# Modeling the Effect of Scrap on the Electrical Energy Consumption of an Electric Arc Furnace

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Melting of Steel Scrap in Liquid Steel

#### 2.1.1. Driving Forces

**Temperature gradients**between the solid scrap and the steel melt is one of the most important driving factors in the melting of steel scrap. The melting rate, in m/s, can be determined by the following equation:

**Alloying element gradients**also contribute to the melting of scrap in a process known as dissolution. In this case, alloying elements migrate to the solid-liquid metal interface. The most dominant alloying element in this process is carbon. However, in the BOF, the carbon concentration difference can exceed 4 wt-% while in the EAF the carbon concentration difference seldom exceeds 1 wt-%.

**freezing effect**occurs in the solid-liquid interface when the scrap first comes in contact with the liquid steel. A solidified shell is formed due to the large temperature difference between the two. This means that the volume of the scrap increases initially. The solidified shell is proportional to the surface area of the scrap that is submerged in the hot metal or molten steel. Hence, the reduction in the steel scrap size does not occur instantly, rather it decreases after the solidified shell has melted.

**stirring**velocity is the velocity of the melt in the boundary layer between the melt and the scrap surface area. Numerous studies have related the stirring velocity to the mass transfer coefficient on scrap in liquid steel. However, there exists a wide range of reported mass transfer coefficient values for scrap in liquid steel under forced convection [12]. Nevertheless, a commonly deduced relationship between the mass transfer coefficient and the stirring velocity may be written as follows:

#### 2.1.2. Scrap Surface-Area-to-Volume Ratio

**Sphere:**$\frac{A}{V}=\frac{4\pi {r}^{2}}{\frac{4\pi {r}^{3}}{3}}=\frac{3}{r}$

**Cylinder:**$\frac{A}{V}=\frac{2\pi rl+2\pi {r}^{2}}{\pi {r}^{2}l}=2(\frac{1}{r}+\frac{1}{l})$

**Cube:**$\frac{A}{V}=\frac{6{l}^{2}}{{l}^{3}}=\frac{6}{l}$

**Square plate:**$\frac{A}{V}=\frac{2{l}^{2}+4lt}{{l}^{2}t}=2(\frac{1}{t}+\frac{2}{l})$

#### 2.1.3. The Steel Plant of Study

#### 2.2. The Electric Arc Furnace

#### 2.2.1. Process

#### 2.2.2. Charging of Scrap Baskets

#### 2.2.3. Parameters Governing the EE Consumption

#### 2.2.4. Non-Linearity

#### 2.3. Statistical Modeling

#### 2.3.1. Inherent Traits

#### 2.3.2. The Abstract Case

- Select statistical model framework. The available hyper-parameters are unique to the statistical model framework and are selected by the modeler.
- Train the model using a set of matching input and output data. Continue the training phase until the accuracy of the model converges.
- Test the model with a set of previously unseen data.
- Evaluate the models practical usefulness using the accuracy on the test data.
- If the accuracy is good enough, deploy the model in a production environment.

#### 2.3.3. Previous Studies

## 3. Method

#### 3.1. Representing Scrap Types

#### 3.1.1. Steel Plant Scrap Yard System

#### 3.1.2. Visual Categorization

#### 3.1.3. Density Categorization

**Light:**0.3–1.0 ton/m${}^{3}$**Light-Mid:**0.56–1.0 ton/m${}^{3}$**Mid:**0.7–1.4 ton/m${}^{3}$**Heavy:**>1.4 ton/m${}^{3}$

#### 3.2. Data Governing the EAF

#### 3.2.1. Variable Selection

#### 3.2.2. Variable Batches

#### 3.3. Data Treatment

#### 3.3.1. Purpose

#### 3.3.2. Domain-Specific Methods

#### 3.3.3. Statistical Methods

#### 3.3.4. Applied Data Treatments

#### 3.4. Modeling the EE Consumption

#### 3.4.1. Statistical Modeling Frameworks

**Artificial Neural Networks:**This model framework uses a fully connected network of nodes to make predictions [21]. The first layer, which is known as the input layer, receives the values from the input variables. The values are then propagated through the intermediate layers, which are known as hidden layers, to the last layer. The last layer is the output layer where the prediction is made. See Figure 5 for an illustration of an arbitrary ANN model.

