Integration of environmental aspects in modelling and optimisation of water supply chains

2018·3.5 MB·English

by  Mariya N. Koleva

ScienceoftheTotalEnvironment636(2018)314–338
ContentslistsavailableatScienceDirect
Science of the Total Environment
journal homepage: www.elsevier.com/locate/scitotenv
Integration of environmental aspects in modelling and optimisation of
water supply chains
MariyaN.Kolevaa,b,AndrésJ.Calderóna,DiZhanga,CraigA.Styanb,LazarosG.Papageorgioua,*
aCentreforProcessSystemsEngineering,Dept.ofChemicalEngineering,UCL,TorringtonPlace,LondonWC1E7JE,UnitedKingdom
bSchoolofEnergyandResources,UCLAustralia,220VictoriaSquare,Adelaide,SouthAustralia5000,Australia
H I G H L I G H T S G R A P H I C A L A B S T R A C T
• The proposed framework encom-
passes climatic, governmental and
sustainabilityconstraints.
• Costandwatersupplyreliabilityare
addressed through multi-objective
optimisation.
• Trade-offbetweencostandreliability
isstudiedthroughgametheoryusing
Nashequilibrium.
• The framework is applied to a case
studybasedonAustralia.
A R T I C L E I N F O A B S T R A C T
Articlehistory: Climatechangebecomesincreasinglymorerelevantinthecontextofwatersystemsplanning.Toolsare
Received10January2018 necessarytoprovidethemosteconomicinvestmentoptionconsideringthereliabilityoftheinfrastructure
Receivedinrevisedform27March2018 fromtechnicalandenvironmentalperspectives.Accordingly,inthiswork,anoptimisationapproach,formu-
Accepted29March2018 latedasaspatially-explicitmulti-periodMixedIntegerLinearProgramming(MILP)model,isproposedfor
Availableonline27April2018
thedesignofwatersupplychainsatregionalandnationalscales.Theoptimisationframeworkencompasses
decisionssuchasinstallationofnewpurificationplants,capacityexpansion,andrawwatertradingschemes.
Editor:SimonPollard
Theobjectiveistominimisethetotalcostincurringfromcapitalandoperatingexpenditures.Assessment
ofavailableresourcesforwithdrawalisperformedbasedonhydrologicalbalances,governmentalrulesand
Keywords:
sustainablelimits.Inthelightoftheincreasingimportanceofreliabilityofwatersupply,asecondobjec-
Hydrologicalbalances
tive,seekingtomaximisethereliabilityofthesupplychains,isintroduced.Theepsilon-constraintmethod
Watersupplychains
SpatiallyexplicitMILP isusedasasolutionprocedureforthemulti-objectiveformulation.Nashbargainingapproachisappliedto
Supplyreliability investigatethefairtrade-offsbetweenthetwoobjectivesandfindtheParetooptimality.Themodels’capa-
Gametheory bilityisaddressedthroughacasestudybasedonAustralia.Theimpactofvariabilityinkeyinputparameters
istackledthroughtheimplementationofarigorousglobalsensitivityanalysis(GSA).Thefindingssuggest
thatvariationsinwaterdemandcanbemoredisruptiveforthewatersupplychainthanscenariosinwhich
rainfallsarereduced.Theframeworkscanfacilitategovernmentalmulti-aspectdecisionmakingprocesses
fortheadequateandstrategicinvestmentsofregionalwatersupplyinfrastructure.
©2018PublishedbyElsevierB.V.
1. Introduction
Concurrent population growth, economic development and cli-
*Correspondingauthor.
E-mail address:[email protected](L.G.Papageorgiou). matechangearethemaindriversfortheacuteandchronicwater
https://doi.org/10.1016/j.scitotenv.2018.03.358
0048-9697/©2018PublishedbyElsevierB.V.
M.N.Kolevaetal. /ScienceoftheTotalEnvironment636(2018)314–338 315
shortages (Dizikes, 2016; Morrison et al., 2009; US Environmental addressanadequatemanagementsystem,thecurrentworkconsid-
Protection Agency, 2016). To mitigate and adapt to the changes, erswatersupplychaindesign,entailingresourcesallocationunder
authoritiesexaminestrategicoptionstoenhancethesupply-demand hydrologicalbalancesandwatertrading,andsupplyreliabilityboth
management for a long term resilience. Planning for 15–35years assingleandmulti-objectiveoptimisationframeworks.
ahead by water industries ensures adequate facilities and infras- Muchattentionhasbeenpaidtooptimisationtechniquesinwater
tructure in place to maintain the security of supplies throughout supply chains (WSC) as they provide a systematic way of making
thoseperiods.Thegapbetweensupplyanddemandcanbefilledby decisionsonfutureinvestments.Rayetal.(2010)addressedtheissue
diversifyingtheportfolioofoptionsforwatersupply.Forinstance, through a linear programming model for the minimum cost con-
alternative sources of water, investing in storage and production figuration of future water supply, wastewater disposal, and reuse
capacities, expanding market participants and water quality grade optionsforthecityofBeirut.Kolevaetal.(2016,2017)proposedlin-
trading options, alongside with interconnectivity and distribution earandnon-linearprogrammingmodelsfortheoptimaldesignof
lossesminimisation,shouldbeincludedinthelist(Pricewaterhouse waterandwater-relatedtreatmentprocesses.Lietal.(2009)devel-
Coopers,2010). opedamulti-stream,multi-reservoirandmulti-periodmixedinteger
Besides the conventional surface water sources diverted from linearprogramming(MILP)modelthatwasintegratedintoaninex-
rivers and lakes, and groundwater extracted from aquifers, non- act multistage joint-probabilistic programming to investigate the
conventionalsourcessuchasseawater,brackishandrecycledwater decision under uncertainty and surplus-flow diversions. Kondili et
have been taking place in the source mix for water provision. al.(2010)presentedamathematicalframeworkforthewatersup-
Althoughtreatedwastewaterisnotthepubliclyacceptedsourcefor ply design taking into consideration various sources and users as
drinking,itisessentialforother,non-potableapplicationsinorderto wellaspossibleconflictingdemandoveratimeperiod.Themodel
meetoveralldemand.Ontheotherhand,non-conventionalsources wasappliedtoacasestudyfortheAegeanIslands.Liuetal.(2010,
require more extensive treatment and therefore, more expensive 2012,2011),Liu(2011),Padulaetal.(2013)andPadula(2015)pro-
purificationtechniques,hence,theyoftenserveasaback-upduring posedmathematicalformulationsfortheminimisationofproposed
prolongdroughts(PricewaterhouseCoopers,2010). installations of plants, storages, pipelines applied on specific case
Asalimitedresource,waterusagebyanentitycanaffectitsavail- studies.Matrosovetal.(2015)lookedatmulti-objectiveoptimisa-
abilitytoanother.Conflictandcompetitionamongentities,whenit tionforwatersuppliesfocusedonLondonandbasedon4-dominance
comestoresources,islikelytoarisehence,acoordinatedallocation non-dominatedsortinggeneticalgorithmsandsimulation.Saifand
system is sought. Such a system is represented by water markets, Almansoori (2014) suggested a multi-period MILP model for the
operatingontheprincipleof‘capandtrade’systemwhere desalinationsupplychainswithdecisionsonlocationsfornewand
extendedplants,storagesandpipelines.Al-NoryandGraves(2013)
• thecapillustratesthewateravailableforsustainableextraction proposed a mathematical programme for the design of desalina-
• market participants hold water abstraction rights or licences tionsupplychainstakingintoconsiderationlocationsofnewplants
whichareapartofthetotalavailablepool installations. Guerra et al. (2016) and Saif and Almansoori (2016)
• the rights and the allocations in every season can be traded integrated water management in different supply chain contexts.
