Effect of strip profile of hot-rolled silicon steel on transverse thickness difference of cold-rolled strip

Abstract

The cross-sectional profile of hot-rolled silicon steel strips has a great impact on the transverse thickness difference of cold-rolled strips. The genetic algorithm - Levenberg Marquardt - backpropagation neural network model was developed for the analysis of heredity effect of the cross-sectional profile of hot-rolled strip. The feature importance was analysed based on random forest and extreme gradient boosting methods. Furthermore, the influence law was analysed by data visualization. The coupling term of the strip crown and thickness , the edge drop and the coupling term of the strip wedge and thickness are recommended as key indexes of the hot-rolled silicon steel strip. As the transverse thickness difference must be controlled below 7 μm, the corresponding , , should be controlled no more than 60, between -10∼10 μm and no more than 7.5 , respectively. Meanwhile, the requirement for the cross-sectional profile indexes is synchronous.

Introduction

Cold-rolled silicon steels are mainly used to manufacture generators, motors and transformers. The cold-rolled silicon steels are punched into pieces, then stacked and pressed, and finally made into the rotors and stators of generators, motors and transformer cores. To improve efficiency and reduce iron loss, the number of silicon steel laminations is increased. Therefore, the requirements for the uniform thickness of silicon steel strips are very high, which presents higher requirements for the cross-sectional profile of silicon steel strips [1].

The transverse thickness difference (TTD) of cold-rolled silicon steel, , is calculated by subtracting the minimum thickness of the cross-sectional steel strip profile, from the maximum thickness, , namely, . The requirement for the shape of silicon steel strip decides to reduce the TTD as much as possible. Techniques to control the cross-sectional profile of silicon steel strips involved mainly two stages: the hot rolling process and the cold rolling process. The crown, wedge and edge drop of the hot-rolled silicon steel strip (HRSS) are the main factors affecting the TTD of the cold-rolled silicon steel strip (CRSS). At present, advanced cross-sectional profile control technologies in hot rolling, such as asymmetry self-compensation work rolls [2], varying contact backup rolls [3], rectangular section control technology [4] and large concave roll technology [1], have reduced the edge drop, central crown and wedge of the hot rolled strips by roll contour configurations, roll shifting strategies, uniform wear of rolls, roll contour retention, etc. The control techniques of the TTD in cold rolling mainly reduced the edge drop of the steel strip in the case of good flatness. The most representative is Kawasaki's work roll shifting (K-WRS) technology [5]. It controls the edge drop of the steel strip by moving the taper work roll and usually needs to be equipped with crown metres and edge drop metres at the inlet and outlet of a tandem UCMW (universal crown mill with middle roll and work roll shifting), respectively, to realize automatic closed-loop control of the edge drop [6]. Compared to the advanced UCMW tandem mill, the UCM (universal crown mill) rolling mill, especially the single-stand rolling mill, has insufficient ability to control the edge drop of silicon steel strip. The taper roll [7] and the variable edge crown roll [8] have been widely reported to perform the control of the edge drop of silicon steel strips. However, whether it is the advanced UCMW mill or the UCM mill, the control for both the edge drop and the flatness of CRSSs have coupling characteristics, for example, the edge drop control deteriorates the flatness of the silicon steel strip [9,10]. In particular, the greater the crown and the edge drop of the incoming HRSS is, the more obvious the control coupling phenomenon in the single-stand UCM mill would be [11], as shown in Figure 1. Bilateral symmetrical quarter buckles easily occur in the edge drop control. Furthermore, the control of edge drop of cold-rolled silicon steel easily causes edge cracks and strip breakages of high-grade silicon steel strips, as shown in Figure 2.

Figure 1.

Edge drop and quarter buckle of silicon steel.

Figure 2.

Edge crack of silicon steel strip.

As mentioned above, the control domain for the TTD of CRSSs is much smaller than that of hot rolling due to flatness defects and edge cracks. The cross-sectional profile of HRSS is key to reducing TTD [4,12]. The hot-rolled cross-sectional profile has an obvious impact on the TTD of CRSSs. The requirements for the cross-sectional profile of HRSSs are higher than those for conventional hot-rolled steel strips. The incoming slabs of a single-stand reversible cold rolling mill of silicon steel are produced by the 1580 mm hot rolling production line. Due to the small output demand for silicon steel, the 1580 mm hot rolling production line conducts silicon steel production in the interval of producing ordinary carbon steel, alloy steel and other steel grades. Generally, the cross-sectional profile requirements of other hot-rolled steel grades are relatively low, and the crown index is usually taken as the control target while taking into account the wedge requirements. The HRSS also takes the crown index as the shape target, but the target value reduces according to the strict requirements for the strip shape. However, under the condition of certain equipment capacities, the control ability for the shape of HRSSs is affected by multiple factors. It is impractical to blindly reduce the control indexes of the silicon steel cross-sectional profile. Inaccurately setting the process parameters will easily cause fluctuations of the self-learning parameters, disturbing the control of the flatness of HRSSs. In turn, it will worsen some cross-sectional profile indexes. Unreasonable indexes may also bring challenges to stable rolling, cause confusion in the rolling scheduling, reduce rolling mileage, and increase unnecessary roll consumption.

