ResourceâTask Network Formulations for Industrial Demand Side

Article pubs.acs.org/IECR

Resource−Task Network Formulations for Industrial Demand Side Management of a Steel Plant Pedro M. Castro,*,† Lige Sun,‡ and Iiro Harjunkoski§ †

Laboratório Nacional de Energia e Geologia, 1649-038 Lisboa, Lisboa e Vale do Tejo, Portugal RWTH University, Templergraben 55, 52062 Aachen, North Rhine-Westphalia, Germany § ABB Corporate Research, Wallstadter Strasse 59, 68526 Ladenburg, Baden-Württemberg, Germany ‡

ABSTRACT: In today’s energy markets, there is a growing effort toward the alignment of the industrial sector and the power grid for the sake of efficient energy distribution and consumption. In this paper, the resource−task network is used to provide a generic modeling framework for production scheduling under energy constraints. Three alternative process models for the energy-intensive melt shop of a steel plant are proposed and linked to a discrete-time formulation. The results show a trade-off between accurate representation of problem data and computational performance. By keeping track of the total energy and power consumption through time, we study the impact of fluctuating energy prices on the scheduling of operations and the economic benefits that can be obtained from the plant’s participation in the price- and incentive-based industrial demand side management programs.

1. INTRODUCTION In the coming decades, the energy sector in Europe will face a fundamental shift toward renewable electricity generation, mostly from wind power. Wind supply is generally unresponsive to need and is associated with a high level of uncertainty due to the volatile nature of wind. It will have price effects on the spot market and will lead to a growing demand for positive and negative balancing power.1 One way to provide the required stability and flexibility to the power system is through demand side management (DSM), which can be categorized as either reducing energy consumption or rescheduling and shifting energy demand to off-peak hours. Large-scale, energy intensive processes may play an important role in this context, and the question that arises in the scientific community is how to balance electricity supply and availability against the underlying profitable conditions for industry. In the industrial demand side management (iDSM) or demand response (DR) grid-consumer interface, the electricity provider gives economic incentives to the industry to alter their electricity usage behavior and there are generally two approaches: • Price-based program. Customers respond to the electricity price structure (e.g., day-ahead market) with voluntary changes in their timing of electricity usage, taking advantage of low-priced periods and avoiding production in high-priced periods. • Incentive-based program. Customers can obtain electricity at a reduced cost, if the electricity consumption is contracted ahead of time (e.g., a few days to weeks in advance). The consumption curve then has to be followed as closely as possible to avoid penalties for over- and underconsumption of electricity. A recent study investigating the potential of iDSM for electricity markets in Germany1 identified that electric arc furnaces in steel manufacturing can provide a significant © 2013 American Chemical Society

positive reserve capacity by decreasing demand when the electricity system falls short on capacity. However, they also recognize that possible disruptions in the process resulting from participation in the spot market may compromise meeting contracted loads in the future, thus reducing the probability for actual calls for energy, to close to zero. Steel manufacturing is recognized as one of the most difficult industrial processes to schedule2 (a large-scale, multistage, and multiproduct batch process involving parallel equipment, nested decisions, and critical production and energy-related constraints). Because of this, steel plant scheduling solutions or teams have traditionally focused on minimizing the makespan or changeover costs or deriving solely a feasible schedule. In order to minimize the energy cost and ensure profitability, the optimization of production planning and scheduling under energy constraints forms an important tool for the steel industry to leverage. Nolde and Morari3 proposed a continuous-time scheduling formulation for electricity load tracking of a steel plant with an identical layout to the one in the work of Harjunkoski and Grossmann.2 The problem consists of scheduling the production tasks such that the energy consumption is kept as close as possible to a prespecified energy curve to avoid penalties from the electricity provider. Power consumption of tasks is given as parameters, but their processing times are allowed to vary between minimum and maximum values to provide scheduling flexibility. Results for the production of 15 identical batches of steel over a day showed that with load tracking scheduling, the plant owner can potentially reduce his energy costs significantly. An improved continuous-time mixedinteger linear programming (MILP) for the same problem has Received: Revised: Accepted: Published: 13046

April 2, 2013 August 13, 2013 August 13, 2013 August 13, 2013 dx.doi.org/10.1021/ie401044q | Ind. Eng. Chem. Res. 2013, 52, 13046−13058

Industrial & Engineering Chemistry Research

Article

Figure 1. Steel supply chain (source: ABB).

more complex than the one featuring two units only in the first stage.2−4 The processing times and power consumption levels are fixed, and transportation times between stages must lie between well-defined lower and upper bounds. Minimum transfer times are important to improve the throughput of the plant and ensure that the next equipment is ready to receive and start processing the heat (batch). They depend on the selection of equipment-pairs, the actual distance between them, and the transport speed (typically the steel is transferred through cranes or rails if the connection is short and direct). In contrast, maximum transfer times are not a function of equipment choices. They are dictated by the cooling effect so as to ensure that the crystallization of liquid steel does not take place, which might compromise the grade quality and would have to be compensated by expensive reheating. Maximum transfer times are interesting constraints that can be modeled approximately, in the context of an RTN process model with the definition of multiple location states11 for the different heats, or rigorously by defining a considerably simpler RTN model coupled with logic constraints. One major goal of this paper will be to show that the selection of the process model is an important decision with respect to computational performance of the resulting MILP problem. The performance of an RTN discrete-time formulation is evaluated for three alternative process models using four test cases. The other novel aspect concerns the incorporation of penalties for underand overconsumption on both power and energy so that deviations from a precontracted load curve for electricity can be accounted for.

been proposed by Häit and Artigues.4 By focusing on the relative position of tasks and intervals, the number of binary variables and complex big-M constraints was reduced leading to a better computational performance. Castro et al.5 addressed the problem of deriving a minimum electricity cost schedule for the grinding stage of a cement plant. A deterministic price profile was given together with hard constraints on maximum power consumption over certain periods of time, addressing a time horizon of one week. A detailed resource−task network6 process model including downstream storage units was proposed to avoid decisions that compromise product delivery to customers at multiple intermediate due dates. RTN-based continuous- and discretetime formulations were tested with the results showing the latter to be capable of tackling problems of industrial relevance to optimality gaps below 1% in under 5 min of computational time. It was later7 combined with the continuous-time model within a rolling horizon8 algorithm to guarantee the generation of practical schedules. The process of crushing rocks in the cement production chain was also addressed by Vujanic et al.9 They proposed a method for obtaining flexible schedules using robust optimization so that delaying the execution of a job does not cause infeasibility problems. This flexibility is aimed to be sold on the market for ancillary services. Mitra et al.10 have also addressed optimal production planning based on predefined hourly prices with a discretetime deterministic model. Industrial case studies on air separation units and cement plants5 were tackled for a time horizon of one week. A novel aspect has been the modeling of transitions between operating modes starting from logic constraints. In this paper, we tackle the scheduling problem of the melt shop of a steel plant under energy constraints. The plant layout features two parallel units in each of the four stages, being thus

2. PROBLEM DEFINITION In the melt shop production process of the steel supply chain (Figure 1), solid metal scrap is molten, further processed to give its steel characteristics, and finally cast into slabs (Figure 13047

dx.doi.org/10.1021/ie401044q | Ind. Eng. Chem. Res. 2013, 52, 13046−13058


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Figure 2. Production stages of the considered steel melt shop.

