Coordinated setting method for bending and shifting of hot rolling work rolls suitable for rolling in a wide range of widths
By constructing a mathematical model and using the NSGA-II algorithm to optimize the position of the shifting rolls and the bending roll force, the wear and shape control problems during the rolling of hot-rolled strips of various widths were solved, achieving uniform wear and precise shape control, thereby improving production efficiency and quality.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- HENAN IRON & STEEL GROUP CO LTD
- Filing Date
- 2025-12-29
- Publication Date
- 2026-06-23
Smart Images
Figure CN122252467A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of metallurgical rolling automation control, and specifically relates to a method for coordinating the setting of bending and shifting of hot rolling work rolls applicable to rolling of various widths. Background Technology
[0002] In the production of hot-rolled strip steel, in order to meet the comprehensive quality requirements of high-end strip steel such as silicon steel under the constraints of narrow process windows, its hot rolling production must adopt multi-width free-range rolling (from wide to narrow, the same width, or from narrow to wide). It is difficult to meet the extreme service conditions of severe uneven wear of rolls and uneven deformation of strip steel during the entire service life of continuous rolling of a large amount of strip steel. This leads to severe uneven groove-shaped "U" wear of work rolls and loss of control of strip cross shape during free-range rolling, which affects the final hot-rolled product's plate shape quality.
[0003] Currently, axial movement of work rolls and adjustment of bending force are core methods for controlling strip shape and roll wear. For example, invention patent application number 202210385450.1 discloses "a method for eliminating local high points on strip in multi-stand work rolls," which effectively eliminates local high points on the edge of the strip by superimposing a random correction amount on the basis of periodic uniform reciprocating roll movement. Invention patent application number 202111007707.1 discloses "an asynchronous roll movement method with double attenuation of work roll movement stroke and roll movement step length," which avoids excessive local wear and improves the uniformity of work roll wear and strip shape by determining relevant parameters, calculating the roll movement stroke, step number, and position. Regarding the quality of the strip, patent application number 202210692366.4 discloses "a method for controlling the shifting of work rolls in hot continuous rolling finishing mills." This invention uses synchronous equal step lengths for work roll shifting in the early stage of rolling, and the step length for shifting rolls in the later stage of rolling is greater than that in the early stage, avoiding harmful areas within the limit value range. This can improve the local high points of the strip in the later stage of rolling and extend the rolling mileage of the finishing rolls. Patent application number 202410968384.X discloses "a method for reducing the shifting of work rolls in strips." This invention determines the rolling unit and wear stand with local high points by tracking, sets the shifting roll interval and step length, and writes it into the model to reduce local wear and local high point defects in the strip.
[0004] While existing methods for optimizing work roll shifting in free-flow rolling can alleviate wear issues under a single width, they do not consider the dynamic adaptation requirements of rolling multiple widths simultaneously and lack real-time response to width changes. Therefore, there is an urgent need for a method for collaborative setting of bending and shifting rolls applicable to rolling multiple widths, in order to achieve uniform wear and precise shape control when rolling strips of different widths. Summary of the Invention
[0005] To address the shortcomings of existing technologies, the present invention aims to provide a method for the coordinated setting of hot rolling work roll bending and shunting rolls applicable to rolling of various widths. By dynamically matching the shunting roll position and bending roll force, it adapts to changes in strip width, thereby achieving uniform work roll wear, improved strip shape adjustment capability, and enhanced rolling stability.
[0006] To achieve the above objectives, the present invention adopts the following technical solution:
[0007] The method for coordinating the setting of bending and shifting work rolls in hot rolling mills applicable to various widths includes the following steps:
[0008] Constructing a mathematical model for hot rolling work roll shifting to adapt to multi-width rolling:
[0009]
[0010] in, i This refers to the quantity of rolled strip steel. The initial direction of the work roll shifting; L The step length of the work rolls during the initial rolling process. mm ; The width of the strip. mm ; This refers to the roll step size during the rolling of the (n+1)th strip. mm ; For the maximum working roll travel, mm ; m The attenuation coefficient of the roller travel is denoted by . ; T The cycle time of the roller; N This represents the number of roll shifting steps within a single roll shifting cycle. q The decay coefficient for the number of roller shifting steps within a cycle. ; φ The initial phase of the roller position;
[0011] Based on the optimization objectives of the mathematical model for hot rolling work roll shifting, multiple objective functions including wear uniformity, thermal crown control, and plate crown accuracy are designed.