How to reasonably set the shape control targets of HRSSs is very important to reduce the TTD of cold-rolled silicon steel strip. It is necessary to deeply explore the heredity effect of the key indexes of the HRSS cross-sectional profile on the TTD of the CRSS to fine-tune the cross-sectional profile index system of HRSSs. The prediction and configuration of plates or strip crowns in hot rolling processes have always been popular in the research of plate and strip profiles [12]. The transcription rate and the heredity coefficient are well used in the research of the heredity from crown to flatness in rolling hot strips [13]. An approximately equal proportional crown [14,15] is a famous heredity principle of the strip profile in hot rolling and is widely applied to set the strip crown at each outlet of the stand in tandem to maintain a good shape. However, with the unceasing rise in the requirements for shape quality, the heredity principle of an approximately equal proportional crown has increasingly highlighted its limitations, and some revised models have been developed. The proportional crown distribution model was improved considering the transverse flow coefficient of the HRSS and raised the shape control effect [16]. The effect function method was developed to calculate the crown effect rate of HRSSs and to analyse the coupling influence of the initial crown with other factors. The initial rate of the crown effect varied with strip width in the type of parabolic curve [17]. With the progress of computing technology, the high-precision prediction of plate crowns has gradually become a popular research topic, which has weakened the research on the heredity effect of cross-sectional profiles in rolling processes. An aspect of research is to establish mechanism models that meet the boundary constraints and predict the strip profile based on the process parameters [18,19]. To reflect the initial crown heredity effect of hot rolling slabs on the strip shape at the outlet of the finished stand, the outlet strip crowns of each hot tandem rolling were calculated based on the influence function method [20]. The result show that the heredity effect of the incoming slab crown of HRSSs that reaches the finished product can be ignored. Tieu et al. analysed the heredity of the hot-rolled coil ridge profile with respect to the ridge-buckle defects of thin cold-rolled strips based on a numerical model [21]. Because rolling plastic deformation belongs to the problem of high coupling, nonlinearity, and large deformation, the conventional mechanism model often has difficulty accurately describing the essence and relationship of cross-section profile heredity.

In recent years, machine learning algorithms have developed rapidly in the prediction of strip crowns and the heredity effect of cross-sectional profiles. Hüseyin Altinkaya et al. employed an artificial neural network to obtain the optimal parameters for a different type of rail-rolling process [22]. Deep neural network (DNN) models for the prediction of the strip crown in hot rolling showed the highest prediction accuracy compared to the artificial neural network (ANN) and optimized ANN using the non-dominated sorting genetic algorithm [23]. Ji et al. [24] proposed an optimized support vector machine (SVM) model for the prediction of strip crowns. Sun et al. [25] established a prediction model for HRSS crowns based on the random forest (RF) method, and it showed stable performance compared to the SVM and regression tree (RT) methods. Wang et al. [26] proposed hybrid forecasting models for strip crown and flatness by combining the genetic algorithm (GA), mind evolutionary algorithm (MEA), principal component analysis (PCA), and multilayer perceptron (MLP) neural networks. To analyse the influence weight of multidimensional input variables, a crown prediction model of HRSSs based on neural networks was proposed [27], and the importance of input features was analysed using sensitivity methods. The Tchaban algorithm for sensitivity analysis was integrated with a backpropagation neural network (BP) to establish the T-GA-BP neural network to predict strip thickness [28], providing a reference for the neural network application research.

Rather than substituting the cross-sectional profile indexes of HRSSs as variables to analyse the influence of incoming HRSS crowns on the subsequent CRSS shape, the above machine learning algorithms are based on hot rolling specifications, equipment parameters, forces and temperatures to predict the HRSS crowns and analyse the influence of hot rolling process parameters on the HRSS flatness. Silicon steel strips require high-quality control of the cross-sectional profile. The typical cross-sectional profile of a silicon steel strip is shown in Figure 3. There are certain correlations, synergies, and nonlinear relationships between the strip cross-sectional profile indexes. In particular, the shape control techniques of silicon steel strips have evolved from the control of crown and flatness to that of edge drop and other comprehensive indexes [13]. The nonlinearity of the heredity effect of the hot-rolled silicon steel cross-sectional profile is more prominent. It is of great significance to reveal the mechanism of influence of the hot-rolled silicon steel cross-sectional profile indexes on the TTD of CRSSs.

Figure 3.

Schematic diagram of cross-sectional profile of hot-rolled silicon steel strip.

To fully study the effect of the heredity of HRSS cross-sectional profiles on the TTD of CRSS, Li et al. [29] analysed the information of hot and cold silicon steel coils based on statistical methods and pointed out that the large crown and wedge of HRSSs would lead to serious edge drop of CRSS. Zhang et al. [30] proposed the evaluation method of pass heredity coefficients, established the matrix relationship between the TTD and the shape control means, and developed a formula for the ratio of the TTD to the comprehensive crown of HRSSs. In the early stage of our investigation, we used the method of numerical simulation and statistical principles to study the profile heredity law of hot and cold-rolled silicon steel strips and presented the goals for finely controlling the parameters of the hot rolling process [31].

Compared with the conventional mechanism model, machine learning methods are powerful tools for describing the essence of nonlinear heredity. Currently, there are few reports on machine learning methods for the profile heredity law of hot and cold rolled steel strips, especially silicon steel strips. A single-layer BP neural network was established to predict the TTD of CRSSs according to the cross-sectional profile indexes of HRSSs, and the ridge regression method was used to analyse the influencing factors [32]. Previous research showed that the profile heredity effect of hot and cold rolled silicon steel strips was non-linear, and ridge regression, as a linear regression method, had a natural defect in dealing with the non-linear heredity effect of strip profiles.

