A Novel Process Modeling Method for Steel Sulphur Content Soft Sensing during Ladle Furnace Steel Refining

Abstract

Steel sulphur content soft sensing is of great importance for optimal control of the desulphurization process during ladle furnace (LF) steel refining. However, the soft sensing models in the literature at present are not able to capture the multi-stage characteristics. For addressing this problem and thereby obtaining satisfactory performance, stage-based modeling is proposed by virtue of sub-models ensemble. The central idea of this method is to establish several individual sub-models in order to focus on the local process property of each stage during desulphurization. Furthermore, soft partition strategy using nonparametric regression is developed for realizing soft handoff among the sub-models of successive stages, by which the close and changing process properties in the stage-to-stage transition region can be accurately described. Finally, the effectiveness of the presented method is validated by practical data. It can be concluded from experiments that the proposed stage-based modeling approach is able to significantly improve the sulphur content soft sensing performance, which makes it helpful in both process monitoring and operations optimization for LF process.

1. Introduction

As a secondary metallurgic unit in steel-making process, ladle furnace (LF) refining aims at producing qualified steel with desired temperature and chemical composition. Desulphurization is one of the core tasks during LF treatment, whose objective is to remove the sulphur content of steel to a certain degree for obtaining favorable material properties.¹⁾ However, the precise control of desulphurization reaction is still a challenged task because desulphurization is typically a complex metallurgic technics involving multiple manipulation variables, nonlinear input-output characteristics and long adjusting period. For attaining operation optimization, it is necessary to access the status of desulphurization process while in progress. However, the sulphur content is obtained through offline sampling in practical production, which brings a relatively large feedback delay for receiving the sample result and inevitably causes a time delay in control. In order to take into account control precision and production efficiency simultaneously, it is of great importance to know well “how the sulphur content changes while in progress”. Soft sensing technology provides a feasible solution, which tries to estimate the sulphur content from process knowledge and hardware measurement.

Soft sensing is a special application of process modeling technique, which aims to estimate the difficult-to-measure quantities by those easy-to-measure variables in the condition that they are highly correlated. Particularly, it was shown that sulphur content can be predicted by related process variables, such as reaction temperature, addition weight of various slags, argon gas stirring intensity, for these easy-to-measure variables in LF provide considerable information about desulphurization process.^2,3,4,5,6) Therefore, soft sensing technique is a promising solution to monitor the change of steel sulphur content in a timely and effective manner.

One way to establish a sulphur content soft sensing model is to use the physical and chemical principle equation associated with desulphurization process. In this way, the reaction behavior can be intensively formulated and sulphur content is therefore estimated. There are several mechanistic models that have been reported for desulphurization reaction description.^3,4,7) These models use reaction thermodynamics and kinetics formulations to explain the basic nature of desulphurization. Thus, they are always good in extrapolation ability. However, desulphurization is so complex a metallurgic reaction that these mechanistic models are commonly not precise enough, generally for the following reasons: (i) It is unable to describe the reaction accurately, for some metallurgical properties are not well known. In fact, different mechanistic models are commonly simplified representations of desulphurization process in varying degrees. (ii) The determination of the parameters in the mechanism models is still a challenge. Although several experimental methods are available for parameter estimation, their predictive capacity is only as good as the structural information and experimental data available. Actually, different steel grades require significant shift in the production conditions so that it is difficult to utilize one simple model to describe such changes. (iii) A certain amount of mechanism models are hardly modified due to the complicated model structure and the lack of detailed knowledge. It is difficult to make precise adjustment on these models when prediction errors occur.

Another way to establish a sulphur content soft sensing model can be realized by extracting information from process data using data-driven modeling technique, resulting in a data model for input-output functional representation. The advantages of such data models are the convenience of modeling and the good self-learning property. There are dozens of data-driven modeling methods available for establishing the soft sensing model, such as partial least square (PLS), neural network (NN) and support vector regression (SVR). These approaches have been intensively researched and widely used in many fields for their low cost, high effectiveness and other characteristics. Some of them have already been introduced to steel-making process^8,9) including the application to endpoint prediction of sulphur content in steel.^2,6) However, a data model built on the training samples collected from a relatively long period of steel sampling is not able to support real-time prediction due to the insufficiency of dynamic information, which is the essence for sulphur content soft sensing. In addition, desulphurization reaction is typically a complex multivariable and nonlinear process as evidenced by mechanistic analysis. A purely data model is short of reliability for practical application due to its heavy dependence on the amount of data.

For obtaining satisfactory performance, previously, we proposed to develop the sulphur content soft sensing models in reference 10) by combination of the advantages of both mechanistic methods and data-driven modeling technique. Experiment result shows that this model is able to obtain better performance compared to other models in literatures. However, LF is a steelmaking process unit that covers multiple stages, including slagging, temperature rising, composition trimming, and depuration etc. With respect to desulphurization, reaction characteristic during slagging stage is obviously different from its following stages. However, the aforementioned model in reference 10) is not able to capture such changes of process properties since it essentially describe the process in a unitary way. In order to improve the soft sensing performance substantially, it should get down to the specifics of each reaction stage to explore the significantly varying characteristics.

In this paper, a stage-based sulphur content soft sensing model is developed to incorporate the changeable characteristics distributed in stages. The whole desulphurization reaction cycle is firstly partitioned into slagging stage and post-slagging stage in chronological order. Accordingly, stage-based modeling method is proposed to equip each stage with a corresponding sub-model for focusing on the local behavior. Thereafter, soft partition strategy is presented to describe the close and changing process properties in stage-to-stage transition region via dynamical weight of the stage sub-models, in which the central ideas of fuzzy membership grade concept and logistic regression method are employed to alter the combination weights with production progress in order to smoothly switch from slagging stage sub-model to post-slagging stage sub-model. In this manner, the gradually changed process properties taking place in the stage-to-stage transition region can be accurately described.