**Random Forest:**This statistical modeling framework is a model made of two, or more, decision trees. See Figure 6 for an illustration of a simple decision tree for prediction purposes. The RF model framework was first reported by L. Breiman [22]. RF belongs to the statistical model group known as ensemble models, which is a group of statistical models that is made up of two or more models that when combined, aim to increase the prediction accuracy.

#### 3.4.2. Parameter Optimization

#### 3.4.3. Selection of Training and Test Data

#### 3.4.4. Model Performance Metrics

#### 3.4.5. Model Selection

- Select all models that pass the following condition: ${\overline{R}}_{max}^{2}-{\overline{R}}_{min}^{2}\le 0.05$.
- Select the model(s) with the highest ${\overline{R}}_{\mu}^{2}$. If the number of models exceeds 1, then proceed to the next step, otherwise select the one model.
- Select the model with $\mathit{min}({\overline{R}}_{max}^{2}-{\overline{R}}_{min}^{2})$.

#### 3.5. Model Evaluation and Analysis

#### 3.5.1. Shapley Additive Explanations (SHAP)

#### 3.5.2. Correlation Metrics

**Pearson correlation:**The Pearson correlation metric that can only detect linear relationships between two random variables [26]. It assumes values between −1 and 1, where the former is a perfect negative relationship between the variables and the latter is a perfect positive relationship, i.e., the variables are identical. A value of 0 indicates that the variables have no relation.

**Distance correlation (dCor):**Although dCor cannot distinguish between positive and negative correlative relationships between variables, it is able to detect non-linear relationships between variables [27]. This is important since some variables governing the EAF process have a non-linear relationship to EE. By using dCor and Pearson in tandem, it is possible to get a clearer picture of the relationships between the variables governing the statistical models.

#### 3.5.3. Charge Types

## 4. Results and Discussion

#### 4.1. EE Consumption Models

#### 4.2. Analysis of the Selected Model

## 5. Conclusions

#### 5.1. Analysis of the Models

- All models using variables from any of the three scrap representations performed better than the models using only the total charged scrap weight as a scrap representation. This consistency provides evidence that more granular scrap representations are important when modeling the EE consumption of an EAF using steel scrap as the main raw material.
- The models using the visual scrap representation had the highest ${\overline{R}}_{\mu}^{2}$-values regardless of cleaning strategy. This provides evidence that a scrap representation based on the shape of the scrap, i.e., surface-area-to-volume ratio, is the most optimal to use. The result agrees well with the surface-area-to-volume ratio and the physico-chemical relationships on the scrap melting rate governed by temperature gradients, alloying gradients, and the freezing effect. In addition, this representation is also intuitive and easy to apply from the perspective of steel plant engineers and scrap yard operators.
- Data cleaning strategies using domain-specific knowledge from either an expert from the steel plant of study or a domain expert provide models with the highest ${\overline{R}}_{\mu}^{2}$-values compared to pure statistical cleaning methods. This further emphasizes the importance of domain-specific knowledge when modeling the EAF.

#### 5.2. Analysis of the Selected Model

- The effects of the most important scrap categories on the EE consumption for the selected charge types A, B, and C are transparently presented by SHAP main interaction values.
- SHAP main interaction values revealed the contribution by each scrap category on the complete prediction space for the selected model. Plate1 and Internal1 were confirmed by the steel plant engineers to agree well with their expectations. However, a more thorough investigation must be conducted to evaluate the reasons behind the effects of the other scrap categories. SHAP interaction values between the input variables is advised as a starting point.
- The experience and knowledge provided by the steel plant engineers were essential to determine the reasons behind the counter-intuitive responses by Injected Carbon, Lance ${O}_{2}$, Burner oil, and Burner ${O}_{2}$ on the EE consumption. This indicates that the participation by steel plant engineers is of outmost importance to evaluate the trustworthiness of a practical model.
- The authors question whether one should use the Carbon Injection as part of EE prediction models in the steel plant of study, given the specific operational practice. It is likely that the Carbon Injection variable creates a model artifact for the selected model.
- The test data coincides with the training data in the SHAP plots. This raises the trust in the model since the model response on previously unseen heats is within the range of SHAP values from the heats used to adapt the model parameters.

