Recent Advances in Sustainable Process Design and Optimization ((With CD-ROM)) || OPTIMAL WASTEWATER NETWORK DESIGN

February 21, 2012 8:25 9in x 6in b933-ch11

OPTIMAL WASTEWATER NETWORK DESIGN

JACEK M. JEZOWSKI∗, GRZEGORZ POPLEWSKI∗,‡and IRINA DZHYGYREY†

∗Department of Chemical and Process EngineeringRzeszow University of Technology, Rzeszow, Poland

‡[email protected]

†National Technical University of UkraineDepartment of Cybernetics of Chemical Technology Processes

Kyiv, Ukraine

1. Introduction

Systematic investigations on water allocation problem began with theseminal paper by Takama et al.1 Due to increasing concerns on limitedwater resources the problem gained great interest in the last 15 years.The soaring number of works is dated from year 1994 when Wang andSmith published two important papers: one on water usage network2 andthe second on wastewater treatment network.3 It is of importance thatthe work of Takama et al.3 treated the problem in holistic manner sincewater usage processes and treatment operations were included into a singlesystem — total water network (TWN). However, the works of Wang andSmith began the division of this total network into two parts: systemof water using processes (with or without regeneration), referred to aswater usage network (WUN) and network of treatment processes, calledwastewater treatment network (WWTN). Many researchers followed thisway and only few more recent papers such as, e.g. Refs. [4–11] dealt withthe total system (the works by Ng et al.10,11 differ from the others in thatthey considered non-mass transfer processes). The division has practicalbackground since treatment plants are usually “somewhat” independentunits in total sites. However, such separation of TWN into two subsystems

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312 J. M. Jezowski et al.

has also evident negative effect if interrelations are not properly accountedfor designing both of them independently. The guidelines from Kuo andSmith12 are helpful to find proper relations among the subsystems thoughthe rigorous solution is to design them simultaneously.

This chapter addresses wastewater treatment network design problem.In particular we will concentrate on two approaches: a hybrid method thatcombines insights from pinch technology with mathematical programmingand a systematic simultaneous approach by solving superstructureoptimization model using stochastic optimization method. The formulationof WWTN problem assumes that effluents streams from WUN are given inregards to flow rates and concentrations of contaminants. Also, wastewatertreatment processes are known. The objective is to design a network thatachieves the best performance in terms of cost or its approximation andensures meeting environmental limits on quality of output streams fromWWTN. Most often operation expenses or total annual cost of WWTNis applied as the goal function. Both costs are dependent on the totalflow rate though the dependence is not linear in case of investmentexpenses. Therefore, some works use total flow rate through operationsas the performance index. Due to this dependence of costs on flow ratea redistributed network is cheaper than traditional centralized plant. Thelatter is illustrated in Fig. 1.

All effluents are mixed together and this total stream passes through alltreatment processes. Redistribution means the use of bypasses and recyclesthat allows reducing the flow rate. For instance bypasses may be appliedsuch as in Fig. 2 around one or more processes.

The flow rate via the process has been reduced thus diminishingoperation and investment expenses. It is necessary to adjust split ratio inthe splitters so as to meet limits on outlet stream/streams’ concentrations.This illustrative case is simplified but shows the core issues — reducing

Wastewater1

Disposalsite

Treatment1

Treatment2

Wastewater2

Fig. 1. Illustration of traditional centralized wastewater treatment network.

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Optimal Wastewater Network Design 313

Wastewater1

Disposalsite

Treatment1

Treatment2

Wastewater2

Fig. 2. Illustration of distributed network with bypasses and recycles.

flow rates complying simultaneously with constraints on concentrations andconditions of running treatment processes.

It is of importance to note that WWTN design requires simultaneoushandling of structure and process parameters. Such problems are specific forprocess systems engineering (PSE). Moreover, TWN, WUN and WWTNproblems are similar to each other and feature close resemblance tomass exchanger network (MEN) and heat exchanger network (HEN). Notastonishingly solution techniques for WWTN have often origins in HENand heat integration — the problems for which exists a vast literature andmany methods.

Generally, more complex structure of connections is needed than simplebypasses such as in Fig. 2 in designing distributed WWTN. Superstructurenotion offers general and convenient frames to formalize and, also, tosolve the problem of finding the best flow structure of WWTN. Whenconstructing this superstructure one should take into consideration thatsequence of processes is not fixed though water cleaning technology imposessome rules on types of processes that can follow each other. Generally,there are three classes of water cleaning processes used in sequence:primary, secondary and tertiary. Each class of treatment processes containsseveral specific technologies. In WWTN design practice, the models oftreatment processes are most often very simple or even too approximated(oversimplified). Two types of such simple design equations are commonlyused: given removal ratio for each contaminant or fixed inlet concentrationof contaminants. In few papers additional conditions are imposed ontreatment operations such as limiting inlet concentrations or upper/lowerlimits on flow rates. The technologies are, thus, distinguishable by treatmentefficiency (or outlet concentrations) and cost functions (parameters in costfunction). The selection of technology increases problem complexity and,thus, is accounted in few approaches. One of them will be addressed in

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Sec. 5. Also, very few approaches allow for using more rigorous model ofprocess/technology. The hybrid method addressed in Sec. 4 belongs to thisgroup.

The structure of the chapter is as follows. Formulation and descriptionof WWTN design problem is addressed in Sec. 2. Next, in Sec. 3, wewill present an overview of literature approaches. In Sec. 4, we will givea description of the hybrid approach with applications. Next, simultaneousmethod will be addressed in Sec. 5.

2. WWTN Problem Formulation and Description

The general formulation of WWTN problem is given below using symbolsdescribed in the section “Symbols.”

Given are:

• Wastewater streams from various sources (s ∈ S) with concentrations ofcontaminants (Ci

s) and flow rates (Fs)• Treatment processes (t ∈ T ) with treatment technologies for each

treatment process (tt(t) ∈ TT (t)). The number of technologies forprocess t is given and denoted byN(t). IfN(t) is equal to 1 the technologyis fixed and process is equivalent to technology. It is important to noticethat the WWNT does not include selection of processes.

• Design model for each treatment process/technology.• Final discharge sites for water streams after treating (e ∈ E) with

environmental limits on concentrations (Ci,maxe ). Notice that usually

there is only one site and limits are imposed only on concentrations.• Technical and technological conditions on operations/technologies and

structure (mainly on connections). The constraints on a structure includeforbidden and must-be connections, upper limits on number of branchesfrom a splitter and so on.

• Cost parameters for calculating goal function

The objective is to design WWTN that minimizes/maximizes certainperformance index.

The most general version of problem formulation met in the literatureassumes that:

• For each treatment process one technology should be chosen from a setof available ones.

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• For each technology (or process) a number of apparatus (units) shouldbe found. However, most often one unit is assumed to perform therequired treatment. Notice, that in fact also arrangement of units(serial, parallel or serial — parallel) should be determined but no workpublished to date considered this.