The rest of this paper is organized as following. In section 2, some preliminaries are briefly reviewed which are used as the basic algorithm for stage-based modeling. The soft stage partition strategy is thereafter studied for constructing the stage-based model in Section 3. In Section 4, the application of the proposed approach to an industrial 120t LF plant is presented and discussions are conducted based on the experiment results. Finally, the conclusions are given in Section 5 to summarize the paper.

2. Some Preliminaries

In this section, the existing hybrid modeling algorithm proposed in reference 10) is briefly reviewed, which is used as a base method in the presented paper to build each stage sub-model. As a beginning, the first principles for desulphurization process are introduced.

2.1. First Principles for Desulphurization Process

In LF process, molten steel is covered by slag. The desulphurization reaction occurs in the slag-metal interface, which is graphically shown in Fig. 1.

Fig. 1.

Schematic diagram for LF desulphurization process.

As shown in Fig. 1, desulphurization process contains three steps.

a) Firstly, the inner sulphur transfers to the steel side adjoining the steel-slag interface.

b) Subsequently, desulphurization reaction occurs in the steel-slag interface.

c) Finally, the resultant sulphide from reaction is absorbed by the top slag.

According to two-film theory, the dynamic behavior of desulphurization process can be mathematically described as follows:^11,12)   

$$[\text{\%}\text{S}]={[\text{\%}\text{S}]}_e+\left\{{[\text{\%}\text{S}]}_0-{[\text{\%}\text{S}]}_e\right\}exp\left(-\frac{t}{\tau }\right)$$(1)

where [%S] is the sulphur content at time t, [%S]₀ is the initial sulphur content in the metal, τ is the reaction time constant that intuitively reflect the equilibrium time would be cost, [%S]_e is the sulphur content that the metal would have in equilibrium. Equation (1) is able to describe desulphurization process that may be controlled either by mass transfer or interfacial chemical reaction, where τ varies its determination factors according to the specified case.^10,12) [%S]_e can be calculated based on the mass balance of sulphur in the metal-slag system:^3,12)   

$${[\%\mathrm{S}]}_e=\left({[\%\mathrm{S}]}_0+{(\%\mathrm{S})}_0\frac{m_{Slag}}{M}\right)/\left(L_S\frac{m_{Slag}}{M}+1\right)$$(2)

where (%S)₀ is the initial sulphur content in the slag, M is the steel mass, m_Slag is the slag mass, L_S is the equilibrium sulphur partition ratio.

It is possible to determine L_S from reaction thermodynamics theory:³⁾   

$$lg{L}_{\mathrm{S}}=-\frac{935}{T}+1.375+lg{C}_{\mathrm{S}}-lg{a}_{\mathrm{O}}+lg{f}_{\mathrm{S}}$$(3)

where C_S is the sulphide capacity of slag, a_O and f_S are the oxygen activity and activity coefficient of sulphur in the metal phase, T is the temperature of the system (K).

So far, the structure of the mechanistic model for sulphur content soft sensing can be established. While the main difficulty focuses on the determination of its unknown parameters C_S and τ.

2.2. Existing Hybrid Modeling Approach for Sulphur Content Soft Sensing

For accurately estimating the unknown parameters C_S and τ, previously, a hybrid modeling approach was proposed in reference 10). Its central idea is to utilize data-driven modeling method to adapt to the hardly determined parameters in the mechanistic model, thereby resulting in a serial hybrid structured model. The principle of this approach is schematically shown in Fig. 2.

Fig. 2.

Illustration of existing modeling approach.

It can be found from Fig. 2 that the data-driven modeling algorithms including data fusion method and neural network (NN) technique are used as parameter estimators to provide the estimation of the hardly determined parameters in the mechanistic model. More concretely, the final estimation of slag sulphide capacity C_S is abstracted from C_S(1), C_S(2), and C_S(3), which by definition, are estimations of C_S based on KTH model,^13,14) optical basicity model^15,16) and IMCT model,¹⁷⁾ respectively. The data fusion algorithm is employed to integrate different estimations for extracting more pertinent and desired estimation of C_S. As for estimating the parameter τ, neural network (NN) technique is used. The input variables for NN model are slag viscosity η (estimated by CSIRO model¹⁸⁾) and argon stirring intensity F, which by practical process experience, are the main influencing factors for determination of reaction time constant τ. Details about this hybrid modeling algorithm refers to reference 10).

3. Stage-based Modeling for Sulphur Content Soft Sensing

During the entire LF treatment period, the desulphurization reaction will go through two successive smelting stages, which possess different physicochemical characteristics in molten slag and steel. As a consequence, each smelting stage shows a local process property in input-output relationship. Furthermore, there is a transition stage stood between these two stages during which the process property gradually transforms from the former stage to the later one. However, the hybrid modeling approach in section 2.2 is not able to take into account such a time-varying characteristic. In fact, it can only capture an average effect (without distinguishing the changes between stages), thereby contributing a modest improvement in soft sensing performance.

For addressing the above problem, in this section, stage-based modeling strategy is introduced. The yielded stage-based model is an ensemble of two sub-models, which are established to respectively capture the local process properties of the above two successive smelting stages. The gradually changed characteristics displayed in the stage-to-stage transition period is described by weight sum of the output of these two sub-models, where the combination weight varies with production progress for synchronous description of the evolving process behavior.

3.1. Available Experience for Stage Partition

The whole LF smelting process covers multiple process steps, including feeding preparation, slags and deoxidizer feeding, temperature rising, alloying, composition trimming, and depuration etc. From the point of view of desulphurization, a LF cycle can be partitioned into slagging stage and post-slagging stage (unite those steps after temperature rising stage into one) in chronological order, which is shown in Fig. 3.

Fig. 3.

The definition of slagging stage and post-slagging stage.

