provides a comprehensive review of the regression modeling approaches employed, including lasso, ridge, elastic-net, and WOA/MARS-based methods. Section 3 presents the results of the WOA/MARS analysis, comparing predicted values with experimental measurements and evaluating the relative importance of the input parameters. Finally, Section 4 summarizes this study’s conclusions and highlights its principal findings.

2. Materials and Methods

2.1. Experimental Setup

The dataset comprises synchronous measurements of nine independent variables and the target variable (energy consumption (EC)), all recorded at 15 min intervals. The dataset for this study comprises 35,040 samples of electric energy consumption and the following process variables: carbon dioxide emissions (tCO₂), lagging current reactive power (LaCRP), lagging current power factor (LaCPF), leading current reactive power (LeCRP), leading current power factor (LeCPF), day of the week (DoW), week status (WS), seconds since midnight (NSM), and load type (LT). The dataset was obtained from the Gwangyang Steelworks, a major industrial facility specializing in the production of steel coils, iron plates, and related steel products [45]. The records span a full year, capturing detailed measurements of key operational variables. The dependent variable in this study is the electric energy consumption (EC), measured in kilowatt-hours (kWh) per 15 min interval. This metric represents the electrical energy used by individual production units or processes within the Gwangyang Steelworks during each sampling period. The 15 min interval aligns with the plant’s data-logging system, capturing high-resolution variations in energy demand associated with operational phases such as smelting, casting, and material handling [46]. The importance of these data comes from their ability to track energy consumption in detail and in real time, providing valuable insights into daily, monthly, and yearly usage patterns [17]. This study does not frame the problem as a time-series forecasting task (e.g., predicting future EC from past values). Instead, the problem is approached as a multivariate regression task, in which EC is estimated from variables measured at the same timestamp. This approach is justified by the industrial need to understand how current operational states (e.g., furnace load and power factor) directly impact energy consumption, without assuming temporal dependencies. Figure 2 shows the experimental setup of the process.

2.2. Variables in Model and Materials

The primary objective of this study was to identify the most influential parameters for estimating energy consumption using the WOA/MARS model. Energy consumption, representing the electrical energy utilized in steel production from iron ore, was designated as the output variable. A total of nine input variables were considered, as summarized in

Table 1. List of the operational variables analyzed in this investigation, gathered with the corresponding standard deviations (STDs) and averages.

Input Variables	Parameter Name	Mean	Standard Deviation	Minimum Value	Maximum Value
Lagging current reactive power (kVarh)	LaCRP	13.035	16.306	0	96.91
Leading current reactive power (kVarh)	LeCRP	3.871	7.424	0	27.76
Carbon dioxide emissions (ton CO2/15 min)	tCO2	0.012	0.016	0	0.07
Lagging current power factor (%)	LaCPF	80.578	18.921	0	100
Leading current power factor (%)	LeCPF	84.368	30.457	0	100
Number of seconds from midnight for each day	NSM	42,750	24,940.534	0	85,500
Week status	WS	-	-	-	-
Day of week	DoW	-	-	-	-
Load type	LT	-	-	-	-
Output Variable
Steelworks’ energy consumption (kWh)	EC	27.387	33.444	0	157.18

:

Lagging current reactive power (kVarh): Indicates an inductive nature with reactive power consumption when the load current peaks at 90° after the voltage peak;

Leading current reactive power (kVarh): Occurs when the current waveform leads the voltage waveform, indicating capacitive loads supplying reactive power;

Carbon dioxide emission (ton/15 min): Emissions from combustion in steel plants and other industrial processes;

Lagging current power factor (%): Reflects the load current peaking after the voltage, with capacitive loads used to compensate;

Leading current power factor (%): Indicates a capacitive load supplying reactive power when current leads the voltage;

Number of seconds from midnight for each day: Time duration from midnight to the specified datetime value;

Week status: Categorical variable indicating whether the day is a “weekend” or “weekday” [47];

Day of week: Categorical variable indicating the day of the week for casting operations;

Load type: Categorical variable with three types of furnace load: “light load,” “medium load,” and “maximum load”.

The operation output variable of this study was as follows:

The Gwangyang Steelworks’ specific electric energy consumption (kWh): The Gwangyang plant is categorized as a high-energy-consumption facility due to the substantial amount of electrical energy required for steel production.

Exploratory data analysis represents a fundamental step in deriving meaningful insights from a dataset. This analysis confirmed that all dataset features were complete, with no missing values. In addition, the interrelationships among the variables were systematically examined. A correlation matrix was employed to visualize the associations between the different variables, and the corresponding heatmap illustrates the strength of the linear relationships among them. Figure 3 depicts the correlations of various variables with respect to energy consumption. The strongest correlation was observed between CO₂ emissions and lagging current reactive power, reflecting their direct relationship with energy consumption.

