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7. The power P wi (1≤i≤N) at each stand is estimated, based on the results of the roll speed and
the roll torque in Steps 5 and 6.
Fig. 1. A typical rolling scheduling procedure for the setup of tandem cold rolling mills.
8. The validity of the rolling parameters is checked to ensure that none of the rolling parameters has exceeded the mill capability; for instance, the limitations of the physical capability of the rolling mill, and electrical requirements.
After the schedule has been determined, the exit strip thickness and speed of each stand, the inter-stand tension stresses, etc., are employed for further calculation of the corresponding actuator references. On the basis of these references, the rolling mill will then be preset to roll the required product.
3. Optimized scheduling
A schedule is a sequence of elements, where every element is considered as the aggregate of all the important system parameters, described as a combination of their respective cost functions. An optimal sequence is arrived at wherever there exists a minimum mismatch between succeeding sequence elements. Optimum rolling scheduling deals with the selection of rolling mill schedule parameters, such as the reductions and motor speeds, which lead to the desired thickness, surface finish and shape for a given product at a minimum cost. In fact, these key schedule parameters fall at corresponding intervals in the data space as described in Section 3.3.1. The objective of the optimal scheduling is to find the best presetting for the parameter combination. The optimization is essentially a nonlinear data-sampling process. Usually, the effective cost functions are first defined, and then optimization techniques are employed to search for the optimal solution. In this paper, an optimization methodology based on GAs is employed to solve the mill-scheduling problem.
3.1. Cost functions
During optimization, one of the constraints should be a constant mass flow at each of the stands. At the same time, the optimization should meet the requirements of all the constraints, and minimize the cost of the rolling operation. The cost functions include the power distribution cost function, the tension cost function and the perfect shape condition.
3.1.1. Power distribution cost
One of the most important objectives in the scheduling is to provide a uniform power distribution and consistent flatness at each stand. Hence, the power distribution is a suitable measurement of how well the current schedule is meeting this requirement. The power distribution cost function is defined as follows:
where K1 is the weighting constant, and N is the total number of stands. The power at each stand is calculated according to the following equation:
Where ω* i and G i are the work roll rotational speed and torque at stand i, respectively.
According to the constant mass flow guideline, Wh iv i=WH I V i=constant, where W is the strip width, Hi and hi are the input and output strip thicknesses, and Vi and vi are the input and output strip velocities, respectively. By considering the forward slip factor (Eq. (3)) and the motor droop (Eq. (4)), the work roll speed can be described as Eq. (5):
where h5 and v5 are the strip thickness and speed at the exit from stand 5. The roll torque formula can now be derived as follows:
Where
Here, t f and t b are the front and back tension stresses at the stand, respectively; k(H) and k(h ) are the yield stresses at entry and exit points of roll bite, respectively; n is Poisson's ratio of the strip; R′ is the deformed work roll radius which is described by Eq.(15) according to Hitchcock's equation; and P is the roll force that can be obtained based on the Bland-Ford-Hill force calculation formula (Bland and Ford,1948, 1952), namely