Data-Driven Batch Process Monitoring for Continuous Annealing of Cold-Rolled Strip Steel

Abstract

The continuous annealing process (CAP) is a crucial process of steel production, which has a significant impact on the uniformity and stability of mechanical properties. A novel batch monitoring process based on kernel dissimilarity (KDISSIM) and Kmeans++ is proposed in this paper, focusing on problems such as unequal sample lengths between batches and nonlinearity between variables. First, KDISSIM is used to describe the dissimilarity between batches. Secondly, Kmeans++ is employed to improve the accuracy of clustering tasks based on historical batches. The largest cluster is considered to be at a relatively stable control level, and these batches are further used as training data. Then, the center batch and boundary batch of the training set are used as the reference batch and monitoring threshold for the monitoring model, respectively. Finally, the effectiveness of the proposed method is verified via the actual CAP data, providing a feasible solution for CAP batch monitoring.

1. Introduction

China is a big steel producer and consumer in the world. As the consumption of steel increases year by year, the requirements for steel properties are also increasing. Heat treatment is a common method to improve the properties of steel. Annealing, as one of the key heat treatment methods, directly affects the final quality of steel. With the development of control technology, the continuous annealing process (CAP) is gradually being adopted in iron and steel enterprises. The process boasts advantages such as high production efficiency, a high degree of continuous production, and a short production cycle. However, it also exhibits characteristics such as nonlinearity, long time delays, and cross-coupling. As the last process of cold rolling, the CAP is a key process in steel production [1]. The microstructure of the strip steel develops a stable grain structure after heating, holding, and cooling in the continuous annealing furnace so as to improve its plasticity and performance [2]. In CAP, the strips pass sequentially through an uncoiler, temper mill, welding machine, electrolytic cell, continuous annealing furnace, shearing machine, coiler, etc., as shown in Figure 1. Although several techniques have been employed to optimize the CAP tension [3] and roller profile [4], the complexity of CAPs makes it increasingly difficult for engineers to analyze the cause of quality anomalies and propose appropriate corrective measures. Additionally, online detection of mechanical properties during the annealing process is not always easy and often involves destructive offline experiments, which cannot serve a preventive or controlling role during production. With the development of intelligent sensor technology, a large number of process parameters can be collected online. The data contain information closely related to the operation status of the production process and the final product quality. It is of great significance to establish an online process monitoring model based on data-driven algorithms, which can make full use of process data to detect abnormalities in real-time and improve the control level.

Recently, some process monitoring technologies have been successfully applied to CAPs. A nonlinear regression algorithm has been proposed to extract output-related changes to realize process monitoring of CAPs [5]. However, the quality information of CAPs is often difficult to obtain, so there are many methods that rely solely on process data for process monitoring. A multilevel process monitoring method based on Principal Component Analysis (PCA) for CAPs has been proposed [6]. However, the principal components in PCA are not really independent, although they are not correlated. Therefore, a nonlinear Independent Component Analysis (ICA) has been proposed to monitor CAPs, which combines the advantages of ICA and reproducing kernel Hilbert space [7]. In fact, the assumptions of each method about the data characteristics can affect the effectiveness of the method. For example, a novel strategy to determine the distribution characteristics of the data using an improved F-straight method has been proposed to select the most appropriate method [8]. CAPs usually have multiple operating modes due to different process systems. Considering both stable and transitional modes, a pattern recognition method based on the similarity of data features has been proposed, and different models have been developed to capture the main trends of process variables [9]. However, the working conditions of CAPs are diverse and complex, and it is difficult to ensure the accuracy of monitoring by studying only two working conditions, so a novel LSTM_GRU network has been proposed to analyze and identify the four working states of CAPs [10].

CAP monitoring belongs to the batch monitoring task of multivariable coupling. Most of the above-mentioned monitoring techniques applied in CAPs are carried out in the time dimension. However, the strip steel in a CAP can be divided into independent steel coils according to the material number, which is particular to a batch production process to a certain extent. Process data are presented in the form of three-dimensional data including a variable dimension, time dimension and batch dimension. Different batches differ in the time dimension due to different production cycles. In actual production, most batches in historical data are in a stable process, and the time series curves of some variables are shown in Figure 2. It can be seen that some batches are unstable due to data deviations from the stable values common to most batches, or large fluctuations. Therefore, it is necessary to monitor process stability via historical data of the steady-state process.

