UNIVERSITY OF CINCINNATI

Transcription

1 UNIVERSITY OF CINCINNATI Date: I,, hereby submit this work as part of the requirements for the degree of: in: It is entitled: This work and its defense approved by: Chair:

2 Path-dependent Approach to Estimate Chlorine Wall Demand Coefficient in Water Distribution System A thesis submitted to the Division of Research and Advanced Studies of the University of Cincinnati in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY in the Department of Civil and Environmental Engineering of the College of Engineering FEBRUARY 2005 By Feng Shang B.S. Tsinghua University, Beijing, China, 1995 M.S. Tsinghua University, Beijing, China, 1998 Committee Chair: James Uber, Ph.D

3 Abstract A novel approach to calibrate pipe wall demand coefficients for chlorine decay using an input-output model framework is proposed. The input-output model of water quality in water distribution system developed in this research is presented as a particle (water parcel) backtracking algorithm, which provides information that is not available using traditional simulation approaches: the various paths that water takes between a particular input source and output node, and their associated time delays and impact on output node water quality. Such information constitutes a complete description of the input-output behavior under typical assumption of first-order chemical decay or production reactions. Therefore the chlorine concentration at downstream locations can be expressed explicitly as a function of concentration at upstream locations, and pipe wall demand coefficients. This not only expedites the computational efficiency of the parameter estimation algorithm, but frees the field study to consider chlorine concentration sampling at any location in a network. Chlorine measurement and hydraulic model errors will cause uncertainty in wall demand coefficient estimates and this should be taken into consideration in sampling design. In this research, sensitivity-based method is developed to select chlorine measurement locations for the purpose of getting reliable estimates of wall demand coefficients. The impact of hydraulic error on wall demand coefficient estimation of different designs is compared using a heuristic representing the sensitivity of modeled chlorine concentration to hydraulic error. Chlorine measurement errors are also considered and a D-optimal design algorithm is applied to select chlorine measurement locations. This

4 proposed approach is suboptimal, yet efficient and practical, and considers both hydraulic and chlorine measurement errors. A field study is conducted in a southern U.S. utility as an application of the parameter estimation approach. A tracer study is implemented and the conductivity measurements are used to calibrate an existing hydraulic model through a manual approach. Chlorine concentrations are monitored continuously both at water source and within the distribution system. The wall demand coefficient is calibrated, assuming all pipes have the same demand coefficient. i

5 Acknowledgements I would like to thank my advisor Dr. James Uber for his inspiration, guidance and encouragement in my graduate pursuit in a fascinating field. My thank also goes to Dr. Marios Polycarpou, my advisor in electrical engineering, for introducing me into multidisciplinary research and his insightful suggestions in my work. I would also like to thank other members of my committee, Dr. Lewis Rossman, Dr. Shafiqul Islam, and Dr. Steven Buchberger, for their valuable advice and helpful discussions. It has been really enjoyable experience to work with current and previous members of our research team: Marco Propato, Mike Tryby, Dr. Dominic Boccelli, Dr. Zhong Wang, and Megan Sekhar. Last but not least, thanks to my wife Kun Dong and my parents for their continuing love and patience. ii

6 Table of Contents Table of Contents List of Tables List of Figures iii vi vii 1 Introduction Modeling Chlorine Decay in Water Distribution System Wall Demand Coefficient Research Objectives Thesis Structure Literature Review Modelling Chlorine Decay in Water Distribution System Purpose of Water Quality Modelling in WDS Governing Equations for Chlorine Transport Modelling Approach Distribution System Model Calibration Sampling Design for Model Calibration Input-Output Model with Particle Backtracking Algorithm Introduction Particle Backtrack Algorithm Overview Particle Backtrack Algorithm without Storage Tanks Example Calculation Particle Backtrack Algorithm with Storage Tanks Water Quality in Common Inlet/Outlet Completely Mixed Storage Tank Algorithm for System with One Tank Implementation Issues iii

7 3.5.1 Efficiency and Accuracy Multiple Water Quantity and Quality Sources Practical Utility of Input-Output Characteristics Disinfectant Residual Feedback Control System Design Water Quality Fault Diagnosis Chlorine Wall Decay Study Model Applications Performance of The Input-Output Model I/O Characteristics and Feedback Control System Design Flow Path Analysis Wall Demand Coefficient Estimation with I/O Model Mathematical Problem Statement and Key Assumptions Multi Input Single Output Water Quality Model Input and Output Flow Path in WDS I/O Model for Parameter Estimation Gauss-Newton Method for Parameter Estimation Model Sensitivity analysis Example Calculation Estimate Uncertainties Analysis under Chlorine Measurement Error Application Network I Network II Observation Sampling Design for Wall Demand Coefficient Estimation Introduction Problem Statement and Assumptions Sampling Design Methodology Sensitivity and D Optimality in Parameter Estimation Sampling Design Method Sensitivity of Model Response to Flow Velocity Application Relationship of H and M with Wall Demand Coefficient Estimation Uncertainty Two Parameters Application Observation iv

