Optimization of Water Network Synthesis for Single-Site and

Review pubs.acs.org/IECR

Optimization of Water Network Synthesis for Single-Site and Continuous Processes: Milestones, Challenges, and Future Directions Cheng Seong Khor,*,† Benoit Chachuat,*,‡ and Nilay Shah‡ †

Chemical Engineering Department, Universiti Teknologi PETRONAS, Bandar Seri Iskandar, 31750 Tronoh, Perak, Malaysia Centre for Process Systems Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom

‡

ABSTRACT: Increasing water demand in the process and its allied industries coupled with global water stress and scarcity has underlined the importance of water as a crucial resource and elevated a need for widespread adoption of water reuse and recycle. This paper provides a state-of-the-art review of the area of water network synthesis focusing on single-site and continuous process problems since its inception in the 1980s. The survey centers around model-based optimization or mathematical programming methods for water network synthesis and covers key findings from the water pinch analysis technique, which are often essential in enhancing model formulations. Major modeling and computational challenges are discussed that explore the issues of nonconvexity, nonlinearity, and uncertainty inherent in water network synthesis problems. The review concludes by providing a perspective of future research directions to be tackled to address the challenges highlighted.

1. INTRODUCTION Water is a key component in industries. It has become a crucial resource particularly in the process industry and its allied industries due to increasingly higher demand for water use that may expose plants to supply disruptions in the future. Furthermore, water stress and increased scarcity in freshwater supply and clean water resources globally have heightened a need for sustainable management of its consumption with minimum water footprint.1 According to a 2008 report by the United Nations Environment Programme (UNEP), industry is the second largest user of freshwater worldwide after agriculture, while freshwater constitutes only 1% of worldwide water resources.2,3 On the other hand, ever more stringent regulations on wastewater discharges to the environment have given rise to greater requirements for plant operating efficiency and optimization. Moreover, according to the International Energy Agency’s World Energy Outlook 2012, water needs for power generation and in oil and gas projects are projected to grow even higher than energy demand itself.4 On top of that, the Intergovernmental Panel on Climate Change has highlighted adverse negative impacts of climate change on global water resources with repercussions for end users, including industries.5 In view of the present situation, it appears timely and warranted by the situation that a review is undertaken on the state of the art of systematic approaches for water recovery. The basic idea of water recovery revolves around conventional schemes that involve water reuse, regeneration (i.e., partial treatment), and recycle, as introduced by Wang and Smith6 and illustrated in Figure 1. Within the realm of the process integration philosophy for minimizing freshwater use and wastewater generation as espoused in the cited work, the notions of water reuse and water recycle are concerned with channeling the effluent from a certain water-using operation to other operations including the operations where it was generated. In further reducing freshwater and wastewater flow rates after exhausting recovery opportunities via direct reuse or recycle, the water regeneration concept can be considered, which © XXXX American Chemical Society

involves performing partial treatment on the effluent by using water treatment and purification technologies such as membrane and steam stripping prior to reuse or recycle. Where regeneration is required, the scheme is referred to accordingly as regeneration−reuse or regeneration−recycle. (The concepts are compiled in section 2.) Taken together, the optimal combination of these schemes gives rise to what is generally termed the “water network synthesis” problem.7 Section 22 attempts to formalize a definition of the problem. Several review papers have appeared in the literature on water network synthesis since the seminal work of Takama et al.8 The first survey published almost 15 years ago by Bagajewicz9 covers water pinch analysis and mathematical programming techniques with an emphasis on the latter, especially in the author’s own work for single and multiple contaminant problems including heat integrated water networks. Almost a decade later, a review by Foo7 focuses exclusively on pinch-based approaches for water network synthesis particularly for single-contaminant fixed flow rate problems, which centers on flow rate targeting and network design methods. The comprehensive survey by Jezowski10 serves as a compendium of work in the area classified alphabetically with key information on the problem scope (model formulation) and solution methods employed. Gouws et al.11 present a review on batch water systems, encompassing both pinch-based targeting and design methods as well as mathematical programming approaches, which may include scheduling consideration. This review largely focuses on the optimization or mathematical programming approach for water network synthesis. The work surveyed is limited to single-site and continuous process variants of the water network synthesis Received: November 21, 2013 Revised: May 15, 2014 Accepted: May 28, 2014

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dx.doi.org/10.1021/ie4039482 | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Industrial & Engineering Chemistry Research

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Figure 1. Schematic diagrams to illustrate conventional water recovery schemes.

network synthesis, which considers both water-using units and water or wastewater treatment operations simultaneously. The basic goal is to synthesize a network that integrates these waterusing and/or water regeneration operations by optimizing a certain objective typically “environomic” in nature, i.e., based on economics (by minimizing cost) and/or environmental sustainability (e.g., by minimizing ecological footprint)16 while complying with constraints on the final discharge limits to the environment. Optimization of water network synthesis aims to develop a systematic approach for optimally synthesizing the alternatives for water reuse, regeneration, and recycle. A list of terms and concepts often encountered in the water network synthesis literature is summarized in Table 1 to provide their conventional definitions. Figure 2 depicts common water network structures or architectures.

problems, although it is noted that the community working in this field has also tackled multisite interplant problems12−14 (besides discontinuous15 and batch water networks). A distinguishing characteristic of the current work is that it strives to undertake an assessment of the overall contribution of the literature on optimization for process systems toward water network synthesis, and how this can further enhance future work in the area.

2. WATER NETWORK SYNTHESIS PROBLEM Water network synthesis is a class of problem in process systems engineering (PSE) that may be considered as a special case of the mass exchanger network synthesis problem.10 Water network synthesis has emerged as a noteworthy area in the PSE community and has been a subject of a considerable amount of research in the past decade or so as is evident in this review. A definition of the problem can be stated as follows. Water network synthesis is the design problem of the synthesis and retrofit of water network for both continuous and batch operations. Depending on the scope of the problem, water network synthesis may involve a consideration of either waterusing units or wastewater treatment operations or both (as will be discussed later). Water-using units comprise both water sources (including external freshwater sources) and water sinks with their corresponding contaminant concentrations. Sources are water supply streams, whereas sinks involve water demand units or equipment including reuse/recycle operations. Wastewater treatment operations act as intermediate regeneration processes, where necessary, before water sources can be subject to reuse/recycle in the sinks. Note that end-of-pipe wastewater treatment operations are considered as sinks. There are three subclasses of problems in water network synthesis: (1) direct reuse/recycle water network synthesis or water-using network synthesis, which considers reuse of untreated wastewater by directly transferring it from a waterusing unit where it is produced to the water-using unit for its use (direct recycle entails reusing water in the same water-using unit where it is produced); (2) regeneration water network synthesis, which considers partial treatment of water or wastewater to facilitate subsequent reuse/recycle; (3) total water

3. APPROACHES FOR WATER NETWORK SYNTHESIS In general, two major approaches for addressing water network synthesis problem can be categorized as insight-based techniques and model-based optimization methods. The former typically involves water pinch analysis (WPA) techniques that offer good insights with low computational burden, yet often at the expense of requiring significant problem simplifications (e.g., see Wang and Smith,6 El-Halwagi et al.,18 and a review by Foo7). On the other hand, the latter approach of optimization allows handling of water network synthesis problems in their full complexity by considering representative cost functions, multiple contaminants, and various topological constraints, although it frequently presents challenges of a high computational burden to achieve optimality. Recent work in this area increasingly has involved the development of optimization models of greater rigor and with real-world features by using mathematical programming (e.g., see Faria and Bagajewicz,17 Ahmetović and Grossmann,19 Khor et al.,20 and Rubio-Castro et al.,14 just to cite a few). The mathematical programming based approach mainly requires constructing a superstructure of all or many possible interconnections of water network elements that lead to a mixed-integer linear programming (MILP) or mixed-integer nonlinear programming (MINLP) model formulation. The formulations often incorporate B