amongparticipants Loucksetal.(2005)andJoshiandJoshi(2016)contributedwithcom-
• thetradingpriceissetbytheparticipantsinthewatermarket prehensive insights into water resources planning, modelling and
management,andadvancesinsupplychain.
Various works on modelling of water resources allocation and
Such water trading schemes exist in Spain, Chile, South Africa, pricinghavebeenpublishedinBrebbia(2015).Heydarietal.(2015)
Australia,UKandsomestatesintheUnitedStatesofAmerica.Water developed an MILP model for the multi-purpose reservoirs opera-
market participants may include users such as industry, irrigation tion. Veintimilla-Reyes et al. (2016) introduced a spatio-temporal
operators,andurbanwaterutilities(AustralianGovernment,2016a). mixedintegerformulationforwaterallocation.Yildiranetal.(2015)
In a regulated market, the availability of water would govern the formulated an MILP model for the short-term scheduling of water
extentoftradeofanentitywithanotherentity.Athoroughwayof reservoirsconsideringday-aheadmarketprices.Heetal.(2015)pro-
assessing water availability is by taking into account the environ- posedan MILPmodelandappliedBendersdecompositionmethod
mentalflows,suchasprecipitation,evaporation,run-off,andinfil- fordynamicresourceallocationandtrafficassignmentinevacuation
trationwhichcanbeexpressedbyaninflow-outflowwaterbalance networks. Hughes (1976) developed an optimisation framework,
foraparticularsysteminaregion. based on integer programming, on solving algorithms for water
Affordableandsecurewatersupplyiscrucialforthedomesticand resources planning problems. Li et al. (2016) presented a stochas-
industrial conductofdailyactivities(Zhuetal., 2015).Watersup- ticquadraticmodelapplicabletodiscrete,fuzzyandrandominput
plyreliabilitycanbedefinedastheprobabilityofmeetingdemand data for water resources allocation with a case study on Heihe
ortheprobabilityofnotmeetingdemandsubtractedbyone(Hawk, Riverbasin,China.Roozbahanietal.(2015)proposedanapproach
2003).Overatimeperiod,reliabilitybecomesthefrequencyorthe and a mathematical model for the allocation of water resources
quantity of supply shortfalls measured against demand. Temporal amongstakeholders.Zengetal.(2014)constructedamodelbased
reliabilitymostofteninvolvessystemfailures,whicharerelatedto on inexact credibility-constrained programming method to inves-
mathematicalproblemswithhighergranularity,forinstance,oper- tigatetheefficiencyofwatertradingundermultipleuncertainties.
ation problems. For designing the water supply chain, the focus Britzetal.(2013)proposedaMultipleOptimisationProblemswith
of interest is a quantification of how much of the demand will Equilibrium Constraints (MOPEC) for hydro-economic river basin
not be met. Hence, the choice of reliability assessment in the cur- modelstoaccountforthedecentralisedaccesstowateruse.Qureshi
rentworkisvolumetric.Governmentsandwaterentitiesarefacing et al. (2013) introduced a mathematical programming model with
numerouschallengesprovidinganadequatesupplyreliability.Such an application on agricultural water use in Murray-Darling Basin,
challenges are climate change, population growth, environmental Australia.Rinaudoetal.(2016)proposedaprice-endogenousmodel
regulations,decayinginfrastructureandcalamities.Enhancingtrad- forthetradingactivityandequilibriumpricesinurbanwatermar-
ing,andexpandingstorageandtreatmentcapacitieswouldincrease kets. Blanco and Viladrich-Grau (2014) analysed the outcomes of
supply reliability (Shamir, 1987; Goulter, 1995; Zhu et al., 2015). irrigatingwatertradingschemethroughanonlinearmathematical
Therefore,satisfactorywatermanagementplanninganddesignhave programmingmodel,appliedtoacasestudyinSpain.Erfanietal.
tobeinplace(CaliforniaUrbanWaterAgencies,2012).Inorderto (2014) presented an optimisation model for short-term pair-wise
316 M.N.Kolevaetal. /ScienceoftheTotalEnvironment636(2018)314–338
spot-markettradingofsurfacewaterrights.Itisbasedonanode-arc waterdistributionnetworksusingmultiobjectiveoptimisationand
multi-commodityapproachfollowingatransactiontrackingmethod gametheory.