Some traditional multivariate statistical analysis methods have been applied to batch process monitoring. Typical methods include Multivariate Statistical Process Control (MSPC) [11] and its improved methods, such as Multiway Principal Component Analysis (MPCA) [12], Multiway Partial Least Squares (MPLS) [13], Multiway Kernel Principal Component Analysis (MKPCA) [14], and Multiway Partial Least Squares Autoencoder (MPLS-AE) [15]. However, there are two problems when these methods are applied in CAPs. First, the length of the time dimension in each batch varies due to product diversity. Second, it is difficult to determine a training data set in a stable state due to insufficient quality information. Dynamic time warping is proposed to implement time series with the same length [16]. In addition, a statistical process monitoring method based on variance analysis of process data, called DISSIM has been proposed, which allows direct quantification of the dissimilarity between two matrices without processing the data [17,18]. This method has several extensions, such as the Extended DISSIM (EDISSIM) for handling batch processing [19], the Sparse DISSIM (SDISSIM) for fault isolation [20], and the Kernel DISSIM (KDISSIM) method for considering nonlinear features [21].

In this paper, a novel batch process monitoring method is proposed for CAPs. The KDISSIM method is proposed to calculate the inter-batch distance, solving the problem of unequal batch lengths and nonlinear correlation of variables in similarity measurement. In addition, an improved Kmeans clustering algorithm (Kmeans++) is employed to improve the accuracy of clustering tasks based on historical batches. According to the actual production situation, the majority class is determined as a normal batch to establish a monitoring model. Finally, the constructed model is used for new batch process monitoring in the CAP, and the process control level is judged via control limits, which improve the quality management level of the CAP.

The rest of the paper is organized as follows. Real continuous annealing data from a steel plant are introduced in Section 2. The KDISSIM-Kmeans++ method is proposed for process monitoring in Section 3. The actual continuous annealing data are used for experiments and verification in Section 4. Finally, conclusions are drawn in Section 5.

2. Materials and Methods

2.1. DISSIM Improved via Kernel Trick

DISSIM belongs to a statistical process monitoring method based on dissimilarity analysis, but it only considers the linear relationship between process variables. In order to quantitatively calculate the nonlinear differences between two data sets, KDISSIM is employed in this paper to extract nonlinear features via projecting the data into a high-dimensional space. The basic principle of KDISSIM is briefly described as follows.

For two data sets ${\mathbf{X}}_1\in {\mathbb{R}}^{N_1\times J}$ and ${\mathbf{X}}_2\in {\mathbb{R}}^{N_2\times J}$, $N_1$ and $N_2$ represent the length of the time dimension in different samples, respectively, and $J$ represents the number of variables. Both ${\mathbf{X}}_1$ and ${\mathbf{X}}_2$ have been normalized.

$${\mathbf{X}}_i^{\mathit{nor}}={(\mathbf{X}}_i-{\overline{\mathbf{X}}}_{1\times J}^i)∕{\mathbf{s}}_{1\times J}^i$$

(1)

where ${\overline{\mathbf{X}}}_{1\times J}^i$ is the mean vector of ${\mathbf{X}}_i$, and ${\mathbf{s}}_{1\times J}^i$ is the standard deviation vector of ${\mathbf{X}}_i$. The mixed covariance matrix of the two data sets is denoted as ${\mathbf{R}}^{\Phi }$:

$${\mathbf{R}}^{\Phi }={\displaystyle \frac{N_1}{N_1+N_2}}{\mathbf{R}}_1^{\Phi }+{\displaystyle \frac{N_2}{N_1+N_2}}{\mathbf{R}}_2^{\Phi }$$