Figure 3. RTN-1 process model, featuring aggregated equipment resources and location independent transfer tasks (task Cast_G1_CC2 omitted for simplification).

two continuous casters (CC). The first three processing stages operate in batch mode, whereas casting consists of pouring a heat (batch) into a tundish that is continuously emptied to form the steel slabs. Critical production constraints apply to the casting process. Due to extreme production conditions, the caster can only be run continuously for a limited number of heats, after which the caster needs maintenance, typically consisting of a caster mold and tundish change. Heats within one casting sequence (group of heats, forming a campaign) must often have the same or very similar grade characteristics, and the casting order is specified by certain rules. Furthermore, a casting sequence must not be interrupted, i.e. when the

2). Products are characterized by grade, slab width, and thickness. Each grade has a given production recipe including strict specifications of chemical composition, processing times, temperature, and energy consumption. The steel industry has typically two types of campaigns: (i) between major product types (e.g., austenitic and ferritic steels), dedicating part of the equipment for the exclusive use of one product type; this is especially important for the electric arc furnaces; and (ii) concerning the continuous casting step, as described below. The steel-making process under consideration consists of four consecutive production stages, featuring two energy intensive electric arc furnaces (EAF), two argon oxygen decarburization units (AOD), two ladle furnaces (LF), and 13048



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Figure 4. RTN-2 process model for steel heat H1, featuring individual equipment resources and location dependent transfer tasks.

casting of a heat finishes, the next heat of the group must be supplied to the caster. Plant electricity schedules are generated on a daily basis, so 1440 min will be the time horizon considered. The electricity price structure is known ahead of time with the prices changing from one hour to the next and is given through the parameter pricehr [€/(MW h)]. Let k ∈ k represent a production stage, u ∈ Uk, a processing unit at stage k, and h ∈ Hg, a heat belonging to casting group g ∈ G. The processing units are characterized by the following: (i) power requirements, powerh,u [MW]; (ii) fixed processing times (duration), dh,u [min]; and (iii) setup times in the case of continuous casters, setupu [min]. Given are also minimum and maximum transportation times between equipment units belonging to consecutive production stages, mintrfu,u′ and maxtrfu,u′ [min]. The objective will be to (i) minimize the makespan, (ii) minimize the total electricity cost over a production day, or (iii) minimize deviations from a precontracted load curve. For the latter, either the contracted energy consumption for hour hr is given, loadEN hr [MW h], or, more generally, the contracted [MW], into power consumption for every time slot t, loadPW t which the time horizon has uniformly been divided. Overconsumption p+ is typically more severely penalized than underconsumption p−. To use a common hourly basis, the units of these penalty factors are Euro per megawatt in the case of energy and Euro hours per megawatt in the case of power.

identical, we mean the same processing time and power consumption for a particular heat and the ability to handle all steel heats. Thus, there is no need to distinguish whether the processing task is going to be executed in unit 1 or unit 2, as long as at least one of them is available at the appropriate time. More specifically, we define aggregate equipment resources named EAFs, AODs, and LFs, and make their initial availability equal to 2 (parameters R0r ). In contrast, continuous casters need to be considered individually (R0r = 1 ∀ r ∈ {CC1, CC2}) due to different processing and setup times. Also it needs to be ensured that all heats belonging to the same casting sequence are processed in the same caster. Thus, the casting tasks need a unit index in conjunction with the heat group index (e.g., Cast_GG_CC1); see Figure 3. Processing tasks interact discretely with corresponding equipment unit resources, consuming the resource at the start, and releasing it back to the system at the end of the task. On the other hand, the tasks consume electricity (EL) continuously. Electricity is actually an aggregate representation of two resources: power and energy, as will be seen in section 4. Also note that the information related to the volatile electricity pricing is not part of the process model. Process model RTN-1 neglects the differences in transportation times between units of consecutive production stages by assuming that they are independent of their locations. More specifically, and as an example, the duration of the transportation time from the electric arc furnaces to the argon oxygen decarburization units (Duration_E−A) is made equal to minu∈EAF,u′∈AOD(mintrfu,u′), leading to a possible underestimation of the transfer times. Transfer tasks change the location of the heat from just after the unit (superscript o) to just before the unit (superscript i), with the superscripts being omitted for the first/last production stages (only a single state), the EAFs, and CCs. To force the immediate execution of the transfer task after completion of a processing task, output states are not allowed to exist (see later on, eq 9). This naturally excludes the outputs from the continuous casters that are our final product resources (H1, ..., HH). On the other hand, heats may wait for the equipment to become available or for a period of lower electricity cost, when in an input state condition (e.g., H1_Ai). While equipment resources such as cranes4 are not considered in Figure 3, it is straightforward to add them in the event they are bottleneck resources. In order to account for the critical production constraints linked to the continuous casting process, casting tasks involve all heats Hg of a given group g. This is possible in the context of a discrete-time formulation that allows events to take place at different times relative to the start of the task.6 Note that the casting tasks of heats of the same group do not necessarily have the same duration. Uninterrupted operation between heats is achieved by consuming the inlet state of heat h at the exact same instant the final state of heat h − 1 is produced; see details

3. RESOURCE−TASK NETWORK PROCESS MODELS The generation of a scheduling model demands a problem representation, which can be viewed as an abstract layer between real plant entities (e.g., heats, equipment, electricity) and model entities (variables and constraints). In this work, we rely on the resource−task network6 representation to describe the sequential steel-making process as a collection of tasks that transform resources. The RTN is preferred over the state−task network12 due to the unified treatment of states, units, and utilities as resources, leading to a more elegant model with a single rather than multiple sets of resource balance constraints.13 Rather than focusing on a unique model, we propose three alternatives in an increasing level of detail, in order to find the most computationally efficient but still representative process model. Note that the decision to explore approximate process models (in the context of phenomenological representation of transportation activities between stages) makes sense in the context of a discrete-time formulation, which in general is unable to consider the exact processing times as will be discussed in section 4.1. 3.1. Aggregated Equipment Resources and Simple Transfer Tasks (RTN-1). The production environment features identical equipment units in the first three stages. By 13049



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Figure 5. RTN-3 process model for heat H1, featuring solely processing tasks, electricity, and equipment resources. Minimum and maximum transfer time constraints are enforced through logic.