[0012] The parameters of the mathematical model for hot rolling work roll shifting were optimized using the NSGA-II multi-objective optimization algorithm.
[0013] The optimal work roll bending force and roll shifting amount are obtained based on the deviation between the target plate crown and the actual crown, combined with the change in strip width.
[0014] Furthermore, the specific steps for constructing the mathematical model for hot rolling work roll shifting adapted to multi-width rolling are as follows:
[0015] In the initial stage of rolling, to ensure the stability of the thermal crown and axial uniformity of the work rolls on each stand, a preset roll shifting step length and stroke are adopted. The mathematical model function for the roll shifting is:
[0016]
[0017] in, For the first rolling unit i Position of the work roll misalignment for strip steel coils, in mm; i This refers to the quantity of rolled strip steel. The initial direction of the work roll shifting; L The step length of the work roll is in mm;
[0018] The mathematical model function for work roll shifting at the start of silicon steel rolling:
[0019]
[0020] in, Maximum work roll travel, mm; m The attenuation coefficient of the roller travel is denoted by . ; T The cycle time of the roller; N This represents the number of roll shifting steps within a single roll shifting cycle. q The decay coefficient for the number of roller shifting steps within a cycle. ; φ The initial phase of the roller position;
[0021] Based on sinusoidal work roll shifting of various widths, the mathematical model function for work roll shifting is obtained as follows:
[0022]
[0023] in, Maximum roller travel, mm;
[0024] The mathematical model for multi-width work roll slippage is as follows:
[0025]
[0026] in, i This refers to the quantity of rolled strip steel. The initial direction of the work roll shifting; L The initial rolling step length is the work roll shift length, in mm; The width of the strip is in mm; For the first n +1. Roller step length during strip rolling, mm.
[0027] Furthermore, based on sinusoidal work roll misalignment of various widths, the specific calculation method for the mathematical model function of work roll misalignment is as follows:
[0028] Since the width of two adjacent strips changes, the center of the axial work roll is set as the zero point of the coordinate system. The contact position distribution between the strip edge and the work roll is obtained, which is expressed by the following formula:
[0029]
[0030]
[0031] in, The contact position between the right side of the nth strip and the work roll is shown in mm. The contact position between the left edge of the nth strip and the work roll is shown in mm. The width of the strip is in mm; The amount of roll shifting for the strip, in mm;
[0032] The contact position between the edge of the work roll and the strip is set as the marker point for periodic reciprocating movement. The strip position at the right contact point is the [missing information]. n The contact position of the steel strip edge is as follows:
[0033]
[0034] in, For the first n +1 The contact position between the right side edge of the strip and the work roll, mm; For the first n +1. Roller step length during strip rolling, mm;
[0035] Rolling n The position of the shifting roll when there is +1 strip steel is as follows:
[0036]
[0037] Therefore, based on sinusoidal work roll misalignment of various widths, the mathematical model function for work roll misalignment is as follows:
[0038]
[0039] in, The maximum roller travel distance is in mm.
[0040] Furthermore, based on the optimization objectives of the mathematical model for hot rolling work roll shifting, multiple objective functions were designed, including wear uniformity, thermal crown control, and plate crown accuracy. The specific steps are as follows:
[0041] In the initial stage of rolling, to ensure the uniformity and stability of the hot roll shape of the work roll during the rolling process, the work roll shape is used as the objective function to optimize the mathematical model of work roll shifting. As shown in the following formula:
[0042]
[0043] in, and The distance from the center and edge of the work roll body, in mm;
[0044] To even out wear on the work roll, the relative value of the cat-ear shape change of the work roll is... Relative value of wear width change The weighted average of the values is used as the objective function to optimize the mathematical model of the work roll shifting. As shown in the following formula:
[0045]
[0046] in, , These are weighting coefficients, ranging from 0 to 1, ensuring... Each was set to 0.5;
[0047] To ensure that the exit crown of the hot-rolled strip reaches the final target plate crown, the objective function is to minimize the deviation of the strip exit crown. As shown in the following formula:
[0048]
[0049] in, For the first i The exit convexity of the frame, in mm; The target convexity of the strip is in mm.