Review

Table 1. Common Terms in the Water Network Synthesis (WNS) Literature term water source wastewater stream water direct reuse/recycle water regeneration/regeneration−reuse/regeneration−recycle water network retrofit (WNR) water-using network (WUN) synthesis (WUNS) wastewater treatment network (WWTN) synthesis (WWTNS) water regeneration network synthesis (WRNS) centralized effluent treatment system (ETS) water pretreatment network synthesis (WPTN) total water network synthesis/retrofit (TWNS/TWNR) complete water system synthesis (CWSS)

definition water stream that can be utilized for reuse/recycle water stream that is generated by a water-using operation, or sent to a wastewater treatment plant, or discharged to environment reuse of untreated wastewater by directly transferring it from the site where it is produced to the operations for its use; direct recycle entails reusing water in the same operations where it is produced partial treatment of water or wastewater to facilitate subsequent reuse/recycle WNS problems dealing with the retrofit of water network systems component of WNS problem dealing with water-using processes only; also called direct reuse/ recycle water network synthesis component of WNS problem dealing with wastewater treatment processes only component of WNS problem dealing with water regeneration for reuse/recycle component of WNS problem dealing with facilities made up of a sequence of technologies for wastewater treatment before discharge to the environment; also called end-of-pipe treatment or waste treatment system; streams may be treated or may bypass the ETS component of WNS problems dealing with pretreatment of freshwater sources17 consists of WUNS, WRNS, and ETS10 consists of WUN, WRNS, ETS, and WPTN;17 transforming TWNS to CWSS requires interconnections of freshwater sources to treatment units for pretreatment operation10

Figure 2. Graphical illustration of some common concepts in the water network synthesis literature.

represents a first attempt at a sequential superstructure-based optimization approach for water network synthesis involving direct reuse/recycle. A sequential procedure is proposed that utilizes a linear programming (LP) approximation as an initial point to solve an NLP. The water-using operations are modeled in terms of fixed mass load for the NLP and fixed maximum outlet contaminant concentration for the LP. To achieve a feasible solution, the concentration balance is relaxed as an inequality. To aid solution convergence, additional constraints are included on maximum wastewater flows and on forbidden stream matches, which eliminate uneconomic small flow rates between operations to simplify the network. However, the approach does not guarantee a global optimal solution. 4.3. Milestone 3: MINLP Formulation Is First Attempted with Consideration for Optimization of Distributed Wastewater Treatment Network by Galan and Grossmann.32 This is the first optimization-based WNS work that employs discrete variables (binary 0−1) in its formulation, giving rise to the first MINLP model in the WNS literature. It is also the first optimization paper that addresses regeneration water network synthesis, specifically the design of the distributed wastewater treatment network, which is known to be more operationally efficient in general. The superstructure is developed based on the work by Wang and Smith6 and does not consider the water-using network. Altogether, this paper presents three formulations: (1) NLP for distributed wastewater treatment network synthesis with nonlinear

physical insights derived from water pinch analysis21−25 as well as concepts from mass integration26,27 and property integration.25,28−30

4. MILESTONES IN OPTIMIZATION-BASED APPROACHES FOR WATER NETWORK SYNTHESIS PROBLEMS This section provides a historical perspective on significant accomplishments in water network synthesis that utilizes optimization-based modeling techniques and computational strategies as summarized in Figure 3. 4.1. Milestone 1: Seminal Work on Water Network Synthesis Using Optimization Approach by Takama et al.8 This pioneering work of applying an optimization approach to water network synthesis presents a nonconvex nonlinear programming (NLP) formulation of the grass-roots design problem. The work addresses the problem of optimal water allocation in a petroleum refinery by integrating the water-using and wastewater treatment processes to account for many possible water reuse and regeneration opportunities. A freshwater source with zero contaminant concentration is considered. It is noteworthy that, for nearly two decades after this seminal work, there was no journal publication reported on an optimization approach to water network synthesis problems until its revival as described in Milestone 2. 4.2. Milestone 2: Sequential Optimization Approach Is First Proposed by Doyle and Smith.31 The work C



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Figure 3. Milestones in optimization-based approaches for water network synthesis [refs 6, 8, 9, 17, 21, 22, 24, 31, 32, 34, 36−39, 47, 58, 59, 61, 65, 69, 72−77, 80].

Grossmann.34 A global optimization algorithm for addressing nonconvexity in MINLP is proposed for a distributed wastewater regeneration network synthesis problem taken from Kuo and Smith.35 The branch-and-bound based algorithm incorporates a bound contraction procedure that eliminates large portions of the search region to address nonconvexity arising from concave univariate, bilinear, and linear fractional terms in the model. The global solution matches the design configuration reported in the original work and that of Galan and Grossmann32 (although the latter does not purport to guarantee global optimality). Lee and Grossmann36 presents an alternative formulation of the generalized disjunctive program (GDP) for this problem and implements a global optimization algorithm to handle nonconvexity arising from bilinear terms in the flows, concentrations, and split fractions.

bilinearities in a mixer unit; (2) MINLP with the 0−1 variables employed for selection of treatment technology; (3) NLP for membrane separation based treatment technologies without considering fixed removal ratios but as design equations (i.e., instead of fixed recovery constants as found in conventional formulations). To handle nonconvexity in the concentration balances, the models are solved using a sequential heuristic search procedure that involves a set of LP relaxation models constructed using McCormick’s33 linear convex underestimators and linear concave overestimators for a tight upper bound on the global optimum. The nonconvex exponential terms are approximated by the linear underestimators of Zamora and Grossmann,34 which is the work highlighted next. 4.4. Milestone 4: Global Optimization of Water Network Synthesis Is First Considered by Zamora and D



