Lansey, Kevin

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22Uncertainty In Water Distribution Network ModelingKevin Lansey Departm ent of C ivil Engin eering an d Eng ineering M echanic s, University of ArizonaIntroductionWater distribution  networ k mod eling requ ires input frommany sources.  Mu ch of this data cann ot be measu reddirectly  and it is rarely known precisely.  Inputimprecision is propagated to uncertain model predictions.This paper d iscusses a tech nique fo r quantif ying theuncertainty of mod el results base d on the  variability inmode l input.  Once the mag nitude of  the unce rtainty is  understood, anengineer can use that inform ation in  several ways.  Basedon the predic tion unce rtainty in pr esent o r futureconditions,  safety factors can be included in design ofnetwork improvements. To enhance m odel predictions,additional field data can be collected w hich will lea d tomore precise model parameters.  Using the predictionuncertainty, the optimal network conditions andmeasurement locations c an be ide ntified.  Finally,different network representations can be evaluated bycomp aring their p redictive u ncertainties. The model input needed to compute the pressure headsand pipe flows are the nodal demands, pump and pipecharacteristics, tank dim ensions, v alve settings, andcoefficien ts describing energy losses in otherappurtenan ces.  Some of this data, such as pipe diameterand length, is acc urately kn own an d available fromconstruction plans.  Oth er inform ation for p ipes, inparticular the pipe roughness, varies over time asencrustation occurs.  N odal dem ands are d ifficult topredict s ince they also change over a short t ime andbecause  they are modeled as lumped at single locationsrather than distribu ted along  a pipe.  A s such, bo thparameters are not measured exactly but estimatedindirectly  from other field m easurements.  This paperfocuses on the un certainties in pipe roug hnesscoefficien ts and no dal dem ands.  Pipe network hydraulicsFlow in a pipe network satisfies two basis principles,conservation of mass and conservation of energy.Conservation of mass states that, for a steady system, theflow into and out of the system must be the same.  Thisrelationship  holds for the entire network and forindividual nodes.  A node is included in a network modelat (1) a demand location and/or (2) a junction where twoor more pipes combine.  One mass balance eq uation iswritten for each n ode in the netw ork as:where Q in and Qout are the flows in pipes entering orexiting the node and Qdemand is the user demand at thatlocation. These d eman ds are un certain sinc e they areestimated from the local user base that cannot bepredicted exactly since they vary nearly continually.  Inaddition, the demand is typically represented as a lumpeddema nd for u sers near th e node. The second governing equation  is a form of conservationof energy that describes the relationship between theenergy loss and pipe flow.  The most commonly usedhead loss equation for water networks, the Hazen-William s equation, will be the only such relationshipconsidered in this paper.  In English units, the equationis written as:where D is the pipe d iameter, L  is the pipe len gth, Q isthe pipe flow, and C is the Hazen-Williams piperoughness  coefficien t.  hl is the head or energy loss in thepipe.  H i and H j are the energy at nodes at the ends of thepipe measured in dimen sions of len gth. Usin g thisequation, conserv ation of en ergy can  be written  inseveral ways.  Most often, it is written for energy lossaround a loop.  The two conservation relationships canbe used  to develop a set of nonlinear equations that canbe solved fo r the pipe flo ws, Q, a nd nodal heads, H,23which due to un certainty  in input ar e inheren tlyuncertain.Quantifying uncertaintyIf the variability in input param eters, C and Qdemand, canbe quantified several techniques exist to determine theresulting uncertainty in model output.  Mon te Carloanalysis  (MC) is a general  enumerative approach forcomp uting the sta tistics of a mo del’s outp ut.  Given the input parameters statistics and probab ilitydistributions, one iteration of Mon te Carlo a nalysisconsists of randomly generating a set of inputparameters.   The numerical model, in this case thehydrau lic relationships, is then solved to determine themodel output.  In this study , KYPI PE (W ood, 19 81) isused to co mpute  the nod al pressure  heads.  To complete a full analysis  consisting of a large numberof iterations, a set o f random  input is gen erated an d theircorresponding output is determined.  The statistics (e.g.  mean,  variance, and standard deviation) of themodel output can then be computed.  If a sufficientnumber of Monte Carlo iterations are completed, theoutput statistics will con verge to th eir actual va lues.  Theprobab ility distribution of the model output can also beexam ined.  To estimate  the predic ted head  uncertain ty, Mon te Carloanalysis  was app lied to seve ral networks and levels ofinput uncertainty in Araujo (1992).   Figure 1 shows oneof the systems considered and Table 1 provides the inputstatistics for pipe roughnesses.  Table 2 summarizes theresults estimates of the predicted head variances.  