8 6 Pilot Field Study Overview Study Area Description Field Test Protocol Brine Solution and Injection Field Water Quality Measurements Bottle Testing Field Measurement Results Data Analysis Bulk Decay Coefficient Hydraulic Model Calibration Wall Demand Coefficient Calibration Pilot Study Summary Summary and Future Work Summary Future Work Appendix A - Markov Random Field Model to Characterize Pipe Heterogeneities 125 Appendix B - WDS Input-Output Model Programmer s Toolkit 130 Bibliography 143 v

9 List of Tables 3.1 Data for PBA Example Calculations Pipe Characteristics of the Example Network Monte Carlo Simulation Cases for Wall Demand Coefficient Estimation Simulation Results for Wall Demand Coefficient Estimation Simulation Results for Wall Demand Coefficient Estimation Simulation Results for Bad Design Impact of Input and Output Measurement Errors Relationship of M and Parameter Estimation Uncertainty under Chlorine Measurement Error Relationship of H and Parameter Estimate Uncertainty under Hydraulic Model Error Relationship of detm and Parameter Estimate Reliability Sampling Design Results and Estimation Reliability Wall Demand Calibration Results vi

10 List of Figures 1.1 An Example Network Chlorine Concentration at Node Backtrack particle in a single pipe PBA Example for Network Without Tanks PBA for Networks With Tanks Pipeline Topology for the Network Application I/O Model and EPANET Comparison for Tank I/O Model and EPANET Comparison for Node Out I/O Characteristics between node Out and Candidate Input Flow Path Analysis in Tank Filling Period Flow Path Analysis in Tank Draining Period Flow Path for Two Sampling Locations in a WDS Travel Segments in Flow Path Parameter Estimation Example Network I for Chlorine Wall Demand Coefficient Estimation Network II for Chlorine Wall Demand Coefficient Estimation Network for Sampling Design Application Value of M and H Relationship of M and Parameter Estimation Uncertainty under Chlorine Measurement Error vii

11 5.4 Relationship between H Value and detm D-Optimal Design Algorithm Results with Different Initial Designs Network Pilot Study Area Network Subarea Network Subarea Continuous Conductivity and Temperature Analyzer and Grab Sample Tap Conductivity Concentration Leaving the Treatment Plant during the 24-hour Injection Period Measured Chlorine Concentration and Modeled Results for Bottle Tests Conductivity Measurements and Simulated Results using Calibrated Hydraulic Model Grab Sample Measurement and Original and Adjusted Analyzer Data Simulated Flow Path from Treatment Plant to AC, AD, and AE viii

12 Chapter 1 Introduction 1.1 Modeling Chlorine Decay in Water Distribution System In north America, chlorine is widely used as a disinfectant in water industries, and one of the most frequently used water quality measures in United States is chlorine residual concentration in treated water. Recent studies [1, 2] emphasize the need to maintain effective chlorine residual in all parts of the drinking water distribution system (WDS) to protect water consumers from microbiological risks. Although enough dosage may be injected into water leaving water treatment plant, chlorine is consumed due to the reaction both within bulk flow and on pipe wall. Currently there are various water quality models that can be used to model chlorine concentration within the WDS and most of them assume first order model for chlorine decay [3, 4, 5]. For a single pipe segment with constant hydraulic and water quality conditions and a constant decay coefficient K, the chlorine concentration at downstream end of the pipe, c o, can be 1

