### DEVELOPMENT OF A MATHEMATICAL MODEL FOR CALCULATION OF THE PASS SCHEDULE FOR A PLATE MILL

#### Companhia Siderúrgica Paulista - COSIPA, Brazil

The steel plate export market is a very competitive and selective one. If a steelmaker wants to keep its share, he must offer products with excellent quality at an affordable price.

One of the keys to achieve this goal is the automation of the plate rolling process. In fact, this is one of the major stages of evolution in a steelplant, as it improves both process and product consistency, minimizes costs and thus promotes a significant increase in the process cost/benefit ratio.

The automation of a plate mill requires the development of several mathematical models for the simulation and quantitative description of the industrial process involved. In this specific case, the model for calculation of the pass schedule during rolling is of utmost importance.

During plate rolling, the attainment of tight tolerances both in thickness and width of the finished plate is a function of several process variables, like the draft at each pass, product dimensions, rolling load, roll diameter, strain rate, temperature, hot flow strength of the steel, mill stretch, and so on.

Philosophy of the model. The adopted strategy for the mathematical modelling of the plate mill pass schedule is the maximization of the draft at each pass. The aim of this philosophy is the minimization of the total plate processing time, that is, maximization of the processing line productivity. Of course, this must to be reached without any harmful effect on the quality characteristics of the final product or overload on plant facilities.

Generally speaking, the pass schedule of a plate mill can traditionally be divided in four stages. For each stage, there is a major constraint that defines the calculation of the pass schedule.

Stage A: Just in the beginning of the rolling process, after the exit of the slab from the reheating furnace and hydraulic descaler, the bite angle is the main constraint for the definition of the roll gap. The most important variables at this moment are the roll diameter, pass-line height and the coefficient of friction between rolls and the rolling stock.

Stage B: During this step, the maximum draft per pass is limited by the maximum torque capacity of the electro-mechanical equipment of the rolling stand. The rolling torque is mainly influenced by the contact length between the rolls and the rolling stock.

Stage C: This stage is characterized by maximum values of rolling loads. For this reason, it is very important that the mathematical model for calculation of this parameter carefully considers the effect of all significant variables that influence the rolling load, as this model becomes a major tool for the determination of the pass schedule.

Stage D: As the rolling process is near completion, the main constraint becomes the precise achievement of the final thickness and a correct plate shape. Now, the parameters to be considered for the calculation of the pass draft are rolling load, roll deflection and plate crown.

Model Outline. The mathematical model for the calculation of the pass schedule for the Companhia Siderúrgica Paulista (COSIPA) plate mill was developed based on the knowledge of the limiting conditions of each step of the rolling process and the specific operational conditions of the mill.

The knowledge required for the modelling of stage A was derived empirically from collected real data during mill operation. Some specific details of the rolling process during this step - like a "shape adjustement pass", plan view control (to increase the "rectangularity index" of plate head and tail) and broadening passes - were incorporated to the general pass schedule model.

A sub-model for calculation of rolling load and torque was developed for use during the stages B, C and D. Many researchers developed mathematical theories for the calculation of hot rolling load. An approach which showed good agreement with real industrial rolling load data can be described by the expression

#### P = b ld K Qp

where P is the rolling load, b is the width of the rolling stock, K is the hot flow strength of the steel, ld is the length of contact between the rolls and rolling stock, and Qp is a geometrical factor.

Ekelund, Tselikov, Orowan, Pascoe and Sims proposed different equations for the calculation of the geometrical factor Qp, based on their own experimental rolling data. An analog situation stands for the hot flow strength K: there are many formulas that permit its evaluation from values of temperature, strain and strain rate. The most known expressions were developed by Tegart, Rossard, Misaka, Ekelund and Shida. These three last models include the influence of carbon and alloy content of steel; the others eventually must be previously fitted using real hot strength data from the specific steel being considered.

One can see that, in fact, there are several combinations of rolling load/hot flow strength models. Considering this situation, it was decided to collect real rolling load data during plate mill operation and to make a statistical comparison between this information and the many combinations of geometrical factor/hot strength expressions. In such a manner, one can find out which of these combinations is the best one to estimate the real hot rolling load values from the process parameters supplied to the model.