In Fig. 3, the first half of the LF treatment cycle is defined as slagging stage, in which various slag materials and alloys are prepared in turn and successively fed into the bath, and argon stirring is thereafter carried out to make these materials fluidified, uniformed and fully reacted in order to obtain molten slag with desired smelting abilities. The duration of this stage varies from a few minutes to more than ten minutes. In slagging stage, desulphurization reaction is greatly suppressed, for the reaction condition in this stage is obviously inferior to the following post-slagging stage.

The time occupying of slagging stage varies with the initial steel temperature before treatment, because initial steel temperature greatly affects the melting and mixing speed of slag material as well as the duration of deoxidation process. Therefore, the stages can’t be divided purely based on process smelting time. It is advisable to partition the stages by simultaneously consideration of the interaction and mutual influence of both smelting time and initial steel temperature.

The empirical stage partition is schematically shown in Fig. 4.

Fig. 4.

Empirical stage partition for LF desulphurization process.

As shown in Fig. 4, the empirical partition idea can be summarized as follows: i) when initial steel temperature is less than 1520°C, slagging stage lasts for about ten minites. ii) when the temperature is between 1520°C and 1540°C, slagging stage lasts for about eight minites. iii) when the temperature is higher than 1540°C, slagging stage lasts for about six minites.

It can be found from above analysis that stage partition based on experience is inevitablely imprecise because the boundary is specified crudely. Therefore, a more reliable and accurate partition method is required.

3.2. Soft Partition for Stage-based Modeling

A straightforward idea for improving empirical partition accuracy is to locate the precise stage boundary by fine tuning. However, it can be found from mechanism analysis and practical experience that, the change of reaction behavior from slagging stage to post-slagging stage doesn’t happen suddenly. There is a transition region around the empirical boundary, in which the state of the molten slag and steel experiences a progressive process from high oxidation to low oxidation (occurs mostly in temperature rising stage and alloying stage) and the process property gradually changes. At the beginning, the reaction behavior is more similar to slagging stage. With the going on of the smelting process, the similarity will gradually shift to post-slagging stage side. Stage partition in a strict way suffers from mismatch problem. Therefore, it is advisable to investigate the stage-to-stage transiting characteristics.

In this paper, stage-based modeling strategy is investigated to describe the desulphurization process that operates in a variety of steady stages and between-stage dynamic transition. The process property in each steady stage is respectively described with a corresponding sub-model in order to focus on the local behavior. Thereafter, the central idea of fuzzy membership grade concept^19,20) and nonparametric regression approach is employed to weight sum of the steady stage sub-models. By dynamically altering the combination weight according to the smelting time, the sub-models ensemble is able to accurately describe the close and changing process property in transition stage.

As a whole, the stage-based modeling method works in three steps:

Step 1: Stages redivision

In this step, we aim to partition a LF cycle into three stages in chronological order, i.e. slagging stage, transition stage and post-slagging stage. By shrinking the empirical boundary to each side, we compress the range of both empirical slagging stage and empirical post-slagging stage to make space for comprising the transition stage, which is shown in Fig. 5.

Fig. 5.

Central idea for stage re-division.

In Fig. 5, the transition stage, as a middle time segment in the whole LF treatment cycle, moves forward or backward in smelting time according to initial steel temperature. Because the transition stage is defined on the region carved out around empirical boundary, its process behavior is considered to exhibit stage-to-stage transiting characteristics. As an illustration, regions corresponding to the above three stages are plotted in Fig. 5. It can be found that the above three stages are segmented according to two attributes, i.e. smelting time and initial steel temperature.

Step 2: Establishment of the sub-models for describing the process properties of slagging stage and post-slagging stage respectively

In this step, we aim to establish two sub-models in order to focus on the local reaction behaviors in slagging stage and post-slagging stage, respectively. The central idea is shown in Fig. 6.

Fig. 6.

Central idea for the establishment of the stage sub-models.

Because the main changes in reaction behaviors occur in the transition stage, the process properties of slagging stage and post-slagging stage are in a relatively steady state. In other words, there is no obvious time-varying characteristic in slagging stage and post-slagging stage. Under such a premise, the hybrid modeling algorithm proposed in section 2.2 makes sense in both slagging stage and post-slagging stage. Therefore, it is used to individually establish the slagging stage sub-model and post-slagging stage sub-model. Details for the modeling process are summarized as follows:

i) On the basis of the stage partition rule in Fig. 5, the whole process data can be divided into three training sample sets according to the attribute values on sampling time and initial steel temperature, which are denoted by ℵ_k={(x_i, y_i), i = 1,…, N_k}, (k = 1, 2, 3) with (x_i, y_i) as the input-output pair of the i-th sample. For details about the constitution information of x and y, see Table 1; ℵ₁ and ℵ₃ are the training sample sets used to fit the slagging stage sub-model and post-slagging stage sub-model, respectively; ℵ₂ is the training sample set that will be used in step 3 for training a soft stage partitioner; N_k (k=1, 2, 3) is the total number of samples contained in each training sample set.

Table 1. Details of the input information about the stage sub-models.

Variable symbol	Variable description	Acquisition of the observations
CS(1)	an estimation of CS calculated by KTH model13,14)	updated when slags feeding
CS(2)	an estimation of CS calculated by optical basicity model15,16)	updated when slags feeding
CS(3)	an estimation of CS calculated by IMCT model.17)	updated when slags feeding
η	slag viscosity estimated by CSIRO model.18)	updated when slags feeding
F	argon stirring intensity	updated when measurement
T	steel temperature	updated when measurement
mSlag	slag mass	updated when slags feeding
t	smelting time	obtained by real-time recording
[%S]0	initial steel sulphur content	obtained by sampling
[%S]	sulphur content at time t	obtained by sampling

ii) The slagging stage sub-model and the post-slagging stage sub-model are established based on ℵ₁ and ℵ₃ individually, where hybrid modeling algorithm in section 2.2¹⁰⁾ is employed. Each stage sub-model is composed by a parameter estimator and a mechanism part. To go along with this, the attributes contained in the input vector of the sub-model are divided into two groups. One group is the input of the parameter estimator, while the rest group together with the output of the parameter estimator (the reaction time constant τ and the estimation of sulphide capacity C_S) are used as the input of the mechanism part.