## 6. Further Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${T}_{HM}$ | Temperature of the molten steel |

${T}_{liq}$ | Scrap melting temperature |

${\rho}_{scr}$ | Density of the scrap metal |

${H}_{s}$ | Heat of melting of scrap |

${c}_{p}$ | Specific heat of scrap |

h | Heat transfer coefficient in the interface of the molten steel and scrap |

$\beta $ | Mass transfer coefficient of carbon |

${C}_{l}$ | Carbon content in liquid steel |

${C}_{i}$ | Carbon content in the solid-liquid interface |

${\rho}_{s}$ | Apparent density of scrap |

${C}_{0}$ | Initial carbon content in the steel scrap. |

${h}_{scr}$ | Mass transfer coefficient under forced convection |

c | Experimentally determined constant |

p | Experimentally determined constant |

u | Average stirring power in the boundary between the melt and scrap surface area |

${R}_{SV}$ | Surface-area-to-volume ratio |

A | The total area of an arbitrary scrap piece |

V | The total volume of an arbitrary scrap piece |

l | The length of an arbitrary scrap piece |

r | The radius of a cylindrical or spherical scrap piece |

t | The thickness of a square plate |

m | The mass of a scrap piece |

T | Temperature of the surface of a body emitting thermal radiation |

${q}_{1}$ | First quartile |

${q}_{3}$ | Third quartile |

$\u03f5$ | Pre-specified constant for outlier extremity |

${x}_{j}$ | An instance of variable j |

$\sigma $ | Standard deviation |

v | Number of input variables |

P | Number of nodes in the previous layer |

${s}_{j}$ | Summation of the input values for j:th node in the current layer |

${w}_{i}$ | Weight of node i in the previous layer |

${x}_{i}$ | Value of node i in the previous layer |

${x}_{k}$ | A data instance to be predicted by a RF |

${y}_{k}$ | The predicted value of data instance ${x}_{k}$ by a RF |

${R}^{2}$ | Coefficient of determination |

${\overline{R}}^{2}$ | Coefficient of determination adjusted for number of data points and variables |

n | Number of data points |

${E}_{i}$ | Regular error for data point i |

${y}_{i}$ | True value of the output variable for data point i |

${\widehat{y}}_{i}$ | Predicted value of the output variable for data point i |

${\overline{R}}_{\mu}^{2}$ | Mean adjusted R-square of the 10 model instances on test data |

${\overline{R}}_{\sigma}^{2}$ | Standard deviation of adjusted R-square of the 10 model instances on test data |

${\overline{R}}_{min}^{2}$ | Minimum of adjusted R-square of the 10 model instances on test data |

${\overline{R}}_{max}^{2}$ | Maximum of adjusted R-square of the 10 model instances on test data |

${\mathrm{\Delta}}_{\mu}$ | Mean error of the mean error of the 10 model instances on the test data |

${\mathrm{\Delta}}_{\sigma}$ | Standard deviation of the mean error of the 10 model instances on the test data |

${\mathrm{\Delta}}_{min}$ | Minimum of the mean error of the 10 model instances on the test data |

${\mathrm{\Delta}}_{max}$ | Maximum of the mean error of the 10 model instances on the test data |

f | An arbitrary statistical model |

${f}_{x}$ | A simplified representation of f |

${\varphi}_{0}$ | The contribution to the prediction by f when all information is absent |

${\varphi}_{i}$ | The contribution to the prediction by f by variable i |

Z | A subset of a set of all input variables |

$\widehat{Z}$ | The set of all input variables |

$\left|Z\right|$ | The number of variables in the subset Z |

$\widehat{Z}\backslash \left\{i\right\}$ | The set of all input variables excluding variable i |

$Z\cup \left\{i\right\}$ | The subset of a set of all input variables including variable i |

$Z\cup \left\{j\right\}$ | The subset of a set of all input variables including variable j |

$Z\cup \left\{\right(i,j\left)\right\}$ | The subset of a set of all input variables including variable i and j |

M | The total number of input variables |

${\varphi}_{i,i}$ | The main interaction value for variable i |

${\varphi}_{j,i}$ | The interaction effect of input variable i imposed on variable j |

${\varphi}_{i,j}$ | The interaction effect of input variable j imposed on variable i |

$F{I}_{j}$ | SHAP feature importance for variable j |

${\varphi}_{j}^{i}$ | Regular SHAP value for variable j and data instance i |

${\rho}_{X,Y}$ | Pearson correlation coefficient between random variables X and Y |