It wasassumed that the roughness coefficients followed anormal probability distribution. As the input un certaintyincreased, the output variability increased at nearly thesame rate.TABL E 1: Pipe roug hness statistics for selected pipes Pipe 1 3 5 8 9Low standard deviation 0.9 1.5 0.4 1.0 1.4Med. standard deviation 4.3 7.7 2.0 5.2 7.1High standard deviation 9.1 15.6 4.1 10.0 13.2TABLE 2: Standard deviations of nodal pressure heads for selected nodesLow pipe standarddeviationMedium p ipe standarddeviationHigh pipe standard deviationNode MC FOSM MC FOSM MC FOSM1 0.04 0.04 0.20 0.21 0.40 0.422 0.40 0.41 2.03 2.10 4.00 4.133 0.08 0.08 0.42 0.44 0.82 0.856 0.03 0.03 0.16 0.17 0.32 0.34Trace* 0.18 0.19 4.74 5.08 18.44 19.65*Trace is the sum of the diagonal elements of the covariance matrix, i.e., variancesAs shown b y Araujo, the d istribution of the pressu reheads followed a normal distribution until a high inputuncertain ty level was evaluated.  This result allows oneto draw conclusions about the confidence level in model24predictions.   For example, the 90 percent confidencelevel that the pressure head at a node is greater than adesired value can be computed.  Thus, when designing asystem expansion, an engineer can consider confidencelevels on pressure h eads as w ell as their  expected valu es.Monte C arlo analysis requires a  large number of modelevaluations and long computer times to determine thedesired statistics.  Alternative approaches th at are lesscomp utationally  intensive have been proposed in thestatistics literature.  On e popu lar class of m ethods isknown as point estimation ap proaches w ith the simplestbeing the  first order second moment approximation(FOSM).   To estim ate the mean and variance of modeloutput,  FOSM requires computing the model output at asingle point and determining the derivative of modeloutput to model input (i.e., change of model output dueto a change in model input).  In the worst case, thesederivatives can be co mpute d by finite d ifferences  withone additional model evaluation for each inputparameter.  The variance of the model output, e.g., thenodal pressure heads, can be expressed as a function ofthe variance  of the m odel para meter, e.g ., Hazen-Williams rou ghness coefficien t, as:where cov(H) is the covariance matrix of the predic tedpressure head and cov(C) is the covariance matrix of theroughness  coefficient .  The derivative terms are thematrix  of representing the change in pressure heads dueto a chang e in roug hness co efficients. Th e key resu lt inthe cov(H) matrix is the diagonal elements that are thevariances of the individual pressure heads.  The meanvalue of the output is assumed to occur when the valueof the input is evaluated.FOSM was used to predict model uncertainties of thenetwork in Figure 1  with results a gain sum marized  inTable  2.   Although FOSM is an approximationtechnique, the results  compare very  favorab ly withMon te Carlo analysis. Howe ver, the diff erence incomputations is dramatic.  Thus, FOSM was used for theanalysis in the following sections.  FOSM, howev er,begins to diverge from Monte Carlo analysis at highlevels of uncertainty.  This result is due to modelnonlinearity  at higher uncertainty levels while theFOSM, by using  only the fir st derivative s, neglects th isnonlinearity.Collection of field data to reduce uncertaintiesIf input uncertainty is very high, a model is likely poo rlycalibrated and add itional field data should be collectedto reduce the input u ncertainty .  Uncertainty analysis canassist in determining where and under what conditionsadditional data should be taken to improve modelcalibration.  Araujo (1992) developed a methodology forcomparing the utility of field  data collected from a set ofpotential deman d cond itions. The  data collectionmethodology is based on lowering the uncertainty levelat one or more nodes through  a reduction in one or morepipe rou ghness c oefficient u ncertainty . Rough ness uncertainty is dep enden t upon fie ldmeasurem ents.  Therefo re, a two lev el gradien t wascomputed.  The first level estimates the change in piperoughness  uncertain ty given a demand pattern andassumed measurement locations.  Extreme demandconditions should b e selected, s uch as simulated fireloads, that stress the network and cause hig h energy loss.These types of demand conditions can be induced byopening one or more fire hydrants and supply usefulinformation for imp roving p aramete r estimates. Theuncertainty analysis is completed using the FOSMmethod assuming that the predicted demand  occursusing:where HM is the measured pressure heads and co v(HM)is the covariance matrix for the measured heads.  Thecovariance of HM varies depending upon the device usedfor measurement and the accuracy of determiningelevations.  The gradien ts are evaluated at the bestestimate of the C v alues and m easured head s.The second level of the methodology computes thepropagation of the expected new pipe roughne ssuncertainties to nodal pressure head predictions.  Thesecond step is identical to the uncertainty predictionsdescribed in the previous section.  By comparing thereductions in pressure head uncertainties for differentdemand patterns, the conditions that provide the m ost25improvement can be id entified.  Th is condition  shouldthen be induced in the field and the network recalibratedwith the new data.  If any nodes continue to haveunacce ptably  uncertainty levels, the process can berepeated to determine other measurem ents that are usefulin further reducing nodal prediction uncertainty.Pipe network representationUncerta inty analysis can also be helpful in identifyinghow a network should be represented in a mathematicalmode l.  