In regards to design model of treatment process/technology, it is oftensimplified to the given removal ratio for contaminants — Eq. (1) — orto fixed outlet concentrations — Eq. (2).

ψi =F inCi,in − F outCi,out

F inCi,in(1)

Ci,out = Ci,fixed. (2)

Additional conditions may impose upper and/or lower bounds on inletconcentrations and also on flow rates.

In regards to performance index several authors used total flow rate viaprocess as approximation of its cost. Hence, the goal function for WWTN isthe sum of flow rates via processes. Total annual cost of WWTN consists of:

TAC = a (investment cost) + operation cost (3)

where a is rate of returnOperation cost is most often calculated from Eq. (4), that is only

expenses on wastewater cleaning are taken into account

Operation cost for treatment/technology = δF (4)

where δ is a parameter.Notice that operating cost of pumping streams can be substantial in

some cases and should also be taken into consideration.Investment costs of treatment operation/technology are calculated

from:

Investment = α+ βF γ (5)

where α, β, and γ are parameters.In some works fixed charges are not accounted for. If cost of

transporting fluids is accounted, then investment on pumps should alsobe included.

In addition to investment on processes also cost of connections(pipelines) is taken into account in a few methods. Most often it is assumedthat cost of a connection between two items of WWTN is known. For

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instance, cost of pipeline from process t to process t′ is equal to ϕt,t′ . Also,approximated goal function is often employed with number of connectionsinstead of their costs. Cost of splitters and mixers is not included into thegoal function similarly to other systems such as MENs and HENs. However,structural constraints can be applied to diminish number of splitters ornumber of branches in a split. Also, forbidden and compulsory connectionscan be imposed. In addition to a cost goal function, an environmentalperformance index was also applied in the paper by Lim et al.13

For the data listed above the problem is nonlinear with continuousvariables (nonlinear programming — NLP). The nonlinearities are causedby mass balances. Cost goal function may also be nonlinear. In caseof fixed charges on processes (see Eq. (5)) and/or cost of connectionsor structural constraints binary variables have to be used. The problembecomes mixed-integer nonlinear programming — MINLP. These featuresare of utmost importance if optimization is applied as solution method.Obviously, problem nonlinearity and mixed integer variables cause serioustroubles for insight based (heuristic) approaches, too.

3. Literature Overview

We will concentrate on papers that address WWTN design problem.Other works that have some connections to this topic such as thosedealing with regeneration in WUN or regeneration/treatment in TWNare listed if necessary for explanation of problem of interest. Also, we willanalyze neither works on designing and simulating wastewater treatmentplant of fixed structure nor distinct treatment processes. As examples werefer only to a few works to give the reader some references for furtherstudies necessary to carry out detailed design of synthesized network.Petrides et al.14 presented application of software EnviroPro Designerby INTELLIGEN Inc. for simulating treatment operations. Additionally,they showed an example of application for revamp design with economicevaluations provided by the program. Gontarski et al.15 applied neuralnetworks to model and simulate WWTN. The optimization of operationof selective membrane separation processes for wastewater treatment wasdescribed in Eliceche et al.16 Similar conceptual designing method byknowledge mining was used in Avramenko et al.17 Treatment plant controlproblems are the scope of the work by Moles et al.18 There are numeroustextbooks and monographs on water reclamation technology and designingtreatment operations such as for instance that by Tchobanoglous et al.19

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To our best knowledge they did not address WWTN as process systemengineering problem.

In regards to approaches for WWTN they can be roughly dividedinto two groups: insight-based and optimization-based. Pinch technologyis a leading technique in the first group similar to WUN problem.Superstructure model optimization is most common technique that ischosen as systematic mathematical way to cope with WWTN design inthe second group of methods.

Pinch technology for WWTN was presented in two basic works, firstby Wang and Smith3 and, then, by Kuo and Smith.20 Also, the paper ofKuo and Smith12 is of interest for the topic since it analyses relation: WUNversus WWTN. However, it did not introduce new elements in regards toWWTN design as a separate system. The objective is how to design WUNso as to minimize cost of WWTN. The water pinch techniques for WWTNwill be addressed in Sec. 4 while describing the hybrid design approach.Similar to WUN problem, design tools for general mass exchange networkproposed by El-Halwagi and co-workers were also adopted to WWTNin Hamad et al.21 The reader is referred to books of El-Halwagi22,23 tostudy the foundations and applications of solution methods for MEN. Theapproach from Hamad et al.21 is aimed at wastewater network retrofit.They developed a sequential method that uses the following techniques:functional analysis, graphical analysis and optimization. The optimizationmodel is linear without binaries since flow rates and parameters of processesare fixed while ratios of recycled, discharged and mixed streams areoptimized. One can notice that there are quite few works with insight-basedor heuristic approaches to WWTN in contrast to WUN.

There are more papers with systematic optimization methods forWWTN. Likely the first one with superstructure concept was that by Galanand Grossmann.24 Notice that in the same year paper by Zamorra andGrossmann25 used also a superstructure to design WWTN but this was onlyan example for novel optimization algorithm. The former authors developedboth NLP and MINLP formulations for WWTN superstructure. The NLPmodel is for minimizing flow rates without treatment technology selection.Also, they applied more detailed model though still of short cut type forone treatment operation — non-dispersive solvent extraction applied towastewater cleaning. To cope with NLP and MINLP complex models Galanand Grossmann24 suggested the use of heuristic multi-start procedure withinitial points generated by solution of linearized problems with LP or MILPsolvers, respectively. Similar superstructure based method was used by

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Hernadez-Suarez et al.26 but with more complex optimization technique.However, they considered standard formulation without technology choiceand with the typical model by Eq. (1). The solution was based ongenerating reduced substructures such that the sequence of units is fixedand there are neither recycles nor recirculations. A substructure modelcan be further simplified to linear one by fixing split ratios. Finally, thesequential sequence of steps was applied, each step with the pair: LP–NLP solutions in cycles. The solution from LP is used as initial pointfor NLP. The change of split fraction was performed with discrete step.Very sharp increase of calculation burden was observed with an increase oftreatment operations number. For instance, in one of the examples for thechange of split ratio with step of 0.1 the number of LP–NLP tasks to besolved soared from 726 for three processes to 208 × 103 for four processes.This makes the CPU time for larger cases almost prohibitive. A noveland complex deterministic global optimization algorithm from Ref. [27]was applied in Ref. [28] to solve WWTN case study from Ref. [24]. Theoptimization procedure was organized in two stages: first level, discretebranch-and-bound finds a feasible choice for disjunctions; second level,spatial branch-and-bound finds an upper bound by closing the gap betweenthe nonconvex constraints and their convex relaxation. CPU time wasreasonable for the example of three treatment operations. In a similar paperon application of a version of global optimization tool for process networksynthesis, Bergamini et al.29 solved two examples also from Ref. [24]. Theoptimization method did not used spatial branch-and-bound technique tolocate global optimum for NLP in contrast to the method of Lee andGrossmann.27,28

The approach was further improved and described in the paper byBergamini et al.30 The case study from Galan and Grossmann,24 whichtook technology selection into account, was solved. The major changes incomparison with the previous method are: solution of bounding MILP toonly feasible point (not the optimum) and a new formulation of piecewiseestimators. CPU time needed for the example was smaller than that for theoriginal method. Also, the important feature of the work is accounting fornumber of units for performing selected wastewater treatment technology.In this respect, the work addresses a general formulation of WWTN butstill with simplified treatment process model.