maxtrfu,u′ = 100 min, we would need to consider 10 tasks with a corresponding duration in number of time slots τ = 1, 2, ..., 10 for each pair (u,u′)). The solver would then select a single task mode for each transfer, which coupled with the additional constraint of eliminating waiting periods for the inlet location resources would make it work. Such an option would prohibitively increase the number of transfer tasks, thus it is discarded. Also because the maximum transfer times have the purpose of avoiding excessive cooling of heats between stages, we consider that they can be approximated for all practical purposes. 3.3. Minimum and Maximum Transfer Times Enforced Through Logic (RTN-3). In the third process model (RTN3), transfer time constraints are derived from logic propositions,15 which have proved quite useful for the derivation of scheduling models.10,16,17 By enforcing minimum transfer times, we ensure that the heats go through processing stages sequentially in the correct order. Thus, it is no longer required to define inlet and outlet location resources for the heats, and with the exception of the final products, resources are limited to the individual equipment units and electricity; see Figure 5. Let Nh,u,t be a boolean variable indicating the processing of heat h in unit u at time t. If Nh,u,t = true, we know that the same heat must be processed in a subsequent unit, u′ (or parallel unit u″), at a time t′ ∈ [t + dh,u + mintrfu,u′, t + dh,u + maxtrfu,u′], where dh,u is the duration of h in u and mintrfu,u′ (maxtrfu,u′) are the minimum (maximum) transfer times from u to u′. Note that we cannot make Nh,u,t ⇔ [∨t′Nh,u′t′] ∨ ... (refer to Figure 5) since it is possible for Nh,u′,t′ = true and Nh,u,t = false if the heat is processed on the parallel unit of u. 3.4. Remarks. The resources and tasks of the previous process models have been generated by a relatively simple algorithm as a function of the number of steel heats to schedule as well as the number of units in each production stage. It is actually flexible enough to cope with plants of different structures in terms of the number of parallel units per stage. As an example, a plant with a single EAF unit and 3 AOD units is handled simply by (i) making R0EAFs = 1 and R0AODs = 3 in RTN-

in Figure 3. The last part of the casting task is a setup to prepare the equipment for the subsequent group. Thus, all heats Hg are produced before completion of the task. 3.2. Individual Equipment Resources and Location Dependent Transfer Tasks (RTN-2). In the process model RTN-2, all equipment units are handled individually so that we can account for different processing times and transfer times between units. Similarly to what is done in design and scheduling formulations,11 transfer tasks are defined for all possible combinations of source and destination units (e.g., Transfer_H1_E1_A1 represents the transfer of heat H1 from electric arc furnace EAF1 to unit AOD1); see Figure 4. As a consequence, inlet and outlet location resources gain an equipment index (e.g., H1_L1i represents the inlet state of heat H1 to ladle furnace L1). Processing tasks belonging to the first three stages also gain a unit index, so overall, there is a substantial increase in the number of tasks and resources when compared to RTN-1. Note that the casting tasks in RTN-2 are the same as in RTN-1, with the exception of differentiating among inlet location resources, i.e. Cast_G1_CC1 consumes H1_C1, while Cast_G1_CC2 consumes H1_C2. It should be noted at this point that RTN-2 can not, by itself, rigorously handle the maximum transfer time constraints. The reason is that the total transfer time is divided in two parts: (i) minimum transfer time, accounted for by the duration of the transfer task, and (ii) waiting time of the inlet location resources. While it is straightforward to enforce maximum waiting times for the inlet location resources (eq 5), we cannot distinguish between, for example, H1_A1i being generated from resource H1_E1 through Transfer_H1_E1_A1 or from resource H1_E2 through Transfer_H1_E2_A1, which becomes relevant in cases where the two transfer tasks have different durations. An alternative to rigorously model transfer times within the RTN process model is to introduce task modes,12,14 each with a different fixed duration ranging from the minimum to the maximum transfer time. Note that the generation of task modes would naturally take into account the discretization of the grid (e.g., for a grid size of δ = 10 min, mintrfu,u′ = 10 min and 13050



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are not multiples of δ. This is particularly problematic when in the presence of sequence dependent changeovers or noninstantaneous transfer times of a different order of magnitude than processing times, for which a coarse time grid will not be sensitive enough to small changes in tasks durations and will thus fail to make a distinction between production sequences and/or equipment pairings. It is also more critical for production time related objectives (e.g., makespan minimization) than economic ones (e.g., total cost minimization), as will be seen in section 5.3. Overall, wherever accuracy becomes a serious issue, continuous-time models should be tried despite being potentially more complex. For this particular case study, a precedence-based continuous-time model performed considerably worse, requiring partial fixing of binary variables in order to generate good solutions23 within an acceptable computational time. 4.2. Model Variables. The proposed formulation requires a subset of the variables that are part of the general discrete-time RTN formulation of Pantelides.6 Since heats in steel production are discrete entities, it suffices to consider the binary extent variables Ni,t, which assign the start of task i to time point t. Note that there is a direct correspondence between the virtual index i and the real plant entities indices defined in section 2. In the case of processing tasks, i is related to single index h, or pair of indices (h, u) or (g, u), depending on the RTN process model (Figures 3−5). In the case of transfer tasks, it can be related to three indices (h, u, u′). Nonnegative continuous variables Rr,t give the excess value of resource r at time t. Resources can also be identified in the RTN graphs, with index r being easily linked to index u (units), h (final products), and indices (h, u, l), where l = {i, o} defines an inlet or outlet location. The RTN graphs also features electricity (EL), which is actually an aggregate representation of two resources: (i) power, r ∈ RPW; (ii) energy, r ∈ REN. Nonnegative continuous variables Πr,t ∀ r ∈ (RPW ∪ REN) are used to calculate the power/energy requirements in slot t. Bear in mind that the additional time resulting from the rounding up errors in the processing times is not taken into consideration when computing the energy consumption over t. During such additional time, there will be no power consumption by the task, thus the calculated power consumption can be seen as a maximum value over t. 4.3. RTN Structural Parameters. The RTN graph is brought into the model by the structural parameters. For this particular problem, we will be using two sets. The first set μr,i,θ gives the discrete interaction of resource r with task i at a time θ relative to the start of the task. The second set πr,t is used to model the interaction with the system boundaries, more specifically, to make equipment units r ∈ REQ temporarily unavailable due to maintenance. In the RTN-based mathematical model, structural parameters come into action in the excess resource balances, which are multiperiod balance equations involving the three types of variables mentioned earlier. An example showing the nonzero values of parameters μr,i,θ and the profile of some related excess resource variables Rr,t for a hypothetical casting task involving a group g of two heats (h and h′) in caster CC1 is given in Figure 7. A few aspects are worth emphasizing: (i) There are three different references for time. First, there is the index t for the time grid, which can be related to an actual hour of the day as a function of δ and the beginning of the schedule horizon. The third reference is the relative time index θ ={0, ..., τi} that is

1; (ii) eliminating resource EAF 2 and tasks of type Melt_Hh_EAF2 (for every heat h) and adding resource AOD3 (R0AOD3 = 1) and tasks Decarb_Hh_AOD3 in RTN-2 and RTN3; (iii) eliminating resources of type Hh_E2 and tasks Transfer_Hh_E2_A1 and Transfer_Hh_E2_A2; adding tasks Transfer_H h _E 1 _A 3 , Transfer_H h _A 3 _L 1 and Transfer_Hh_A3_L2 and adding resources Hh_A3i and Hh_A3o, for RTN-2. For RTN-3, it is also required to change the logic constraint in Figure 5 to reflect the new topology.

