Water Quality Model Calibration under Unknown DemandsP. M. R. Jonkergouw1; S.-T. Khu2; Z. S. Kapelan3; and D. A. Savić4
Abstract: It has often been cited that a water distribution system �WDS� hydraulic model needs to be highly accurate before it may beused in combination with a water quality model �WQM� to simulate the dispersion and decay of a residual disinfectant. However, evena well-calibrated WDS hydraulic model may not have data relating to the specific water demands during a given period, which mayimpede WQM calibration. This study examines using residual disinfectant data to calibrate a WQM under unknown or uncertain demandsby calibrating a residential demand multiplier pattern �DMP� in tandem with the WQM parameters. Two artificial scenarios and one realcase study are investigated. The artificial scenarios are used to �1� verify the proposed methodology under ideal conditions and �2� validatethe proposed methodology when the hydraulic model and calibration data contain realistic errors. The real case study uses residualchlorine data and a WDS model for which a hydraulic and WQM calibration had been performed previously. The estimated demands fromthe real case study are validated using tracer test data. Results from the artificial case studies may be summarized as follows: �1� theproposed methodology can estimate the demands and calibrate WQM parameters correctly, although increasing model and calibration dataerrors adversely affect calibration results; �2� the calibrated WDS models reproduce the true residual chlorine concentrations with verylittle error. Results from the real case study indicate that the original WQM calibration was performed using underestimated WDSdemands. Tracer test data confirm that the calibrated DMP provides good hydraulic velocities. The calibrated WDS model from the realcase study is in good agreement with measured residual chlorine concentrations. The mean absolute error between the simulated chlorineconcentrations from the calibrated network model and the observed values is 0.059 mg /L.
DOI: 10.1061/�ASCE�0733-9496�2008�134:4�326�
CE Database subject headings: Water quality; Calibration; Water distribution systems; Hydraulic models.
Introduction
Chlorine is widely used as a drinking water disinfectant becauseof its applicability, relatively low cost, effectiveness, and abilityto provide disinfectant residual throughout a water distributionsystem �WDS�. However, a major drawback of chlorine as a dis-infectant is the formation of harmful disinfection by-products andpossible odor problems. Therefore, the use of chlorine as a disin-fectant becomes a trade off: An increased chlorine dosage in-creases disinfection by-product formation, while a lower chlorinedosage has diminished disinfection efficiency. In addition, chlo-rine undergoes chemical reactions in the bulk of the water and atthe pipe wall, which causes the concentration to decay over time,thereby necessitating even higher chlorine dosages at the watertreatment plant in order to maintain a residual chlorine concentra-
1Research Student, Center for Water Systems, Univ. of Exeter, ExeterEX4 4QF, U.K. E-mail: [email protected]
2Senior Lecturer, Center for Water Systems, Univ. of Exeter, ExeterEX4 4QF, U.K. E-mail: [email protected]
3Senior Lecturer, Center for Water Systems, Univ. of Exeter, ExeterEX4 4QF, U.K. E-mail: [email protected]
4Codirector, Center for Water Systems, Univ. of Exeter, Exeter EX44QF, U.K. E-mail: [email protected]
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tion capable of preventing microbial regrowth and limiting bio-film augmentation throughout the WDS.
The complex behavior of chlorine dispersion and decaythroughout a WDS can be modeled using a chlorine decay modelcoupled with a hydraulic WDS model. Results from water qualitymodel �WQM� simulation�s� may be used to optimize the chlorinedosing at the water treatment plant or to determine optimum chlo-rine booster locations �Cozzolino et al. 2005�. However, a modelrequires calibration before useful results may be obtained �Walski1983�. While the calibration of water quantity WDS models hasbeen researched extensively �Kapelan et al. 2003, 2006�, only alimited number of studies have been dedicated to calibratingWQM parameters. Vasconcelos et al. �1997� calibrated globalwall correlation coefficients for various WDS models throughtrial and error, Maier et al. �2000� used a reverse-calibrationmethod, and Munavalli and Mohan Kumar �2003, 2005� appliedan automated Gauss-Newton minimization technique to calibrateWQM parameters under dynamic state conditions. The preferredmethod for WDS WQM calibration requires an extended periodsimulation �EPS� and, thus, one or more demand multiplier pat-terns �DMPs� �assuming demand-driven WDS modeling soft-ware�. Previous studies have assumed that the underlyinghydraulic model flows and demands from the WDS are correctprior to WQM calibration. It has been shown that the reliability ofresults from WQM parameter calibration is not only largely de-pendent on the quality of the calibration data �Maier et al. 2000�,but also on the accuracy of the underlying hydraulic model �Vas-concelos et al. 1997�. Thus, errors in a WDS hydraulic modelcould result in the incorrect calibration of WQM parameters.While nothing may be done about the quality of the residualdisinfectant data, the accuracy of the underlying hydraulic model
may be improved upon through calibration of demands using re-
sidual disinfectant data when flow data is unavailable. This studyinvestigates results from the simultaneous calibration of WQMparameters and demands using water quality data and a previ-ously calibrated hydraulic WDS model. WDS simulations are per-formed using the EPANET 2 WDS simulation softwaredistributed by the United States Environmental Protection Agency�Rossman 2000�.
The optimization algorithms, network models, calibration datasets, and methodology used are discussed in the following sec-tions. Results are presented and discussed and conclusions aredrawn in the subsequent sections.