A special global optimisation deterministic technique with piecewiselinear reformulation–linearisation was also employed to attack WWTN

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problem in the paper by Meyer and Floudas.31 The superstructure with noloops was used and MINLP model with detailed cost function that includedoperation and investment cost of pipes and apparatus. The formulationconsidered also treatment technology and apparatus arrangement asvariables. The authors noticed that WWTN is a case of generalized poolingproblem — one of the most common in chemical industry. As the examplefor illustrating the application of the method a large case with 10 treatmentprocesses but with fixed technology was applied. In spite of heuristicproblem reduction to four processes the CPU time needed to locate thesolution close to the global optimum was high, amounting to 79 h.

In parallel to sophisticated global optimization technique severalauthors tried to omit local optima traps by developing initialization schemesthat generate feasible or good (close to optimum) starting point. The worksby Galan and Grossmann24 and Hernando-Suarez et al.26 described inthe preceding went along this line. Also, the works of Martin-Sistac andGraells,32 Lili et al.,33 Statyukha et al.34,35 and Kvitka et al.36 belong tothis class. The work by Martin-Sistac and Graells32 applied NLP modelbecause flow rate minimization is the objective and treatment technology isfixed. The discrete changes are made on variables and simulation controlledfeasibility of solutions. The same assumptions were imposed in the paperby Lili et al.32 The number of variables was reduced before optimization byapplying pinch rules from Wang and Smith.3 It is important to notice thatthe removal ratio of biological treatment was calculated from Monod andAndrews plug-flow reactor model. The use of water pinch technology frompapers by Wang and Smith3 and Kuo and Smith20 to find initial point wasa basis of the approach which was first presented in Statyukha et al.35 Theapproach addressed there was limited to typical simple model of treatment.Then, the approach was extended for application of more detailed modelsand some examples were presented in short contributions in Statyukha etal.34 and Kvitka et al.36 The full extended approach with industrial casestudies will be given in Sec. 4.

An alternative to finding feasible initial points for deterministic NLPor MINLP algorithms is to use stochastic or meta-heuristic approaches.More recently this became popular way of solving process network synthesisproblems. Let us mention WUN synthesis with genetic algorithms appliedin papers by: Tsai and Chang,5 Prakotpol and Srinophakun37 and Lavricet al.37 to mention a few. Also other stochastic optimization techniqueswere used to WUN such as particle swarm optimization (PSO) in Hulet al.,39 Tan et al.40 or adaptive random search (ARS) in Poplewski and

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Jezowski,41 Jezowski et al.42 Somewhat astonishingly only ARS was used toWWTN. This was presented in brief conference contribution of Poplewskiand Jezowski.43 In this chapter, we will present it in detail in Sec. 5.

All the approaches reviewed to this point applied cost or flow rate asWWTN performance index. The work by Lim et al.13 differs in that point.This is a part of a sequence of papers44,45 aimed at estimating of waternetworks by environmental criteria. The paper of Lim et al.13 comparesthe traditional and distributed WWTN on the basis of life cycle analysis(LCA) and life cycle costing (LCC). The most interesting conclusion drawnout from the comparison is that the life cycle cost of distributed network islower than that of non-distributed one — the difference was 10.1% in thecase study.

In addition to methods aimed at WWTN formulation from Sec. 2, thereare some works that deal with similar problems. The problem addressed firstin conference paper of San Roman et al.46 and, then, in journal paper byBringas et al.47 differs from WWTN formulation in two points:

1. there is only a single effluent stream — groundwater,2. treatment units (emulsion pertractation technology) are used to clean

groundwater, and, also to recover substances from emulsion.

The consequence of (2) is that: (i) two kinds of operations are performedin the units and there are two networks — the networks have commontreatment operations, (ii) there are environmental conditions on permissibleconcentration of groundwater after treatment, and, also additionalcondition on minimum concentration of substances to be recovered (theconcentration should be higher than the minimum). Treatment technologyis fixed — emulsion pertractation with hollow fibre modules but the numberand type of arrangement (serial, parallel or serial-parallel) are variables.Also, a rigorous model of treatment operation was applied. Simultaneousapproach by superstructure optimization was used. The superstructureembeds all possible arrangements of treatment units but the number ofunits is fixed. Thus, superstructure NLP model has to be solved for varyingnumber of units, the selection is then carried out to choose the bestfrom generated solutions. A global optimization method by spatial branchand bound technique with Lagrangean decomposition was employed witha feasible starting point calculated by simplifications of superstructuremodel.

Yet another version of WWTN problem is considered in two papersby Saif et al.48,49 This is the design of optimal reverse osmosis network

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(RON) applied to clean wastewater. Thus, only one technology is used butwith apparatus organized in stages. Each stage has several units in parallel.Both number of stages and number of modules are variables. In additionto reverse osmosis units there are also other processes that have to beincluded into the superstructure: pumps to raise pressure and turbines torecover kinetic energy. The superstructure model presented in Saif et al.48

is the extension of the original one developed in El-Halwagi.50 The solutiontechnique is identical to that developed by Galan and Grossmann.24 Foursubnetworks were generated from the general superstructure and solvedwith a sequence of the pair LP–NLP problems until convergence. Pulp andpaper small case study was solved in a short CPU time. The second paper —Ref. [49] — addresses the same problem but solved by another optimizationalgorithm. Generally this was a branch-and-bound technique. Tight lowerbounds were developed to approximate nonconvex terms with under-and-over-estimators. The approach was employed to designing desalination ofa single water stream. The solution reported was superior to that fromEl-Halwagi50 by 14.8%.

The most important conclusion from the investigations on WWTNcarried out to date is that the problem is difficult (particularly incomparison to WUN) for existing optimization tools, both deterministicand stochastic as well. This is most likely caused by highly constrainedfeasible space. We have performed some numerical experiments for threeliterature examples by randomly generating N points for decision variableschosen in the way so as to obtain zero degrees of freedom for WWTNsuperstructure model — see point 5. Then, we checked if the point isfeasible — the calculation procedure is described in Sec. 5. For 2 × 109

points no feasible solutions were found for the examples. Thus, a methodof generating feasible points is expected to be nontrivial.

Finally, we also list some other conclusions and remarks from literatureoverview:

• Systematic (optimization, mathematical) approaches are prevailing incomparison with WUN (TWN). Likely it results from the fact thatmultiple contaminant cases were considered.

• Deterministic optimization techniques were preferred over stochasticoptimizers.

• Deterministic optimization algorithms used either one step procedurewith “global” optimizer or apply initialization scheme to find feasiblestarting points for local optimizers.