(Erfani et al., 2013). Peng et al. (2015) proposed an optimisation To the best of the authors’ knowledge, there is no work which
modelforwatertransferdecisionmakingprocessconsideringshort- integratesindetailtheconceptsofsupplychains,allocations,reli-
agesinreservoirsofboth,recipientsanddonors. ability and game theory. The current work addresses this gap by
Reliabilityofwatersupplyhasincreasinglybecomethefocusofa not only combining all the aforementioned separate concepts but
numberpublications.Damelinetal.(1972)firstintroducedthecon- alsoconsideringtheentirewatercyclewithlegislativeregulations
cept of reliability of water supply in a simulation context. Barlow altogetherinamixedintegerlinearprogramming(MILP)formula-
(1984)presentedahistoricalangleofmathematicaltheoryofrelia- tion.Thepaper,thus,aimsatinvestigatinghowtoconsolidatethose
bility.Pengetal.(2015)presentedamathematicalformulationfor multiple-aspects into a single optimisation framework. A multiple
waterallocationsaccountingforreliability.Reliabilityhasalsobeen number of sources, users, trading and time periods are geograph-
thefocusofnumerousworkswhichconsideritalongsidecalamities icallyconsidered.Thelocationsandcapacitiesforsurface,ground-
andchangingclimate(WangandAu,2009;Simonitetal.,2015;Clark water, seawater plants and the trading volumes of each source
etal.,2015;Yooetal.,2016). amongregionsaretobeoptimised.Then,amulti-objectiveoptimi-
Multi-objectiveoptimisationapproacheshavebeenthefocusofa sationisformulatedforthesimultaneousminimisationoftotalcost
largenumberofliteratureworks.Pokharel(2008)wasoneofthefirst andmaximisationofreliabilityusing4-constraintmethod.Inorder
workstousemulti-objectiveoptimisationinsupplychainnetwork to identify the fairest operating point along the Pareto curve, we
design where two-objective decision-making model for the choice propose the implementation of game theory, specifically, the bar-
ofsuppliersandwarehousesforasupplychainnetworkdesignwas gainingapproach.Aglobalsensitivityanalysis(GSA)isimplemented
proposed.Amodeoetal.(2009)integratedevolutionaryalgorithms toaddresstheimpactofuncertaintyassociatedwithinputdataon
andsupplychainsimulationforthemaximisationofcustomerser- thewatersupplychain.Fourparameterswereselectedforthispur-
vicelevelandthetotalinventorycost.LiuandPapageorgiou(2013) pose as follows: rainfalls, capital investments, facilities efficiency,
developed an MILP model for cost, responsiveness and customer anddemand.Therestofthisworkisstructuredasfollows:Section2
servicelevelusing4-constraintmethodandlexicographicminimax sets out the problem statement whose mathematical formulation
methodassolutionapproaches.ChenandAndresen(2014)applieda is presented in Section 3. The applicability of the model is inves-
weighted-sumapproachminimisingcosts,emissions,andemployee tigated in a case study, described in Section4, followed by results
injuriesinasupplychain. anddiscussioninSection5.Finally,concludingremarksaremadein
Whenmorethanonefactordeterminesthedesignofthewater Section6.
managementsystem,besidesanoptimal,afairstrategicdecisioncan
betakenbyapplyinggametheory.Gametheorycanbeutilisedfor
variousapplications,suchasengineering,lifesciences,management 2. Problemstatement
and economics. Games can be collaborative, when the best strate-
gies for the players are to cooperate, and competitive, when the The supply chain problem at hand entails strategic decisions
players can maximise their outcome if they do not take into con- for the allocation of water resources, procurement and treatment
siderationtheoutcomesoftherestoftheplayers.Gamescanalso ofsourcestypes,locationsandcapacitiesexpansionsfordamsand
besimultaneousandsequential,whenthedecisionsoftheplayers treatmentplants,tradingdirectionsandvolumes.
are taken at the same time or one after another, like in a leader- Ageographicalareaisconsideredwherewaterdemandscanbe
followertypeofgame.Theformeroftenimpliestheinformationis metbysurfacewater,groundwaterandseawater.Optionssuchas
not well known and in cases of the latter, normally the follower reclaimed water and individually collected rain water are disre-
makes a decision based on the action of the leader. This leads to gardedinthiswork.Theareaisdividedintosub-regions,orstates,
dealingwithperfectandimperfectinformationgames.Recentworks basedontheirfederalgovernanceandautonomy.Thewaterdemand
on game theory in mixed integer programming have been classi- for each territory is estimated according to the population predic-
fiedqualitativelybasedontheaforementionedapplications.Atypical tionsandconsumptionpatternspercapita.Additionally,thewater
leader-follower game is the Stackelberg game which has been the demand varies seasonally, peaking in summer and plummeting in
chosenstrategyindifferentliteraturesources(YueandYou,2014; winter.Springandautumnseasonsarecharacterisedwithmoder-
Bard et al., 2000; Yang et al., 2015; Pita et al., 2010; Yin, 2013). ateconsumptionvolumes.Regulatedwaterservicesofeveryregion
Zhang et al. (2013) developed mathematical models for fair elec- areprovidedbywatersupplierstomeettheurbanwaterdemand,
tricity pricing microgrid, scheduling, planning, and in Zhang et al. whichoccursfromresidential,commercial,municipalandindustrial
(2017) - carbon capture and storage following cooperative Nash usage.Astatemightnotbeabletomeetitsregionaldemandconse-
approach(Nash,1950).Nashequilibriumhasbeenappliedinsupply quently,itshouldidentifyastrategyfordealingwithwaterdeficit.In
chainsandscheduling(Zamarripaetal.,2013;Gjerdrumetal.,2002; casesourcewaterisindeficit,tradingamongregionsisconsidered.
Banaszewskietal.,2013;PiraandArtigues,2016;Ortiz-Gutierrezet Onlysurfaceandgroundwatercanbetraded.Ontheotherhand,if
al.,2015;Tusharetal.,2014).Generalsupplychaingametheoryand storageorproductioncapacitiesarenotsufficient,optimaldecisions
transferpriceshavebeencoveredbySimchi-Levietal.(2004)and forthecapacityandlocationfortheexpansionofexistingplants,and
Rosenthal (2008). Additionally, Shelton (1997) and Tambe (2012) installationofnewdamsandplantsaremade.