Methodology
Water Quality Modeling
This section will discuss and explain the equations behind a com-monly used WDS chlorine decay model. More comprehensivedescriptions of these and other chlorine decay models can befound in Hua et al. �1999�, Powell et al. �2000�, Rossman �2000�,Clark and Sivaganesan �2002�, and Boccelli et al. �2003�, to namebut a few. The equations shown in this section for calculating thechlorine concentration in a simulated water distribution systemare those used by EPANET 2.
Because chlorine decay in WDS pipes is said to be causedboth by aqueous constituents �e.g., ammonia and natural organicmatter� as well as pipe wall materials �e.g., the corrosion of ironand biofilm�, the decay of chlorine is commonly modeled as oc-curring at two physically distinct sites, viz., in the bulk of thewater and at and near the wall of the pipe. These models aresuperimposed on a transport model to calculate the chlorine con-centration in a given pipe as follows:
�C
�t= − u
�C
�x− �kb + Rw�C �1�
where C=chlorine concentration �mass/volume�; u=flow velocity�length/time�; x=position in the pipe; kb=first order rate constant�1/time� of the bulk reaction; and Rw=flow-dependent wall reac-tion coefficient �1/time�. Eq. �1� demonstrates the dependency ofthe chlorine concentration on the rate of the decay reaction andthe flow velocity of the water.
When zero-order chlorine wall decay kinetics are assumed, thepipe wall reaction rate is limited by the rate of mass transfer to thepipe wall and becomes a function of the chlorine concentration
Rw =4
dmin� kw
C,Mf� �2�
where d=diam of the pipe �length�; kw=wall decay rate constantof the pipe �mass/area/time�; and Mf =flow-dependent mass trans-fer coefficient of the pipe �length/time� �Rossman 2000�.
Mixing at junctions is assumed to be instantaneous and com-plete and the chlorine concentration leaving junction k will be theflow weighted average of the chlorine concentrations from allinflowing sources �from the upstream pipes and/or an externalsource�, as follows:
Ck,out =�i�Ik
�Qi,inCi,in� + Qk,extCk,ext
�i�Ik�Qi,in� + Qk,ext
�3�
where Ck,out=chlorine concentration leaving junction k �mass/volume�; Ik=set containing the pipes with water flowing to junc-
tion k; Qi,in=flow into junction k from pipe i �volume/time�;
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Ci,in=chlorine concentration of water entering junction k frompipe i; Qk,ext=external flow into junction k; and Ck,ext=chlorineconcentration of the water entering junction k from an externalsource �e.g., source concentration of a reservoir�. Water with anonzero chlorine concentration that enters the WDS model has tobe specified by the user and will be referred to as the input orsource concentration.
Objective Function
The objective of the calibration is to obtain a good correlationbetween the simulated and observed chlorine concentrations overthe entire simulation period. Here, this is achieved by minimizingthe sum of squared errors �SSE� between observed and simulatedchlorine concentrations, as follows:
Minimize �p�Pk
�j=1
mp
�Cp,tj
obs − Cp,tj
sim�2 �4�
where Pk=set containing the junctions with measurement data,mp=number of observations for junction p; and Cp,tj
obs andCp,tj
sim=observed and simulated chlorine concentrations at junctionp and time tj, respectively. It may happen that the time of obser-vation tj does not coincide with the simulated time, in which caseCp,tj
sim is interpolated from the computed chlorine concentrations atjunction p just before and after time tj.
Optimization Algorithms
This study uses a stochastic optimization algorithm followed by aderivative-based optimization algorithm to minimize the objectivefunction value and calibrate model parameters. In particular, thisstudy uses a modified version of the shuffled complex evolution�SCE� algorithm, originally developed by Duan et al. �1993�, todetermine an appropriate starting point for the Levenberg-Marquardt optimization algorithm. While it will be shown hereinthat these algorithms can be used successfully to calibrate themodel parameter values, it is not the objective of this study tolimit the methodology to these algorithms. Instead, it is the pur-pose of this study to demonstrate a practical use for residual chlo-rine data, which has not been considered before. As such, anysuccessful stochastic algorithm may be used in conjunction with aderivative-based algorithm to determine the appropriate param-eter values. Nevertheless, for completeness, a summary ofthe modified version of the original SCE algorithm is given asfollows:1. Define the number of complexes q, the number of samples
per complex m, the number of evolutionary iterations percomplex g, and the minimum number of objective functionevaluations n as a termination criterion. Here, a complex isdefined as a subset of m samples taken from the population.
2. Initialize a random population of q�m samples and evaluatethe objective function values.
3. Randomly select q complexes of m samples.4. Evolve each individual complex, as follows:
a. Sort the complex according to objective function value;define the lowest �best� objective function value f1;draw a random value b from a uniform random distri-bution between 0 and 1; determine the weighted cen-
where xi= ith sample in the complex.b. Produce a new sample xnew, as follows:
xnew = 2 � xce − xm
c. Evaluate the objective function value of the newsample fnew. If f1 / fnew�b, continue to step 4g. Other-wise, continue to the next step.
d. Determine the standard deviation of the parameter val-ues in the complex �x. Produce a new sample xnew, asfollows:
xnew = N�x1,�x2�
e. Evaluate the objective function value of the newsample fnew. If f1 / fnew�b, continue to Step 4g. Other-wise, continue to the next step.
f. Produce a new sample xnew, as follows:
xnew = 0.5 � �xce + xm�
g. Replace the worst sample xm with the new sample xnew.h. Repeat Steps 4a–4g g times.