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4.5. Milestone 5: A Formal Attempt at Incorporating Water Pinch Analysis in Optimization of Water Network Synthesis Is Proposed by Alva-Argáez et al.21,22,24 The authors present a first formal attempt at systematically integrating physical insights derived from water pinch analysis within an optimization-based framework. An MINLP model is presented for multicontaminant total water network synthesis with extension to retrofit applications. The superstructure postulated includes all possibilities for water regeneration− reuse/recycle. The objective function is augmented with an increasing penalty term to drive reduction of problem infeasibilities. The formulation is nonconvex due to bilinear terms in the contaminant mixing constraints and sizing equations and cost functions for the water-using units. A water pinch analysis based targeting heuristic is incorporated to achieve maximum water reuse in the water-using network. Besides reducing the number of variables and constraints, the use of water pinch analysis supports a decomposition-based solution procedure involving a sequence of MILP relaxations. The bilinearity relaxations employ a projection strategy coupled with an LP relaxation of the mass balances and an iterative solution scheme of MILP−LP sequence to approximate a nonconvex NLP version of the original MINLP by fixing the 0−1 variables corresponding to an alternative structure. An approach is also postulated to eliminate the presence of regeneration recycling, which is deemed undesirable due to possible contaminant accumulation (although it promotes a reduction in freshwater demand). However, global optimality is not guaranteed. 4.6. Milestone 6: Fixed-Flow-Rate Representation Is First Adopted by Huang et al.37 This work introduces a fixed-flow-rate representation for water-using units within an NLP formulation for multicontaminant total water network synthesis with distributed wastewater treatment network. The formulation extends the superstructure of Takama et al.8 in three ways: by incorporating multiple water sources and multiple sinks, by accounting for water losses, and by considering both process water and utility water for improved plantwide integration. To generate initial feasible points for solving the NLP, the authors use water pinch analysis or fix certain key design variables; hence a global optimal solution is not guaranteed. Fixed regenerators are considered; therefore their selection is not optimized. Additionally, a heuristic involving repeated water treatment units is proposed to represent the effect of recycling wastewater that requires further additional regeneration. 4.7. Milestone 7: Rigorous Proofs for the Necessary Conditions of an Optimal Direct Reuse/Recycle Water Network Synthesis Is Presented for a Single-Contaminant Case by Savelski and Bagajewicz.38 Mathematical proofs are presented for the necessary conditions of optimality for an LP formulation of direct reuse/recycle water network synthesis with a single contaminant. The conditions involve enforcing maximum allowable concentrations for all freshwater-using processes while obeying concentration monotonicity. The LP degenerate solutions are also analyzed. The authors provide an extension of the proof for a multicontaminant case in a later work39 in which it is shown that at least one contaminant achieves its maximum allowable concentration at the outlet of a freshwater-using process while the condition of concentration monotonicity only applies to certain key contaminants. 4.8. Milestone 8: First Review Paper on Optimization of Water Network Synthesis by Bagajewicz.9 A review is

presented of survey work published up to 2000 on water network synthesis and retrofit for both optimization-based and water pinch analysis approaches with an emphasis on the former. The review focuses mainly on the author’s recent work on modeling and solution strategy for interactions between the water allocation problem (involving reuse) and the decentralized distributed wastewater treatment problem. While the author postulated that the area is moving toward the use of optimization techniques because the graphically driven water pinch analysis (WPA) is deemed tedious and often requires heuristics-based modifications particularly in handling multicontaminant systems, there has been a recent resurgence of WPA-based work as espoused in the review by Foo.7 A more recent review is available from Jezowski.10 For batch water network synthesis, extensive review materials are available in the article by Gouws et al.11 The book by Mann and Liu40 is devoted to water network synthesis and mainly covers early work in the area by researchers at the University of Manchester Institute of Science and Technology (UMIST) (now part of the University of Manchester). Substantial coverage on the subject can also be found in books by El-Halwagi,41,42 Sikdar and ElHalwagi,43 Rossiter,44 Smith,45 and Foo.46 4.9. Milestone 9: MILP Formulation Is First Presented for Water Network Synthesis by Bagajewicz and Savelski.80 The 0−1 variables are used to represent interconnections between processes in a network. An iterative procedure involving the solution of a sequence of LP and MILP is developed. For a water-using network without regeneration, a sequential two-step procedure is proposed in which the LP solution with a minimum freshwater objective value is subsequently fed into the MILP that minimizes the number of interconnections. A similar approach is postulated for a waterusing network with regeneration with an additional step of the MILP feeding into another LP to determine the minimum water through the regeneration network. A proof for the necessary conditions of optimality for a water-using network with decentralized regeneration is provided. The work also proposes optimality conditions on a water regeneration network without recycle. 4.10. Milestone 10: Uncertainty in Water Network Synthesis Is First Addressed by Koppol and Bagajewicz.47 The work presents a first attempt at addressing uncertainty in water network synthesis for a single-contaminant problem coupled with a novel consideration for financial risk management. A bounded uniform distribution is adopted in a discrete scenario generation approach to represent the uncertain contaminant mass load parameter. A scenario reduction technique is proposed to mitigate the curse of dimensionality caused by an exponential increase in the number of variables due to admitting a large number of scenarios. The authors advocate that it is not possible to mitigate risk when the operating cost is much larger than the capital cost because a design with a minimum expected operating cost usually poses minimum risk, but when capital cost is comparable to operating cost, reuse of wastewater is amenable to reducing risk. 4.11. Milestone 11: Alternative Approaches for Water Network Synthesis Problems Is First Considered by Tan and Cruz.48 Although the focus of this review is on deterministic mathematical programming based optimization methods for water network synthesis, it is noteworthy that work applying other methods particularly stochastic optimization-based techniques has emerged in the literature. Tan and Cruz48 first applied a fuzzy optimization technique to water E



Review

strategy is able to generate very tight lower bounds that reduce the number of nodes enumerated in a spatial branch-and-bound search tree but at the expense of a large computational load due to an increase in the number of 0−1 variables used. 4.14. Milestone 14: Alternative Formulations for the Piecewise-Affine Estimators Are Presented by Wicaksono and Karimi.61 This work proposes up to 15 equivalent alternative representation schemes of the piecewise-affine relaxation underestimators for bilinearities. It is demonstrated that the relaxations developed by Karuppiah and Grossmann58 and Meyer and Floudas59 can be enhanced through these alternate mathematical formulations that are developed by adopting big-M, convex hull, and incremental cost approaches. Later, Gounaris et al.62 compared the computational performances of these relaxation variants for univariate partitioning (i.e., partitioning on only one variable instead of both variables appearing in each bilinear term) besides suggesting five additional formulations. Subsequently, Hasan and Karimi63 investigate the numerical efficiencies for the case of bivariate partitioning. A technically similar scheme has been designed by Pham et al.64 for large-scale problems using a branch-andbound method through partition-refining techniques that promote a fast computational time but without guaranteeing global optimality. 4.15. Milestone 15: Membrane Separation Based Water Regenerators Are First Considered for Total Water Network Synthesis by Tan et al.65 This work purports to optimize the synthesis of water networks with a partitioning regenerator, which functions by splitting a contaminated feedwater stream into a regenerated lean permeate stream (i.e., the permeator) and a low-quality retentate stream (i.e., the rejector). A primary example of such a class of regenerators involves membrane separation based processes. The model considers a partitioning regenerator with fixed contaminant removal ratio and fixed liquid phase recovery factor (i.e., the ratio of feed that exits as the permeator). A centralized single partitioning regenerator is integrated within a source−sink superstructure consisting of fixed-flow-rate water-using units. The formulation gives rise to a nonconvex NLP due to bilinear terms in the regenerator concentration balances, and it is solved using a branch-andbound method via Lingo (version 10) with a global optimization solver. The effects of the regenerator physical parameters on the resultant optimal reuse/recycle network are investigated. The work of Khor et al.20,66 extends the formulation to a network with multiple types and units of both membrane- and nonmembrane-based technologies. 4.16. Milestone 16: An Alternative to Economics-Based Objective Function Is First Considered by Ku-Pineda and Tan.16 An objective of maximizing water recovery thereby minimizing freshwater use and wastewater generation can entail increased materials and energy use (e.g., due to piping and pumping requirements), thus outweighing the water saving benefits. Hence, it may be worth considering potentially more representative alternative objective functions. Ku-Pineda and Tan16 propose a single contaminant LP that adopts an objective called the “sustainable process index”,67 in which the solution obtained can lead to reduced environmental impact as compared to one with a minimum freshwater use objective. In another contribution, Tan et al.68 postulate an MILP that considers the total resource consumption impact of water, electricity, and capital goods material for a water network through minimizing a metric called “emergy”, which represents