${\sigma}_{X}$ | Standard deviation of random variable X |

${\sigma}_{Y}$ | Standard deviation of random variable Y |

${V}_{1}$ | Random variable |

${V}_{2}$ | Random variable |

$dCor({V}_{1},{V}_{2})$ | Distance correlation between ${V}_{1}$ and ${V}_{2}$ |

$dCov({V}_{1},{V}_{2})$ | Distance covariance between ${V}_{1}$ and ${V}_{2}$ |

$\sqrt{dVar\left({V}_{1}\right)}$ | Distance standard deviation for ${V}_{1}$ |

$\sqrt{dVar\left({V}_{2}\right)}$ | Distance standard deviation for ${V}_{2}$ |

## Abbreviations

EE | Electrical Energy |

EAF | Electric Arc Furnace |

BOF | Basic Oxygen Furnace |

DRI | Direct Reduced Iron |

DMS | Demand Side Management |

MLR | Multivariate Linear Regression |

PLS | Partial Least Squares regression |

ANN | Artificial Neural Network |

RF | Random Forest |

KS | Kolmogorov–Smirnov |

dCor | Distance Correlation |

SHAP | SHapley Additive Explanations |

TTT | Tap-to-Tap Time |

TCB2 | Time to Charging of Basket 2 |

HM | Heavy Melting Scrap |

VB | Variable Batch |

## Appendix A

Computer model | Dell Latitude E5570 |

CPU | Intel Core i7 2376 MHz |

RAM | 16,203 MB |

Purpose | Software/Package | Version |
---|---|---|

Operating system | Microsoft Windows 7 Professional | 6.1.7601 Service Pack 1 Build 7601 |

Programming language | Python 3 | 3.7.1 |

Python distribution | Anaconda 3 | 4.6.7 |

Data handling | pandas | 0.23.4 |

numpy | 1.17.4 | |

Statistical modeling | scikit-learn | 0.20.1 |

SHAP values | shap | 0.8.1 |

Distance correlation | dcor | 0.3 |

Visualization | matplotlib | 3.0.2 |

seaborn | 0.9.0 |

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**Figure 1.**The Electric Arc Furnace (EAF) process in the steel plant considered in the current article. It shows the process divided into six parts. (

**I**) The first scrap basket is fully charged with up to 10 layers of scrap. The furnace shell contains hot heel. (

**II**) The scrap from the first basket has been charged into the furnace whereupon the lowermost scrap layers intermix with the hot heel. The furnace arcs are bored down into the scrap and the transformer is powered on. (

**III**) After the first melting phase the furnace contains a steel scrap-melt mixture and heaps of partially solid scrap. The second scrap basket is charged in as much to fully fill the furnace in the next step. (

**IV**) Charging of the second basket. (

**V**) Refining of the molten steel where injected carbon and oxygen lancing facilitate an even layer of foaming slag. (

**VI**) The heat is ready for tapping.

**Figure 2.**The relationship between the available scrap and the scrap contribution to the Electrical Energy (EE) consumption.

**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, $SC{R}_{Aggr}$ and $Aggregate$, for the plant scrap representation and visual scrap representation, respectively.

**Figure 4.**The distributions for four variables governing the EAF under study highlighting the absence of the Gaussian distribution. All values are normalized, and the dashed lines indicate the mean values. (

**A**): TTT. (

**B**): Charged weight of internal scrap. (

**C**): Total charged weight of raw materials. (

**D**): Charged weight of shredded scrap.

**Figure 5.**An Artificial Neural Network (ANN) for predicting an output value based on two input values [2]. It has one hidden layer with three nodes. The lines between the nodes illustrate that the ANN is fully connected and the forward flow of calculations in the network.

**Figure 6.**A simple decision tree sorting points on the $\{x,y,z\}$ coordinate system. Points satisfying the conditions on each node proceed on the left branch, the rest proceed to the right branch. The points $A=\{0,42,5\}$, $B=\{-10,-10,-10\}$, $C=\{31,4,0\}$, and $D=\{2,4,61\}$, are sorted in the decision tree.

**Figure 7.**An arbitrary RF model consisting of 6 decision trees. The prediction, ${y}_{k}$, of the data instance, ${x}_{k}$, is determined by averaging the sum of the outputs from the decision trees. The filled nodes show the path ${x}_{k}$ has taken in each decision tree.

