**2. Methodology**

The consumed volume of oxygen during the converter steelmaking process is mainly related with the following four types of factors:


In this study, the exact amount of oxygen that will be consumed in the not-started steelmaking process is forecasted. This task is usually realized by static prediction models (before blowing) which obviously differ with dynamic prediction models (during blowing). Hence, the control parameters are not selected in this study. Since the final carbon target of the similar steel grades is very close, prediction models can be trained independently with different carbon target. The equipment conditions cannot be directly observed, so we assume they are constants within a short period. Therefore, only the input materials are considered in our proposed methodology stated in the following subsections.

### *2.1. Reaction-Based Linear Model*

To describe the converter steelmaking process, reaction Equations (1)–(9) are defined where the symbols [·], (·) and {·} indicate metal, slag and gas phases, respectively. As shown in (1)–(6) oxidation is the most important chemical reaction mainly carried out on the hot metal, which converts carbon to carbon oxide, silicon to silica, manganese to manganous oxide, phosphorus to phosphate, and sulphur to sulfur dioxide.

$$\mathbf{I}\left[\mathbf{C}\right] + \frac{1}{2}\{\mathbf{O}\_2\} = \{\mathbf{CO}\}\tag{1}$$

$$\text{'}\left[\text{C}\right] + \left|\text{O}\_{2}\right\rangle = \left|\text{CO}\_{2}\right\rangle \tag{2}$$

$$\text{'}\left[\text{Si}\right] + \left|\text{O}\_{2}\right\rangle = \left(\text{SiO}\_{2}\right) \tag{3}$$

$$\{Mn\} + \frac{1}{2} \{O\_2\} = \left(MnO\right) \tag{4}$$

$$2\left[P\right] + \frac{5}{2}\left|O\mathbf{2}\right\rangle = \left(P\_2O\_5\right) \tag{5}$$

$$\mathbf{S}\begin{bmatrix} \mathbf{S} \end{bmatrix} + \begin{Bmatrix} \mathbf{O}\_2 \end{Bmatrix} = \begin{Bmatrix} \mathbf{SO}\_2 \end{Bmatrix} \tag{6}$$

Unfortunately, a little iron is also combined with oxygen in addition to these chemical reactions, and produces *FexOy* as follows.

$$2\left[Fe\right] + \frac{3}{2}\left|O\_2\right\rangle = \left(Fe\_2O\_3\right) \tag{7}$$

*Processes* **2019**, *7*, 352

$$\text{E}\left[\text{Fe}\right] + \frac{1}{2}\text{[O}\_2\text{]} = \left(\text{FeO}\right) \tag{8}$$

In addition, the liquid slag releases a little oxygen, and acts as an oxidizer to produce some by-products:

$$\text{[[S]} + \text{(CaO)} \rightarrow \text{(CaS)} + \text{[O]} \tag{9}$$

It should be noted that, a number of other oxygen-related reactions occur in the steelmaking process. For instance, the post reaction {*CO*} + <sup>1</sup> <sup>2</sup> {*O*2} → {*CO*2} will consume an amount of oxygen, and the *C* element in the converter lining will also absorb oxygen. Specially, when the phenomenon of rephosphoration and remanganization occur with rising temperatures and low *FeO* contents in the slag, the reduction of the (*P*2*O*5) and (*MnO*) with the solved [*C*] in the steel droplets in the slag/gas emulsion will release a little oxygen. However, these reactions fail to be directly observed and recorded. Therefore, the consumed and released oxygen during the reaction or blowing are identified as constants or random noises.

We classify these input materials (*x*) into two sets: The materials consumed oxygen *O*<sup>+</sup> and the materials released oxygen *O*−. To estimate the value of oxygen consumption (*y*), we assume there is linear relationship between *x* and:

$$y = f(\mathbf{x}) = \sum\_{i=1}^{m\_1} w\_i \mathbf{x}\_i + \sum\_{i=1}^{m\_2} w\_i (-\mathbf{x}\_i) + w\_0 = \sum\_{i=1}^{m\_1} w\_i \mathbf{x}\_i - \sum\_{i=1}^{m\_2} w\_i \mathbf{x}\_i + w\_0 \tag{10}$$

where *x* = (*x*1, ··· , *xm*) represents the materials reacting with oxygen, *wi* denotes their reactions coefficient, *w*<sup>0</sup> is a constant term determined by learning from data, *m*<sup>1</sup> and *m*<sup>2</sup> respectively represent the size of *<sup>O</sup>*<sup>+</sup> and *<sup>O</sup>*−, and *<sup>m</sup>*<sup>1</sup> + *<sup>m</sup>*<sup>2</sup> = *<sup>m</sup>*. To determine the values of *<sup>w</sup>* = (*w*0, *<sup>w</sup>*1, ··· , *wm*), the pre-defined loss function Equation (11) is minimized.

$$f(w) = \frac{1}{2m} \sum\_{j=1}^{m} \left( f(\mathbf{x}^{(j)}) - y^{(j)} \right) \tag{11}$$

where *x*(*j*) and *y*(*j*) respectively represent the input and output values of *j th* sample data. The loss function *J*(*w*) can be solved by the least square or gradient descent least angle method [22].

However, the suggested multiple linear regression (MLR) model is an ideal theoretical model, because the converter steelmaking process in nature is a complex system with multi-component, multi-phase and multi-reaction, the detailed process of each reaction is impossible to be precisely formulated. Additionally, in actual production environments, considerable number factors played in or affected the reactions fail to be observed. Therefore, the MLR model based on these reactions always suffers from low precision and low robustness in the actual production process. To overcome this shortage, this study develops the data-driven prediction model in Section 2.2.