13 expressed as a function of K, water travel time in the pipe, t, and concentration at upstream end, c i. c o = c i exp( Kt) (1.1.1) A model based on mass transfer [4] is used in this study to predict chlorine decay in WDS. The model assumes first order reactions of chlorine both in bulk flow and on pipe wall. The wall reaction rate is a function of mass transfer rate of chlorine to pipe wall and is therefore dependent on pipe geometry and hydraulic conditions. Under certain hydraulic conditions, the overall chlorine decay coefficient in a pipe, K, is K = K b + K f K w R H (K f + K w ) (1.1.2) where K f = flow dependent mass transfer coefficient; K b and K w = bulk decay and wall demand coefficient, respectively; and R H = hydraulic radius. The bulk decay coefficient K b is a function of temperature, initial chlorine concentration, and organic content in bulk water [6] and can be determined in the lab. In this study, it is assumed that K b is known or determined separately by bottle test and chlorine concentration measured in field is used to calibrate only wall demand coefficients K w. 1.2 Wall Demand Coefficient The impact of wall demand coefficients on chlorine loss within distribution system is shown through simulations. The wall demand coefficients tested here are within the wide range reported in literatures [4, 7], which is from 0 to 5.0 ft/day. In the network shown in Figure 1.1, chlorine concentration at the source is constantly 2 mg/l and 2

14 the bulk decay coefficient is 0.5 1/day. The modeled chlorine concentrations at node 22 as a function of uniform wall demand coefficient are shown in Figure 1.2. It can be seen that wall demand coefficients can cause large loss of chlorine within distribution systems. 27 k Source Figure 1.1: An Example Network Pipe wall demand coefficient may depend on pipe characteristics[4], such as material, age, and degree of corrosion, which can differ significantly from pipe to pipe even in the same network. Although wall demand coefficient can significantly affect water quality modeling results, study on it is very limited. In most water quality modelling efforts so far, it is common to ignore wall demand, or to use wall demand coefficients as calibration parameters; explicit estimation of K w using measured data and dedicated experiments is rarely reported. Among the limited research that has 3

15 2 Chlorine Concentration (mg/l) Time (hours) Kw=0.0 Kw=0.5 ft/day Kw=1.0 ft/day Kw=1.5 ft/day Figure 1.2: Chlorine Concentration at Node 22 been published are least-square estimation for a long pipe based on the chlorine concentration measured at both ends [8] and trial and error calibration using chlorine data sampled throughout a WDS [9]. However, both of these approaches are impractical for system-wide estimation of pipe wall demand coefficients. A trial and error method does not identify the pipes that affect sampled water quality and adjusts wall demand coefficients of all pipes in the study region in order to get model response to parameters. Water quality modeling of the WDS is needed for each combination of possible parameter values. Such an approach can be computational demanding, especially for a large WDS with many parameters to estimate, and it is not affordable to estimate uncertainties caused by chlorine measurement and hydraulic model error through Monte-Carlo simulation analysis. The single pipe segment approach requires that the selected pipe be long enough to make sure the chlorine measurement errors 4

16 are not significant compared to chlorine decay between pipe ends. Pipes that have approximate length and are typical within a WDS will be difficult to locate. 1.3 Research Objectives Accurate modeling of chlorine concentration throughout WDS over time is necessary to understand water quality issues within distribution systems. As mentioned before, pipe wall demand may cause significant chlorine decay in WDS. Unfortunately, little work has been done to study the pipe wall demand coefficients and the relationship between coefficient values and pipe characteristics. One important step to improve the understanding of wall demand coefficients is to develop an efficient, practical and robust approach to estimate this coefficient within a water distribution system, which is the primary goal of this research. A novel parameter estimation algorithm is developed to calibrate pipe wall demand coefficient so that the difference between measured and modeled chlorine concentration can be minimized. This parameter estimation algorithm is computationally more efficient than trial and error method and is more flexible to implement than the single pipe segment approach: the chlorine measurement locations can be placed almost anywhere in the network. In reality, a WDS is usually very complicated and consists of thousands of nodes where chlorine concentration can be measured. Since errors exist in chlorine measurement, the selection of sampling locations can affect the estimation reliability. For 5

17 example, if the chlorine concentration difference between the upstream and downstream node is small compared to the chlorine measurement error, the wall demand coefficient estimation will be inaccurate. On the other hand, the hydraulic model, which is the basis of chlorine concentration simulation, also contains uncertainties. For example, hydraulic parameters such as water demand at different nodes over time and pipe roughness are usually estimated in hydraulic calibration process using meter data, measured tank water level, and/or tracer concentration. Since these data are not complete or accurate, hydraulic model parameters have uncertainties and such hydraulic uncertainties will lead to estimation error of wall demand coefficients. This research investigates sampling design approaches to select chlorine measurement locations that are more likely to produce reliable wall demand coefficient estimates. The proposed wall demand coefficient estimation algorithm and sampling design method are tested in simulation. In simulation analysis, true wall demand coefficients and statistical distribution of chlorine measurement and hydraulic uncertainties are assumed to be known. The sampling design method will be used to select simulated chlorine measurement locations and Monte-Carlo simulation employed to study the uncertainties of wall demand coefficients estimated using the proposed estimation algorithm. The calibration uncertainties are compared to test the sampling design method. A network is selected for a pilot field study, in which both hydraulic data and chlorine concentration are collected, and the hydraulic model is calibrated before pipe wall demand coefficients are estimated. 6