network synthesis to handle uncertainty in the contaminant mass load of the water-using units. A fuzzy programming based approach has also been adopted by Tan and co-workers to water network retrofit49 and game-theoretic problems involving multiple plants.13,50−52 Tan et al.53 utilize a Monte Carlo simulation approach to analyze the sensitivity of water pinch analysis solutions that account for uncertain mass load. Artificial intelligence based metaheuristic algorithms have also been implemented to water network synthesis problems including the Luus−Jaakola method,54 particle swarm optimization,55,56 and adaptive random search.57 4.12. Milestone 12: Piecewise-Affine Relaxations of Nonconvex Terms Is First Proposed by Karuppiah and Grossmann.58 This is the first definitive paper on the use of ab initio domain partitioning and MILP-based piecewise-affine (or semilinear) relaxations of nonconvex bilinear terms for augmenting a global optimization procedure. A deterministic spatial branch-and-contract algorithm is employed with branching on the continuous variables of individual contaminant flows. The bilinear terms are relaxed using McCormick’s33 convex and concave envelopes in generating a lower bound for the original NLP formulation, while the concave terms on equipment costs in the objective function are relaxed by underestimators derived from secants of these terms. Piecewiseaffine underestimators obtained from partitioning of the flow variables are then exploited to construct tighter envelopes and concave underestimators; importantly, the main idea here is to yield an improved tighter lower bound by shrinking the variable domain. Linear equalities on overall contaminant balances are incorporated to further strengthen the lower bound. Ultimately, the model is solved using disjunctions by employing a generalized disjunctive programming (GDP) framework. To reduce the number of disjunctions, a heuristic is obeyed in which the variable of a bilinear term that participates in a larger number of constraints is elected for partitioning. Logic cuts are incorporated to enhance convergence, for example, by enforcing that two identical flow variables should fall within an interval. The algorithm then proceeds with a simplified variant of the bound contraction procedure used by Zamora and Grossmann.34 It is found that a higher number of partitions promotes tighter lower bounds but at the cost of greater computational expense. 4.13. Milestone 13: Global Optimization Based on Piecewise Reformulation−Linearization Technique (RLT) Is First Postulated by Meyer and Floudas.59 This work attempts to address bilinearities in pooling problems encompassing water network synthesis by applying the RLT approach of Sherali and Alameddine60 in a piecewise-affine manner to promote a greater level of tightness in the relaxation. In the reformulation stage, new nonlinear RLT constraints are derived by multiplication of valid constraints from the original formulation. The new RLT constraints are redundant to the original problem but may not be so in the convex relaxation. The continuous space of one of the variables participating in a bilinear term is partitioned into several intervals by using these RLT-based redundant constraints. The resulting linearization stage produces an MILP that generates lower bounds by substituting a bilinear term with a new variable and adding new constraints derived by multiplying inequality constraints on the bounds. A lower and upper bounding scheme ensues through which the gap between the lower and upper bounds is reduced by augmenting the lower bounding problem using a set of 0−1 variables to partition the continuous space. On the whole, the F



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number of 0−1 variables that scale linearly with the number of segments in the variable partitioning. In addition, the algorithm also incorporates relaxation strategies using McCormick’s33 convex and concave envelopes and the RLT besides customized branching strategies and bounds tightening that further exploit the problem structure. 4.21. Milestone 21: A Planning Model for Water Network Synthesis Is First Proposed by Faria and Bagajewicz.74 This is the first and hitherto only paper that considers planning issues in the grass-roots design and retrofit of water network synthesis. The model allows for capacity expansion due to a mass load increase in existing water-using units or installation of new units. A fixed-load direct reuse/ recycle water network synthesis formulation is presented that incorporates time-varying decisions using a discrete time approach. 4.22. Milestone 22: Global Optimization with PiecewiseAffine Relaxations Enhanced with Bound Contraction Is Proposed by Faria and Bagajewicz.75−77 This work further advances the line of work on exploiting piecewise-affine relaxation schemes in attaining global optima. The strategy is performed by enhancement through bound contraction via an interval elimination technique. A new idea postulated is that the procedure does not necessarily have to be applied to the set of partitioned variables. Bound contraction based on successive execution of linear lower bounds is performed to eliminate subspaces of the feasible region by temporarily forbidding certain intervals. The procedure is repeated with new bounds until convergence is attained or contraction is exhausted, but without requiring the addition of new 0−1 variables. A resultant tighter lower bound is obtained typically in fewer iterations than previous methods, which provides an initial point for solving the original MINLP to generate an upper bound. The associated algorithm proceeds by increasing the number of intervals or by applying branch-and-bound search with bound contraction at each node. A summary of the work discussed in the milestones and other important work in the literature on single-site and continuous process water network synthesis problems is reviewed in Table 2 with emphasis on the modeling approaches (column 5) and solution strategies (column 6).

the cumulative life-cycle solar energy inputs to a system. The results reveal that a minimum total impact network is achieved by prioritizing energy and material conservation over water recovery. 4.17. Milestone 17: A First Attempt at OptimizationBased Automation of a Pinch Analysis Technique Is Presented by Ng et al.69 This work represents the first of a series of papers on automating the pinch analysis based targeting method using optimization.25,70 The paper proposes an LP formulation for the targeting of a fixed-flow-rate water network synthesis problem as inspired by an analogous targeting method for mass exchange network synthesis.71 The optimization-based automated targeting provides a role similar to that of a conventional pinch analysis method in devising feasible design targets prior to detailed design. The automated technique can cater to an objective function of either minimizing freshwater flow rate or total annualized cost, whereas a conventional pinch analysis tool is restricted to the former objective. In addition to its flexibility, the proposed method can handle multiple external freshwater sources with contaminants. 4.18. Milestone 18: Consideration of Pretreatment in Water Network Synthesis Is First Proposed by Ng et al.72 and Faria and Bagajewicz.17 Ng et al.72 account for pretreatment within an optimization-based automated targeting formulation for a multicontaminant total resource conservation network. Ng et al.25 expand the proposed formulation to more extensive applications including for plantwide integration. Faria and Bagajewicz17 introduce the concept of complete water system synthesis (CWSS) that encompasses the water-using network, water regeneration network, and centralized effluent treatment system as well as a novel consideration for the water pretreatment network. The authors also undertake a systematic study on superstructure development and the associated model formulations for which a discharge stream recycle is considered that may give rise to a network with zero liquid discharge. 4.19. Milestone 19: An Industrial-Scale Water Network Synthesis Problem Is First Addressed by Ng et al.72 The work applies the aforementioned automated targeting approach (with pretreatment consideration) to a palm oil milling process to optimize water recovery through direct reuse in a separation process using a clay bath system. The authors also implement a similar approach to minimize freshwater use in synthesizing a total water network for an industrial wafer fabrication process. Khor et al.66 address an industrial-size total water network synthesis problem based on an actual operating petroleum refinery with a detailed membrane regenerator model that employs heuristic-based logic cuts to aid convergence to global optimality. A more generalized large scale total water network synthesis formulation that considers multiple technology types of both membrane- and nonmembrane-based regenerators is presented by Khor et al.20 4.20. Milestone 20: Formulation for Water Network Synthesis with a Reduced Number of Bilinear Terms Is Postulated by Misener and Floudas.73 This paper presents a nonconvex mixed-integer quadratically constrained quadratic program (MIQCQP) pooling problem formulation, which is applicable to water network synthesis that promotes a reduction in the number of bilinear terms. The solution strategy implements a branch-and-bound global optimization algorithm that exploits the aforementioned recent advances in the design of piecewise-affine relaxations of bilinearities. Suitable under- and overestimators are activated using a

