Ind. Eng. Chem. Res. 2009, 48, 8551–8565
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Sequential Methodology for Scheduling of Heat-Integrated Batch Plants Iskandar Halim† and Rajagopalan Srinivasan*,†,‡ Institute of Chemical and Engineering Sciences, A*STAR (Agency for Science, Technology and Research), 1 Pesek Road, Jurong Island, Singapore 627833, and Laboratory for Intelligent Applications in Chemical Engineering, Department of Chemical and Biomolecular Engineering, National UniVersity of Singapore, 10 Kent Ridge Crescent, Singapore 119260
Optimal scheduling of tasks to be performed in different unit operations is of paramount importance in the batch processes. The pressure toward sustainable operation has warranted more attentions to be paid to the issues of heat integration to reduce the utility consumption. This paper presents a new approach for incorporating heat integration in batch process scheduling. The method is based on a sequential framework, where the overall problem is decomposed into two sequentially solved problems of scheduling and heat integration. First, the schedule is optimized to meet the economic objective such as makespan or profit. Next, alternate schedules are generated through a stochastic search-based integer cut procedure that adds further constraints to the scheduling formulation. Such schedules may be the alternate optima of the scheduling problem. Finally, heat integration analysis coupled with the time average model (TAM) and time slice model (TSM) is applied to each of the resulting schedules to establish the minimum utility targets. The approach is based on the precept that opportunities for heat integration between tasks are higher in near-optimal schedules where tasks are temporally clustered together. The method differs from other sequential approaches published in the literature in that the heat integration problem is solved with the intent to retain the optimality of the scheduling solution. The proposed method has the ability to handle problems too complex to be solved using simultaneous optimization approaches. It has been applied to two literature case studies. The first one involves profit maximization as the objective function. The second one entails a more complex scenario involving makespan and utilities. The results from both case studies demonstrate the effectiveness of the approach. 1. Introduction Today, about half of industrial chemical products are manufactured using batch processes.1 Unlike in continuous processes, batch processes are intrinsically flexiblesa variety of products can be produced through sharing the same equipment in the plant. The batch processing mode is favorable for the production of high-value-added chemicals such as pharmaceuticals, foods, fine chemicals, and agrochemicals. As demands for such products are highly seasonal and low in volume, a batch process with its inherent flexibility is often preferred. However, this flexibility leads to extra complexity in the design and operation of the plant. As multiple tasks can be performed in the same equipment, optimal task scheduling becomes exceedingly crucial for meeting the production in a cost-effective manner. In the mean time, the continuing global concern over carbon emissions has pressurized the chemical industry to reduce its energy footprint. As a proven energy minimization technique, heat integration has now become an integral part of chemical process design. However, heat integration has for long been the privilege of continuous rather than batch processes.2 As hot and cold streams flow intermittently during the batch operation, opportunities for heat integration between such streams become restricted. In the past, batch chemical industries could tolerate such energy intensive operations due to the high values of final products, which outstripped the energy costs. However, the continuing global competition for raw materials and energy together with the pressure on environmental sustainability now provides a strong impetus for the batch chemical industries to * To whom all correspondence should be addressed. E-mail:
[email protected]. Fax: +65 67791936. Phone: +65 65168041. † A*STAR (Agency for Science, Technology and Research). ‡ National University of Singapore.
