Optimization of Primary Steelmaking Purchasing and Operation under

Article pubs.acs.org/IECR

Optimization of Primary Steelmaking Purchasing and Operation under Raw Material Uncertainty David Gerardi, Thomas E. Marlin, and Christopher L. E. Swartz* Department of Chemical Engineering, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada, L8S 4L7 ABSTRACT: A centralized optimization strategy is proposed to determine optimal raw material purchasing and plant operation practices as applied to primary steelmaking in the steel processing industry. Raw materials are purchased on the open market and include coal, iron ore pellets, and scrap steel. There are many raw material vendors, providing products varying in quality and price. It is desired to determine the least costly method of both purchasing and processing the raw materials to make steel of acceptable quality. A model for primary steelmaking is developed using a combination of mass balances and empirical relationships. The model, in addition to process constraints, is combined with an economic objective function and the resulting optimization problem solved using a mixed-integer nonlinear programming (MINLP) solver. Case studies illustrate the strong connection between plant sections, and the significant impact that the carbon, volatile matter, and phosphorus content of the coals and pellets have on raw material selection. Raw material uncertainty is incorporated using two-stage stochastic programming. The results indicate that by making a slightly more expensive raw material purchase, the frequency of constraint violation during processing can be significantly reduced.

1. INTRODUCTION Raw material purchasing is a very important aspect of steel production as it comprises the vast majority of overall operational costs. Decisions must be made for various subareas of the plant with regard to both what types of material should be purchased as well as how to subsequently operate the plant. Due to the interactions among subareas, cost-effective decisions are very difficult to make. This is compounded by the fact there are often numerous raw material suppliers offering products of differing price and quality. In this paper, optimization of raw material purchasing and plant operation is studied, based on application to the steel processing industry. The flow sheet is broken down into three subareas, cokemaking, ironmaking, and steelmaking, each with an individual raw material to purchase, specifically coal, iron ore pellets, and scrap steel. Each area is modeled using a combination of mass balances and empirical relationships. An economic objective function is developed and combined with the models and constraints in a centralized optimization formulation which is solved using a mixed-integer nonlinear programming (MINLP) solver. The goal is to determine the minimum purchasing and operation cost in order to produce a specified quantity of steel in a single time period. Uncertainty with respect to raw material quality further complicates the decision process as variation in composition can have detrimental effects on plant feasibility and subsequent production costs. A method to incorporate uncertainty is developed in order for effects of composition variance to be included in the optimization solution. The focus of this study is on primary steel processing comprising three subareas of cokemaking, ironmaking, and steelmaking, illustrated in Figure 1. Scheduling of downstream operations such as casting is not included, as it is beyond the scope of this study. The first process step involves production of coke for the blast furnace, where iron ore is reduced, to produce liquid hot metal. From the blast furnace outlet, the hot © XXXX American Chemical Society

metal is transported to a Basic Oxygen Furnace (BOF) where oxygen is blown through the vessel to oxidize nonferrous impurities. The Electric Arc Furnace (EAF) operates in parallel to the BOF and produces liquid steel from scrap metal alone. The BOF and EAF batches are then taken to the Ladle Metallurgy Facility (LMF) where product-specific alloys are added to satisfy customer orders. Finally, this hot steel is processed in the casters where it is cooled and drawn into slabs of finished steel. In the remainder of this section, reviews are presented for previous work on modeling, optimization, and dealing with uncertainty in primary steel processing. The production of coke is extremely important for the operation of the blast furnace. It provides a permeable support in the blast furnace through which iron and gases can pass; the carbon content of coke provides a majority of the energy input to the blast furnace; and it supplies carbon which reacts and produces gases required in the reduction of iron oxides.1 Diez et al.2 cite the physical strength of coke to be the most important factor in production. In this review paper, the authors identify good coke strength by testing for Coke Strength after Reaction (CSR) and discuss models that have been developed to predict this characteristic. The models referenced consider the impact of coal petrographical, rheological, and chemical parameters on the final coke strength. Petrographical properties are those which are studied on a microscopic level and include the coal reflectance and mineral content. Rheological properties are related to the fluidity of coal such as the dilation, contraction, and temperature at which it begins to soften. Zhang et al.3 develop a CSR model and Special Issue: John MacGregor Festschrift Received: December 20, 2012 Revised: May 31, 2013 Accepted: June 5, 2013

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dx.doi.org/10.1021/ie3035543 | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Industrial & Engineering Chemistry Research

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Figure 1. Steel production flow diagram. [Icons used by permission of the American Iron and Steel Institute.]

include maximum fluidity, volatile matter content, inert content, and mineral catalysis index as predictor variables. Todoschuk et al.4 include the conditions of the coking process in the creation of a linear CSR model. These include factors such as the bulk density of the coke oven after the coals are charged and the final oven temperature. They also provide models for two additional coke parameters not given much attention in the literature. The first is that of coke stability which is similar to CSR but measures strength under far lower temperatures. The second is coke oven wall pressure which measures the exertion of forces by the coal mass on the oven walls as it transforms into coke. These two models involve only petrographical coal properties and oven operating conditions as predictor variables. Pearson5 reviews several CSR models and remarks that, despite effort in the cokemaking community, there is no universally accepted model. In this previous work, modeling attempts with respect to cokemaking are focused on the prediction of CSR, while other coke properties such as stability and oven wall pressure have not been developed in great detail. Several models for blast furnace operation have been proposed. A simplified blast furnace model is presented by Davenport and Peacey6 where steady state mass and energy balances are applied. The model was validated using data from blast furnaces operating at several geographic locations. Making assumptions about coke, exit hot metal composition, and inlet air temperature, the model becomes linear and can be included in a linear optimization program. The authors then include physical constraints such as maximum air flow and oil consumption to show how optimization can effectively reduce the overall cost of production. Ertem and Gürgen7 collected data from a single blast furnace over an entire month in order to develop complete mass and energy balances. This included measurements of blast air intake, moisture contents, temperatures, and chemical analysis of exit hot metal. The objective was to hold production as close to steady state as possible to investigate possible energy recovery from operation. Austin et al.8 present a two-dimensional steady-state blast furnace model that considers four material phases comprising gas, lump solid, liquid, and powder. Chemical reactions, motion, and heat transfer are included in the model. Takatani et al.9 describe a three-dimensional dynamic model for a blast furnace based on mass, momentum, and energy balances. de Castro et al.10 and Nogami et al.11 consider five phases in a blast furnace model. The phases are as defined in Austin et al.,8 with the liquid phase subdivided into molten metal and liquid slag phases. Timevarying behavior is considered, with the model based on mass,

momentum, and energy balances. Jindal et al.12 developed an ironmaking model that comprises all reactions and components involved inside the blast furnace. Due to complexities involved in modeling the various zones within the furnace, the authors assume that the furnace can be partitioned into two areas: raceway and shaft. The shaft represents the furnace from where raw material is charged at the top, until it is liquefied near the bottom. The raceway is located at the bottom and involves the zone where blast air is blown into the furnace and ascends upward through the falling mass charge. The authors combine shaft and raceway models and use blast furnace plant data to estimate parameters such as heat transfer coefficients. Udea et al.13 present a review of mathematical modeling of blast furnaces, which includes an extensive discussion on discrete element methods. The steelmaking process centers around BOF and EAF operation. Johansson et al.14 provide a model to estimate the decarburization rate inside the BOF. The goal of this process is to achieve a specific carbon content after blowing oxygen through the furnace. The proposed model is tested in openloop yielding suboptimal performance. An analysis of the BOF off-gas is then used to provide feedback and is incorporated using a nonlinear observer. Only the components carbon and silicon are included in the model. A comprehensive dynamic EAF model is reported by MacRosty and Swartz15 where four zones are used to represent the different mass phases inside the furnace. A sensitivity analysis was conducted to identify a subset of key parameters to be included in rigorous parameter estimation using several batches of plant data. In a subsequent paper,16 the model was embedded within an economics-based dynamic optimization framework to determine the input trajectories that optimize an economic objective. By contrast, Miletic et al.17 developed partial least-squares (PLS) models using historical batch data from EAF plant operation. Modeling of the scrap purchasing practice was also conducted in order to capture the various scrap grades available for production. These models were then cast into an optimization framework with the objective of minimizing the overall cost and electricity usage. The optimization inputs are the steel demand, internal scrap inventory quantities, and vendor scrap pricing for a single time period. Upon solving, a solution is provided that determines the sources from which scrap should be ordered and the total quantity that is purchased. Optimization of scrap usage was also studied by Bernatzki et al.18 based on BOF operation. Here, a simplified BOF model is based on heuristics for scrap to hot metal addition to the furnace. B