havepublishedexhaustivecompilationbooksongametheory,secu- Waterisdivertedfromlakesandreservoirs,andabstractedfrom
rity and markets. Madani (2010) compiled a literature review on aquiferstakingintoaccounttheseasonalhydrologicalcyclesandsus-
gametheoryconceptsappliedtowaterresourcesmanagement.The tainableyields(withdrawals)withintheterritories.Waterbalances,
research suggested non-cooperative game theory can be a power- or budgets, are performed over the total regional available water
fultoolformanagingrealconflictswithoutthenecessityofaccurate storages. Reservoirs, dams, ponds, lakes and groundwater aquifers
quantitative information. Sechi et al. (2011) suggested a decision are referred to as storages. The inflows into the storages are the
makingtoolusinggametheorytodeterminefairwaterpricingwith seasonalprecipitations,run-off,streamflowsandrechargewhilethe
sustainability principles. Daumas (2009) proposed a mathematical outflows consist of evaporation, discharge and diverted/abstracted
modelforthetheoryofcooperativegamesfortransferableutilities volumes. Precipitation refers to the rainfall that falls directly onto
(TU). Souza Filho et al. (2008) investigated game theory on water the storage area. Run-off represents excess of moisture turning
users’strategicbehaviour.NikjoofarandZarghami(2013)simulated intothestreamflowfromthecatchmentsordrainagebasinstothe
M.N.Kolevaetal. /ScienceoftheTotalEnvironment636(2018)314–338 317
storage.Streamflowsrefertotheriverflowingintothestorage.Evap- Thus,theproblemdescriptionwithkeyparameterscanbestated
oration is the direct evaporation from the storage surface, while below.
dischargereferstotheriverstreamleavingthesystem.Diversions
Given:
arethewaterflowswithdrawnforhumanusage.Groundwaterdis-
chargeorseepageisignoredduetobeingaminorcomponentand • geographicaldivisionsintoregions/states/territories
due to the scarce historical data available. As a matter of conve- • planningtimehorizon,e.g.a25-yearhorizon
nience,thestreamflowsfordifferentriversystemsinaregionhave • water sources, i.e. surface water (sw), groundwater (gw),
been summed up. It is assumed that dead storage comprises 10% seawater,ordesalinatedwater(dw),etc.
of the water storage capacity. Further, by a rule of thumb, 10% of • final water uses, i.e. urban (uw), rural (rw), etc., and sea-
the rainfall in drainage basins infiltrates to become groundwater sonaldemandoverplanninghorizon
inflow. Climatic data is extracted for the entire planning horizon • regionalandseasonalclimaticdata,i.e.precipitation,evap-
reflecting fluctuations in the weather conditions and therefore, el oration,run-off,streamflows
NiñoandlaNiñaevents,whichoccurevery5–7years.Oceansand • initialwaterstoragesindrainagebasinsandreservoirs
seas are not taken into account in water budgets due to their • geographicaldistribution,capacities,operatingefficiencies,
abundance. andoperatingandcapitalcostparametersofexistingand
In every region there are rights for maximum water sources potentialdamsandplants
diversions/extractions.Theyarecalledtargetallocations,orentitle- • maximumallocatedwatersourcesperend-use,i.e.entitle-
ments, and apply for surface water and groundwater. In a season ments
whenavailabilityinstorageissufficient,theallocationsinaregion • tradingtopologyandpricesoptions
can reach target allocations. The amount of water that has been • inflationanddiscountfactors
allocated for withdrawal and has not been used in a given year • regionalsustainablediversions/abstractions
can be rolled to the next year unless regulations oblige a return • penaltycostsfornotmeetingdemand
to the abstraction basin. After the resources are withdrawn, they
areprocessedinsurfacewatertreatmentplants(SWTP),groundwa- Determine:
tertreatmentplants(GWTP)andseawaterdesalinationplants(SDP) • availablewatersourcesfordiversion/abstraction
whichoperatewithdifferentefficiencies.Then,theproductwateris • procurementrateforeachwatersourceandend-usewater
distributed for urban usage, after which it is assumed 60% of that productionrate
water is collected as sewerage. The wastewater is then treated in • trading and carry-over flowrates of surface and
wastewatertreatmentplants(WWTP)andreturnedasrecharge.It groundwater
mustbenotedthatnodecisionsaremadewithrespecttoWWTPs • watersupplyreliability
and the concept is introduced merely to close the water cycle. • location and capacities of new dams and plants installa-
Ascheme,representingtheproblem,isillustratedinFig.1. tions,andexistingplantsexpansions
Fig.1. Schematicrepresentationofthewaternetworksystem.
318 M.N.Kolevaetal. /ScienceoftheTotalEnvironment636(2018)314–338
So as to minimise the annualised total cost for operating and whereRain isthedirectrainfalltothereservoir,Runoff istherun-off
igtq igtq
buildingthenetworkproposedsubjecttoenvironmental,opera- seepingintothereservoirsandRiverrepresentsthestreaminflowsto
tional,logicalandeconomicconstraints.Initially,thesupplychain igtq
thestorage.Itisassumedthatrun-offoccurringinoneregionfills
problemisformulatedasaspatially-explicitmulti-periodMixed
thereservoirsinthesameregionandnootherneighbouringregions.
IntegerLinearProgramming(MILP)model.Then,asecondobjec- A proportion of the rainfall which falls onto the mainland, LRain,
tiveisaddedforthemaximisationofreliability,hence,themodel igtq
infiltratesintothegroundandbecomesaninflowforaquifers.
becomesmulti-objective.
Rigtq=rinfl • LRaigitnq, ∀i=“gw”,g,t,q (2)
3. Mathematicalformulation
where rinfl is the fraction of the rainfall that infiltrates. Simultane-
In this section the mathematical formulation is presented first, ously,thetotaloutflowsfromthesystemaretheevaporationlosses,
as a single objective problem, in Section 3.1, where key con- Ligtq,outflows,Oigtqandallocatedwater,Aigtq.Itisassumednoaddi-
straints are the water cycle balances, procurement and produc- tionallossesoccurforboth,surfacewaterandgroundwatersystems.
tionconstraints,logicalconstraints,supplyreliability,andoperating TheinflowsandoutflowsareillustratedinFig.2.Theseasonaland
and capital expenditures. The objective is to minimise the total yearlyformulationsareshowninEqs.(3)and(4),respectively.
cost for the entire region for the planning horizon. Next, reliabil-
ity of supply is added as a second objective function and solved DSigtq|i=“sw”+WSigtq=DSigt,q−1|i=“sw”+WSigt,q−1+Rigtq+RCigtq
usingan4-constraintmethodinSection3.2.1,andgametheoryin +AC −L | −A
igtq igtqi=“sw” igtq
Section 3.2.2. A multi-objective optimisation for minimising total
−O | , ∀i∈LW,g,t,q>1 (3)
cost and maximising reliability of supply is going to reveal the igtqi=“sw”
extentthesupplychainnetworkdesignisinfluencedbybothfactors.