4. MATHEMATICAL FORMULATION Following the description of the process through alternative RTN graphs, the next step is to link them to a RTN-based mathematical formulation featuring variables with resource r, task i, and time t, indices. We start by choosing the time representation concept, before defining the model variables and discussing how structural parameters can be generated, and end with the model constraints. 4.1. Discrete-Time Representation. The RTN discretetime formulation proposed by Pantelides6 gives rise to simple, elegant, and very tight MILP models that may allow the solution of problems involving up to tens of thousands of binary variables and constraints to near optimality. It is perhaps the most general approach, being shown capable of handling problems of industrial relevance. Examples of industrial scheduling problems that have been tackled by RTN discretetime models include the following: (i) recipe optimization of a blending process,18 (ii) handling limited steam availability in the cooking process of a pulp plant,19 (iii) equipment selection in a fine chemicals plant,20 (iv) grinding process of ceramic tile21 and cement plants,5 and (v) a liquid waste treatment network.22 The main advantage of a discrete-time representation results from knowing the exact location of every time point t in the grid, leading to a straightforward modeling of intermediate events. Examples that are relevant to the present case study are the change in electricity pricing and power availability, as well as consumption/production of resources beyond the starting and ending times of tasks. The latter possibility was already explored in the casting tasks of Figure 3, which can be viewed as aggregated tasks that sequentially process all the heats that are grouped together. The discrete-time formulation is linked to a single uniform time grid featuring t ∈ T slots of width δ [min] that span 24 h; see Figure 6. Parameter δ is chosen by the modeler, effectively

Figure 6. Uniform time grid used by discrete-time formulation.

setting the approximation level of the problem data. While an exact model can be obtained with δ equal to the greatest common factor among the duration of all tasks (e.g., 1 min), this option is rarely used in practice for reasons related to problem tractability. Instead, processing times are rounded to a multiple of δ, leading to the calculation of parameters τi = ⌈di/ δ⌉ that give the duration of task i in number of time slots. In other words, the discrete-time representation can not, in general, accurately model events occurring at time instants that 13051



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Figure 7. Illustration of structural parameters and excess resource balances for a casting task from RTN-2.

point t is equal to that at the previous event t − 1 adjusted by the amounts produced/consumed by all tasks ending or starting at t and also by the external inputs/outputs. Note that the initial resource availability term R0r , given in Figures 3−5, only appears for t = 1. To avoid power and energy from propagating from one slot to the next, such that the external variables Πr,t come into play at the right time, the Rr,t−1 term is not written for these resources.

related to the start of task i. (ii) Due to rounding errors, resource Hh′ is only produced at t = 12 (17:30) in the schedule rather than at 17:24. (iii) Power and energy resources interact continuously with the task (solid instead of dashed lines) meaning that these are consumed at multiple discrete points θ ={0, ..., τi − 1}. (iv) Whenever dh,u ≠ [dh,u/δ]δ, meaning that the task will actually end before the end of the slot, the energy consumption in the task’s last active slot is no longer given by powerh,uδ and a correction factor needs to be added (e.g., while requiring the same power and lasting one time slot, 7 MWh are required for casting heat h but h′ needs just 6.3 MW h). (v) In order to keep track of electricity requirements, the exact value consumed must be supplied exclusively by variables Πr,t, which is ensured by making RPW,t = REN,t = 0 ∀ t. 4.4. Model Constraints. The RTN discrete-time formulation for the short-term scheduling of the melt shop of a steel plant under volatile electricity pricing and power availability is given below. Resource availability over the time grid is managed by the excess resource balances, in which the excess resource at event

R r , t = R r0|t = 1 + R r , t − 1|r ∉ (RPW ∪ REN) + Π r , t |r ∈ (RPW ∪ REN) τi

+ πr , t |r ∈ REQ +

∑ ∑ μr ,i ,θ Ni ,t− θ i

θ=0

∀ r, t (1)

Operational constraints are used to ensure that processing and transfer tasks are executed in the proper amount. Assume that subset Ih,u holds the task corresponding to the processing of heat h in unit u (k ≤ 3) and that subset Ig,u gives the task linked to heat group g in continuous caster u (k = 4). Equations 13052



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2 and 3 ensure that all heats are processed exactly once in each stage.

∑ ∑ ∑ Ni ,t = 1 u ∈ Uk i ∈ Ih , u

h

∀ h , k = 1, ..., 3

∀ g, k = 4

t

(3)

Similarly, assume that subset Ih,u,u′ gives the transfer task of heat h from unit u, belonging to stage k, to unit u′, belonging to stage k + 1 . Equation 4 ensures that a single transfer task of heat h is executed between consecutive stages and applies to both RTN-1 and RTN-2. Maximum transfer time constraints are then approximated (recall the discussion in section 3.2) through eq 5. It states that the inlet resource r ∈ RIL h,u′ of heat h to unit u′, can be available (Rr,t = 1) in at most as many slots as the stage’s maximum possible transfer time.

∑ ∑ ∑ ∑ Ni ,t = 1 u ∈ Uk u ′∈ Uk + 1 i ∈ Ih , u , u ′

∑ ∑ ∑ R r ,t ≤ ⌈ u ′∈ Uk + 1 r ∈ R hIL, u ′

∀ h, k ≠ 4 (4)

t

t

u ∈ Uk u ′∈ Uk + 1

MS ≥ (5)

min

(6)

(7)

∑

t + τi + mxi , i ″

Ni ′ , t ′ +

t ′= t + τi + mni , i ′

∑

Ni ″ , t ′

t ′= t + τi + mni , i ″

∀ i , i′, i″ , t

(8)

Returning back to constraints common to all process models, eq 9 prevents existence of resources linked to output states, power, and energy, while eqs 10 and 11 set upper bounds on the maximum power and energy consumption, respectively. ∀ r ∈ (ROL ∪ RPW ∪ REN), t

R r ,t = 0

Π r , t ≤∑ ∑ powerh , u h

u

(13)

Πr ,t

t ∈ Thr

(14)

4.4.1.3. Minimizing Penalties in Incentive-Based Program. In the incentive-based program, the precontracted load curve must be followed as closely as possible to avoid penalties. This can be done over, e.g., hourly slots (with loadEN hr being the precontracted energy consumption for hour hr) or generically in δ-min slots (loadtPW gives the precontracted power consumption for time slot t). The appropriate measurement frequency will be set by the contract, with the latter option corresponding to a more general approach. Note that Nolde and Morari3 and Häit and Artigues4 have used 15-min intervals. Equation 15 computes the positive Δ+r,hr and negative Δ−r,hr deviations for energy resource r during hour hr, which are defined as nonnegative continuous variables that have power units [MW]. For the objective function, we choose to minimize the total energy cost in addition to the penalties for over and under consumption; see eq 16. While we are assuming the price profile of the price-based program, which can be information from the day-ahead market, deviations from the precontracted load may lead to inteaction with the intraday or real-time markets. This can be modeled quite easily by disaggregating variables Πr,t among the different markets or electricity providers and pricing them differently in the objective function.