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plant were modeled using a combination of material balances, energy balances, and empirical relationships and a NLP formulated to minimize the specific cost of production of liquid steel. Ghanbari et al.30 developed a NLP formulation to analyze the economic potential of integrating steel production with polygeneration that included methanol production. A configuration in which CO2 stripped gases were recycled to the blast furnace was included. Some studies have considered uncertainty inherent in steel production. Anandalingam31 modeled seven integrated units in steel production using linear empirical relationships developed from industrial data. The cost parameters were considered to be uncertain, with known probability distributions, and an investigation was conducted to determine how implementing different energy technologies can minimize the impact of cost fluctuations. Scenarios were generated using Latin hypercube sampling from the cost probability distributions, and the model was solved for each scenario. The significant variance in the overall purchasing and production costs indicated the necessity of incorporating parameter uncertainty in the model. Rong and Lahdelma32 use fuzzy programming to incorporate scrap steel quality uncertainty into an optimization framework for EAF production. Fuzzy programming uses a membership function to represent the extent to which a parameter is associated with a particular value. Optimization was conducted for various constraint tolerances and a trade off between quality and purchasing cost established. Shih and Frey33 employ chance constraint programming in a coal blending problem. In this approach, constraint violations are allowed up to a certain tolerance level based on a probability distribution. The problem studied by Shih and Frey considers uncertainty in coal quality with respect to the concentrations of sulfur and ash. Chance constraints are written to limit the sulfur and ash violation probabilities to 5% and 10%, respectively. The overall blend cost is a function of coal quality, and the authors seek to minimize both the expected value and variance of cost. Miletic et al.17 also impose chance constraints where impurities in scrap, used as a feed to the EAF, are assumed to be uncertain and follow standard normal distributions. An optimization problem is formulated to select the least costly EAF scrap blend that satisfies the chance constraints. In this paper, a mathematical model of a primary steel processing operation is developed and employed within an optimization framework. A two-stage stochastic programming formulation is used to account for raw material quality uncertainty. In this approach, decision variables are partitioned into first stage and second stage variables. First stage variables are those that are decided prior to uncertainty realization, while second stage variables represent recourse decisions that can be made once the uncertain parameters can be measured. The recourse flexibility captures actual operation well, as, in practice, the steel plant operation can be altered in response to measured raw material properties. The paper is organized as follows. Section 2 describes the development of the mathematical model used in the optimization, and Section 3 describes nominal and stochastic programming optimization problem formulations for the steel processing system under consideration. Case studies that demonstrate the application of the optimization formulations are presented in Section 4 and results discussed. Finally, conclusions are presented in Section 5.

Several studies have taken an integrated approach that considers the interaction between subprocesses in steel production. Dutta and Fourer 19 present a review of optimization applications in the steel industry that includes areas of raw material blending, planning, scheduling, and inventory management. In pioneering work by Fabian,20 modeling was done for the blast furnace, basic oxygen furnace, and rolling mills downstream of the casters. Mass balances for oxygen and silicon are neglected as well as energy related to reduction of impurities such as silicon, manganese, and phosphorus. Fabian21 improved upon his previously developed steel model and included balances on silicon, oxygen, and a more comprehensive energy balance. The model is cast as a linear program (LP) with the total cost of raw materials minimized. Lawrence and Flowerdew22 developed a model that integrated the blast furnace and BOF. The authors note that complete mass and energy balances are not included, and several heuristics are used to express missing relationships. Optimization is carried out using the model in order to minimize raw material and operating costs. The results provide the burden mix charged to the blast furnace, as well as the proportions of hot metal and scrap to the BOF. Bandyopadhyay23 considered a BOF and open-hearth furnace operating in parallel to produce steel. Steady state mass and energy balances are written for both furnaces creating two linear models. Optimization is conducted minimizing utility, raw material, and labor costs. However, the results in this study are limited to steelmaking only and do not consider any costs with regard to cokemaking or the blast furnace. Gao and Tang24 modeled an integrated steel plant by using simple heuristics. The objective of this study was to investigate raw material vendor selection based on factors of quality, price, and delivery time. These three parameters were provided for seven vendors, based on data from a real steel plant, and a linear program was formulated using a weighted objective function. Chen and Wang25 enlarged the scope of the integrated plant to include transportation of raw materials delivered on-site and to the distribution of product to customers. Heuristics were used to represent production in the plant; no component or energy balances were included. The supply chain was modeled as an LP with the objective of maximizing profit based on steel demand from various geographical locations supplied by the steel facility. Steel inventory was studied by Zanoni and Zavanella26 where focus was placed on the integration of EAF steelmaking and the continuous casters. A mixed-integer linear programming (MILP) formulation was developed in order to minimize holding, production, and back order costs over a planning period of one month. While the formulation considers the connection between steelmaking and casting, it does not include any information about costs or impact from ironmaking or cokemaking. Several modeling and optimization studies in steel production have been motivated by environmental considerations. Larsson and Dahl27 formulated a MILP to optimize energy use for steel production. The process model comprised mass and energy balances, combined with empirical relationships. Sutherland and Haapala28 considered electric arc furnace steelmaking and formulated nonlinear programming (NLP) problems to analyze optimal operation under objectives of electrical energy, slag, and off-gas minimization, subject to a number of specification constraints. Helle at al.29 investigated the effect of recycling blast furnace top gas to the combustion zones after CO2 stripping. Process units in an integrated steel C



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2. PROCESS MODEL The following sections provide an overview of how each subarea within primary steelmaking is modeled, with key relationships and assumptions indicated. The reader is referred to the thesis of Gerardi34 for the complete model. 2.1. Cokemaking. Coke is the most important fuel feed to the blast furnace as its combustion provides more than half the required energy input.6 Coke is produced from subjecting raw coals to high temperatures in a coking oven. First, different coal types are blended and charged into the oven. Coke composition significantly impacts blast furnace operation and is dependent upon the composition of the charged coal blend. Coal composition can be broken down into four areas: ash, carbon (C), water (H2O), and volatile matter (V.M.). The carbon value is perhaps the most important component of a coal. High coal carbon content yields a coke with high carbon content and provides a higher heat of combustion per mass of coke. Ash is an undesirable coal component as it represents impurities that must be removed at an additional cost in the blast furnace and BOF. After the coals have been charged to the coke oven, they are heated using a combination of hot recycled gases and/or burned natural gas. As the oven temperature rises, this reducing environment, void of air, causes volatile matter in the coals to be released into the off-gas. The coals then begin to decompose and form plastic-like layers, at the coke oven wall, which then begin to grow toward the middle of the charge until all the coal has transformed into coke. Upon completion, most of the coal ash and fixed carbon remain with only trace amounts of the initial volatile matter. The gas released during coking is considered to be a revenue stream. Aromatic compounds such as benzene, toluene, and xylene are present in small amounts and can be recovered and sold. The remaining volatiles produce gas with a high calorific value that can be reused as fuel in other areas of the plant, including the production of more coke. Final coke strength is very important as it provides a physical packed bed support structure for the iron ore pellets and fluxing agents that are added to the blast furnace.2 Coke strength is measured by two variables: coke strength after reaction (CSR) and stability. CSR is a measure of the coke’s ability to hold up at temperatures similar to those experienced in the lower half of the blast furnace. Stability is measured at room temperature which simulates the environment at the very top of the blast furnace where the iron pellets and fluxes are added. Coke oven wall pressure is another important part of coke production. The plastic layers of coke that first develop against the oven walls immediately begin to exert pressure outward. Over time, this pressure buildup can cause serious damage to the oven lining and may require repair. The cokemaking process begins with the charging of various coal types to the oven, with the total mass given by Mcoal,in =