To fully understand the highly nonlinear heredity effect of HRSS cross-sectional profiles on the TTD of CRSSs and to establish optimal and scientific control objectives of HRSSs, BP neural networks and genetic algorithms were used to establish a prediction model to fit the TTD according to the cross-sectional profile indexes of HRSSs, and the importance of the feature was discussed by means of sensitivity analysis and verified with grouping prediction of the neural network. Then, the influence law of the HRSS cross-sectional profile indexes on TTD was analysed through data visualization. Finally, reference was given to the control standard of the HRSS profile.

Description of silicon steel strip profile and analysis of data characteristics

Cross-sectional profile indexes of hot-rolled silicon steel profile

The cross-sectional profile indexes of HRSSs mainly include crown, wedge, and edge drop among others, as shown in Figure 3. The crown represents the difference in thickness between the middle and the two sides of the strip. Wedge and edge drop are the differences in thickness of the two sides of the strip and the two positions near one side of the strip edge, respectively. The symbols , and are the crowns at positions of 100, 40, and 25 mm from the strip edge, respectively. , and are the wedges at positions of 100, 40, and 25 mm from the strip edge, respectively. , and represent the edge drops between positions of 100 mm and 25 mm, 100 mm and 40 mm, and 40 mm and 25 mm from the strip edge, respectively.

where is the thickness of the centre of the strip. , , and are the thicknesses at positions of 100, 40, and 25 mm from the strip edge at the operating side (OS), respectively. , , and are the thicknesses at positions of 100, 40 and 25 mm from the strip edge at the driving side (DS), respectively.

According to the definition of crown and edge drop, the following relationship can be obtained:

Analysis of data characteristics

The data used in the neural network modelling are the measured cross-sectional profile indexes with a total of 3176 groups of hot and cold rolled silicon steel strips from a plant. The objective-dependent variable is the TTD, , of the CRSS. The data tracked in this study were obtained from a single stand reversible 6-high cold rolling mill. The cold rolling production line adopts the sampling method to measure the TTD of CRSS discretely, and the profilometry accuracy is 1 μm. The samples were selected from the dividing positions in the middle of the strip coils in the finishing process to avoid abnormal shape defects at the head and tail of the strip. Most of the data groups were collected from different strip coils. The 6-high single-stand reversing mill was only used for silicon steel rolling, the process is relatively mature, and the parameters almost have little change. That is to say, the settings for shifting positions and bending forces of both work rolls and intermediate rolls are almost unchanged, and the actual process parameters have a small fluctuation. Meanwhile, all TTDs were obtained by sampling under the condition that all roll contours remain unchanged.

In fact, it is almost impossible to point-to-point correspond the cold rolling process data, the cross-sectional profile indexes of HRSSs, and the TTD of the sampling point of CRSSs, e.g. the real-time wear of the rolls can hardly be recorded and correspond to the TTD of sampling position. Therefore, in the situation of small fluctuation range of process parameters and large number of statistical data, it is assumed that the influence of cold rolling on the TTD basically remains at a certain level.

This study only takes the cross-sectional profile indexes of HRSSs as input variable to study the influence on the TTD of CRSSs. As shown in Table 1, the neural network inputs have 26 feature variables, which are mainly divided into six types that include steel grade, specification, crown, wedge, edge drop, as well as the coupling and high-order variables of the crown, wedge, and edge drop. Except for the steel grade,

, is artificially numbered by 1–12, while the other five types of data are originally collected data or reprocessed data. The strip specification mainly includes the width

, and thickness

, of the HRSS; the thickness

, and width

of the CRSS; the thickness ratio of the hot and cold rolled strip

; and the crowns

, the wedges

and edge drops

of the HRSS are the initial features. According to the results of previous research [31], there was mutual influence between the cross-sectional profile indexes of HRSS. To explore the coupling influence of the crown and wedge with other indexes on the TTD, second-order variables of features and other derived features are also used as inputs, include the coupling terms of crown and wedge

, the coupling terms of crown and crown

the coupling terms of wedge and wedge

, the coupling terms of crown and thickness

, the coupling terms of wedge and thickness

. Specific input features are shown in Table 1. The crowns

present obviously nonlinear distributions due to the nonlinear edge drop as shown in Figure 3, so do the wedge. Thus, the indexes derived from the above indexes are also nonlinear. The TTD sample sizes of the collected data groups are shown in Table 2.

Table 1.

Input features.

Features	Grade	Specifications	Crown	Wedge	Edge drop	Coupling terms and high-order variables
Symbols	(1∼12)

Table 2.

Sample size of TTD of collected data groups.

TTD/μm	Sample size	TTD/μm	Sample size
1	12	10	351
2	13	11	190
3	350	12	97
4	17	13	59
5	109	14	17
6	248	15	11
7	555	16	9
8	557	17	4
9	575	18	2

Modelling of neural networks

Data pre-processing

The following data pre-processing and neural network modelling programming were performed in Matlab and Python. The feature data are divided into four categories: normal distribution, gamma distribution, exponential distribution, and no obvious regular distribution. (1)

For feature data without an obvious statistical distribution, the box chart is used to clean abnormal data. It uses the quartile to describe the discrete distribution of the data in the box chart. The description range is limited by Equation (1), and data outside the range are abnormal values. (1) where , and are the lower quartile, upper quartile, and interquartile distance, respectively.

The statistical features that are relatively concentrated or without obvious distribution are the grade, _, the thickness and width of HRSSs, and , the thickness ratio of the hot and cold rolled strip, , the thickness and width of CRSSs and , respectively. The data groups containing abnormal data are formulated according to Equation (1).