The material selected for this experiment was a continuously cast, carbon-manganese, 40 kgf/mm2 grade steel. Data was collected from approximately 50 plates. The comparison between real and calculated rolling loads showed that the combination of Sims' equation (for the geometrical factor) and Misaka's formula (for the hot flow strength) is the best approximation with the real world. Its fitness was even improved after performing a linear regression between the calculated and real rolling load data. The mean difference between real and calculated data was 43±13 t, with 95% confidence, and the maximum amplitude difference reached 211 t. These results were considered very satisfactory for a first approximation.

An important factor to be considered during the calculation of the pass schedule is the mill stretch. It was developed a semi-empirical model for the calculation of this parameter from the calculated rolling load and the associated deformations of each structural component of the rolling stand, rolls and bearings. The final result was a polynomial regression equation that must be periodically re-adjusted in order to express precisely the real condition of the stand.

The probability of occurence of shape defects in the rolling stock, like turn-up or turn-down of plate head, must be considered during the definition of the pass schedule. A previous work delimited the specific draft conditions that increase the susceptibility of turn-up apparition. It was then verified that this kind of shape problem is more likely to occur in a given range of values of an adimensional factor m, or "strain penetration index". This parameter is defined as the length of contact between the rolls and rolling stock, divided by the weighted average of plate entry and exit thicknesses during a given pass. This expression was incorporated to the pass schedule model, and so the calculation of draft values that could produce such defect in the rolling stock can be avoided. Initially, this calculation was performed through an equation derived by polynomial regression, but latter it was substituted by a relatively simple artificial neural network that enabled a more precise calculation.

The supression of the formation of waves in the plate depends, on an appreciable extent, of the application of a correct pass schedule. It was shown by Shohet and Townsend that a good plate shape can be achieved through the observation of a constant crown difference/thickness difference ratio between two subsequent passes. A previous experiment performed at COSIPA's plate mill determined the permissible variation range of this parameter during the three last passes, while keeping a good plate shape. Only these passes of the schedule revealed to be decisive for the definition of plate shape.

Each pass in the range D of the schedule must be calculated taking care of the good shape of the rolling stock. This can be done through the calculation of the plate crown from rolling load, roll/plate crown and dimensions. Then a check is made to verify if the resulting plate crown and the corresponding roll gap satisfy the Shohet-Townsend relationship. If not, the value of the draft is calculated again and again, until this specific condition is satisfied.

In addition to these models, some ancillary algorithms were developed to complement plate mill operation control. They include models for the calculation of temperature evolution of the slab during its reheating process, temperature profile along plate thickness during the rolling process and the optimized pass schedule to be applied in the hot leveller, just after plate rolling.

The calculated temperature profile along slab thickness during its reheating process can be compared with real data collected using a digital data logger enclosed in a special refractory box filled with ice and water. This set is assembled in a special test slab, which is submitted to the reheating process. This kind of instrumentation permits to verify the effect of several parameters on the soaking of slabs, like the variation of furnace set points, slab dimensions, fuel types, and so on.

Model Operation. The execution of the pass schedule model for plate rolling is intended to be fully automatic. First, general data about the raw slab and final product (steel composition, dimensions) are transfered from the mainframe computer (which controls the global production flow) to the IBM microcomputer that controls mill screw, main motors, rolling load, temperature and thickness sensors. This machine then calculates the pass schedule, using the mathematical models described herein. The superficial temperature of the rolling stock is measured at each pass. This input data is used for the calculation of a plate equivalent temperature, that is, a weighted mean of the plate temperature profile along its thickness. This correction is made using the plate temperature profile model. This paramter was carefully checked through the comparison of calculated and previously measured values of rolling loads, as it is an important input parameter for the rolling load model.

The stages A, B and C are executed without any checking of the real rolling stock thickness. Immediately before the beginning of stage D , plate thickness is measured and, if necessary, the model adjusts the subsequent pass schedule in order to eliminate any eventual deviation.

During the last step of the pass schedule, plate thickness is measured every two passes. The model then calculates the next two drafts, based on that input data, assuring in such way the precision of the finished plate thickness.