2.3. Techniques for Mathematical Modeling

2.3.1. Multivariate Adaptive Regression Splines (MARS) Method

The MARS approach provides a versatile non-parametric approach to regression problems [18,19,20,21,22,23,24,25,30,31]. This technique can be considered as extending both stepwise linear (SL) regression and classification and regression tree (CART) decision trees [30,48,49], while addressing their respective limitations. The primary objective of MARS is to foretell a continuous dependent output variable’s results $\mathit{y}\left(n\times 1\right)$, relying on a multitude of independent input features, collectively referred to as $\mathit{X}\left(n\times p\right)$. The basic principles underlying the MARS methodology can be explained by a specific mathematical formulation [18,19,20,21,22,23,24,25,30,31]:

$$\mathit{y}=f\left(\mathit{X}\right)+\mathit{e}$$

(1)

where

f: This component represents an aggregation of basis functions, each of which depends on the input variable set $\mathit{X}$. These functions are combined by a weighted summation process.

$\mathit{e}$: This term denotes the error vector, which is characterized by its n-dimensional nature, where n corresponds to the number of observations in the dataset. It is typically represented as a column vector with dimensions $\left(n\times 1\right)$.

The MARS approach does not necessitate a predetermined functional relationship between the output and input features. Mathematically, it can be articulated as an assemblage of segmented polynomial expressions of order q, referred to as basis functions, with their parameters entirely determined through a detailed analytical process involving $\left(\mathit{X},\mathit{y}\right)$.

The methodology of MARS is implemented through the adjustment of specific ranges of the predictor variables using these basis functions. In particular, two-sided truncated power functions, or splines, are the fundamental building blocks of the MARS model. These functions, called hinge functions, can be expressed mathematically as follows [18,19,20,21,22,23,24,25,30,31]:

$$\begin{gathered} {\left[-\left(x-t\right)\right]}_{+}^q=\left\{\begin{array}{ccc}{\left(t-x\right)}^q& \mathrm{if}& x<t\\ 0& \mathrm{if}& \mathrm{otherwise}\end{array}\right\} \\ {\left[+\left(x-t\right)\right]}_{+}^q=\left\{\begin{array}{ccc}{\left(t-x\right)}^q& \mathrm{if}& x\ge t\\ 0& \mathrm{if}& \mathrm{otherwise}\end{array}\right\} \end{gathered}$$

(2)

The hinge function knots in the MARS model are data-derived thresholds optimized to minimize prediction error. These values reflect empirical breakpoints in the relationship between predictors and energy consumption, rather than physical constants.

The power q (where $q\left(\ge 0\right))$ determines how smooth the resulting approximated function is and, at the same time, also determines the category of splines employed. Specifically, when $q=1$, the model uses linear splines, which are the focus of the present investigation. Quadratic splines are generated when $q=2$, while cubic splines are generated when $q=3$, with this pattern continuing for higher values of q. To illustrate this concept, Figure 4 provides a visual representation of two splines in the case where $q=1$, with the node (also referred to as the knot) positioned at $t=3.5$. The purpose of this graphical representation is to demonstrate the behavior of linear splines at a particular node location. The choice of q has significant implications for the flexibility of the model and its capacity to identify nonlinear patterns in the data. Linear splines $(q = 1)$ offer a balance between model simplicity and the ability to approximate nonlinear functions, making them a popular choice in many applications. Higher-order splines $(q > 1)$ can provide greater smoothness but can also introduce additional complexity into the model. The MARS algorithm’s capability to automatically select appropriate basis functions and node locations contributes to its versatility in different areas of scientific investigation.

Unlike regularized regression methods, the implementation of MARS does not require predictor scaling or centering [50]. The algorithm’s hinge functions adapt to the natural scale of each variable, while the forward/backward selection process ensures parsimony by excluding irrelevant predictors (e.g., unused categorical levels).

The MARS approximation can be characterized as an ensemble of M piecewise linear multivariate splines of a dependent variable derived from M basis functions, built using the hinge functions. The mathematical formulation is as follows [18,19,21,22,23,24,25,30,31]:

$$\widehat{\mathit{y}} = {\widehat{f}}_M\left(\mathit{x}\right) = c_0 + \sum _{m=1}^M{c}_m{B}_m\left(\mathit{x}\right)$$

(3)

where the MARS method’s predicted output variable is $\widehat{\mathit{y}}$ and

• $c_0$ represents a constant coefficient, commonly referred to as the intercept;
• $B_m\left(\mathit{x}\right)$ signifies the m-th basis function;
• $c_m$ corresponds to the coefficient related to the basis function $B_m\left(\mathit{x}\right)$.

To obtain the optimal collection of basis functions for the comprehensive method, the MARS methodology uses the technique of generalized cross-validation (GCV) [18,19,20,21,22,23,24,25,30,31]. GCV is calculated as the mean-squared residual error divided by a penalty factor, the latter being directly proportional to model complexity, and can be expressed mathematically as [18,19,21,23,24,25,30,31,50,51]

$$GCV\left(M\right)={\displaystyle \frac{\frac{1}{n}{\sum }_{i=1}^n{\left(y_i-{\widehat{f}}_M\left({\mathit{x}}_i\right)\right)}^2}{{\left(1-C\left(M\right)/n\right)}^2}}$$

(4)

The more basis functions that are employed in the MARS model, the higher the complexity penalizing term, $C\left(M\right)$. This term is expressed as [18,19,20,21,22,23,24,25,30,31,50,51]

$$C\left(M\right)=\left(M+1\right)+dM$$

(5)

where M is the number of model basis functions in Equation (3), and d is the model penalty parameter for each basis function.