consider heat integration as a means to minimize energy consumption. Traditionally, scheduling and heat integration of batch processes have been considered as two separate problems. Batch scheduling entails determining an optimal sequence of tasks over limited resources (raw materials and process units) for a given objective, such as minimizing makespan or maximizing profit.3 Such optimal operation is crucial as it improves the economics of the process through productivity improvement and resource savings. A review of different scheduling methodologies is reported by Janak and Floudas.4 In general, batch scheduling is formulated as a mixed integer linear programming (MILP) problem. In terms of time representation, the methodologies can be differentiated into two main categories: discrete-time and continuous-time. One of the early approaches was that of Kondili et al.,5 who discretized the entire time horizon into a number of equal duration intervals and associated different task events to the boundaries of the intervals. However, this approach can lead to a large number of binary variables and hence limits its applications.6 Continuous time-based approaches solve the inherent limitation of the discrete-time approach by allowing task events to take place at any point in time. Networkrepresented continuous-time approaches employ a state-task network (STN)5 or resource-task network (RTN)7 to represent the production recipe. The network-represented approach can be further categorized into global event and unit-specific event models. In the global event model, a set of specific events referred to as time slots is used to determine the timing and sequencing of all tasks in the units. The unit-specific event model introduces the concept of event points, i.e., a sequence of time instances located along the time axis of a unit to represent the beginning of a task or utilization of a unit.8 Both the global event and the unit-specific event model can be differentiated
10.1021/ie900367j CCC: $40.75 2009 American Chemical Society Published on Web 08/13/2009
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Ind. Eng. Chem. Res., Vol. 48, No. 18, 2009
Table 1. Comparison between Simultaneous and Sequential Optimization Approach methodology
advantages
simultaneous approach • scheduling and heat integration are taken into account simultaneously • can lead to global optima
sequential approach
• simpler as the problem is split into two distinct parts: scheduling and heat integration • does not require simplifying assumptions such as one-to-one heat exchange and preinstalled heat exchanger unit
disadvantages • utility demands are represented as a linear model • limiting temperature constraint for heat transfer is not considered • heat exchange superstructure network needs to be specified a priori • may or may not lead to global optima
into synchronous and asynchronous representations.9 The synchronous model involves synchronizing the slots across all units in the plant,3,9,10 while in the asynchronous model, the slots differ from one unit to another.11-13 While these methodologies are useful, they are limited in their applications to scheduling of tasks within the constraints of raw material and process unit availability. Heat integration is often not addressed as it would introduce yet another layer of complexities to the scheduling formulation. Historically, another reason for the limited application of heat integration to batch scheduling was the relatively small energy savings that can be derived. However, it is well-known that batch processing in food, dairy, brewing, and biochemical industries incur substantial energy consumptions.14 Successful application of heat integration to these processes would therefore contribute significantly to their sustainability standing. As a matter of fact, up to 20% savings in energy costs and 10% reduction of carbon emissions in these industries have been reported.1 The methodologies for incorporating heat integration in batch scheduling can be distinguished as simultaneous and sequential frameworks. Both approaches have been demonstrated in the literature and are known to have their own advantages and disadvantages (see Table 1). The simultaneous framework involves formulating an integrated mathematical model comprising of heat integration and process scheduling and solving them simultaneously for maximum economic benefit. The sequential framework on the other hand decomposes the problemsthe scheduling part is solved first followed by heat integration. Using a simultaneous approach, Papageorgiou et al.15 extended the discrete-time scheduling formulation of Kondili et al.5 by embedding a heat integration model within the scheduling formulation. However, this approach requires a large number of binary variables. Lee and Reklaitis16 addressed the problem of task scheduling for maximum energy recovery. However, to avoid generating a complex formulation, the pairing between the hot and cold streams was limited to one-hot-onecold matching. Zhao et al.17 removed the restriction of one-toone heat exchange policy and applied heat cascade analysis. The result was a mixed integer nonlinear programming (MINLP) formulation which was solved by assuming constant heat exchange time between process streams. A graphical approach
was introduced by Adonyi et al.18 to solve the scheduling and heat integration problem simultaneously. Their algorithm was based on a branch-and-bound framework, where the branching procedure was applied for scheduling of the process units and the bounding procedure was applied for determining the bounds of the utility costs. A discrete-time approach for design of a multipurpose plant with direct heat integration was proposed by Pinto et al.19 In their approach, a heat exchange network in the form of a superstructure was configured and solved for the optimal structure. However, the number of binary variables used in the formulation made it a large MILP problem. A continuoustime model was proposed by Majozi2 for simultaneous scheduling and direct heat integration. A key drawback common to the above-mentioned simultaneous methods is their oversimplification of the heat integration problem. For instance, the availability of a suitable temperature difference as a driving force for heat transfer is generally presumed in these methods. In addition, instead of energy balance, a simple mathematical correlation is usually applied to calculate the utility (steam or cooling water) demands posed by process units. Further, in these approaches, the heat exchange network in the form of superstructure configuration needs to be specified a priori. In the sequential framework, although the scheduling part is generally solved using a dedicated optimization approach, the heat integration part can be solved either through pinch analysis or mathematical optimization. The former considers the availability of a fixed production schedule and applies pinch analysis to reduce the amounts of utilities. Further, it assumes a pseudocontinuous behavior of batch operations. Some of the early works on the batch pinch analysis method include those of Obeng and Ashton,20 Kemp and MacDonald,21,22 and Kemp and Deakin23 who performed heat cascade analysis for heat recovery network design. In these, the process was divided into a set of temperature and time intervals within which heat could be cascaded. Wang and Smith24 introduced a graphical representation of time pinch analysis. Their approach involved plotting the energy composite curve in the form of heat transferred versus time. However, time was taken as the primary constraint that limited the amount of heat to be recovered from the hot streams. Having exhausted the possibilities of cascading through time, the excess heat was cascaded to the next temperature interval. Another attempt was by Uhlenbruck et al.,25 who sought to establish heat integration from the pragmatic view of heat exchange network synthesis. Unlike in the previous pinch approaches which were based on the thermodynamic insights of the process, in their approach, the heat exchange between the hot and cold stream was constrained to a one-toone match so as to arrive at a simple network. The mathematical optimization technique can also be used for the heat integration step in the sequential methodology. Vaselenak et al.26 explored the possibility of heat recovery between vessels over a predefined production schedule. They solved the formulation using a combined heuristic and MILP optimization. Similarly, Corominas et al.27 solved the problem of direct heat transfer in multiproduct plants with a prespecified campaign mode. However, their formulation was only applicable to the specific case where each product follows the same sequence of process units. Vaklieva-Bancheva et al.14 considered a more general process scheme wherein each product was allowed to pass through a subset of equipment stages. Their objective was to determine the minimum costs (capital and operating) of preinstalled heat exchange units. Bozan et al.28 developed a two-part approach for optimizing the cost of a heat exchange network. In the first stage, a product campaign was
Ind. Eng. Chem. Res., Vol. 48, No. 18, 2009
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Figure 1. STN model for the simple batch production.