5. Reinsert the evolved complexes into the population.6. If the current number of objective function evaluations is less
than the minimum number of sample evaluations n, continueto Step 3. Otherwise, stop the algorithm.
The original SCE algorithm is a population-based evolutionaryalgorithm that was developed with the strengths and concepts ofexisting algorithms such as the simplex procedure �Nelder andMead 1965�, controlled random search �Price 1978�, and competi-tive evolution �Holland 1975� in mind. Full details of the shuffledcomplex evolution algorithm may be found in Duan et al. �1993�.The modified algorithm was found to consistently outperform theoriginal algorithm in terms that it found solutions with lowerobjective function values and was found to be less sensitive tovariations in the initial population.
Finally, the widely used Levenberg-Marquardt algorithm isused to improve the best solution found by the modified SCEalgorithm. The resultant solution is a calibration solution that rep-resents a minimum with respect to the chosen minimum change inthe parameter values. The Jacobian matrix used in the Levenberg-Marquardt algorithm is calculated as discussed in the followingsection.
Parameter Uncertainty
Parameter uncertainty is often cited in terms of the 95% confi-dence intervals, which can be determined from the parametervariance �k
2 obtained from the diagonal elements of the parametervariance-covariance matrix �Kapelan et al. 2003�
Cov = s2�JTJ�−1 �5�
where s2 is taken to be the �nonweighted� variance of the chlorineerror residuals; and J=generalized sensitivity �Jacobian� matrixand is defined as
J =�Ci
�ak�i = 1, . . . ,ns;k = 1, . . . ,np� �6�
where ak=parameter k; ns=number of data measurements, andnp=number of parameters. The Jacobian matrix has np columns
and ns rows. The 95% confidence interval of each parameter k is
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defined as two standard deviations �2�k=2�Cov�k ,k�� above andbelow the calibrated parameter value.
The objective function described above is relatively time con-suming because both a WDS hydraulic and a water quality simu-lation are required for each evaluation. As a result, there is a limitto the number of evaluations that can be performed to determineparameter sensitivities and some simplifying assumptions have tobe made. In particular, the elements of the Jacobian matrix areapproximated using the central finite differences method becauseof the relatively small number of function evaluations that arerequired �2�np+1�. A description of the central finite differencemethod can be found in Ames �1977�.
Calibration Data
In order to avoid potential interference during calibration fromincorrect initial nodal chlorine concentrations, all values were setequal to zero at the start of the simulation �except for water leav-ing the reservoir�. This also meant that observational data thatwere dependent on the initial chlorine concentrations of nodes inthe WDS model and not the chlorine concentration of water leav-ing the reservoir should be removed from the calibration data set.For this reason, the transport times of water from the source at thestart of the simulation to the sampling locations were estimatedand incremented by 1 h and any observational data relating to atime before these values were removed from the data set.
Calibration Parameters
Three different types of parameters are calibrated and each isexplained in the following sections.
Demand Multiplier Pattern ParametersEPANET 2 is a demand-driven water distribution network mod-eling software that uses temporal demand multiplier patterns�DMPs� to periodically vary demands �and thereby flows� of agroup of nodes over the course of an EPS. At each hydraulic timestep, the relevant value in the pattern is multiplied with the basedemands of the nodes in the group to determine the actual de-mand from the node. The base demand of a node is a constantvalue that represents the relative demand of the node with respectto the other nodes in the group. The assumption is that junctionswithin a given group exhibit the same relative temporal variationin demand.
The main residential DMP will be calibrated in order to repro-duce the correct nodal demands in a WDS model. It is assumedthat there is no a priori knowledge of the values in the pattern,except that the minimum and maximum values are within pre-defined limits, here 0 and 2.5, respectively. Preliminary tests in-dicated that the proposed methodology provided acceptableestimates when each parameter value represents a base demandmultiplier for 2 consecutive h.
Wall Correlation Coefficient ParametersSeveral studies have suggested that pipes may be grouped accord-ing to geographical location for water quality modeling purposesand calibration �Zierolf et al. 1998; Mallick et al. 2002�. In thisstudy, the network pipes are manually divided into several groupsaccording to the distance from the source. Each zone is assignedone correlation coefficient parameter that defines the wall decay
where kw,i=wall decay constant of pipe i, CHW,i=Hazen-WilliamsC-factor of pipe i; Fz=wall correlation coefficient of zone z; andIz=set of pipes in zone z. The Hazen-Williams C-factor is anindicator of pipe material and age and the above equation, there-fore, relates the decay of chlorine at or near the pipe wall to thesame qualities. A relationship between pipe material, roughnessand age, and the extent of chlorine consumption at and near thepipe wall has been observed previously �Kiéné et al. 1998;Hallam et al. 2002�, which suggests that Eq. �7� is a practicalrelationship.
It has been observed that the overall �bulk and wall� pipedecay rate increases with decreasing pipe diameter �Sharp et al.1991�. This observation may be explained by the increase in theratio between the pipe wall surface and the internal pipe volumein smaller diam pipes and the subsequent �relative� increase inchlorine demand by the pipe wall. For this reason, some studieshave grouped network pipes according to pipe diam �Munavalliand Mohan Kumar 2005�. However, this form of grouping is un-necessary when the relationship between the pipe surface/pipevolume ratio and the wall decay coefficient is used �as shown inEq. �2��.
