Effect of Uncertainty on Water Distribution System Model Design Decisions

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Effect of Uncertainty on Water Distribution SystemModel Design Decisions

Derya Sumer1 and Kevin Lansey2

Abstract: The goal of calibrating a water distribution model is to establish a numerical model that represents the real system. Given theuncertainty in field data, recent studies have examined model parameter and predictive uncertainties and how those uncertainties can guidefuture data collection experiments. However, a model is to be used in some decision making process and those decisions can be influencedby the model uncertainty. Therefore, a methodology is presented here to evaluate the impact of uncertainty in pipe roughness values ondecisions that are made using the model for a system expansion design using a steady state hydraulic model. To complete the analysis,model parameter uncertainty is evaluated using a first order second moment �FOSM� analysis of uncertainty. Parameter uncertainties arethen propagated to model prediction uncertainties through a second FOSM for a defined set of demand conditions. Finally, modelprediction uncertainties are embedded in an optimal design model to assess the effect of the uncertainties on model-based decisions. Ifuncertainty levels are large, the monetary benefits of reducing uncertainties from additional data collection can be addressed directly byexamining the change in the design cost with additional data. For demonstration, the methodology is applied to a small literature network.Results suggest that the cost reductions are related to the convergence of the mean parameter estimates and the reduction of parametervariances. The impact of each factor changes during the calibration process as the parameters become more precise and the design ismodified. Identification of the cause of cost changes, however, is not always obvious.

DOI: 10.1061/�ASCE�0733-9496�2009�135:1�38�

CE Database subject headings: Water distribution systems; Calibration; Uncertainty principles; Optimization; Algorithms; Design.

Introduction

A mathematical model of a water distribution system should becalibrated to ensure that the model truly represents the real net-work. Early work on the calibration problem assumed that avail-able data were exact, and by varying the system parameters�generally pipe roughness and nodal demands but may includeminor loss coefficients�, the differences between observed valueson the field and the calculated values from the model were mini-mized. Later studies considered the errors in field measurementsas a source of uncertainty. While all parameters in a networkmodel are in fact uncertain, it is difficult to work with all of therandom variables. Therefore, common practice is to assume thatparameters that are of less importance or more difficult to esti-mate are deterministic. In general, roughness coefficients are usedas calibration parameters while nodal demands are assumed to beknown with certainty.

In practice, common measures of a satisfactory calibratedmodel are that the predicted values are within several feet ormeters of the measured pressure heads or that tank levels trackmeasured values. Tank levels are not significantly affected bychanges in roughness coefficients so this metric generally fails to

1Water Resources Engineer, CH2M HILL, 2485 Natomas Park Dr.,Ste. 600, Sacramento, CA 95833. E-mail: [email protected]

2Professor, Dept. of Civil Engineering and Engineering Mechanics,Univ. of Arizona, Tucson, AZ 85718. E-mail: [email protected]

Note. Discussion open until June 1, 2009. Separate discussions mustbe submitted for individual papers. The manuscript for this paper wassubmitted for review and possible publication on January 18, 2008; ap-proved on March 10, 2008. This paper is part of the Journal of WaterResources Planning and Management, Vol. 135, No. 1, January 1, 2009.
©ASCE, ISSN 0733-9496/2009/1-38–47/$25.00.
38 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT

J. Water Resour. Plann. Man

provide a true test of the calibration. Most commonly, pressuresare not measured during stressed periods so the response to aroughness parameter is very small and a large range of parametervalues give nearly the same pressures. Here, engineering logiccan reasonably provide satisfactory parameter estimates. How-ever, rather than to simply match field measurements, more oftena model’s goal is to accurately predict pressure conditions underrarely occurring fire or high demands. In these cases, when theeffect of parameters is significant, predictions from coefficientsbased on judgment may be far from correct. Thus, systematiccalibration assessment and processes are necessary.

To that end, the Lansey et al. �2001� approach is extended hereto evaluate the impact of the uncertainties on model decisions.Within this methodology, a modeler can assess the value of addi-tional field data on the basis of the impact on the model-baseddecisions and their cost. The focus in this paper is a network thatis undergoing a system expansion to supply areas outside theexisting network.

Background

Once uncertainty is incorporated in the calibration process, thefirst questions that arise are how to assess the accuracy of thecalibration, what accuracy is needed, and what conditions shouldbe sampled to improve model accuracy. Other heuristic schemesfor selecting measurement locations and conditions have alsobeen developed. De Schaetzen et al. �2000� used three differentapproaches for sampling design: two are based on shortest pathsand the third used sensitivity and entropy. Meier and Barkdoll�2000� assumed deterministic and steady state flow conditions in
a different optimal sampling measurement condition design ap-
© ASCE / JANUARY/FEBRUARY 2009

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proach. They presume that a pipe’s roughness only affects systempressures if it causes an observable head loss. As an indirect mea-sure, they judged that if the pipe velocity exceeds a thresholdvalue, then it will cause the observable head loss and maximizedthe total length of pipes that have nonnegligible velocities��0.328 m /s �1.0 ft /s�� by changing the nodal demands.

To assess the model calibration, Bush and Uber �1998� sug-gested using the uncertainty �variance� of the model parameters�pipe roughness coefficients� known as D-optimality as the crite-rion for assessing calibration accuracy. The authors proposedthree statistical measures related to D-optimality and comparedresults from the alternative measures. D-optimality is computedby taking a first-order approximation of the covariance matrix ofthe pipe roughness coefficients based upon the gradients of thesystem equations for the measured conditions. The sum of thediagonal terms of the covariance matrix �known as the matrixtrace� is used as the D-optimality metric.

Lansey et al. �2001� used D-optimality to calculate the prob-ability distribution of roughness coefficients. They then extendedthe approach by propagating the uncertainty in measurements toestimated roughness coefficients and predicted heads for otherconditions, and calculated the model prediction errors using a firstorder second moment �FOSM� approach. They argued that uncer-tainties in predicted conditions, either using individual nodal headvalues or the sum of diagonals of the covariance matrix, are moreintuitive and interpretable. Predictive uncertainty �also known asI-optimality� was used to guide the sampling design. Assuming aset of measurements are readily available, a new sampling condi-tion is selected among potential alternative conditions as the onethat causes the largest reduction in predictive uncertainty.