determined through a heuristic procedure to specify the locations of heat exchange units. In the next step, the resulting heat exchange network was optimized for minimization of the total energy costs (the operating costs of hot and cold utilities plus the annualized capital costs of heat exchange units). The key advantage of the sequential approaches lies in their ability to solve heat integration problem without resorting to simplifying assumptions that are characteristics of the simultaneous methods. However, their drawback is they could result in suboptimal solutions.18 The two-step optimization in the sequential methodology does not necessarily imply a poorer solution compared to the simultaneous approach. Although it has not received as much attention in the literature, multiple (alternate) solutions occur widely in batch scheduling problems.29 Finding the alternate schedules of the scheduling problem can allow the process engineer the flexibility to choose a solution that optimizes another (secondary) objective without deterioration in the primary objective function. For instance, among the different scheduling solutions which have the same makespan value as the objective function, it would be possible to select one that requires minimum utilities. In this paper, a sequential method that exploits alternate schedules in the batch scheduling is proposed. The method has been developed based on a threestep procedure. The scheduling problem is first optimized to meet the economic objective such as makespan or profit. Next, alternate schedules are generated through a stochastic searchbased integer cut procedure that adds further constraints to the formulation. Such schedules may be the alternate optima of the scheduling problem. Finally, heat integration analysis is applied to each of the resulting schedules to establish the minimum utility targets. The method proposed here differs from other sequential methods published in the literature in that the energy integration problem is solved with the intent to retain the optimality of the scheduling solution since it engenders temporal overlap among tasks and thus enables better heat integration. Another important feature of the method is its ability to tackle problems too complex to be solved using simultaneous optimization approaches. For instance, most if not all of the optimization problems solved using the simultaneous method involves only single objective function (e.g., maximum profit). In the proposed approach, multiple objectives such as minimum makespan and utilities can be used if necessary. Further, in the simultaneous approach, the utility requirement for each batch is usually calculated by assuming a simple linear correlation without considering the temperature constraints for heat transfer. Such simplification is obviated here. The rest of this paper is organized as follows: in the next section, we begin with a detailed problem definition. In section 3, we present the scheduling formulation used here followed by the heat integration model in section 4. In section 5, we propose the heatintegrated scheduling framework and illustrate it using two case studies.
