최적제어 기법을 이용한 철강공정의 일정계획

박 형진, 문 성득, 이 동엽, 박 선원

한국과학기술원 화학공학과

Scheduling of Primary Steelmaking Processes with an Optimal Control Approach

Hyungjin Park, Sungdeuk Moon, Dong-Yup Lee, Sunwon Park

Dept. of Chemical Engineering, KAIST

Introduction

Numerous scheduling methodologies that are important to process industries have been developed because the effective scheduling of operations has the potential for high economic return. (Bok and Park, 1998). However, most algorithms would not be a viable solution from the practitioners point of view due to the fact that they do not leave enough flexibility to react to unpredictable events.

The optimal control approach focused as a useful methodology for very complex scheduling problems (Kogan and Khmelnitsky, 1996) is to find acceptable control functions. In order to use the optimal control approach for the scheduling problem, a two-level hierarchical approach is required (Brandimarte et al., 1995). The first level is to produce reference trajectories by a relaxed model of the scheduling problem. In the second level, the optimization problem is solved to track the reference trajectory to enforce the actual model in real time. Brandimarte et al. (1995) presented that the optimal control approach could overcome the defects usually faced by classical job shop scheduling methods, such as computational complexity, limited ability to model all the features of real manufacturing environments and to cope with unpredictable events. Park et al.(1999) applied this method to operation scheduling of refinery processes that convert crude oil mix into a variety of marketable products. The scheduling problem in steel plants is known to be one of the most difficult of industrial scheduling problems. In this paper, we apply the optimal control algorithm to the short-term scheduling of primary steelmaking process, as a middle stage of the whole steelmaking process.

Problem Definition

The overall steelmaking process can be classified into three stages: ironmaking, primary steelmaking and finishing. Since many literatures (Tamura et al, 1998) have described detail operations for each steelmaking process, attention will be focused on primary steelmaking for the purpose of this paper.

The primary steelmaking stage accepts the supply of hot iron from the BF and transforms it into semi-finished products in a variety of grades (specific metallurgical compositions of steel), shapes and dimensions, as shown in Figure 1. At the beginning of the primary steelmaking process, hot iron is combined with scrap steel and further heated in a basic oxygen furnace (BOF).  Molten steel is transferred via ladles from the BOF, to a ladle metallurgical facility (LMF).  If the desired carbon content was not achieved at the BOF, additional heat can be applied at the LMF to further reduce the carbon. Also, additional alloys can be added to the heat to adjust its chemical composition. Molten steel from the LMF next moves to the casting step, where the grade-specific steel first takes solid form. Steel from the tundish flows into a long series of molds that simultaneously contain it, cool it, and move it forward. The formed steel section solidifies from the outside surface inward and exits from the series of molds as a fully solid shape. After steel has been cast into its initial solid shape, it is hot-rolled to the shape and size that will represent the final product of the primary steelmaking process.

In scheduling problems for the primary steelmaking process, the processing time should depend on the amount of the material being processed by the specific equipment. And there are some assumptions for the operation rules and model formulation.

Model Formulation

The general mathematical model for semi-continuous processes with variable size of a production is considered. The primary steelmaking process is similar to a sequential multipurpose batch process except containing continuous processing units, such as CC. Therefore, a continuous time representation described in Moon and Hrymak (1999) is adopted here. In their model, time slots are used to simultaneously assign products and its production paths. The detail model equations are described in Lee (2000).

In the case when product i is assigned to event slot k, a production path r corresponding to product i should be simultaneously allocated to the same slot. In order to consider the assignment product i and production path r to time slot k, integer variables Xikr is defined as

And the objective can be formulated to minimize the makespan, MS.

Objective: Minimization of MS

(1)

In eq. (1), one of the completion times for continuous casters, heavy plate mill (HPM), hot strip mill (HSM), and the rod rolling (RR) for a given product i is considered as makespan of the primary steelmaking process because one of these units may take the longest processing time to complete operations for product i at the final event slot. On the basis of the above notation and the objective function, the model involves assignment constraints. And the additional constraints should be required to schedule this system, such as time horizon, capacity, material balance, duration, demand, and sequence constraints. (Lee, 2000)

Optimal Control Approach

1. First step: Relaxed model and reference trajectory.

For production scheduling problems, a suitable relaxed model may be obtained by approximating the discrete material flow by a continuous flow based on the assumption that parts are infinitely divisible. Xikr are assumed as continuous variables with range from 0 to 1. We add then the bounds for Xikr as follows:

(2)

By assuming Xikr as continuous variables, the problem can be changed to LP problem. And the solution of this relaxed model problem is used as the reference trajectory in the second step.

2. Second step: Tracking reference trajectory

Tracking the reference trajectory for steel production flow is accomplished by penalizing any deviations from the reference trajectories of Xikr. Therefore, the new objective function consists of penalty terms of binary variable deviations.