18 In summary, the objectives of the research are: Develop a novel algorithm to efficiently estimate pipe wall demand coefficients in water distribution systems. Develop a sampling design approach to select chlorine measurement locations that are more likely to produce robust estimates of pipe wall demand coefficients. Test the wall demand coefficient estimation algorithm and sampling design approach through Monte-Carlo simulations. Conduct a pilot filed study to estimate pipe wall demand coefficients. 1.4 Thesis Structure The thesis is organized as follows. In Chapter 2, previous work on water quality modeling and parameter estimation in WDS is reviewed. Application of sampling design methods for distribution system model calibration is also surveyed. Chapter 3 presents the development of an Input-output water quality model for distribution system analysis, which is used as the basis for the parameter estimation algorithm described in Chapter 4. In Chapter 5, sampling approach to select chlorine measurement locations are proposed and tested through Monte-Carlo simulations. A pilot field study conducted in an utility is summarized in Chapter 6. Chapter 7 summarizes major findings of this research and discusses future research needs. 7

19 Chapter 2 Literature Review 2.1 Modelling Chlorine Decay in Water Distribution System Purpose of Water Quality Modelling in WDS A water distribution system consists of pipes, pumps, valves, and storage tanks that are used to deliver water from source points to consumer nodes. Water quality deteriorates in the network and the deterioration depends on the pipe characteristics and detention time, and therefore is path dependent. Different nodes have different flow paths from the sources and the water quality can differ significantly at different locations. Additionally, due to the looped configuration of the pipe network and continuous change of the water demands, both flow path and travel times from source points to consumer nodes vary in time. Water quality at nodes depends on network hydraulics and it is common to observe high variability in water quality at a node over time. The existence of storage tanks may cause more significant change in water 8

20 quality. When tanks are refilled, consumer nodes receive fresh water directly from sources; while when tanks are emptied, much older water is delivered to some consumer nodes. Usually water from the tanks gas a high residence time and may have relatively poor quality. Since the switch from tank filling and draining is rapid, water quality at some nodes can change dramatically during a short period of time. Even medium-sized cities may have thousands of miles of pipes in the WDS and it is not feasible to monitor water quality throughout the whole network. Even when extensive field monitoring is implemented, it is not possible to experience all possible operation scenarios during a certain sampling duration. Water quality prediction under different conditions is desirable to improve the understanding and operation of WDS. For these reasons, mathematical modelling of water quality behavior has become an efficient way to study temporal and spatial variations of water quality in WDS, and due to the increasing concern over drinking water quality, has attracted more attention both from academic researchers and engineers Governing Equations for Chlorine Transport Conservation of chlorine mass during transport along a link is described by onedimensional advection equation with reaction. For any link in a network C t = µ C x + R(C) (2.1.1) where C=chlorine concentration in the link at location x and time t; µ=flow velocity in the pipe; and R(C)=reaction rate as a function of concentration. In 2.1.1, it is assumed that longitudinal dispersion is negligible. 9

21 2.1.3 Modelling Approach Various computer models [10, 11, 12, 13, 14, 15, 16, 17, 18] have been developed to predict the change of disinfectant species concentration in water distribution systems. These simulation models are either steady state or dynamic. Steady state models determine the final spatial distribution of chlorine residual under constant hydraulic conditions. These models can only provide intermittent assessment capabilities and prediction of water quality over time is not feasible. Dynamic models determine the movement and reaction of chlorine under time varying conditions. Since in WDS water demand and network operations (pump, valve and tank control) are time dependent, dynamic models are required for accurate and realistic description of water quality behavior. Dynamic models of water quality can be classified spatially as either Eulerian or Lagrangian[19, 20]. Eulerian models divide the network into a series of fixed control elements and record the changes at the boundaries and within these elements, while Lagrangian models track changes of discrete parcels of water as they travel through the network. Eulerian Finite Difference Method (FDM). FDM is an Eulerian approach that approximates the derivatives in with their finite difference along a grid of points in space and time. In [20], second order accurate Lax-Wendroff scheme was used: C t = Ct+ t s Cs t t C x = 1 + α Cs t Cs 1 t + 1 α 2 x 2 C t s+1 C t s x (2.1.2) (2.1.3) 10