2. Problem Statement In a multipurpose batch plant, in addition to sharing limited resources such as process units and materials, it is common for some of the operations to require utilities for heating or cooling purposes. A suitable recipe model is thus necessary for representing this in an effective manner. In this paper, the state-task network (STN) of Kondili et al.5 is used to represent the batch production recipe. An STN model consists of three basic components: state nodes, task nodes, and arcs. The state nodes (shown graphically as circles) model the material states of the feeds, intermediates, and final products. The task nodes (notated by rectangles) describe the operations (tasks) that transform materials from their input to the output states. In addition, they show information about the plausible units that can be used for the operations. Connecting the state and the task nodes are arcs, which define the precedence of tasks. The STN model of Kondili et al.5 has been more recently adapted to include the heating and cooling requirements of tasks. This is done by assigning vertical arrows to the corresponding task nodessan upward arrow represents tasks that require heating and downward arrow for tasks that require cooling.18 This representation is illustrated in Figure 1 which shows a modified STN model for a simple production process.9 The plant produces chemical D by converting raw material A to two intermediates B and C. The entire production involves three sequential tasks. First is a reaction from A to B involving two plausible units (unit 1 and unit 2). As this is an exothermic reaction, cooling is required throughout the duration. The next task involves a reaction from B to C in unit 3 with cooling, followed by the production of D in either unit 4 or unit 5. The latter task is endothermic and requires heating. The problem to be addressed in this paper can be stated as the following: Given information on a multipurpose batch plant in the form of (i) production recipe (STN model), (ii) equipment configuration (flowsheet), (iii) task processing times, (iv) capacity limits of process units, (v) material heat capacity, (vi) costs and values of raw materials, utilities, and products, and (vii) scheduling horizon (makespan minimization problem) or production demand (profit maximization problem), determine (i) the production schedule (i.e., timing of all tasks, batch sizes, and allocation of tasks to units) and (ii) the heat exchange configuration, so as to maximize the economics of the plant. The following conditions are assumed: (i) All materials are stored in a storage facility; (ii) Material transfer and unit setup times are lumped into processing times; (iii) Different batch sizes can be processed by tasks;
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Ind. Eng. Chem. Res., Vol. 48, No. 18, 2009 K
min makespan )
∑ SL
or max profit )
k
∑g I
m mK
m
k)1
(3) where K is the total number of slots, ImK is the inventory of material m at the final time slot TK, and gm is the cost or value of unit mass of material m. The objective function is subject to a number of constraints, involving the status of unit, processing time, mass balance, and material inventories. This is described as follows. A balance on the status of a unit j can be written as yijk ) yij(k-1) + Yij(k-1) - YEijk
Figure 2. Flowsheet for the simple batch production.
(iv)
The processing time of task i on unit j is a constant or varies linearly with batch size; (v) An unlimited number of heat exchangers is available; (vi) Only direct heat integration with a cocurrent scheme is considered, i.e., heat storage is not allowed; (vii) Heat transfer occurs in a pseudocontinuous mode; and (viii) Heat losses are negligible. To illustrate the problem, consider again the case study described by the STN model of Figure 1. The equipment configuration for this process is shown in Figure 2. The objective here is to schedule the production in such a way that profit is maximized. Figure 3 shows one possible way for minimizing the operating cost through cocurrent heat exchange.16 In this scheme, materials from units 1 and 4 are heat exchanged and then returned to the same vessel. Similarly, heat is exchanged between materials from units 2 and 5. The result is reduced consumption of utilities (cooling water and steam). 3. Batch Scheduling Formulation In general, batch scheduling involves a decision making process to determine when, where, and how to produce a set of products under given process requirements, production recipes, and limited resources in a specific time horizon.6 Given the complex nature of batch scheduling, a mathematical formulation that effectively models the numerous interconnections between the determining factors is essential. The mathematical model used in this work is the continuous-time synchronized-slot model reported by Sundaramoorthy and Karimi.9 This section describes some of the key elements of the model. The reader is referred to the work of Sundaramoorthy and Karimi9 for more details. The model involves splitting the batch horizon H into K (k ) 1, 2, ..., K) number of slots, as shown in Figure 4. The slots are synchronized on all units (j ) 1, 2, ..., J). A time point Tk is defined as a time at which slot k endssT0 signifies the beginning of the horizon (T0 ) 0). As shown in the figure, a task starting at Tk-1 can finish before, at, or after time Tk according to the following relation: Tk ) Tk-1 + SLk
(1)
where, SLk is the variable slot length whose sum cannot exceed the time horizon H, i.e.,
where, Yijk, yijk, and YEijk are binary variables described as follows: Yijk ) yijk )
{ {
1 if unit j begins task i at time Tk 0 otherwise
∑ SL
k
eH
k)1
The objective function can be expressed as follows:
(2)
i ∈ I j, 0 e k < K
1 if unit j is continuing to perform task i at time Tk 0 otherwise i ∈ Ij, 0 e k < K
YEijk ) 1 if unit j ends task i and releases its batch at time Tk 0 otherwise i ∈ Ij, 0 e k e K
{
Here, Ij represents a set of tasks which unit j can perform. Equation 4 states that whenever a task i is not under progress on unit j at time Tk, yijk is zero. It becomes one only after a task has begun on the unit and zero when it ends at Tk. On the basis of the binary Yijk, the following binary variable Zjk is introduced: Zjk )
{
1 if unit j begins a task (including i ) 0) at time Tk 0 otherwise 0ekeK
In this case, if a task i merely continues on unit j at Tk, then Zjk ) 0. Since only one task can start on a unit j at any time Tk, the following expression can be written: Zjk )
∑Y
ijk,
0ek