(2)

where ${\mathbf{R}}_i^{\Phi }={\displaystyle \frac{1}{N_i}}\tilde{\Phi }{\left({\mathbf{X}}_i^{\mathit{nor}}\right)}^{\mathrm{T}}\tilde{\Phi }\left({\mathbf{X}}_i^{\mathit{nor}}\right)$, and $\tilde{\Phi }\left({\mathbf{X}}_i^{\mathit{nor}}\right)$ are the mapped sets. ${\mathbf{R}}^{\Phi }$ is subjected to eigenvalue decomposition to obtain the orthogonal matrix ${\mathbf{P}}_0$ denoting the main projection direction, and $\mathsf{\Lambda }$ denoting a diagonal matrix with diagonal elements as eigenvalues of ${\mathbf{R}}^{\Phi }$. The original matrix $\tilde{\Phi }\left({\mathbf{X}}_i^{\mathit{nor}}\right)$ is converted to ${\mathbf{Y}}_i^{\Phi }\in {\mathbb{R}}^{N_i\times R}$.

$${\mathbf{Y}}_i^{\Phi }=\sqrt{{\displaystyle \frac{N_i}{N_1+N_2}}}\tilde{\Phi }\left({\mathbf{X}}_i^{\mathit{nor}}\right){\mathbf{P}}_0^R{{\mathsf{\Lambda }}_R}^{-1/2}$$

(3)

where R indicates that there are R directions of change retained in ${\mathbf{P}}_0$, and so does $\mathsf{\Lambda }$, which are denoted as ${\mathbf{P}}_0^R$ and ${\mathsf{\Lambda }}_R$, respectively. The covariance matrix of ${\mathbf{Y}}_i^{\Phi }$ is ${\mathbf{S}}_i^{\Phi }$.

$${\mathbf{S}}_i^{\Phi }={\displaystyle \frac{1}{N_i}}{{\mathbf{Y}}_i^{\Phi }}^{\mathrm{T}}{\mathbf{Y}}_i^{\Phi }={\displaystyle \frac{N_i}{N_1+N_2}}{{\mathsf{\Lambda }}_R}^{-1/2}{{\mathbf{P}}_0^R}^{\mathrm{T}}{\mathbf{R}}_i^{\Phi }{\mathbf{P}}_0^R{{\mathsf{\Lambda }}_R}^{-1/2}$$

(4)

To obtain ${\mathbf{P}}_0^R$ and ${\mathsf{\Lambda }}_R$ without directly calculating the eigenvalues of ${\mathbf{S}}_i^{\Phi }$, an $N\times N$ Gram matrix $\mathbf{G}=\tilde{\Phi }\left(\mathbf{X}\right){\tilde{\Phi }\left(\mathbf{X}\right)}^{\mathrm{T}}$ is defined. The kernel trick is used to avoid dot product operations in the feature space, that is $k(\mathbf{x},\mathbf{y})=〈\Phi \left(\mathbf{x}\right),\Phi \left(\mathbf{y}\right)〉$. The most commonly used kernel function is the Radial basis kernel: $k(x,y)=\exp \left(\right(-{‖x-y‖}^2)/\mathrm{c})$. The unknown ${\mathbf{P}}_0^R$ and ${\mathsf{\Lambda }}_R$ in ${\mathbf{S}}_i^{\Phi }$ are obtained via the kernel method.

$$\begin{gathered} {\mathit{P}}_0^R={\tilde{\Phi }\left(\mathit{X}\right)}^T{\widehat{\mathit{P}}}_0^R{{\widehat{\mathit{\Lambda }}}_R}^{-1/2} \\ {\mathit{\Lambda }}_R={\displaystyle \frac{{\widehat{\mathit{\Lambda }}}_R}{N_1+N_2}} \end{gathered}$$