(2)

The statistical features that approximate the normal distribution include the HRSS crown , , , and , as shown in Figure 4(a–e). According to the independent additivity of the normal distribution, the edge drop , , and also follow the normal distribution.

Figure 4.

Features approximately obeying normal distribution.

The abnormal data with a normal distribution are formulated according to the principle of Equation (2). That is, outliers are identified as low probability events. (2) where and are the average value and standard deviation, respectively. (3)

The feature data approximately obeying the gamma distribution include the second-order variables of the HRSS crown, , , and the coupling term of the HRSS crown . The first three are shown in Figure 5. The feature data approximately obeying an exponential distribution include the HRSS wedge , and , the second-order variables of the HRSS wedge and crown , , , the coupling terms of the HRSS wedge and thickness and . Some of them are shown in Figure 6.

Figure 5.

Features approximately obeying gamma distribution.

Figure 6.

Features approximately obeying exponential distribution.

For data approximately obey these two distributions, the data within 99.7% of the cumulative density will be retained, and the other portion would be labelled as abnormal data. That is, the data is greater than calculated at the value of , as shown in Equation (3) will be deleted. (3)

where , and are known estimated parameters.

Finally, 2929 data groups were obtained after the abnormal data was deleted.

Normalization of data

As shown in Table 3, the many features have a wide range of values, large averages and variances, such as HRSS width

, thickness

, crown

and

. In addition, the numerical dimensions are different, such as mm, mm² and μm. When these values with large differences are directly input into the neural network, they will cause large oscillations in the training and fitting of the neural network, with the weights obviously changing. Therefore, to eliminate the influence of the dimensional difference and the wide value ranges of the variables, it is necessary to preprocess the data. The maximum and minimum values are adopted to normalize all kinds of data and convert them to the interval of 0∼1, as shown in Equation (4). It not only reduces the quantity of calculations but also improves the efficiency of the neural network operation. (4)

where

and

are the values before and after conversion, respectively.

Table 3.

Features of partial features.

Dimension	Average	Variance
mm	1250.33	5.28
mm	2.36	0.15
μm	37.74	8.09
μm²	1490	327.57

Modelling of neural networks

The BP neural network is a feedforward neural network that uses the BP algorithm to train multilayer networks. In practical applications, BP neural networks are widely used in classification, regression prediction, function approximation, etc. Therefore, in this paper, a BP neural network and appropriate computing methods are used for optimization.

Activation function and loss function

Among them, the activation function is used to complete the non-linear transformation of the data. Commonly used activation functions include , , , etc. The expressions of the specific activation function are shown in Equation (5). The function is used as the activation function of the hidden layer. Generally, the output layer is set with a linear activation function . (5)

The loss function is used mainly to judge and update the threshold and weight. And the loss function of the regression problem generally selects the mean square error (MSE) function. Here, the mean square error function is selected as the loss function. (6) where is the sample size, is the predicted value and is the real value.

Number of neurons and number of neuron layers

The number of neurons has a great influence on the fit of the neural network. If the number is too small, the model is simple, and the fitting effect will be poor; if the number is too large, the model is complex and difficult to fit, resulting in an excessively long fitting time. However, to date, there is no definite formula for the number of neurons in the hidden layer. The number of neurons in the BP neural network can be determined by an empirical formula, as shown in Equation (7): (7) where is the number of neurons in the input layer , is the number of neurons in the output layer , is a constant between 0 and 10, and is the number of neurons in the hidden layer, with .

Based on the two empirical formulas, the number of neurons in the hidden layer is adjusted by multiple experimental comparisons. When increasing the number of neurons has little effect on the fitting effect, it is necessary to consider increasing the number of layers of the neural network to improve the fitting of the neural network. Table 4 shows the MSE of the different numbers of neurons in the BP neural network with a hidden layer structure.

Table 4.

MSE performance of BP neural network under different numbers of neurons (test set).

Neuron number	5	8	11	13	15	18	20	25	30	35
MSE/	34.1	39.1	35.6	27.4	36.6	42.3	33.2	32.8	30.5	33.4

The number of layers of neurons plays an important role in the non-linear fitting of the neural network. In existing research, there is no better method to determine the number of neuron layers, which can only be determined under actual debugging. The MSE of the three hidden layers is relatively small as shown in Table 5, showing good performance. The layer structure of the BP neural network obtained after final debugging is shown in Figure 7. The network structure is 26-13-18-25-1; that is, the network structure includes 26 neurons in the input layer, 13 neurons, 18 neurons, and 25 neurons in each hidden layer, and 1 neuron in the output layer.

Figure 7.

BP network structure.

Table 5.

MSE corresponding to different numbers of hidden layers.

Number of hidden layers	1	2	3	4
MSE /	27.4	20.6	15.3	18.3

Training method

The training function is used mainly to update and understand weight and bias in neural networks. The limited-memory Broyden Fletcher Goldfarb Shanno (limited-memory -BFGS) and Levenberg-Marquardt (LM) algorithms are typical second-order optimization algorithms. The loss values, iterations, and running time of LM methods for the BP neural network in this project are shown in Table 6.

Table 6.

Results of optimization methods for BP neural network.

Training method	Training set loss	Verification set loss	Iterations
LM	0.0197	0.0225	24

In this paper, the LM optimization algorithm is selected to optimize the neural network. The process is shown in Figure 8.

Figure 8.

LM algorithm flow chart for TTD prediction.

