2.3.2. The Abstract Case

Supervised statistical models will be used in this paper which means that each row of input data, i.e., variables, has a corresponding output data point. This framework can be explained by the following steps:


In a practical context, the model hyper-parameters (step 1) are chosen based on a comprehensive hyper-parameter search, also known as grid-search. During the parameter search, several models are trained for each combination of hyper-parameters. The combination with the highest and most stable accuracy is the most optimal hyper-parameter selection.

Non-linear supervised statistical models should be used when predicting the EE consumption of an EAF. This is because some important input variables governing the EAF process are non-linearly related to the EE consumption. The statistical model framework should always be chosen based on the nature of the prediction problem.

Although non-linear statistical models are excellent at learning complex relations between variables, these types of models are susceptible to overfitting. Overfitting means that the statistical model has adapted itself too well to a particular set of data, i.e., the training data, in such a way that it cannot predict well on future data. Combating this phenomenon is important since the relations between the variables are expected to change from the training data to test data. This is the natural course of any steel plant process. The strategies to reduce overfitting will be explained further in Section 3.4.1 where the specific model frameworks used in the numerical experiments are presented.

### 2.3.3. Previous Studies

Statistical models have previously been used as a tool to predict the EE consumption of the EAF. A comprehensive review of the subject has recently been published [1]. However, only four of the previous studies have used some representation of scrap types as part of the input variables in the models and as part of the model analysis [3–6].

The first study used the weight of shredded scrap as the only scrap type variable for a Multivariate Linear Regression (MLR) model [6]. The coefficient for this variable is negative, which indicates that less EE is needed than what is normally required when more shredded scrap is added. The model was then used on data from 5 different EAF, all of which used various amounts of other scrap types which were not taken into consideration by the model.

The second study used response graphs to investigate the total EE prediction response by each scrap type in the first and second baskets [4]. However, a response graph only displays the total EE prediction when varying one single input variable and does not reveal the specific contribution by each input variable.

The third study used Partial Least Squares (PLS) regression to model the EE consumption of two different EAF [3], one of which is the steel plant governing the data in the current study. However, the significance of each scrap type representation was only given by an ad-hoc subjective measure, as indicated by the descriptive words *low* and *high*.

The last study used the statistical modeling frameworks ML, RF, and Artificial Neural Networks (ANN) to predict the EE consumption [5]. The effect of the scrap types on the EE was only reported for the MLR model since the model coefficients reveal the impact of each scrap type on the EE consumption. The values of the coefficients were then compared with experience-based values. However, some assumptions were made regarding the comparison since the MLR model used kWh/t charged scrap and the experience-based values were reported in kWh/t tapped steel.

A recent study, published after the review, used Kolmogorov–Smirnov (KS) tests and correlation metrics to highlight the change of all input variables between the training and test data [2]. In addition, permutation feature importance was used to investigate the importance of each input variable to the model prediction for both data sets. In combination, the KS tests and permutation feature importance produced evidence that some of the input variables had the main influence in the performance reduction of the model. However, the focus was never to investigate the specific effects of the scrap types on the EE consumption of the selected models.

#### **3. Method**

#### *3.1. Representing Scrap Types*

The scrap types will be represented in three distinct ways. The first representation will use the scrap codes from the steel plant of study, i.e., scrap type. The second representation is based on a visual categorization of each scrap code with the aim to provide an intuitive categorization for the steel plant engineers as well as to minimize the number of distinct scrap categories. The third representation is based on the estimated apparent density of each scrap code. The estimated apparent densities are both from established technical specifications as well as from estimations conducted by the plant engineers. The relations between the three scrap representations can be seen in Figure 3. Henceforth, the term *scrap representation* will refer to either of the three distinct scrap representations and the term *scrap category* will be used to specify a specific category in either of the visual or apparent density categorizations. The term *scrap type* will be used to refer to a specific scrap type as defined by the steel plant coding system.

**Figure 3.** The relationship between the steel plant scrap types and the two scrap representations based on visual and estimated apparent density properties, respectively. The bold underlined scrap types and categories occur in less than 10% of the heats and are therefore bundled together into two aggregate variables, *SCRAggr* and *Aggregate*, for the plant scrap representation and visual scrap representation, respectively.

Each of the three representations are further described in detail.