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4. Hybrid Approach for WWTN

4.1. Overview of the approach

Insight-based methods for process system synthesis, particularly those fromPinch Technology are conceptually simple and user driven. The maindrawback is that they have some inherent simplifications with attendantapproximations such as the use of exergy losses as the goal function,which can result in inferior solutions in regards to cost, function inparticular. On the other hand, optimization methods required advancedcomplex algorithms and/or feasible initial points. The idea of the hybridapproach presented in basic version in Statyukha et al.35 was to use waterpinch based technique to locate feasible starting point for an optimizationprocedure. The method was strongly directed at industrial applicationsin Ukraine and other east-central Europe countries. This caused somespecific requirements. First, retrofit applications were of more interestthan synthesis. Next, it was strongly required not to use commercialoptimizer (and other software as well) but simple “in-house” subroutine.Application of water pinch technology met the first condition since userdriven philosophy is very useful in retrofitting systems. Also it usuallygenerates not only feasible but also, good solution (local optimum).Thus, even simple optimizer should be able to cope with optimizationproblem.

A hybrid sequential approach for WWNT was developed, whichconsists of three stages: targeting stage, structure development stage andoptimization stage.

Techniques applied in the first two steps are based on “wastewaterpinch” concepts developed by Wang and Smith3 and improved by Kuo andSmith.20 They are able to generate such WWTN, which minimizes exergylosses. However, the optimization step of the hybrid approach is aimedat cost optimization. It is important to note that the hybrid approach isnot limited to parameter optimization of a network with fixed topology.The solution from stage (2) is treated as the initial point for nonlinearoptimizer. Two alternative ways can be applied when formulating theoptimization problem. Firstly, the designer can apply whole general WWTNsuperstructure such as described in the next section. Secondly, insertingnew potential connections into the network from pinch technology stepscan create case specific small superstructure. This option is very useful inretrofit scenario. It is necessary to note that at present the hybrid approachis limited to NLP models due to limitations of optimizers. However, this

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is only the problem of using an appropriate optimization procedure. It isplanned to adapt stochastic optimizer from Sec. 5.

In the following, we will present the stages of the hybrid approach.We will describe the targeting stage in more detail since its importance inWWTN design independently of a solution technique. Both other stagesare presented briefly due to space limitation. The reader is asked to seekdetails, if necessary, in original papers.

4.2. Targeting stage

Wang and Smith3 and, then, Kuo and Smith20 developed the graphicalprocedure for targeting the minimum flow rate in order to minimizetreatment cost. Water pinch analysis for wastewater treatment systemdesign includes three variants of design procedures: (1) for singlecontaminant and single treatment process (TP), (2) for single-contaminantand multiple treatment processes/technologies, and (3) for multiplecontaminant and multiple TPs. We will describe them in this sequence.

(1) Single contaminant and single TP

Once the problem has been put in the framework of “concentration-load” diagrams, a composite curve representing all wastewater streamscan be constructed.3 This can be achieved by first plotting all ofthe effluent streams on the same concentration versus mass load axes.The concentrations of a contaminant in the wastewater streams defineconcentration intervals. Within each concentration interval the rate ofchange of concentration against mass load is constant, i.e., this is straightline graphically. By combining mass loads within concentration intervals acomposite curve of the effluent streams is obtained — piecewise straightline. Having constructed the composite curve for the wastewater streams,a treatment line is then drawn against it. The reciprocal of the slope of thetreatment line is the wastewater flow rate treated. This results directly fromEq. (6) defining mass load of contaminant i transferred to water stream ina treatment operation.

Li = F (Ci,in − Ci,out). (6)

Notice that flow rate was assumed constant since mass load value isrelatively small in all treatment processes.

The start and end points of the treatment line segment correspond tothe inlet concentration and the outlet concentration from the TP. If thetreatment cost decreases continuously with decreasing flow rate, then, for

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a given set of wastewater streams, cost is minimized when the flow rate tothe TP is minimized. A minimum treatment flow rate can be determinedin straightforward manner depending whether Eq. (1) or Eq. (2) is to beused. Graphically, the slope of the treatment line should be maximizedto minimize the treatment flow rate. However, the slope of the treatmentline is limited by both the composite curve of the wastewater streams andthe performance (Eq. (1) or Eq. (2)) of the TP. Rotating the treatmentline around limiting treatment point, anticlockwise until it touches thecomposite curve of the wastewater streams, minimizes the treatment flowrate. Limiting treatment point is located according to the specified removalratio given by Eq. (1) or defined by Eq. (2). It is of highest importance tonote that a pinch is created at the point where the treatment line touchesthe composite curve. The procedure is illustrated by Fig. 3 for the treatmentoperation defined by fixed outlet concentration Cout,fixed.

The composite curve plot gives not only the minimum total flowrate but also shows the designer what streams require treating andwhat streams do not. This information results from pinch location. Thewastewater streams with contaminant concentration level higher thancontaminant concentration in the pinch point must pass through treatment.The wastewater streams with contaminant concentration level equal to

C

C

Lt L

composite curve

treatment line

out,fixed

Fig. 3. Illustration of graphical procedure for determining the minimum flow rate inWWTN.

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contaminant concentration in the pinch point partially bypass treatment.Finally, the wastewater streams with contaminant concentration lower thancontaminant concentration level in the pinch point should bypass cleaningoperation. These conclusions are of utmost importance in insight-basedapproaches. Also, they can be great help for reducing a search space inoptimization methods.

(2) Single contaminant and multiple TP

Again the problem is put in the framework of “concentration-load” diagramwith a composite curve and treatment lines of TPs. Wang and Smith3

suggested that the maximum mass load should be removed by the cheaperTP. This is true only if the ratio of costs for TPs is very high. Therefore, theprocedure was improved by Kuo and Smith.20 They proposed to optimizethe mass load removed by each TPs. The target flow rate for each TPallows the total cost to be estimated through cost functions of treatmentflow rates. The optimization allows obtaining the distribution of streams inTPs for single contaminant.

(3) Multiple contaminant and multiple TP

Wang and Smith3 suggested an approach, in which the design formultiple contaminants is carried out by first designing a network for eachcontaminant separately and, then, merging the sub-networks. Kuo andSmith20 noticed drawbacks of the approach. First, in case of multipleTPs, they showed that the original method by Wang and Smith3 assumesserial connections of processes. However, parallel configuration can beadvantageous in certain cases. Hence, Kuo and Smith20 developed thesolution scheme that requires re-drawing the composite curve plot afterselection of each TP based on thermodynamics.

4.3. Structure development optimization stage

Information from composite curve plots are used by the designer toconstruct a structure of the system and to estimate flow rates. In principles,this is relatively easy and straightforward in case of a single contaminantand one type of TP. The technique is the same as for developing WUN. Todeal with multiple contaminants Kuo and Smith20 applied the concept ofwastewater degradation to eliminate unnecessary mixing. Exergy losses ofmixing are the measure of the degradation. The network should minimizeexergy losses. Thus, instead of merging the networks for single contaminantsas suggested by Wang and Smith,3 a superstructure is built from the

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subnetworks. The optimal structure is found by choosing such sequenceof the processes, connections and flow rates that minimize exergy losses.