+ (Ni ″ , t + τi + mni ,i″ + ··· + Ni ″ , t + τi + mxi ,i″) ≥ 1

Ni , t ≤

∑ ∑ price hr ∑

r ∈ REN hr

1 − Ni , t + (Ni ′ , t + τi + mni ,i′ + ··· + Ni ′ , t + τi + mxi ,i′)

t + τi + mxi , i ′

t

4.4.1.2. Energy Cost Minimization in Price-Based Program. The objective of minimizing the total energy cost is given by eq 14, where the energy consumption during hour hr is calculated by adding the energy consumed in all slots t belonging to hr (t ∈ Thr), a subset that can easily be generated as a function of the discretization factor δ.

∀ h , k ≠ 4, t , u ∈ Uk , u′

∀ i , i′, i″ , t

∑ ∑ Ni ,t(t − 1 + τi − ⌈setupu /δ ⌉)δ

∀ g , k = 4, u ∈ Uk

Ni , t ⇒ (Ni ′ , t + τi + mni ,i′ ∨ ... ∨ Ni ′ , t + τi + mxi ,i′) ∨ (Ni ″ , t + τi + mni ,i″ ∨

∈ Uk + 1 , u″ ∈ Uk + 1 , i ∈ Ih , u , i′ ∈ Ih , u ′ , i″ ∈ Ih , u ″

(12)

i ∈ Ig , u

For process model RTN-3, transfer tasks do not need to be defined since the minimum and maximum transfer time constraints are enforced by relating the binary extent variables of processing tasks belonging to consecutive stages. Starting with the logic proposition of Figure 5 and after replacing the pair of indices (h, u) with task index i and expressing the processing and transfer times in number of time slots (mxi,i′ = ⌈maxtrfu,u′/δ⌉ and (mni,i′ = ⌈mintrfu,u′/δ⌉), eq 6 results. The logic proposition can then be converted to MILP format using De Morgan’s Theorem and a truth table,15,17 leading to eq 7 and finally to eq 8.

... ∨ Ni ″ , t + τi + mxi ,i″)

(11)

u

min MS

max (maxtrfu , u ′ − mintrfu , u ′)/δ ⌉

∀ h, k ≠ 4

∀ r ∈ REN , t

4.4.1. Objective Functions. Four alternative objective functions will be considered ranging from makespan minimization to total cost minimization from the price and incentive-based programs. It is straightforward to implement weighted sums of different criteria. For instance, Moon and Park24 have combined the minimization of makespan and timedependent electricity costs when tackling a scheduling problem from a manufacturing company with a hybrid genetic algorithm. 4.4.1.1. Makespan Minimization. Makespan is defined as the completion time of the last heat to be processed in the continuous casters, eq 12. Since the last group is not known a priori (the last heat within a group is given), we have to consider all possibilities. The domain of eq 13 considers all groups g and last stage units u, with the summation over i identifying the single task belonging to Ig,u. The starting time of the task is given by Ni,t(t − 1)δ (refer to Figure 6), with the completion time obtained after adding the processing time and subtracting the setup time.

(2)

t

∑ ∑ ∑ Ni ,t = 1 u ∈ Uk i ∈ Ig , u

Π r , t ≤∑ ∑ powerh , uδ /60

(9)

∑

∀ r ∈ RPW , t

+ − Π r , t = loadEN hr + Δr ,hr − Δr ,hr

∀ hr, r ∈ REN

t ∈ Thr

(10)

(15) 13053




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r ∈ REN hr

Article

Table 4. Minimum Transfer Times [min]

Πr ,t

t ∈ Thr

∑ ∑ (p+ ·Δ+r ,hr + p− ·Δ−r ,hr )

+

(16)

r ∈ REN hr

If penalties are measured in δ-min slots, index t replaces index hr in the deviation variables of the power resource. Equation 17 features a correction factor to make the overall penalties independent of the chosen value of δ (recall that p+ and p− are the penalties over 1 h). In this more general case, we may also have penalty free tolerances Φt (defined as continuous variables) below a certain given value ϕt; see eqs 18 and 19.


min

r ∈ REN hr

r ∈ RPW

Πr ,t

(17)

t

Π r , t = load tPW + Δ+r , t − Δ−r , t + Φt

∀ t , r ∈ RPW (18)

∀t

−ϕt ≤ Φt ≤ ϕt

(19)

5. COMPUTATIONAL STUDIES The performance of the discrete-time formulations for the three alternative process models and different objective functions is Table 1. Steel Heat/Group Correspondence group g