**Figure 8.**${\overline{R}}_{\mu}^{2}$-values for each variable batch (VB 1-8 on the abscissa) and cleaning strategy.

**VB 1–2:**Without scrap representation.

**VB 3–4:**Steel plant scrap representation.

**VB 5-6:**Visual scrap representation.

**VB 7–8:**Density scrap representation.

**Figure 9.**SHAP main interactions on EE by Plate1, Internal1, HM, and Shredded scrap categories. The three probability density plots below each plot show the distribution of the scrap category for each of the charge types A, B, and C, respectively. See Section 3.5.3 for an explanation of the different steel types. The y-axis of the scatter plot shows the SHAP main interactions on EE for each variable and the y-axes of the probability density plots show the frequency of each charge weight. The x-axes of both the SHAP plot and the probability density plot for each charge type are the amount charged of the scrap category. The values on the x-axes has been omitted due to proprietary reasons. The values governing the plots are from both the training and test data.

**Figure 10.**SHAP main interaction effects for the scrap categories for the selected model. The y-axes show the main interaction effect on EE while the x-axes show the amount of each charged scrap category. The values on the x-axes has been omitted due to proprietary reasons. The grey dots represent values from the training data and the black dots represent values from the test data.

**Figure 11.**SHAP Main interaction effects for the base variables for the selected model. The y-axes show the main interaction effect on EE while the x-axes show the increasing amount of each variable. See Table 2 for details about the definition of each variable. The values on the x-axes has been omitted due to proprietary reasons. The grey dots represent values from the training data and the black dots represent values from the test data.

**Figure 12.**SHAP feature importance for the selected model as defined in Equation (15). The bars show the feature importance on the training data and the dark dots show the feature importance on the test data.

Energy Factor | % of Total Energy Sources or Energy Sinks | |
---|---|---|

In | Electric | 40–66% |

Oxidation of alloying elements | 20–50% | |

Burner fuel | 2–11% | |

Out | Liquid steel | 45–60% |

Slag and dust | 4–10% | |

Off-gas | 11–35% | |

Cooling | 8–29% | |

Radiation and electrical losses | 2–6% |

Variables | Unit | Definition |
---|---|---|

Electrical Energy (EE) | $\mathrm{kWh}$ | The electrical energy consumption for the heat. |

Total Weight | $\mathrm{kg}$ | The sum of all charged scrap types. |

Tap-to-Tap time | $\mathrm{min}$ | The time between the end of the tapping from the previous |

(TTT) | heat to the end of tapping of the current heat. | |

TCB2 | $\mathrm{min}$ | Time between start of heat until charging of the second basket. |

Burner oil | $\mathrm{kg}$ | Total amount of oil added by burner. |

Burner ${O}_{2}$ | $\mathrm{N}{\mathrm{m}}^{3}$ | Total amount of oxygen added by burner. |

${O}_{2}$-lance | $\mathrm{N}{\mathrm{m}}^{3}$ | Total amount of oxygen added by lance. |

Injected carbon | $\mathrm{kg}$ | Total amount of carbon injection. |

Lime and dolomite | $\mathrm{kg}$ | Total lime and dolomite added. |

No. Charges | − | Total number of scrap baskets added. |

$SC{R}_{101}$ | $\mathrm{kg}$ | Above $3\phantom{\rule{3.33333pt}{0ex}}\mathrm{mm}$ thick plate. Apparent density above $1.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{ton}/{\mathrm{m}}^{3}$. |

$SC{R}_{103}$ | $\mathrm{kg}$ | Thin plate, cuttings of rolled thin plate. Apparent density above $1.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{ton}/{\mathrm{m}}^{3}$. |

$SC{R}_{212}$ | $\mathrm{kg}$ | Heavy cuttings of internal scrap. Apparent density above $1.4\phantom{\rule{3.33333pt}{0ex}}\mathrm{ton}/{\mathrm{m}}^{3}$. |

$SC{R}_{333}$ | $\mathrm{kg}$ | Anthracite (carbon). Apparent density $0.72\phantom{\rule{3.33333pt}{0ex}}\mathrm{ton}/{\mathrm{m}}^{3}$. |

$SC{R}_{400}$ | $\mathrm{kg}$ | Purchased rebar and plates. Apparent density $0.56\u20131.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{ton}/{\mathrm{m}}^{3}$. |