Details of the input-output information about the stage sub-model are listed in Tables 1 and 2.

Table 2. Details of the output information about the stage sub-models.

Variable symbol	Variable description	Acquisition of the observations
[%S]	sulphur content at time t	obtained by sampling

Besides the input attributes that have been listed in Table 1, there are some other attributes that are required for the mechanism part, including initial sulphur content in slag (S)₀, steel weight M, oxygen activity a_O, activity coefficient of sulphur in the metal phase f_S. The values of these process variables are specified with some constants according to steel grades or calculated by empirical process models.

On the basis of the modeling strategy described above, each stage is equipped with its own sub-model. As a result, the local process properties in slagging stage and post-slagging stage can be represented contrapuntally.

Step 3: Ensemble of the sub-models for representing the process property of transition stage

In this step, we aim to seek an accurate description of the process property of transition stage. An ensemble model that consists of slagging stage sub-model and post-slagging stage sub-model is proposed to cope with the reaction behavior transition. A soft stage partitioner, which is established based on the information provided by the training sample set ℵ₂, is used to produce the dynamical combination weight for this ensemble.

As shown in Fig. 7, the gradually changed characteristics displayed in the transition stage are represented by weight sum of the output of the slagging stage sub-model and the post-slagging stage model. The combination weights w^slagging and w^{post-slagging}, which are produced by a soft stage partitioner, vary with production progress for synchronous description of the evolving process property. It should be noted that, the combination weights used for the purpose of model integration are usually being known as fuzzy membership grades in the field of fuzzy theory. In general, fuzzy theory introduces some “fuzziness” into the formulation of some real-world problems that possess some “ambiguity” properties, such as fuzzy clustering and fuzzy inference. Taking clustering problem as an example, the boundary between clusters could be fuzzy rather than crisp; that is, a data point could belong to two or more clusters with different degrees of membership. In many cases, this formulation is closer to reality and better performance may be expected. Therefore, such an idea is used to describe the stage-to-stage transition characteristics displayed in the transition stage.

Fig. 7.

Central idea for description of the process property of transition stage.

Based on the analysis above, it can be known that the key point lying in this step is “how to establish a soft stage partitioner with sufficient accuracy”. This paper proposes to represent the stage-to-stage transition property by making use of the changes information of the prediction accuracy of the above two stage sub-models. More specifically, the accuracy of slagging stage sub-model performed in the transition stage will degrades with the going on of the transition process in general. While the accuracy performance of the post-slagging stage sub-model present the opposite case.

Firstly, each training sample (x_i, y_i) in ℵ₂ is augmented with two attributes, i.e. the stage membership grades w^slagging and w^{post-slagging}, so as to support the modeling of the soft stage partitioner. w^slagging and w^{post-slagging} are scored as follows:   

$$\{\begin{array}{l}{w}_i^{\text{slagging}}={\text{err}}_2\text{(}{\mathit{x}}_i)/\left({\text{err}}_{\text{1}}\text{(}{\mathit{x}}_i)+{\text{err}}_{\text{2}}\text{(}{\mathit{x}}_i)\right)\\ \hspace{1em}\hspace{1em}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }={\displaystyle \frac{\left|y_i-y_i^{post-slagging}\right|}{\left|y_i-y_i^{slagging}\right|+\left|y_i-y_i^{post-slagging}\right|}}\\ w_i^{\text{post-slagging}}=1-w_i^{\text{slagging}}\end{array}$$(4)

where output response errors using absolute error are adopted as the criterion of accuracy. For any input x_i, $y_i^{slagging}$ and err₁(x_i) are the output response and the output response error of the slagging stage sub-model, $y_i^{post-slagging}$ and err₂(x_i) are the output response and the output response error of the post-slagging stage sub-model. It can be concluded from Eq. (4) that, the smaller response error performed, the larger membership grade is assigned to the corresponding sub-model. Accordingly, the process property is considered to be more similar to the corresponding stage.

Secondly, data-driven modeling technique is used to establish the soft stage partitioner. Because its outputs, i.e. w_i^slagging and w_i^{post-slagging}, are physically mutual determined (the sum of them is equal to 1), the soft stage partitioner can be represented using a multi-input single-output model with either w_i^slagging or w_i^{post-slagging} as the output. Because such an output is constrained within the interval [0,1] instead of freely valued, the nonparametric regression based data-driven modeling methods cannot be adopted directly. In order to addressing this problem, output transform strategy using “log odds ratio” treatment is exploited. The so-called “log odds ratio” treatment is to take logarithm of the odds ratio. “log odds ratio” treatment is commonly used in logistic regression to make the connection with the binomial distribution probability for which the probability value is limited in [0,1]. In this paper, the log odds ratio is analogically defined as following   

$$\text{Log\_OR}=log\left(\text{odds ratio}\right)=log\left(\frac{w^{\text{slagging}}}{w^{\text{post-slagging}}}\right)$$(5)

By Eq. (5), we are able to seek a Log_OR model by nonparametric regression, and obtain w_i^slagging and w_i^{post-slagging} via backward deduction:   

$$\{\begin{array}{l}{w}^{\text{slagging}}={\displaystyle \frac{1}{1+exp\left(-\text{Log\_OR}\right)}}\\ w^{\text{post-slagging}}={\displaystyle \frac{exp\left(-\text{Log\_OR}\right)}{1+exp\left(-\text{Log\_OR}\right)}}\end{array}$$(6)

