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Fuzzy Optimization Model for Source-Sink Water Network Synthesis with Parametric Uncertainties Raymond R. Tan* Chemical Engineering Department, De La Salle University, 2401 Taft Avenue, 1004 Manila, Philippines ABSTRACT: Source-sink problems occur in various process systems engineering applications, such as the design of water reuse systems and resource conservation networks. Such problems may be solved using alternative approaches, including graphical pinch analysis and linear programming. In most cases, it is normally assumed that the stream flow rates and quality levels are deterministic. This paper presents a fuzzy mathematical programming model for the synthesis of water networks when the model parameters exhibit fuzzy uncertainties. The modeling approach is illustrated for three water network case studies adapted from literature. These illustrative examples cover concentration- and property-based stream quality indices, as well as single- and multiple-plant water integration.
’ INTRODUCTION In recent years, environmental considerations have become integral to the design of chemical processes. The trend was initially driven simply by the need to comply with increasingly stringent environmental legislation; however, nowadays sustainability issues are increasingly integrated into decision-making in industry. For example, climate change is now recognized by the majority of the scientific community as the single most important environmental issue facing the modern world. This has led to significant shifts in industry toward the use of energy efficient technologies and low-carbon energy sources. Similarly, water resource consumption has recently been identified by Rockstrom et al.1 as one of the important global environmental issues of the 21st Century. The current level of global freshwater use, now at 2600 km3/year, is rapidly approaching the sustainable limit, which they estimated at 4000 km3/year. At the same time, the availability of freshwater of acceptable quality is also closely linked with some of the other environmental issues identified, namely, chemical pollution, climate change, and the global nitrogen and phosphorus cycles. In particular, climate change may result in water stress in some parts of the world as a result of changes in local precipitation levels. Thus, it is evident that the efficient use of freshwater in the process industries is essential to achieving sustainable development. Industrial water conservation can be achieved through a number of different strategies. One approach is to make fundamental changes in processes to make these less water intensive. For example, air may be used in place of water as a coolant in thermal systems. Alternatively, process integration techniques can be used to conserve water without making any changes in the process units themselves. In such an approach, the key is to design and install a water network for the optimal reuse, recycle, and regeneration of water streams within a process plant or an eco-industrial park (EIP). Such networks are designed to maximize opportunities for water recovery through the identification of suitable process matches. Processes within a plant may act as water sources, water sinks (i.e., demands) or both. Network r 2010 American Chemical Society
topologies may then be established through systematic sourcesink matching. Such source-sink problems occur extensively in process systems engineering literature. In these problems, a process network is assumed to consist of a set of black boxes acting as stream sources and sinks, with each stream having fixed flow rate and quality specifications. Most early applications involve the design of water reuse networks using pinch analysis or equivalent optimization models.2-9 Note that this approach represents a departure from earlier water network research, which were predominantly based on a special form of mass exchange network (MEN). Bagajewicz10 provides a survey of those early approaches, while a more recent review11 focuses on the source-sink representation. The source-sink framework has also led to extensions that are not possible with the older, mass-transfer-based representation. For example, in property integration, stream quality is indicated not by concentration levels of contaminants, but by functional physical properties.12-17 Furthermore, direct analogies have led to applications in industrial gas integration18,19 and energy planning with various environmental constraints.20-23 The basic structure of the source-sink problem is shared by all the above applications. The equivalent linear programming (LP) model is described by El-Halwagi et al.5 However, in most cases it has been asssumed that stream flow rates and quality levels are known precisely. In practice, process data may be subject to uncertainty or variability, which may be probabilistic (because of randomness of operating conditions) or epistemological (arising from lack of complete knowledge or understanding of a process). The integration of such uncertainties is essential for the design or robust processes.24,25 A number of techniques have been developed Special Issue: Water Network Synthesis Received: May 4, 2010 Accepted: August 19, 2010 Revised: July 30, 2010 Published: September 02, 2010 3686