∑ mccoal c

where MH2O,removed, MS,removed, and MV.M.,removed are the amounts of water, sulfur, and volatile matter removed and wquench is the amount of cooling quench water added to the dry coke. The inert feed gas is not included, as it would appear on both sides of the equation. It is assumed that all water initially present in the coal is driven off during coking; however, not all volatile matter and sulfur are removed. A nonlinear regression was conducted on an experimental data set in order to determine the appropriate fractions of sulfur and volatile matter removed during coking, yielding 25.85% of sulfur and 98.33% of volatile matter. After water, sulfur, and volatile matter are removed, the coke composition can be related to the coal mass fractions via xqcoke =

coke Mout

∀q (3)

where q = {Al2O3, C, P2O5, S, SiO2, Volatile Matter}. The mass fraction of water in the coke after quenching is assumed to be 0.06.35 Limits are imposed in the optimization formulation on the coke strength, stability, and pressure exerted on the oven wall during coking. Developing mechanistic models for these three variables would be very challenging, and therefore, alternative empirical models have been built in order to make predictions. Raw data were obtained from testing conducted in a pilot plant coke oven, containing a total of 77 batch runs. During the experiments, 11 coal blend properties and 11 oven operating conditions were recorded, in addition to the final coke CSR, stability, and oven wall pressure. Coal properties include chemical, rheological, and petrographical measurements, whereas operating conditions are related to such factors as heating rate and coal charge density. Partial Least Squares (PLS) regression was used to obtain the regression coefficients for a separate model for each of the three desired response variables using Prosensus Multivariate 10.02, a commercial software package. The basic premise of PLS regression is that raw data points are projected onto a lower dimensional subspace, called the latent variable space, found from the correlation between variables. The total number of dimensions in the latent variable space is equal to the number of orthogonal directions required to satisfactorily explain the raw data variance in the latent space. Each orthogonal dimension is denoted as a latent variable with the total number being smaller for more correlated data sets. PLS model performance can be evaluated by examining its Q2 value representing the percentage of variance explained, with a higher Q2 value indicating better predictive capability. The PLS algorithm successively adds latent variables until Q2 is maximized. Another useful evaluation tool is the Variable Importance to Projection (VIP) plot, which measures the influence of each variable in explaining both the X and Y data space.36 The greater the value for a particular variable in the VIP plot, the more important that variable is to the model prediction capability. Variables with a VIP score of 1 or greater are generally considered significant to the model. It was first assumed that wall pressure, stability, and CSR are each a linear function of the coal blend properties and oven operating conditions. A model building procedure was followed in which raw data were normalized and mean centered, a PLS regression model built, outliers removed by examining residual prediction plots and the model rebuilt, and variables with small VIP values removed and the model rebuilt. The steps from

(1)

where mcoal represents the mass of coal type c added to the coke c oven. An overall mass balance for a single coke batch can be written as Mcoal,in = M H2O,removed + MS,removed + MV.M.,removed coke + (Mout − wquench)

coal ∑c (xccoal , q · mc ) − Mq ,removed

(2) D



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important consideration in cokemaking, and using a poor quality model would diminish the value of any results. 2.2. Ironmaking. The blast furnace operation comprises the entirety of ironmaking production. The main input stream to the blast furnace is iron ore pellet addition in which iron is present in the form of various oxides. Undesirable gangue components are also present. Silica, magnesium oxide, manganese, and calcium oxide are present in significant quantities, all of which are undesirable and must be removed. Trace amounts of phosphorus and sulfur are introduced to the blast furnace via iron pellets as well. The main objective of the blast furnace is to provide a heating environment that can reduce iron oxide into elemental iron. Furthermore, it is important to separate this iron from the undesired impurities by creating a slag which floats on top of the hot iron. Slag formation is promoted by controlling the acid to base ratio of impurities. Acidic components consist of alumina and silica entering through the coke and pellets. Basic components are magnesium oxide and calcium oxide which are present in the pellets only. In order to control this ratio, fluxing agents are added as inputs to the furnace in the form of lime and dolomite. The feed components spend a significant residence time inside the blast furnace before reaching the exit. For efficient operation, it is important that the pellets, coke, and fluxing agents are fed in accurate proportions and properly layered. As these solids descend through the furnace, the coke begins to combust and pellets and flux begin to reduce and liquefy. Heat is provided from several different sources. These include coke combustion, hot blast air addition, and oil combustion (if required). Blast air is fed from the bottom of the furnace at a temperature of around 1000 °C. This air is important to provide oxygen for coke combustion in the furnace. The iron blast furnace is a continuous operation, and an overall mass balance is expressed by

outlier removal to removal of variables with small VIP values were repeated until the Q2 value of model was maximized. Table 1 gives a summary of the final results, including the performance and total number of variables used in each case for Table 1. Coke PLS Model Summary Statistics input variables in final model

model

removed outliers

latent variables

Q

0 3 2

4 3 2

0.72 0.81 0.75

CSR stability wall press.

2

R

coal property variables

oven condition variables

0.87 0.83 0.77

5 4 4

3 5 4

2

predictions. The number of outliers removed and number of latent variables used are also included. The model forms for the prediction of CSR, stability, and wall pressure are given by yCSR = [1, u blend(Inerts), u blend(Ro), u blend(V.M.), u blend(H 2O), z oven(Oil), z oven(BD), z oven(Grind), z oven(900)]T βCSR

(4)

y Stab. = [1, u blend(Ash), u blend(Ro), u blend(V.M.), u blend(H 2O), z oven(Oil), z oven(BD), z oven(Grind), z oven(900), z oven(Soak)]T βStability

(5)

ln(y W.P. ) = [1, u blend(FSI), u blend(Ro), u blend(V.M.), u blend(H 2O),z oven(Oil), z oven(BD), z oven(Grind), z oven(900)]T βW.P.

(6)

in out Ṁ = Ṁ

where the model input variables are defined in Table 2.

(7)

where Table 2. Classification of Coke Property PLS Model Inputs model variable ublend(FSI) ublend(Ro) ublend(H2O) ublend(V.M.) ublend(Ash) ublend(Inerts) zoven(Grind) zoven(BD) zoven(900) zoven(Oil) zoven(Soak)

in coke pell dol lime oil steam Ṁ = Ṁ + Ṁ + Ṁ + Ṁ + Ṁ + Ṁ air + ρair V̇

description free swelling index: a cross-sectional profile measurement of the coal after coking measure of the coal reflectance mass percent of water in the coals mass percent of volatile matter in coals mass percent of ash in the coals percent of inert material in coals measure of coal coarseness bulk density of coals charge heating rate to 900 °C oil added to coal charge time after charge reaches 950 °C

out hot slag topgas Ṁ = Ṁ + Ṁ + ρtopgas V̇

The inputs include coke Ṁ coke, pellets Ṁ pell, dolomite Ṁ dol, limestone Ṁ lime, oil Ṁ oil, steam Ṁ stream, and blast air V̇ air. Solid inputs exit the furnace as elements of the slag Ṁ slag or hot metal Ṁ hot streams. Gases resulting from combustion and other reactions leave in the top gas V̇ topgas. All variables are expressed in mass flow units except for the top gas and blast air streams which are volumetric flows. The pellet addition rate, Ṁ pell, is the major mass input and is equal to the sum of the individual pellet type mass rates, ṁ pell p . Similar to cokemaking, different quality raw materials can be mixed and charged to the furnace, represented by

The accuracy of the models is fairly good, as indicated by the Q2 values, with the linearity assumption for CSR and stability models justified. The initial PLS for wall pressure resulted in a Q2 value of 0.50, which coupled with additional diagnsotic analysis indicated that the assumed linear structure was not valid in this case. By using the logarithmic variable transformation indicated in eq 6, the Q2 increased to 0.75. While this adds more nonlinearity to the formulation, the model improvement is too large to ignore. Wall pressure is an

pell Ṁ =

∑ ṁ ppell p

(8)

where p indicates the pellet type. Hot metal leaving the blast furnace is comprised of several components, the main ones being carbon, iron, manganese, phosphorus, silicon, and sulfur. The summation of these E



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Figure 2. Observed vs predicted results for blast furnace model.

individual masses, ṁ hot i , is equal to the total hot metal leaving the blast furnace: hot Ṁ =