The training samples used for nonparametric regression is generated by augmenting each sample in ℵ₂ with an expected output item Log_OR. The resulting sample set is denoted by $\tilde{\aleph }$={($\tilde{\mathit{x}}$_i, Log_OR_i), i = 1,…, N₂}, where $\tilde{\mathit{x}}$_i is the input vector. According to the stages re-division rule and Fig. 5, $\tilde{\mathit{x}}$_i is composed of smelting time and initial steel temperature. Because the training samples are corrupted with noise, a reasonable definition of Log_OR model can be described via the equation   

$$\text{Log\_OR}=f\left(\tilde{\mathit{x}}\right)+\epsilon$$(7)

without loss of generality, the random noise ε is assumed to be subject to normal distribution with zero mean and some variance σ². f(•) is undefined that needs to be estimated. In this paper, LSSVM model, which is a well-known nonparametric regression method, is used for its simplicity and good prediction performance.^21,22,23) Thus the representation of f($\tilde{\mathit{x}}$) can be   

$$f(\tilde{\mathit{x}})={\omega }^T\cdot \phi \left(\tilde{\mathit{x}}\right)$$(8)

where ω is the weight vector, φ(∙) is a nonlinear mapping function that transfer input $\tilde{\mathit{x}}$ to a high dimensional feature space H. The inner product between two feature vectors in H can be   

$${⟨\phi \left(\tilde{\mathit{x}}\right),\phi \left({\tilde{\mathit{x}}}^{\prime }\right)⟩}_{\mathbf{H}}={⟨k\left({\tilde{\mathit{x}}}^{\prime },\cdot \right),k\left({\tilde{\mathit{x}}}^{\prime },\cdot \right)⟩}_{\mathbf{H}}=k\left(\tilde{\mathit{x}},{\tilde{\mathit{x}}}^{\prime }\right)$$(9)

where k(∙,∙) is a kernel function. It can be found from Eq. (9) that the inner product of two feature vectors in the high dimensional feature space can be easily computed without knowing the exact function of φ(∙), which is usually called the kernel trick. In this paper, Gaussian kernel $k(\tilde{\mathit{x}},{\tilde{\mathit{x}}}^{\prime })=exp\left(-{\Vert \tilde{\mathit{x}}-{\tilde{\mathit{x}}}^{\prime }\Vert }^2/{\nu }^2\right)$ is used, where ν is a prior scale parameter.

The noise assumption in Eq. (7) together with the model shown in Eq. (8) directly gives rise to the likelihood, i.e. the probability density of the observations given the parameters, which is factored over cases in the training sample set   

$$\begin{array}{l}L\left(\omega \right)=\prod _{i=1}^{N_2}p\left({\text{Log\_OR}}_i|{\tilde{\mathit{x}}}_i,\omega \right)\\ \hspace{1em}\hspace{1em}=\prod _{i=1}^{N_2}{\displaystyle \frac{1}{\sqrt{2\pi }\sigma }}exp\left(-{\displaystyle \frac{{({\text{Log\_OR}}_i-\phi {\left({\tilde{\mathit{x}}}_i\right)}_{}^T\omega )}^2}{2{\sigma }^2}}\right)\\ \hspace{1em}\hspace{1em}={\displaystyle \frac{1}{{\left(\sqrt{2\pi }\sigma \right)}^{N_2}}}exp\left(-{\displaystyle \frac{1}{2{\sigma }^2}}\sum _{i=1}^{N_2}{({\text{Log\_OR}}_i-\phi {\left({\tilde{\mathit{x}}}_i\right)}_{}^T\omega )}^2\right)\end{array}$$(10)

Putting a zero mean Gaussian prior on the weight vector as ω~N(0, λI), we can obtain the posterior of ω as follows   

$$\begin{array}{l}p\left(\omega |{\left({\tilde{\mathit{x}}}_i,{\text{Log\_OR}}_i\right)}_{i=1}^{N_2}\right)\\ \propto exp\left(-\frac{1}{2{\sigma }^2}\sum _{i=1}^{N_2}{({\text{Log\_OR}}_i-\phi {\left({\tilde{\mathit{x}}}_i\right)}_{}^T\omega )}^2\right)\cdot exp\left(-\frac{1}{2\lambda }{\omega }^T\omega \right)\\ \propto exp\left(-\frac{1}{2{\sigma }^2}{\left(\mathbf{L}\mathbf{o}\mathbf{g}\mathbf{\_}\mathbf{O}\mathbf{R}-{\mathrm{\Phi }}^T\cdot \omega \right)}^T\left(\mathbf{L}\mathbf{o}\mathbf{g}\mathbf{\_}\mathbf{O}\mathbf{R}-{\mathrm{\Phi }}^T\cdot \omega \right)-\frac{1}{2\lambda }{\omega }^T\omega \right)\\ \propto exp\left(-\frac{1}{2}{\left(\omega -\overline{\omega }\right)}^T\left({\mathrm{\Phi }}^T\mathrm{\Phi }+\frac{{\sigma }^2}{\lambda }\cdot \mathrm{I}\right)\left(\omega -\overline{\omega }\right)\right)\end{array}$$(11)

where $\mathrm{\Phi }=\left[\phi \left({\tilde{\mathit{x}}}_1\right),\cdots \phi \left({\tilde{\mathit{x}}}_{N_2}\right)\right]$, Log_OR=[Log_OR₁, ···,Log_OR_N2]^T, N₂ is the sample size of $\tilde{\aleph }$, $\overline{\omega }={\left(\mathrm{\Phi }{\mathrm{\Phi }}^T+\frac{{\sigma }^2}{\lambda }\cdot \mathbf{I}\right)}^{-1}\mathrm{\Phi }\cdot \mathbf{L}\mathbf{o}\mathbf{g}\mathbf{\_}\mathbf{O}\mathbf{R}$. Thus the maximum a posteriori estimation of f($\tilde{\mathit{x}}$^*) is given by   

$$\begin{array}{l}f\left({\tilde{\mathit{x}}}^{*}\right)={\phi }^T\left({\tilde{\mathit{x}}}^{*}\right)\overline{\omega }\\ ={\phi }^T\left({\tilde{\mathit{x}}}^{*}\right){\left(\mathrm{\Phi }{\mathrm{\Phi }}^T+\frac{{\sigma }^2}{\lambda }\cdot \mathbf{I}\right)}^{-1}\mathrm{\Phi }\cdot \mathbf{L}\mathbf{o}\mathbf{g}\mathbf{\_}\mathbf{O}\mathbf{R}\\ ={\phi }^T\left({\tilde{\mathit{x}}}^{*}\right)\mathrm{\Phi }{\left({\mathrm{\Phi }}^T\mathrm{\Phi }+\frac{{\sigma }^2}{\lambda }\cdot \mathbf{I}\right)}^{-1}\cdot \mathbf{L}\mathbf{o}\mathbf{g}\mathbf{\_}\mathbf{O}\mathbf{R}\end{array}$$(12)