G1

G2

G3

G4

G5

G6

Hg

H1−H4

H5−H8

H9−H12

H13−H17

H18−H20

H21−H24

Table 2. Power Consumption [MW] (Independent of Steel Heat h) unit u

EAF1

EAF2

AOD1

AOD2

LF1

LF2

CC1

CC2

powerh,u

85

85

2

2

2

2

7

7

Table 3. Processing Times [min] dh,u

EAF1

EAF2

AOD1

AOD2

LF1

LF2

CC1

CC2

H1−H4 H5−H6 H7−H8 H9−H12 H13−H14 H15−H16 H17 H18 H19 H20 H21−H22 H23−H24

80 85 85 90 85 85 80 80 80 80 80 80

80 85 85 90 85 85 80 80 80 80 80 80

75 80 80 95 85 85 85 95 95 95 80 80

75 80 80 95 85 85 85 95 95 95 80 80

35 45 20 45 25 25 25 45 45 30 30 30

35 45 20 45 25 25 25 45 45 30 30 30

50 60 55 60 70 75 75 60 70 70 50 50

50 60 55 60 70 75 75 60 70 70 50 60

AOD1

AOD2

EAF1 EAF2 AOD1 AOD2 LF1 LF2

10 20

20 10

LF1

LF2

4 20

20 8

CC1

CC2

10 10

20 15

The mathematical formulations give rise to MILP problems. They were implemented in GAMS 23.8.2 and solved by CPLEX 12.4 using a single thread and default options up to (i) relative optimality tolerance = 10−6 and (ii) maximum computational time = 7200 CPUs. The hardware consisted of a desktop with an Intel i7 950 (3.07 GHz) with 8 GB of RAM running Windows 7. 5.1. Problem Data. All four problems share the same problem data besides the number of heats considered. Data for the problem with H heats corresponds to the first H heats in the list. The members of a particular heat group (only applicable to the continuous casters) are given in Table 1. Power consumption values are part of Table 2, where it can be seen that almost 90% of the power consumption occurs in the electric arc furnaces. The processing times are shown in Table 3, while the minimum transfer times are provided in Table 4. Maximum transfer times between stages k and k + 1 are equal to 240, 240, and 120 min, respectively for k = 1, ..., 3. The setup times are setupCC1 = 70 and setupCC1 = 50 min. Finally, the electricity price profile is given in Table 5, where it should be emphasized that hour 0 of the scheduling horizon does not necessarily correspond to midnight, e.g. it can be 14:00 (early afternoon). 5.2. Comparison of RTN Process Models. The RTN-3 process model provides the most accurate representation of the production environment with the rigorous modeling of minimum and maximum transfer times but is it computationally efficient? On the other hand, RTN-2 handles minimum transfer times rigorously but can only approximate the maximum transfer times. It can be viewed as a classic RTN process model, featuring many more tasks and resources than those required when deriving specific constraints from logic propositions, as was done for RTN-3. So which one of these two conceptually different approaches is better? In the third option, RTN-1, focus is on the processing tasks since the duration of transfer tasks is approximated. The advantage is the significant reduction in the number of tasks and resources that may considerably improve computational performance. How important is this approximation in the context of a discrete-time grid that forces us to incur rounding errors for the processing times? To answer these questions, Tables 6 and 7 show the computational statistics for the objectives of total cost (section 4.4.1.2) and makespan minimization (section 4.4.1.1) for δ = 15 min. The results in Table 6 clearly show that the most rigorous RTN-3 approach is computationally by far the most inefficient, being unable to solve to global optimality even the simplest 12heat problem (it should be highlighted that by removing the maximum transfer constraint, second term in the right-hand side (RHS) of eq 8, RTN-3 led to the same results as RTN-2, thus validating the approach). More importantly, no feasible solution could be found within 2 h of computational time for 24 heats. The fact that RTN-2 is significantly more efficient

r ∈ Thr

∑ ∑ (p+ ·Δ+r ,t + p− ·Δ−r ,t )δ /60

+

mintrfu,u′

now illustrated through the solution of four test cases generated from real industrial data. Problem complexity is related to the number of heats to produce (12, 17, 20, and 24), which also gives an indication on how far the plant is operating below maximum capacity when minimizing the total energy cost. Results are generated for three different values of δ (15, 10, and 5 min) to evaluate the influence of the rounding errors on the value of the objective function. 13054



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Table 5. Electricity Price Profile [€/(MW h)] hour

0

1

2

3

4

5

6

7

pricehr hour

209.06 8

199.48 9

166.29 10

146.04 11

151.89 12

130.88 13

97.82 14

84.87 15

pricehr hour

73.76 16

60.81 17

54.12 18

49.50 19

50.36 20

59.69 21

75.07 22

78.90 23

pricehr

88.78

103.88

120.89

140.41

153.94

169.68

184.76

199.18

Table 6. Computational Results for Total Electricity Cost Minimization (δ = 15 min) heats

model

resources

tasks

binary variables

total variables

equations

RMIP (k€)

MIP (k€)

CPU (s)

12

RTN-1 RTN-2 RTN-3 RTN-1 RTN-2 RTN-3 RTN-1 RTN-2 RTN-3 RTN-1 RTN-2 RTN-3

91 166 22 126 231 27 147 270 30 175 322 34

78 222 78 110 314 110 130 370 130 156 444 156

5357 15320 5360 7416 21280 7420 8781 25118 8786 10588 30286 10594

14379 31617 7689 19833 43882 10234 23235 51503 11891 27758 61715 14087

8939 16214 7154 12380 22565 9605 14445 26376 11142 17198 31457 13223

104.337 104.337 101.352 171.637 171.637 166.773 222.346 222.363 216.881 298.018 298.020 292.242

104.394 104.394 104.932b 171.782 171.785 173.950c 222.667 222.671 228.688d 298.540 298.560 no solutione

7.9 33.6 7200a 25.9 263 7200a 223 3916 7200a 587 5353 7200a

17

20

24

a

Maximum computational time. bRelative optimality gap (gap) = 1.3%. cGap = 3.1%. dGap = 4.4%. eBest possible solution = 294.490.

Table 7. Computational Results for Makespan Minimization (δ = 15 min) heats

model

binary variables

total variables

equations

RMIP (min)

MIP (min)

CPU (s)

12

RTN-1 RTN-2 RTN-3 RTN-1 RTN-2 RTN-3 RTN-1 RTN-2 RTN-3 RTN-1 RTN-2 RTN-3

5357 15320 5360 7416 21280 7420 8781 25118 8786 10588 30286 10594

14379 31617 7689 19833 43882 10234 23235 51503 11891 27758 61715 14087

8944 16219 7159 12387 22572 9612 14454 26385 11151 17209 31468 13234

335.0 334.9 304.7 400.8 400.7 357.7 433.0 433.0 390.0 475.1 475.1 423.2

810 810 810 1035 1035 1035 1155 1155 1230b 1335 1335 1395c

13.0 122 1457 193 1452 4156 480 3766 7200a 1656 8480d 7200a

17

20

24

a

Maximum computational time. bGap = 20%. cGap = 35%. dMaximum computational time constraint was relaxed to allow closing the gap.

Table 8. Computational Results as a Function of δ for Total Electricity Cost Minimization δ = 10 min heats

model

binary variables

equations

MIP (k€)

CPU (s)

binary variables

equations

MIP (k€)

CPU (s)

12

RTN-1 RTN-2 RTN-1 RTN-2 RTN-1 RTN-2 RTN-1 RTN-2

8243 23555 11344 32524 13453 38449 16308 46608

13307 24182 18428 33653 21501 39336 25598 46913

103.418 103.420 168.825 168.927b 216.656 217.029c 286.137 286.202d

106 3681 1080 7200a 4291 7200a 3037 7200a

16674 47538 23042 65909 27292 77824 33002 94106

26411 48086 36572 66917 42669 78216 50798 93281

102.577 102.577 167.105e 166.928f 213.411g no solution 283.502h no solution

248 3811 7200a 7200a 7200a 7200a 7200a 7200a

17 20 24 a

δ = 5 min

Maximum computational time. bGap = 0.25%. cGap = 0.39%. dGap = 0.07%. eGap = 0.47%. fGap = 0.40%. gGap = 0.22%. hGap = 0.07%

always prove optimality, with RTN-1 being roughly 1 order of magnitude faster. This, when coupled with the fact that the differences in total electricity cost are almost negligible, leads to the selection of RTN-1 as the most suitable process model. The results for makespan minimization in Table 7 show the exact same solutions for the three process representations

than RTN-3 while requiring about 3 times as many binary variables and 2.5 times as many equations, is particularly surprising. It really attests to the power of the excess resource balance constraints, which make the formulation tighter (see linear relaxation RMIP column) and seem better at driving the search in the right direction. Models RTN-1 and RTN-2 could 13055