[0050] Furthermore, the parameters of the mathematical model for hot rolling work roll shifting are optimized using the NSGA-II multi-objective optimization algorithm;
[0051] Set parameters such as population size, crossover rate, mutation rate, and maximum number of iterations, initialize the population individuals, and ensure that all constraints are met;
[0052] Non-dominated sorting and crowding calculation: Perform non-dominated sorting on individuals in the population and calculate individual crowding to ensure solution set diversity;
[0053] Selection, crossover, and mutation operations are performed; a binary tournament selection method is adopted, and simulated binary crossover and polynomial mutation operators are used to generate new individuals and update the population;
[0054] Termination condition determination; determine whether the maximum number of iterations has been reached, and output the optimized working roll shifting model parameters.
[0055] Furthermore, the specific steps for obtaining the optimal work roll bending force and roll shifting amount based on the deviation between the target plate crown and the actual crown, combined with the change in strip width, are as follows:
[0056] Initialize the rolling process parameters and set the work roll shifting method;
[0057] Based on the deviation between the target plate crown and the actual crown, the optimal bending roller force is dynamically searched within the bending roller force adjustment range;
[0058] By combining the changes in strip width and iterative optimization conditions, the optimal work roll bending force and roll shifting amount are output.
[0059] Furthermore, at the start of the silicon steel rolling process, in order to balance uniform work roll wear and rolling stability, a larger step size is used at the center of the rolling process and a smaller step size is used at the extreme positions to increase the frequency of contact between the strip and the work roll.
[0060] The beneficial effects of this invention are as follows: This invention provides a method for the coordinated setting of bending and skewing rolls in hot rolling processes applicable to various widths, possessing significant technical advantages and engineering application value. By introducing a multi-width sinusoidal attenuation skewing roll model, it can effectively adapt to the rolling requirements of strips of different widths, significantly improve the problem of uneven wear on work rolls, and avoid the generation of "cat-ear" or "box-shaped" wear. Combined with the NSGA-II multi-objective intelligent optimization algorithm, it achieves coordinated optimization of skewing roll strategy and bending roll force, taking into account multiple key indicators such as thermal crown control, wear uniformity, and plate crown control, thereby improving the accuracy of plate shape control and system stability. Actual industrial application results show that this method can significantly improve the plate crown hit rate, reduce bending roll force fluctuations, and extend the service life of work rolls, providing reliable technical support for the high-quality and high-efficiency production of high-end strip and plate products such as silicon steel, and has good prospects for widespread application. Attached Figure Description
[0061] Appendix Figure 1 This is a flowchart of the hot rolling work roll bending and shifting roll coordinating setting method applicable to rolling of various widths according to the present invention;
[0062] Appendix Figure 2 This is a schematic diagram showing the frequency distribution of contact between the work roll and the strip at different roll positions, provided by an embodiment of the present invention.
[0063] Appendix Figure 3 A flowchart for optimizing the work roll shifting strategy based on the NSGA-II method provided in this embodiment of the invention;
[0064] Appendix Figure 4 A flowchart illustrating the optimized control strategy for multi-width work roll bending and slipping rolls provided in this embodiment of the invention;
[0065] Figure 5(a) is a schematic diagram comparing the results before and after the optimization of the work roll shifting amount of the F6 frame provided in the embodiment of the present invention;
[0066] Figure 5(b) is a schematic diagram comparing the results before and after the optimization of the bending force of the F6 frame work roll provided in the embodiment of the present invention. Detailed Implementation
[0067] The present invention will now be further described with reference to the accompanying drawings and specific embodiments:
[0068] This embodiment applies the hot rolling work roll bending and shifting collaborative setting method provided by the present invention, applicable to rolling of various widths, to a large industrial hot continuous rolling mill production line, such as... Figure 1 The diagram shown is a flowchart illustrating a specific implementation of the present invention, which includes the following steps:
[0069] Step 1: Based on on-site measured data and the characteristics of work roll shifting, construct a mathematical model for hot rolling work roll shifting suitable for multi-width rolling:
[0070]
[0071] Where i is the number of rolled strips; The initial direction of the work roll shifting is shown; L is the initial step length of the work roll shifting in the early stage of rolling, in mm; The width of the strip is in mm; The step length of the hopping roll during the rolling of the (n+1)th strip, in mm; is the maximum working roll travel, in mm; m is the attenuation coefficient of the roll travel. T represents the cycle period of the skewed roller; N represents the number of skewed roller steps within a single cycle; q represents the attenuation coefficient of the number of skewed roller steps within a cycle. φ represents the initial phase of the shifting roller position;
[0072] like Figure 2 This is a schematic diagram showing the frequency distribution of contact between the work roll and the strip at different roll positions, as provided in an embodiment of the present invention.
