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3. The coefficients of the cost function are assigned to 0.3333 for K1, 0.3333 for K2, and 0.3333 for K3:
In the numerical calculation, the assumed rolling condition is: the work roll diameter is 500 mm, the backup roll diameter is 1300 mm, the roll face width is 1426 mm, the strip width is 960 mm, the strip entry gauge is 2.5 mm, and the strip exit gauge is 0.5 mm.
3.3.2.2. Optimization results.
The optimized results are listed in Table 1 in comparison with the empirically based rolling schedule. The distributions of roll force, power, work roll speed, tension force and torque at each of the five stands are shown in Figs. 5-9 for the purposes of comparison between the schedules under the empirical method and the GA-based optimized approach. The data describing the costs of power, tension and shape, and the total cost in Table 1 have been normalized.
As defined in Eq. (2), the power is the product of the work roll rotational speed and torque. Fig. 7 shows the work roll speeds for all the five stands under different schedules, including the semi-empirical schedule, with three optimized schedules. Figs. 6 and 9 demonstrate the power distribution and torque for each stand under the different schedules. From Fig. 6, conclusions can be drawn that the power distributions under optimized schedules are more uniform than those under the semi-empirical schedule. The roll force shown in Fig. 5 is an important factor during rolling, in addition to the results for the power, the tension and the strip flatness. Although the tension distributions under the different schedules in Fig. 8 are not obviously similar, the tension distributions under the three optimized schedules are likely to be kept midway between the lower and upper limits for each neighboring pair of mill stands, which is also numerically demonstrated in Table 1
Table 1
Comparison of an empirical rolling schedule and the optimized schedules
Fig. 5. Roll forces for different schedules.
Fig. 6. Power distributions for different schedules. Fig. 8. Tension forces for different schedules.
Fig. 7. Work roll speeds for different schedules. Fig. 9. Torque at each stand for different schedules.
3.3.3. Discussion of the results
As illustrated in Table 1, the optimized schedules result in a lower cost of power, more uniform distributions of roll forces at the different stands, and better shape. The tension forces are more likely to be kept midway between the upper and lower limits. For example, the power distribution cost value generated by the GA approach is reduced by about 33.4% in comparison with the schedule based on the semi-empirical formula when the weight coefficient of the power cost function is assigned to a value of 0.4. The cost of power increases from 0.6673 to 0.9714 when the weight coefficient is assigned a smaller value from 0.4 to 0.3333. The cost of tension is significantly reduced when the weight coefficient of the tension cost function is changed from 0.2 to 0.3 and then to 0.3333. The cost of tension in the optimized schedule 1 is slightly greater than that in the empirical schedule. This is the result of a small weight coefficient of the tension cost function, the value 0.2, which means that less attention is paid to the tension distribution. With the change of the weight coefficient of the shape cost function from 0.3333 to 0.4, the shape cost is slightly decreased from 0.9986 to 0.9963. Although the shape cost is slightly reduced, this is an important move towards better flatness. The optimized schedules shown in Table 1 demonstrate that the weight coefficient can be assigned a certain value, based on the different significance considerations given to the shape, power and tension distributions. Generally, the weight coefficients of the power, tension and shape costs should be based on the significance given to the shape, power and tension distributions, respectively.