Wastewater degradation is unavoidable but it makes sense to minimizeit as much as possible whilst selecting the TP sequence. Possible sub-networks are estimated under this criterion and the one with minimummixing exergy loss is chosen for the first TP placement. Then, the remainderof the problem is re-targeting until whole TP sequence will be obtained.Entire conceptual design procedure including targeting and structuredevelopment stages is as follows:

Step 1. Targeting sub-networks for all TPs and contaminants.Step 2. Estimation of mixing exergy losses for contaminants and TP

operating in parallel.Step 3. Selection of TP with minimum mixing exergy loss.Step 4. Re-targeting sub-networks for remaining TPs.Step 5. If whole TP sequence is obtained then the procedure is finished,

else back to Step 1 for remaining TPs.

Wastewater degradation is the thermodynamic function, and, thus itcannot guarantee optimality in regards to cost. So the conceptual approachdoes not guarantee optimality of the final design in terms of operation ortotal cost. Therefore, the third stage was added in hybrid method.

4.4. Final optimization stage

As we have mentioned the solution from stage (2) is the basis for buildinga superstructure by the designer. For retrofit case the superstructure iscreated by adding potential splitters, mixers and branches. An optimizationmodel for WWTN superstructure is presented in detail in Sec. 5 onsimultaneous method. For that approach, the model is crucial. Hence, weomit its presentation here and limit ourselves to most important specificfeatures for its application in the hybrid approach. As we have mentionedthe optimization model should be NLP. The lack of binary variables causesthat goal function cannot involve fixed charges on technologies. Also, somestructural issues as minimizing number or cost of connections cannot beaccounted for. The designer should exclude forbidden connections whiledeveloping the superstructure. To solve NLP model Statyukha et al.35

used the random search optimization technique — Improving Hit-and-Run algorithm from Zabinsky et al.55 The investigations on developingsimple robust optimizer for the MINLP are carried on. Generally, other

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direct search optimization techniques can be applied since starting pointis a local optimum. This is also the reason that more rigorous modelsof processes can be applied. Researchers from National University ofUkraine developed such the models for various operations. Some of themare given in Statyukha et al.34 and Kvitka et al.36 They are based onheuristic design procedures used widely in former USSR for designingwastewater treatment facilities. Some modifications were performed tochange calculation procedure to simulation-oriented mode in order toinsert them into optimization methods. Notice, that indirect or stochasticoptimization techniques suit better for these functions due to their irregularand even discontinuous character. The example in the following shows anindustrial case study.

4.5. Example of application

Here, we will present industrial case study — a meatpacking plant fromUkraine. First a solution will be presented for simplified model with Eq. (1)for all treatment processes. Then, a more rigorous model will be given andthe solution of WWTN for retrofit case.

Wastewater of the meatpacking plant belongs to the strong sewage withhigh content of organic contaminants. Three wastewater streams, namelyindustrial fat wastewater (stream 1), industrial foul wastewater (stream2) and society’s wastewater (stream 3) generated by the packing plantare treated in the existing central facility with serial arrangement withoutrecycles and bypasses. Streams are contaminated with suspended solids,fats, BOD5 and chlorides. The environmental limits on the concentrationsof the four contaminants are 0.75, 2, 3 and 350 ppm, respectively. Fourtreatment processes are applied: settling (TPI), pneumoflotation (TPII),electrocoagulation (TPIII) and biofiltration (TPIV). Tables 1 and 2 givethe required data. Notice, that the removal ratios applied are typical forthese classes of processes. Superstructure and performance index for theoptimization stage were consulted with plant management and engineers.

Table 1. Wastewater streams data for the meatpacking house case study.

Contaminant concentration (ppm)

Stream number Flow rate (t/h) Suspended solids Fats BOD5 Chlorides

1 232.5 710 380 690 12002 103.8 430 40 320 4503 53.8 90 10 190 375

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Table 2. Treatment process data for the meatpacking house case study.

Removal ratios (%)

Treatment processes Suspended solids Fats BOD5 Chlorides

TPI 35 63 25 0TPII 96 95 60 0TPIII 95 97 85 65TPIV 80 0 90 35

Treatment flow rates for each of the units were 390.1 t/h in the existingfacility. The solution of the second stage features the following treatmentflow rates: 232.5 t/h for TPI, 303.1 t/h for TPII, 390.1 t/h for TPIII andTPIV. After consultation the total flow rate was applied as the goalfunction. The factory management agreed to apply the superstructureembedding all potential splitters, mixers and branches. The optimizationstage requires solution of NLP problem with 4 constraints and 18 variables.The final solution in Fig. 4 shows that the flow rate in pneumoflotationprocess TPII can be further reduced, in comparison with pinch technologysolution, to 296.9 t/h without changes of treatment system structure.

Reduction of the treatment flow rate was 40% for TPI and 24% forTPII, respectively, in comparison with the existing treatment system. Theoutlet concentration for suspended solids is 0.7 ppm, for fats — 0.3 ppm,for BOD5 — 3 ppm, and for chlorides — 202ppm. They all satisfy givenrequirements.

Then, mathematical models of primary settler, pneumatic flotationunit, electrocoagulator and biological aerated filter were used at theoptimisation stage. Let us consider as the example, the model of particularbiofiltration unit: biofilter with plastic packing. The model of biofilterrequires the following input parameters: input treatment flowrate Q, t/h,

1

2

3

TP I

232.5 t/h

TP II TP IVTP III

103.8 t/h

53.8 t/h

39.4 t/h

232.5 t/h 269.9 t/h 309.1 t/h

Fig. 4. Final solution for meatpacking case study for simplified models.

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inlet BOD5 concentration C in, ppm, maximum inlet BOD5 concentrationC in,max, ppm, diameter of biofilter sections DS, m, number of sectionsNS, height of biofilter H , m, specific surface of filter bed S, m2/m3, filtermedia porosity P , %, wastewater average winter temperature T , ◦C. InletBOD5 concentration is constrained by maximum allowable inlet BOD5

concentration:

C in ≤ C in,max (7)

Biofilter model consists of the following relations — see also Dolyna51 formore details:

(1) Hydraulic loading, m2/(m3· h):

qH =4 ×Q

H ×D2S ×NS × π

. (8)

(2) Wastewater average winter temperature coefficient:

KT = 0.2 × 1.047T−20. (9)

(3) Criterion “complex” (specific criterion of Voronov design approach)

η =P ×H ×KT

Cin × qH × S. (10)

(4) Outlet BOD5 concentration, ppm

Cout = 10(2.18−0.385)η for Cout ≥ 11 (11)

and

Cout = 10(1.23−0.066)η for Cout < 11. (12)

(5) BOD5 removal ratio by biofiltration:

ψ =Cin − Cout

Cin. (13)

By simple algebraic manipulation one obtains the final equation forremoval ratio (for Cout ≥ 11):

ψ =Cin − 102.18−0.385a

Cin× 100%, (14)

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1

2

3

TP I

232.5 t /h

TP II TP IVTP III

103.8 t /h

53.8 t/h

23.9 t/h

220.4 t /h 300.3 t /h 378 t/h

Fig. 5. Final design for the meatpacking plant case study with the use of rigorousmodels.

where

a = 0.05 × P ×H2 ×D2S ×NS × π × 1.047(T−20) × S

Cin ×Qin.