With the purpose of ensuring the robustness of the findings, it is essential to consider additional factors alongside the aforementioned GCV. These include the following [18,19,21,22,23,24,25,30,31,50,51]:

RSS (residual sum of squares): This measure quantifies the difference between the observations and predictions;

N-subsets: The factor that specifies the total count of groups within the model encompassing all variables.

2.3.2. Ridge Regression and the Least Absolute Shrinkage and Selection Operator (Lasso) Regression Methods

Both regression procedures represent regularization techniques employed in this study to predict the energy consumption (EC) of Gwangyang Steelworks [20,21,22,30,31]. These methods are particularly effective in addressing multicollinearity, which frequently arises in linear regression models with numerous predictors. Overall, they introduce a controlled level of bias in exchange for enhanced efficiency and stability in parameter estimation.

Ridge regression (RR) was developed as a potential solution to address inaccuracies in least squares estimators arising from multicollinearity among independent variables in linear regression models. By providing variance and mean square estimates that are often lower than those obtained via ordinary least squares, RR enables more precise estimation of model parameters. Consequently, it is also employed to predict the energy consumption (EC) of Gwangyang Steelworks [20,21,22,30,31,52,53].

By adding positive components, the problem of a quasi-singular moment matrix ${\mathit{X}}^T\mathit{X}$ is mitigated by lowering the condition number of the diagonals. The following formula for the ordinary least squares estimator can also be used to find the simple ridge estimator [20,21,22,30,31,52,53]:

$${\widehat{\beta }}_{RR}={\left({\mathit{X}}^T\mathit{X}+\lambda \mathit{I}\right)}^{-1}{\mathit{X}}^T\mathit{y}$$

(6)

When $\mathit{y}$ is the regressand, $\mathit{I}$ represents an identity matrix, with $\mathit{X}$ as the design matrix, and the constant that moves the diagonals of the moment matrix is called the ridge parameter $\lambda \ge 0$, sometimes dubbed as the regularization or complexity parameter. It can be demonstrated that this estimator may be used to solve the following least squares problem with the restriction $‖\beta ‖\le t$ (a predetermined free parameter t determines the degree of regularization), whereby a Lagrangian can be used to represent it [25,26,27,31,32], as follows:

$$L^{RR}=\underset{\beta }{min}{\left(\mathit{y}-\mathit{X}\beta \right)}^T\left(\mathit{y}-\mathit{X}\beta \right)+\lambda {‖\beta ‖}_2^2\mathrm{subject} \mathrm{to}{‖\beta ‖}_2^2\le t$$

(7)

where ${‖\beta ‖}_2={\left({\sum }_{i=1}^N{\beta }_i^2\right)}^{1/2}$. This proves that $\lambda$ is just the Lagrange multiplier of the restriction. The ridge estimator turns into ordinary least squares when $\lambda =0$, demonstrating that the restriction is not enforceable. Typically, $\lambda$ is chosen heuristically, which means that it will not exactly meet the requirement. Therefore, the principal advantage of ridge regression over standard linear regression lies in its capacity to balance the trade-off between bias and variance. Specifically, an increase in bias is accompanied by a corresponding reduction in variance, and vice versa.

The lasso regression (LR) method, also applied in this study to address the aforementioned issue, shares several key similarities with ridge regression (RR). The loss function of lasso regression is formally defined as follows [25,26,27,31,32,54,55]:

$$L^{LR}=\underset{\beta }{min}{\left(\mathit{y}-\mathit{X}\beta \right)}^T\left(\mathit{y}-\mathit{X}\beta \right)+\lambda {‖\beta ‖}_1\mathrm{subject} \mathrm{to}{‖\beta ‖}_1\le t$$

(8)

so that ${‖\beta ‖}_1={\sum }_{i=1}^N\left|{\beta }_i\right|$ and where the data determines the exact relationship between t and λ.

To sum up, LR can reduce coefficients to zero, whereas RR cannot [20,21,22,30,31,52,53]. This distinction arises from the difference in the shapes of their constraint boundaries. Although they are subject to different constraints—${‖\beta ‖}_1\le t$ for lasso regression and ${‖\beta ‖}_2^2\le t$ for ridge regression—both ridge and lasso regression can be interpreted as minimizing a common objective function.