Kapelan et al. �2003a,b� used a multiobjective genetic algo-rithm �GA� to solve the sampling design problem. One objectivewas to maximize the calibrated model accuracy and the other wasto minimize the sampling cost. Given the number of measure-ments, three different calibration accuracy criteria are compared:D-optimality and A-optimality from Bush and Uber �1998� andprediction uncertainty from Lansey et al. �2001�. A-optimality isthe average of standard deviations of parameters. They concludedthat the first and third calibration measures are preferred over theA-optimality. Since D-optimality is computationally easier, it wasrecommended. Kapelan et al. �2005� applied a Bayesian calibra-tion approach that does not require assumptions on the normalityof the parameters. According to their results, the parameters’probability density functions are similar to normal distributionexcept in the tails of the distribution. They calculated and com-pared parameter and predictive uncertainties.

Calibration Methodology

To provide guidance on the impact of uncertainties, the calibra-tion methodology in Fig. 1 has been developed in an attempt toassess the effect of model prediction uncertainty on model-baseddecisions. This information can then be fed back to the data col-lection process to determine conditions for collecting additionalinformation. In this paper, the model goal is to assist in designinga system expansion. Although all parameters contain some uncer-tainty, only pipe roughness coefficients are considered as calibra-tion parameters here. The steps taken in the methodology aredescribed in the following subsections.

The process begins with a set of collected data that will beused to calibrate the hydraulic model. Field data consist of head
measurements from locations throughout the network. These mea-
JOURNAL OF WATER RESOURCES PLANN


surements can be from fixed supervisory control and data acqui-sition locations or from mobile pressure gages. The demandcondition, that is assumed to be known with certainty, is alsoavailable. Field data are collected under a range of measurementconditions. A measurement condition is a steady state demandcondition with known nodal demands and measurement locations.The following steps are then completed within an iteration of thecalibration process.

Parameter Estimation

Using the data defined above, an implicit parameter estimationscheme is applied to minimize the sum of squared difference be-tween the set of measured and calculated head values �Lansey andBasnet �1991�. This optimization problem can be written as

Minimize J = �k=1

LOADS

�i=1

FM

�Hi,km − Hi,k

p �2 �1�

Subject to: G�HP,C� = 0 �2�

C� � C � C̄ �3�

where Hm and Hp=vectors of field measured and model predictedheads, respectively; and FM=number of field measurements fordemand pattern k. In the application presented here, the “mea-sured” values are computed by a deterministic hydraulic modeland random errors are added to introduce field uncertainty. Sincethe heads are all of similar magnitude, the terms are not normal-ized in the objective function. The conservation of mass andenergy relationships are represented by a set of functions �G� ofthe pressure heads and the unknown pipe roughness coefficients,C �Eq. �2��. This set can represent one or more measurementconditions.

The pipe equations form of the system of equations usesthe head loss for each pipe as conservation of energy. TheHazen-Williams equation is used to calculate frictional losses.Alternatively, the pipe’s relative roughness as it affects the Darcy-Weisbach friction factors can be considered as the model param-eter. Note that the C value directly affects pressure heads through

Fig. 1. Calibration/data collection methodology flowchart

the headloss term. The roughness coefficients are bounded with

ING AND MANAGEMENT © ASCE / JANUARY/FEBRUARY 2009 / 39

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values defined by engineering judgment �Eq. �3��. A GA schemedeveloped at the University of Arizona is used to solve thisproblem.

Parameter Uncertainty Estimation

Assuming that the C values computed in the calibration step aretheir mean values, the covariance matrix of the calibrated pa-rameters can be estimated by a first-order approximation�D-optimality measure� by

Cov�C� = �H2 �� Hp

�C�T�Hp

�C�−1

�4�

where Cov�C�=covariance matrix of the pipe roughnesses; and�H

2 =variance of the measured nodal heads. The derivatives arecomputed using the Xu and Goulter �1998� analytical scheme andare evaluated at the solution point of the calibration problem si-multaneously for all field measurement conditions. The gradientscan also be estimated numerically by a difference equation. Notethat only the gradient terms for measured locations are included.

Model Prediction Uncertainty

To estimate the uncertainty that propagates to model predictionsdue to uncertainty in model parameters �Lansey et al. 2001�, asecond first order analysis is completed. Araujo and Lansey�1991� and Xu and Goulter �1998� showed that a first order analy-sis of uncertainty to compute the variances of model predictednodal heads is accurate except for large parameter uncertaintiesand, in some cases, in the extreme ranges of the nodal heads. Atruncated Taylor series expansion is used for the first order ap-proximation of the nodal heads or

Cov�HD� =�HD

�CCov�C�� HD

�C�T

�5�

The gradient terms are evaluated analytically at the mean param-eter values for some design demand condition�s�. In this study, a“design” demand condition is used for evaluation of the system’sability to supply flow under extreme conditions that cannot beinduced and measured. The diagonal terms in Cov�HD� are thenodal pressure head variances under the design demand conditionand can be computed for all nodes in the network.