Additionally,gametheorywillprovideafairtrade-offbetweenthe
twoobjectives.
DSigtq|i=“sw”+WSigtq=DSig,t−1,q|i=“sw”,q=4+WSig,t−1,q|q=4+Rigtq
3.1. Monolithicapproach +RCigtq+ACigtq−Ligtq|i=“sw”−Aigtq
−O | , ∀i∈LW,g,t,q=1 (4)
igtqi=“sw”
3.1.1. Hydrologicalbalances
Theestimationofwateravailabilityinstoragerestsontheinflows
whereLWisasetcontainingtheinlandwatersources,i.e.surface
into the system, R , recharges, RC and the total storage from
igtq igtq waterandgroundwater.Surfacewaterstoragesincludedams,and
thepreviousseason,Sigt,q−1.Rigtqrepresentsasummationofrainfall, naturalstorages,i.e.lakesandwetlands.Inthiswork,acumulative
run-offandstreamflowsforsurfacewater,andinfiltratedrainfallfor
termtorefertoboth,human-madeandnaturalstorages,isstorage
groundwater,showninEqs.(1)and(2).
or reservoir. Aquifers are the only storage for groundwater which
occursinitsnaturalform.Thesumofdams’storage,DS ,andnat-
Rigtq=Raigitnq+Ruigntqoff +Riivgteqr, ∀i=“sw”,g,t,q (1) uralstorage,WSigtqadduptothetotalstorage,Sigtq,showigntqinEq.(5).
Fig.2. Inflows(rainfall,run-off,riverstreamflows,recharges)andoutflows(evaporation,withdrawals,outflows)fromareservoirsystem.
M.N.Kolevaetal. /ScienceoftheTotalEnvironment636(2018)314–338 319
Fig.3. Visualisationofyearandseasonaltimediscretisation.Thesequenceofseasonsqdependsonthestartandendofthefiscalyeartagovernmentuses.
Sigtq=DSigtq|i=“sw”+WSigtq, ∀i∈LW,g,t,q (5) gradewaterw,Pwgtq,tomeetpopulatedcentresdemand,Dwgtq.Italso
takesintoaccountQigg(cid:5)tqandQig(cid:5)gtq,whicharethetradedflowssent
toandreceivedfromotherregions,respectively.
where at t = 0 and q = 0 the storage is the summation of the
initial reservoirs’ and lakes’ storages. A representation of the time (cid:2) (cid:2)
discretisationinyears,seasonsandtheirsequenceisdemonstrated Digtq+ Qigg(cid:5)tq−PDigtq=Pigtq+ Qig(cid:5)gtq, ∀i,g,t,q (7)
inFig.3. g(cid:5)∈gigg(cid:5) g(cid:5)∈gig(cid:5)g
Themaximumnaturalstoragecapacityperregion,WSmax,should
ig
not be exceeded in any year t and season q in order to prevent
overflows(Eq.(6)). where gigg(cid:5) and gig(cid:5)g define the allowed directions of flow from
region to region. Regions with hydrological or physical connectiv-
ity are selected for trading. When a demand cannot be met by
WSigtq≤WSimgax, ∀i=“sw”,g,t,q (6) the treatment plants, water flows, PDigtq, are allowed to compen-
sate for the shortage. These flows are penalised in the objective
function.
3.1.2. Supply-demandbalances
Fig.4delineatesthewatersupplychainflowsforgivenregionsg 3.1.3. Procurementconstraints
andg(cid:5). Theamountsofdivertedsurfacewaterandextractedgroundwa-
Eq.(7)representsthoseinteractionsthroughaglobalmassbal- teraredeterminedbytheallocatedwaterrights,orallocationsA ,
igtq
anceequationwhichentailsthewatertypeflowsi(I = S∪W)at aregiongisallowedtowithdrawinyeartandseasonq.Thecarry-
every node of the WSC: withdrawals of raw water s, Psgtq, accord- overvolumes,ACigtq,aretheamountsrolledoverfromoneseason
ingtopurificationplantintakedemand,Dsgtq,andproductionoffinal tothenextafterensuringenoughwaterissetasideformeetingthe

Fig.4. Asupply-demandflowdiagram,includingwithdrawals,production,distributionandtradingbetweenregiongandregiong(cid:5).
320 M.N.Kolevaetal. /ScienceoftheTotalEnvironment636(2018)314–338
regionaldemand.Theseasonalandannualexpressionsareshownin 3.1.5. Capacityconstraints
Eqs.(8)and(9),respectively. At a given time t, every region g and plant p have produc-
tion capacity, TCAPgpt, which is a limiting factor for the plant feed
ACigtq=ACigt,q−1+Aigtq−Pigtq, ∀i∈LW,g,t,q>1 (8) flow,Vigptq.Therefore,theeffluentshouldnotexceedthetotalplant
capacity,demonstratedinEq.(17).
ACigtq=ACig,t−1,q|q=4+Aigtq−Pigtq, ∀i∈LW,g,t,q=1 (9) (cid:2) (cid:2)4ii(cid:5)·Vigptq≤TCAPgpt/Qmax,∀g,p∈PGg,t,q (17)
i∈S∩SPpi(cid:5)∈W
Thecentralprinciplebehindcarry-overisthatunusedwatercan
becarriedover,butitmustnotdisplaceinflowsthatsupportnew
where4ii(cid:5) istheproductionyield,dependingonwatersourcei.PGg
allocations.OnlyinlandwaterLWisaffectedbyallocationrulesdue
is a subset of g which contains the operating plant p in region g,
totheassociatedfinitenesswiththoseresources.Theallocationsare and Qmax is the number of seasons used to obtain seasonal capac-
calculatedonthebasisofthewaterrights,ortheentitlementsent ,
igt ity.Providedmorewaterhastobeprocessedtomeetdemand,the
whicharethemaximumwatervolumesthatcanbewithdrawnin
total capacity will increase by installing new plants or expanding
yeart.
old ones. A binary variable I is assigned for the installation of
gplt
(cid:2) newplantpwithcapacitylinregiongattimet.Whenanincrease
Aigtq≤entigt, ∀i∈LW,g,t (10) incapacityisnecessary,Igplt isactivatedandequals1,otherwiseit
q equals0.Anotherbinaryvariable,E ,isassignedfortheexpansion
gplt
ofexistingplants,whichoperatesonthesameprinciple.