∑ ṁ ihot

Further relationships are required to complete the blast furnace model, which could in principle be determined through mass balances on hydrogen, oxygen, and carbon, together with an energy balance. However, the number of reactions involved is numerous, and therefore, energy and mass balances on hydrogen, oxygen, and carbon are very difficult to develop from first principles. The strategy for obtaining the required relationships for use in this work was to generate them from an existing blast furnace model used by our industrial partner. By manipulating the various inputs and solving for the outputs, a data set of blast furnace operating scenarios was generated from the “black box” model and regression applied to obtain a simplified, empirical model. A 213−6 design of experiments was generated for the 13 input IV variables considered. A total of 128 sets of conditions were entered into the “black box” model, and in each case the output variables were recorded. PLS regression was applied to yield a model of the form

i = {C, Fe, Mn, P, Si, S} (9)

i

Iron enters through pellet addition and is assumed to exit entirely in the hot metal stream represented by ṁ hot Fe . This iron balance is completed by multiplying the iron mass fraction in pellet p, xpell p,Fe, by the mass of that particular pellet added to the furnace:

∑ (xppell,Feṁ ppell) = ṁ Fehot (10)

p

Phosphorus enters the furnace through the coke and pellets and exits in the hot metal represented by ṁ hot P . In coke, phosphorus is in the form of P2O5 and is reduced to elemental phosphorus inside the furnace. The parameter αP converts the phosphorus oxide mass into an elemental phosphorus mass, and the variable xcoke P2O5 is the mass fraction of phosphorus oxide in the coke fed to the furnace. αPx Pcoke Ṁ 2 O5

coke

+

y BF = (r BF)T βBF

where

∑ (xppell,P ṁ ppell) = ṁ Phot

model T ̇ model , Vtopgas ̇ model , ṁcoke y BF = [Vair ]

(11)

p

model model model model r BF = [1, ṁ pellet , ṁ Silica , mAsh , xcoke, ̇ model , ṁ iron C, model model model model model model ṁ steam , ṁ dol ṁ lime , ṁoil , ṁ hot,P , ṁ hot,Si , model model T ṁ hot,C , ṁ hot,Mn]

Silica enters the furnace through both the coke and pellets. Some of this silica leaves as part of the slag, ṁ slag SiO2, while the remainder is reduced to silicon and exits in the hot metal stream represented by ṁ hot Si . In eq 12, the parameter αSi converts the silicon dioxide mass into solely silicon. coke ̇ αSixSiO M 2

coke

In the optimization formulation, the model variables are related to the process variables defined earlier. The regression results, using Prosensus Multivariate 10.02, are shown in Figure ̇ model 2 for the prediction of ṁ model coke and Vair . In these figures, the “observed” values are from the detailed “black box” model, and the predicted values are from the linear model in eq 14. The same modeling procedure as described for the coke property model development was applied. The Q2 value for this model is 0.982 meaning the predictive capability is excellent. The accuracy of the predictions shows that the linearity assumption is valid for the range of data generated from the “black box” model. Before entering the steelmaking process, the hot metal is processed in a Desulphurization Facility. The only effect on the hot metal considered is the removal of sulfur to achieve the sulfur specification. The associated material balance is

slag pell hot + αSi ∑ (xppell ,SiO2ṁ p ) = ṁ Si + αSiṁ SiO2 p

(12)

Similar component balances are completed for alumina, calcium oxide, magnesium oxide, manganese, and sulfur. As with silica, these components are all partially removed in the slag stream. The ratio between the amount of a component transferred to the slag versus hot metal is specified using industry heuristics.35 The variable ṁ slag w represents the individual mass flows of components transferred to slag, and the summation over the set w defines the total slag flow rate Ṁ slag, slag Ṁ =

∑ ṁ wslag w

w = {Al 2O3 , CaO, MgO, Mn, S, SiO2 }

(14)

hot metal De. Ṁ = Ṁ + Ṁ S

(13) F

(15)



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̇ metal is the where Ṁ De S is the mass rate of sulfur removed and M mass rate of hot metal transferred to steelmaking. 2.3. Steelmaking. Hot metal exiting the Desulphurization Facility is transported to the Basic Oxygen Furnace (BOF) which is operated as a batch process. The objective of the final primary steelmaking stage is to oxidize and remove remaining impurities to very low levels yielding steel of high purity iron. Furthermore, this stage is used to adjust the liquid steel temperature in preparation for casting into steel slabs. Oxidation of impurities is done through the addition of oxygen to the system. Similar to the blast furnace operation, fluxing agents of lime and dolomite are added in order to form slag and remove the oxidized impurities. Steel grades produced in the BOF differ mainly with respect to final carbon content. The three grade carbon contents considered in this paper are denoted as GA, GB, and GC. The lower the carbon content, the longer the batch duration and thus more oxygen that is consumed. Other factors that differentiate grades of steel include the temperature at which they exit the BOF and the alloys added to the batch at the downstream Ladle Metallurgy Facility (LMF). The hot metal that enters the furnace heats up considerably in the presence of exothermic oxidation and combustion reactions. In order to control the exit temperature to meet specifications for casting, scrap steel is added to the vessel. Not only is the utilization of scrap an effective recycling technique, but also the added cold steel melts and acts as a heat sink. Depending on the hot metal quality and steel grade to be produced, an appropriate scrap quantity is added to meet the target exit temperature. The effect of the quality of the scrap on the steel composition is not considered in this study because of the dilution that occurs when hot metal is added to the BOF. Since the hot metal charge is over three times that of scrap metal, the impact of these impurities is considered to be negligible. Similar to the blast furnace, BOF operation involves many reactions, thus deriving mass and energy balances is very challenging. To overcome this difficulty, a simplified empirical model is built from an already existing “black box” model.37 A design of experiments is employed to generate a data set from which the simplified empirical model is derived. The model inputs and outputs are listed in Table 3. The “black box” model differs depending on the steel grade to be produced, defined by the hot steel temperature, alloy addition, and carbon content. For each of the steel grade models, these three variables were kept constant while the various inputs were manipulated in the design of experiments used to generate data. Using this procedure, three simplified

models were built, one for each grade in the set g. The models can be used to make predictions of the outputs, based on the inputs, subject to the constant parameters that define a particular steel grade. The linear model structure is as follows, ygBOF = (rgBOF)T βg ∀ g = product grades A, B, C

where BOF BOF BOF BOF BOF T ygBOF = [Mhot, g , M scrap, g , VO2 , g , Mdol, g , Mlime, g ]

manipulated inputs xhot Si xhot P xhot Mn xhot C xsteel Si,g xsteel P,g xsteel Mn,g

model outputs hot metal mass scrap mass

MBOF hot,g

oxygen volume dolomite mass lime mass

VOBOF 2,g

MBOF scrap,g

MBOF dol,g

(18)

rBOF g

Absent from Table 3 and the input vector are the hot metal and steel mass fractions of sulfur and iron. Sulfur is not included in the “black box” model due to its insignificant impact on the overall energy and mass balances. While it is present in the slag and steel exit streams, its concentration is extremely low since it has already passed through the Desulphurization Facility. At this point, sulfur is no longer a concern for the steelmaking process, and accordingly, the model considers its presence to be negligible. Iron is not included simply because its mass fraction can be determined directly from the components specified in vector rBOF g ; thus, including the iron content of the steel and hot metal streams is unnecessary. A single model was built to predict the variables in eq 17 using PLS-2 regression and the Prosensus Multivariate 10.02 software package. The model fit for all prediction variables is very good, indicated by the large overall Q2 scores of 0.997 for each grade. This is expected since it is known that the “black box” model structure is linear. The number of batches of steel grade g to be produced during the optimization period are denoted by DBOF,g. This is an integer value that is input into the formulation before optimizing and represents the steel demand for grade g. The required production rate from the blast furnace depends on the number of BOF batches to be made and the mass of metal added to each steel batch represented by MBOF hot,g. Equation 19 ensures that the blast furnace produces enough hot metal for the total steel demand to be satisfied. metal Ṁ =

xBOF lime,g

BOF ∑g Mhot, g DBOF, g

Tlength

(19)

3. OPTIMIZATION PROBLEM FORMULATION The purpose of this planning optimization is to purchase the best raw materials. Therefore, a steady-state model of a fixed time will be analyzed; in this paper, the time is taken to be one week. Many coke and BOF batches will occur during the week, and all three steel product grades will be produced. The optimization goal is to minimize the net cost of steel production based on satisfying the defined steel demand quantity. Both raw material purchasing and plant operation variables are manipulated to achieve this goal. We consider here two problem formulations: nominal optimization where the parameters are assumed to be perfectly known, and optimization under uncertainty, where uncertainty in raw material compositions is considered. 3.1. Nominal Optimization. The optimization problem takes the following general form,

constant inputs hot metal temp. hot steel temp. alloy addition steel C content steel output