Additionally, Speight et al. �2004� investigated different pipestratification criteria for water quality characterization and foundthat spatial stratification with distance from the source as acriterion provides more reliable results than pipe diameter andmaterial.
During calibration, the value of the grouped wall correlationfactors Fz are allowed to vary between −21,500 mg /m2 /day and0. Such a large range ensures that the maximum wall decay ratecan be reached, viz., the diffusion-limited rate of decay, as shownin Eq. �2�.
Source Chlorine Concentration ParametersThe variation of the source chlorine concentration has to beknown and specified a priori and is often measured. However,there is no guarantee that the measured values are correct andmay contain large errors that will be progressed throughout theWDS water quality simulation. For this reason, the source con-centration at each hydraulic time step will be allowed to vary upto 0.3 mg /L from the measured value. For example, if themeasured value at a given hour is 0.7 mg /L, the calibrationsearch range is bound between 0.4 and 1.0 mg /L. Given thatcommon chlorine measurement techniques usually have a stan-dard deviation of less than 0.1 mg /L �e.g., N,N�-diphenyl-p-phenylenediamine �DPD� colorimetric method�, these boundsshould cover more than 99% of the errors.
Case Studies
In total, three scenarios are investigated. All three studies usevariants of the same WDS model, details of which are given inthe following section. The first two scenarios use artificially gen-erated data sets and are used to verify and evaluate the proposedmethodology when the true parameter values are known. Thethird scenario uses measured data obtained by Vasconcelos et al.�1996�, for which true model parameter values are unknown. Allscenarios calibrate the same parameters, as discussed previously.
After several trials with different settings, the parameters ofthe modified shuffled complex evolution optimization algorithm
used during this study are given as follows: number of complexes
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q=20; number of samples per complex m=25; number of itera-tions per complex g=25; total number of sample evaluationsn=50,000.
For every case study, five independent calibration runs wereperformed. Initial populations were created by selecting valuesfrom a uniform random distribution over the given parameterranges. For every calibration run, the modified SCE algorithmwas used to obtain a starting point for the Levenberg-Marquardtalgorithm �similar to the calibration model described in Kapelan�2002��, as discussed above. Finally, the best �lowest objectivefunction value� solution from the five independent calibrationruns was selected as the calibration solution.
The calibration solution was used to determine parameter sen-sitivities and uncertainties following the central finite differencesapproach. The demand multiplier pattern parameter values, sourcechlorine concentration values, and wall correlation coefficientswere perturbed by 0.005, 0.005 mg /L, and 50 mg /m2 /day aboveand below the calibrated values, respectively, to determine theelements of the Jacobian matrix. These perturbations representapproximately 0.5–1% of the model values and were found to besufficient to avoid significant round-off errors.
Water Distribution Network Model
Fig. 1 shows the pipe layout of the WDS model used in this study.The WDS model is based on a portion of a larger system that wasstudied previously in �Vasconcelos et al. 1996, 1997�.
The duration of the extended period simulation represents 35 hwith the hydraulic and pattern time interval set to 1 h and thewater quality time interval to 5 min. Four 24 h demand multiplierpatterns are used to represent the relative temporal demand fluc-tuations �N.B. the EPANET 2 simulator repeats a pattern wherethe duration of an extended period simulation exceeds the dura-tion of the pattern�, as follows: �i� a residential DMP; �ii� a com-mercial DMP; �iii� a school DMP; and �iv� a constant �sampling�DMP. The DMP that describes relative residential demand fluc-tuations is applied to approximately 75% of junctions with non-zero demand. The commercial and school DMPs are applied to a
Fig. 1. Schematic pipe layout of the water distribution system modelused in this study
small number of nodes. The constant demand multiplier pattern is
used to represent demand at sampling �hydrant� locations. Exceptfor the residential pattern, it is assumed that DMPs �ii�, �iii�, and�iv� are known a priori and are not included for calibration.
The bulk decay order of the WQM was set to 1 and the pipebulk decay rate was set to −0.232 day−1 as determined by mul-tiple bottle tests �Vasconcelos et al. 1996�. Altogether, the mea-sured data set, obtained here through personal communication�Rossman, personal communication, 2004�, is comprised of 1,024residual chlorine values spread over 31 locations �Vasconceloset al. 1996�. The calibration data set used for the network modelin this study consisted of 26 locations and 673 sampling valuesafter separating part of the calibration data as outlined above. Thesampling locations are indicated in Fig. 1.
For calibration of the pipe wall correlation coefficients, threegroups were selected manually after evaluating results from sev-eral trials with different pipe groupings. The pipe groups indi-cated in Fig. 1 are those used during calibration and provided themost reliable and accurate results. Manual selection of appropri-ate pipe groups is a time-consuming process and may be im-proved upon in the future, perhaps by using results from asensitivity analysis. Pipe grouping is a separate study subject thatwill not be discussed further herein.
Preliminary results indicated that the value of the source chlo-rine concentration at the 35th simulation hour did not affect theobjective function value due to the lack of calibration data andwas, therefore, excluded from calibration. In total, three differenttypes of model parameters resulting in 49 variables were cali-brated simultaneously: 12 DMP, three wall correlation coefficient,and 34 source chlorine concentration parameters.