5. CHALLENGES IN WATER NETWORK SYNTHESIS Based on the foregoing survey, this section outlines gaps in the literature as presented by the three main challenges in the modeling and computation of water network synthesis problems, namely, the issues of nonconvexity, nonlinearity, and uncertainty. 5.1. Handling the Challenge of Nonconvexity. A major challenge in water network synthesis problems is the presence of nonlinear nonconvex bilinear terms in the concentration balances as a result of contaminant mixing in the water regeneration system. Addressing this hurdle calls for the implementation of global optimization techniques; otherwise, a reliable solution cannot be guaranteed. As is evident in section 4 (particularly Milestones 12, 13, 14, and 22), recent advances in global optimization techniques to handle bilinearities can be largely framed around developments of the following three techniques: 1. The first technique is MILP-based piecewise-affine relaxation schemes, in which a general framework of such a procedure is elucidated in the flowchart in Figure 4.58,61,62,73,75−77,88,91,93−98 G


H

single

single and multiple

multiple

Koppol and Bagajewicz47

Koppol et al.81

Chang and Li82

multiple

Huang et al.37

single

multiple

Alva-Argáez et al.22

Bagajewicz and Savelski80

multiple

Zamora and Grossmann34

multiple

multiple

Galan and Grossmann78

Benko et al.79

multiple

single

single/multiple contaminants

Doyle and Smith31

Takama et al.

8

author

NLP with nonconvex bilinearities

MILP

MILP with financial risk management

LP and MILP

MINLP with nonconvex bilinearities

NLP

MINLP with nonconvex bilinear, linear fractional, and concave univariate objective function terms MINLP with nonconvex bilinearities

NLP and MINLP with nonconvex bilinearities and exponential objective function terms


NLP with nonconvex bilinearities and nonlinear cost-based objective

model type

fixed flow rate

fixed load

fixed load

fixed load

fixed load

fixed flow rate

fixed load

fixed load

fixed load

fixed outlet concentration

fixed load

water-using unit representation model formulation

TWNS; superstructure incorporates additional design options and repeated treatment units; inequality concentration constraints on unrecoverable contaminants

WUNS under uncertainty in mass loads of water-using units (represented using discrete scenarios) TWNS with zero water discharge; regenerators with fixed removal ratios; considers interconnections between regenerators

TWNS with distributed WWTNS; extends superstructure of Takama et al.;8 does not optimize selection of regenerators TWNS; considers rates of different contaminants as interdependent and not independent of their concentrations TWNS and WUNS; uses 0−1 variables to represent interconnections between units

TWNS; incorporates WPA-based targeting heuristic

distributed WWTNS

WUNS; superstructure based; includes constraints on maximum wastewater flows and forbidden stream matches distributed WWTNS; considers membrane treatment technologies without fixed removal ratios

TWNS; superstructure based; uses split ratios on streams to determine network structure

Table 2. Summary of Important Work in Water Network Synthesis for Continuous Operations

for single contaminant systems: sequential procedure for freshwater use followed by decentralized treatment; for multicontaminant systems: branch-and-bound search for all possible maximum reuse structures heuristic for good initial point to solve NLP by manipulating network structural properties

branch-and-bound with CPLEX

recursive decomposition-based procedure involving a sequence of MILP relaxations uses WPA tools to generate initial feasible points for solving NLP and for fixing key design variables; heuristic for recycling wastewater requiring further regeneration recursively eliminates units or streams from superstructure with zero results based on optimal NLP solution (cover-and-eliminate approach) iterative two-step solution procedure of MILP−LP sequence

sequential heuristic search procedure involving LP relaxations via McCormick’s33 convex/concave envelopes for bilinear terms and linear underestimators of Zamora and Grossmann34 branch-and-bound-based global optimization algorithm with bound contraction procedure

uses the complex method; reformulates into a sequence of unconstrained problems by appending corresponding penalty functions uses LP approximation to initialize solution of NLP

solution strategy

does not guarantee global optimum

provides proofs for necessary optimality conditions for WUNS that is decentralized and without recycle does not consider total water integration and risk management with regeneration network does not provide rigorous proofs to verify effects of proposed network simplifications




reported global optimum matches that of the original work

proposed heuristic obtains the global optimum of Zamora and Grossmann34 although no such guarantee is provided

no guarantee of global optimality

utilizes simulation for fixed split ratios to minimize objective function with penalty on discharge constraint

remark

Industrial & Engineering Chemistry Research Review


multiple

multiple

multiple

Meyer and Floudas59

Karuppiah and Grossmann58

Alva-Argáez et al.21,24

I

multiple

single

single and multiple

Putra and Amminudin86

Tan et al.65

Faria and Bagajewicz87

multiple

multiple

Al-Redhwan et al.84

Karuppiah and Grossmann85

multiple


Gunaratnam et al.83

author

Table 2. continued

NLP and MINLP with nonconvex bilinearities


MILP; NLP with nonconvex bilinearities

NLP and MINLP (two-stage stochastic programs) with nonconvex bilinearities and concave objective function terms

MINLP incorporated with WPA

NLP and MINLP with nonconvex bilinearities and concave objective function terms


two-stage stochastic NLP with nonconvex bilinearities


model type

fixed load

fixed flow rate

fixed load

fixed load

fixed flow rate

fixed load

fixed flow rate

fixed load

fixed load

water-using unit representation model formulation

TWNS with local recycle; computes multiple (suboptimal) solutions; incorporates practical considerations for varying piping complexity TWNS with partitioning regenerators specifically membrane separation based processes TWNS/TWNR using different objective functions (minimum freshwater, maximum net present value NPV, or maximum return on investments ROI) and discount factors in NPV calculations

TWNS/TWNR under uncertainty in mass loads of water-using units and removal ratios of regenerators, which is represented using discrete scenarios

TWNS/TWNR

TWNS for regenerators with fixed removal ratios

TWNS under uncertainty in mass loads of water-using units, which is represented using discrete scenarios TWNS for regenerators with fixed removal ratios

TWNS; controls network complexity by specifying minimum allowable flow rates, maximum streams for mixing, and piping costs

solution strategy

sequential procedure of maximizing net savings, then computes NPV and ROI; determines range of feasible freshwater consumptionminimum is given by optimal value of minimum freshwater model while maximum is given by no reuse model

global optimization involving branchand-bound search

global optimization involving MILPbased piecewise-affine relaxation based on reformulation− linearization technique (RLT) global optimization involving spatial branch-and-contract procedure with MILP-based piecewise-affine relaxation; includes cuts on overall contaminant balance decomposition-based procedure involving MILP and LP relaxations; iterative MILP−LP sequence to approximate nonconvex NLP version of original MINLP global optimization involving relaxation via McCormick33 convex/concave envelopes for bilinear terms and linear overestimators from secants for concave terms in objective function; spatial branch-and-cut scheme with Lagrangean relaxation (with BARON and DICOPT) sequential MILP and NLP procedure that corresponds to pinch-based targeting for structural optimization and design for parameter optimization, respectively

iterative procedure with material balance relaxation by fixing maximum outlet concentrations and introducing slack variables; MILP solution on flow rates gives starting point for solving LP relaxation, which then gives new concentration values as data for MILP in subsequent MILP−LP solution sequence local optimization using generalized reduced gradient with GAMS/ CONOPT2

remark

shows that optimal solution depends on objective function used; studies effects of freshwater flow rates and capacities of new regeneration process on network structure

does not optimize over piping interconnections cost

does not guarantee global optimality

proposed procedure applicable only to fixed-load formulations (requires nontrivial customization for fixed-flowrate formulations)

does not guarantee global optimum; does not consider local recycle

lower bound-tightening cuts are largely applicable only to fixedload formulation

reformulation as MILP requires additional 0−1 variables

considers only a single regenerator

eliminates regeneration recycle to avoid contaminant accumulation; no guarantee of global optimality