2.3.3. Elastic-Net Regression (ENR)

Elastic-net regression (ENR) combines the principles of both lasso and ridge regressions. The elastic-net Lagrangian function is formulated by integrating the penalty terms of these two methods [20,21,22,30,31,52,53]:

$$L^{EN}=\underset{\beta }{min}{\left(\mathit{y}-\mathit{X}\beta \right)}^T\left(\mathit{y}-\mathit{X}\beta \right)+{\lambda }_1{‖\beta ‖}_1+{\lambda }_2{‖\beta ‖}_2^2$$

(9)

In fact, lasso regression is obtained if ${\lambda }_2=0$, while ridge regression is obtained if ${\lambda }_1=0$. The pair of parameters $\lambda$ can be substituted with the $L_1-ratio$ parameter, which determines the ratio of the L₁ penalty to $\lambda$. Since most computational codes use the number $L_1-ratio=0.5$ the most, it is used in the calculations here. In this regard, the elastic-net regression’s final Lagrangian is provided by [21,22,31,32,34,35]

$$L^{EN}=\underset{\beta }{min}{\left(\mathit{y}-\mathit{X}\beta \right)}^T\left(\mathit{y}-\mathit{X}\beta \right)+\lambda \left(1-L_{ratio}\right){‖\beta ‖}_1+\lambda L_{ratio}{‖\beta ‖}_2^2$$

(10)

2.3.4. Whale Optimization Algorithm (WOA)

The Whale Optimization Algorithm (WOA) was originally proposed by Mirjalili and Lewis [28]. It is an optimization technique inspired by the complex hunting behavior of humpback whales. In a process known as bubble-net feeding, whales generate bubbles to encircle their prey during the hunt. Typically, the whales dive approximately 12 m before creating a spiral of bubbles around the target and subsequently ascend in pursuit. The WOA operates through three primary phases [26,27,28,29]:

Exploration phase: During this phase, each search agent (whale) explores the search space randomly to locate the optimal solution (prey). To update the position of a given search agent, a randomly selected agent is used instead of the current best-performing agent. Mathematically, this behavior is expressed as follows:

$$\begin{gathered} \mathit{D}=\left|\mathit{C}\cdot {\mathit{X}}_{rand}\left(t\right)-\mathit{X}\left(t\right)\right|, \\ \mathit{X}\left(t+1\right)={\mathit{X}}_{rand}\left(t\right)-\mathit{A}\cdot \mathit{D}, \end{gathered}$$

(11)

where ${\mathit{X}}_{rand}$ is the position vector of a randomly selected prey, t denotes the actual iteration, and $\overrightarrow{X}$ indicates the whale position vector The coefficient vectors are defined as $\mathit{A}=2\mathit{a}\cdot {\mathit{r}}_1-\mathit{a}$ and $\mathit{C}=2{\mathit{r}}_2$, with $\overrightarrow{a}$ decreasing linearly from 2 to 0 during the iterations, and ${\mathit{r}}_1, {\mathit{r}}_2$ are random vectors in the range [0, 1]. Exploration is promoted when $\left|\mathit{A}\right|>1$, forcing whales to move away from a given agent and thereby enhancing the global search ability of the algorithm.

Encircling prey: During the hunting process, humpback whales surround their prey. Other candidate solutions adjust their positions to converge toward the best-performing agent, with the current optimal solution regarded as the global best. Mathematically, this behavior is represented as follows:

$$\begin{gathered} \mathit{D}=\left|\mathit{C}\cdot {\mathit{X}}_p\left(t\right)-\mathit{X}\left(t\right)\right|, \\ \mathit{X}\left(t+1\right)={\mathit{X}}_p\left(t\right)-\mathit{A}\cdot \mathit{D}, \end{gathered}$$

(12)

where $\mathit{X}$ denotes the whale’s position vector, and ${\mathit{X}}_p$ denotes the position vector of the best solution found so far (the prey). In this case, exploitation is promoted when $\left|\mathit{A}\right|<1$, ensuring that whales move toward the best solution identified.

Exploitation phase: The objective of this stage is to employ a bubble-net strategy to capture the prey. The bubble-net approach combines two mechanisms: (1) the shrinking encircling mechanism, in which the coefficient vector decreases from 2 to 0 over successive iterations; and (2) the spiral updating position mechanism, which calculates the distance between the whale and the prey and updates the position along a logarithmic spiral:

$$\mathit{X}\left(t+1\right)={\mathit{D}}^{\prime }{e}^{bt}\mathit{cos}\left(2\pi t\right)+{\mathit{X}}^{\ast }$$

(13)

where $\mathit{D}\prime =\left|{\mathit{X}}^{\ast }\left(t\right)-\mathit{X}\left(t\right)\right|$ is the distance between the whale and the prey, b is a constant defining the shape of the spiral, t is a random number in [−1, 1], and ${\mathit{X}}^{\ast }$ is the prey’s position.

To integrate these two strategies, the WOA assigns a 50% probability of applying either the shrinking encircling mechanism or the spiral updating mechanism in each iteration, which can be formalized as follows:

$$\mathit{X}\left(t+1\right)=\left\{\begin{array}{ccc}{\mathit{X}}^{\ast }\left(t\right)-\mathit{A}\cdot \mathit{D}& \mathrm{if}& p<0.5,\\ {\mathit{D}}^{\prime }{e}^{bt}\mathit{cos}\left(2\pi t\right)+{\mathit{X}}^{\ast }& \mathrm{if}& p\ge 0.5,\end{array}\right\}$$

(14)

where p ∈ [0, 1] is a random probability.