Article

higher (20% for the 20 heats problem vs 4.4% for cost minimization). 5.3. Influence of Time Grid Discretization (δ Value) in Solution Quality. As discussed in section 4.1, the size δ of the time slots that make the discrete-time grid is related to data accuracy since the real processing times are rounded up to multiples of δ. As a consequence, some tasks may be unnecessarily delayed, which might compromise the value of the objective function. As the value of δ decreases, the data becomes more accurate and the effect will become less important. The results in Tables 8 and 9 show that changing δ has a greater impact on makespan minimization (between 5.6 and 7.4% reduction for RTN-1, from δ = 15 to 10 min) than in cost minimization (between 0.9 and 4.2%). A further decrease from δ = 10 to 5 min is not as significant (0.8−1.5% for RTN-1), which was expected since the rounding errors are even smaller. Notice that RTN-1 can still prove optimality for δ = 10 min, regardless of the objective function, and returns solutions for δ = 5 min below 0.5% gap, proving itself as a very efficient model. RTN-2, on the other hand, returns very similar solutions for δ = 10 min (total cost minimization), again confirming that underestimating the transfer times is not particularly relevant. 5.4. Total Electricity Cost vs Makespan Minimization. While makespan minimization ensures the most efficient use of the available equipment resources, it may lead to a major increase in the energy bill. If one computes the energy costs for the makespan schedules using the given price profile (6th column in Table 9) and compares them to the minimum costs that can be obtained using the same process model and δ value (5th column in Table 8), one can estimate the potential savings that can be obtained by taking the electricity costs into consideration and switching the energy intensive tasks from high- to low-cost periods. It should however be highlighted that the given price profile is particularly damaging for this process since the first stage comprises the most energy intensive tasks and the first two hours have the highest tariffs. The potential savings naturally decrease as we reduce the excess capacity by increasing the number of heats, but even for the 24 heats problem, which takes up 87% of the production time, a reduction of 12% can be achieved. Figure 8 shows the optimal schedule for total cost minimization for process model RTN-1. Since it considers aggregate equipment units in stages 1−3, it was required to distribute the heats over the two parallel units, which was done by giving preference to unit 1. Note also that the background pattern is related to the electricity price to facilitate interpretation (highest for dark red and lowest for dark green). Not surprisingly, the energy intensive electric arc furnaces (EAF) operate around the green areas; whereas for makespan minimization, they are shifted toward the left to ensure that the steel heats are produced as fast as they can be; see Figure 9. A far more interesting observation is that the power profile for total cost minimization has fewer spikes, which is undoubtedly a more desirable operation mode for the electricity provider. 5.5. Tracking Contracted Load in Incentive-Based Program. Unforeseen events may force the plant to adopt a different schedule than the one previously planned, which might have been used to define a precontracted load curve with the electricity provider. We are now going to assume that equipments EAF2 and CC2 require maintenance between hours [8, 12] and [12, 16], respectively, in order to derive schedules

Table 9. Computational Results for Makespan Minimization (δ = 10 min) heats

model

RMIP (min)

MIP (min)

CPU (s)

cost (k€)

potential savings

12

RTN-1 RTN-2 RTN-1 RTN-2 RTN-1 RTN-2 RTN-1 RTN-2

310.7 310.7 376.9 376.9 408.1 408.1 443.3 443.3

750 750 960 960 1090 1280 1250 no solution

42.1 408 716 5019 6758 7200a 6187 7200a

216.398 217.717 253.380 253.534 272.638 278.952 323.526

52% 52% 33% 33% 21% 22% 12%

17 20 24

a

Maximum computational time.

Figure 8. Optimal schedule for total electricity cost minimization (RTN-1, 24 heats, and δ = 5 min).

Figure 9. Optimal schedule for makespan minimization (RTN-1, 24 heats, and δ = 10 min). Note that the results for δ = 5 min were worse (makespan = 1365 min at the time of termination).

whenever the models can be solved to global optimality. It is an indication that the assumption made for RTN-1 of the transfer time being equal to the stage’s minimum transfer time is a valid one, at least for the steel heats that define the bottleneck. Another interesting result is that while this is a more complex objective for RTN-1 and RTN-2, RTN-3 actually performs better, being able to prove optimality for the 12 and 17 heats problems. On the other hand, the optimality gap is considerably 13056



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Table 10. Computational Results for Minimizing Penalties from Under- and Overconsumption for RTN-1 δ = 10 min heats

CPU (s)

cost (k€)

12 17 20

48.9 2426 1972

175.732 256.455 309.102

12 17 20 a

7139 7200a 7200a

173.073 253.100c 305.847d

δ = 5 min incrementb tracking energy curve 70% 52% 43% tracking power curve 67% 50% 41%

CPU (s)

cost (k€)

incrementb

81.2 1618 7200a

178.072 257.267 307.795e

74% 54% 44%

7200a 7200a 7200a

173.291f 251.406g 303.887h

69% 50% 42%

Maximum computational time. bWith respect to results in Table 8. cGap = 0.18%. dGap = 0.74%. eGap = 0.10%. fGap = 1.28%. gGap = 0.19%. Gap = 1.15%.

h

of total cost minimization for RTN-1 and δ = 5 min. Penalties for over- and underconsumption are equal to p+ = 120 and p− = 60, and the penalty free tolerance is ϕt = 5 ∀ t [MW]. The results in Table 10 show that there is a significant decrease in computational performance when switching from tracking the energy curve in 1-h intervals to tracking the power curve in 10- and 5-min intervals (part of it is due to the additional continuous variables that are required in the latter case). In fact, one can barely solve the 12 heats problem in 2 h of computational time for δ = 10 min when tracking the power curve; whereas, only the 20 heats problem for δ = 5 min can not be solved to global optimality when tracking the energy curve. The difference in cost between the two objective functions is not significant, with the lower values from tracking the power curve resulting from the penalty free tolerance. It is also relevant to note the higher costs obtained for δ = 5 min when compared to δ = 10 min for the 12 and 17 heats problem, which can be explained by the energy load curve for δ = 10 min being easier to follow (this was confirmed experimentally), an effect that compensates the typically higher value from the first term of eq 16, as previously discussed when analyzing Table 8. The schedule for the base case is given in Figure 10, with the equipment maintenance constraints forcing us to anticipate the start of production by at least 2 h; see Figure 11. Despite a few

Figure 10. Optimal schedule for total electricity cost minimization (RTN-1, 20 heats, and δ = 5 min).

that minimize the penalties from deviating from the precontracted load (see details in section 4.4.1.3). Since these constraints make it impossible to produce 24 heats in one day, the analysis is limited to the other three cases. The precontracted loads are taken from the results for the objective