J

single and multiple

multiple

Hung and Kim90

multiple

Khor et al.66

Feng et al.89

multiple


multiple

multiple

Ahmetović and Grossmann19

Faria and Bagajewicz75−77

multiple

Misener et al.88

multiple

single and multiple

73



Misener and Floudas

author

Table 2. continued

multistage (multiperiod) MINLP

NLP and MINLP with nonconvex bilinearities


MINLP with nonconvex bilinearities and concave and fractional objective function terms




MINLP with nonconvex bilinearities and concave objective function terms


model type

fixed load

fixed load

fixed load and fixed flow rate (for separate models)

fixed flow rate

fixed load

fixed load

fixed flow rate


fixed flow rate

water-using unit representation

TWNS with flexible operation under uncertainty in mass load, which is represented using discrete deviation from nominal values; uses 0−1 variables to select streams with flows to be adjusted under disturbance WUNS under uncertainty in inlet flow rate and mass load of water-using units, which is represented using average values

TWNS; regenerators with fixed removal ratios; include constraint to eliminate small flow rates

complete water system synthesis with local recycle in units; multiple freshwater sources; accounts for trade-offs between cost and network complexity by limiting piping connections using 0−1 variables planning with capacity expansion due to mass load increase in existing or new water-using units; handles time-varying decisions using discrete time formulation large scale industrial-size TWNS with detailed membrane regenerator models

TWNS using pooling problem formulation with connections between pools of regenerators; includes local recycle introduces complete water system synthesis that considers water pretreatment; proposes systematic approach for superstructure development and formulations for various network configurations TWNS using pooling problem formulation with connections between pools of regenerators

model formulation

decomposition-based iterative solution of sequence of MILP and LP relaxations

global optimization using spatial branch-and-bound augmented by logic cuts on design and structural specifications to accelerate convergence global optimization involving piecewise-affine relaxation enhanced with bound contraction using interval elimination; bound contracting the variables one at a time allows gap reduction between the bounds obtained from solving the original problem (MINLP) local optimization with Lingo; heuristically adjusts stream flows while maintaining nominal network structure with minimum freshwater use

global optimization involving bound contraction procedure of Faria and Bagajewicz75,76

global optimization involving piecewise affine relaxation of bilinear functions; compares linear and logarithmic scaling in number of 0−1 variables used with number of segments in variable partitioning global optimization with cuts on overall contaminant balance to tighten lower bounds generated by LP relaxation; uses rigorous equation-based variable bounds

global optimization involving MILPbased piecewise-affine relaxation to strengthen lower bounds in spatial branch-and-bound scheme outer approximation algorithm with initial values given by LP relaxation of bilinear terms via McCormick33 envelopes and concave terms via linear underestimators

solution strategy

does not guarantee global optimality

heuristic adopted for flexible operation does not guarantee optimal solution

algorithm proceeds by increasing number of intervals or applying branch-and-bound with bound contraction at each node; proposed bound contraction does not add new extra 0−1 variables

does not consider multiple membrane regeneration technologies

does not consider time variation in mass load or removal ratios

lower bound-tightening cuts are largely applicable only to fixedload formulation

excludes feasible solutions involving cycles between regenerators

considers discharge stream recycle to achieve zero liquid discharge

does not consider membrane regenerators

remark



does not consider a multiobjective formulation that offers a Pareto curve of solutions

Figure 4. General framework for incorporating piecewise-affine relaxations into a global optimization algorithm within a spatial branch-and-bound framework.

fixed flow rate

2. The second is application of RLT-based reduction constraints, which are linear constraints redundant to an original nonconvex problem but not so for its convex relaxation.99−102 3. The third technique is the use of constraint propagation that originates from the constraint programming paradigm.103−106 The first two techniques are typically implemented within a spatial branch-and-bound scheme to achieve bound-strengthening relaxations. 5.2. Handling the Challenge of Nonlinearity. Water network synthesis falls under the class of pooling problems,95 and total water network synthesis in particular presents the additional complexity of modeling the mixing pools of wastewater treatment operations to perform regeneration for reuse/recycle. For this purpose, more advanced tertiary treatment using membrane processes have been considered in addition to nonmembrane-based technologies.20,66,78 In this regard, membrane regenerators such as ultrafiltration and reverse osmosis have found extensive industrial applications.107,108 Although regeneration technologies are inherently nonlinear, to the best of our knowledge, there have been few works to date that consider a detailed nonlinear regeneration model for water network synthesis. Recent works in process synthesis have increasingly involved rigorous optimization-based models particularly for the separation section,109−112 but a relatively small numbers of works have been conducted that incorporate detailed mechanistic or phenomenological models on the subsystems of a water network.113,114 While there is an appreciably rich literature on rigorous design models for wastewater treatment technologies (e.g., for reverse osmosis),41,115−120 there is a gap in the water network synthesis literature on papers that simultaneously optimizes their interactions with multiple water-using units by adopting such detailed models. A related issue is the development of an appropriate superstructure representation for a water network synthesis problem. Earlier work surveying systematic methods for superstructure generation can be found in Yeomans and Grossmann121 and

multiple Khor92

MINLP with nonconvex bilinearities as well as concave and fractional objective function terms

multiple Khor et al.20


fixed flow rate

large scale industrial-size TWNS with both membrane- and nonmembrane-based regenerators large scale industrial-size TWNS with both membrane- and nonmembrane-based regenerators under nominal and uncertain conditions

global optimization involving logic cuts using BARON and GloMIQO with stepwise procedure to improve tractability of two-stage stochastic MINLP

considers multiple membrane regeneration technologies

tighter lower bounds can be generated within a subspace of a partitioned feasible region

global optimization involving search and elimination of subspaces; partitions feasible region of variables into intervals in boxed subspaces; relaxation (MILPbased) to obtain a lower bound; determines a sequence of subspaces of the partitioned variables where global optimum may exist; bound contraction through subspace fathoming global optimization involving logic cuts using BARON TWNS; p-formulation for pooling problems multiple