[0073] Step 2: Based on the optimization objectives of the mathematical model for hot rolling work roll shifting, design multiple objective functions including wear uniformity, thermal crown control, and plate crown accuracy;
[0074] The objective function of the mathematical model for multi-width work roll shifting is as follows:
[0075]
[0076] in, and The distance from the center and edge of the work roll body, in mm. , These are weighting coefficients, ranging from 0 to 1, ensuring... They were each set to 0.5. For the first i The exit convexity of the frame, in mm; The target crown of the strip, in mm;
[0077] Specifically, the constraints of the mathematical model for multi-width work roll shifting are shown in Table 1;
[0078] To avoid excessive reduction in roll travel during the later stages of rolling, which would cause the roll travel value to concentrate in the middle of the roll body, then... m The minimum stroke value after attenuation of the maximum roller travel of 100mm must not be less than 40mm.
[0079] It is required to ensure that the number of roll shifting cycles in a single rolling unit is greater than 1, and that the number of roll shifting cycles N in a single cycle is less than the maximum value of 150 in a single rolling unit.
[0080] q This is to avoid the position of the skewed rolls being concentrated at the extreme position in a single cycle during the middle and later stages of rolling. However, its coefficient cannot be reduced too much, otherwise the number of skewed roll steps in the next cycle will be reduced too much and the distribution of the skewed roll positions will be uneven.
[0081] To avoid repeated roller shifting at the zero point, the initial phase is set... φ The value cannot be too large, otherwise the initial roll shifting position will deviate from zero. Therefore, the final multi-width work roll shifting strategy parameters are shown in Table 1.
[0082] Table 1. Parameters to be optimized for the roll shifting strategy
[0083]
[0084] Step 3: Optimize the parameters of the mathematical model for hot rolling work roll shifting using the NSGA-II multi-objective optimization algorithm;
[0085] like Figure 3 This is a flowchart of the optimization process for the work roll shifting strategy based on the NSGA-II multi-objective optimization method provided in an embodiment of the present invention.
[0086] Specifically, the mathematical model of the work roll slippage was optimized using the NSGA-II algorithm. The initial parameters of the NSGA-II algorithm were set as follows: crossover probability of 0.5, mutation probability of 0.001, population size of 100, and maximum number of iterations of 200. The optimal parameters of the work roll slippage model were obtained through iteration, as shown in Table 2.
[0087] Table 2 Optimization parameters for the roll shifting strategy
[0088]
[0089] Specifically, the optimized multi-width work roll shifting model function is as follows:
[0090]
[0091] in, i This refers to the quantity of rolled strip steel. The initial direction of the work roll shifting; L The initial rolling step length is the work roll shift length, in mm; The width of the strip is in mm; For the first n +1. Roller step length during strip rolling, mm.
[0092] Step 4: Based on the deviation between the target plate crown and the actual crown, and combined with the change in strip width, obtain the optimal work roll bending force and roll shifting amount, set the work roll bending and shifting collaborative control mode for multiple widths of the downstream hot rolling stand, and conduct industrial verification and stable application.
[0093] like Figure 4 A flowchart illustrating the optimized control strategy for multi-width work roll bending and slipping rolls provided in this embodiment of the invention;
[0094] Specifically, data from a typical complete rolling unit was selected for verification in this embodiment, and some rolling data are shown in Table 3.