(17)

hot steel steel steel T rgBOF = [1, xMn , x Phot , xSihot , xChot , xMn, g , x P, g , xSi, g ]

Table 3. BOF “Black Box” Model Inputs and Outputs

metal Si content metal P content metal Mn content metal C content steel Si content steel P content steel Mn content

(16)

xsteel C,g MBOF steel,g

G



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between x and z variables in the problem solution, although the distinction is useful for engineering analysis and essential for results implementation in the plant. In the sequel, we discuss the objective function, inequality constraints, and optimization problem structure. The equality constraints comprise model equations of the type discussed in Section 2. 3.1.1. Objective Function. Steel production rates of all three grades are a fixed requirement and directly specified. In this problem formulation, the weekly rates of production of all grades are predefined to achieve plant-wide production goals. Therefore, revenue generated from selling the final product is constant and can be removed from the objective function; however, a variable revenue stream does exist with regard to the collection of off-gas during coke production. The cost objective function therefore takes the form

min ϕ(x , z , δ) z, δ

s.t. h(x , z , δ) = 0 g (x , z , δ ) ≥ 0

where x ∈ 9 n are continuous model response variables z ∈ 9 m are continuous decision variables δ ∈ {0, 1}l are binary variables h is an n-dimension vector of equality constrains g is an r−dimension vector of inequality constrains It is convenient to view x as model response variables, driven by the decision variables and related to them through the model equations, h. The model formulation and optimization employ the open-form approach. Therefore, there is no distinction

coal coal δcusemmin ≤ mccoal ≤ δcusemmax ∀c

with decision variables

(20)

The interpretation of δuse c is explained as

coke lime dol K = [δcuse , Bccoal , mccoal , Bppell , ṁ ppell , Ṁ , Ṁ , Ṁ ,

⎧ 0 if coal c is not used in the coal blend δcuse⎨ ⎩1 if coal c is used in the coal blend

steam oil air steel steel steel scrap Ṁ , Ṁ , V̇ ,xMn, ] g , xP , g , xSi, g , B

In the above, the c coefficients represent unit costs and p*By‑prod. represents the unit revenue from coking byproducts. For the BOF, costs are incurred through high purity oxygen and fluxing agent addition. During the cokemaking process, volatile matter is driven off from the coals and collected for reuse in the plant. The mass that can be collected from each batch is represented by the variable MV.M.,removed. Depending on the number of coke batches made, Acoke, the total revenue can be calculated using the market selling price, p*By‑prod., of the byproducts. The costs of the blast furnace operation are represented by steam, blast air, oil, dolomite, and limestone usage, as well as a cost for sulfur removal. The blast furnace operating cost variables are written as mass or volumetric flows. In order to determine the total amount used, each term must be multiplied by Tlength which is the length of specified time period. Coal, iron ore pellets, and scrap steel are bought on the open market in quantities represented by the variables Bcoal c , scrap Bpell , and B ; the total purchasing cost can be calculated from p these three variables. 3.1.2. Inequality Constraints. Cokemaking. The amount of each coal type c added can be zero or between minimum and coal maximum limits represented by the parameters mcoal min and mmax. Adding this constraint ensures that the solution does not call for an addition of an impractically small amount. The binary variable, δuse c , is used to enforce this constraint.

In this formulation, it is assumed that there is no initial coke inventory and no coke at the end of the week; therefore, any coke used must also be made during the same period in question. The mass of coke used is defined as the coke addition rate to the continuously operated blast furnace, Ṁ coke. The coke mass produced can be calculated from the size of one coke coke batch, Mcoke . The out , and the total number of batches made, A constraint that the total coke used in steelmaking during the optimization period, Tlength, cannot exceed the total coke production mass thus becomes coke coke coke Ṁ Tlength ≤ Mout A

(21)

Acoke is treated as a continuous variable in the optimization implementation, which can be justified due to the large number of coke batches produced during the optimization time period. Based on the total number of coke batches made, the total mass of coal type c used during the time period must not be more than mass purchased represented by Bcoal c . mccoal Acoke ≤ Bccoal ∀ c

(22)

Upon delivery, different coal types are organized by separating them into individual inventory piles. There is only enough room on site for up to Ncoal piles. H



∑ δcuse ≤ N coal c

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Equations 28 and 30 are added to the formulation in order to ensure the optimization does not obtain a solution which predicts CSR, stability, and wall pressure outside of their respective model regions of validity. Ironmaking. Silica and alumina are removed with the slag which is formed by the addition of magnesium and calcium oxides. In order for this slag to float on top of the exiting blast furnace hot metal, it must be of a certain basicity.6 Slag basicity measures the ratio of basic to acidic components and is desired to be between specified limits, blow and bup:6

(23)

Additional constraints must be written relating to the oven operation and the final coke strength as described in Section 2.1. The final coke strength, yCSR, and stability, yStab., must meet minimum requirements. Likewise, there is an upper limit on the pressure exerted on the oven wall during coking which is represented by yW.P.. CSR yCSR ≥ ymin

y

Stab.

y

W.P.

(24)

≥

Stability ymin

(25)

≤

Pressure ymax

(26)

Th = (rh) Wh ∀ h

∑ p ṁ ppell a

∑ i=1

Th , i 2 sh , i 2

≤ Tmax , h 2 ∀ h

∑ p ṁ ppell f

(27)

i

≤ bup (31)

≤ pup (32)

≤ pup (33)

Sulfur is removed at the Desulphurization Facility with the total mass removed denoted by the variable Ṁ De. S . The extent of sulfur removal must be such that the exit hot metal content is less than sup. De.

ṁ Shot − Ṁ S metal Ṁ

≤ s up

(34)

Limits on the hot metal composition of carbon, manganese, phosphorus, and silicon are also imposed: ximin ≤

ṁ ihot ≤ ximax ∀ i metal ̇ M

(35)

where (28)

i = {C, Fe, Mn, P, S, Si}

While the hot metal manganese, phosphorus, and silicon can also be removed downstream in the steelmaking process, there are limitations as to the amount of each impurity that can be extracted at this latter stage, and for this reason, it is desired that the hot metal from ironmaking not exceed limits as specified by eq 35. Oil combustion could account for any energy deficit not provided by coke addition in the blast furnace, though this is only true to a limited extent. Coke must still provide a physical support for the descending pellets, and thus it is important that an oil to hot metal ratio not be exceeded. This is enforced by eq 36 which requires that the oil consumption to hot metal rate not exceed a ratio of roil.

(29)

The matrix Ph is obtained from the PLS model. The squared difference between the original (rh) and estimated point (r̂h) represents the new point’s orthogonal distance off the model plane and is called the squared prediction error of x (SPE-X), also known as the residual error vector. This residual distance can be constrained by limiting it to a user selected upper control limit of εh, to avoid extrapolation too far off the model plane.

∑ (rh,i − rĥ ,i)2 ≤ εh ∀ h

f

∑p ṁ ppell

In the above, Th,i is the ith component of vector Th, sh,i2 is computed from the latent variable model and is equal to the variance of the latent T-scores used in the building of model h, and nlv is the number of latent variables used in the model. The new data point’s T-score can also be used to determine its orthogonal distance off the model plane. This T-score, in combination with loading coefficients for the model, Ph, is used to estimate the value of rh, represented by r̂h.

(rĥ )T = ThPhT ∀ h

a

∑p ṁ ppell

where rh represents the centered and scaled input variables for the empirical model for response variable h and matrix Wh represents loading values of model inputs for each latent variable. The T-score, Th, is first used to determine the distance of the new data point from the center of the latent space. This score represents the projection of the rh vector onto the latent space. The Hotelling T2 value represents the distance from the center of the latent variable space to this projection. For any rh vector, this distance is restricted to a maximum user selected distance Tmax,h2 to avoid extrapolation too far along the model plane: nlv

slag ṁ Sislag ̇ 2O3 O2 + mAl

Operators prefer that flux and acid pellets are always mixed during processing and at most a fraction, pup, can be added.

