Integrated scheduling of rolling sector in steel production with consideration of energy consumption under time-of-use electricity prices
Introduction
The rolling sector is the major and most profitable sector in steel production, where semi-finished coils are further processed to various types of highly individualized finished products with high added-value. The average sale price of galvanized product (one of main products in rolling sector) during December 12-15, 2017 in China was 5409 CNY/ton, which is 19.8% higher compared to 4516 CNY/ton, the price of hot rolled product (data from China Iron & Steel Association). Therefore, cost reduction is becoming increasingly important to the steel enterprise, especially when facing supply-front reform which aims at promoting lean production. Nowadays, many steel enterprises are encountering decreasing profit margins due to the rising prices of electric power and raw material, which makes it critical to control costs to remain competitive. From the various methods to reduce production costs, optimizing production management is an effective approach, for which no additional investment is required.
Usually, a rolling sector consists of acid rolling, annealing, rewinding and a series of metal coating sections. Fig. 1 shows a simplified distribution of the rolling sector, which includes some typical sections. Each section uses its own criterion for scheduling the production and most of time they organize the production independently without coordination with other sections. Usually, this mode leads to some undesired situations such as unbalanced material flow, shortage of feedstock at downstream section and losses in production efficiency, which often yields extra production cost and decreased profit. This implies the need for proper integrated scheduling over all sections in the rolling sector to guarantee optimal production and to reduce production costs.
Another factor that represents a rather considerable part of production cost is electricity consumption. Steel production involves several energy-intensive processes, acid rolling for example. In a small to medium size steel company in China with 3.5–4 million tons of output, the monthly average electricity consumption in acid rolling sector is 7630 MWh, which consumes nearly 70% of the whole energy consumption of acid rolling. A major operation in acid rolling section is to roll the thick steel strip into a much thinner one. All the rolling mills are driven by electricity and it consumes a large amount of electricity to generate rolling pressure during production. Due to the rapid increase of power demand, the power grid is now using time-of-use pricing strategy on industry customers to improve the utilization of electricity power on the demand side and keep stability of the power supply. Pricing of electricity can significantly affect the production mode and profitability of steel production. In this context, the steel enterprise can take advantage of production flexibility and pricing scheme to gain potential benefits, and reduce energy cost by organizing production on proper pricing time period.
Production scheduling has become a major optimization problem of industrial significance. Harjunkoski et al., 2014, Maravelias and Sung, 2009, Méndez et al., 2006 reviewed the scope for industrial applications of scheduling models and solution methods. As for the steel industry, most of the literature focuses on the upstream process of steel production involving steel making and casting process. Harjunkoski and Grossmann (2001) addressed a decomposition approach to solve complex scheduling problem in steel making process. Tang et al. (2000) proposed a multiple traveling salesman model for hot rolling scheduling. Pacciarelli and Pranzo (2004) developed a model of steelmaking-continuous casting production based on alternative graph formulation with detailed constraints that are relevant for the scheduling problem. Tang et al. (2001) gave a comprehensive analysis of scheduling of integrated steel production. Li et al. (2012) addressed a multi-stage scheduling problem of steelmaking process with batching decision at the casting stage. A unit-specific event-based continuous time MILP model is developed and a modified rolling horizon method is proposed to solve the problem. Compared with much research on steelmaking scheduling, not much attention has been paid to the integrated scheduling of the rolling sector, which includes multiple finishing sections of steel production. The rolling sector is characterized by different production structure from steel making and continuous casting process. Production in the rolling sector has features of low-volume and high-variety, and more complex supply networks between the associated sections. Therefore, it is challenging to schedule the processing tasks for all sections in the rolling sector with respect of technological constraints while seeking overall optimal profit.
From the viewpoint of time representation, models can be classified into continuous-time model and discrete-time model. Compared with discrete time, continuous time models can account for timings more accurately and are closer to the real production (Floudas and Lin, 2004). The approaches to formulate continuous time models include the time-slot concept (Erdirik-Dogan and Grossmann, 2008, Pinto and Grossmann, 1996a, Pinto and Grossmann, 1996b, Pinto and Grossmann, 1994), order precedence and event based concept. The slot concept and order precedence methods are suitable for sequential processes, while the event based concept is more suitable for network processes (Floudas and Lin, 2004). According to the characteristics of the considered scheduling problem, the slot concept is adopted in this paper.