By introducing kernel trick, f($\tilde{\mathit{x}}$^*) can be rewritten in the following way   

$$\begin{array}{l}f\left({\tilde{\mathit{x}}}^{*}\right)=\mathit{k}{({\tilde{\mathit{x}}}^{*})}^T\cdot {\left(\mathbf{K}+\tau \mathbf{I}\right)}^{-1}\mathbf{L}\mathbf{o}\mathbf{g}\mathbf{\_}\mathbf{O}\mathbf{R}\\ \hspace{1em}\hspace{1em}\hspace{0.5em}=\mathit{k}{({\tilde{\mathit{x}}}^{*})}^T\cdot \beta =\sum _{i=1}^{N_2}{\beta }_i\cdot k({\tilde{\mathit{x}}}^{*},{\tilde{\mathit{x}}}_i)\end{array}$$(13)

where ${\left(\mathit{k}({\tilde{\mathit{x}}}^{*})\right)}_i=k\left({\tilde{\mathit{x}}}^{*},{\tilde{\mathit{x}}}_i\right)$, (K)_ij= k($\tilde{\mathit{x}}$_i, $\tilde{\mathit{x}}$_j), i, j = 1,…, N₂. β=[β₁, ∙∙,β_N2]^T=(K+τI)⁻¹∙Log_OR. τ=σ²/λ. For more details about Eq. (13), refer to 24).

Once the estimation of the Log_OR of a testing point $\tilde{\mathit{x}}$^* is computed by Eq. (13), the w^slagging and w^{post-slagging} can be given by   

$$\{\begin{array}{l}{w}^{\text{slagging}}={\displaystyle \frac{1}{1+exp\left(-f({\tilde{\mathit{x}}}^{*})\right)}}\\ w^{\text{post-slagging}}=1-w^{\text{slagging}}\end{array}$$(14)

Table 3 summarizes the calculation process for the stage-based modeling algorithm.

Table 3. Calculation process for stage-based modeling.

Algorithm: stage-based modeling
Initialization
Training samples {(xi, yi), i = 1,…, N};
Scale parameter ν;
Regularization parameter τ.
Step 1) Training data partition
The training samples are divided into three parts according to the stage re-division rule
ℵ1={(xi, yi), i = 1,…, N1}→ training sample sets for slagging stage;
ℵ2={(xi, yi), i = 1,…, N2}→ training sample sets for transition stage;
ℵ3={(xi, yi), i = 1,…, N3}→ training sample sets for post-slagging stage.
Step 2) process modeling for slagging stage and post-slagging stage
Exploit hybrid modeling algorithm in section 2.2 to develop the stage submodels
i) Establish the slagging stage submodel S_fun#1 on ℵ1;
ii) Establish the post-slagging stage submodel S_fun#2 on ℵ3.
Step 3) process modeling for transition stage
Ensemble of the above two sub-models for representing the process property of transition stage
i) For each sample in ℵ2, score the stage membership grades wslagging and wpost-slagging using Eq. (4), and generate the transformed output Log_OR using Eq. (5);
ii) Training the soft stage partitioner using Eq. (13).
Output
i) slagging stage submodel: S_fun#1;
ii) post-slagging stage submodel: S_fun#2;
iii) transition stage model: S_fun#1·wslagging+S_fun#2·wpost-slagging (wslagging and wpost-slagging are generated by Eq. (14), which will vary with production progress).

In Table 3, the scale parameter ν and regularization parameter τ need to be specified in advance, for which some model selection methods are available. The central idea of model selection is to make a trade-off between data fitting and the so-called “model complexity” for the sake of obtaining the optimal generalization performance on testing data instead of training data. The most frequently used methods are cross-validation. For more details about model selection, refer to 25).

4. Experiments

In this section, the performance of the proposed stage-based modeling algorithm is tested by practical data collected from a 120t LF. The data samples are preprocessed in order to improve their quality. Firstly, experiential method is used. The illogical data samples with the attribute values that are out of the scales of operation are discarded. Thereafter, k-Means is used to exclude the outliers that are hard to discriminate by experience. Finally, a total of 980 samples are obtained. Among these samples, 500 samples are randomly selected for training and the rest are utilized for testing. For determination of the scale parameter ν and regularization parameter τ of the soft stage partitioner, 10-fold cross validation is used. The optimal value of ν and τ are selected as 5.5 and 0.01, respectively.

For testing the prediction performance of the proposed stage-based model, a soft sensing model without using stage partition is established for validating the effectiveness of stage-wise modeling strategy. This model is designated as “Unitary model” for convenience of discussion. Furthermore, another stage-based model using empirical stage partition strategy as shown in Fig. 4 is also constructed for testing the potency of the proposed soft stage partition method. This model is abbreviated as empirical stage-based model for convenience. In this model, there is no transition stage considered in the partition framework. As a result, the training and the using of the stage sub-models are both different compared to the proposed stage-based model. Firstly, the whole process data is divided into two parts instead of three parts. As a result, the training samples sets for sub-model establishment in the empirical stage-based modeling scheme are different from that in the proposed stage-based modeling method, and consequently different slagging stage sub-model and post-slagging stage sub-model are produced. Secondly, the reaction behavior in the stage-to-stage transition period is differently described in a soft manner and a rigid manner. The proposed stage-based model utilizes a dynamically weighted scheme, while the empirical stage-based model uses a rigid switching mode. Therefore, the usage of the stage sub-models is also different.