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Industrial & Engineering Chemistry Research particularly for water network design problems. For example, Tan et al.26 proposed the use of Monte Carlo simulation to assess alternative water network configurations to variable mass loads. Al-Redhwan et al.27 developed an approach based on sensitivity analysis and stoachastic programming to develop flexible and resilient process water networks. Stochastic optimization approach for the design of mass-transfer based total water networks (composed of water-using processes, regenerators, and treatment units) under uncertainty have also been developed.28 Zhang et al.29 proposed new numerical indices to quantify the resiliency of water network designs, while Riyanto and Chang30 made use of a flexibility index to develop revamp strategies in water networks. Tan and Cruz31 developed a fuzzy approach for water network design that introduces parametric uncertainty in the right-hand side of the mathematical model constraints. The formulation used in that early paper is based on the concept of fuzzy optimization, which is suited to handling uncertainties arising from ambiguity or vagueness of information, as opposed to probabilistic noise.24,32,33 The fuzzy approach has also proven to be applicable to water network problems with multiple objectives34 or multiple decision-makers.35-37 However, the model of Tan and Cruz31 is limited to uncertainties occurring in the quality tolerances of the sinks or demands. This work generalizes the previous approach by allowing parametric uncertainties in all stream quality levels to be included in the formulation. The rest of the paper is organized as follows. A specific problem statement is first given, followed by a description of the model itself. Three illustrative case studies are then presented. The first two examples involve single-plant water integration, with and without topological constraints. The third example illustrates a further extension of the model for the case of property-based interplant water integration. Finally, conclusions and prospects for further research are given at the end of the paper.
’ PROBLEM STATEMENT The problem may be formally stated as follows. We consider a system with a set of internal sources and sinks that may be matched to meet the objective of minimizing the requirement to import an external, high-value resource. A fuzzy resource consumption goal is specified in terms of upper and lower bounds, corresponding to the least and most desirable freshwater usage levels. Each internal source is characterized by a flow rate and one or more quality indices, which are defined in terms of fuzzy intervals. Likewise, each sink is defined in terms of fuzzy flow rate and fuzzy quality limits. The model combines the approaches of Zimmermann33 and Carlsson and Korhonen38 and covers a range of solutions ranging from optimistic (i.e., high-risk) and conservative (i.e., low-risk) extremes. Solving the model identifies an optimal solution which achieves the best compromise between the need to conserve freshwater on one hand, and the need to avoid adverse process effects on the other. The final source-sink network must satisfy all the flow balances, quality restrictions and resource conservation goal defined by the fuzzy intervals. ’ FUZZY OPTIMIZATION MODEL Indices i = index for sources (i = 1, 2, ...m) j = index for sinks (j = 1, 2, 3, ...n)
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k = index for stream quality (k = 1, 2, 3, ...p) Variables λ = overall degree of fuzzy constraint satisfaction bij = binary variable indicating the existence of internal reuse stream allocated from source i to sink j fj = external resource requirement of sink j rij = internal reuse stream allocated from source i to sink j yj = binary variable indicating interplant water stream to sink j Parameters DU i = upper bound of the stream flow rate requirements of sink j DLi = lower bound of the stream flow rate requirements of sink j FU = upper limit of the total external resource requirement of the network FL = lower limit of the total external resource requirement of the network FU A = upper limit of the total external resource requirement of Plant A FLA = lower limit of the total external resource requirement of Plant A FU B = upper limit of the total external resource requirement of Plant B FLA = lower limit of the