Design-Based Data Collection

The magnitude of model prediction uncertainty provides an intui-tive indicator on the quality of a model calibration. However, abetter gage is to understand the impact on the cost of that uncer-tainty. As noted, here the hydraulic model is assumed to be usedin a system expansion design. Similar steps would be taken forother model purposes such as system operation. Under the givenuncertainty in predicted heads, the best design for a desired con-fidence level is determined using a chance constrained optimiza-tion formulation or

Minimize cost: F�HD,D� �6�

Subject to: G�HD,D� = 0 �7�

¯
D � D�D� ,D� �8�


Prob�HD � H� � � �, for all nodes �9�

The objective function �Eq. �6�� is to minimize the cost of the newpipes that is a function of new pipe diameters D, while satisfyingthe constraints that are essentially the conservation of mass andenergy as functions of pipe sizes, pressure heads, HD, and nodaldemands �Eq. �7��. Pipes are selected from a defined set of diam-eters �Eq. �8��. A genetic algorithm optimization method is usedto solve this problem and the pipe diameter decisions are discretevariables. The design pressure heads must be greater than theminimum head requirement H, with some confidence level � �Eq.�9��. This constraint incorporates the uncertainty in estimatednodal pressure heads that propagates from field measurements.Assuming that the pressure heads follow a known distribution�here the normal distribution�, this constraint can be converted toa deterministic equivalent of the form as

H̄D − F−1��HD � H� �10�

where H̄D=mean estimates of the nodal heads that are computedfrom solving the set of conservation of mass and energy equationsunder the design demand and the present estimate of mean rough-ness coefficient values. The variances of predicted heads are com-puted as described in the previous section using Eq. �5�. F−1��=inverse probability distribution for the assumed distribution atconfidence level �. The assumption of normality is justified fromwork by Kang et al. �2008� and others that demonstrated thatpressure heads are normally distributed given normally distrib-uted roughness coefficients. Only the pipe roughness coefficientsare assumed to be uncertain in Eq. �10� in this application. Futuredemands and changes in roughness over time �from the calibratedvalues� could be considered in a more complex chance constraintform.

Eq. �10� is appended to the objective function in a penaltyterm to convert the constrained problem to an unconstrainedform. The penalty term is

penalty = − M � min��H̄D − F−1��HD − H� �,0� �11�

where M=vector of penalty weights. Their values are chosen tocause the penalty term to be larger than the expected cost term.

It is important to note that Eq. �10� includes a term related to

the mean parameter values H̄D, that are estimated with measureddata and are expected to approach their true values during thecalibration process. The second term, F−1��H, represents theuncertainty in those parameters that should decrease with addi-tional field data. As demonstrated below, the mean and varianceof the predicted parameters both affect the optimal design.

Collecting Additional Field Data

The last three sections provided background on the mechanicalsteps that comprise the overall data collection process. Additionaldata should be collected with the goal of reducing the design costby reducing the parameter and model prediction uncertainties. Asnew field measurements are collected, the uncertainty in predictedheads is expected to decrease �Eqs. �5� and �10��. Since the vari-ance of HD decreases, Eq. �10� is relaxed and the optimal costwill decrease. The modeling question is: what conditions shouldbe induced in the field to provide additional information? Theapproach taken is to solve the least cost system expansion forseveral potential alternative demand and sampling conditions.

The data collection goal is to identify the measurement condi-
tion that provides the most information causing the largest de-

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crease in design cost prior to actually expending the money andmanpower to collect the data. Thus, potential new measurementsconditions are identified and added to those conditions alreadymeasured. Assuming that the present estimate of the mean valueis correct, Eq. �4� can be solved for the new smaller variance ofC. This estimate can then be used in Eqs. �5� and �10� in thedesign problem and a new design determined that approximatesthe conditions after the additional measurement condition data isavailable. The benefits in terms of a lower expected cost designcan then be assessed directly. Other data collection conditions canthen be evaluated and the benefits between conditions can becompared. Note that the measurement conditions can be alterna-tive demand scenarios and/or alternative measurement locations.

This approach allows the modeler to evaluate if it is cost ef-fective to collect additional data by comparing the benefit of ad-ditional measurement loads with the cost of collecting the data.This criterion or a budget limitation would stop the looped datacollection procedure.

Application

To demonstrate the methodology, the example from Lansey et al.�2001� is revisited �Fig. 2�. The network contains 16 pipes and 12nodes. Pipe characteristics are listed in Table 1. It is assumed thatthe primary goal of the network model is to be used in evaluatingsystem expansion designs �Fig. 2� under the design demand con-dition of 1.8 times the normal load �shown in Fig. 2�. It is as-sumed that data have been collected for four measurementconditions described as base loads:1. Normal �N�.2. Low �L�=0.4*N.3. High �H�=1.4*N.4. Fire �F�=0.8*N+127 l /s at nodes 3 and 8.

Pressure heads were measured at nodes 3, 5, 6, 9, and 11 foreach demand. Note that measurements do not necessarily have tobe at the same location for all measurement loads. Measurementerrors were introduced by adding normally distributed randomcomponents with 0 mean and 0.372 m2 �4 ft2� variance to the trueheads. Although the normal and low demand conditions willlikely not have a significant impact on the calibration, they areincluded as the conditions are easy to measure since they do notrequire additional field effort.

A set of five possible additional measurement conditions are

12

11

109

8

57

2

1

43

6

(1)

(7)

(6)

(5)

(4)(3)

(2)

(11)

(10)

(9)

(8)

(14)

(13)

(12)

(16)

(15)

[50.3,41][48.7,44]

[45.7,0]

[45.7,31]

[48.7,37]

[44.2,0]

[47.2,24]

[39.6,27]

[42.7,0]

[41.1,22]

[44.2,0]

[39.6,17]

100.0 m

[elevation (m) , demand (lt/s)]

14

13

(17)

(18)

(19)

(21)

(20)

(22)

(23)

(24)

[50,40]

[51,40]

(2)[45.7,31]

12

11

109

8

57

2

1

43

6

(1)

(7)

(6)

(5)

(4)(3)

(2)

(11)

(10)

(9)

(8)

(14)

(13)

(12)

(16)

(15)

[50.3,41][48.7,44]

[45.7,0]

[45.7,31]

[48.7,37]

[44.2,0]

[47.2,24]

[39.6,27]

[42.7,0]

[41.1,22]

[44.2,0]

[39.6,17]

100.0 m

[elevation (m) , demand (lt/s)]

14

13

(17)

(18)

(19)

(21)

(20)

(22)

(23)

(24)

[50,40]

[51,40]

(2)[45.7,31]

Fig. 2. System layout for application network; solid lines representexisting system; dashed lines represent the new pipes, nodes, anddemands to be added in the design condition

proposed and the sequence of best conditions will be determined



using the methodology shown in Fig. 1. The new loads are:

• Qp1 :0.8*N+fire at node 1 �+127 l /s�;



• Qp4 :0.8*N+fire at nodes 4 and 5 �+127 l /s each�; and

• Qp5 :0.8*N+fire at nodes 2 and 9 �+127 l /s each�.