Detailed mathematical models of treatment processes have been appliedinstead of fixed removal ratios. Notice that the ratios from Table 2 areapplied in initialization scheme. The final solution is shown in Fig. 5.

Treatment flow rate in settling process does not change, but the removalratio of suspended solids was obtained by the model of continuous-operatedsettler and amount to 50% against initial 35% (Table 2). Additionallyflowrate of the residues of settling process can be calculated. Models ofpneumoflotator and electrocoaculator cannot give us information aboutremoval ratios changes. But they let us to compute some process unitparameters, e.g. flotation time for flotator and current density, electricityrate etc. for electro-coagulator. Application of biofilter model gives nextBOD5 removal ratio change: 92% instead initial 90%. So the final design isnot cheaper than that with simple models but close to the real operationconditions as it was checked by measurements in the plants.

5. Simultaneous Approach with the Use of StochasticOptimization Method

5.1. Superstructure and optimization model

In the short conference paper Poplewski and Jezowski43 described asimultaneous approach by single-stage solution of WWTN superstructureoptimization model. The superstructure consists of given wastewatersources, treatment operations and disposal sites. Here we limit to a single

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disposal site in order to simplify the model. All possible connectionsamongst the basic items of WWTN are embedded into the superstructureby applying splitters and mixers. Each effluent source (s) has one splitter toredistribute the stream to all treatment processes (Fs,t, t ∈ T ) and disposalsite (Fs,e). Likewise, one splitter is attached at the outlet of each treatmentoperation (t). Such splitter can redistribute outlet stream to other processes(Ft,t′), to the same process and to disposal site (Ft,e). One mixer is attachedto each treatment process at its inlet to gather streams from the sourcesand other processes. Also, there is a mixer for the disposal site. This isillustrated by Fig. 6. Flow rates in the model have the following meaning:

Ft — flow rate through process tFs,t(s ∈ S, t ∈ T ) — flow rate from source s to treatment operation tFs,e(s ∈ S) — flow rate from source s to disposal siteFt,t′(t, t′ ∈ T ) — flow rate from treatment operation t to treatmentoperation t′

Ft,e(t ∈ T ) — flow rate from treatment operation t to disposal site

For clarity sake we will limit here to a case without technology selection.This will be explained together with example in Sec. 5.3. Thus, binaryvariables are applied only to identify connections. In general they aredenoted by

yi,j =

{1 if connection between element i and j exists,

0 if not.(15)

Indices i, j are used here instead of indices t ∈ T ; s ∈ S; e.

Sources

Disposalsite

Treatment processt

e,sF

e,tF

1,sF2,sF

t,sF

e,sFe,F1e,F2

...

...

...

...

...

...e,F1

e,F2e,tF

t,F1t,F2

t,sF

t,F1t,F2

t,tF 1,tF2,tF

t,tF

(a)

(b)

(c)

Fig. 6. Elements of WWTN Superstructure: (a) source; (b) arrangement: mixer-treatment process-splitter; (c) mixer of disposal site.

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Notice that all possible connections are embedded into thesuperstructure. The model consists of the goal function as well as balancesand design equations of all elements of the superstructure. The goalfunction — total annual cost of WWTN — consists of two items: costof treatment operations and cost of pipelines (connections):

TAC =

{a

∑t∈T

(αt + βtFγt) +

∑t∈T

δtFt

}

+

{∑s∈S

∑t∈T

φs,tys,t +∑s∈S

φs,eys,e +∑t∈T

∑t′∈T

φt,t′yt,t′ +∑t∈T

φt,eyt,e

}.

(16)

The constraints are as follows:

(1) Overall mass balances for splitters of sources (isothermal process):

Fs − Fs,e −∑t∈T

Fs,t = 0; s ∈ S. (17)

(2) Overall mass balances of treatment operations:∑s∈S

Fs,t +∑t′∈T

Ft′,t − Ft,e −∑t′∈T

Ft,t′ = 0; t ∈ T. (18)

(3) Mass balances of contaminants of treatment processes are:∑s∈S

Cis × Fs,t +

∑t′∈T

Ci,outt′ × Ft′,t −

∑t′∈T

Ci,outt × Ft,t′ − Ci,out

t × Ft,e

= Ci,int × Ft − Ci,out

t × Ft; i ∈ I, t ∈ T. (19)

(4) The design equation for the process is very simple and is the definitionof contaminant removal ratio:

ψit =

Ci,int × Ft − Ci,out

t × Ft

Ci,int × Ft

; i ∈ I, t ∈ T. (20)

Notice, that the ratios are given in data. By combining Eqs. (19) and(20), Eq. (21) can be simply obtained:∑

s∈S

Cis × Fs,t +

∑t′∈T

Ci,outt′ × Ft′,t −

∑t′∈T

Ci,outt × Ft,t′ − Ci,out

t × Ft,e

= ψit × Ci,in

t × Ft; i ∈ I, t ∈ T. (21)

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These equalities are used in the model instead of Eqs. (19) and (20).(5) Mass balances of contaminants for mixers of treatment processes:∑

s∈S

Fs,t × Cis +

∑t′∈T

Ft′,t × Ci,outt − F in

t × Ci,int = 0; i ∈ I, t ∈ T.

(22)

(6) Mass balances of contaminants for mixer of the disposal site:

Fe × Ci,ine −

∑s∈S

Fs,e × Cis −

∑t∈T

Ft,e × Ci,outt = 0; i ∈ I. (23)

(7) Definitions of flow rates through processes. Notice, that they are notnecessary since the basic variables for flow rates among elements ofWWTN can be used directly in the goal function. Here, we apply themin order to make the presentation clearer. Also, the Ft parameters areconvenient to use for the case of technology selection.

Ft =∑s∈S

Fs,t +∑t′∈T

Ft′,t. (24)

(8) Logical conditions which force flow rates to zero if appropriate binariesare zero.

Fi,j < yi,jFmax. (25)

Fmax is a great number such that ensure that conditions (25) areinactive if binary variables are equal to one. Alternatively, values areFmax can be estimated for each connection to tight on upper boundsand, thus, to increase robustness of solving MINLP problem.

(9) Inequality constraints on contaminant concentrations to the disposalsite. They ensure that the concentrations are not higher than the givenenvironmental limits.