Optimization method

The initial weight and threshold of the BP neural network are generally set by directly assigning random numbers. Initial weight and threshold play a very important role in the training and final fitting of the neural network. The genetic algorithm (GA) is one of the algorithms used to solve optimization problems. In this paper, a GA is used in the neural network to optimize the initial weight and threshold. The process of BP neural network optimization by GA is shown in Figure 9.

Figure 9.

Flow chart of GA-BP algorithm for prediction of TTD.

Results and discussions

Performance of GA-LM-BP neural network

After correcting the abnormal data, the cross-validation method is used for data grouping; that is, it is randomly divided into a training set, verification set and test set according to the ratio of 8:1:1. A total of 2343 groups of data are used for neural network training, 293 groups of data are used to verify the neural network to avoid overfitting, and another 293 groups of data are used to predict and verify the generalization ability of the models.

The regression results after training are shown in Figure 10. The abscissa is the actual value, the ordinate is the predicted value, the solid line is the fitting line between the target value and the predicted value, and the fitting line equation is listed on the lower right side of the figure. Figure 10(a,b) are the regression results of the training set, and the test set, respectively. The corresponding fitting line equations are and , respectively. The corresponding Pearson's correlation coefficients are 0.811 and 0.817, respectively. According to the reliability interval method [30], the error interval of the TTD, that is, the difference in the interval between the predicted TTD and the actual TTD, is set as (–2, + 2) in Figure 10. The error bar graphs of the train set and the test set in Figure 11(a,b) show that the proportion of train samples falling into the reliability interval is 86.77%, and the proportion of the test set is 89.76%. That is, the accuracy of the model is 89.76%. respectively.

Figure 10.

Regression results after training (a) training set, (b) test set.

Figure 11.

Sample error bar graph: (a) train set and (b) test set.

The mean absolute error (MAE), mean absolute percentage error (MAPE), root mean square error (RMSE), determination coefficient ( ) and adjusted determination coefficient ( ) are used to evaluate the overall performance of the regression model. The criteria calculation formulas are as follows:

where is the sample size, is the number of features, is the predictive value, and is the actual value.

Table 7 shows the MAE and MAPE between each target value and the predicted value of TTD. According to the data distribution in Table 1 and Table 7, the larger the sample size of the TTD used for training, the smaller the MAE and MAPE between the regression value and the measured value. The MAE and MAPE of the test sample are 0.923 and 13.49%, respectively. Furthermore, the MAPEs of the 157 and 95 data are approximately 10% and 5%, accounting for 53.58% and 32.42% of the total sample, respectively. In practice, a large amount of TTD is mainly between 7 μm and 11 μm. For these data, both MAE and MAPE are relatively small, with MAE and MAPE not more than 1 and 12%, respectively. It should be noted that the 3 μm TTDs are mainly caused by excessive edge cutting loss to meet the high requirements of customers. Therefore, the MAPE of prediction results based on hot rolling shape data has reached 40.26%.

Table 7.

MAE and MAPE of each TTD.

TTD			TTD
3	1.2129	40.26%	9	0.7765	8.63%
5	1.2332	24.66%	10	0.8588	8.58%
6	1.2601	21.00%	11	0.9161	8.33%
7	0.8077	11.54%	12	1.8325	15.27%
8	0.6503	8.13%	13	2.6482	20.37%

Moreover, we further established the training model based on the random forest (RF) and extreme gradient boosting (XGBoost) methods, performed regression prediction analysis, and compared the importance of input features. RF is an algorithm that integrates multiple decision trees and belongs to the category of integrated learning. It is a very flexible algorithm with high accuracy. It can run through large datasets without reducing its performance and process high-dimensional features datasets without reducing its dimensionality. Another function is evaluated to demonstrate the importance of the features. Its evaluation principle is to apply noise interference to each feature. The greater the range of change in the evaluation performance, the greater the importance of the feature to which noise is applied.

XGBoost is a kind of integrated learning method. Its core idea is to fit the error of the last prediction by continuously adding decision trees. After training an amount of trees, the corresponding scores of leaf nodes in each decision tree are added to be the predicted value of the sample. The advantage of the XGBoost algorithm is that the score that indicates the importance of each feature can be obtained directly after the tree is created. In general, the score weighs the importance of features in the construction of the decision tree model. The more a feature is used to build a decision tree in the model, the higher its importance.

The , , and are used to measure the performance of the GA-LM-BP neural network, RF, and XGBoost methods. In an overall view of Figure 12, the RF model obtains the optimal training model, but a relatively poor prediction result for the test set. This shows that the RF model is considerably over-fitting. The GA-LM-BP neural network had not only excellent performance for the training model, but also optimal performance for the test set.

Figure 12.

Performances of training set and test set based on three methods.

According to the analysis of the regression results, the correlation between the cross-sectional profile indexes of the hot-rolled silicon steel strip and the TTD of the CRSSs is very high, showing that the cross-sectional profile indexes of the HRSSs are an important factor affecting the TTD. Relatively speaking, as a large number of data are used for analysis and mining, the effect of the cold rolling process on the TTD is limited. In general, the neural network method is relatively superior in analysing the effect of the HRSS cross-sectional profile on TTD.

Importance of feature

To fully verify and analyse the importance of the features, we applied the sensitivity analysis methods of both the RF and the XGBoost to classify the importance of the feature. The results are shown in Table 8 and Figure 14. It can be seen from the results in Table 8 that the ranking results obtained by the two methods are similar to each other in the first three features, especially in the order of the steel grade

, the coupling term of the crown and thickness

and the thickness

. In addition, in the first 13 variables, the coupling term of the crown and thickness

, the crown

, the wedge

, the edge drop

also appear in the two importance ranking. These features mainly include crown indexes

and

, the edge drop index

, the wedge index

and HRSS thickness

Table 8.