Based on the pipe physical characteristics, the Hazen-Williamscoefficient has a limited range and the large uncertainties are notrealistic. Modeler judgment can provide additional information.In this case study, a limit of 20 is defined for the maximumstandard deviation of pipe roughness. Following the data collec-tion procedure, parameter uncertainty is propagated to predictionuncertainty within the optimal design model. Piping and installa-tion costs vary with pipe diameter and whether the pipe is locatedin an existing �pipes 22–24� or the newly developed area �pipes17—21� �Table 2�. The minimum pressure requirement was de-

Table 1. Physical Data for Application Network

Pipenumber

Length�m�

Diameter�mm�

RoughnessC

1 3,048.0 610 110

2 1,524.0 457 110

3 1,524.0 406 100

4 1,676.4 356 100

5 1,066.8 305 120

6 1,676.4 356 120

7 1,371.6 305 90

8 762.0 152 90

9 1,066.8 305 90

10 670.6 381 90

11 1,981.2 457 110

12 1,524.0 356 100

13 1,676.4 305 120

14 914.4 356 100

15 1,219.2 306 100

16 1,219.2 406 90

17 914.4 — 120

18 914.4 — 120

19 1,371.6 — 120

20 1,066.8 — 120

21 1,676.4 — 120

22 4,145.3 — 120

23 2,072.6 — 120

24 182.9 — 120

Table 2. Pipe Cost as Function of Diameter

D �mm�Cost �$/m�

�pipes 22–24�Cost �$/m�

�pipes 17–21�

152 4.0 3.6

203 8.2 7.4

254 13.9 12.4

305 22.7 20.4

356 34.4 30.9

406 49.3 44.3

457 67.8 61.0

508 90.8 81.7

610 147.3 132.7


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fined as 30 m. A chance constraint was added to the GA in theform of Eq �11� assuming that pressure heads are normally dis-tributed. A confidence level imposed was 99% in the probabilisticconstraints. The elements of M in the penalty term were fixedat 108.

During model calibration, pipes of similar age and material areassigned the same pipe roughness coefficient. Here, to analyze theeffect of the amount of data, the 16 pipes were divided into three-,six-, and eight-pipe groups. The network was calibrated using thesame data set for each set of pipe groups. This next section dem-onstrates the methodology for the eight-group case and some gen-eral results using the 8-pipe group case as an example. This isfollowed by a comparison of calibration process results for thethree pipe group cases.

General Results

Identification of Data Collection Loading ConditionDuring an iteration of the calibration procedure, parameters areestimated using a set of existing measurements that are describedas base loads to estimate the mean values of roughness coeffi-cients by solving the optimization problem �Eqs. �1�–�3��. Thebase load set increases with the additional measurement condi-tions as the calibration process continues. Using the calibratedparameters from the base loads, additional loads are considered toestimate their impact on the parameter and predictive variancesand optimal costs.

At the beginning of the first iteration, data from the four baseloads are used to estimate the mean and standard deviations of thepipe roughness for each pipe group. The goal is then to identifythe best conditions to introduce in the field from the candidate set.Each possible measurement condition is added independently tothe four initial available measurement demands. The prospectivenew measurement condition has not been induced in the field soduring the evaluation, the mean parameter values from the previ-ous iteration do not change since no new information is available.However, the gradient terms for the additional measurement con-dition are computed at those mean values and are appended to thematrix in Eq. �4� to update the covariance matrix of the piperoughness coefficients. Table 3 lists the traces of the resultingmatrices for each new potential measurement condition andshows that the trace of the parameter uncertainty matrix decreasesfor each additional measured demand condition. The greatest im-provement results by adding measurement load Qp

4.The pipe roughness covariance matrices are then passed to Eq.

�5� to compute the predicted pressure head means and variances

Table 3. Expected Changes in the First Iteration of the Calibration Proc

Loads

Piperoughnessuncertaintysum ��C

2 �

Pressurehead

predictionuncertaintysum ��H

2 � Cost

Base 5,713 8.64 $530,078

Base+Qp2

1 5,531 7.91 $530,932

Base+Qp35,550 8.21 $529,669

Base+Qp45,659 8.88 $534,403

Base+Qp54,801 7.98 $534,029

Base+Qp 5,085 6.88 $534,029

Deterministic 0 0.00 $428,323

within the optimization problem �Eqs. �6�–�9��. The system is



reoptimized with the new data. Values shown in Table 3 are thetraces of the predictive uncertainty matrices at the optimal solu-tion. Since the resulting flow distribution may have small flows inhighly uncertain pipes and have little impact on the predictedheads, the change in the Tr�Var�C�� is based on all pipe coeffi-cients and is not directly translated to the prediction uncertainty.As seen in Table 3, predictive head uncertainty suggests that load5 should be induced due to the decrease in its value from8.64 to 6.88 m2, rather than load 4 for parameter uncertainty.Note also that predictive uncertainty is computed after the systemis optimized, which is different than its application in Lansey etal. �2001�. At the optimal design, the flow distribution may sig-nificantly change between iterations as different pipe sets are in-cluded in the system expansion. With a different flow pattern, thepredictive uncertainty does not necessarily move consistentlywith parameter uncertainty and may even increase as seen forload 3 �discussed below�.

The new measurement condition that caused the largest ex-pected cost reduction was load 2. Thus, it is selected to be addedto four base conditions to form the base load set for the nextiteration, Base2. The methodology �i.e., parameter estimation, pa-rameter uncertainty, predictive uncertainty, and optimization� isrepeated in this case for all potential measurement conditions.

Increases in Expected CostIn general, one would expect that the lower variance in predictedpressure heads would result in smaller pipe diameters since thesecond term in Eq. �10� is reduced and the constraint is less re-strictive. However, some costs actually are expected to increasecompared to the initial base condition �Table 3�. For those cases,it was confirmed that the optimal solution computed using thebase load information was infeasible for the base load data thatwas supplemented with the new demands.