(5)

where ${\widehat{\mathbf{P}}}_0^R$ and ${\widehat{\mathsf{\Lambda }}}_R$ are the diagonal matrix with the principal projection direction and diagonal elements as eigenvalues of G that retain the respective R principal components. The value of R is determined by the cumulative explained variance. Therefore, the eigenvalue problem of ${\mathbf{R}}^{\Phi }$ is transformed into the eigenvalue decomposition problem of the Gram index. We then substitute Equation (3) into Equation (5):

$${\mathbf{S}}_i^{\Phi }={{\widehat{\mathsf{\Lambda }}}_R}^{-1}{{\widehat{\mathbf{P}}}_0^R}^{\mathrm{T}}\left[\begin{array}{cc}{K}^{1i}{K}^{i1}& K^{1i}{K}^{i2}\\ K^{2i}{K}^{i1}& K^{2i}{K}^{i2}\end{array}\right]{\widehat{\mathbf{P}}}_0^R{{\widehat{\mathsf{\Lambda }}}_R}^{-1}$$

(6)

where ${{\rm K}}^{ij}=k\left({\mathbf{X}}_i, {\mathbf{X}}_j\right)$, and then the eigenvalue decomposition of ${\mathbf{S}}_i^{\Phi }$ is used to obtain the eigenvalues ${\lambda }_i^j$ and ${\omega }_i^j\left(j=1,2,\dots ,R\right)$. We define a monitoring indicator $D^{\Phi }$:

$$D^{\Phi }={\mathit{diss}}^{\Phi }({\mathbf{X}}_1,{\mathbf{X}}_2)=\mathit{diss}\left(\tilde{\Phi }\right({\mathbf{X}}_1),(\tilde{\Phi }\left({\mathbf{X}}_2\right))={\displaystyle \frac{4}{R}}\sum _{j=1}^R{({\lambda }_j-0.5)}^2$$

(7)

When the two data sets are very similar to each other, the eigenvalue ${\lambda }_j$ is close to 0.5, so $D^{\Phi }$ is close to 0. When the two data sets are not similar to each other, $D^{\Phi }$ will be close to 1.

2.2. Clustering Based on Kmeans++

The Kmeans algorithm is widely used in industry and has been proven to have relatively good results. However, the randomness of initial clustering center selection may not only cause the clustering results to vary greatly from the data distribution, but also reduce the convergence speed of the algorithm. The Kmeans++ algorithm, which optimizes the Kmeans method to randomly select the initial cluster centers, has been proposed to address this problem [22]. The specific steps of the Kmeans++ algorithm are shown in Figure 3.

The number of clusters k in the Kmeans++ algorithm needs to be determined artificially. The clustering results can be evaluated by the silhouette coefficient, and the silhouette coefficient of the ith sample is defined as shown in Equation (8):

$$s_i={\displaystyle \frac{b_i-a_i}{\max (a_i,b_i)}}$$

(8)

where $a_i$ is the average distance between the ith sample and other samples in the same cluster, i.e., the intra-cluster dissimilarity, and the inter-cluster dissimilarity $b_i=\min (b_{i1},b_{i2},\cdots ,b_{\mathrm{ij}},\cdots ,b_{\mathit{ik}})$, $b_{\mathit{ij}}$ represents the average distance between the ith sample and all samples of the cluster $c_j$, and the mean value of s_i is the silhouette coefficient of the overall clustering result.

$$s_c={\displaystyle \frac{\sum _{i=1}^n{s}_i}{n}}$$

(9)

where n is the number of clusters. The closer s_c is to 1, the more reasonable the clustering result is.

It is important to select the best value of k to improve the silhouette coefficient, but when the Kmeans++ is used for CAP monitoring, the actual conditions of different process modes should also be considered to determine the value of k to achieve the best clustering effect.

2.3. Batch Process Monitoring Procedure

The KDISSIM-Kmeans++ method is used to construct the reference space by clustering the stable controlled batches in a large amount of historical data. Then the central batch and multiple boundary batches representing the whole reference space are selected as the reference batches to build a monitoring model.