(3)

In this equation, PENT is the penalty constant for the deviations with binary variables and reference trajectory. We can obtain the tracking results by solving the MILP problem with new objective function. Using TOT as the objective function requires for less number of iterations than using MS as the objective function. Because according to the penalty term, the search area of the scheduling problem can be restricted to a specific region including the reference trajectory. As the result, we can obtain the optimum or sub-optimum with less calculation time than when we solve the model (1) directly.

Fig. 1. Process flow for the large scale scheduling problem of primary steelmaking process.

Results

Several examples are provided to illustrate the capabilities of the proposed scheduling method and it will be extended to a large scale primary steelmaking process similar to real steel plants. The results of a large scale example is presented in this paper.

The modeling system GAMS (Brooke et al., 1998) is used for setting up the optimization models and the problems are solved by CPLEX 6.5 on a SUN Ultra I.

In order to address a scheduling problem for the primary steelmaking process, we consider a primary steelmaking process, which consists of two BOFs, five LMFs (three different types), four CCs, and three rolling mills, as illustrated in Fig. 1. Moreover, this process involves three different grade steels, Products A, B, and C, and five shapes for each product. We assumed that every product could be manufactured by using one of five LMFs. Table 1 lists the process data for this problem.

Table 2 compares the computational results of the traditional MILP formulation with the optimal control approach proposed in this paper. Fig. 2. shows the reference trajectories and tracking trajectories of product amounts of the case of A type plate and coil. In this system the solution obtained by the proposed method shows the same value as the global optimum obtained by traditional MILP formulation. However, the proposed algorithm shows excellent performance for a complex problem that has been shown to be very difficult to solve. In this sense, the optimal control approach can deal with real time-disturbances for the very complex scheduling problems.

Conclusions

The short-term scheduling problem arising in the primary steel making process has been addressed in this paper. A new scheduling algorithm based on the optimal control theory is proposed. The LP-based branch and bound method was applied to the model with the optimal control approach. An example of the primary steelmaking process are presented to show the performance of the optimal control approach. Although the proposed algorithm may not guarantee the global optimal solution, it reduces computational time by an order of magnitude, as shown in the example.

Table 1. Processing data for a primary steelmaking process problem.

	Equipment
	Prod	BOF1 (u1)	BOF1 (u2)	LMF1 (u3)	LMF2 (u4)	LMF3 (u5)	LMF4 (u6)	LMF5 (u7)	CC1 (u8)	CC2 (u9)	CC3 (u10)	CC4 (u11)	HPR (u12)	HSM (u13)	RR (u14)
Processing Time (sec/ton)	A	8.0	8.0	10.0	10.0	11.0	11.0	12.0	10.0	10.0	30.0	30.0	32.0	18.0	42.0
	B	8.0	8.0	11.0	11.0	12.0	12.0	13.0	10.0	10.0	30.0	30.0	32.0	18.0	42.0
	C	8.0	8.0	12.0	12.0	13.0	13.0	14.0	10.0	10.0	30.0	30.0	32.0	18.0	42.0
Demand of Product (ton)	A	-	-	-	-	-	-	-	-	-	-	-	200	600	300
	B	-	-	-	-	-	-	-	-	-	-	-	200	600	300
	C	-	-	-	-	-	-	-	-	-	-	-	200	600	300

Fig. 2. Tracking reference trajectory of product amounts.

Table 2. Computational results of the scheduling problem for the primary steelmaking process.

	Continuous Variables		Binary Variables	CPU time (s)	Iterations	Total Profit	Optimum (MS)
Trad. MILP		2,487	110	85,782.7	18,757,753	153,841	45,900
Proposed Method	1 step	2,597 3,687	0 110	9.4 431.3	41,429	155,873	45,900
	2 step

References

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Brandimarte, P., Ukovich, W. and Villa, A., □Continuous flow models for batch manufacturing: a basis for a hierarchical approach, Int. J. Prod. Res., Vol. 33, No. 6, 1635~1660, 1995

Brooke, A., Kendrick, D. and Meeraus, A., □GAMS: Users Manual, Scientific Press, Redwood City, CA, 1998

Kogan, K. and Khmelnitsky, E., □An optimal control model for continuous time production and setup scheduling, Int. J. Prod. Res., Vol. 34, No. 3, 715~725, 1996

Lee, D. Y., Environmentally Friendly Scheduling for Primary Steelmaking Processes, M. Thesis, KAIST, 2000

Park, H., Bok, J. and Park, S., Scheduling of Refinery Processes with An Optimal Control Approach, The 8th APCChE Congress, 315-318, 1999

Tamura, R, Nagai, M., Nakagawa, Y., Tanizaki, T. and Nakajima, H., Synchronized Scheduling Method in Manufacturing Steel Sheets., Int. Trans. Opl. Res., 5, 189,.1998