Higher costs are due to an increase in predictive uncertainty atone or more nodes. For example, in the first iteration, standarddeviation of predicted head at node 13 increased when load 3 isadded to the base loads and caused a reduction in calculated pres-sure at that node causing the pressure head after adjusting for thechance constraint to fall below minimum allowable pressure head�Eq. �10�� using the previous optimal solution.

The change in parameter uncertainty covariance matrix �Eq.�5�� is due to the engineering judgment bound on the maximumroughness coefficient standard deviation. This bound is intro-duced by calculating the correlation coefficient between eachgroup of roughness coefficients and setting diagonal terms in theparameter covariance matrix to 400 if they are a larger value. The

Eight-Pipe Groups

Selected diam �mm� for new pipes

18 19 20 21 22 23 24

152 305 406 152 406 610 508

152 305 406 152 406 610 610

152 305 406 152 406 610 457

152 305 406 152 406 610 406

152 305 406 152 406 610 356

152 305 406 152 406 610 356

203 356 406 152 305 610 406

ess for

17

203

203

203

254

254

254

152

off-diagonal elements are then calculated using the same correla-


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tion coefficients computed before the adjustment. This results inhigher and lower off-diagonal terms than the original parametercovariance matrix and alters the resulting prediction uncertaintymatrix. This problem is largely due to limited data in the earlyiterations to correctly estimate the roughness coefficients.

Expected and Actual Cost ReductionsThe expected design cost reduction that is computed from theoptimization model during the load identification step does notresult from measurement errors and changes in the mean C val-ues. The difference between the predicted improvement and ac-tual result may be significant, particularly early in the calibrationprocess. For example, the expected cost after adding load 4 initeration 1 was $529,669, while the actual decrease after measur-ing the condition �with its associated measurement errors� andsolving the optimization problem, the new optimal was $529,748.When adding condition 5 to the Base2 loads in iteration 2, thenew cost is expected to be $509,201 �Table 4� and the actual costdecreases to $477,864. At the next iteration �iteration 3�, the costwas expected to drop to $469,511, but it increased to $513,912after the model was recalibrated with the new “field” data.

Since the field measurements contain errors, the estimatedmean pipe roughness coefficients change at each iteration and,

Table 4. Expected Changes in the Second Iteration of the Calibration Pr

Loads

Piperoughnessuncertaintysum ��C

2 �

Pressurehead

predictionuncertaintysum ��H

2 � Cost

Base2 5,565 8.58 $529,748

Base2+Qp3

1 5,407 7.56 $525,675

Base2+Qp44,907 11.58 $544,825

Base2+Qp54,502 9.06 $539,852

Base2+Qp 4,719 5.64 $509,201

Deterministic 0 0.00 $428,323

Table 5. Flow Values and the Standard Deviations of Roughness Coeffic

Pipe number

Pipe flow rates �l/s�

Base2

Base2

+Qp1

Base2

+Qp3

Base2

+Qp4

1 267 341 341 468

2 148 118 194 262

3 63 50 99 139

4 22 17 66 106

5 6 4 43 24

6 37 29 68 127

7 21 17 13 8

8 6 5 9 14

9 28 23 27 32

10 74 59 106 160

11 119 95 147 206

12 41 33 60 88

13 16 13 26 25

14 34 27 66 94

15 10 8 47 75

16 17 14 14 14

Note: Negative flow for pipe 5 denotes change in flow direction from Base2.



with the parameter variances, affect the optimal solution depend-ing if increase or decrease relative to the previous estimates. Inlater iterations and when using fewer model parameters, the esti-mated cost, are better predictors since the mean parameter valuesare close to their true values and the cost change is dominated bychanges in the parameter and prediction variances rather thanchanges in the mean parameter values.

Analysis of Measurement Condition SelectionCorrelation exists between high flows and parameter uncertain-ties. Larger flows in pipes are expected to result in smaller rough-ness coefficient uncertainty as noted by several modelers.Parameter uncertainty becomes smaller as additional informationis collected and can be explained mathematically by examiningEq. �4�. If the imposed load results in larger flows in some pipes,larger changes in the uncertainty of roughness coefficients willresult due to the larger values of �HP /�C. This effect can be seenby comparing the number of pipes with higher flow rates foriteration 2 for the eight-pipe group case �Table 5�. Measurementcondition 4 increased flow in nearly all pipes while Qp

5, the opti-mal measurement condition, increased flow rates in nine pipes.Loads Qp

3 and Qp1 increased flow in fewer pipes. Thus, the number

for Eight-Pipe Groups

New pipe diam �mm�

18 19 20 21 22 23 24

203 305 406 152 406 610 406

254 254 356 152 406 610 406

152 356 457 152 406 610 356

152 305 406 152 406 610 406

152 356 508 152 356 610 406

203 356 406 152 305 610 406

or Iteration 2

Pipe roughness coefficients standarddeviations

e2

p5 Base2

Base2

+Qp1

Base2

+Qp3

Base2

+Qp4

Base2

+Qp5

8 5 3 4 3 3

7 5 3 4 3 3

9 20 20 20 16 15

6 20 20 20 16 15

4 20 20 15 15 20

1 20 20 15 15 20

7 20 20 20 20 20

2 20 20 20 20 20

9 20 20 20 20 20

7 20 20 20 20 16

0 18 14 15 15 14

6 20 20 20 16 15

0 20 20 15 15 20

6 20 20 20 20 20

7 20 20 20 20 20

4 20 20 20 20 20

ocess

17

152

254

152

305

254

152

ients f

Bas+Q

46

26

5

2

−

2

11

3

4

13

20

4

2

1


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of pipes with high flow rates, as proposed by Meier and Barkdoll�2000�, does reflect the general improvement trend. However,since the goal of the calibration problem is to find the least costsystem expansion, the selection of new measurement conditions isdriven by that optimization problem and not simply improving thesensitivity to some pipes.

