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This paper presents an investigation into an optimal scheduling for tandem cold rolling mills based on GAs. The scheduling process includes the following steps:
1. A rolling model is set up to establish the relation-ships among the rolling parameters.
2. The constraints in the GA-based optimization, i.e., the cost functions and validity checks, are defined. The cost functions include the power distribution cost function, the tension cost function and the optimum flatness condition. The validity checks include the roll force and torque limitations, work roll speed references, strip exit thickness, and threading conditions and tension force limitations. Once the effective cost functions have been properly constructed, the optimum scheduling problem is transformed into a nonlinear optimization problem.
3. The GAs are used to optimize the schedule. A total cost function is employed as the fitness function of the GA during the search for optimized rolling parameters.
4. The optimal rolling parameters generated from the GA optimization procedure are further checked against the practical rolling constraints, to ensure that the optimal rolling parameters would not result in unrealistic settings.
The paper is organized as follows: the basic principles of tandem cold rolling mill scheduling are described in Section 2; the GA-based optimization for scheduling is addressed in Section 3, followed by numerical experiments and conclusions.
2. Basic scheduling for tandem cold rolling mills
The basic procedure for the scheduling of tandem cold rolling mills is usually based on past experience, on trials or on rules of thumb. A typical scheduling procedure for the setup of tandem cold rolling mills is illustrated in Fig. 1.
As an illustration, the following steps describe such a semi-empirical scheduling procedure.
1. Based on the coil data and semi-empirical stand thickness-reduction patterns R′ i(1≤i≤N); N is the number of stands), a reduction at each stand is allocated. The coil data includes strip material grade, strip width W, strip entry thickness H and strip exit gauge h. The stand thickness-reduction patterns are empirically generated on the basis of providing uniform power distribution and consistent flatness at each stand. The patterns vary with total mill reduction rate and last stand conditions (namely, shot-blast or bright rolling mode).
2. According to the empirical equations, inter-stand tension stresses (including front tension stress t f and back tension stress t b ) are determined. These tension stresses depend on the strip width, nominal yield stress deviation of the coil material, and the mill exit thickness at each stand, generated in Step 1.
3. Based on the Bland-Ford-Hill force formula, the roll force Pi（1≤i≤N）at each stand is calculated. The formula is a function of variables including the deformed work roll radius R′w, the average yield stress k-, the stress state coefficient Q p, the front and back tensions determined in Step 2, the strip width, strip entry and exit gauges at each stand, and the friction coefficient at each stand.
4. The forward slip ratio ƒ i (1≤i≤N) at each stand is calculated from the Bland-Ford forward slip formula, hence determining the location of the neutral point of roll bite.
5. Based on the torque formula derived by Bryant, the roll torque G i (1≤i≤N) for each stand is determined. The torque formula is a function of the work roll diameter D w, the angle of contact, the yield stresses at the roll bite entry k(H) and exit k(h ), and the front and back tension stresses.
6. According to the constant mass flow principle, the roll speed ω i (1≤i≤N) at each stand is
calculated. The forward slip factor and the motor droop are considered in the calculation.