$SC{R}_{407}$ | $\mathrm{kg}$ | Heavy melting mix (HM1). Apparent density above $0.7\phantom{\rule{3.33333pt}{0ex}}\mathrm{ton}/{\mathrm{m}}^{3}$. |

$SC{R}_{409}$ | $\mathrm{kg}$ | Shredded scrap. Apparent density above $1.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{ton}/{\mathrm{m}}^{3}$. |

$SC{R}_{412}$ | $\mathrm{kg}$ | Internal scrap. Apparent density above $1.4\phantom{\rule{3.33333pt}{0ex}}\mathrm{ton}/{\mathrm{m}}^{3}$. |

$SC{R}_{416}$ | $\mathrm{kg}$ | Turnings. Apparent density above $0.4\phantom{\rule{3.33333pt}{0ex}}\mathrm{ton}/{\mathrm{m}}^{3}$. |

$SC{R}_{451}$ | $\mathrm{kg}$ | Purchased scrap (HM1 and HM2). Apparent density above $0.7\phantom{\rule{3.33333pt}{0ex}}\mathrm{ton}/{\mathrm{m}}^{3}$. |

$SC{R}_{452}$ | $\mathrm{kg}$ | Internal scrap. Apparent density above $1.4\phantom{\rule{3.33333pt}{0ex}}\mathrm{ton}/{\mathrm{m}}^{3}$. |

$SC{R}_{455}$ | $\mathrm{kg}$ | Turnings. Apparent density above $0.4\phantom{\rule{3.33333pt}{0ex}}\mathrm{ton}/{\mathrm{m}}^{3}$. |

$SC{R}_{492}$ | $\mathrm{kg}$ | Mixed internal scrap. Apparent density above $0.7\phantom{\rule{3.33333pt}{0ex}}\mathrm{ton}/{\mathrm{m}}^{3}$. |

$SC{R}_{633}$ | $\mathrm{kg}$ | Si-rich plate. Apparent density $0.7\u20131.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{ton}/{\mathrm{m}}^{3}$. |

$SC{R}_{677}$ | $\mathrm{kg}$ | Incineration scrap. Apparent density above $0.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{ton}/{\mathrm{m}}^{3}$. |

$SC{R}_{686}$ | $\mathrm{kg}$ | Grinding swarfs and grinding swarf briquettes. Apparent density $0.4\u20131.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{ton}/{\mathrm{m}}^{3}$. |

$SC{R}_{698}$ | $\mathrm{kg}$ | Skulls. Apparent density above $0.7\phantom{\rule{3.33333pt}{0ex}}\mathrm{ton}/{\mathrm{m}}^{3}$. |

$SC{R}_{699}$ | $\mathrm{kg}$ | Grinding swarfs and grinding swarf briquettes. Apparent density $0.3\u20131.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{ton}/{\mathrm{m}}^{3}$. |

$SC{R}_{711}$ | $\mathrm{kg}$ | Incineration scrap. Apparent density above $0.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{ton}/{\mathrm{m}}^{3}$. |

$SC{R}_{Aggr}$ | $\mathrm{kg}$ | Sum of scrap types charged in less than 10% of the heats. |

Incineration | $\mathrm{kg}$ | Scrap representation based on the shape of the scrap type. See Section 3.1.2 and Figure 3. |

Heavy melting scrap (HM) | $\mathrm{kg}$ | |

Plate1 | $\mathrm{kg}$ | |

Plate2 | $\mathrm{kg}$ | |

Internal1 | $\mathrm{kg}$ | |

Internal2 | $\mathrm{kg}$ | |

Shredded | $\mathrm{kg}$ | |

Swarfs | $\mathrm{kg}$ | |

Turnings | $\mathrm{kg}$ | |

Carbon | $\mathrm{kg}$ | |

Skulls | $\mathrm{kg}$ | |

Aggregate | $\mathrm{kg}$ | |

Heavy | $\mathrm{kg}$ | Scrap representation based on the reported and estimated apparent density ranges or values of each respective scrap type in the steel plant. See Section 3.1.3 and Figure 3. |

Mid | $\mathrm{kg}$ | |

Light-Mid | $\mathrm{kg}$ | |

Light | $\mathrm{kg}$ | |

Hot heel | − | 1 if hot heel is present at the start of the heat, else 0. To account for the heat transfer by the hot heel. |

Furnace shell number | − | An ordinary variable counting the number of heats since the last furnace barrel maintenance. |