The entitlements are treated as parameters based on the
(cid:2)
aInsstuhmispwtioornk,thitaitstahsesyumareedauthtoamtmataicxaimllyumren1e0w0%edoffrtohmeeynetairtlteomyeenatrs. TCAPgpt=TCAPgp,t−1+ icappl•Igpl,t−ictp|p∈NP
(cid:2) l
canbereceivedinayeart.
Additionally,alimitationisconsideredthataregiongcantrade + ecappl•Egpl,t−ectp, ∀g,p∈PGg,t (18)
waterwithotherregionsg(cid:5)onlywhengsatisfiesitsregionaldemand l
fromtheseasonalallocationsavailable(Eqs.(11)–(13)).
where ictp and ectp are the respective construction and expansion
(cid:2) times for plant p, ecap represents the available options of capac-
Qigg(cid:5)tq≤entigt•Bigtq, ∀i∈LW,g,t,q (11) ityexpansionlandicappl -thecapacityinstallationoptionsofnew
pl
g(cid:5)∈gigg(cid:5) plants(NP)throughouttheplanninghorizon.
Atmostonecapacitylevellfromagivennumberofoptionscan
Pigtq≤Digtq+entigt•Bigtq, ∀i∈LW,g,t,q (12) bechosen(Eq.(19)).
(cid:2)(cid:2)
Pigtq≥Digtq−entigt•(1−Bigtq), ∀i∈LW,g,t,q (13) Igplt≤1,∀g,p∈PGg∩NP (19)
l t
where B is a binary variable, equal to 1 when demand for a
igtq
sourceiinaregiongislowerthanthesupply.Further,thediverted Egplt isabinaryvariablethatisactivewhenaplantisexpanded
or abstracted volumes should be within the limit of sustainable whichcanhappenuptoEmaxnumberoftimes(Eq.(20)).
withdrawals,recommendedbylocalgovernments(Eq.(14)). (cid:2)(cid:2)
(cid:2) Egplt≤Emax,∀g,p∈PGg (20)
P ≤SPmax, ∀i=LW,g,t (14) l t
igtq igt
q
Onlyonecapacitylevellfromagivennumberofoptionscanbe
whereSPmaxisthemaximumsustainablewaterdiversionorabstrac- chosentobeexpandedatatime(Eq.(21)).
igt
tioninayeart.Theselimitsareinplacetoensurethewithdrawnvol- (cid:2)
umeswouldnotexhibitadetrimentalimpactontheenvironment. Egplt≤1,∀g,p∈PGg,t (21)
l
3.1.4. Reliabilityofwatersupply
It is a common practice in many cities to be under agreed vol- Expansions can occur only after a plant has been installed,
umetric or temporal water restrictions, or an agreed reliability of imposedbyEq.(22).
supply. The reliability of water supply is calculated based on the (cid:2) (cid:2) (cid:2)
volumetricshortage,whenthedemandexceedssupply.Thewater Egplt≤ Igplt(cid:5),∀g,p∈PGg∩NP,t (22)
supplyreliability,WSRigtq,isshowninEq.(15). l l t(cid:5)≤t−ictp
WSR =1−PD /dem ,∀i∈W,g,t,q (15) Thesurfacewaterthatiskeptindams’storageshouldbelessor
igtq igtq igtq
equalthanthecurrentexistingdams’capacity.
wherePD istheimportinaperiodofwatershortageanddem
istheurbaigntqwaterdemand.Thenormalisedreliabilityfortheentiigrteq DSigtq≤DAMgt,∀i=“sw”,g,t,q (23)
countryisanaverageoftheregionalreliabilities(Eq.(16)).
(cid:2)(cid:2)(cid:2)(cid:2) InEq.(23),DAMgt isthetotalcapacityofdamsineveryregiong
WR= WSRigtq/(Gmax•Tmax•Qmax) (16) attimet.AsDSigtqisthewatervolumerelatedtotheabilitytowith-
drawwaterfromit,thevolumeofwaterwhichstagnatesshouldbe
i∈W g t q
considered.Hence,simultaneously,thestorageshouldnotfallbelow
whereWRisthenormalisedreliability,Tmaxistheplanninghorizon thedeadstorageofthereservoir,expressedinEq.(24).
period,Qmax-thenumberofseasonsandGmaxisthetotalnumberof
regions. DSigtq≥dsf•DAMgt,∀i=“sw”,g,t,q (24)
M.N.Kolevaetal. /ScienceoftheTotalEnvironment636(2018)314–338 321
wheredsfisafactorforthetypicaldeadstoragewhichremainsin wherecaplant andcapexp arethecapitalcostsassociatedwitha
pl pl
dams.Thetotaldamscapacityinaregionequalsthecapacityofthe plantpanditsrespectiveinstalledorexpandedcapacityl.Thetotal
existingdamsandthenewlybuiltdams.Thedecisionofbuildinga capitalcostisthesumofallthecapitalcostcomponentsandisshown
newdamisexecutedthroughabinaryvariableID . inEq.(33).
glt
(cid:2)
DAMgt=DAMg,t−1+ l idamgl•IDgl,t−dct,∀g,t (25) CAPEXt=CDAMt+CPLt, ∀t (33)
where dct is the time for dam construction and idamgl represents whereCAPEXtisthecapitalexpenditureattimet.
theoptionlforcapacityinstallationofdamsinregiong.Onlyone
capacityoptionlcanbeselectedinaregiong,giveninEq.(26). 3.1.8. Operatingexpenditureconstraints
(cid:2)(cid:2) The operating expenditures are calculated in a similar manner.
IDglt≤1,∀g (26) Theoperatingcostsformaintainingthedams,ODAMt,arecalculated
l t byEq.(34).