[0095] Table 3 Process parameters within a complete hot rolling unit
[0096]
[0097] Compared to traditional work roll shifting strategies, this patent ensures a more uniform distribution of shifting roll positions throughout the entire rolling cycle (e.g., Figure 5a The bending force of the work rolls on stands F4-F7 was reduced by 11.15%, 13.47%, 33.21%, and 11.02%, respectively (e.g., Figure 5b This significantly improves the adjustment efficiency of the bending force of the work rolls. Measurements of the roll profile of the F6 stand were made and compared with those of the roll profiles in adjacent rolling units. The worn roll profile of the roll profile showed a flat rolling area with no obvious cat-ear-shaped wear, which can improve the severe uneven wear of the work rolls and ensure the high-precision plate shape quality of silicon steel in the later stages of the complete rolling unit, fully utilizing the plate shape control capability of the downstream stands of the hot strip mill.
[0098] In summary, the hot rolling work roll bending and skewing collaborative setting method proposed in this embodiment, applicable to rolling of various widths, achieves uneven wear of work rolls on downstream hot rolling stands without affecting production and with minimal equipment modifications, significantly improving the high-precision plate shape quality of hot-rolled silicon steel.
[0099] Furthermore, it should be noted that, in this document, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or terminal device that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or terminal device. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or terminal device that includes said element.
[0100] Finally, it should be noted that the above description represents a preferred embodiment of the present invention. It should be pointed out that although preferred embodiments have been described, those skilled in the art, once they understand the basic inventive concept of the present invention, can make various improvements and modifications without departing from the principles described herein. These improvements and modifications should also be considered within the scope of protection of the present invention. Therefore, the appended claims are intended to be interpreted as including both the preferred embodiments and all changes and modifications falling within the scope of the embodiments of the present invention.
Claims
1. A method for coordinated setting of bending and shifting work rolls in hot rolling mills applicable to various widths, characterized in that, The specific steps are as follows: Constructing a mathematical model for hot rolling work roll shifting to adapt to multi-width rolling: Where i is the quantity of rolled strip; d r The initial direction of the work roll shifting; L is the initial step length of the work roll shifting during the initial rolling process, in mm; B n Se represents the width of the strip, in mm. n+1 S represents the roll step length during the rolling of the (n+1)th strip, in mm; max The maximum working roll travel is given by f(m), measured in mm; m is the attenuation coefficient of the roll travel, f(m) = S. max ·m T-1 T is the cycle period of the skewed roll; N is the number of skewed roll steps within a single skewed roll cycle; q is the decay coefficient of the number of skewed roll steps within a cycle, f(q)=N·q T-1 ; The initial phase of the roller position; Based on the optimization objectives of the mathematical model for hot rolling work roll shifting, multiple objective functions including wear uniformity, thermal crown control, and plate crown accuracy are designed. The parameters of the mathematical model for hot rolling work roll shifting were optimized using the NSGA-II multi-objective optimization algorithm. The optimal work roll bending force and roll shifting amount are obtained based on the deviation between the target plate crown and the actual crown, combined with the change in strip width.
2. The method for coordinated setting of bending and shifting work rolls in hot rolling mills applicable to multiple widths, as described in claim 1, is characterized in that... The specific steps for constructing a mathematical model for hot rolling work roll shifting adapted to multi-width rolling are as follows: In the initial stage of rolling, the mathematical model function for work roll shifting is: S(i)1=S(i-1)+d r L Where S(i) is the position of the work roll shifting of the i-th strip within the rolling unit, in mm; i is the number of strips rolled; d r The initial direction of the work roll shifting is L; the step length of the work roll shifting is L, in mm. The mathematical model function for work roll shifting at the start of silicon steel rolling: Among them, S max The maximum work roll travel is denoted as , in mm; m is the attenuation coefficient of the travel, f(m) = S. max ·m T-1 T is the cycle period of the skewed roll; N is the number of skewed roll steps within a single skewed roll cycle; q is the decay coefficient of the number of skewed roll steps within a cycle, f(q)=N·q T-1 ; The initial phase of the roller position; Based on sinusoidal work roll shifting of various widths, the mathematical model function for work roll shifting is obtained as follows: Among them, S max Maximum roller travel, mm; The mathematical model for multi-width work roll slippage is as follows: Where i is the quantity of rolled strip; d r The initial direction of the work roll shifting; L represents the initial step length of the work roll shifting during the initial rolling process, in mm; B n Se represents the width of the strip, in mm. n+1 The step length of the roll during the rolling of the (n+1)th strip is in mm.