Ci,ine ≤ Ci,in,max

e . (26)

(10) Other technological case specific constraints as for instance lowerlimits on flow rate via piping sections:

Ft,t′ ≥ Fmint,t′ yt,t′ ; t, t′ ∈ T (27)

Fs,t ≥ Fmins,t ys,t; s ∈ S, t ∈ T. (28)

Parameters Fmint,t′ and Fmin

s,t have to be given in data.(11) Non-negativity conditions on continuous variables.

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The model is nonlinear because of bilinear terms in mass balances (21–23)and, also the nonlinear goal function. The variables are flow rates via pipingsections and contaminant concentrations at inlets and outlets of treatmentprocesses as well as at the mixer of disposal site. Additionally all binariesare also variables. This gives a large-scale complex problem particularlyfor higher number of treatment processes. Notice that such or very similarsuperstructure and optimization model were used in other works.

5.2. Overview of solution approach

The complex nonlinear optimization model with both equality andinequality constraints and numerous variables has been solved by ARSoptimization method. The algorithm applied is a version of originalLuus–Jaakola algorithm from Ref. [52]. The detailed presentation ofthe optimization method together with results of tests is given inRef. [53]. In this chapter, we limit ourselves to Appendix with the basicalgorithm of applied ARS method. Since the technique is efficient ratherfor unconstrained optimization tasks with small or medium number ofcontinuous variables53,54 some problems had to be solved to achievesufficient robustness and efficiency of solving WWTN model. The problemsare as follows:

(1) How to deal with binary variables?(2) How to deal with constraints?(3) How to find feasible starting point (points)?

The binary variables for connections were treated as dependent parameterscalculated in a solution procedure on the basis of decision variable — flowrates by connections. The simple statements were used in ARS method:

If Fi,j ≤ Fmin THEN Fi,j := 0 and yi,j := 0. (29)

The value of Fmin was assumed at 2 t/h. However, it can be problemdependent and one can use for instance 0.0.

As for inequality constraints so called “death penalty” was applied.It means that the solutions generated by the ARS algorithm infeasiblein regards to the inequalities are simply rejected. This mechanism workswell in ARS optimization — see e.g., Ref. [53] for benchmark optimizationproblems and Jezowski et al. (2007) for application of ARS to WUNproblem. Equality constraints are hard problem for all types of stochastic

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optimization approaches. We applied the direct solution of equalities withinoptimization procedure.

Such scheme performs well if the constraints are linear which is not thecase in WWTN model.

Hence, we applied a method that linearizes the equality constraints. Itgenerates equations that can be solved sequentially as sets of simultaneousequations or sets of single equations which are linear in regards to certainvariables, called dependent ones. This requires a division of all variables inthe optimization problem into two subsets, and, appropriate sequencing ofequality constraints. It is worthwhile noting that such method cannot beused in frames of deterministic optimization approaches since they solvethe equality constraints as one set of simultaneous equations. First, thevariables were splitted into two groups: decision variables generated by ARSprocedure and dependent variables that are calculated from the equations.Such decision variables have been chosen, based on problem analysis, whichcause the equations become linear in respect to dependent variables forgiven values of decision variables. In the case of WWTN problem, thedecision variables are the flow rates of all streams except of those sent to thedisposal site. All other variables are calculated from equality constraints,that is, the variables are dependent variables.

The solution procedure for calculating dependent variables for knownvalues of independent variables generated by the stochastic optimizer is asfollows:

(1) Calculate the flow rates to disposal site from Eqs. (17) and (18). Checkif the calculated values are nonnegative. If they are negative apply thedeath penalty.

(2) Calculate the concentrations (Ci,outt ) of all contaminants at the outlets

from the water using processes from Eq. (21).(3) Calculate the concentrations (Ci,in

t , Ci,ine ) of all contaminants at the

inlets to the water using processes and to the disposal site fromEqs. (22) and (23). Check if the concentrations to the disposal sitemet the upper limits. If they are infeasible apply the death penalty.

It is worthwhile noting that due to the method described above the numberof all variables was reduced to the number of degrees of freedom of theWWTN model.

Though for many nonlinear optimization problems ARS technique doesnot need feasible starting point, this is not the case for the WWTN modeldue to a small space of feasible solutions and large number of variables. We

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have tested some initialization methods and found that the following simplesolution close to that from Galan and Grossmann24 performs well in allcases we have solved to date. The initial starting points for the solver are allpossible centralized treatment networks for the given treatment processes.The networks differ as for the sequence of treatment operations. Hence, forN processes there exists N ! possible sequences. For illustration purposes wepresent in Fig. 7 two initial solutions for the case of two processes. Notice,that in industry the number of treatment processes is limited and, thus,the number of initial points is acceptable. It should be noticed that thereare cases where these starting solutions are not feasible.

5.3. Examples of application

We will present here two examples; one without technology selection andthe second with this choice. Example 1 is taken from Hernando-Suarezet al.26 The problem has seven wastewater sources with five contaminants.The data for sources and limiting concentrations at disposal site are givenin Table 3. The last row of the table gives values of environmental limits on

Treatmentprocess 1

Treatmentprocess 2

Treatmentprocess 2

Treatmentprocess 1

streams fromall sources

to disposalsite

Initial solution 1

Initial solution 2streams fromall sources

to disposalsite

Fig. 7. Initial structures for the solution algorithm in case of two treatment processes.

Table 3. Data for wastewater sources and disposal site for example 1.

Ci (ppm)

Source F [t/h] A B C D E

1 18 1390 10 250 200 400

2 25 14, 000 110 400 600 28003 50 25 100 1350 2500 31154 60 8550 800 45 220 2305 36 500 300 600 500 5006 12 50 1500 400 200 1007 8 2300 12, 500 200 1000 200

Ci,in,maxe 150 200 140 175 200

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Table 4. The removal ratios for treatmentoperations for example 1.

ψ (%)

Treatment processes A B C D E

1 99 70 80 60 552 90 88 55 85 90

s1

s6

s4

s5

s2

s3

s7

t1

t2

e

18

25

50

60

36

8

10.33

7.67

5.79

19.21

60

50

36

1.09

8

10.91

26.8450.37

160.92

209

12

Fig. 8. Optimal solution for example 1.

contaminants. The values of the removal ratios for two treatment operationsare given in Table 4. The objective is to minimize total flow rate viatreatment operations.

The solution obtained by the approach is identical to that calculatedby Hernando-Suarez et al.26 It has the goal function of 238.13 t/h and thestructure shown in Fig. 8. Average CPU time for a single run of the solver

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amounts to approximately 270 s for processor Intel Centrino 1.5GHz. It isnecessary noting that some runs are necessary for the stochastic solver. Itis important to note that the optimization model applied in our methoddoes not eliminate certain connections. This is necessary in the approach ofHernando-Suarez et al.26 However, for this case study the elimination doesnot influence the optimum.

The second example is taken from Galan and Grossmann24 and itwas also solved in Bergamini et al.29,30 This case study includes theselection of treatment technology for every process. The data for wastewatersources together with the environmental limits on the concentrations ofcontaminants A, B and C to disposal site are given in Table 5.