**Table 3.**The domain-specific variable batches. The variables present in each variable group are shown in Table 4.

Variable Batch | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
---|---|---|---|---|---|---|---|---|---|

Variable Group | Base | x | x | x | x | x | x | x | x |

Plant scrap category | x | x | |||||||

Visual scrap category | x | x | |||||||

Density scrap category | x | x | |||||||

Furnace related | x | x | x | x |

Variable Group | Variables | No. Variables | Variable Group | Variables | No. Variables |
---|---|---|---|---|---|

Base | Total Weight | 9 | $SC{R}_{686}$ | ||

TTT | $SC{R}_{698}$ | ||||

TCB2 | $SC{R}_{699}$ | ||||

Burner oil | $SC{R}_{711}$ | ||||

Burner ${O}_{2}$ | $SC{R}_{Aggr}$ | ||||

${O}_{2}$-lance | Visual scrap | Incineration | 12 | ||

Injected carbon | representation | HM | |||

Lime and dolomite | Plate1 | ||||

No. Charges | Plate2 | ||||

Plant scrap | $SC{R}_{101}$ | 20 | Internal1 | ||

representation | $SC{R}_{103}$ | Internal2 | |||

$SC{R}_{112}$ | Shredded | ||||

$SC{R}_{333}$ | Swarfs | ||||

$SC{R}_{400}$ | Turnings | ||||

$SC{R}_{407}$ | Carbon | ||||

$SC{R}_{409}$ | Skulls | ||||

$SC{R}_{412}$ | Aggregate | ||||

$SC{R}_{416}$ | Density scrap | Heavy | 4 | ||

$SC{R}_{451}$ | representation | Mid | |||

$SC{R}_{452}$ | Light-Mid | ||||

$SC{R}_{455}$ | Light | ||||

$SC{R}_{492}$ | Furnace | Hot heel | 2 | ||

$SC{R}_{633}$ | related | Furnace shell number | |||

$SC{R}_{677}$ |

**Table 5.**Parameter combinations used for the RF models. Each value is separated by a comma. m is the number of input variables. Each combination of parameters represents one model type.

Parameter | Variations | #Combinations |
---|---|---|

Number of trees | 10, 30, 50, 70, 90, 110, 130, 150, 170, 190, 210, 230, 250 | 13 |

Max tree depth | 2, 3, 4, 5, 6, 7, 8, 9, 10, Unlimited | 10 |

Max features in split | $v,\sqrt{v}$ | 2 |

Cleaning strategies | See Section 3.3 | 4 |

Variable batches (domain-specific) | See Section 3.2.1 | 8 |

Total: | 8320 |

**Table 6.**Parameter combinations used for the ANN models. Each value is separated by a comma. The topology (z) and (z,z), indicate one and two layers with z nodes in each layer, respectively. Each combination of parameters represents one model type.

Parameter | Variations | #Combinations |
---|---|---|

Activation function | Hyperbolic tangent function, Logistic sigmoid | 2 |

Learning rate | 0.1, 0.01, 0.001 | 3 |

Topology | (z) and (z,z) in z ∈ 1, 3, ..., 29 | 30 |

Cleaning strategies | See Section 3.3 | 4 |

Variable batches (domain-specific) | See Section 3.2.1 | 8 |

Total: | 5760 |

**Table 7.**The adjusted-${R}^{2}$ and error metric variants that are used to evaluate the performance of the aggregated model instances.

Symbol | Definition |
---|---|

${\overline{R}}_{\mu}^{2}$ | Mean adjusted R-square of the 10 model instances on the test data |

${\overline{R}}_{\sigma}^{2}$ | Standard deviation of adjusted R-square of the 10 model instances on the test data |

${\overline{R}}_{min}^{2}$ | Minimum adjusted R-square of the 10 model instances on the test data |

${\overline{R}}_{max}^{2}$ | Maximum adjusted R-square of the 10 model instances on the test data |

${\mathrm{\Delta}}_{\mu}$ | Mean error of the mean error of the 10 model instances on the test data |

${\mathrm{\Delta}}_{\sigma}$ | Standard deviation of the mean error of the 10 model instances on the test data. |

${\mathrm{\Delta}}_{min}$ | Minimum error of the mean error of the 10 model instances on the test data. |

${\mathrm{\Delta}}_{max}$ | Maximum error of the mean error of the 10 model instances on the test data. |

**Table 8.**The performance and meta data of the best models, and the meta data for the four cleaning strategies. The values in parentheses show the values from the ANN models.