In summary, the WOA technique is a very effective tool for solving difficult optimization issues in a variety of fields.

2.4. Goodness of Fit

Nine input variables were employed in the development of the WOA/MARS model, with energy consumption (EC) serving as the dependent variable. The objective was to predict the output variable (SEC) based on these nine process variables and to identify the model that best aligns with the experimental data. The coefficient of determination, R², was adopted as the primary metric to assess model performance [52,53]. For each data point $t_i$ in the dataset, there exists a corresponding model-estimated value $y_i$. The observed values correspond to the actual measurements, whereas the predicted values are derived from the model. The variability within the dataset is quantified using the following sums of squares [52,53]:

• First item, total sum of squares ($S{S}_{tot}$): $S{S}_{tot}={\sum }_{i=1}^n(t_i-\stackrel{-}{t}{)}^2$. This measure is proportional to the sample variance.
• Regression sum of squares ($S{S}_{reg}$): $S{S}_{reg}={\sum }_{i=1}^n(y_i-\stackrel{-}{t}{)}^2$, sometimes dubbed as the total of squares described;
• Residual sum of squares ($RSS$): $RSS={\sum }_{i=1}^n{\left(t_i-y_i\right)}^2$.

In these equations, $\stackrel{-}{t}$ is the arithmetic mean of the n data points in the experimental dataset and is calculated as

$$\stackrel{-}{t}={\displaystyle \frac{1}{n}}\sum _{i=1}^n{t}_i$$

(15)

The coefficient of determination $R^2$ is defined as [52,53]

$$R^2\equiv 1-{\displaystyle \frac{RSS}{S{S}_{tot}}}$$

(16)

As the $R^2$ statistic approaches unity, it indicates a diminishing disagreement between the experimental and projected data, thus signifying an improved model fit. Similarly, the mathematical expressions for the other two statistics used in this study, root-mean-square error (RMSE) and mean absolute error (MAE), are as follows:

$$RMSE\equiv \sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{\left(t_i-y_i\right)}^2}}$$

(17)

$$MAE\equiv {\displaystyle \frac{{\displaystyle \sum _{i=1}^n\left|t_i-y_i\right|}}{n}}$$

(18)

In order to optimize estimation accuracy (R²), this study employs various approximation techniques. These methods model the EC as the dependent variable using nine input factors from experimental samples [47,54]. While the WOA-tuned MARS model was optimized using 10-fold cross-validation to handle its nonlinear structure and hyperparameter tuning, the linear baseline models—ridge regression (RR), linear regression (LR), and elastic-net regression (ENR)—were evaluated on the same independent test set without cross-validation. Given the inherent simplicity of these models and the absence of complex hyperparameters (beyond fixed regularization terms for RR/ENR), a single train–test split (80-20) was deemed sufficient to establish a fair and computationally efficient performance baseline for comparative purposes.

The efficacy of the MARS approximation is heavily contingent upon the selection of its hyperparameters [18,19,21,22,23,24,25,30,31], namely,

(a)

Maxfuncs: the highest amount of hinge functions (MF).

(b)

The degree to which variables interact (D).

(c)

The penalty parameter (d), a GCV parameter imposing a penalty on each node, linked to model complexity (Equation (4)). When d = 0, terms are penalized, but nodes are not; when d = −1, no penalty applies. The common value is d = 2.

(d)

The pruned model’s maximum term count (P). The backward pruning stage eliminates ineffective terms to improve model generalization after the forward phase’s overfitting.

To optimize these hyperparameters, this study employs the whale optimization algorithm (WOA) [26,27,28,29]. The WOA is a metaheuristic search algorithm that emulates the hunting behavior of humpback whales in nature, efficiently exploring the hyperparameter search space to identify optimal configurations.

The methodology for implementing the WOA/MARS model is as follows:

Data partitioning: The dataset is bifurcated; twenty percent of the data is employed for testing, while the remaining eighty percent is employed for training.

Model training and hyperparameter optimization:

(a)

The training set is utilized to construct the WOA/MARS model.

(b)

To adjust the MARS model parameters, a k-fold cross-validation method is used, taking the value k = 10 [52,53,55].

(c)

The WOA algorithm is applied within this cross-validation framework to identify the optimal hyperparameter configuration.

Model construction: Once the best parameters are determined, the full training set is employed to construct the final model.

Model evaluation:

(a)

The forecasted results are then evaluated against the actual values.

(b)

The observed values are then contrasted with these forecasts.

(c)

This comparison is used to assess how well the model fits the data.

In this study, several key parameters were defined to implement the WOA and guide the optimization process. The search space parameters are summarized in

Table 2. Search limits of MARS hyperparameters for WOA tuning.