3. The method for coordinated setting of bending and shifting work rolls in hot rolling mills applicable to multiple widths, as described in claim 2, is characterized in that... Based on sinusoidal work roll misalignment of various widths, the specific calculation method for the mathematical model function of work roll misalignment is as follows: By setting the center of the axial work roll as the zero point of the coordinate system, the contact position distribution between the strip edge and the work roll is obtained, which can be expressed by the following formula: in, The contact position between the right side of the nth strip and the work roll is shown in mm. B represents the contact position between the left edge of the nth strip and the work roll, in mm. n S represents the strip width in mm. n The amount of roll shifting for the strip, in mm; The contact position between the edge of the work roll and the strip is set as the marker point for the periodic reciprocating movement. The contact position of the strip at the right contact point, and the contact position of the edge of the (n+1)th strip are as follows: in, The contact position between the right edge of the (n+1)th strip and the work roll, in mm; Se n+1 The step length of the hopping roll during the rolling of the (n+1)th strip, in mm; The position of the shifting rolls during the rolling of the (n+1)th strip is given by the following formula: Therefore, based on sinusoidal work roll misalignment of various widths, the mathematical model function for work roll misalignment is as follows: Among them, S max The maximum roller travel distance is in mm.
4. The method for coordinated setting of bending and shifting work rolls in hot rolling mills applicable to multiple widths, as described in claim 1, is characterized in that... Based on the optimization objectives of the mathematical model for hot rolling work roll shifting, multiple objective functions are designed, including wear uniformity, thermal crown control, and plate crown accuracy. The specific steps are as follows: In the initial stage of rolling, the work roll shape is used as the objective function to optimize the mathematical model of work roll misalignment. The objective function f1 is as follows: Where, x w and x d The distance from the center and edge of the work roll body, in mm; To even out work roll wear, the relative value H of the cat-ear shape change of the work roll is... c / H c0 The relative value of wear width change B w / B w0 The weighted average of the values is used as the objective function to optimize the mathematical model of the work roll shifting. The objective function f2 is as follows: Where c1 and c2 are weighting coefficients, with values ranging from 0 to 1, ensuring that c1 + c2 = 1, and are respectively set to 0.5; To ensure that the exit crown of the hot-rolled strip reaches the final target plate crown, the objective function f3 is to minimize the deviation of the strip exit crown, as shown in the following formula: Among them, C exit C represents the exit convexity of the i-th frame, in mm; target The target convexity of the strip is in mm.
5. The method for coordinated setting of bending and shifting work rolls in hot rolling mills applicable to multiple widths, as described in claim 1, is characterized in that... The parameters of the mathematical model for hot rolling work roll shifting are optimized using the NSGA-II multi-objective optimization algorithm. Set parameters such as population size, crossover rate, mutation rate, and maximum number of iterations, initialize the population individuals, and ensure that all constraints are met; Non-dominated sorting and crowding calculation: Perform non-dominated sorting on individuals in the population and calculate individual crowding to ensure solution set diversity; Selection, crossover, and mutation operations are performed; a binary tournament selection method is adopted, and simulated binary crossover and polynomial mutation operators are used to generate new individuals and update the population; Termination condition determination; determine whether the maximum number of iterations has been reached, and output the optimized working roll shifting model parameters.
6. The method for coordinated setting of bending and shifting work rolls in hot rolling mills applicable to multiple widths, as described in claim 1, is characterized in that... The steps for obtaining the optimal work roll bending force and roll shifting amount based on the deviation between the target plate crown and the actual crown, combined with the change in strip width, are as follows: Initialize the rolling process parameters and set the work roll shifting method; Based on the deviation between the target plate crown and the actual crown, the optimal bending roller force is dynamically searched within the bending roller force adjustment range; By combining the changes in strip width and iterative optimization conditions, the optimal work roll bending force and roll shifting amount are output.
7. The method for coordinated setting of bending and shifting work rolls in hot rolling mills applicable to multiple widths, as described in claim 2, is characterized in that... At the start of the silicon steel rolling process, in order to balance uniform wear of the work rolls and rolling stability, a larger step size is used at the center of the rolling process and a smaller step size is used at the extreme positions to increase the frequency of contact between the strip and the work rolls.