The management of power consumption in industrial processes has recently received much attention. Mitra et al. (2012) investigated production planning of continuous process with respect of time-sensitive electricity prices based on discrete time representation. Castro et al. (2011) addressed optimal scheduling of continuous plants with energy constraints based on discrete time representation. Castro et al. (2013) proposed RTN formulations for industrial demand side management of a steel plant based on discrete-time representation. Zhang et al. (2017) proposed scheduling models based on RTN formulations and discrete time representation that incorporate the EAFs’ flexibilities to reduce the electricity cost. For the models with discrete time representation, it is easier to calculate the electricity consumption due to the fixed time interval for both electricity price and processing task. In contrast, for continuous time models, it is more difficult since the relevant times of event points or time slots are variables to be determined. Castro et al. (2009) proposed a new continuous time representation for handling variable electricity cost based on a resource-task network (RTN) representation. Nolde and Morari (2010) addressed the electrical load tracking scheduling of a steel section and proposed a general approach with 6 binary variables to represent task-time interaction relationships for continuous time formulations, which has been used and extended by other researchers. This approach is adopted by Hadera and Harjunkoski (2013) and modified by Hadera et al. (2015). Hadera et al. (2015) addressed the scheduling of the melt shop section of a stainless steel production plant, where a 2-binary variable approach was proposed compared with the 6-binary variable approach. Apart from steel production, Castro et al. (2014) applied the concept in Nolde and Morari (2010) to optimize the maintenance scheduling in a gas engine plant. They formulated the concept as GDP constraints and found a tighter model for accounting for electricity consumption. In this paper, we improve these GDP constraints in Castro et al. (2014) to deal with electricity consumption. While in Castro et al. (2014), the number of time slots is known a-priori, in this paper the number of time slots is a variable to be determined that results from the campaign decisions in the acid rolling section of the steel plant.
In this paper, we address a new practical integrated scheduling problem with demand-side management consideration, which is derived from the rolling sector of steel production. The main contribution of this paper is the first attempt to determine an integrated scheduling of multiple finishing sections in steel production, and to achieve coordination of the scheduling with energy consumption. Due to the new features and distinctions with other scheduling problems in steel production, former formulations cannot be easily adapted to the problem considered in this paper. A new hybrid MINLP/GDP model is proposed in this paper to address the scheduling to the problem based on the continuous time concept. To take advantage of MILP solvers, the MINLP model is transformed into an MILP model by linearization and hull reformulation.
The remaining paper is organized as follows. Section 2 gives a brief description of the production in the rolling sector and the statement of the integrated scheduling problem under consideration. In Section 3, an MINLP model is established with GDP constraints. Section 4 presents the reformulation of the MINLP model, where the non-linear constraints are linearized, and the GDP constraints are reformulated as MILP constraints. Next, in Section 5, numerical tests are conducted and analyses of the results are made. Section 6 draws the conclusion of this paper and describes future work.
Section snippets
Problem Statement
To explain the considered scheduling problem, a brief description of the production in rolling sector is introduced first. In this paper, a series of typical sections are considered, including acid rolling, continuous annealing, galvanizing, tin plating, and rewinding as shown in Fig. 2. There is intermediate storage between the associated sections. These sections give rise to a multi-stage process. The acid rolling section is the bottleneck process of the rolling sector, which usually has
Mathematical formulation
The proposed MINLP/GDP formulation addresses a multi-stage process in coil production with consideration of various operational constraints and electricity consumption management. The formulation is a hybrid of the slot-based continuous-time concept (see Fig. 3), the immediate precedence concept and GDP. The immediate precedence formulation is known to represent sequence-dependent relationships, which are used to account for changeover costs in this paper. Since the model is based on the
Reformulation
The scheduling model given by Eqs. (1)–(26), (29) and (30) includes non-linear constraints and GDP constraints. We reformulate the model as an MILP by the following steps.
- •
Linearization
To linearize Eq. (9), a set of auxiliary continuous variables (Glover, 1975) are introduced in this paper. Let , then we obtain Eqs. (31)–(35). When binary variable , then Eqs. (32) and (33) are activated and enforce , which is consistent with
Case study and numerical experiments
To test the performance of the integrated scheduling model, numerical experiments are carried out on instances based on typical data of practical production. The models resulting from linearization and hull reformulations of the GDP constraints were implemented in GAMS 24.7.3 and solved with CPLEX 12.6.3 solver with four threads, and default options up to relative optimality tolerance = 0.0001 and 0.01, respectively. The hardware consisted on a laptop with an Intel i7-6500 U (@2.50 GHz) with
Conclusion
In this paper, we have considered the production in the multiple finishing sections of steel industry, where the target is to obtain schedules of all these sections under various technical constraints. Especially for the energy intensive sections, electricity consumption is also optimized based on demand-side management techniques. The integrated scheduling problem of the rolling sector with consideration of energy consumption under time-of-use electricity prices was proposed to optimize the
Acknowledgments
The authors gratefully acknowledge the financial support from National Key Research and Development Program of China (2016YFB0901900), the Major International Joint Research Project of the National Natural Science Foundation of China (Grant No. 71520107004), the Fund for Innovative Research Groups of the National Natural Science Foundation of China (Grant No. 71621061), the Fund for the National Natural Science Foundation of China (Grant No. 61374203), the 111 Project (B16009), the Center for
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