total external resource requirement of Plant B LU jk = upper bound of the kth quality load of sink j LLjk = lower bound of the kth quality load of sink j M = arbitrary large number QU ik = upper bound of the kth quality index of the stream from source i QLik = lower bound of the kth quality index of the stream from source i QFU k = upper bound of the kth quality index of the external resource QFLk = lower bound of the kth quality index of the external resource rmin = minimum acceptable flow rate for reuse or recycle stream SU i = upper bound of the available stream flow rate from source i SLi = lower bound of the available stream flow rate from source i The fuzzy optimization model is as follows. The objective function is to maximize the overall degree of fuzzy constraint satisfaction max λ
ð1Þ
This objective is subject to the following constraints. The fuzzy degree of satisfaction lies within the interval [0, 1], where a value of 0 corresponds to unsatisfactory, a value of 1 is completely satisfactory, and fractional values indicate partial satisfaction 0eλe1
ð2Þ
In fuzzy optimization using “max-min” aggregation, every fuzzy constraint should be satisfied partially at least to the degree λ.33 A fuzzy goal for freshwater resource consumption is specified for the network using the linear membership function as follows 0P 1 fj - F U B j C ð3Þ @ L Agλ F - FU 3687
dx.doi.org/10.1021/ie101025p |Ind. Eng. Chem. Res. 2011, 50, 3686–3694
Industrial & Engineering Chemistry Research Note that the total freshwater usage approaches FU when λ = 0, and becomes FL when λ = 1. Then, the water balance for each source is given as X rij eSUi þ λðSLi - SUi Þ "i ð4Þ j
Equation 4 means that the amount of water to be reused to various process sinks is less than or equal to the amount available from the source, with any excess being discharged from the network as effluent. It can easily be seen that when λ = 0, the amount of water available from source i is SiU, while when λ = 1, the amount available is SiL. This relationship means that lower values of λ are inherently more optimistic (and thus riskier), while higher values of of λ are inherently more conservative. This concept is described in more detail by Carlsson and Korhonen.38 The objective function (eq 1) thus forces the model toward a more conservative solution that yields a robust network. If the water available from source i is known precisely, then SiU = SiL, and the fuzzy interval has zero width. The water balance for each sink or demand is given as X rij ¼ DLj þ λðDUj - DLj Þ "j ð5Þ fj þ i
Equation 5 means that each sink is satisfied by freshwater resource combined with reused water coming from different process sources. It can also be seen here that the water demand of sink j is DjL when λ = 0 and DjU when λ = 1; the logic used is similar to the one previously described, with higher values of λ denoting more conservative (or less risky) conditions. Also, if the water demand of sink j is known precisely, the fuzzy interval will also have zero width. The quality load balance for each sink is then given as fj ½QFLk þ λðQFUk - QFLk Þ þ
X
rij ½QikL þ λðQikU - QikL Þ
i
eLUjk þ λðLLjk - LUjk Þ "j, k ð6Þ As in conventional formulations, the quality indices are assumed to be inverse (i.e., lower values indicate higher stream quality or purity) and follow a linear mixing rule. These assumptions hold true for concentration-based quality indices that are used in conventional water integration problems. In the case of propertybased integration, appropriate property operators may be used to linearize quality indices to conform to these assumptions.12-17 These variants are illustrated in the case studies that follow. Equation 6 states that the kth quality load for the mixed stream entering a given sink j must not exceed the load tolerance of the said sink. Note that for λ = 0, the streams entering the sink are of the best quality (QFkL and QikL), which corresponds to an optimistic solution that maximizes opportunties for water reuse. However, the water recovery potential must counterbalanced against considerations of data uncertainty. Thus, λ = 1 corresponds to the most conservative solution, which in turn assumes that the available streams are of the poorest quality (QFkU and QikU). Likewise, inspection of the right-hand side of eq 6 shows that the least conservative solution (λ = 0) assumes maximum sink tolerance to quality load (LjkU), while the most conservative solution (λ = 1) assumes minimum tolerance (LjkL). In summary, optimistic solutions correspond to maximizing opportunities for water reuse, at the expense of the added risk of having excess