Feature importance ranking.

RF				XGBoost
Order	Feature	Order	Feature	Order	Feature	Order	Feature
1		14		1		14
2		15		2		15
3		16		3		16
4		17		4		17
5		18		5		18
6		19		6		19
7		20		7		20
8		21		8		21
9		22		9		22
10		23		10		23
11		24		11		24
12		25		12		25
13		26		13		26

Figure 13 shows the importance weight of the ranking features obtained by the RF and XGBoost methods. The importance weights of the first three features, the thickness , the coupling term of the crown and the thickness and the steel grade , are obviously higher than those of the other features, and the first seven features show a relatively important role.

Figure 13.

Importance ranking order of features.

Among these features, the coupling terms of crown and thickness have more conspicuous effects. , and all rank relatively top, demonstrating the importance of . It is consistent with its significance as an important control target for the cross-sectional profile of silicon steel strip at present. Although all three features have similar constraints for the HRSS profile, ranks higher, indicating that it is one-sided to employ the crown as the single control and evaluation index. More attention should be paid to the coupling effect of the crown and thickness . The edge drop is also an important factor affecting the TTD of CRSSs. Compared with the edge drop at 25∼40 mm from the strip edge, the edge drop at 40∼100 mm shows a more obvious influence on the TTD of CRSSs. Another process parameter that has a great influence is the thickness of silicon steel, the HRSS thickness and the thickness ratio of the hot and cold rolled strip . In fact, there are only two national standard thicknesses of CRSSs in the collected data, 0.5 mm and 0.35 mm. Thus, the influence weights of the thickness and the thickness ratio are also consistent. Compared with crowns, edge drops and thickness, the influence of wedge index is relatively weak, and the wedge is also affected by the thickness of hot-rolled silicon steel strip. should be considered when evaluating its influence.

Based on the results of the sensitivity analysis, we halved the input variables of the GA-LM-BP model to further verify and focus the importance of the features. The original data features were trained with the same network topology model. Then we judged the importance of input feature variables by comparing the prediction results with those of the original neural network. The features in Group I trained by the neural network are roughly included in the set of the top 10 important features obtained by the RF and XGBoost methods. Except for the steel grade in Group I, Group I and Group II both include steel strip specification, crown, wedge, edge drop, so do the coupling and high-order variables of crown, wedge, and edge drop as shown in Table 9.

Table 9.

Features in two groups.

Group I	Group II

The target feature of the two groups is the TTD, and the performance evaluation indexes of the total samples and the test sets trained by Group I and Group II are compared in Figure 14. The , , and of the model trained with the features from Group I are closer to the test results of the total samples than those of Group II, while those performance indexes of the model trained with the features from Group II decrease significantly. This indicates that the collection of features from Group II achieves better prediction results. Therefore, the features of Group I can be considered to be more important. It proves that the results of the sensitivity analysis results are of reference value.

Figure 14.

Performance evaluation indexes of test sets trained by different groups.

Based on the above analysis, we focus on the key indexes of hot rolled strips. To further analyse the influence law of the key indexes, we carried out some experiments and visualized the collected data.

Effect of thickness ratio

The changes in both the thickness ratio and thickness indexes are similar and synergistic since there are few types of thickness specifications for the finished CRSS. A greater thickness ratio means a smaller or larger . The top ranking clearly shows that the thickness ratio is an important factor affecting the TTD of CRSSs, which has been verified in practice.

We found that the TTD of the 0.5 mm thick CRSS was larger than that of the 0.35 mm thick strip. To verify the effect of the thickness ratio, we carried out experiments in a rolling unit. The thickness of HRSS in the unit was increased to 2.6 mm and then cold rolled to a 0.5 mm thick strip ( ). Compared to the conventional rolling with a HRSS 2.2 mm thick ( ), the average TTD of the CRSS in the experiment was 2–3 μm lower than in the same stage, as shown in Figure 15(a). The percentages of TTDs less than 8 and 10 μm are 50%∼70% and 30%∼40% higher than those of conventional rolling, as shown in Figure 15(b). To verify the importance of the feature of the thickness ratio , we used the key feature as coordinates to visualize the distribution law of the TTD of the CRSSs varying with key features of the HRSSs. The coupling term of crown and thickness , the edge drop and the coupling term of wedge and thickness of the HRSS, as shown in Figure 15(c), are selected as the x, y, z coordinate axes, respectively, due to the importance of , and in the sensitivity analysis. To avoid the point being too scattered or too clustered, so that the law is not obvious, we divide the abscissa by the same weight. In Figure 15(c), the ball and cube represent the experimental data and conventional data, respectively. The grey and black dots on the XY plane, YZ plane and ZX plane represent the projections of conventional data and experimental data, respectively. The ball and cube sizes vary with TTD, and the larger the TTD, the greater the ball or cube size and the warmer the colour. The result shows that the TTDs in the experiment are obviously smaller than those of conventional rolling. Nevertheless, the distribution range of the cross-sectional profile indexes of the experimental HRSS roughly coincides with those of the conventional rolling, and there is no significant advantage in the indexes of HRSSs except that the distribution is relatively concentrated. All of this further indicates that the proper increase of the thickness ratio is conducive to reducing the TTD of CRSSs.

Figure 15.