Results and Discussion
Artificial Scenario 1: Methodology Verification
In this scenario, the calibration data set is generated using a WDSsimulation with a given �2 hourly� residential DMP, source chlo-rine concentration pattern, and single wall correlation coefficientusing a zero-order chlorine wall decay model. The given WDSmodel is calibrated using the generated data set and the purposeof this scenario is to verify the proposed methodology under idealconditions.
As described previously, the parameter search space for thevalues in the source chlorine concentration pattern are definedwith an initial estimate of the pattern, usually measured values.Here, rather than using the true pattern, an initial estimate isgenerated by introducing random errors ��0.3 mg /L� to the truevalues. This ensures that the true values are not in the center ofthe search space, which could have benefited the proposedmethodology.
Table 1. Summary of Calibration Results from Artificial Scenario 1
Parameters
Mean error Maximum error
AbsoluteRelativea
�%� AbsoluteRelativeb
�%�
DMP 0.01 0.7 0.03 2.2
RCCPc 0.004 mg /L 0.4 0.01 mg /L 1.3
Fz 77 mg /m2 /day 1.1 158 mg /m2 /day 2.3aWith respect to the mean of the true values.bWith respect to the true value.c
Reservoir chlorine concentration pattern.
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Calibration results from the first artificial scenario aresummarized in Tables 1 and 2 and shown graphically in Figs. 2and 3. It may be seen from Table 1 that the proposed calibrationmethodology resulted in good estimates of the model parametervalues. All errors are �2.3% of the true value and the averagerelative �absolute� error is 0.58% of the true value. Table 2 showsthe absolute values of the calibrated and true wall correlationcoefficients and the 95% confidence intervals. Figs. 2 and 3 showthe calibrated and true residential demand multiplier pattern andthe reservoir chlorine concentration pattern, respectively,including the 95% confidence intervals.
The standard deviation between the calibration data set and thecalibrated WDS model simulation output is s=8.5�10−3 mg /L.Due to this low value, the 95% confidence intervals are almostreduced to zero �see Eq. �5�� and do not generally include the trueparameter values. The latter is likely due to the nonlinearity of themodel equations and the assumption of linearity when calculatingthe parameter sensitivities. Unfortunately, there is no viablealternative than to assume linearity and use approximate methodswhen the objective function is time consuming. Nonetheless, thesensitivity analysis demonstrates that the objective function issensitive to all parameters. Not unsurprisingly, the greatest 95%confidence interval �0.5% of the true value� is associated with thesource chlorine injection parameter at the 34th h �the last value inthe source chlorine concentration pattern�. This may be seen inFig. 3 where the gray lines above and below the calibrated valuesrepresent the 95% confidence intervals.
The results demonstrate that the best solution found by thestochastic algorithm did not provide a starting point that allowedthe Levenberg-Marquardt to converge to the global optimum,which highlights the complexity of the model equations and thecalibration problem at hand. Nevertheless, the combination of thestochastic algorithm with the gradient-based algorithm hasprovided good estimates of the parameter values in an idealized
Table 2. Artificial Case Studies: Zero-Order Wall CorrelationCoefficients and 95% Confidence Bounds ��2��
z
Fz �mg /m2 /day�
True Scenario 1 Scenario 2
1 −6,997 −7,154 �5.8 −6,583 �86
2 −6,997 −6,977 �6.4 −7,075 �110
3 −6,997 −7,052 �7.0 −7,014 �146
Fig. 2. Artificial Scenario 1: comparison of the true and calibrated24 h residential demand multiplier pattern and 95% confidenceintervals �2��
case study. The differences between the calibrated and originalWDS model are negligible and overall agreement is very good.
Artificial Scenario 2: Methodology Evaluation
In this scenario, the artificial calibration data were generated froma WDS model with an hourly residential DMP, an hourly sourcechlorine concentration pattern, and a single wall correlation coef-ficient using the same measurement locations and times as in thereal chlorine concentration data set of the WDS model. A zero-order wall decay model was used. Various error distributions wereadded to several WDS model parameters as well as the calibrationdata set, exact details of which are given in Table 3, in order tocreate a realistic calibration scenario where true parameter valuesare known. Apart from the effect on the hydraulic performance ofthe WDS model, the normally distributed noise added to all theHazen-Williams coefficients ensures that a single wall correlationcoefficient for multiple pipes can never reproduce the same walldecay constants used to create the artificial data set. However,since the errors in the Hazen-Williams coefficients are normallydistributed around the mean �i.e., the true values�, the optimalwall correlation coefficients for each pipe group should be thesame as �or very similar to� the values used to create the calibra-tion data set �shown in Table 2�. The normally distributed errorsin all nodal base demands represent the over- and underestimationof the average demands, similar to what may be expected in realwater distribution system models.