Hyperparameters	Lower Limit	Upper Limit
MF	3	100
D	1	4
d	1	30
P	−1	4

. The algorithm was configured with a population size of 40 and a maximum of 50 iterations as the stopping criterion. Unlike certain other optimization methods, this WOA implementation does not use a convergence threshold to terminate the process, relying instead on the predetermined number of iterations.

Figure 5 provides a schematic representation of the WOA/MARS-based approximation process utilized in this investigation.

Predictive modeling frequently uses cross-validation to determine the true coefficient of determination $R^2$ [52,53]. In this investigation, where k = 10, the k-fold cross-validation technique was utilized to assess the predictive capacity of the WOA/MARS-based approximation [52,53,55].

The computational experiments were conducted on an iMac equipped with 24 GB of RAM and an Intel^® Core^TM i5-4570 CPU operating at 3.20 GHz with 4 cores. The software environment included the Ubuntu 22.04 operating system, R version 4.5.1, and several key R packages outlined below:

MARS approximation: earth package, version 5.3.4 [56,57];

WOA optimizer: metaheuristicOpt package, version 2.0.0 [28];

RR, LR, and elastic-net regression: glmnet package, version 4.1-9 [58].

The solution space’s variation intervals used in this investigation are displayed below in Table 2.

The WOA metaheuristic optimizer was employed to fine-tune the MARS model parameters, determining optimal values for the upper bound of hinge functions, the level of variable interactions, the penalty parameter, and the maximum number of terms in the pruned model by evaluating cross-validation error differences. The four-dimensional search space corresponds to these four parameters under optimization. The root-mean-square error (RMSE) was adopted as the primary fitness metric and objective function [52,53], providing a robust measure of predictive accuracy for effective parameter evaluation and optimization.

3. Results and Discussion

Table 3. Optimized hyperparameters of the most accurately fitted MARS model, as determined by the WOA optimizer, for predicting electric energy consumption (EC).

Hyperparameters	Optimal Values
MF	17
D	3
d	24
P	0

presents the best optimized hyperparameters for predicting electric energy consumption using the MARS-based approach, as found by the WOA optimizer.

The primary basis functions and associated coefficients for the optimally adjusted WOA/MARS-based model that was utilized to forecast EC are listed in

Table 4. List of hinge functions and the associated $c_i$ coefficients in the optimally adjusted WOA/MARS-based model for predicting energy consumption.

Bi	Definition	ci
B1	1	$50.5129$
B2	$h\left(15.16-\mathrm{LaCRP}\right)$	$-2.3608$
B3	$h\left(\mathrm{LaCRP}-15.16\right)$	$1.8091$
B4	$h\left(0.03-{\mathrm{tCO}}_2\right)$	$-748.1992$
B5	$h\left({\mathrm{tCO}}_2-0.03\right)$	$28.3717$
B6	$h\left(94.79-\mathrm{LaCPF}\right)$	$-3.0689$
B7	$h\left(\mathrm{LaCPF}-94.79\right)$	$3.0469$
B8	$h\left(29.74-\mathrm{LaCRP}\right)\times h\left(0.03-{\mathrm{tCO}}_2\right)$	$10.8099$
B9	$h\left(\mathrm{LaCRP}-29.74\right)\times h\left(0.03-{\mathrm{tCO}}_2\right)$	$-12.4142$
B10	$h\left(0.03-{\mathrm{tCO}}_2\right)\times h\left(\mathrm{LaCPF}-94.36\right)$	$-73.7558$
B11	$h\left(0.03-{\mathrm{tCO}}_2\right)\times h\left(94.36-\mathrm{LaCPF}\right)$	$95.9375$
B12	$h\left({\mathrm{tCO}}_2-0.03\right)\times h\left(\mathrm{LaCPF}-88.40\right)$	$138.3409$
B13	$h\left({\mathrm{tCO}}_2-0.03\right)\times h\left(88.40-\mathrm{LaCPF}\right)$	$-97.0837$
B14	$h\left(\mathrm{LaCPF}-94.79\right)\times h\left(\mathrm{LeCPF}-71.15\right)$	$0.0932$
B15	$h\left(\mathrm{LaCPF}-94.79\right)\times h\left(71.15-\mathrm{LeCPF}\right)$	$-0.0051$
B16	$h\left(\mathrm{LaCRP}-29.74\right)\times h\left(0.03-{\mathrm{tCO}}_2\right)\times h\left(\mathrm{LaCPF}-86.80\right)$	$10.6701$
B17	$h\left(\mathrm{LaCRP}-29.74\right)\times h\left(0.03-{\mathrm{tCO}}_2\right)\times h\left(86.80-\mathrm{LaCPF}\right)$	$-0.1697$

. It is important to note that a basis function is defined using the hinge functions that follow the expression [18,19,21,22,23,24,25,30,31,58]:

$$B\left(x\right)=\left\{\begin{array}{ccc}x& \mathrm{if}& x>0\\ 0& \mathrm{if}& x\le 0\end{array}\right\}$$

(19)

Fundamentally, MARS represents a type of nonparametric regression technique and can be considered an extension of linear regression. To systematically capture nonlinearities and interactions among input variables, MARS utilizes a weighted sum of the hinge functions described previously [18,19,21,22,23,24,25,30,31].