As noted above, the best induced measurement condition forimproving the system design is not always the condition withlargest reduction in either global parameter or prediction uncer-tainty. In the optimization problems, costs are reduced if the bind-ing �active� pressure bound constraints are relaxed by reducingpredicted pressure head variances at critical nodes �rather than theglobal prediction uncertainty as measured by the trace ofCov�HD�� is reduced. The difficulty in determining the measure-ment condition to induce is in identifying pipes that will reducethe predictive head variance as �generally� all pipes in the net-work affect all nodes. The number of pipes with lower uncer-tainty, their distribution in the network, the number and locationof nodes with higher pressures all affect the selection of measure-ment conditions. If the C value correlations were negligible, therelative contributions to the active node’s pressure head uncer-tainty could be found from the FOSM analysis, but this is not thecase so the problem is more complex. Table 5 lists the pipe pa-rameter standard deviations for iteration 2. The differences be-tween pipes and measurement conditions are not very large.

Table 6. Nodal Head Standard Deviations for Optimal Designs in Iterati

Standard

Node Base2 Base2+Qp1

1 0.46 0.41

2 0.86 0.73

3 0.51 0.57

4 0.76 0.71

5 0.71 0.66

6 0.73 0.67

7 0.75 0.69

8 1.16 1.14

9 1.11 1.11

10 0.21 0.21

11 0.90 0.89

12 0.17 0.18

13 0.78 0.63

14 1.04 0.92

Table 7. Calibrated Roughness Coefficients and Design Cost for Eight-P

Groupnumber Pipes Base Base2

1 8 104.6 104.6

2 10 88.4 87.4

3 11 119.7 117.3

4 1, 2 109.6 109.6

5 14, 15 110.2 106.2

6 3, 4, 12 101.6 102.4

7 5, 6, 13 96.4 102.7

8 7, 9, 16 80.0 87.7



The critical nodes in this problem are nodes 13 and 14 withhigher elevations at locations far from the source. The nodal headstandard deviations �Table 6� are telling and explain the differ-ences in the expected optimal costs. Measurement conditions 1and 5 reduced node 13 and 14 uncertainties and had lower ex-pected costs. Other measurement conditions lowered uncertaintiesthroughout the network. The result in Qp

5 is that the longest newpipe �pipe 22� could be reduced from 406 to 356 mm. Since thispipe carries significant flow, several other pipes’ diameters wereincreased since they had less impact on the critical nodes. Identi-fication of these combinations would be extremely difficult with-out a structured framework like that presented here.

Convergence of Mean Roughness CoefficientsThe calibrated roughness coefficients approach their true values ifthe relevant pipes carry significant flow, which is consistent withthe Meier and Barkdoll �2000� hypotheses and engineering judg-ment. For example in the eight-group case, groups 1 �pipe 8� and7 �pipes 5, 6, and 13� have the poorest convergence �Table 7�.Pipe 8 carries a small flow, thus, pressures are less sensitive tochanges in its roughness coefficient. Since it is the only pipe inthe group, it does not benefit from information on other pipes.Random components that are added for field simulations also af-fect the convergence of C values to true values.

ions �m�

Base2+Qp3 Base2+Qp

4 Base2+Qp5

0.47 0.46 0.39

0.82 0.64 0.62

0.61 0.50 0.39

0.97 0.83 0.64

0.96 0.82 0.66

0.98 0.83 0.61

0.97 0.78 0.63

1.27 1.19 0.83

1.19 1.13 0.79

0.23 0.21 0.25

0.90 0.90 0.93

0.18 0.18 0.22

1.07 0.87 0.66

1.20 1.09 0.79

oups after Each Iteration

3 Base4 Base5 Base6

Truevalue

1 92.7 97.8 101.4 90

7 88.4 88.4 85.4 90

7 108.5 110.2 109.3 110

8 111.1 110.2 110.2 110

1 98.2 90.2 103.9 100

4 99.5 101.6 103.8 100

1 108.6 109.7 108.1 120

0 90.2 89.2 89.3 90

on 2

deviat

ipe Gr

Loads

Base

114.

92.

118.

107.

117.

107.

101.

86.


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During iteration 4, the mean parameter estimates were muchcloser to their true values, particularly for pipes carrying highflows �Table 5�. It may be such that the limited amount of dataprior to iteration 4 was not sufficient to provide good parameterestimates. This result points to the need for more rigorous statis-tical analysis or identifiability studies to provide guidance whensufficient data are not in place to provide stable estimates. An-other possibility is to apply the Bayesian approach �Kapelan et al.2005� to determine parameter uncertainty and carry those distri-butions and statistics in the framework presented here.

Results for Different Pipe Group SizesThe methodology is applied to the same model for different num-bers of parameters by grouping the pipes into eight, six, and threegroups. Figs. 3 and 4 show the changes in the traces of the piperoughnesses and predictive head covariance matrices and the op-timal design cost for all three cases. As expected, the uncertaintiesin model parameters and model predictions monotonically de-crease with more information. The optimal design results, how-ever, are more complicated. Although the trace measure forpredictive uncertainty decreases monotonically for all parametersets, cost fluctuations occur.

As noted above, optimal costs change as the chance constraintrelationship �Eq. �10�� is altered in two ways. First, as more in-formation is provided during the calibration process, the mean

parameter values change that, in turn, cause H̄D to change. Pa-rameter values can increase or decrease depending upon the newfield data. As the estimated mean C of a critical group increases,the headlosses in these pipes decrease and pipes with smaller

0

1000

2000

3000

4000

5000

6000

1 2 3 4 5 6Iterations

Trace[Var(C)]

8 groups 6 groups 3 groups

Fig. 3. Variation of parameter and model prediction uncert

$400,000

$450,000

$500,000

$550,000


DesignCost

8 groups 6 groups3 groups Deterministic (8 groups)Deterministic (6 groups) Deterministic(3 groups)

Fig. 4. Variation of design cost through all iterations for all threecase studies



diameters can be chosen in the design problem. The optimal costmay even fall below the deterministic optimal cost if coefficientvalues are overestimated. The opposite occurs when C values arelower than their true values.