In most m odeling  efforts, limited  field data iscollected to determine model parameters.  Typically, thedata is insufficient to estimate parameters for individualpipes.  As such to improve the confidence in thecalibrated pipe roughness coefficients, pipes are groupedand all pipes within a group are assumed to have thesame roughness value.  Groupings may be madearbitrarily  or using judgement after examining pipematerial and installation records. By reducing thenumber of parameters, confidence in the estimatedvalues increases and the parameter uncertainty or errordecreases.Although the parameter uncertainty decre ases, the grouproughness  coefficients are not individual pipe values andcan be considered as averaged or surrogate modelparameters.   Thus as fewer pipe groups are considered,the mathem atical model is less representative of the truesystem and a m odel erro r is introduced.  The tradeoffbetween errors can be seen in  Figure 2.  As the numberof parameters increases the model error decreases (forthe reason above) and the parameter error increases sincemore parameters a re being estimated from the sameamount of  information.  An envelope is shown sinceseveral pipe groupings can have the same number ofparameters but have different er rors.  For ex ample if   allpipes in a pipe group have similar field roug hnesscoefficients, the model error will be smaller than a pipegroup ing that the  pipe rou ghnesse s are very  different.   The key to modeling is identifying the tradeoff betweenthe two uncertainties.  Mallick and Lansey (1994)developed a technique to compare these errors.  Thetechnique is an extension of the work by Yeh and Yoon(1981) for groundwater mode ling.  For a se t of field dataand a network model with a defined pipe grouping, theoptimal parameters can be identified using one of severalapproaches (Basnet and Lansey, 1989) .  Mode l error isdefined as the sum  of the squares of  the differencesbetween the field measurements and the modelpredictions after calibratio n.  At the o ptimal parametervalues, parameter error can be computed using FOSM asdescribed  in the prev ious section s.  A number of com binations of pipe groupings arepossible for even the smallest pipe network.  Tocompare  pipe groupings, model versus parameter errorcan be plotted for all groupings (Figure 3).  Both typesof errors are small for points  close to the origin.  The sepipe grouping combinations are the best modelrepresentations.ConclusionsImprecise  field measurements result in uncertain modelpredictions for all types of models including those forwater distribution networks. High uncertainties requireincreased safety factors and result in more costlydesigns.  Thus, an understanding of the magnitude ofprediction uncertainties is critical when making planningand operation decisions based on numerical models.Rapid  evaluation of prediction uncertainties can becompleted for pipe networks using FOSM  analysis.Model prediction prob ability distributions were alsodetermined which allow confidence levels to beestimated.Because  of its speed, FOSM can be u sed as a blo ck inother networ k analysis stu dies.  Tw o applica tions, datacollection and netw ork repr esentation , were de scribed inthis paper.  In addition, the uncertainty approach can beused in reliability analyses as described by Xu andGoulter (1996).ReferencesAng, A.H-S. and Ta ng, W .H., Proba bility Con cepts inEngineering Planning and Design: Volume 2, JohnWiley and sons, New York, 1984.Araujo, J.,   “A statistically based procedure forcalibration of water distribution systems,” Ph.D.dissertation, Oklahoma State Un iversity, Stillwater,Oklahoma, 1981.Araujo, J., and K. Lansey, "Uncertainty Quantificationin Water Distribution Parameter Estimation," presentedat the 1991 National Hydraulic Engineering Conference,ASCE, Nashville, TN.26Lansey, K., and C. Basnet, "Parameter Estimation forWater Distribution Systems," Journal of WaterResources Planning and Management, ASCE , Vol.  117,No. 1, Jan. 1991, pp. 126-144.Mallick, K. and K. Lansey, "Determining OptimalParameter Dimen sions for W ater Distr ibution NetworkMod els," presented at the 1994 ASCE conference onWater Resources Planning and Manag ement, Denver,May.Wood, D.J., “Computer Analysis of Flow in PipeNetworks Including Extended Period  Simulatio ns,”User’s manual, University of Kentucky, Lexington,Kentucky, 1981.Xu, Chengchao an d I. Goulter, “Uncertainty analysis ofwater distribution networ ks,”  in Stochastic Hydraulics‘96, Proceedings of the 7th IAHR InternationalSymposium, Mackay, Australia, 29-31 July 1996,Balkema, Rotterdam, 1996.Yeh, W. W -G. and  Yoon , Y., “Aqu ifer parameteri d e n t i fi c a t io n  wi th  opt imu m dime nsion  inparam eterization,”  Water Resources Research, Vol. 17,n. 3, June 1981, pp. 664-672.Kevin  Lansey is an Associate Professor in theDepartment of Civil Engineering and EngineeringMechanics at the University of Arizona.  He is an activeresearcher in applying systems analysis appro aches towater resources systems with emphasis on waterdistribution systems.

Uncertainty in Water Distribution Network Modeling

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Uncertainty in Water Distribution Network Modeling

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