(cid:2)
3.1.6. Productionconstraints ODAMt= vodt•DAMgt,∀t (34)
The water flows that are withdrawn to be processed in plants g
must equal the demand for raw sources, D , calculated from
igtq
Eq.(27). wherevodt arethevariableoperatingcostsofdamsattimet.The
(cid:2) operating costs of plantsconsist of fixed, fopplt and variable, vopplt
D = V ,∀i∈S,g,t,q (27) costs(Eq.(35)).
igtq igptq
p∈SPi∩PGg (cid:2)(cid:2) (cid:2) (cid:2) (cid:2)
OPLt= fopplt+ fopplt•IPgplt
The above equation applies only to raw sources, S, i.e. surface
water,groundwaterandseawater.Theproductionofwaterforusage, p∈O(cid:2)P l(cid:2) (cid:2) (cid:2)g p(cid:2)∈PGg(cid:2)∩NP l
P , equals the summation of the effluents from plants treating + vopplt•4ii(cid:5)•Vigptq, ∀t (35)
diigftfqerentrawwaters(Eq.(28)). g p∈PGg l i:p∈SPii(cid:5)∈W q
(cid:2) (cid:2)
Pigtq= 4i(cid:5)i•Vi(cid:5)gptq,∀i∈W,g,t,q (28) calcTuhleatpeednbaylisEeqd.(c3o6s)t.fornotmeetingtheproductwaterdemandsis
i(cid:5)∈Sp∈SPi(cid:5)∩PGg
(cid:2)(cid:2)(cid:2)
Theregionaluserdemand,demigtq,mustbemetandthiscondition OPent= pc•PDigtq,∀t (36)
isenforcedfromthefollowingequation: i∈W g q
(1−dffg)•Digtq=demigtq,∀i∈W,g,t,q (29) wherepcisapenaltycostrate,significantlyhigherthantradingcosts.
The total OPEX is a summation of the operating dams’ and plants’
costsandtradingcosts,expressedfromEq.(37).
wheretheequationappliesonlyforfinalproductwaterpurposeW.
Theparameterdff accountsforthedistributionlossesduetobro-
g
ken pipes and leakages varying regionally. The recharge volumes, OPEXt=ODAMt+OPLt+OTRt,∀t (37)
RC ,areestimatedbytheamountofwaterthathasbeencollected
igtq
insewerage,treatedbywastewatertreatmentplantsandreturned
tosurfaceandgroundwaterstorages(Eq.(30)). 3.1.9. Objectivefunction
(cid:2) Thetotalcost,TC,representstheadditionofthecapital,operating
RCigtq= up•4i(cid:5)i•Di(cid:5)gtq,∀i∈LW,g,t,q (30) costsandpenaltyovertheplanningtimehorizon(Eq.(38)).
i(cid:5)∈W
(cid:2)
whereupisthewaterutilisationpercentage,i.e.thefractionofdis- minimiseTC= (cdft•CAPEXt+odft•OPEXt+odft•OPent) (38)
t
tributedwaterwhichiscollectedassewage,and4i(cid:5)iistheoperating
efficiencyofthewastewatertreatmentplant.
wherecdf andodf arediscountfactorsofthecapitalandoperating
t t
costs,respectively.Theobjectivefunctionissubjectto
3.1.7. Capitalexpenditureconstraints
Next, the capital expenditure for dams, plants installation and
• hydrologicalandsupply-demandbalancesEqs.(3)–(7)
expansionisdiscussed.TheCAPEXofdams,CDAMt,dependsonthe • procurementconstraintsEqs.(8)–(14)
capacityoptionlforinstallationselectedandthecorrespondingcost
• reliabilityconstraintsEqs.(15)and(16)
forconstruction,capdam (Eq.(31)).
l • capacityconstraintsEqs.(17)–(26)
(cid:2) (cid:2) • productionconstraintsEqs.(27)–(30)
CDAMt= capdaml•IDglt, ∀t (31) • capitalexpenditureconstraintsEqs.(31)–(33)
g∈ND l • operatingexpenditureconstraintsEqs.(34)and(37)
The capital cost of plants, CPLt, is a summation of the costs for
Inthelightoftheincreasingimportanceofreliability,theobjec-
installationsandexpansions,andiscalculatedfromEq.(32).
tiveisnolongertoonlyminimisecostbutalsoensurethesystem
(cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) is reliable to an economically adequate level while shortfalls are
CPLt= caplantpl•Igplt+ capexppl•Egplt, ∀t brought to minimum. Therefore, a multi-objective optimisation is
g p∈PGg∩NP l g p∈PGg l appliednextfortheminimisationofthetotalcostandthemaximi-
(32) sationofthereliability.
322 M.N.Kolevaetal. /ScienceoftheTotalEnvironment636(2018)314–338
3.2. Multi-objectiveoptimisationformulation Similarly,aparameter,kkisassignedforthelogarithmreliability
difference,showninEq.(44).
3.2.1. 4-constraintmethod
An4-constrainedmethodisappliedforthesolutionofthemulti- kk=ln(WRk−WRquo),∀k (44)
objectiveoptimisationwherethefirstobjectiveistominimisethe
totalcostforthesupplychainandthesecondobjectiveistomax-
AnSOStype2variable,X ,isusedtorepresenttheselectionofthe
k
imisethereliability.Thereliability,however,isimplicitlyrelatedto
cost,shownasfollows:
thepenalty,OPent,whichisthereasonithastobeexcludedfromthe
(cid:2) (cid:2)
objectivefunctionofthetotalcost.Hence,
TCk•Xk= (cdft•CAPEXt+odft•OPEXt) (45)
(cid:2) k t
Objective1: minimise TC= (cdft•CAPEXt+odft•OPEXt) (39)
t Thesamevariableisusedforthereliability,expressedasfollows:
(cid:2) (cid:2)(cid:2)(cid:2)(cid:2)
whichissubjectto WRk•Xk= WSRigtq/(Gmax•Tmax•Qmax) (46)
k i∈W g t q
Objective2: WR≥WR∗ (40)
Uptotwoconsecutivekpointscanbeselectedthroughaspecial
setorderset(SOS2)requirement.Theactiveoptionkshouldaddup
and
to1,representedinEq.(47).
• hydrologicalandsupply-demandbalancesEqs.(3)–(7) (cid:2)
• procurementconstraintsEqs.(8)–(14) Xk=1 (47)
• reliabilityconstraintsEqs.(15)and(16) k
• capacityconstraintsEqs.(17)–(26)
• productionconstraintsEqs.(27)–(30) Andtheauxiliaryvariablehastobepositive.