solution strategy

Review

Faria and Bagajewicz

91

author

Table 2. continued


model type

water-using unit representation

model formulation

remark


K



Review

Caballero and Grossmann122 that categorize superstructure types into a state−task network (STN), state−equipment network (SEN), resource−task network (RTN), and an intermediate representation with features between that of an STN and an SEN. These superstructures give rise to an MILP or MINLP. More recently, Quaglia et al.123 provided a survey of approaches for superstructure generation that are classified as the following three methods: (1) The alternative collection technique generates a superstructure by considering all known alternative configurations based on past experience and the literature, which has been applied to water network synthesis, e.g., in Khor et al.66,124 A drawback of the approach is that a superstructure generated may not consider innovative alternatives previously unexplored or not discovered. (2) The combinatorial synthesis technique generates a superstructure by considering all possible interconnections involving all relevant materials (states) and operations (tasks), and hence may overcome the drawback of the foregoing approach. However, this may lead to a computationally expensive model due to a large search space as shown in the water network model of Karuppiah and Grossmann.58,85 (3) The insight-based synthesis technique, as inspired by the means−ends analysis strategy of Siirola and Rudd125 and applied in water network synthesis by Quaglia et al.,126 is able to generate a superstructure that includes innovative alternatives while discarding options deemed infeasible or impractical by current knowledge. The insight-based method for superstructure generation may appear to be superior to the others, but it requires more effort to implement. On the other hand, combinatorial synthesis is suitable in dealing with new problems in which sufficient knowledge may not yet be available to eliminate potentially infeasible alternatives.123 It is evident that care is required in postulating an appropriate superstructure representation while accounting for the resulting model type and complexity, besides the underlying assumption of a problem (to exemplify the latter consideration, the superstructure of Khor et al.66 implicitly assumes that there is no interaction between the input of a sink and its output, which is a source). 5.3. Handling the Challenge of Uncertainty. A third major challenge in addressing water network synthesis concerns handling uncertainty posed by the data and parameters in the problems. As reflected in the milestones in section 4, previous work in water network synthesis largely involved handling the uncertain contaminant mass load data53,84,85,89,90,127,128 with an extension to financial risk management.47 In retrospect, the process systems engineering (PSE) community has long emphasized the importance of considering uncertainty in optimization-based decision-making procedures. This has also given rise to a need for risk management in enhancing the robustness of optimization under numerous possible scenarios. Various approaches are applicable for optimization under uncertainty in the PSE domain, which mainly include scenario-based two-stage and multistage stochastic programming with recourse,129,130 chance-constrained optimization,131 fuzzy programming,132 parametric programming,133 flexibility analysis,134 and robust optimization.135−137 Early efforts in optimization under uncertainty have undergone substantial developments in both theory and algorithms.138,139 An uncertain parameter can be assumed to obey a discrete probability distribution, in such a way that the uncertainty is amenable to a representation via a finite number of possible realizations termed as scenarios.140−143 In this manner, an

optimization problem can be equivalently reformulated as a multiscenario model. However, it is well-recognized that a large number of scenarios are typically required to gain a meaningful representation of a problem, which results in a computationally expensive model (due to the so-called curse of dimensionality). Thus, a particular challenge in handling uncertainty entails circumventing a high computational burden due to a need to consider prohibitively many scenarios in formulating a wellposed water network synthesis problem under uncertainty. Three possible approaches can be considered for formulating optimization models for water network synthesis under uncertainty: (1) stochastic programming, (2) robust optimization, and (3) stochastic optimal control.139,144 A specific methodology within a stochastic programming formulation is a two-stage model with recourse that divides the decision variables into two stages. The first-stage so-called “here-andnow” variables are those that have to be decided before future realization of the uncertain parameters. Then, the second-stage “wait-and-see” variables are those used as corrective measures or recourse against any infeasibilities arising during the unveiling of the uncertainty.145−147 Thus, it can be envisaged that the role of the second-stage variables is to accommodate any actual realization of the uncertain parameters, which can conveniently take the form of discrete scenarios (as indicated earlier). Such a two-stage structure is applicable in the context of water network problems under uncertainty: the maximum allowable water flows for reuse/recycle are determined in the first stage of design. Then, as effects of uncertainty in the system performance is established in the second stage, operating decisions are made to meet the actually realized reuse/recycle in the sinks through adjusting freshwater supply and wastewater flows. For multiperiod problems, the two-stage framework can be extended to a multistage paradigm to incorporate future recourse actions.148 It is noteworthy, however, that the multiscenario two-stage stochastic program suffers numerically from the curse of dimensionality due to an exponential increase in size with both number of scenarios and number of uncertain parameters. Robust optimization or the robust counterpart optimization technique has received increasing attention in the past decade for addressing problems under uncertainty.135,137,149−153 An original variant of the approach assumes the underlying description of the uncertain parameters to be captured by ellipsoids,136 while extensions have been proposed that cover other continuous and discrete probability distributions.153,154 However, such representations of uncertainty may turn out to be a limitation for water network problems whose random variables can often be adequately described using a finite set of known values in the form of multiple discrete scenarios provided from historical data or through expert experience. Stochastic optimal control, also known as the Markov decision process, describes a sequential decision-making problem that involves selecting an action in a certain state at a particular time according to a decision rule. While such a paradigm may fit the bilevel sequence of design and operation for water network problems under uncertainty, the approach is more suitable for formulating multiperiod problems in which decisions are required on a real-time basis. This feature is in contrast to stochastic programming models in which, typically, decisions are required to be taken less frequently due to the relatively lower information level available regarding the uncertainty. As such, stochastic optimal control problems are L



Review

case studies20,66 entails a need for future work that enables faster numerical solutions of nonlinear nonconvex problems. We recommend developing customized strategies for handling the nonconvex bilinear terms in a water network synthesis model as well as other sources of nonconvexity. In this respect, a noteworthy strand of work in global optimization techniques that has gained a lot of attention recently involves the use of MILP-based piecewise-affine relaxation schemes to handle bilinearities. This solution method has been surveyed earlier in section 5.1. The development of these relaxation schemes is motivated by the fact that the tightness of the convex or concave envelopes of a bilinear term can be improved by shrinking the domain under consideration. Thus, a basic idea underlying such approaches is to subdivide the original domain into multiple subdomains and to subsequently recombine the convex or concave envelopes that are constructed on these subdomains. Drawing inspiration from the vigorous development of this class of relaxations, which are also benefiting tremendously from the excellent MILP solvers available now (notably CPLEX),174 potential future work is to employ the reformulation of the bilinear terms as separable univariate quadratic terms prior to generating the piecewise-affine relaxations, as inspired by the work of Beasley.175 We postulate that such an approach may potentially require fewer auxiliary variables, particularly the harder to handle 0−1 variables, in achieving a similar level of tightness on the lower bounding problem, compared to existing piecewiseaffine relaxation schemes. Indeed, the ability to generate tight relaxations faster for bilinear terms holds promise for applying such a relaxation scheme in spatial branch-and-bound algorithms to obtain a global optimum in reduced computational time and fewer nodes explored to achieve convergence. Another avenue for future work will involve investigations using other MINLP solvers, besides a state of the art tool such as BARON, for problems of industrial significance. In recent years, significant advances in solvers for the class of pooling problems, which water network synthesis belongs to, have been attained. This achievement is evidenced in the improved performances of various commercial solvers and academic codes such as LindoGLOBAL,176 SCIP,177 Couenne,178 and SBB.179 6.2. Development of More Meaningful OptimizationBased Formulations. It is noteworthy that the detailed nonlinear membrane regenerator modeling approach employed by Khor et al.66 is not limited to only a single unit or one specific type of water treatment technology. The proposed formulation can be extended to multiple treatment technologies with multiple numbers of discrete units connected in series or parallel. A plausible configuration is one that involves a sequence of an ultrafiltration unit and a reverse osmosis unit, which has found practical applications in the industry.180 As expected, the main complexity will arise from optimizing the topology of the two or more technologies considered along with determination of the intermediate contaminant concentrations. The two-stage stochastic programming framework for addressing uncertainty with risk management can be reformulated as a stochastic multiobjective MINLP. Such a formulation caters more explicitly to environmental performance metrics commonly found in lifecycle analysis (LCA) studies as related to freshwater use and wastewater generation instead of the present largely economic-based objective.181−183 Using a Pareto-based approach, the model solution will yield a curve