22 where x=distance between each spatial grid point; t=water quality time step; C t s=concentration at grid point s at time t; and Courant number α = µ t x. This difference scheme is called explicit because the concentration at a future time can be computed using the present time values. In [17], Islam and Chaudhry used a four point implicit finite difference scheme, which is also second order accurate, to approximate 2.1.1, C t = β 1 C t+ t s Cs t t C x = (1 β 2) Ct s+1 Cs t x + (1 β 1 ) Ct+ t s+1 C t s+1 t Cs+1 t+1 Cs t+1 + β 2 x (2.1.4) (2.1.5) where β 1 and β 2 =weighting parameters in the temporal and spatial directions, respectively. The finite difference approximation of the derivatives in leads to a series of algebraic equations for the entire network. These equations can be solved explicitly by marching forward in time and down the length of each pipe, or implicitly using numerical routines, such as Gauss-Jordan method. The accuracy of FDM depends on the size of water quality time step. It is also well known that FDM is subject to numerical dispersion, which can be minimized by keeping the Courant number close to unity. Eulerian Discrete Volume Method (DVM). Application of DVM to WDS was first described by Grayman et al. [12] and refined by Rossman et al. [14]. In this method, each pipe is divided into a series of segments with equal size and each segment is assumed to be completely mixed and have uniform chlorine concentration. At the end of each water quality time step, the following steps are done: 11

23 Concentration within each segment is adjusted according to reaction law. Concentration within each segment is transferred downstream to adjacent segment or node. Concentration at each node is updated based on the inflow. Concentration at each node is transferred to first segment of the outflow pipes. The above sequence of steps is repeated until hydraulic conditions of the pipe network change, at which time pipes are re-segmented and concentration in each segment is re-calculated based on the mass distribution. The size of the segments in a pipe is determined by the travel time and water quality time step: the number of segments is the largest integer less than or equal to the travel time divided by the water quality time step. The accuracy of DVM is affected by the water quality time step. Within each hydraulic period, DVM is not subject to numerical dispersion because the contents among adjacent segments are not blended together. A certain level of mixing occurs at the start of a hydraulic period when pipes are re-segmented and chlorine mass is re-distributed to segments. Lagrangian Time-Driven Method (TDM). This method tracks the concentration and size of a series of non-overlapping segments of water that fill each pipe of the network. The size of the upstream segment of most links increases as water enter the links while equal loss in size occurs for the downstream segment of these pipes as water leaves. The size of segments between the upstream and downstream ones 12

24 remains unchanged. At the end of each water quality time step, the following steps are done: Concentration within each segment is adjusted according to reaction law. Segments are moved downstream. Mass and flow are accumulated at each node. Node concentration is updated based on the incoming flow and mass. Flow and mass accumulated at nodes are released to the downstream segment. If node concentration differers from downstream concentration by a specified tolerance, new segments are generated at the start of all flowing pipes. Lagrangian method avoids any numerical dispersion within the interior links. At the start of each hydraulic period, the order of segments in links experiencing flow reversal is switched; no other adjustment is required. When more than the leading segment of a pipe is consumed at its downstream node, mixing between different segments can happen. The accuracy of this method depends on water quality time step and concentration tolerance to limit the generation of new segments (see last step above). Lagrangian Event-Driven Method (EDM). EDM [15, 16]is similar to TDM except that it updates the network only when the leading segment in some pipe is consumed at the pipe s downstream node. It maintains an ordered list of projected lifetime of the leading segment in each pipe. The lifetime is calculated based on the pipe flow velocity and segment size. Whenever the leading segment with the shortest 13

25 lifetime ( event segment ) is consumed at the downstream node, an event occurs and the followings are done: Flow and mass of the event segment is accumulated at downstream node ( event node). The event segment is destroyed. The segment next to event segment is specified as leading segment. Concentration of the node where the event segment is consumed is updated. Flow and mass accumulated at event node are released to the downstream segment. If node concentration differers from downstream concentration by a specified tolerance, new segments are created at the start of all pipes with flow leaving the node. Projected lifetime of all leading segments are adjusted and the event list is reordered. This iterative process continues until the end of the hydraulic period, when segment positions and concentrations are updated and segment order in pipes with flow reversal is updated. Then a new event list is generated and the sequence of event processing continues. EDM is free of any numerical dispersion and is not dependent on water quality time step. Like the TDM, the concentration tolerance to limit the generation of new segments is a source of error. 14