Since the models for coke strength, stability, and wall pressure comprise empirical correlations, their use in optimization should be appropriately restricted. We address this through constraints designed to prevent model extrapolation, during the optimization, beyond the variable range supported by data used in model building.38 In the PLS framework, each data point is first projected onto the model’s latent variable space according to eq 27. T

slag slag ṁ CaO + ṁ MgO

blow ≤

oil Ṁ ≤ r oil hot Ṁ

(36)

Equations 31−36 are nonlinear in this formulation due to the ratio term but can be easily transformed into linear inequality constraints by multiplying the inequalities by the denominator appearing in the ratio of variables.

(30) I



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The blast air intake also presents a physical process constraint because there is a maximum volumetric rate, V̇ max, that can be supplied to the furnace.

V̇

air

≤ V̇

max

Table 4. Variable and Equation Size of Final Formulation total continuous variables discrete variables equality constraints •linear •nonlinear inequality constraints •linear •nonlinear

(37)

Finally, the total mass of pellet type p used during the time period should be less than mass purchased, represented by Bpell p . ṁ ppell Tlength ≤ Bppell ∀ p

(38)

Steelmaking. There are specifications for the quality of steel produced in each BOF batch which depend on the steel grade type g being made. These are enforced through constraints, steel steel steel X min ,Mn, g ≤ x Mn, g ≤ X max ,Mn, g ∀ g

(39)

steel steel steel X min ,P, g ≤ x P, g ≤ X max , P , g ∀ g

(40)

steel steel steel X min ,Si, g ≤ xSi, g ≤ X max ,Si, g ∀ g

(41)

⎧ 7 → Blast furnace inputs and scrap purchase: ⎪ coke lime dol steam oil air , Ṁ , V̇ , ⎪ Ṁ , Ṁ , Ṁ , Ṁ ⎪ Bscrap ⎪ ⎪ 3C → Coal usage and purchasing: ⎪ ⎪ mcoal , δ use , Bcoal c c c D.O.F. ⎨ ⎪ ⎪ 2P → Pellet usage and purchasing: ⎪ ṁ ppell , Bppell ⎪ ⎪ 3G → Desired end steel specifications: ⎪ steel steel steel ⎪ xMn, g , x P, g , xSi, g ⎩

The amount of dolomite and limestone added to each batch is constrained by upper limits, given by BOF BOF,max Mdol ≤ Mdol ∀g g

(42)

BOF BOF,max M lime ≤ Mlime ∀g g

(43)

In this study, 7 coals (C = 7) and 3 pellets (P = 3) are available for purchase on the open market. Three steel grades are produced during each optimization period (G = 3), which is consistent with that of a typical steel plant.37 The formulation was coded in GAMS 23.3.3, employing the commercial MINLP solver SBB Level 006 on a Dell XPS 420 computer with Intel 2.40 GHz CPU and 4GB RAM. SBB is a local MINLP solver which uses CONOPT3 to solve relaxed NLP problems in a branch and bound algorithm. Computation times are modestsolution times for the two-stage stochastic formulation (described in the next section) with 50 scenarios in the application study considered varied from approximately 1 to 10 minutes. Solution of the nominal case with no uncertainty was significantly faster. As the problem is nonconvex, global optimality is not guaranteed. 3.2. Optimization under Uncertainty. Stochastic twostage optimization partitions the variables into two categories: first and second stage. First stage variables are those which must be implemented immediately, while second stage decisions can be delayed until the uncertainty is realized and measured. A common approach in the formulation and solution of twostage stochastic programming problems is to consider a set of discrete values of the uncertain variables. The objective function comprises a deterministic component involving only first-stage variables and a second component involving secondstage variables, where an expected value over the uncertain parameter realizations is used. We note that second-stage variables may assume different values for each uncertain parameter realization; thus a set of second-stage decisions is defined for each uncertain parameter realization (scenario). The stochastic two-stage formulation is given below39

Scrap metal is the only raw material associated with BOF production, and the total mass used must not exceed the amount purchased, represented by Bscrap: BOF scrap ∑ Mscrap, g DBOF, g ≤ B

(44)

g

144 + 2C + 2P + 8G C 137 + 5G 121 + 5G 16 28 + 2C + P + 8G 21 + 2C + P + 8G 7

3.1.3. Problem Structure and Solution Method. The formulation is both nonlinear in structure and contains integer variables. It is therefore classified as a mixed-integer nonlinear program (MINLP). Nonlinearities exist in numerous equations but are only of three different types: bilinear, conic, and logarithmic. The prediction model for coke oven wall pressure is of a logarithmic relationship. Second order conic constraints limit the domain in which the coke prediction models could be used. Bilinear terms were introduced in equations where component mass balances were written. It is possible to employ a linearization technique to the logarithmic nonlinearity by introducing more integer variables. However, the bilinear mass balance equations cannot be linearized with great accuracy and therefore all equations are left nonlinear. This problem is relatively small in size and is nonconvex in structure due to the bilinear nonlinearities. A breakdown of the total number of variables and equations which comprise the formulation is given in Table 4. The values are given with respect to the number of coals available (C), pellets available (P), and grades of steel to produce (G). A degrees of freedom analysis can be conducted by subtracting the total number of variables from the total number of equality constraints. D.O.F. = (144 + 3C + 2P + 8G) − (137 + 5G) = 7 + 3C + 2P + 3G

The degrees of freedom are broken down into the categories below and are equivalent to the decision variables specified earlier in vector K. J



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Figure 3. Two-stage stochastic separation of variables. [Icons used by permission of the American Iron and Steel Institute.]

If coal c is not purchased, meaning δbuy is equal to zero, the c purchasing amount must also be equal to zero. This is enforced by the constraint below.

Subject to, h(x1) = 0 g (x1) ≥ 0 hs(x1 , x 2s) = 0

∀s

gs(x1 , x 2s) ≥ 0

∀s

Bccoal ≤ δcbuyBcmax ∀ c

In this equation, Bmax represents the upper purchasing limit for c coal c. In this problem, no maximum purchase amount exists, and so, the exact value is not important as long as it is sufficiently large. The remaining first stage decisions are the purchase amounts of coal, iron pellets, and scrap, giving the first stage decision vector:

where Nscen. corresponds to the number of scenarios considered, x1 are the first stage variables, and x2s are the second stage variables associated with scenario s. In the above case, dividing by Nscen. in the second stage cost term attributes equal probability to each scenario. Since uncertainty in this case study is considered only for the pellet and coal properties, it is logical to categorize the raw material purchasing variables as first stage and operation-related variables as second stage. This division is illustrated in Figure 3. This categorization of first and second stage variables is consistent with actual plant operation. In practice, raw materials are purchased knowing only the approximate quality values. Once the raw materials are received and the material properties are measured, the operations can be manipulated in order to account for the actual raw material properties. For example, if iron ore pellets arrive with a higher than expected silica content, more dolomite would need to be added in order to maintain the slag composition within target specification limits. This would in turn require a greater energy input to the furnace, thereby raising the coke and hot blast air inputs. The formulation in the previous section is now transformed into the two-stage stochastic framework. 3.2.1. Decision Variables and First Stage Constraints. A new first stage binary variable, δbuy c , is introduced which decides whether or not coal c is to be purchased. This variable is required to enforce the constraint that at most four coals can be stored in the plant:

∑ δcbuy ≤ N coal c

(46)

K1st = [(δcbuy )T , (Bccoal )T , (Bppell )T , Bscrap]T

The vector Ks contains all second stage decision variables for scenario s. coke T

T coal T ̇ Ks = [(δcuse , s ) , (mc , s ) , (Ms dol

steam T

(Ṁ s )T , (Ṁ s

lime T

T ̇ ) , (ṁ ppell , s ) , (M s

oil

) ,

air

steel T ) ,(Ṁ s )T , (Vṡ )T , (xMn, g , s) ,

T steel T T (x P,steel g , s) , (xSi, g , s) ]

For a total of Nscen. scenarios, the complete second stage decision variable vector K2nd can be written: K 2nd = [K1T , K 2T , ..., K NT scen.]T

The first and second stage decision variables are combined to give the complete decision variable vector Kstoch: K stoch = [(K1st)T , (K 2nd)T ]T

(45)

3.2.2. Objective Function and Second Stage Constraints. The objective function for the two stage stochastic programming problem gives equal probability to each scenario s occurring, and is written as

This constraint was previously enforced by eq 23 with the use binary variable δuse c . In the two-stage approach, the variable δc use becomes a second stage variable δc,s which is still needed to ensure a minimum coal mass is added to each blend. K



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N scen.

min stoch

K

listed in Table 5 and a demand of 35 batches for each of the three steel product grades, optimization is completed to select the appropriate raw material mix and plant operating settings. This case study is denoted as Problem 1a, and the optimization result yields an overall scaled cost of $23.30 per tonne of steel produced. The solution for coal purchasing and coke production is given in Table 6.