Cleaning Type | Expert | Domain-Specific | Tukey | Tukey-Domain-Specific |
---|---|---|---|---|

Train/test split | 2571/187 | 3625/241 | 3179/212 | 3089/207 |

% cleaned train/test | 36.2%/28.9% | 10.1%/8.4% | 21.2%/19.4% | 23.4%/21.3% |

% total cleaned data | 35.8% | 10.0% | 21.0% | 23.3% |

Variable batches | 6 (6) | 6 (6) | 6 (6) | 6 (6) |

No. Variables | 21 (21) | 21 (21) | 21 (21) | 21 (21) |

${\overline{R}}_{\mu}^{2}$ | 0.461 (0.490) | 0.430 (0.457) | 0.381 (0.380) | 0.377 (0.379) |

${\mathrm{\Delta}}_{\mu}$ | 408 (238) | 369 (209) | 252 (214) | 239 (126) |

(kWh/heat) | ||||

${\mathrm{\Delta}}_{\sigma}$ | 1938 (1863) | 2173 (2063) | 2062 (1995) | 2042 (1976) |

(kWh/heat) | ||||

${\mathrm{\Delta}}_{min}$ | −4496 (−5316) | −7064 (−7756) | −5061 (−5464) | −5079 (−4262) |

(kWh/heat) | ||||

${\mathrm{\Delta}}_{max}$ | 7113 (5699) | 7833 (7735) | 7810 (7608) | 7942 (7162) |

(kWh/heat) |

**Table 9.**Correlation between the input variables and the EE consumption as defined in the transformer system. The input variables are ordered by dCor in the training data set. The values in parentheses are the Pearson correlation coefficients.

Input Variable | Training | Test | Input Variable | Training | Test |
---|---|---|---|---|---|

Carbon injection | 0.39 (0.40) | 0.33 (0.31) | TCB2 | 0.12 (0.13) | 0.14 (0.07) |

Total weight | 0.38 (0.36) | 0.35 (0.37) | HM | 0.10 (0.10) | 0.22 (0.21) |

TTT | 0.28 (0.31) | 0.33 (0.31) | Lance O2 | 0.09 (0.07) | 0.16 (0.13) |

Burner oil | 0.24 (0.24) | 0.33 (0.29) | Anthracite | 0.09 (0.05) | 0.23 (0.17) |

Plate1 | 0.20 (−0.17) | 0.34 (−0.31) | Internal2 | 0.08 (0.05) | 0.34 (0.06) |

Turnings | 0.20 (0.20) | 0.28 (0.24) | Swarfs | 0.07 (0.04) | 0.28 (0.20) |

Skulls | 0.18 (0.15) | 0.35 (0.34) | Shredded | 0.05 (−0.03) | 0.22 (0.22) |

Charges | 0.17 (0.19) | 0.20 (0.21) | Lime+Dolomite | 0.04 (0.01) | 0.13 (−0.13) |

Internal1 | 0.16 (0.16) | 0.22 (0.13) | Plate2 | 0.04 (−0.02) | 0.07 (−0.01) |

Burner O2 | 0.15 (0.13) | 0.14 (0.0) | Aggregate | 0.04 (−0.04) | 0.10 (−0.07) |

Incineration | 0.15 (0.10) | 0.33 (0.30) |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Carlsson, L.S.; Samuelsson, P.B.; Jönsson, P.G.
Modeling the Effect of Scrap on the Electrical Energy Consumption of an Electric Arc Furnace. *Processes* **2020**, *8*, 1044.
https://doi.org/10.3390/pr8091044

**AMA Style**

Carlsson LS, Samuelsson PB, Jönsson PG.
Modeling the Effect of Scrap on the Electrical Energy Consumption of an Electric Arc Furnace. *Processes*. 2020; 8(9):1044.
https://doi.org/10.3390/pr8091044

**Chicago/Turabian Style**

Carlsson, Leo S., Peter B. Samuelsson, and Pär G. Jönsson.
2020. "Modeling the Effect of Scrap on the Electrical Energy Consumption of an Electric Arc Furnace" *Processes* 8, no. 9: 1044.
https://doi.org/10.3390/pr8091044