The reason for the reduction in the DMP resolution for cali-bration compared to that used for the creation of the artificial dataset is twofold. First, the reduction in resolution attempts to repre-sent the same simplification that applies to any DMP. To clarify,WDS model DMPs do not capture the true and continuous varia-tions in the aqueous demands, but instead average �often hourly�values are used. Similarly, in this study, a 1 h resolution DMP isused to create an artificial data set, whereas a 2 h DMP is usedduring calibration. Second, a 2 h resolution was found to producebetter results than a 1 h resolution DMP, for reasons that will be
Table 3. Artificial Error Distributions Used in Scenario 2
Error distribution
Junction based demands �N�1,0.0025�Hazen-William C-factors �N�1,0.01�Source chlorine pattern +N�1,0.01�Calibration data +N�0,0.0064�
Fig. 3. Artificial Scenario 1: comparison of the true and calibrated temintervals �2��
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discussed later. The objective of this scenario is to evaluate theapplicability and usefulness of the proposed methodology underrealistic conditions.
Calibration results from the second artificial case study aresummarized in Table 2 and shown in Figs. 4 and 5. The standarddeviation between the artificial calibration data set and the cali-brated WDS model is s=0.077 mg /L and the square of Pearson’scorrelation coefficient is R2=0.917. Importantly, both these valuesare slightly better than those obtained when the correct WDSmodel is used in combination with the data set with the artificiallyintroduced errors �s=0.08 mg /L and R2=0.915, respectively�.Consequently, the presence of significant errors in the WDSmodel and the calibration data can result in an optimal solutionwhere the parameter values differ from the true parameter values.Logically, it is very likely that the amount of deviation from thetrue parameter values depends on the extent of errors that arepresent in the calibration data set and the hydraulic WDS model.In this scenario, where significant and realistic errors are intro-duced into the calibration data set and the WDS model param-eters, the calibration solution provides parameter values that aresimilar to the true values.
All three calibrated wall correlation coefficients correspondvery well with the true values, as shown in Table 2. The table alsoincludes the 95% confidence intervals. It may be seen that theintroduction of �artificial� errors results in an increase in para-meter uncertainty compared to results from the first artificialscenario.
The original and calibrated models have different flowsthroughout the distribution network due to the errors in the nodal
variation in the reservoir chlorine concentration and 95% confidence
Fig. 4. Artificial Scenario 2: comparison of the true and calibrated24 h WDS demand and 95% confidence intervals �2��
base demands and the Hazen-Williams C-factors. As a result, adirect comparison of the calibrated and original DMP should notbe made, but, instead, the total temporal demand variation �fromthe reservoirs� is compared in Fig. 4. Any additional errors be-tween the true and calibrated flows throughout the water distribu-tion system are the result of the errors in the hydraulic parametersand depend on the quality of a prior hydraulic WDS model cali-bration. It may be seen that the correlation between the reservoirdemand from the calibrated and the correct WDS model �that wasused to create the artificial chlorine data� is reasonably good, butthere are some large differences. On average, the calibrated WDSdemand is overestimated by 19.9 m3 /day over 35 h, which is2.10% of the average WDS demand. Although not conclusive,this suggests that even in the presence of significant errors, theoptimal solution requires a good estimate of the water balance.
Furthermore, this provides a plausible reason for why calibra-tion results were found to be more reliable when a DMP with a2 h, rather than 1 h, resolution was used. When a 1 h resolution isused, there is greater flexibility in the model output and, thus,more opportunity to deviate further from the true parameter val-ues �since it has already been shown that the solution with lowestobjective function does not necessarily provide the true parametervalues�. However, it appears that a good solution is defined moreby its agreement with the overall average demand from the res-ervoir than by temporary �and potentially large� deviations fromthe true demand. Hence, for this study, a 2 h resolution may havebeen the appropriate trade off between accuracy and reliability.
The standard deviation between the calibrated and true sourcechlorine concentration pattern is just under 0.05 mg /L, not unlikethe error that may be expected from standard colorimetric aque-ous chlorine concentration measurements. Thus, unless it isknown with sufficient certainty that a measured source chlorineconcentration pattern is accurate, the proposed methodology canhelp to establish the true input concentration.
Similar to the first scenario, the parameter 95% confidenceintervals, indicated by the gray lines above and below the cali-brated values in Figs. 4 and 5, do not always include the trueparameter values. The reason for this is likely to be the same asoutlined for the first artificial scenario. Nevertheless, it may beseen that the parameter uncertainty has increased, which will bepartially due to the increase in the resultant variance between theartificial data set and the simulated values. Furthermore, the sen-sitivity analysis results indicate that the final two values of thesource chlorine concentration pattern cannot be calibrated reliablydue to the relatively large uncertainty in the final parameter values
Fig. 5. Artificial Scenario 2: comparison of the true and calibrated temintervals �2��
�as shown in Fig. 5�.
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Finally, it may be seen from Fig. 6 that despite the errors in theWDS hydraulic model and the WDS water quality model, thesimulated residual chlorine concentration from the calibratedWDS model and the true chlorine residuals are in excellent agree-ment. The standard deviation between the correct residual chlo-rine data set �prior to the addition of artificial errors� and thesimulated chlorine residuals from the calibrated WDS model isjust under 0.03 mg /L and the correlation is R2=0.99. This sug-gests that the demands in a WDS model do not need to be as exactas might have been expected in order to provide useful and accu-rate results from a WDS water quality model simulation. Impor-tantly, it suggests that the use of average demand values issufficiently accurate for water quality modeling purposes. How-ever, more research would be required in order to confirm this.
variation in the reservoir chlorine concentration and 95% confidence
Fig. 6. Artificial Scenario 2: comparison of the true and calibratedresidual chlorine concentrations at stations OH02, OH12, and OH20
The methods described above are applied using a real WDSmodel and residual chlorine concentration data set. The WDSmodel, calibration data set, and calibration parameters are as de-scribed above. The calibration results are summarized in Figs. 7and 8, and Table 4.
