1. Introduction
Aluminum can be made into alloys with various metals; it is widely used in automotive, aviation, and military industries due to its good ductility, plasticity, recyclability, and oxidation resistance. A regenerative aluminum smelting furnace is important for the aluminum smelting process, in which the real-time measurement and control of liquid aluminum temperatures influence the quality of the aluminum. However, on industrial sites, there are many influencing factors, such as the aging of temperature-measuring thermocouples and fluctuations in the operating voltage, which bring difficulties to the real-time measurements of the aluminum liquid temperature. Hence, it is essential to develop a modeling method to predict the liquid aluminum temperature for quality improvement of the aluminum. The aluminum smelting process is a typical complex industrial furnace production process. In recent decades, many studies on industrial furnaces have been performed (regarding ‘mechanism modeling’) [
1,
2,
3]. Although the physical meaning of ‘mechanism modeling’ is clear, there are some problems, such as complicated calculations for industrial furnace systems. At the same time, mechanism models may not be reliable enough since they usually make simplified assumptions. The furnace temperature, airflow rate, etc., fluctuate greatly in different working states due to the intermittent working characteristics of the regenerative aluminum smelting furnace. The real-time update of the model for the regenerative aluminum smelting furnace is also a problem that needs to be considered.
To overcome the shortcomings of mechanism modeling, a soft-sensor that makes full use of the industrial data is proposed [
4]. There are many researchers working on the data-driven modeling of industrial furnaces and similar processes, such as partial least squares (PLS) [
5], the kernel principal component regression (KPCR) [
6], and kernel partial least squares (KPLS) [
7], which have been successfully applied with good results. However, these methods are generally considered to be global modeling (and trained offline). Moreover, after these models are put into application, they will face problems, such as difficulties in model updating. Consequently, to deal with the adaptive update problem of the model, the moving window technique [
8,
9], recursive models [
10,
11], and the just-in-time learning (JITL) strategy [
12,
13] are usually used as online adaptive update strategies. The JITL strategy trains an online local model to predict the query samples by selecting similar samples from historical samples, so it is more suitable for processes such as industrial furnaces with state mutations. For example, Chen et al. [
14] proposed a least squares support vector machine temperature prediction model based on JITL to deal with large temperature change lags in roller kilns. Dai et al. [
15] combined the moving window technique and the JITL strategy as an update strategy to select similar samples in both time and space dimensions, and they verified the effectiveness of the proposed method on an industrial kiln. In [
16], a locally weighted partial least squares regression (LWPLS) model was proposed by JITL-based local modeling. In LWPLS, the samples most similar to the query sample are assigned different weights and selected for local modeling. The current model will be discarded when the next query sample is available. Then, a new local PLS model will be established for the model’s online update. However, LWPLS only considers the sample similarities, not the variable correlations. The data of the aluminum smelting process often present high-dimensional characteristics and each input variable has a different degree of influence on the liquid aluminum temperature. Hence, except for the sample similarities, it is necessary to consider the variable correlations [
17,
18,
19]. Furthermore, the accuracy of the JITL strategy depends on the quality of the selected samples. However, the traditional similarity measurement criteria, such as Euclidean distance and Mahalanobis distance, only consider the input information without considering the output information, and often cannot obtain accurate similar samples. Thus, investigating new similarity measurement criteria is important for the JITL strategy.
In recent years, artificial intelligence algorithms, such as long short-term memory networks (LSTM) [
20,
21,
22] and extreme learning machine(s) (ELM) [
23,
24,
25,
26] have also been used in soft sensor modeling. The basic assumption for LSTM is that process data are sampled at even and unified frequencies; it is very difficult to meet these conditions for ‘process data measurements’ in industrial processes, especially for quality variables. Hence, LSTM is unsuitable for some processes with irregular sampling frequencies. ELM is a single hidden layer neural network with a low algorithm complexity, which does not need backpropagation to solve iteratively, and has been used in the temperature prediction of regenerative aluminum smelting processes. Huang et al. [
27] proposed an extreme learning machine furnace temperature prediction model based on the kernel principal component analysis and showed that ELM has a better effect than the traditional BP neural network. Liu et al. [
28] proposed an ELM model optimized by the restricted Boltzmann machine (RBM) to solve the random initialization of the input weights and biases in the ELM. Moreover, ELM has a fast learning speed and is suitable as an online prediction model. For example, Li et al. [
29] built a local online ELM model in combination with a JITL strategy, allowing the online prediction of polyethylene terephthalate (PET) viscosity without relying on time-consuming laboratory analysis procedures. However, this ELM-based online prediction model neither considers sample similarities nor variable correlations, which is unreasonable in local modeling. Moreover, the original ELM runs the risk of model overfitting. Hence, a regularized extreme learning machine (RELM) [
30] was proposed to solve the model’s overfitting problem.
Although some research studies have been carried out on ELM, there are few discussions about sample similarities and variable correlations in RELM, especially in temperature prediction. Based on the above discussions, a soft sensor modeling method of the JITL-based triple-weighted regularized extreme learning machine (JITL-TWRELM) was proposed to solve the above problems. Compared with the traditional data-driven modeling method described above, the method proposed in this paper not only allows real-time updating of the model but also obtains more accurate local modeling samples due to the use of the WJITL strategy, which uses correlation information between the input and output variables in the sample selection stage. Meanwhile, in the local modeling stage, the proposed method overcomes the shortcomings of the traditional local modeling method, which only considers the sample similarities and analyzes the variable correlations, highlighting the influences of different variables on the output. The remainder of this article is structured as follows. Firstly, the regenerative aluminum smelting furnace is briefly introduced. Secondly, the regularized extreme learning machine (RELM), sample weighted regularized extreme learning machine (SWRELM), and variable weighted regularized extreme learning machine (VWRELM) are introduced, respectively. Then, the JITL-based triple-weighted regularized extreme learning machine (JITL-TWRELM) is described. Next, the flexibility and effectiveness of the proposed method are validated in the industrial aluminum smelting processing. Finally, we present the conclusions.
5. Conclusions
This paper mainly deals with the estimation of the liquid aluminum temperature in the regenerative aluminum smelting furnace. A JITL-TWRELM soft sensor modeling method is proposed. In this method, both the sample similarities and the variable correlations are considered in RELM to deal with the differences between samples and variables. Each modeling sample is assigned different weights according to the similarity calculation, and each dimension of the sample is also assigned a corresponding weight according to the correlation analysis, which improves the accuracy of the modeling compared with the original RELM. Furthermore, a weighted similarity measurement criterion is proposed for JITL to select similar samples for local modeling. Compared with the original JITL strategy, more similar modeling samples are selected for each query sample, enhancing the accuracy and reliability of the local modeling dataset. The flexibility and effectiveness of JITL-TWRELM were validated through the industrial aluminum smelting process. The industrial applications show that the proposed method can effectively deal with the nonlinear and time-varying problems in the regenerative aluminum smelting process and achieve a higher accuracy of temperature prediction compared with the other five methods.
For each query sample, the model needs to be updated once, although some adjacent query samples do not need to update the model so frequently. Selective updating of the model will improve the modeling efficiency. Therefore, developing a selective update strategy will be the focus of future work.