The second reason for an optimal cost change is that, as moredata are collected, the parameters and model prediction variancesdecrease. With stable, reasonably accurate parameter estimates,smaller pressure head variances cause the chance constraints to beless restrictive, lowering optimal design costs. The hypothesis thatthe value of new information will reduce cost and can be used toestimate the value of the information is based on this reasoning.This rationale is also the basis for using D- or I-optimality as thecalibration process measure.

Three-Pipe GroupsFig. 5 plots the mean parameter values versus the optimal cost forthe three-parameter case in which the 13 pipes are grouped intothree sets. Recall that an increase in a C value corresponds to asmoother pipe and less head loss. Therefore, an increase in a Cvalue will cause a decrease in the design costs. Changes in group1’s mean parameter values do not have a visible effect on thedesign cost. In iteration 2, group 2’s C value increases slightlyand the predictive variance falls abruptly. The design cost alsodecreases considerably, likely due to the decrease in predictive

0.01.02.03.04.05.06.07.08.09.010.0


Trace[Var(H)]

8 groups 6 groups 3 groups

uring the calibration process for three-pipe grouping cases

80

90

100

110

120


C

$460,000

$480,000

$500,000

$520,000

$540,000

$560,000

Group 1 Group 2Group 3 Design Cost

Fig. 5. Change in design cost and estimated mean roughness coeffi-cients for three-pipe groups; �group 1 pipes: 1, 5, 6, 13 with truevalue 120; group 2 pipes: 2, 4, 11, 12, 14, 15 with true value 100;group 3 pipes: 3, 7, 8, 9, 10, 16 with true value 90�

ainty d


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uncertainty �Fig. 3�a��. At the third iteration, the predictive uncer-tainty decreases slightly and group 3 roughness increases. Eitherof these changes or some combination of them could cause theobserved optimal cost decrease. At iteration 4, the cost increases.Since the prediction uncertainty falls, the only explanation is achange in a parameter mean. Group 2’s parameter is the only onewith a higher value compared to iteration 3, so its parametercontrols the change in design cost. After this iteration, the costdecreases slightly while pipe groups 2 and 3 are flat or decrease.Pipe group 1, that until these iterations did not appear to influencethe optimal cost, does go up about two units. Most probably, thedecline in the predictive variance is the controlling factor.

Thus, in the three-parameter case, the mean values are rela-tively constant and converge to their true values. With the smallset of pipe groups, this small network is likely especially sensitiveto individual parameter estimates. However, with little variabilityin the means, parameter uncertainty �the second factor for loweroptimal costs� plays an important role in the design cost exceptfor iteration 3. As seen, the predictive variance decreases mono-tonically and the variance reduction at node 13 �the critical designnode� occurs rapidly in the three-parameter case. A modeler mightstop after the fourth iteration if the cost of sampling was morethan the decrease in design cost. After iteration 3, the valueof new information is small and the design cost is relativelyconstant.

The data collection load sequence �Qp1, Qp

4, Qp5, Qp

3, and Qp2�

shows that it is not always intuitively clear for an engineer tochoose the sampling location. The first load added Qp

1 does notinclude loads that are the largest or most distant from the sourcethat are typical engineering judgment criteria.

Six-Pipe GroupsFor the six-pipe group case, three additional parameters are cali-brated with the same amount of collected data as the three-pipegroup analysis. During this calibration process, the mean esti-mated C values are more variable and convergence to their truevalues is poorer than the three-parameter case �Fig. 6�. Somecorrelation is seen between roughness coefficients and the costfluctuation.

To better understand the relation, a sensitivity analysis is car-

80

90

100

110

120


C

$440,000

$460,000

$480,000

$500,000

$520,000

Group 4 Group 6Design Cost

Fig. 6. Change in design cost and estimated mean roughness coeffi-cients for six-pipe groups �only the groups that appear correlated withthe design cost are shown�; �group 4 pipes: 3, 4, 12 with true value100; group 6 pipes: 7, 8, 9, 10, 16 with true value 90�

ried out in order to find the critical pipe group that has the most



effect on the predicted heads at the critical node with pressure ator nearly equal to its lower bound in the design problem �node13�. This group is identified by evaluating the predictive variancewith only one pipe group as being uncertain and all other assumedto be known with certainty. Here, group 4 is identified as thecritical group for all iterations for all alternative loads for thesix-parameter case.

Although the roughness value for group 4 is relatively con-stant, through iteration 3, the cost decreases. Other parametersexcept for group 5 do not vary significantly. Group 5 does in-crease at iteration 2 and to some degree tracks the objective func-tion changes later in the process. Thus, the decrease in cost islikely caused by a combination of parameter and predictive un-certainty reduction. In iteration 4, predictive uncertainty continuesto drop but cost increases. The group 4 parameter estimate is theonly C value that decreases, so it is the controlling parameter.Although group 3’s value changes dramatically during this itera-tion, its effect is very small since these pipes carry little flow inthe existing and expanded systems.

In iteration 5, the cost decreases. Mean values decrease forgroups 1 and 4 and increase slightly for other groups. Since theincreases are not large, the decrease in cost is most probably dueto decreasing predictive variance. Finally in iteration 6, the costincreases and groups 2, 5, and 6 parameter values decreaseslightly as does the prediction uncertainty �Fig. 3�. As the esti-mated values approach their true values, the cost appears to bemore sensitive to minor changes in the parameter means and vari-ances in this small network.