For comparative analysis, the lasso, ridge, and elastic-net regression techniques were also employed in this study. Their accuracy is contingent upon the parameter $\lambda$ [30,32,33,34,35].

Table 5. Optimized λ parameters for lasso, ridge, and elastic-net regression models, as determined by WOA optimization.

Optimal λ	SEC
RR	3.2982
LR	0.0939
ENR	0.1711

displays the parameter λ’s ideal value for these regression models, as established by the WOA optimizer.

To ensure the robustness and validity of the WOA-optimized MARS model, diagnostic analyses of the residuals were performed. Figure 6a presents the residuals versus fitted values plot, which exhibits a random scatter around zero without discernible patterns, confirming the absence of heteroskedasticity and supporting the appropriateness of the model’s functional form. Figure 6b displays the Q–Q plot of the residuals, indicating minor deviations from normality, a phenomenon commonly observed in industrial datasets due to operational outliers. Importantly, these deviations do not impair the model’s predictive performance, as MARS is a nonparametric method that does not rely on the assumption of normally distributed residuals.

To further assess the interpretability of the model, partial dependence plots (PDPs) were generated for the three most influential variables (CO₂, LaCRP, and LaCPF), as presented in Figure 6. These plots depict the relationships between the input variables and the predicted energy consumption, incorporating both first- and second-order terms derived from the WOA-optimized MARS model, thereby elucidating their individual and interactive effects. In Figure 7a, EC (ordinate) is plotted against lagging current reactive power (abscissa), with four other factors kept constant. Similarly, Figure 7b,c show EC versus carbon dioxide emissions and the lagging current power factor, respectively, keeping eight variables fixed. Figure 7d presents a 3D plot of EC (applicate) against carbon dioxide emissions (abscissa) and lagging current reactive power (ordinate), with other variables constant. Likewise, Figure 7e,f depict 3D plots of EC versus carbon dioxide emissions and the lagging current power factor, and lagging versus leading current power factors.

Table 6. Statistical performance indicators (RMSE, MAE, R², and r) of regression models for EC prediction on the test dataset.

Model	RMSE (kWh)	MAE (kWh)	R2	r
WOA/MARS	1.7962	1.1801	0.9972	0.9985
RR	4.5150	2.6336	0.9820	0.9911
LR	4.4017	2.5274	0.9829	0.9914
ENR	4.4088	2.5413	0.9821	0.9912

summarizes the values of the coefficient of determination (R²), correlation coefficients, and RMSE for various regression models applied to the EC output variable using the testing dataset. The models evaluated include the WOA/MARS-based approach [18,19,21,22,23,24,25,30,31], as well as ridge regression (RR), lasso regression (LR), and elastic-net regression (ENR) [30,32,33,34,35].

The most recent statistical evaluation indicates that the optimally fitted MARS model achieves an R² value of 0.9972 and a correlation coefficient (r) of 0.9985 for the EC variable. These results demonstrate a strong agreement between the experimental data and the MARS predictions, indicating a high level of reliability in the model’s goodness of fit. Consequently, this model was selected as the most suitable approach for estimating the dependent variable (EC).

These results are now contextualized by discussing comparable data-driven models reported in the literature. Mubarak et al. [59] introduced a stacked ensemble approach, Stack-XGBoost, which combines multiple machine learning models to predict active and reactive energy consumption in the steel industry. Their methodology employs extra trees regressor (ETR), adaptive boosting (AdaBoost), and random forest regressor (RFR) as base models, with an extreme gradient boosting (XGBoost) algorithm serving as the meta-learner. This approach demonstrated superior accuracy for short-term forecasting horizons. Nevertheless, this MARS model also exhibits strong performance, particularly with respect to the coefficient of determination, indicating an excellent fit to the data. Furthermore, a key advantage of the MARS model is its ability to quantify the importance of each independent variable, providing deeper insights into the factors driving energy consumption predictions.

As a supplementary outcome of these analyses,

Table 7. Ranking of the relevance of the operation variables in the best-fit WOA/MARS model for the EC as stated in the GCV criterion.

Input Variable	Nsubsets	GCV	RSS
Carbon dioxide emission (tCO2)	16	100.0	100.0
Lagging current reactive power (LaCRP)	14	14.4	14.4
Lagging current power factor (LaCPF)	12	12.1	12.1
Leading current power factor (LeCPF)	12	12.1	12.1

and Figure 7 illustrate the hierarchical importance of the process variables (input factors) in predicting the EC (output dependent factor) [58,60] for this complex investigation.