It is necessary to note that the concentration value for source no. 3and contaminant B (CB

3 ) is from Bergamini et al.30 This concentration inGalan and Grossmann24 was given as 100 ppm. The analysis of the solutionfrom the former work showed that the concentration CB

3 of 1000ppm looksmore probable since calculated by us concentration of contaminant B to theenvironment (CB,in

e ) in the solution from Bergamini et al.30 is close to thelimit, i.e., to 100ppm. This is characteristic feature of cost optimal solution.We were not able to to perform such the examination for the solution fromGalan and Grossmann24 since they did not present the network.

Parameters of available treatment technologies for three treatmentprocesses are gathered in Table 6.

In order to explain how we accounted for treatment technology selectionlet us introduce new sets of indices for each process t ∈ T :

TT(t) = |tt(t) : tt(t) is treatment technology number for process t|.

The total number of treatment technologies for process t is denotedby tt(t)f .

Table 5. Data for wastewater sources and environmental limits oncontaminants’ concentrations for example 2.

Concentrations of contaminants (ppm)

Source F [t/h] A B C

1 20 1100 300 4002 15 300 700 15003 5 500 1000 600

Ci,in,maxe 100 100 100

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Table 6. Removal ratios and costs for treatment processes andtechnologies in example 2.

ψ (%) Costs

Treatment Treatment Investment Operationprocesses technology A B C ($) ($/h)

1 90 0 40 3840 F 0.7 0 F1 2 50 70 0 469 F 0.7 10 F

3 0 80 0 26 F 0.7 F1 0 90 0 726 F 0.7 0.0089 F

2 2 0 99 0 1260 F 0.7 0.018 F3 50 99 80 5000 F 0.7 5.8 F1 80 0 60 320 F 0.7 6 F

3 2 0 0 80 58 F 0.7 15 F3 0 0 40 10 F 0.7 F

Next, a new binary variable for every treatment technology is definedas follows:

Ytt(t) =

{1 if technology tt(t) is chosen;

0 if not.(30)

These variables are used in the goal function and Eqs. (21).Finally, continuous variable x(t) has to be used for every process t ∈ T .

The variable is from the range: 0 < x(t) < tt(t)f

The selection procedure developed for the ARS algorithm is as follows:

(1) generate randomly from the uniform distribution a value of xt from thegiven range

(2) calculate binary variable Ytt(t) from the relation:

if xt ∈ (tt − 1, tt) then Ytt(t) = 1 else Ytt(t) = 0

It is worthwhile noting that the region of searching for xt value is continuallydiminished, similarly to search regions for other independent variables,according to the algorithm given in the Appendix. Hence, the search forthe binary variable is not chaotic throughout the overall calculations butgradually concentrates around a certain value that improves goal function.It is also important that binary variables Ytt(t) are treated as dependentones similarly to other binary variables applied in the basic algorithm —see relation (29).

Galan and Grossmann24 calculated the optimum network for example2 of the goal function of 1.69×106 [$/year]. The technologies selected are as

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s3

s1

s2

t1tt1

t2tt3

e

5

15

4020

t3tt3

2.4

7.26

12.74

37.6

24.86

Fig. 9. Optimum network for example 2 from Bergamini et al.30

follows: technology 1 for process 1 and technology 3 for processes 2 and 3.The network, i.e., structure and parameters, had not been given in thework by Galan and Grossmann.24 Bergamini et al.30 presented the optimalsolution shown here in Fig. 9. They did not give the goal function and wecalculated it as 1,692,760 [$/year]. The technologies chosen are identical tothose from Galan and Grossmann.24

Using small number of function evaluation (NEL = 1000, NIL =10,000), that requires only 66 CPU seconds per run at PC with processorIntel Centrino 1.5GHz, we obtained the better solution with the goalfunction equal to 1,650,578 [$/year]. For larger CPU time we were able tocalculate the network shown in Fig. 10 with the cost of 1,647,392 [$/year].Both solutions use the same treatment technologies as those in the networkby Bergamini et al.30 The analysis reveals that the improvement has beenachieved due to recycle P1 → P3 → P1, which do not exist in the solutionof Bergamini et al.30

The example shows that simple stochastic approach was able to findthe better solution than the sophisticated global optimization algorithmapplied by Bergamini et al.30 and also better than the solution of Galanand Grossmann.24

Several other problems from the literature were also calculated withthe simultaneous method. Only in one case we haven’t reached theglobal optimum network though the solution differs only slightly from theoptimum.

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s2

s1

s3

t1tt1

t2tt3

e

15

5

7.263

40

20

t3tt3

12.737

27.262

19.44

32.177

Fig. 10. Optimal solution to example 2 calculated by ARS-based approach with goalfunction of 1,647,392 [$/year].

Symbols

C, concentration of contaminantF , flow rateL, contaminant mass load

Superscripts

min, minimummax, maximum

i, contaminant

Subscripts

e, disposal sitein, inlet

out, outlets, source of wastewatert, treatment

tt, treatment technology

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Abbreviations

ARS, adaptive random searchMILP, mixed-integer linear programming

MINLP, mixed-integer nonlinear programmingNLP, nonlinear programmingTP, treatment process

TWN, total water networkWWTN, wastewater treatment network

WUN, water usage networkHEN, heat exchanger network

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Appendix

This appendix gives the brief description of the algorithm of basic version ofadaptive random search optimization applied to WWTN. We will explainit using standard NLP problem without equality constraints.

min GF(X)s.t.

xli ≤ xi ≤ xu

i ; i = 1, . . . , P

gj(X) ≥ 0; j = 1, . . . , J

Given: initial point X0; final search regions δfi ; initial search region δ0i =xu

i −xli; number of external loops — NEL; number of internal loops — NIL.

The main steps of ARS algorithm are:

(1) Calculate from data search region contraction parameter βi:

βi =(∂f

i

δ0i

)1/NEL

(A.1)

(2) Set external loop counter k at 1(3) Set internal loop counter l at 1(4) Calculate Xl from (A.2) (with X∗ = X0, δk

i = δ0i for k = 1)

xki = x∗i + riδ

k−1i ; i = 1, . . . , p (A.2)

where ri is random number of uniform distribution from the range(−0.5, 0.5)

(5) Increase counter l by 1. If l = NIL + 1 go to (9).(6) Check the feasibility of inequality constraints gj(X). If at least 1

constraint is not met go to (4).(7) If GF(Xl) is better than GF(X∗), update X∗ = Xl.(8) Go to (4).(9) Update δk

i according to (A.3)

δki = βiδ

k−1i (A.3)

(10) Increase counter k by 1 up to NEL value and go back to (3).

Notice that this ARS algorithm differs from the original one by Luus andJaakola52 mainly in definition of parameter βi that contracts search regiongradually. We proposed to calculate it from (A.1) instead of fixed value

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identical for all variables. Contraction parameters are variable dependent.They depend on search region sizes δ0i , δ

fi . Final region sizes δfi can be

estimated by the user based on physical interpretation of correspondingvariables. Also, good results are obtained using δfi of order 10−2/10−4 forall variables.

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