The prediction performance on testing dataset of the above three soft sensing models are shown in Fig. 8. The measured sulphur content is the sample value taken from testing sample, while the predicted value is evaluated from soft sensing model. For completeness, different error evaluations between predicted value and measured value are listed in Table 4, where ±30 ppm hite rate denotes the percentage of the testing samples on which the absolute prediction error is less than 0.003% (mass%). ±30 ppm hite rate is commonly used for evaluation in practical LF production process.

Fig. 8.

Comparison on overall prediction performance of above soft sensing models. (a) prediction by unitary model. (b) prediction by empirical stage-based model. (c) prediction by the proposed stage-based model.

Table 4. Comparison between above three soft sensing models on prediction errors.

Soft sensing models	root mean square error (ppm)	mean absolute error (ppm)	±30 ppm hite rate
Unitary model	26.2	20.6	72.7%
empirical stage-based model	23.5	18.4	80.0%
proposed stage-based model	19.9	15.7	84.4%

It can be found from Fig. 8 and Table 4 that the prediction performance of the two stage-based models generally outperform the unitary model, validating that the process property of the whole desulphurization process exhibit different internal characteristics in different stages. The stage-wise modeling strategy is feasible to capture such a diverse properties for the sake of improvement of the sulphur content soft sensing accuracy. It can also be found that the proposed stage-based model is superior to empirical stage-based model on all the prediction performance evaluation items due to the consideration of the stage-to-stage transition property. The above result validates the effectiveness of the proposed soft partition method, which is crucial for the improvement of the sulphur content soft sensing performance.

For intensive investigation, experiment is carried out to test the performance of above three soft sensing models at different smelting stage. As a beginning, each sample in the testing dataset is assigned to a data subset among slagging stage testing data, transition stage testing data and post-slagging stage testing data, according to partition scheme as shown in Fig. 5. As an outcome, the whole testing dataset are divided into three data subsets, on which the prediction performance are respectively evaluated.

The prediction performance on slagging stage testing data are shown in Figs. 9–10.

Fig. 9.

Comparison on slagging stage testing data. (a) prediction performance of unitary model. (b) prediction performance of empirical stage-based model. (c) prediction performance of the proposed stage-based model.

Fig. 10.

Histogram of prediction error distribution on slagging stage testing data. (a) histogram for unitary model. (b) histogram for empirical stage-based model. (c) histogram for the proposed stage-based model.

The prediction performance on post-slagging stage testing data are shown in Figs. 11–12.

Fig. 11.

Comparison on post-slagging stage testing data. (a) prediction performance of unitary model. (b) prediction performance of empirical stage-based model. (c) prediction performance of the proposed stage-based.

Fig. 12.

Histogram of prediction error distribution on post-slagging stage testing data. (a) histogram for unitary model. (b) histogram for empirical stage-based model. (c) histogram for the proposed stage-based model.

It can be found from Figs. 9, 10, 11, 12 that, both in slagging stage and post-slagging stage, the prediction accuracy of stage-based models are obviously higher than that of unitary model, verifying again the effectiveness of stage-wise modeling strategy. While the prediction precision between empirical stage-based model and the proposed stage-based model is quite close. This result is as expected because both of empirical stage-based model and the proposed stage-based model work in the same way, i.e. specifying slagging stage and post-slagging stage with their own sub-model for focusing the different properties in each stage. The subtle differences of these two stage-based models come from their training process, for there is a slight discrepancy between their training samples due to the different data partition scheme. Furthermore, it can be found that there exist a large overall bias of unitary model. The most obvious deviation occurs on post-slagging stage testing data subset, where the distribution of prediction error is significantly displaced from the central zero point. The reason may be that the unitary model tries to maintain an overall symmetry and has no ability to hold the local symmetry in each stage. With respect to the two stage-based models, this bias problem is corrected to a certain extent, for they can focus on their attention on the local behavior. Therefore, development of the sulphur content soft sensing model in a stage-wise way relieves a lot of bias problem.

The prediction performance on transition stage testing data are shown in Figs. 13–14.

Fig. 13.

Comparison on transition stage testing data. (a) prediction performance of unitary model. (b) prediction performance of empirical stage-based model. (c) prediction performance of the proposed stage-based model.

Fig. 14.

Histogram of prediction error distribution on transition stage testing data. (a) histogram for unitary model. (b) histogram for empirical stage-based model. (c) histogram for the proposed stage-based model.

It can be found from Figs. 13 and 14 that, the proposed stage-based model earns the first place on transition stage testing data. While for the rest ones, the empirical stage-based model shows poor performance with respect to unitary model. The reason for this phenomena may be that the empirical stage-based model describe the process property of transition stage by hard switching from slagging stage sub-model to post-slagging stage sub-model, which is not able to accurately capture the stage-to-stage transition property and therefore bring some instabilities on prediction performance. The result validates the effectiveness of the proposed soft partition method for stage-based modeling.

Subsequently, testing is carried out on the soft stage partitioner. In the present paper, a soft stage partitioner is developed to produce the combination weight for ensemble of the stage sub-models, which is represented as a log odds ratio model that is obtained by regression on the scored samples. In our framework, the input vector of this model, denoted as $\tilde{\mathit{x}}$, is defined on two attributes, i.e. smelting time and initial steel temperature. This definition is proposed according to Fig. 5. However, “are these two items enough to regress the log odds ratio by the data” is required to investigate. Therefore, an experiment is carried out. In this experiment, additional attributes, including slag viscosity η, argon stirring intensity F, slag mass m_Slag and initial steel sulphur content [S]₀, are attached in $\tilde{\mathit{x}}$. Details for obtaining the values of these additional attributes are listed in Table 1. The yielded input vector is denoted as $\tilde{\mathit{x}}$_new, by which a new soft stage partitioner is trained. This new partitioner is compared with the original one by evaluating their prediction performance on the transition stage testing data. The statistical results are listed in Table 5.