BOF lime BOF ∑ [∑ (c O VOBOF,g ,s + c dolMdol, Mlime, g , s) g ,s + c 2

2

s=1

g

DBOF, g − p*By‐prod. AscokeMVMM,removed, s lime dol air steam + (c limeṀ s + c dolṀ s + c airVṡ + c steamṀ s oil

De.

+ c oilṀ s + c De.Ṁ S , s )Tlength]/N scen. +

∑ cccoalBccoal c

+

∑

cppellBppell

+c

Table 6. Cokemaking Results for Problem 1a

scrap scrap

B

coal blend

p

Coal Coal Coal Coal

The constraints defined for the nominal optimization formulation in Section 3.1 are retained in the two-stage stochastic formulation, but as second stage constraints, and thus are written for each scenario s in the full set S.

4. CASE STUDIES Two case studies are presented to illustrate the application of the proposed optimization formulations. The first considers nominal optimization only and illustrates the effect of raw material availability and quality on the optimal solution. The second considers uncertainty in raw material quality and compares the two-stage stochastic approach to nominal optimization. Raw material is available from a number of different vendors. Each coal vendor supplies a unique coal type along with specifications relating to its chemical, rheological, and petrographical qualities. The case studies involve seven coal types, three different iron pellets which include two flux pellets (Pellet A and Pellet B) and one acid pellet (Pellet C), and scrap steel. Costs for these raw materials are listed in Table 5. All values are scaled by dividing by a constant factor, which is not

parameter

scaled cost

Coal A Coal B Coal C Coal D Coal E Coal F Coal G Pellet A Pellet B Pellet C Scrap

ccoal A ccoal B ccoal C ccoal D ccoal E ccoal F ccoal G cpell A cpell B cpell C cscrap

$5/tonne $6/tonne $5/tonne $5/tonne $4.9/tonne $4.7/tonne $4.3/tonne $13/tonne $13/tonne $12.7/tonne $20/tonne

coke composition 0% 19.1% 38.5% 42.4%

Al2O3 C P2O5 S SiO2 volatile matter

2.54% 90.50% 0.0503% 0.0415% 4.99% 0.50%

The optimizer chose to purchase Coals B, C, and D, thus mixing the most expensive coal with the cheapest coals. The resulting coke composition has a fairly high carbon percentage of 90.5% meaning it has good fuel value with respect to blast furnace operation. The P2O5 content is high at 0.0503%, which can be detrimental to other decisions in this case. Phosphorus cannot be removed in the blast furnace; therefore, the only means of controlling the exit hot metal content at this stage is by using low phosphorus pellets. The subsequent pellet blend is given in Table 7. Table 7. Ironmaking Results for Problem 1a pellet blend percentage Pellet A Pellet B Pellet C

Table 5. Primary Steelmaking Raw Material Scaled Cost Values cost item

A B C D

a

0% 37.6% 62.4%

blast furnace operating settings Ṁ coke 521.9 kg/thma pell ̇ M 1.52 tonnes/thm Ṁ dol 85.9 kg/thm V̇ air 919.9 (kN m3)/thm slag Ṁ 195.6 kg/thm

Tonne of hot metal.

Pellet C constitutes the majority of the blend with Pellet B being the remainder. Since Pellet C is 2.3% cheaper than the other two options, using it in such a great quantity is clearly cost-effective. Pellet B has a lower iron content than Pellet A meaning more has to be purchased per tonne of hot metal produced. The reason for using this pellet, however, is the fact that both pellets A and C have high phoshporous contents. Pellet B addition is required to lower the input phosphorus amount in order to satisfy the exit hot metal upper limit. Table 7 also lists the blast furnace operating conditions for this case study. The results show that producing one tonne of hot metal requires just over two tonnes of pellets and coke. Results relating to BOF steelmaking are summarized in Table 8. Of note is the small amount of scrap metal added to each

reported for proprietary reasons. By scaling all costs in the objective function by the same factor, the optimal values for the variables are not influenced. Since the objective function is scaled, the improvements achieved via optimization are evaluated by the percent improvement in the objective. The number of coal types that can be purchased during the optimization period considered is limited to four. As previously explained, the optimization formulation considers a single lumped planning period under a quasisteady-state assumption, typical of planning models. 4.1. Problem 1with No Uncertainty. 4.1.1. Problem 1a. The problem is first formulated by considering that only Coals A, B, C, and D are available for purchase. For the costs

Table 8. Steelmaking Results for Problem 1a steel grade g

L

variable

GA

GB

GC

MBOF scrap,g MBOF dol,g MBOF lime,g VOBOF 2,g

278.4 kg/thm 18.6 kg/thm 20.4 tonnes/thm 63 m3/thm

354.4 kg/thm 6.3 kg/thm 16.2 kg/thm 62 m3/thm

348.2 kg/thm 4.3 kg/thm 17.3 kg/thm 62 m3/thm



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batch of steel grade GA. This is due to the fact the exit temperature requirement of this grade is very high. Since the scrap mass acts as a heat sink, less is added to the batch. As a result, more hot metal is charged, requiring more dolomite and limestone to float off the impurities. 4.1.2. Problem 1b. The problem parameters from Problem 1a remain unchanged except a new raw material, Coal E, can be bought at a scaled cost of $4.90/tonne. This coal contains nearly 32% volatile matter and so is very profitable with respect to collecting the resulting volatile matter. To its detriment, Coal E has poor strength characteristics and using only this coal would result in very low CSR and stability values. After increasing the maximum allowable purchase for Coal E from zero to an unlimited quantity, the problem is resolved and denoted as Problem 1b. In comparison to Problem 1a, the overall scaled cost was reduced by 2.5% from $23.30/tonne to $22.71/tonne. The optimizer made many changes to purchasing and operation in order to achieve this improvement. The two biggest changes made were those of the selected coal and pellet blends; these are shown in Figure 4.

The cost reduction of 2.5% from Problem 1a to 1b is largely due to the cheaper coal blend that was used and the chain reaction it had on the process. Coal E allowed for the very expensive Coal B to be eliminated from the blend. As a result, the coke carbon content was raised and thus the amount of coal consumed per tonne of hot metal was lowered by 4.6 kg. The coke phosphorus content was decreased significantly which, as mentioned, allowed for the two pellets with the highest iron contents to be used. The ash contents of Pellets A and C, namely, silica and manganese, are higher than that of Pellet B. Therefore, in Problem 1b, the dolomite addition was increased as well as the subsequent slag mass flow. 4.2. Problem 2with Uncertainty. In this case study, uncertainty is considered in a number of coal and pellet compositions, and the two-stage stochastic programming formulation described in Section 3.2 is applied. The uncertain mass fractions are assumed to belong to independent standard normal distributions. The standard deviations are given in Table 10 and are assumed to be the same for all the coal and Table 10. Standard Deviation of Uncertain Components in Coals and Pellets raw material coal

pellets

The new coal blend is comprised of Coals A, D, and E. Only Coal D remains from the initial blend. The optimizer took full advantage of the newly available cheap coal and decided to use it as the major component in the coke batches. The pellet blend also changed significantly as the maximum acid addition of 80% Pellet C was used with the remainder being Pellet A. Low phosphorus Pellet B was not used in the new solution since the new coke composition was of a far lower phosphorus oxide content. The major differences in solution variables between Problem 1b and 1a are shown in Table 9 to reflect how plant operation and purchasing change according to the addition of a single raw material. Table 9. Comparison of Problem 1b and Problem 1a Results problem 1b

problem 1a

$22.71/tonne steel 91.22% 0.0388%

$23.30/tonne steel 90.50% 0.0503%

517.3 kg/thm 91.6 kg/thm 198.3 kg/thm

521.9 kg/thm 85.9 kg/thm 195.6 kg/thm

σ, wt %

ash H2O P2O5 volatile matter Mn P SiO2

0.8% 0.5% 0.001% 1% 0.3% 0.001% 0.3%

pellet types. In this respect, it is assumed the raw material suppliers provide an equally uncertain product. In order to ensure the mass fractions add to one, the sum of mass fractions are set equal to 1 by adjusting the carbon content for coal and iron content for pellets. For this study, a total of 50 uncertain scenarios are considered (side studies confirmed that expanding to 100 scenarios did not significantly affect the results). The compositions of the raw materials are generated randomly from their respective standard normal distributions. The random values are allowed to be no further than three standard deviations away from the respective means of each component. This sampling procedure is repeated 50 times, building a scenario database. Seven coals and three pellets are available for purchase. The optimization solution is given in Table 11 for the first stage variables only. Also listed are the purchasing decisions for the nominal case in which uncertainty is ignored. All mass quantities are given in kg of raw material per tonne of steel produced and scaled costs given per tonne of steel produced. The reported costs represent the raw material costs discounted by the revenue that could be accrued through the coal volatile matter byproduct. The purchasing decision for each raw material is different when uncertainty is considered. Coal E is very profitable with respect to generating byproduct off gas. Only about half of Coal E in the nominal case is purchased in the stochastic case. This is replaced by purchasing more of Coals F and G as well as previously unused Coal C. Coals F and G have excellent strength attributes and therefore are blended in greater quantities in order to guard against violating the CSR and