Eight-Pipe GroupsThe results with eight-pipe groups show that the mean C valuesvary considerably throughout the calibration process �Fig. 7�. Itappears that changes in cost for the eight-pipe groups can berelated to changes in mean C values. Groups 1, 2, 3, 5, and 6 Cvalues are plotted in Fig. 7 and generally show an inverse rela-tionship with the design cost, whereas this link is not apparent forthe remaining groups. As seen, groups 1 and 6 mirror changes incost during all iterations. Except for iteration 5, the cost decreaseswith increasing group 5 C values and vice versa. Thus, it appearsthat in this model representation that the mean values are poorlyestimated due to the limited field data and cost is sensitive to theirchange. In the last iteration, 45 measurements have been made to

80.00

90.00

100.00

110.00

120.00

1 2 3 4 5 6

Iterations

C

$460,000

$480,000

$500,000

$520,000

$540,000

Group 1 Group 2Group 3 Group 5Group 6 Design Cost

Fig. 7. Change in design cost and estimated mean roughness coeffi-cients for eight-pipe groups �only the groups correlated with the de-sign cost are shown�; �group 1 pipes: 8 with true value 90; group 2pipes: 10 with true value 90; group 3 pipes: 11 with true value 110;group 5 pipes: 14, 15 with true value 100; group 6 pipes: 3, 4, 12 withtrue value 100�

estimate eight parameters. However, the spatial distribution af-


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fects the independence and data value. Tools, such as identifiabil-ity, to quantify the value of information are needed to betterdesign data collection experiments for water networks.

Overall, it is possible to interpret qualitatively why the designcost changes. Controlling factors are the sensitivity to the param-eters and the stability of their estimates. Design cost is affectedprimarily by the mean values of parameters in early calibrationprocess iterations. After they are stabilized, their and the result-ing prediction variance begins to play a larger role in the costoptimization.

Conclusions

Based on the premise that a hydraulic model is constructed toaddress one or more engineering issues, a methodology is pre-sented for assessing the value of alternative sampling and experi-mental designs on the changes in the model-based decisions andtheir cost. A detailed examination of alternative calibration crite-ria is also performed. During the data collection process, changesin parameter uncertainty, predictive uncertainty, and the designcost do not follow the same pattern. If the parameter uncertaintyor the predictive uncertainty is used as the calibration criterion,improvement in these criteria by collecting more data does notnecessarily result in a monetary benefit.

Sampling design using engineering judgment is the commonpractice, however, as the results suggest, a water distribution sys-tem is very complex and the best sampling location is not alwaysintuitively obvious. In general, the goal is to increase energylosses and parameter sensitivity. However, identifying the pipesthat are most important may not be clear, particularly during de-sign as the calibration and optimal decisions are strongly affectedby the flow distribution in the network. As new data are collected,the calibrated parameters can change and alter the flow distribu-tion. In addition, new pipe sizes that are selected during the de-sign process may alter the flows to a greater extent.

After mean parameters have converged to near their true val-ues, changes in costs will be monotonic. At this point in thecalibration process, judgments for collecting additional data canbe based on the expected cost reduction compared to the cost forcollecting additional data.

A number of calibration/modeling issues remain and can beaddressed using the approach presented herein. Pipe grouping re-quires knowledge of the system and engineering judgment. Thespatial distribution of pipes within a group and the flow throughthose pipes can influence the calibration process and accuracy. Inthis work, measurement demands are assumed to be known withcertainty. The demand uncertainty will affect the model calibra-tion and should be incorporated in this formulation.

Measurement conditions and locations have begun to be ad-dressed in the paper but more complex sampling designs can be



considered. Meters can be placed at different nodes that are iden-tified as critical to the load applied. The optimal sampling loca-tions will be different for each physical network layout. A moredetailed study can be developed to optimize the loading locations,metering locations for each load, number of groups, and the typeof pressure gauges used.

From a statistical analysis perspective, the methodology usessome fairly rudimentary tools. It can be improved by examininghow to estimate the probability distribution function of the piperoughness and predicted nodal heads. It is possible that modelsdeveloped for different purposes will require different levels ofmodel accuracies. A follow up study will test the formulatedmethodology for different model purposes such as pump opera-tions and system rehabilitation/cleaning/expansion. Finally, alter-native data sources are available for pipe network modeling, suchas pressure transients. This approach can be extended to incorpo-rate that data source.

References

Araujo, J., and Lansey, K. �1991�. “Uncertainty quantification in waterdistribution parameter estimation.” Proc., Hydraulic Engineering, ’91,ASCE, New York, 990–995.

Bush, C., and Uber, J. �1998�. “Sampling design methods for water dis-tribution model calibration.” J. Water Resour. Plann. Manage.,124�6�, 334–344.

De Schaetzen, W. B. F., Walters, G. A., and Savic, D. A. �2000�. “Opti-mal sampling design for model calibration using shortest path, geneticand entropy algorithms.” Urban Water, 2�2�, 141–152.

Kang, D. S., Pasha, M. F., and Lansey, K. �2008�. “Approximate methodsfor uncertainty analysis of water distribution systems.” Urban Water,in press.

Kapelan, Z. S., Savic, D. A., and Walters, G. A. �2003a�. “A hybridinverse transient model for leakage detection and roughness calibra-tion in pipe networks.” J. Hydraul. Res., 41�5�, 481–492.

Kapelan, Z. S., Savic, D. A., and Walters, G. A. �2003b�. “Multiobjectivesampling design for water distribution model calibration.” J. WaterResour. Plann. Manage., 129�6�, 466–479.

Kapelan, Z. S., Savic, D. A., and Walters, G. A. �2005�. “Advancedcalibration of water distribution models using the Bayesian type pro-cedure.” Proc., Computing and Control in the Water Industry: WaterManagement for the 21st Century, Vol. 1, 167–172.

Lansey, K., and Basnet, C. �1991�. “Parameter estimation for water dis-tribution networks.” J. Water Resour. Plann. Manage., 117�1�, 126–144.

Lansey, K., El-Shorbagy, W., Ahmed, I., Araujo, J., and Haan, C. �2001�.“Calibration assessment and data collection for water distribution net-works.” J. Hydraul. Eng., 127�4�, 270–279.

Meier, R. W., and Barkdoll, B. D. �2000�. “Sampling design for networkmodel calibration using genetic algorithms.” J. Water Resour. Plann.Manage., 126�4�, 245–250.

Xu, C., and Goulter, I. �1998�. “Probabilistic model for water distribution
reliability.” J. Water Resour. Plann. Manage., 124�4�, 218–228.

age. 2009.135:38-47.

Documents

Effect of Uncertainty on Water Distribution System Model Design Decisions