In the MARS model, variable importance is estimated using three distinct criteria. First, the nsubsets criterion counts how many model subsets include the variable, with variables appearing in more subsets considered more important. Here, “subsets” refer to the subsets of terms generated during the backward pass of the model-building process. Second, the RSS criterion calculates the decrease in residual sum of squares (RSS) for each subset relative to the previous subset during the backward pass. Variables causing larger net decreases in the RSS are deemed more important. Third, the GCV criterion operates similarly to RSS but uses generalized cross-validation (GCV) instead of the RSS. This criterion was employed to determine variable importance. Both the GCV and RSS columns are normalized so that the largest net decrease is set to 100, facilitating comparison.

Within the MARS framework, carbon dioxide emissions (tCO₂) were identified as the most influential predictor of the output variable EC. This is followed, in descending order of importance, by the lagging current reactive power (LaCRP), the lagging current power factor (LaCPF), and the leading current power factor (LeCPF). It is noteworthy that the remaining input variables were not included in the model as explanatory factors.

Figure 8 illustrates the proportional significance of process variables in the optimally fitted WOA/MARS-based approach for EC prediction, determined by the generalized cross-validation (GCV) criterion.

In steel plants, electrical energy consumption plays a critical role in production efficiency. Among the influencing factors, CO₂ emissions are the most significant, as they exhibit a direct correlation with energy usage, particularly in high-temperature operations and fossil fuel combustion processes [61]. The generation of CO₂ serves as an indicator of both energy consumption and process efficiency.

Lagging current reactive power, the lagging current power factor, and the leading current power factor exert a secondary influence, primarily affecting power quality and equipment efficiency rather than total energy consumption. While reactive power impacts supply stability, its effect is less pronounced than that of CO₂ emissions [62]. Similarly, the power factor influences energy losses but does not directly quantify total energy usage [63]. In contrast, CO₂ emissions provide a direct measure of both consumption and process efficiency, establishing them as the most relevant indicator of energy performance in steel mills.

Industries incur significant costs due to reactive energy, which is consumed without performing useful work, thereby increasing electricity expenses. Large energy consumers, particularly those operating machinery with transformers, tend to generate higher amounts of reactive energy. A leading power factor indicates that voltage lags behind current (capacitive load and negative reactive power), whereas a lagging power factor signifies that current lags behind voltage (inductive load and positive reactive power). Low power factors necessitate higher currents to maintain the same active power (P), which, in turn, requires larger conductor sizes.

Although reactive power and the power factor influence operational efficiency, CO₂ emissions remain the primary determinant for optimizing electric energy consumption in steelworks. Reducing CO₂ emissions is critical for managing energy usage and minimizing associated emission costs [64].

Energy efficiency plays a crucial role in reducing industrial greenhouse gas emissions, with the iron and steel sector accounting for approximately 5% of global anthropogenic CO₂ emissions [65]. The implementation of energy-efficient technologies in steel plants decreases energy consumption (EC), thereby reducing operational costs and mitigating environmental impacts.

This study effectively estimates EC using the WOA/MARS-based model, demonstrating strong agreement with observed data. Figure 9 presents a comparison of experimental and predicted EC values obtained from the LR, RR, ENR, and WOA/MARS models. The MARS model provides the most accurate regression fit, achieving the highest R², thereby confirming the WOA/MARS approach as the optimal solution.

4. Conclusions

This study demonstrates the effectiveness of a WOA-tuned MARS model for estimating energy consumption in steel production, achieving exceptional predictive accuracy (R² = 0.9972). The model’s superiority over linear alternatives (RR, LR, and ENR) stems from its ability to capture nonlinear interactions between critical process variables, which align with established steelmaking physics:

• CO₂ emissions and energy consumption (EC) exhibit a strong coupling due to shared dependencies on production volume and combustion intensity.
• Lagging reactive power (LaCRP) reflects inductive load dominance, where a poor power factor increases the apparent power demand and, thus, EC.
• Lagging current power factor (LaCPF) extremes indicate suboptimal electrical efficiency, correlating with higher energy losses.

In conclusion, the WOA-tuned MARS model presents a robust methodology for estimating electrical energy consumption in the steel industry. Given the absence of analytical equations capable of accurately predicting steelworks’ electrical energy consumption (SEC) from experimental data alone, alternative diagnostic approaches become indispensable. This methodology demonstrates adaptability to a broad spectrum of manufacturing processes, though its implementation must account for the specific operational characteristics and conditions of individual steelworks.

Limitations and future work: While the proposed methodology demonstrates high accuracy, several limitations and opportunities for future research should be acknowledged:

• Dataset specificity: The current analysis focuses on data from Gwangyang Steelworks (South Korea). Future studies should validate the model’s generalizability across diverse geographical regions and steel plant configurations.
• External factors: Incorporating market demand fluctuations, regulatory changes, and seasonal variations could refine the model’s adaptability to dynamic industrial conditions.
• Temporal validation: Employing time-aware data splits (e.g., training on historical data and testing on recent periods) would further assess the model’s robustness to operational evolution.
• Potential methodological extensions include the following:
  • Dynamic hyperparameter tuning using expanding/rolling window strategies to capture long-term trends.
  • Hybrid approaches (e.g., MARS + ARIMA) to balance interpretability and predictive power for time-series forecasting.