equally susceptible to the curse of dimensionality, particularly when implemented as dynamic programs.144 A closely related research strand considers the notion of risk in handling optimization under uncertainty. An early work in PSE by Bok et al.,155 which drew inspiration from the Nobel Prize winning work of Markowitz’s mean−variance model156,157 and the robust stochastic programming approach of Mulvey et al.,158 involves risk management using variance as applied to capacity expansion planning of chemical processing networks. Ahmed and Sahinidis159 propose use of the upper partial mean (UPM) as an alternative measure of variability with the aim of eliminating the nonlinearities which arise due to adopting variance. An additional attraction of using the UPM is that it presents an asymmetric measure of risk that only penalizes cases of unfavorable risk, as opposed to the shortcoming of minimizing variance that also undesirably penalizes cases of favorable risk. Alternatively, Applequist et al.160 adopt a risk premium defined as an increase in the expected return in exchange for a given amount of variance in order to evaluate risk and uncertainty for chemical manufacturing plants. In a more recent work, Khor et al.161 applied mean-absolute deviation as a risk measure for petroleum refinery planning under uncertainty. Koppol and Bagajewicz47 address uncertainty in water network synthesis by simultaneously considering risk management. In fact, this work precedes a series of papers by Bagajewicz and co-workers in proposing the probabilistic financial risk metric that is defined based on cumulative probability distributions. The concept gives rise to the notion of a risk curve as a proxy for representing risk, in which the authors emphasize the importance of considering an entire spread of such a risk curve in decision-making.143 The approach has been extensively applied to various problems within PSE, for example, in offshore oilfield planning and scheduling,162 oil refinery planning,163,164 and batch plant scheduling.165 While the technique overcomes the drawback of symmetric penalization in using variance, it comes at the expense of greater computational load due to the introduction of additional 0−1 variables. In our recent work,166 we further hedge against uncertainty by employing two risk measures with origins in the fields of insurance and finance, namely, value-at-risk (VaR) and conditional value-at-risk (CVaR). CVaR is a risk measure initially intended for use in reducing the probability that an investment portfolio will incur high losses. It is closely related to value-at-risk (VaR), which measures the maximum expected loss in the value of a risky entity at a certain confidence interval over a given period under normal conditions. CVaR is the expected loss given that the actual loss exceeds the VaR threshold at the same confidence level.167,168 Within the PSE domain, CVaR has found increasing applications to address uncertainty, with recent work reported for planning and scheduling of research and development activity pipeline management,169 design and planning of oilfield development and chemical production,170 operational planning of large-scale multipurpose multiproduct batch plants,171 and strategic supply chain planning for the petroleum industry.172 It has also been used as a postoptimality measure in the capacity investment planning of multiple vaccines.173

6. POTENTIAL FUTURE DIRECTIONS 6.1. Enabling Faster Numerical Solution. The huge computational expense in solving realistic industrially relevant M



Review

into a deterministic equivalent form particularly for problems with uncertain left-hand-side constraint coefficients. Such a transformation is also applicable for stochastic programming,187,188 but there is no notion of the intuitive recourse decisions in the other two techniques. Nonetheless, an advantage in the absence of the scenario-dependent recourse variables is that both robust optimization and chanceconstrained programming typically require lower computational resources. It is noteworthy that if information on the underlying probability distribution is not available, a fuzzy programming approach can be used that deals with the uncertain parameters as fuzzy numbers and their associated constraints as fuzzy sets.209,210 6.4. Extension to Resource Recovery Systems. As defined in section 2, total water network synthesis involves a simultaneous consideration of two components: water-using units and wastewater treatment operations. For the latter component, in view of the ever momentous drive for sustainability, a potentially important extension of our proposed approach will be to append it with the emerging paradigm of resource recovery systems synthesis.211,212 The extended network will involve sources and sinks in which materials and energy, in addition to water, can be recovered. As substantiated by Guest et al.,213 technologies are readily available for harnessing and optimizing value from the renewable resources present in wastewater-handling operations. These technologies will be incorporated as regenerators in a conventional water network synthesis formulation. Such an approach will optimize not only the quality and availability of water but also minimize violation of sustainability-related constraints pertaining to use of materials and energy. Hence, the latter considerations ought to be appropriately considered in the proposed extension to a multiobjective formulation as delineated in section 6.2.

that examines trade-offs between the typically conflicting objectives of expected profit and the associated risk on one hand, and the environmental-based criteria on the other hand. As well, such a formal multiobjective approach rigorously addresses a more representative range of values for the risk factors. A well-reported technique to solve a multiobjective problem is the epsilon constraint method.184−188 A more rigorous alternative involves its reformulation as a parametric programming problem.189 A related aspect is to investigate the use of risk measures other than CVaR that also offer attractive computational properties, for instance, downside risk190,191 and mean-absolute deviation (MAD),160,161,192,193 which may be potentially more suitable for risk management of water network synthesis problems. There are also incentives in applying alternative formulations and conducting computational comparison for incorporating risk measures to hedge against uncertainty in water network synthesis problems, which with regard to CVaR include techniques such as an LP-based approximation by Rockafellar and Uryasev194 and later work by Krokhmal et al.195 and Alexander et al.196 Another extension worthy of consideration is to reformulate the two-stage framework as a more general multistage stochastic programming formulation.169,197 This approach will give rise to a multiperiod formulation in which the decision variables and constraints are disaggregated into groups of corresponding temporal stages. An advantage of such a model is that each new stage unveils new different information about the uncertain parameters, thus contributing to richer first-stage variables, which are typically the more important decisions. It is anticipated that implementing formal scenario reduction techniques may become necessary in solving the more challenging multistage problems.186,198−202 On a more conceptual level, although the focus of this review is on mathematical programming techniques for single-site water network synthesis problems, it can also benefit from concepts in related subareas within the domain (and vice versa). For example, work on hybrid mathematical programming and water pinch analysis techniques21,22,24,69 have shown that the latter (i.e., pinch) can enhance a formulation and reduce model size as well as be exploited in devising solution strategies. In this regard, there is potential that approaches for interplant water integration12−14,50−52 may help formulate problems in a more meaningful way in terms of incorporating the geographic aspects that are applicable to single-site problems for large industrial plants. 6.3. Alternative Methods for Optimization under Uncertainty. An approach that has gained increasing attention within the PSE community in dealing with optimization under uncertainty is robust optimization.151−153,171,203−206 Another alternative method is chance-constrained programming.187,188,207 Robust optimization seeks to determine an optimal solution that is the best possible with respect to the original objective function (in the nominal or deterministic formulation) while ensuring the constraints are feasible for all realizations of the uncertain parameters at a specified probability level. On the other hand, solution of a chance-constrained program may not be feasible for all uncertainty outcomes, even for the nominal parameter values, except for at least some highly desirable (but not entirely essential) probabilistic constraints.208 Although the two methods differ in consideration for feasibility, in practice, chance-constrained programming can provide a convenient mathematical framework to reformulate probabilistic constraints

7. CONCLUDING REMARKS This review provides a survey of representative work on water network synthesis problems focusing on single-site and continuous processes. We also elucidate three major modeling and computational issues that are inherent in water network synthesis, which pertain to nonconvexity, nonlinearity, and uncertainty of the problem formulation. Future research directions are discussed from four encompassing themes that are of direct consequence in handling the foregoing challenges.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

REFERENCES

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