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quality load (i.e., poor stream quality) being allocated to the process sinks; conversely, the most conservative solutions provide a safety margin and reduce recovery opportunties by using the worst-case assumptions regarding the streams. The fuzzy +model allows a compromise solution to be identified between these two extremes. It can also bee seen that the left-hand side of eq 6 includes bilinear terms, thus making the entire model a nonlinear programming (NLP) problem, except in the special case where the stream quality levels are known precisely and the fuzzy intervals have zero width. In the latter case, eq 6 reduces to a conventional quality load balance. Finally, all flow rates in the system are non-negative fj g 0
"j
ð7Þ
rij g 0
"i, j
ð8Þ
Three case studies are given in the succeeding sections to illustrate the use of the fuzzy optimization model. Additional constraints are given in two of the case studies, in order to account for case-specific design considerations. The case studies are solved using the commercial optimization software Lingo (version 11), which is equipped with a global optimization toolbox for nonlinear problems.39
’ CASE STUDY 1 This case study involves the design of a single-component water network with four sources and four sinks, where contaminant concentration serves as the measure of stream quality. The objective is to minimize the external fresh water requirement for the system. The data used is adapted from Polley and Polley,3 but fictitious uncertainty margins have been added to the concentration levels to demonstrate the fuzzy model to be used. These uncertainties arise from various sources within the process. For example, in practice, the concentration limits for the sinks are often estimated subjectively by plant engineers based on their working knowledge of the processes; likewise, concentrations of streams emanating from the sources may be estimated from historical plant data and can be subject signficant variations as well. An uncertainty margin is also given for the flow rate required by sink 4, which for instance can reflect seasonal variations because of evaporation losses or changes in raw material composition. The process data are given in Table 1. The lower and upper limits of the fuzzy goal from fresh water consumption are based on the target established by Polley and Polley3 of 70 t/h, and the total freshwater demand of 300 t/h without water recovery, respectively. Solving the model yields the network shown in Figure 1, where it can be seen that the fresh water requirement is 112.6 t/h, which is fed only to sinks 1, 2, and 3. The corresponding level of satisfaction of the fuzzy limits is λ = 0.815. There are also a total of six water reuse or recycle streams in the network. Note that sink 4 uses only reused water from other processes to supply its demand, which ranges from 70-80 t/h. At the optimal value of λ = 0.815, the actual allocation of water to sink 4 is 78.2 t/h (i.e., 70 t/h þ 0.815 (80 - 70 t/h)). As a result, the corresponding wastewater flow rate is 94.4 t/h. By comparison, the solution to the crisp (i.e., nonfuzzy) problem is 70 t/h of freshwater and 50 t/h of wastewater.3 The increase in resource demand (and effluent generation) is the penalty incurred to provide a margin of safety based on process mass balances, particularly with regard to contaminant load. 3688
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Table 1. Process Data for Case Studies 1 and 2 (Adapted from Polley and Polley3) source
flow rate (t/h)
concentration (mg/L)
flow rate (t/h)
sink
concentration (mg/L)
1
50
(50, 60)
1
50
2
100
(100, 110)
2
100
(10, 20) (30, 50)
3
70
(150, 175)
3
80
(80, 100)
4
60
(250, 300)
4
(70, 80)
fresh water
(70, 300)
(150, 200)
(0, 5)
Figure 1. Optimal network for case study 1.
The optimal value of λ may be interpreted as follows. For any given process, the parameters may be defined in terms of a fuzzy interval, which ranges from an optimistic (or risky) value to a conservative (or low-risk) value. For example, the concentration limit for sink 1 ranges from 10 to 20 mg/L. The lower limit of this range corresponds to a conservative or safe assumption, while the upper limit entails more risk by assuming that the process is able to physically tolerate more contaminant. In other words, using 10 mg/L as the limit for this sink is equivalent to setting λ = 1, while using 20 mg/L assumes that λ = 0. As the optimization model allows λ to be adjusted until the optimal value of 0.815 is found, it follows that the corresponding concentration limit for this solution is 11.85 mg/L (i.e., 20 mg/L - 0.815 (20 - 10 mg/L)). Note that this result is much closer to the conservative assumption than to the risky one. A similar argument applies for all the fuzzy parameters in the model.
’ CASE STUDY 2 This case study makes use of the same process data as in Case Study 1. It can be seen in the solution to the previous example (see Table 2) that the resulting optimal network includes two reuse streams of relative small flow rate (