26 2.2 Distribution System Model Calibration Mathematical model of WDS are formulated and used to predict the performance of a pipe network under different conditions. Before a network model can be used, it must be ensured that the model would predict, with reasonable accuracy, the behavior of the system. Cesario and Davis [21] define model calibration as the process of finetuning a model until it simulates field conditions for a specific time horizon (such as maximum-hour conditions) to an established degree of accuracy. Hydraulic simulation models are now widely used for design, analysis, and, to a lesser degree, operation of water distribution systems. Calibration of WDS hydraulic models has been investigated for about 20 years, starting with the calibration scheme of Tom Walski [22], who defines calibration as a two step process consisting of: (1) Comparison of pressures and flows predicted (by the model) with observed pressures and flows for known operating conditions, i.e., pump operation, tank levels, pressure reducing valve (PRV) settings; and (2) adjustment of the input data for the model to improve agreement between observed and predicted values. The numerous hydraulic model calibration approaches developed in the past can be grouped into three categories: (1) trial and error method; (2) explicit method; and (3) implicit method. The first group of hydraulic model calibration methods are based on specifically developed, analytical equations. Walski [22, 23] developed methods to assist in deciding how to adjust water use and C-factors to achieve good agreement between modeled and observed heads. It is suggested that (1) calibration accuracy be described in terms 15

27 of the ratio of observed to predicted head losses, rather than the difference between observed and predicted head losses; (2) The heads at boundary nodes (pumps, tanks, pressure reducing valves) be known when pressures are measured in the system; (3) Observed and predicted head be compared at multiple loadings to reduce the possibility of errors in one parameter compensating for errors in another parameter. An example illustrated that it is possible to accurately calibrate a hydraulic model for a distribution systems with hundreds of nodes and pipes, if adequate and accurate data are collected. Bhave [24] proposed a calibration method based on a modification of Walski s equations. He developed an iterative calibration procedure that adjusts pipe resistance coefficients and nodal demands. This method also considers multiple loading conditions and uses the ratios of head losses predicted to observed to judge calibration accuracy. Although these methods are straightforward, they are approximate and can not ensure high level of convergence. The major benefit from the development of these methods is the establishment of engineering guidelines for WDS hydraulic model calibration. The second group of calibration methods [25, 26] treat model parameters as dependent variables and explicitly solve a set of steady-state mass balance and energy equations. These algorithms are non-iterative and thus computationally efficient. However, they can be used for model calibration only when the number of unknown parameters equals the number of equations formulated, and measurement errors and stability of the resulting estimated values are ignored. Implicit calibration methods formulate and solve model calibration problems as 16

28 optimization problems. In these procedures, an objective function is formulated to measure the difference between measured and model predicted variables such as pressures, heads and flows, and parameter estimates are iteratively updated to minimize the objective function. Ormsbee [27] combined a simulation model and an extended version of Box complex method, which is a nonlinear optimization algorithm, to adjust pipe roughness, source grades, and nodal demands. The objective function in this calibration model is subject to both system and bound constraints. Both steady state and extended period loading conditions may be considered. Lansey et al. [28] developed a nonlinear programming algorithm to solve the model calibration problems based on data from multiple loading conditions. The model has an objective of minimizing the sum of the squares or absolute values of the difference between observed and predicted values of pipe flows and nodal heads. The estimated parameters include pipe roughness coefficients, valve settings, and nodal demands. The gradients required by the optimization model are computed by solving sets of linear sensitivity equations. The impacts of model and measurement errors on parameter estimates have also been considered. Datta et al. [29] proposed a weighted-least-square method using sensitivity analysis technique to calibrate Hazen-Williams coefficients for pipes. The effect of choice of weights, effect of initial estimates, and uncertainty in parameter estimates are discussed. Reddy et al. [30] used weighted-least-square method based on Gauss-Newton minimization techniques to estimate different network parameters, such as pipe resistances and nodal consumptions. The influences of different choice of weights on parameter estimation are considered and an iteratively re-weighted scheme 17

