COPING WITH UNCERTAINTY IN THE RELIABILITY EVALUAT

Proceedingsofthe10thAnnualWaterDistributionSystemsAnalysisConferenceWDSA2008,VanZyl,J.E.,Ilemobade,A.A.,Jacobs,H.E.(eds.),August17-20,2008,KrugerNationalPark,SouthAfrica.COPINGWITHUNCERTAINTYINTHERELIABILITYEVALUATIONOFWATERDISTRIBUTIONSYSTEMSG.Darvini1,P.Salandin2andL.DaDeppo21IstitutodiIdraulicaeInfrastruttureviarie,UniversitàPolitecnicadelleMarche,Ancona,Italy;email:[e-mail]2DipartimentoIMAGE,UniversitàdegliStudidiPadova,Padova,Italy;emails:[e-mail],[e-mail]ofthereliabilityanalysistechniquestothewaterdistributionsystemsaimstoproperlyachieveinprobabilistictermsthefulfilmentofthenodaldemandinaconceptualandphysicalcontextsubjecttodifferentcausesofuncertainties.Theavailabilityofsystemcomponents,likepumpsorpipessubjectedtofailure,theuncertaintyinthespatial–temporalbehaviourofnodaldemand,thepiperoughness,thereservoirlevelaswellastheavailabilityofsupplyresourcescontributetothedefinitionofthesystemreliability.Neverthelessthecomplexityofareliabilitymodelincreaseswiththenumberofdifferentuncertainties,andusually,foreachspecificcase,onlyfewparametersareassumedasrandomwhiletheremainingarefixedasdeterministic.Thisfactalwaysimpliesmoreorlesslimitationsinthesystemdescriptionalsowhendeterministicassumptionsarebasedonreasonablehypotheses.Toovercomethisshortcoming,amixedanalytical–numericalapproach,abletotakeintoaccountboththemechanicalfailureofsystemcomponentsandtherandomspatial–temporaldistributionofnodaldemands,isheredeveloped.ThegoalisachievedbyjoiningaMonteCarlonumericalapproachbasedonthehead-drivensimulationwithaFirstOrderSecondMoment(FOSM)closedformsolution(XuandGoulter,1998).TheMonteCarloapproachletstoanalyzetheseriesofpartialshutdownsrelatedtotherenewalprocessofpipesandthetimeevolutionofthedemand,whilethespatialvariabilityofthelatterandtheuncertaintyinthepiperoughnessaredescribedbytheFOSMapproach.Anillustrativeexampledevelopedinthewell-knowncaseoftheAnytownnetwork,demonstratesthattheuncertaintyrelatedtothespatialnodaldemandandpiperoughnessmayhavearelevantimpactonthereliabilityevaluationofadistributionnetworkwhosepipesaresubjecttothemechanicalfailure.1.INTRODUCTIONThetechniquesforassessmentthereliabilityofwaterdistributionnetworksconcernwithmeasuringtheabilityofthesystemtosupplytheconsumerrequirementsunderdifferentoperatingconditions.Thesystemconditionsaredependingonseveralfactorsasthenetworkgeometry,thesizeandroughnessofpipes,themechanicalfailureofsomecomponents(pipes,valves,pumps,etc.)andthetemporaryshortfallofwatersources.Thenodaldemandalsoisvariablebothinspaceandintimeandwithothersrandomvariablescontributestolimitthevalidityofthewellknowndeterministicsolutions,fromtimetotimerepresentativeofaparticularstateorconfigurationofthewaterdistributionnetwork,butwithouttakingintoaccounthisprobabilityoccurrenceinthesystemlifetime.Ontheotherhand,duetothetheoreticaland/orcomputationalcomplexity,somesourcesofuncertaintyonly,chosenonthebasisoftheanalyzedaspect-design,managementormaintenanceproblem-aregenerallytakenintoaccountbytheprobabilisticapproach,whiletheremainingfactorsareassumedasdeterministic.Avarietyofreliabilityalgorithmsforwaterdistributionnetworkshavebeendevelopedoverrecentyears.ForinstanceQuimpoandShamsi(1991)andKansaletal.(1995)analyzedthereliabilityaspectsarisingfromthemechanicalCopyrightASCE2009WaterDistributionSystemsAnalysis2008Downloaded09Nov2011to221.212.176.62.RedistributionsubjecttoASCElicenseorcopyright.Visit(1990)consideredasrandomthenodaldemandsintermsoftheflowstobesuppliedandtherangeofpressureatwhichthoseflowsmustbeprovidedandthepiperoughness.ThecaseoftherandomfailureofnetworkcomponentsforafixedflowdemandconfigurationistakenintoaccountbyGuptaandBhave(1994;1996),whileanattempttointegratethecontributionsofreliabilityfrombothcomponentfailureanddemandvariationisdevelopedbyGoulterandBouchart(1990)andBouchartandGoulter(1991).XuandGoulter(1998)suggesthandlingtheuncertaintyarisingfromthenodaldemands,thevariationofthereservoirlevels,andthepiperoughnessbyaprobabilisticmodelbasedonthelinearizedmassbalanceandtheenergyconservationequations,bytakingintoaccountatthesametimetheprobabilityofoccurrenceofpipemechanicalfailure.Themainlimitationofthisapproachisrelatedtothefluctuationssizeofdemand.Bythelinearizedtechniquesonlysmallfluctuationsofthenodaldemandcanbeanalyzed,sothatcannotbetakenintoaccounttheflowvariationsthataremetinpracticalapplications.Toovercomethislimitationanumericalmethodforevaluationofthesystemreliabilityincorporatingthecomponentsfailureandthedemandvariation(SalandinandBertola,1996;DarviniandSalandin,2004;SalandinandDarvini,2004)iscoupledwiththeclosedformsolutionlimitedtothefirst-orderinthefluctuatingtermssuggestedbyXuandGoulter(1998).Bythismethodthespatialfluctuationofflowdemandandthetimevariabilityofmeandemandareconsideredseparately.Theformerisgenerallyunknownwhilethelattercanbederivedbythehistoricaldataofsuppliedflow,andsincethespatialfluctuationsaroundthemeanvalueofwaterdemandarelimited,