• capitalexpenditureconstraintsEqs.(31)–(33)
• operatingexpenditureconstraintsEqs.(34),(35)and(37) Xk≥0,∀k (48)
The obtained solutions will be Pareto optimal and any of them Then,theobjectivefunctionbecomes
canbechosentoplanthewatersupplychain.TheNashbargaining (cid:2)
approach,however,canprovidetheexactpointontheParetocurve maximise tˆ = [(nk+kk)•Xk] (49)
wherethetwostrategiescanco-existatequilibrium. k
whichissubjectto
3.2.2. Nashbargainingapproach
Acooperativegameisconsideredtoobtainthebeststrategiesfor • hydrologicalandsupply-demandbalancesEqs.(3)–(7)
expendituresandsupplyreliabilityusingNashbargainingapproach. • procurementconstraintsEqs.(8)–(14)
Itisaimedtominimisetotalcountry’scostbyincreasingthediffer- • reliabilityconstraintsEq.(15)
encebetweenthestatusquopointandtheoptimisationvariable.On • capacityconstraintsEqs.(17)–(26)
the other hand, it is aimed to maximise the reliability by increas- • productionconstraintsEqs.(27)–(30)
ingthedifferencebetweenthevariableanditsstatusquopoint.By • capitalexpenditureconstraintsEqs.(31)–(33)
maximisingtheproductofallthestrategies’deviations,afairsolu- • operatingexpenditureconstraintsEqs.(34),(35)and(37)
tiondistributionisensuredwherenostrategycanbeimproved.The • gametheoryconstraintsEqs.(45)–(49)
dependencyisexpressedinEq.(41).
The applicability of the mathematical model is investigated
maximise t¯ =(WR−WRquo)•(TCquo−TC) (41) throughacasestudy,presentedinthefollowingsection.
where TCquo is the upper cost bound for the country, which is 4. Illustrativeexample
obtainedinacasenoinfrastructureisplannedandnotradingoccurs.
Then,TC≤TCquo.WRquoisthecorrespondingpointforthestatusquo The applicability of the proposed framework is investigated
pair.Then,WR≥WRquo.Eq.(41)resultsinanon-linearformulation through the implementation of a case study based on Australia.
whichcanresultinlocaloptima.Therefore,itfollowstobefurther The objective is to minimise the total country’s cost for obtaining
linearised.Eq.(41)isexpressedasaseparablefunctionbytakingthe an optimal water network by meeting the regional urban water
logarithmofbothhandsidesandusinglogarithmicproperties: demands. In this section, the major data on regional divisions,
waterdemands,efficiencyfactors,hydrologicaldata,installationand
lnt¯ =ln(WR−WRquo)+ln(TCquo−TC) (42) expansioncapacities,andcostfactorsarepresented.
4.1. GeographicalrepresentationofAustralianregions
Each of the logarithmic terms on the right hand side of the
equation contains a continuous variable, which means they are Australia is divided into 8 internal state and territory gov-
stillnon-linear.Hence,thetotalcost,TC,andwaterreliability,WR, ernments, namely: Queensland (QLD), New South Wales (NSW),
domains are discretised into k number of points, which necessi- Victoria(VIC),SouthAustralia(SA),WesternAustralia(WA),North-
tatestheintroductionoftheTCkandWRkparameters.Anadditional ernTerritory(NT),AustralianCapitalTerritory(ACT)andTasmania
parameter,nkisintroducedtoequalthelogarithmofthecostdiffer- (TAS). Each state has itslocal state governmentwhichownsall or
ence(Eq.(43)). mostofthewaterprovidersoperatingwithinthestate(Australian
Government, 2017). Australian water providers can supply urban
nk=ln(TCquo−TCk),∀k (43) and rural areas with drinking as well as different quality grades
M.N.Kolevaetal. /ScienceoftheTotalEnvironment636(2018)314–338 323
water.Duetothelargenumberofprovidersandtheavailabledata consideredforbrackishwaterandhence,itspurificationcansome-
on regional water demand and hydrological balances, the spatial timesbereferredtoasbrackishwaterdesalination.Statesthatcount
discretisation is performed on a state basis. Besides water supply, on groundwater availability are Western Australia and the North-
majority of the suppliers offer sewerage management, too. A map ern Territory due to their remoteness from the main river basin -
showingtheconsideredregionsisshowninFig.5. Murry-Darling.Fig.5depictsthelocationsofthelargergroundwater
treatmentplantsinAustralia.
4.2. Existingplantsanddamsforprovidingurbanwatersupply Damscanbedefinedas“anartificialbarrierthathastheability
toimpoundwater,wastewater,oranyliquid-bornematerial,forthe
Eachstatepossessesassetsforthetreatmentofanyofthethree purposeofstorageorcontrolofwater”(InternationalCommissionon
sourcesconsidered:seawater,surfacewaterandgroundwater. LargeDams,2016).Theycanvaryimmenselyinsizeandshape,from
The prolonged lack of rainfall from 2000 to 2010 in Australia smalldamsthatserveforwateringfarmstolargedamsthatcanpro-
necessitated finding alternative sources of water supply. Seawater videthestorageforurbancentres.InAustraliatherearealtogether
desalination,althoughanexpensiveoption,hasbeenconsideredas more than 600 dams numbering a total capacity of approximately
theleadingsolutiontowatershortages.Currently,ineverystatebut 80,000GL.AspatialrepresentationofallAustraliandams’locations
TAS,NTandACTexistsatleastonelargecapacityseawaterdesalina- isshowninFig.5.Thecumulativecapacityofalldamsinastateis
tionplant(Fig.5).Theirlocations,capacitiesandconstructioncosts reportedinTable2.
are summarised in Table 1. The plants are in operation as a non- Surface water in Australia is diverted from lakes, rivers and
conventionalmeasureindroughtperiods,whenthereisinsufficient streamsanddams,anditsabstractionvolumesdependonthepre-
freshwaterinthestates’storages.In2016alldesalinationsiteswere cipitationintheterritories.Tasmaniapossessessufficientamountsof
producingdrinkingwater. freshwaterwhereasSA,VIC,QLDandNSWrelypredominantlyonthe
GroundwaterinAustraliaisextractedfromundergroundaquifers availabilityinMurry-DarlingBasin(MDB).Theavailabilityoffresh-
and after the appropriate treatment it can be used for water sup- waterinWAandNTislimited.Surfacewatertreatmentplantsmay
ply,agricultureandindustry.Itssalinitycanbehighenoughtobe involvefulltreatmentwithcoagulationandfiltrationormembrane
Fig.5. Dams,majorplantsandurbanwaterdemandandsourcemixinAustralia.