Figure 11. Optimal schedule for precontracted load tracking (RTN-1, 20 heats, and δ = 5 min). Results for energy on the left and power on the right. In bottom charts, dashed lines give precontracted load and solid lines give the rescheduled load. 13057



Article

(5) Castro, P. M.; Harjunkoski, I.; Grossmann, I. E. New Continuous-Time Scheduling Formulation for Continuous Plants under Variable Electricity Cost. Ind. Eng. Chem. Res. 2009, 48, 6701. (6) Pantelides, C. C. Unified Frameworks for the Optimal Process Planning and Scheduling. In Proceedings of the Second Conference on Foundations of Computer Aided Operations; Cache Publications: New York, 1994; pp 253. (7) Castro, P. M.; Harjunkoski, I.; Grossmann, I. E. Optimal scheduling of continuous plants with energy constraints. Comput. Chem. Eng. 2011, 35, 372. (8) Dimitriadis, A. D.; Shah, N.; Pantelides, C. C. RTN-based rolling horizon algorithms for medium term scheduling of multipurpose plants. Comput. Chem. Eng. 1997, 21, S1061. (9) Vujanic, R.; Mariéthoz, S.; Goulart, P.; Morari, M. Robust Integer Optimization and Scheduling Problems for Large Electricity Consumers. 2012 American Control Conference, Montreal, Canada, June 27−29, 2012; pp 3108−3113. (10) Mitra, S.; Grossmann, I. E.; Pinto, J. M.; Arora, N. Optimal production planning under time-sensitive electricity prices for continuous power-intensive processes. Comput. Chem. Eng. 2012, 38, 171. (11) Castro, P.; Barbosa-Póvoa, A.; Novais, A. Simultaneous Design and Scheduling of Multipurpose Plants Using Resource Task Network Based Continuous-Time Formulations. Ind. Eng. Chem. Res. 2005, 44, 343. (12) Kondili, E.; Pantelides, C. C.; Sargent, R. A General Algorithm for Short-Term Scheduling of Batch Operations - I. MILP Formulation. Comput. Chem. Eng. 1993, 17, 211. (13) Sundaramoorthy, A.; Maravelias, C. T. A General Framework for Process Scheduling. AIChE J. 2011, 57, 695−710. (14) Sundaramoorthy, A.; Maravelias, C. T. Computational Study of Network-Based Mixed-Integer Programming Approaches for Chemical Production Scheduling. Ind. Eng. Chem. Res. 2011, 50, 5023. (15) Biegler, L. T.; Grossmann, I. E.; Westerberg, A. W. Systematic Methods of Chemical Processing Design; Prentice Hall: Upper Saddle River, NJ, 1997. (16) Lima, R. M.; Grossmann, I. E.; Jiao, Y. Long-term scheduling of a single-unit multi-product continuous process to manufacture high performance glass. Comput. Chem. Eng. 2011, 35, 554. (17) Castro, P. M.; Grossmann, I. E. Generalized Disjunctive Programming as a Systematic Modeling Framework to Derive Scheduling Formulations. Ind. Eng. Chem. Res. 2012, 51, 5781. (18) Glismann, K.; Gruhn, G. Short-Term Planning of Blending Processes: Scheduling and Nonlinear Optimization of Recipes. Chem. Eng. Technol. 2001, 24 (3), 246. (19) Castro, P.; Matos, H.; Barbosa-Póvoa, A. P. F. D. Dynamic modelling and scheduling of an industrial batch system. Comput. Chem. Eng. 2002, 26, 671. (20) Castro, P. M.; Novais, A. Q.; Carvalho, A. Optimal Equipment Allocation for High Plant Flexibility: An Industrial Case Study. Ind. Eng. Chem. Res. 2008, 47, 2742. (21) Duarte, B.; Santos, L.; Mariano, J. Optimal Sizing, Scheduling and Shift Policy of the Grinding Section of a Ceramic Tile Plant. Comput. Oper. Res. 2009, 36, 1825. (22) Wassick, J. Enterprise-wide optimization in an integrated chemical complex. Comput. Chem. Eng. 2009, 33, 1950. (23) Hadera, H.; Harjunkoski, I. Continuous-time Batch Scheduling Approach for Optimizing Electricity Consumption Cost. Comput.Aided Chem. Eng. 2013, 32, 403−408. (24) Moon, J.; Shin, K.; Park, J. Optimization of production scheduling with time-dependent and machine-dependent electricity cost for industrial energy efficiency. Int. J. Adv. Manuf. Technol. 2013, 68, 523−535.

differences in the schedules for energy and power load tracking, being the most noticeable the relative position of the groups processed in continuous caster CC1, the rescheduled load curves are quite similar. Minimizing deviations from the precontracted power profile does however lead to a smoother and preferable profile around 4:30, when there is not a sudden and short-lived drop in power to zero. Recall that such spikes are diluted when keeping track of energy consumption over the hour, whereas in power tracking they are measured every time slot. Overall, the computational results in Table 10 show an increase in cost around 40% for 20 heats that increases to 70% as the number of heats decreases. Since we can find solutions within a 1.28% gap of the global optima, high penalties are incurred even when using a powerful optimization model to best react to unforeseen events. It is a clear indication of the importance of the proposed decision-making tool.

6. CONCLUSIONS This paper has shown that a resource−task network discretetime formulation can efficiently capture the most important scheduling constraints of the melt shop of a steel plant. It can be used as an important tool for industrial demand side management or demand response, a concept in which the plant adapts its operational behavior by changing the timing of electricity usage from on-peak to off-peak hours for the collective benefit of society. When compared to the traditional optimization scope of minimizing the makespan, minimizing the total electricity cost can lead to savings of 12% for high capacity utilization up to 52% for low capacity utilization. The optimization model was also able to track precontracted load curves with the actual energy and power consumption of the steel plant, penalizing under- and overconsumption. Future work will investigate efficiency improvements to solve larger problems instances, the impact of uncertainty in the problem data, and continuous-time models that can efficiently account for variable processing times.

■

AUTHOR INFORMATION

Corresponding Author

*Tel.: +351-210924643. E-mail: [email protected] (P.M.C.). [email protected] (I.H.). Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS P.M.C. gratefully acknowledges financial support from Fundaçaõ para a Ciência e Tecnologia through project PTDC/EQU-ESI/118253/2010.

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REFERENCES

(1) Paulus, M.; Borggrefe, F. The potential of demand-side management in energy-intensive industries for electricity markets in Germany. Appl. Energy 2011, 88, 432. (2) Harjunkoski, I.; Grossmann, I. E. A Decomposition Approach for the Scheduling of a Steel Plant Production. Comput. Chem. Eng. 2001, 25, 1647. (3) Nolde, K.; Morari, M. Electrical load tracking scheduling of a steel plant. Comput. Chem. Eng. 2010, 34, 1899. (4) Häit, A.; Artigues, C. On electrical load tracking scheduling for a steel plant. Comput. Chem. Eng. 2011, 35, 3044. 13058


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