2.3.1. Central Batch Selection

Assuming a total of n batches of data, the KDISSIM index from each batch to all other batches is calculated and noted as ${\mathit{kd}}_{\mathit{ij}}(i,j=1,2,\dots ,n)$, obtaining a distance matrix D of order $n\times n$.

$${kd}_i=\mathrm{sum}\left({kd}_{i1},{kd}_{i2},\cdots ,{kd}_{ij}\right)$$

(10)

$$k{d}_m=min\left\{k{d}_1, k{d}_2, \cdots , k{d}_i\right\}$$

(11)

$$\mathbf{D}=\left[\begin{array}{ccc}{kd}_{11}& \cdots & {kd}_{1j}\\ ⋮& \ddots & ⋮\\ {kd}_{i1}& \cdots & {kd}_{\mathit{ij}}\end{array}\right]$$

(12)

where the diagonal element of matrix D is zero. The sum of KDISSIM indicators kd_i between each batch and the other batches is calculated. A smaller kd_i value indicates a smaller relative distance between the batch and all other batches. Therefore, the batch with the minimum sum of indicators is the mth batch named ${\mathbf{X}}_{J\times N_{\mathit{centre}}}^{\mathit{centre}}$, defined as the central batch of the reference space.

2.3.2. Boundary Batch Selection

In order to obtain the control limits for batch process monitoring, it is necessary to find the boundary batches representing the boundary of the whole training set. The boundary samples distributed on the boundary of the reference space are the furthest batches from the central batch. Therefore, the kd values between the boundary batches and the central batch are larger than those between other batches from the central batch. The probability density values of all batches from the center batch are estimated via the kernel density estimation method [23], which ensures that the boundary batches are representative and universal.

Kernel density estimation is a commonly used method to describe the distribution of data. Assuming that $x_1, x_2, \dots , x_n$ satisfies the uniform distribution requirement, its density function f(x) is unknown, and the f(x) needs to be estimated from the known data samples. The empirical distribution function of the data samples is F(x).

$$F\left(x\right)={\displaystyle \frac{1}{n}}\left\{x_1,x_2,\dots ,x_n\right\}$$

(13)

The estimated density function can be obtained as $f_h\left(x\right)$.

$$f_h\left(x\right)={\displaystyle \frac{\left[F_n\left(x+h_n\right)-F_n\left(x-h_n\right)\right]}{2h}}={\int }_{x-h_n}^{x+h_n}{\displaystyle \frac{1}{h}}K\left(\frac{t-x}{h_n}\right)\mathrm{d}{F}_n\left(t\right)={\displaystyle \frac{1}{n{h}_n}}\sum _{i=1}^n\left({\displaystyle \frac{x-x_i}{h_n}}\right)$$

(14)

where $h_n$ is the bandwidth to maximize the likelihood estimate after cross-validation, and the value $K(\ast )$ denotes the kernel function, which is usually taken as a symmetric singlet probability density function. With a given confidence level $\delta$, the corresponding control limits $X_{\mathit{hm}}$ can be solved by the integral formula of the probability density function.

$${\int }_{-\infty }^{X_{\mathit{hm}}}{f}_h\left(x_i\right)\mathrm{d}\left(x_i\right)=\delta$$

(15)

The $\delta =90\%$ of control limit kd_limit is obtained according to Equations (13)–(15).

The specific steps for batch process monitoring based on KDISSIM-Kmeans++ are shown in Figure 4 and are divided into two steps: offline training and online monitoring.

Offline training process

(1)

The historical batch data X_I×J×K are used as the input to the model, where I, J and K denote the number of the batch, variable and sample, respectively.

(2)

Normalize the X_I×J×K according to Equation (1).

(3)

Calculate the distance between each batch and other batches to obtain the distance matrix ${\mathit{kd}}_{\mathit{ij}}(I\times I)$, as in Equation (7).

(4)

Cluster ${\mathit{kd}}_{\mathit{ij}}$ with the Kmeans++ method, and then the silhouette coefficient calculated from Equations (8) and (9) is combined with the actual situation to choose the reference batches from the largest cluster.

(5)

Determine the central batch ${\mathbf{X}}_{J\times N_{\mathit{centre}}}^{\mathit{centre}}$ of reference batches using Equations (10)–(12).

(6)

Estimate the probability density function of the distance from the reference batch to the central batch with the kernel density estimation method using Equations (13)–(15). The 90% control limit kd_limit is determined by integration as threshold.