Table 5. Comparison between soft stage partitioners with different input attributes.

Soft stage partitioner	root mean square error (ppm)	mean absolute error (ppm)	±30 ppm hite rate
Partitioner with $\tilde{\mathit{x}}$	19.9	15.7	84.4%
Partitioner with $\tilde{\mathit{x}}$new	20.3	15.5	83.8%

It can been found from Table 5 that, the soft stage partitioner with $\tilde{\mathit{x}}$ shows weak advantage on the items of RMSE error and ±30 ppm hite rate. While the performance of the one with $\tilde{\mathit{x}}$_new outstands on the item of mean absolute error. In general, the overall performance of these two soft stage partitioners are quite similar. In other words, additional attributes don’t contribute significant improvement on prediction performance. Therefore, we can conclude that the definition of $\tilde{\mathit{x}}$ is reasonable.

Finally, the output of the soft stage partitioner for two case studies is shown in Fig. 15.

Fig. 15.

The output of the soft stage partitioner for two case studies. (a) the combination weight as a function of the smelting time when the initial steel temperature is 1520°C. (b) the combination weight as a function of the smelting time when the initial steel temperature is 1540°C.

Figure 15 shows that, in both cases, w^slagging decreases with smelting time, while w^{post-slagging} presents the opposite trend. There is a small imperfection that the combination weights at the stage boundary are close to 1 (or 0) instead of strictly equal to 1 (or 0). However, the trend for the changes of w^slagging and w^{post-slagging} as a function of the smelting time agrees well in both cases. Furthermore, it can be found that there is a distinct handover process for case (a) and case (b) when the initial steel temperature is different.

5. Conclusion

In this paper, stage-based modeling strategy is studied for establishing the steel sulphur content soft sensing model in LF. By focusing on learning the local behavior in each stage, the proposed soft sensing model is able to capture the evolvement of process properties from one stage to another. Experiment result shows its superior in describing both the diverse stage property and the stage-to-stage transition characteristics. It can be concluded that the proposed stage-based modeling algorithm for sulphur content soft sensing is not only able to obtain good predictive performance, but also can be used to investigate into the inner reaction behavior under different technical condition, which is helpful for improving the LF process control level.

Acknowledgments

This work was supported by National Natural Science Foundation of China (Nos. 61503068 and 51634002) and National Key R&D Program of China (Nos. 2017YFB0304100).

References

• 1)   N. S.  Cyril and  A.  Fatemi: Int. J. Fatigue, 31 (2009), 526.
• 2)   J. H.  Feng,  X. J.  Li and  X. H.  Zhu: J. Iron Steel Res., 35 (2007), 33 (in Chinese).
• 3)   M.  Andersson,  M.  Hallberg,  L.  Jonsson and  P.  Jonsson: Ironmaking Steelmaking, 29 (2002), 224.
• 4)   M.  Hallberg,  T. L. I.  Jonsson and  P. G.  Jonsson: ISIJ Int., 44 (2004), 1318.
• 5)   W.  Lv: Master thesis, Northeastern University, (2009), http://202.118.8.24/docinfo.action?dbid=72&docid=59805, (accessed 2009-07-10).
• 6)   H. S.  Zhang,  D. P.  Zhan and  Z. H.  Jiang: Iron Steel, 42 (2007), 30 (in Chinese).
• 7)   P. G.  Jonsson and  L. T. I.  Jonsson: ISIJ Int., 41 (2001), 1289.
• 8)   W.  Lv,  Z. Z.  Mao and  P.  Yuan: Steel Res. Int., 83 (2012), 288.
• 9)   H. X.  Tian and  Z. Z.  Mao: IEEE Trans. Autom. Sci. Eng., 7 (2010), 73.
• 10)   W.  Lv,  Z.  Xie,  Z. Z.  Mao,  P.  Yuan and  M. X.  Jia: Neural Comput. Appl., 25 (2014), 1125.
• 11)   C. J. B.  Fincham and  F. D.  Richardson: Proc. R. Soc. Lond. A, 223 (1954), 40.
• 12)   J. Y.  Choi,  D. J.  Kim and  H. G.  Lee: ISIJ Int., 41 (2001), 216.
• 13)   J.  Bjorkvall,  D.  Sichen and  S.  Seetharaman: Ironmaking Steelmaking, 28 (2001), 250.
• 14)   M. M.  Nzotta,  D.  Sichen and  S.  Seetharaman: ISIJ Int., 38 (1998), 1170.
• 15)   D. J.  Sosinsky and  I. D.  Sommerville: Metall. Trans. B, 17 (1986), 331.
• 16)   R. W.  Young,  J. A.  Duffy,  G. J.  Hassall and  Z.  Xu: Ironmaking Steelmaking, 19 (1992), 201.
• 17)   C. B.  Shi,  X. M.  Yang,  J. S.  Jiao,  C.  Li and  H. J.  Guo: ISIJ Int., 50 (2010), 1362.
• 18)   L.  Zhang,  S.  Jahanshahi and  S.  Sun: JOM, 54 (2002), 51.
• 19)   L. A.  Zadeh: Inf. Sci., 178 (2008), 2751.
• 20)   L. A.  Zadeh: A Theory of Approximate Reasoning, Halstead Press, New York, (1979), 1.
• 21)   J. A. K.  Suykens and  J.  Vandewalle: Neural Process. Lett., 9 (1999), 293.
• 22)   V. N.  Vapnik: Statistical Learning Theory, John Wiley and Sons, New York, (1998), 1.
• 23)   A. J.  Smola and  B.  Scholkopf: Stat. Comput., 14 (2004), 199.
• 24)   C. E.  Rasmussen and  C. K. I.  Willianms: Gaussian Processes for Machine Learning, The MIT Press, London, (2006), 7.
• 25)   M. W.  Browne: J. Math. Psychol., 44 (2000), 108.