Effect of thickness ratio on TTD in experimental and conventional rolling (a) average TTD (b) percentage of TTD grade (c) comparison of cross-sectional profile indexes (d) thickness ratio and TTD distribution.

In order to further intuitively display the effect of thickness ratio on the TTD of CRSSs, we replaced the wedge index with the thickness ratio , and obtained the spatial visual distribution figure of the TTD varying with coupling term of crown and thickness , the edge drop and the thickness ratio of the conventional CRSS as shown in Figure 15(d). The thickness ratio in Region I is evidently greater than or equal to 6.2 (2.2 mm/0.35 mm), and the corresponding TTDs are mostly distributed between 2∼9 μm. The thickness ratio of in Region II is less than or equal to 5.2 (2.6 mm/0.5 mm), and a large number of TTD exceeds 10 μm. The law further proves that an appropriate large thickness ratio is beneficial in reducing the TTD of CRSSs.

In fact, there are always specified heavy gauge transition coils in the conventional hot rolling unit, as shown in Figure 16(a,b). Zone I contains the specified transition coil of silicon steel, and the silicon steel is located in Zone II. Hot-rolled commercial strips or carbon steel strips are arranged before Zone I and after Zone II. In Zone I, the thickness of the silicon steel strip changes from 2.8 to 2.2 mm to adapt to the gradual stability of the roll thermal crown. Although the crown target of the silicon steel strip in Zone I is the same as those in Zone II, statistics show that the TTD of the CRSSs corresponding to these HRSSs is not particularly small. In fact, the actual crown and edge drop of the strip in Zone I of Figure 16(a) are larger than those with a thickness of 2.2 mm in Zone II. In particular, the flatness and crown of the strip steel in Zone I fluctuate greatly, resulting in a high probability of wedge indexes fluctuating, as shown in Figure 16(b). Although the thickness ratio of the hot and cold rolled strips in Zone I is large, the TTD of cold-rolled silicon steel is not small due to the large indexes of wedges, crowns and edge drops.

Figure 16.

Strip crown and wedge vary in a rolling unit (a) crown indexes (b) wedge indexes.

The fact that a proper large thickness ratio is conductive to reducing the TTD to a certain extent may be affected by the work hardening of strips. The larger the thickness of the incoming slab, the greater the accumulated deformation and the more obvious the hardening of the steel strip. Then the deformation resistance of the steel strip increases. Finally, the edge drop in the last pass decreases in the case of roughly equal rolling force. However, increasing the thickness ratio will increase the production cost, which needs to be optimized according to the user demands, profits, and costs.

To verify the results of the sensitivity analysis and the importance of the feature, we use the key feature as coordinates to visualize the distribution law of the TTD of the CRSSs varying with the key features of the HRSSs, and the results are shown in Figure 17. According to the results of sensitivity analysis, the key features mainly include steel grade , strip thickness , the coupling term of the strip crown and thickness , , , the crowns and , the edge drop , , the coupling term of the strip wedge and thickness , the thickness ratio . Among these features, the cross-sectional profile indexes of HRSSs mainly involve the crown , , the edge drop , , and the coupling items , , . These features , , have coordinated constraints on the cross-sectional profile, while has a greater weight, so it is selected as the key evaluation index of strip crowns that affect the TTD of CRSSs. Although and are both edge drop indexes, the reflects the edge drop at 25–40 mm from steel strip, which is at the region cut in the finishing process. Therefore, is selected as the key evaluation index of edge drop that affects the TTD of CRSSs. It takes into account the influence of on the TTD. The wedge is calculated by the difference between the thicknesses at two sides of the strip, reflecting the shape quality and rolling process stability. It fluctuates obviously due to the simple calculation method. As the cross-sectional profile is closer to the edge of the steel strip, the non-linear drop in thickness is obvious. The severer the non-linear decline of thickness, the more obvious and random the fluctuation of wedge degree, the more difficult it is to reflect the influence of the wedge on the TTD in cold rolling. That is to say, the middle thickness of the strip is relatively stable, and the wedge away from the strip edge can better reflect the quality of the hot rolled strip. Therefore, the sensitivity analysis shows that the most influential wedge index is , but is not related to and . Thus, we choose as the wedge index.

Figure 17.

Effect of , and on TTD of cold-rolled strip: (a) point-to-point, (b) and (c) average-to-point from different perspectives.

Effect of crown or wedge and thickness coupling term and edge drop

The distributions of the scatter cloud were established by taking , and as coordinates, and the sizes of the scattered points represent the TTD of CRSSs. The larger the TTD is, the larger the ball size and the darker the colour. To avoid that the points are too scattered so that the law is not obvious, we divide the abscissa by the same weight to make the change to . Figure 17 shows the scatter cloud distributions of both point-to-point (Figure 17 (a)) and average-to-point (Figure 17(b,c)).

The scattered points are distributed along a spatial slope A in Figure 17(a). There are two typical regions I and II. In Region I, all TTDs are distributed in the range of 5∼10 μm, and the corresponding indexes , and are distributed between 0∼35 , −10 ∼ 10 μm and 0∼10 , respectively. In region II, some of the TTDs are distributed between 12 and 17 μm. The corresponding is between 40∼100 and are obviously scattered between 10 and 15 μm. Meanwhile, in the dimension of , the higher the , the higher the probability that the TTD of the CRSS increases. That is to say, the distribution law of point-to-point data of cold and hot rolling shows that the decline of and is obviously conducive to reducing the TTD, as marked with the black arrow, and the increase of will increase the probability of the excess TTD.