Online evaluation process

(1)

Obtain new batch data ${\mathbf{X}}_{J\times N_{\mathit{new}}}^{\mathit{new}}$.

(2)

Normalize the ${\mathbf{X}}_{J\times N_{\mathit{new}}}^{\mathit{new}}$ as in Equation (1).

(3)

Calculate the ${\mathit{kd}}_{\mathit{centre}}^{\mathit{new}}$ between the new batch ${\mathbf{X}}_{J\times N_{\mathit{new}}}^{\mathit{new}}$ and the central batch ${\mathbf{X}}_{J\times N_{\mathit{centre}}}^{\mathit{centre}}$ using Equation (7).

(4)

Compare the ${\mathit{kd}}_{\mathit{centre}}^{\mathit{new}}$ with the control limit kd_limit. If it exceeds the control limit, the batch is considered as a batch with an unstable control level; otherwise, the batch is considered as a batch with a stable control level.

3. Results

3.1. Data Preparation for the CAP

During the continuous annealing, the strip is preheated and heated to the required process temperature and then cooled after a certain period of time. The traditional continuous annealing furnace includes eight heat treatment stages, including the preheating section (PHS), heating section (HS), soaking section (SS), slow cooling section (SCS), rapid cooling section (RCS), overaging section 1 (OAS1), overaging section 2 (OAS2), and final cooling section (FCT), as shown in Figure 5. During the annealing process, the internal stress of the strip is eliminated, the plasticity is enhanced, and the quality of the strip is further improved.

The quality of the strip in the CAP is determined by the eight annealing stages, so the temperatures of each annealing stage have closely coupled relationships. The eight annealing temperatures shown in

Table 1. Annealing process parameters for batch monitoring.

No.	Parameter Name	Parameter Description
1	PHT	Preheating temperature
2	HT	Heating temperature
3	ST	Soaking temperature
4	SCT	Slow cooling temperature
5	RCT	Rapid cooling temperature
6	OAT1	Overaging temperature 1
7	OAT2	Overaging temperature 2
8	FCT	Final cooling temperature

are taken as the research objects of this paper, and the annealing temperature curve of each batch is collected at a certain frequency. The strip passes through each furnace section in sequence, resulting in different annealing temperatures of the same strip section not being aligned in the time dimension. Additionally, changes in the running speed of the steel strip may lead to differences in the lengths of the annealing temperature curves for the same steel coil. In order to obtain equal length data for the strip in each furnace section, the annealing temperature curve needs to be converted from the time dimension to the length dimension. The length is expressed as the integral of the velocity in time, that is, $L=\int Vt$, where $t$ is the sampling interval time and $L$ is the length of the sampling interval. This converts the time interval to the length interval between two samples. When the length of the same section of strip is fixed, the same length of data can be obtained in different furnace sections. The conversion process is shown in Figure 6. ${∆t}_i=t_i^{\prime }-t_i$ represents the time taken for a strip to pass through the ith furnace section, where $t_i^{\prime }$ represents the moment when the strip leaves the ith section, $t_i$ represents the moment when the strip enters the ith section, and $∆l=l^{\prime }-l$ indicates the length of a strip, where $l^{\prime }$ represents the end position of the strip, and $l$ represents the head position of the strip.

The preprocessed continuous annealing data form a third-order tensor data set containing the batch dimension, variable dimension and length dimension. However, the data have problems such as unequal lengths between batches along the length dimension and nonlinear coupling along the variable dimension. Additionally, the lack of quality information hinders building a model via normal batch data, which brings challenges to process monitoring.