29 is recommended for situations with noisy measurements. In contrast to hydraulic model calibration, water quality model calibration has not been studied until recently. Least-square method has been used to estimate chlorine decay parameters for a long pipe using chlorine concentration measured at both ends [8]. A 1.3 mile long pipe was constructed specifically for wall demand coefficient estimation. A procedure to correct chlorine meter errors was introduced and applied and difficulties in fitting the parameters were noticed. Trial and error calibration using chlorine data sampled throughout a WDS has also been used [9], and wide variations in both bulk decay coefficient and pipe wall demand coefficients were observed among several water utility sites. Munavalli et al. [31] used an inverse model to estimate separately or simultaneously wall demand coefficients and source chlorine concentration under steady state. Weighted-least-square method based on Gauss-Newton minimization technique is used to identify parameter estimates. 2.3 Sampling Design for Model Calibration The objective of sampling design for model calibration is to collet quality input data and identify accurately the physical parameters of the system. Several researchers have investigated the problem of sampling design for WDS hydraulic model calibration. Walski [22] was one of the first to suggest where to observe pressure heads and flows in order to collect data for model calibration. Walski [32] also discusses the impact of measurement errors and provides guidelines for collecting good data to achieve meaningful model calibration. Meier et al. [33] used a genetic algorithm (GA) 18

30 to find a fixed number of flow test locations that cause water to flow at a noneligible velocity through as much of a pipe network as possible. Sensitivity-based heuristic methods have also been proposed to design sampling plans for the calibration of water distribution hydraulic model. Bush et al. [34] developed three simple, yet efficient methods for sampling design: the max-sum, weighted sum, and max-min methods. These methods are inspired by D-optimality criteria [35] and rank the sampling locations and types of measurement for parameter estimation. Lansey et al. [36] developed a three-step sampling design procedure that considers measurement uncertainties, their impact on model parameter estimates, and their impact on model predictions. The first-order second-moment analysis is used to assess model output uncertainties given the input uncertainties. One of the general conclusions of these two and similar studies is that measurements should be taken at sensitive locations (locations where the model responses are sensitive with respect to parameters to be estimated). The significant impact of sensitivities on parameter estimation has also been observed by researchers working on groundwater systems [37]. Kapelan et al. [38] formulated the sampling design problem as a constrained two-objective optimization problem. The objectives are to maximize the calibration accuracy by minimizing relevant uncertainties that are based on sensitivity analysis and to minimize total sampling costs. The calibration accuracy objective can based on either prediction uncertainty or parameter uncertainties based on D-optimality or 19

31 A-optimality criteria [35]. The optimal sampling design problem is solved by multiobjectives genetic algorithm based on Pareto ranking, niching, and restricted mating. The application finds that the optimal set of N monitoring locations is not always a superset of the optimal solution for N 1 locations. 20

32 Chapter 3 Input-Output Model with Particle Backtracking Algorithm 3.1 Introduction Traditional WDS models are efficient in providing output information at all nodes over time, but forget the internal details of the links between input and output (or, at least, they are not available in a useful form). These internal details - the inputoutput (I/O) information - include the number of flow paths, their time delays, and their impact on water quality. Such I/O information is critical to certain applications. One such example is the design of feedback control algorithms used to regulate disinfectant injections based on measurements at a distributed set of sensor locations [39]. Such an application assumes that disinfectant injection can be related to output concentration, using either an I/O or state space model. For water distribution systems, the state space dimension can preclude real-time use of a state space model for computational reasons. Furthermore, prior estimation of I/O characteristics for 21

33 a particular application (the time delays and impacts associated with each flow path between injector and sensor) would be critical for successful design of feedback control systems and algorithms. Input-output information is also important when water quality deteriorates at a particular location, because fault diagnosis can then focus on important flow paths between the source and problem area. More generally, I/O information is required to explain water quality transformation processes that are path-dependent, such as those that involve pipe material interactions. Zierolf et al. [40] first developed an I/O model for chlorine transport in networks without storage tanks. Their algorithm tracks the travel of water particles in networks in reverse time. This model can find all the paths from the input to the output and the corresponding delays and was used to calibrate pipe chlorine reaction rates offline. The particle backtrack algorithm (PBA) presented here extends the work of Zierolf et al. to consider the existence of tanks and to allow multiple water sources and quality inputs. Unlike Zierolf et al., who track each particle individually and sequentially from output to input, the PBA tracks a large number of water particles, moving them simultaneously along their paths (in reverse time). This change results in a simpler algorithm to describe, and one that is likely more efficient because the algorithm moves in parallel with the time variation in the hydraulic solution. This chapter is organized as follows. An overview of the PBA first introduces key concepts and definitions. Following this, the PBA is described for the simplest case of a network with one water quality source and no storage tank. Detailed example calculations are provided to illustrate this basic algorithm. The PBA is then 22