Results from Artificial Scenario 2 indicated that although thecalibrated demand may temporarily deviate from the true demand,the calibrated and true average demand should be in close agree-ment in order to obtain good water quality simulation results.According to the calibrated WDS demand, the original demandwas underestimated by 192 m3 /day, or 21.7%, on average. Themost significant differences between the calibrated and originalWDS demand are the two hours following 7 a.m., 1 p.m., 1 a.m.,and 5 a.m. While the high demands from 5 a.m. until 9 a.m. andbetween 1 p.m. and 3 p.m. can be explained by elevated earlymorning and afternoon water demands, respectively, the increaseddemand between 1 a.m. and 3 a.m. is probably an incorrect esti-mation due to the presence of errors in the WDS model and thecalibration data. Due to the surprisingly large difference betweenthe two WDS demand variations, the true demand is very likely tobe greater than the original estimate. However, as shown by theresults from Artificial Scenario 2, the calibrated WDS demandsare not likely to be highly accurate and should be considered asimproved estimates only.
The calibrated and original source chlorine concentration pat-terns are in fairly good agreement, which indicates that the source
Fig. 7. Real case study: comparison of the original �Vasconceloset al. 1996� and calibrated �this study� 24 h WDS demand and 95%confidence intervals �2��
Fig. 8. Real case study: comparison of the original �as used by Vascthe reservoir chlorine concentration and 95% confidence intervals �2
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concentration pattern may not have required calibration or adjust-ment. The standard deviation and mean error between the twopatterns are 0.08 and 0.00 mg /L, respectively.
Table 4 shows that the calibrated wall correlation coefficientsdecrease �i.e., the decay rates increase� with distance from thesource. The high negative value of the third wall correlation co-efficient F3 suggests that wall decay in that zone is often limitedby the diffusion of chlorine from the bulk of the water to the pipewall. One reason for increasing wall decay rate constants withdistance from the source is that the buildup of biofilm and otherparticulates might be greater where chlorine concentrations areconsistently lower, such as further from the source, resulting in anincrease of chlorine-demanding materials. Wable et al. �1991� ob-served that the wall decay rates of a Paris WDS chlorine decaymodel were greater where the quality of the water contained moreorganic content. These results may be related to a similarphenomenon.
Results from Artificial Scenario 2 have already shown thatthe proposed methodology can reproduce the “true” residualchlorine concentrations with relatively low error, even in the pres-ence of realistic errors in the calibration data and the WDS model.The correlation between the calibration data and the simulationresults from the calibrated network is R2=0.97% compared toR2=0.93% for the original WDS model. Likewise, the standarddeviation of the residual errors between the calibrated model andthe calibration data is 0.078 mg /L with mean −0.005 mg /L, ver-sus 0.106 mg /L and 0.012 mg /L for the original network, respec-tively. Fig. 9 shows the simulated chlorine residual by thecalibrated WDS model from this study and from Vasconceloset al. �1997�, as well as the measured chlorine concentrations atthree sampling locations in the network �OH02, OH12, andOH20�. The figure shows that the observed chlorine concentra-tions and the simulated residuals from the calibrated network inthis study are in good agreement. In particular, the figure showsthe improvement in correlation between the observed and simu-
Table 4. Real Case Study: Zero-Order Wall Correlation Coefficients and95% Confidence Bounds ��2��
z
Fz �mg /m2 /day�
Vasconcelos et al. �1997� This study
1 −6,994 −6,795 �140
2 −6,994 −8,031 �204
3 −6,994 −12,782 �280
s et al. 1996, 1997� and calibrated �this study� temporal variation in
lated chlorine concentrations at station OH12 as compared to theoriginal calibration results, due to the changes in timing of theincreasing and decreasing of the chlorine concentration. This isvery likely to be the result of an improvement in hydraulic flowsrather than any improvement in calibrated wall decay constants.On the other hand, the results from station OH20 show that theaverage simulated chlorine concentration by the calibrated net-work model from this study is slightly closer to the average of theobserved concentration as opposed to the original calibration re-sults, which is more likely to be the result of an improvement inthe chlorine decay modeling results, rather than in the hydraulicmodel.
Finally, Fig. 10 shows the correlation between measured fluo-ride concentrations obtained during a tracer test �performed whilethe chlorine measurements were also taken� and the simulatedfluoride concentrations using the original WDS model �Vasconce-los et al. 1996� and the calibrated WDS model from this study. Itserves to demonstrate that the calibrated WDS model is producingwater travel times that are compatible with tracer test observa-tions. Fluoride measurements were taken at 26 locations and theresults shown in Fig. 10 are typical.
Conclusions
This study proposes a novel approach to WDS water qualitymodel calibration. In particular, a calibration model is proposedthat uses residual chlorine measurements to calibrate a residentialDMP and WQM parameters simultaneously when the true de-mands are unknown �or uncertain�. The model combines the useof a stochastic algorithm with the Levenberg-Marquardt algo-
Fig. 9. Real case study: comparison of the original �Vasconcelos etal. 1997� and calibrated �this study� residual chlorine concentrationsat stations OH02, OH12, and OH20
rithm to find the optimal calibration solution. Furthermore, the
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Jacobian matrix was used to calculate parameter uncertainties,where the elements of the matrix were determined using the cen-tral finite difference approach.