3.2. Batch Monitoring Results

The proposed monitoring method is applied to a CAP for strip steel in this section. A total of 562 batches of varying lengths were collected on-site, with batch lengths ranging from 27 to 144, and each batch containing eight annealing temperatures from temp1 to temp8. All the batches of steel strips were produced under the same annealing regime, allowing process monitoring to be conducted via a unified model. The 531 batches were divided into one cluster by the proposed KDISSIM-Kmeans clustering algorithm and were considered to be normal batches under stable control. Among them, 431 batches were used as training sets to construct the reference space, and the remaining 100 batches and 31 abnormal batches with unstable control constituted the testing set for verification. A radial basis kernel, $k(x,y)=\mathrm{e}\mathrm{x}\mathrm{p}\left(\right(-{‖x-y‖}^2)/c)$, was used, with c empirically chosen as 10m, where m is the number of process variables, that is eight [24]. The final testing results are shown in Figure 7.

The false alarm rate (FAR) and fault detection rate (FDR) are defined as follows for a more intuitive comparison.

$$\begin{gathered} FAR=NN/TNN \\ FDR=AN/TAN \end{gathered}$$

(16)

where NN is the number of normal batches judged as faulty batches, and TNN is the total number of normal batches. AN represents the number of abnormal batches judged as abnormal batches, and TAN represents the total number of abnormal batches. The FDR and FAR of the testing set were 84% and 8% respectively, which were calculated via the monitoring results in Figure 7, illustrating that the proposed batch process monitoring method based on KDISSIM-Kmeans++ can achieve accurate identification of unstable control batches based solely on historical batch process data.

To further validate the conclusions in terms of each variable, the trends of the testing batches with the largest and the smallest KDISSIM metrics are compared with the central batch. The sample size of each batch is different, with the minimum batch sample size of the indicator being 55, the maximum batch sample size of the indicator being 109, and the central batch sample size being 37. Therefore, the sample sizes of the eight variables are unified to 109 using the interpolation method for clear observation, and the results are shown in Figure 8.

In each heat treatment stage of continuous annealing, the batch with the minimum kd from the central batch is more similar than the batch with the maximum kd, both in terms of data pattern and absolute distance. From the perspective of temperature control in the CAP, this indicates that the control levels of batches with smaller kd values are closer to the majority of batches under stable control.

4. Discussion

This study aims to investigate the issue of batch monitoring in the continuous annealing process and proposes a novel KDISSIM-Kmeans++ method. The experimental results show that 26 out of 131 test batches are determined to have process anomalies, corresponding to an anomaly detection rate of 84%. A visual analysis of the abnormal batches in Figure 8 reveals significant deviations in the annealing parameters, including preheating section temperature, slow cooling section temperature, overaging section 1 temperature, and final cooling section temperature. In actual production, the in-furnace time should be appropriately increased to ensure that the steel strip reaches the desired temperature in the preheating section. Additionally, it is recommended to adjust the atmosphere temperature in each section of the annealing furnace to keep the slow cooling section temperature, overaging section temperature 1, and final cooling temperature within appropriate ranges. Finally, the running speed of the strip should be kept stable to reduce the temperature fluctuation within the strip batch in the heating section and the soaking section, and thereby achieve the goal of improving the quality stability of the continuous annealing product.

5. Conclusions

A batch process monitoring method combining multivariate statistics and clustering, called KDISSIM-Kmeans++, has been proposed in this paper, aiming to identify unsteady strip steel in CAPs. The main contributions are as follows:

(1)

A space–time transformation method based on velocity and time integration is proposed so as to solve the problem of aligning multivariate time series data caused by different annealing temperature measurement locations in the CAP.

(2)

The KDISSIM algorithm is proposed to extract the nonlinear features of multiple annealing temperatures and further measure the similarity between steel coils with different lengths.

(3)

The improved clustering model is employed to find batches with stable temperature control in historical data, and these batches are used as the training set to construct a reference space, monitoring the temperature control levels of different batches of steel coils in the CAP.

The proposed KDISSIM-Kmeans++ algorithm represents a new attempt for batch process monitoring of strip steel in CAPs. It can fully utilize historical production data to perform real-time monitoring of the complex annealing process in the presence of quality information, providing data support to ensure the process stability and quality evaluation of steel strips. However, the selection of kernel parameters in the KDISSIM algorithm can affect the inter-batch dissimilarity and the clustering results, leading to a decrease in monitoring accuracy, so further research is worthwhile.