Figure 17(b,c) are the distribution law of the TTD of the CRSSs with the mean value of , and of the HRSSs in different perspectives. Figure 17(b) shows that the scatter points are distributed along the spatial slope B, and there are also two typical spatial Regions I and II, in which the distribution law is similar to that in Figure 17(a). The average values of the indexes of the whole coil of strip steel mask some local feature information, and the average value must be in the middle of the extreme value range. Therefore, compared to point-to-point scatter points, some points with a TTD greater than 11 μm fill between Region I and Region II in Figure 17(b,c). Meanwhile, the distribution of is relatively centralized, and the value is small.

Although the regularity of average-to-point data is slightly worse than that of point-to-point data, in general, the scatter-point distribution laws corresponding to the two methods show that the effects of the coupling term of strip crown and thickness , the edge drop and the coupling term of strip wedge and thickness on the TTD of CRSSs are clear and the correlation is relative high. Compared to the traditional crown-based evaluation method of , the above three indexes of the cross-sectional profile of hot-rolled silicon steel strips are more comprehensive.

Recommended standards of key index values

According to the analysis of the importance of the feature obtained by the GA-BP and the RF and the XGB model, we have determined three key features,

, and one specification parameter

that affect the TTD of CRSSs. Then the visualization distribution relationship among the key coupling term

, the edge drop

and the coupling term

and the TTD is organized as shown in Figure 17. As the TTD of the CRSSs is required to be controlled below 9 μm, the visual data distributions, especially in Figure 17(a), show that the corresponding

should be no more than 80

(Figure 17(a,c)), distributed between −10∼10 μm and no more than 10

, respectively. Meanwhile, the spatial slope distribution of the scatter points determines that the requirement for the cross-sectional profile indexes of the HRSS is synchronous. It may not be possible to obtain the ideal TTD of the CRSS by restricting the single index of the HRSS. Finally, the crown and thickness coupling term

, the edge drop

and the wedge and thickness coupling term

are recommended as the key indexes of the hot-rolled silicon steel strip. On the basis of the above analysis, we give the reference of the profile index values as shown in Table 10.

Table 10.

Recommended standards for profile index values of hot-rolled silicon steel strip.

TTD		μm
≤9 μm	≤80	−10∼10	≤10
≤7 μm	≤60	−10∼10	≤7.5
≤5 μm	≤45	−10∼10	≤5

Conclusions

To develop the control indexes of hot-rolled silicon steel strips and predict the TTD of CRSSs according to the HRSS information, the GA-LM-BP neural network was developed for the regression of data and compared with the performance of the RF and XGBoost methods. Next, the sensitivity analysis method was used to classify the importance of the input features. The key features were discussed and verified by group-dividing training through the neural network. Then, visual analysis of cold and hot rolling data was performed based on the key features obtained, and the influence law of the HRSS key profile indexes on the TTD was analysed. Finally, key index values of the cross-section profile of the hot rolled strip on the TTD were recommended. Our conclusions are summarized as follows:

On the basis of the GA-BP neural network algorithm, the heredity effect model of the cross-sectional profile of the HRSS on the TTD of the CRSS was established. Compared to the RF and XGBoost methods, the GA-LM-BP neural network makes the regression result of the neural network more accurate.

The importance of the input features was ranked based on the sensitivity analysis of the RF and XGBoost methods, and the key features were discussed and verified by group-dividing training through the neural network. The results show that, in addition to the inherent features of the steel grade and the process parameter of thickness ratio , the coupling term of the strip crown and thickness , the edge drop and the coupling term of the strip wedge and thickness are recommended as key indexes of the hot-rolled silicon steel strip.

Data visualization indicate that an appropriate large thickness ratio is beneficial in reducing the TTD of CRSSs, and the smaller all the , and , the more likely the TTD is to be reduced.

As the TTD of the CRSS is required to be controlled below 7 μm, the corresponding , , should be controlled at no more than 60 , between −10∼10 μm and no more than 7.5 , respectively. Meanwhile, the requirement for the cross-sectional profile indexes of the HRSS is synchronous.

Footnotes

Disclosure statement

No potential conflict of interest was reported by the author(s).

Author contribution

Xiaobao Ma: Data acquisition, Investigation, Problem analysis, Writing original draft, Resources, Model, Funding acquisition. Bao Ma: Review, Model, Investigation, Writing original Draft, Jiangjiang Li: Editing, Test, Validation. Peng Chen: Resources, Review & Editing. Yongheng Peng: Investigation, Editing. Zhongkai Ren: Investigation, Writing & Editing, Supervision.

Declarations

Ethics approval: The paper follows the guidelines of the Committee on Publication (COPE).

Consent to participate: The authors declare that they all consent to participate in this research.

Consent for publication: The authors declare that they all consent to publish the manuscript.

Data availability statement

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

ORCID

Xiaobao Ma

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TTD/μm	Sample size	TTD/μm	Sample size
1	12	10	351
2	13	11	190
3	350	12	97
4	17	13	59
5	109	14	17
6	248	15	11
7	555	16	9
8	557	17	4
9	575	18	2

TTD/μm	Sample size	TTD/μm	Sample size
1	12	10	351
2	13	11	190
3	350	12	97
4	17	13	59
5	109	14	17
6	248	15	11
7	555	16	9
8	557	17	4
9	575	18	2

TTD/μm	Sample size	TTD/μm	Sample size
1	12	10	351
2	13	11	190
3	350	12	97
4	17	13	59
5	109	14	17
6	248	15	11
7	555	16	9
8	557	17	4
9	575	18	2