High-precision online prediction model of plate crown based on multi-factor interaction mechanism

: The crown is a key quality index of strip and plate. Up to now, many plate crown prediction models have been established, however, the calculation accuracy of these traditional models is not very high. This is because these models only consider the effect of various influencing factors on the outlet plate crown under the basic process parameters, but these factors are not independent of each other. Therefore, in order to improve the accuracy of the plate crown prediction model, a high-precision prediction model of plate crown is established by introducing the evaluation coefficient that represents the influence of single factor on plate crown and the correction coefficient that represents the effect of factor a on the evaluation coefficient of factor b . This paper takes the steel SPHC as an example, calculates the plate crown without considering the correction coefficient and the plate crown considering the correction coefficient. Then, the two calculation results are compared with the calculation results of the coupled model and the accuracy of the new model is improved after considering the correction coefficient. In general, compared with the traditional crown prediction model, the accuracy of the new model is greatly improved.


Establishment of a high-precision online plate crown prediction model 2.1 Traditional plate crown prediction model and its limitations
Based on the basic plate crown, the overall calculation model of plate crown considering the influence of various factors [27] can be expressed as: Where C is the calculation value of the plate crown. n is the number of influencing parameters(unit rolling force, bending roll force, work roll diameter, backup roll diameter, work roll crown, backup roll crown, load distribution, etc.). For example, for different reductions, the influence of roll shifting will inevitably change and the effect of bending force will be different for different roll diameters [28].
Suppose the final outlet crown can be calculated by the following function: Where i x is the factor affecting the plate crown. Assuming that the outlet plate crown function has derivatives for each influencing factor. When the above formula expand with Taylor series, it can be expressed as: The zero-order term in the equation is the basic value of plate crown. The primary term in the formula is the correction amount under the effect of each influencing factor alone. The quadratic term in the formula is the correction amount of the mutual influence between the two factors. The third term in the formula is the correction amount of two factors to the third factor, and so on. It can be seen that there is only primary term in the traditional model. In order to improve the accuracy of the calculation and ensure the calculation efficiency, this paper proposes the calculation method of the quadratic term, and ignores the third-order and higher-order terms.

Evaluation coefficient and correction coefficient of the effect of influencing factors
In the case of neglecting the thermal crown and the wear of the roll, the main process parameters affecting the outlet plate crown includes: plate width, reduction, roll shifting, bending force, inlet plate crown, work roll diameter and backup roll diameter [29]. In order to analyze the influence of these factors on the plate crown, this paper defines the Where max C  is the maximum change in the outlet plate crown when a certain factor changes.
h is the outlet plate thickness.
Taking the effect of the roll shifting value on the plate crown as an example to analysis and calculation. Figure 1a) shows the variation of the outlet plate crown with the width of the plate when the roll shifting value is taken as two limit values. Figure   1b) shows the evaluation coefficient of the roll shifting value at different plate widths. a) Outlet plate crown b) Evaluation coefficient As can be seen from Figure 1 In order to quantitatively analyze the influence of a certain factor a on the evaluation coefficient of another b , this paper introduces a correction coefficient, as shown in the following: It can be seen that when the correction coefficient , ab m is less than 1, the change of the factor a makes the effect of the factor b smaller. When the correction coefficient is more than 1, the change of the factor a makes the effect of the factor b larger.

Establishment of modified model
is the reconstruction plate crown caused by bending force. Similarly, the reduction, the roll shifting quantity, the inlet plate crown, the work roll diameter, and the backup roll diameter can all be corrected by all other influencing factors. The final outlet plate crown can be expressed as: Where i j m , is the correction coefficient for the effect of the factor j on the factor i . In order to verify the calculation accuracy, this paper combines the actual situation of Baosteel's 2050 hot rolling line. Takes the steel SPHC as an example, the plate width is 1150mm, and the outlet thickness is 20mm. The calculation example is calculated using a predictive model that considers the interaction between the influencing factors and a predictive model that does not consider the interaction between the influencing factors. The main process parameters are shown in Table 1 below: Reduction/mm 15 15 The finishing mills of Baosteel's 2050 hot strip continuous rolling mill consists of 7-stands four-high CVC mills. In order to calculate the plate crown influencing rate i K and the correction coefficient , ab m of each factor, this paper takes the F1 stand as an example, the outlet thickness of the first pass is 20mm, the basic process parameters used in the calculation are shown in Table 2:

Calculation of the basic plate crown
Where () ai is the fifth-degree polynomial fitting coefficient of the basic plate crown. B is the plate width.
The curve of the basic plate crown with the width of the plate is shown in Figure 5  The fifth-degree polynomial fitting coefficient () ai of basic plate crown are shown in Table 3: Table 3 Fit coefficients of the basic center crown

Calculation of the influencing rate of plate crown
The influencing rate of the plate crown of each influencing factor varies with the plate width as shown in the Figure 6 a The relationship between each influencing factor and the plate width can be fitted with a fifth degree polynomial, as show in the following 5 0 ( ) Where () ai is the fitting coefficient. B is the plate width.
The fitting coefficients of each influencing factors are in the Table 4   Table 4 The fitting coefficients of each influencing factors

Correction coefficient of bending force
When the other influencing factors take the base value, the evaluation coefficient of the bending force is

Correction coefficient of work roll diameter
When the other influencing factors take the base value, the evaluation coefficient of the work roll diameter is  The relationship between the evaluation coefficient of the work roll diameter p N and the plate width can be fitted with a fifth degree polynomial, as show in the following

Correction coefficient of reduction
When the other influencing factors take the base value, the evaluation coefficient of the reduction is

Correction coefficient of roll shifting value
When the other influencing factors take the base value, the evaluation coefficient of the roll shifting value is  The bending force and work roll diameter has very little effect on the roll shifting value, because the three curves almost coincide.
The relationship between the evaluation coefficient of the roll shifting value p N and the plate width can be fitted with a fifth degree polynomial, as show in the following:

Correction coefficient of inlet crown
When the other influencing factors take the base value, the evaluation coefficient of the inlet crown is  The bending force and work roll diameter has very little effect on the inlet crown, because the three curves almost coincide.
The relationship between the evaluation coefficient of the inlet crown p N and the plate width can be fitted with a fifth degree polynomial, as show in the following: The fitting coefficients of

Correction coefficient of backup roll diameter
When the other influencing factors take the base value, the evaluation coefficient of the backup roll diameter is

Comparison of plate crown calculation accuracy
According to Eq. (6), this paper calculates the change in plate crown caused by each influencing factor, as shown in Table 11. The basic plate crown is 347µm and when the bending force and the work roll diameter is changed, the plate crown calculated by coupled model is 320µm. To further verify the accuracy of the modified model, Table 12 below gives a comparison of the calculation results for five sets of identical materials (SPHC) and different process parameters. It can be seen from Table 12 that the prediction model considering the correction coefficient has higher calculation accuracy under different process parameters, which is most obvious when the board width is 1800mm. The error of calculation without the correction coefficient is 11 μm, but only 2 μm after considering the correction coefficient. Because the process parameters in this example are all deviated from the base value, and the variation of the evaluation coefficient generally increases with the increase of the width.