Figure 4. Problem 1a and 1b raw material selection comparison.

scaled cost xcoke C xcoke P2O5 Ṁ coke Ṁ dol Ṁ slag

component

M



Article

Table 11. First Stage Purchasing Comparison of Stochastic and Nominal Solutions raw material

two-stage purchase (kg/tonne steel)

nominal purchase (kg/tonne steel)

Coal C Coal E Coal F Coal G Pellet A Pellet B Pellet C Scrap purchasing scaled cost

38 183 160 188 460 223 970 251 $21.87/tonne

0 370 139 90 74 171 980 255 $21.72/tonne

stability coke constraints. Coal C has a fairly low phosphorus content, and its addition is likely decided upon by the optimizer to back-off from the phosphorus upper limit constraint in the blast furnace. A greater amount of low phosphorus and silica Pellet B was purchased in the two-stage solution. This low ash pellet reduces the dolomite consumption in the blast furnace and also contributes to the back-off from the phosphorus upper limit constraint. The impact on BOF production is positive, and thus, less phosphorus is required to be oxidized and removed. A reduction in phosphorus oxidation means less scrap needs to be added to the BOF batches in order to control the final steel temperature; hence, less scrap was purchased in the two-stage solution. The first stage cost of the nominal solution is 0.71% cheaper than that of the two-stage approach. This makes sense because the decisions become more conservative when considering uncertainty. Table 12 shows the difference in production if the raw materials were used and nominal raw material compositions were encountered.

Figure 5. Uncertainty method comparison approach for nominal and two-stage operation.

then reoptimized with the purchasing decisions fixed and the constraints relating to stream compositions and coke strength softened. The cost of the resulting operation in dollars and each constraint violation are recorded, and the process is repeated with a new random scenario. By “softened”, we mean that the original inequality “hard” constraints are changed to goal programming constraints with penalties for violation. This modification is used at the “simulate ” stage in Figure 5 where the plant operation is adjusted to the specific raw material compositions for the scenario. The degree of constraint violation is added to the objective function and assigned a weight. The results below correspond to a penalty cost of 106, with constraint violations in units of % for coke CSR and stability, lb/in2 for wall pressure, and 0.001% for blast furnace exit hot metal phosphorus content. The procedure outlined in Figure 5 was simulated a total of 500 times with both the constraint violations and cost recorded. The result using the two-stage stochastic solution would be dependent on the particular set of scenarios chosen. This procedure was therefore repeated for seven different two-stage stochastic programming solutionsfive of which used a scenario size of 50, with the remaining cases corresponding to a scenario size of 100, 125, and 150, respectively. The results are summarized in Table 13. The average of the two-stage solutions yields an overall cost of $22.83/tonne in comparison to an overall cost of $23.08/ tonne for the nominal solution. No two-stage solution exceeded

Table 12. Nominal and Two-Stage Production Comparison if No Uncertainty Were Present two-stage purchase

nominal purchase

xcoke P2O5

0.038%

0.043%

coke stability slack coke CSR slack hot metal P slack

5.4% 6.8% 0.0122%

0% 0% 0%

The conservativeness of the two-stage solution is illustrated in Table 12. The optimizer essentially backs off from the CSR and stability constraints as well as the upper limit on exit hot metal phosphorus content by altering the coal and pellet purchasing decisions. As a result, a different coal blend is chosen, highlighted by the coke phosphorus composition in Table 12 . In order to determine the benefits of the more conservative two-stage approach, the solution, along with the nominal solution, must be implemented on the plant. Figure 5 shows the approach taken to compare these two methods. The proposed testing method fixes the purchasing decisions for both the nominal and two-stage solutions after the initial optimization. The materials are then delivered to the plant for storage. A random scenario (values for the uncertain raw material properties) is generated based on the standard normal distributions of the raw material components. The problem is

Table 13. Simulation Results Using Nominal Optimal and Two-Stage Stochastic Programming Solutions average constraint violations problem nominal optimal two-stage stochastic N

average cost ($/tonne)

CSR

stability

wall press.

P

$23.08 $22.83

46% 0%

51% 0%

0% 0%

0% 0%



Article

nominal solution exceeded that of the two-stage stochastic solution, due to the high cost of plant operation changes required to limit stream quality constraints.

any inequality constraints. By contrast, the nominal optimal case resulted in coke CSR constraint violations in 46% of the simulations and coke stability constraint violations in 51% of the simulations. In most of the 500 simulation cases, the hot metal phosphorus constraint was controlled by changing the total amount of coke added to the blast furnace. To make up for the energy deficit normally provided by coke, oil consumption was increased. Since oil is costly, this action raised the overall cost to the point where it was greater than that of all the two-stage solutions. The CSR and stability violations could not be avoided since these are a function of coal blend types only. If the wrong coals (or coal ratios) are purchased initially, the plant cannot order a different slate of coals and instead must use what is in the first stage decision to somehow minimize the extent of constraint violation. Being able to place a precise dollar value on constraint violation would help in the analysis of optimization under uncertainty. In most industrial operations, determining these penalties is difficult, and therefore, plant personnel should be consulted in developing the weights to be used in the objective function.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support for this research through the McMaster Advanced Control Consortium (MACC) is gratefully acknowledged. The authors also wish to express their gratitude to their industrial partner, and acknowledge in particular the valuable contributions by Vit Vaculik, Laura Ronholm, and Harry Gou.

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REFERENCES

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5. CONCLUSION An integrated model of primary steelmaking was presented that connects the three areas of cokemaking, ironmaking, and steelmaking. Cokemaking involved writing component mass balances for the transformation of coals to coke. Additionally, PLS models were built in order to predict the final coke stability, CSR, and resulting oven wall pressure using an experimental data set of trial runs conducted on a pilot plant coke oven. Ironmaking modeling was conducted using a combination of rigorous component mass balances and multivariate empirical models. Steelmaking involved building empirical models in order to capture the Basic Oxygen Furnace operation. Connecting the models for each subarea results in a single primary steelmaking model which can be used to investigate effects such as operational and raw material changes on the resulting process and which can be incorporated within an optimization framework. Two optimization formulations were presentedone for nominal optimization and a two-stage stochastic formulation that accounts for raw material uncertainty. Optimization results demonstrated the strong connection between cokemaking, ironmaking, and steelmaking. The optimizer responds to changes to a single raw material availability by modifying several optimization decisions simultaneously. Changes in purchasing decisions led to changes in active constraints, which subsequently required operation to be altered. It is apparent that the optimization can answer questions about raw material purchasing and operation that are far from obvious to a human decision maker. Results of the two stage stochastic optimization formulation show that a more conservative set of purchasing decisions must be made in order to accommodate the fluctuations in raw material composition. The benefits of this approach are observed when applying the purchasing solution to plant operation. In this situation, the operation could be adjusted so that none of the constraints were violated in any of the simulation studies. By comparison, implementing the nominal solution resulted in coke CSR and stability constraint violations approximately 50% of the time due to the limited ability of the plant to effectively compensate for uncertainty under the nominal first-stage decisions. Moreover, the overall cost of the O



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Optimization of Primary Steelmaking Purchasing and Operation under

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