The simultaneous calibration of a residential DMP and WQMparameters was performed for two artificial scenarios and one realcase study. The first artificial scenario considers idealized situa-tion where the WDS model and calibration data contain no errors.Although the model parameters are not exactly reproduced by theproposed calibration model, the errors between simulation resultsfrom the calibrated WDS model and the true chlorine concentra-tions are negligible. Results from the scenario demonstrated thatthe proposed methodology can determine the true parameter val-ues with very little error ��2.3% � under idealized conditions.
Results from the second artificial scenario indicate that distrib-uted errors in the calibration data and the hydraulic WDS modelcan result in an optimal calibration solution where the estimatedparameter values differ from the true parameter values. Neverthe-less, results also demonstrated that the calibrated WDS modelreproduced the true water quality simulation results with verylittle error ��0.03 mg /L�, despite the presence of errors in theWDS model. Furthermore, it appeared that an accurate waterquality simulation solution required a good agreement with thetrue average demand over the whole simulation period �35 h�,while temporary deviations from the true hydraulic flows wererelatively unimportant. In other words, a temporary overestima-tion of demands is acceptable, in terms of water quality simula-tion results, provided it is followed �or preceded� by a similarunderestimation of demands. As a result, substantial errors in thecalibration data do not allow an accurate calibration of the DMP,but average demands can still be determined with high precision�2.1% error in the study considered here�.
Fig. 10. Real case study: comparison of the simulated residual fluo-ride concentration at stations OH02, OH12, and OH20 from this andthe original study �adapted from Vasconcelos et al. �1996��
Other results from the second artificial case study demon-
strated that the wall decay correlation coefficients can be cali-brated with the proposed methodology when significant errors arepresent in the calibration data set and the WDS model. Here, thecalibrated wall correlation coefficients were all within 6% of thetrue parameter values.
Results from the two artificial case studies indicate that mea-sured residual disinfectant data can be used to estimate demandsand WDS calibrate WQM parameters simultaneously, but the ac-curacy of the results are dependent on the quality of the calibra-tion data and the accuracy of the underlying WDS hydraulicmodel.
Results from the real case study indicated that the originalreservoir demand was underestimated by roughly 20% and fluo-ride data were used to show that the calibrated reservoir demandproduced aqueous velocities that are compatible with observation.
The results from this study have shown that the wall decay rateconstants can still be calibrated when demands are uncertain orunknown. Furthermore, these results have demonstrated that re-sidual chlorine concentration data contain valuable informationthat is traditionally unused. The results presented here suggestthat water quality data can be used in combination with hydraulicdata such as pressure and/or flow data to calibrate hydraulic andWQM parameters simultaneously. Additionally, in the absence ofmore appropriate data �or to increase parameter certainty�, mea-sured residual chlorine concentrations in a particular area of awater distribution system could be used to estimate average localdemands.
Acknowledgments
The writers thank Lewis Rossman of the United States Environ-mental Protection Agency for providing the water distributionmodel and measured chlorine residual and fluoride data, as wellas comments on the first version of this manuscript. The com-ments by the anonymous reviewers are also gratefully acknowl-edged.
Notation
The following symbols are used in this paper:ak � parameter k;b � random value from uniform random distribution
U�0,1�;C � chlorine concentration �mass/volume�;
Cp,tj
obs � observed chlorine concentration at time t andjunction j �mass/volume�;
Cp,tj
sim � simulated chlorine concentration at time t andjunction j �mass/volume�;
CHW,i � Hazen-Williams coefficient of pipe i;Ci,in � chlorine concentration entering a junction from
pipe i �mass/volume�;Ck,out � chlorine concentration leaving junction k
�mass/volume�;Ck,ext � chlorine concentration of water entering the
WDS through junction k �mass/volume�;di � diameter of pipe i �length�;Fz � wall correlation coefficient of zone z;f i � objective function value of sample xi in complex
in modified SCE algorithm;
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g � number of evolutionary steps per complex in themodified SCE algorithm;
Ik � set of pipes with water flowing to junction k;Iz � set of pipes in zone z;J � parametric Jacobian matrix;
kb,i � chlorine bulk rate constant of pipe i �1/time�;kf ,i � mass transfer coefficient of pipe i;kw,i � zero-order chlorine wall rate constant of pipe i
�mass/area/time�;m � number of samples per complex in modified
SCE algorithm;mp � number of measurement samples for junction p;
n � minimum number of objective functionevaluations;
np � total number of parameters;ns � number of measured chlorine concentration
samples;Pk � set of junctions with measured data;Qi � flow through pipe i �volume/time�;Qk � external flow into junction k �volume/time�;
q � number of complexes in modified SCE algorithm;Rw,i � flow-dependent wall reaction coefficient of pipe i
�1/time�;s � standard deviation between simulated and
observed chlorine concentration values;ui � flow velocity through pipe i �length/time�;x � position along the length of a pipe �length�;
xce � weighted centroid of a complex;xi � parameter set i in a complex in the modified
SCE algorithm;z � pipe zone number;
�k � standard deviation of parameter k from covariancematrix; and
�x � standard deviation of parameter x in a complex.
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