This model was tested during real plate mill operation. The equipment characteristics are depicted in Table I. The material processed during these trial runs was the same kind of steel used for the evaluation of the geometrical factor and hot flow strength models used for the calculation of the rolling load, as described before. The finished plates had thicknesses and widths in the range of 8,0 to 40,0 mm and 2000 to 3000 mm, respectively. The slab reheating parameters and start rolling temperatures (1100oC) were kept constant during all these tests.

Type4-High, Reversible Rougher
ManufacturerMitsubishi Heavy Industries
Nominal Power2 x 6000 HP, D.C.
Rolling Thickness Range6 to 150 mm
Table Width4100 mm
A.G.C.Electrical
Gauge MeterGamma Ray
Roll Speed40 to 100 rpm
Work Roll
MaterialCast Iron
Diameter 970 to 1070 mm
Back-up Roll
MaterialForged Steel
Diameter1800 to 2000 mm
Mill Stretch700 t/mm
Table I: Main characteristics of the COSIPA's plate mill.

The trials showed that the performance of the pass schedule model was very promising. The mean difference between the scheduled and real thicknesses was 0,08±0,03 mm, with 95% confidence; the maximum deviation amplitude was 0,19 mm. The respective results for width were 24±5 mm and 40 mm. The variation ranges were considered satisfactory, since they are within the present normal operational conditions of plate rolling at COSIPA.

Model refinement and upgrading. These favourable results encouraged further refinement of the plate mill pass schedule model developed at COSIPA. The first development made, still now in progress, was the extension of the use of this model to the processing of other steel grades, including microalloyed alloys. A research project is under way, with the aim of determining the hot flow strength of the most processed steel grades at COSIPA's plate mill, using torsion tests. That work includes an extended comparison of mathematical models for the calculation of hot flow strength from temperature, strain and strain rate for each alloy studied. Up to now, seven models are being tested: Misaka, Shida, Tarokh, Hajduk, Tegart, Rossard and Jäckel. However, the most promising solution for this modelling is an inedit approach: the use of artificial neural networks, particularly for microalloyed steels, despite their relative greater mathematical complexity. It is also intended to use this resource to describe quantitatively the influence of alloy content on the hot flow strength of steel. Additional parameters that are being determined for microalloyed steels include non-recrystallization temperature (Tnr) and the temperatures of ferrite transformation start (Ar3) and finish (Ar1).

The use of artificial neural networks is being tested too regarding the direct calculation of the pass schedule. The great advantage of the artificial neural networks is the fact that there is no need to develop an algorithm to calculate the parameter being modelled, in that case, the pass schedule. They can extract the necessary knowledge (that is, the philosophy of the pass schedule) only from the presentation of real data. They can "learn" the specific rules automatically from a data set containing the association of the real applied rolling process parameters and the real plate characteristics obtained. The first trials showed impressive results and, in fact, pass schedules can be fastly calculated using artificial neural networks, yielding dimensionally precise products, since they were correctly trained with genuine process and product data. Another advantage stands from the fact that, as neural networks can be periodically trained, this kind of model can be dynamically retrofitted during plate mill operation, assuring the precision of the calculated pass schedule. In such a manner, this kind of model can become immune to periodical and unavoidable process variations, like inhomogeneities during the slab reheating process, mill stretch alterations, alloy content oscilations, and so on. The effect of these deviations can be included in the model during the periodically performed "re-training".

A possibility to be considered in a near future is the development of expert systems for the calculation of the pass schedule. The aim of this project is to compare the performance of different Artificial Intelligence approaches in the calculation of the plate mill pass schedule.

Finally, last but not the least, COSIPA's plate mill is to be revamped in the near future. This modernization will include the upgrade of all automation and instrumentation systems, including a more sophisticated data collection system. Special attention is being given to the specification of temperature sensors and flatness meters, as these systems play a vital role in the definition of the final product quality. Of course, systems for plate shape control are being considered also.

All these developments in the field of plate mill automation and instrumentation are only a portion of the global work that COSIPA is developing in order to meet consistently the more and more stringent requirements of the export market of steel plate, keeping the satisfaction of its customers.

 Last Update: 14 May 1996 © Antonio Augusto Gorni