34 extended to include completely mixed storage tanks, again for networks with a single water quality source. Next, the issues of multiple water quantity and quality sources are discussed, along with the required (and relatively straightforward) modifications to the PBA. An implementation of the PBA based on the EPANET programmer s toolkit [41]is then used to apply the PBA to various scenarios that illustrate the PBA accuracy as well as some applications where it is useful to know explicit input-output characteristics. 3.2 Particle Backtrack Algorithm Overview When chlorine decay is assumed to be governed by a first-order reaction, the inputoutput model concisely describes the output concentration c o as a linear function of the input source strength. It is expressed as: N c o (T o ) = γ j c ij (3.2.1) where N is the number of travel paths between input and output, T o is the output time, c ij = water quality source input at time T o t j, t j = time delay for travel path j, and γ j is the impact coefficient for travel path j. The impact coefficient γ j = c o / c ij corresponds to the coefficients of the discretized impulse response function [42] and quantifies the effect of source input on output concentration. In the description of PBA in this chapter, it is assumed that there is only one water quantity source, which is the only water quality source. The units of input source strength c i and impact j=1 coefficient γ j depend on the source model used (e.g., a specified concentration or mass rate source); this issue is discussed in a later section, along with modifications 23

35 required for multiple water quality and quantity sources (these modifications are implemented in the I/O programmer s toolkit algorithms included as Appendix B). The relationship (3.2.1) is exact for first-order water quality kinetic models, as shown by Zierolf et al. [40] and (via a linear superposition argument) Boccelli et al. [43]. The objective of the particle backtrack algorithm (PBA) is to calculate the number of paths N and their time delays and impact coefficients. The PBA is a Lagrangian model and runs in reverse time. First, an extended period network hydraulic simulation is performed to compute the time series of link flow rates (e.g., [44]). The algorithm time T a is initialized to zero at the beginning of the PBA (corresponding to the output time T o ) and runs opposite to the hydraulic simulation time T s (T a = T o T s ). The water particle reaching the output at time T o is transported upstream ( backtracked ), dividing into more particles at junctions, until all particles arrive at a water source. Each particle represents one travel path with an associated time delay and impact coefficient, the impact coefficient being adjusted dynamically in the backtrack process to reflect dilution at junctions and first-order chlorine decay in pipes and storage tanks. In addition, the pipes and storage tanks included on each travel path may be recorded as the algorithm proceeds. If the simulation time is long enough so that each particle reaches a chemical source, the solution of the PBA depends only on source boundary conditions and initial conditions are not needed. 24

36 3.3 Particle Backtrack Algorithm without Storage Tanks First, the basic algorithm is described for networks without storage tanks. The network hydraulic solution provides a time series of link flow rates that are assumed constant within specified time periods, separated by hydraulic events (changes in water demands, pump or valve status, and the like). Within each hydraulic period, the following quantities are known for each pipe: Upstream node u and downstream node d. Length L. Flow velocity µ 0 directed from upstream to downstream node. Composite first-order decay coefficient k, including both the effects of bulk and wall reactions [20]. Each water particle is associated with a particle characterized by four main quantities: the pipe that contains it, its position along the pipe 0 x L as measured from the upstream node u, its path travel time t from the source node, and its path impact coefficient γ that quantifies the contribution of the particle path to the output concentration (see Figure 3.1). In algorithm 1 below, the backtrack algorithm is described for any particle within a pipe during a single hydraulic period. For clarity, subscripts identifying a specific particle have been omitted. Two lists of particles are stored: the active list A, and the 25

37 u γ, t µ, k d x L Figure 3.1: Backtrack particle in a single pipe completed list C. List A includes particles on their way upstream to a water source, while list C includes particles that have reached the source. Time T is the algorithm time that corresponds to the simulation time at the beginning of the current hydraulic period. Algorithm 1 propagates the particle upstream, and updates the associated characteristics t and γ, until one of two conditions is met: the particle reaches the upstream node u, or the algorithm time equals T (the beginning of the hydraulic period). If the particle reaches the upstream node and the upstream node is not the source then it must be split among all inflows (see algorithm 2 below), while if the beginning of the hydraulic period is reached then flow conditions in the pipe must be updated to reflect the new hydraulic solution (see algorithm 3 below). PBA Algorithm 1: Backtrack Particle Within a Pipe During Single Hydraulic Period 0. Given γ, x, t, T, u, µ, k for current particle/pipe and hydraulic period 1. Calculate time particle remains in pipe and under current hydraulic condition, δt: (a) If µ(t t) < x (particle remains in pipe at time T ), δt = T t (b) Else (particle reaches node u before time T ), δt = x/µ 2. Update particle characteristics: 26

38 (a) x = x µδt (b) t = t + δt (c) γ = γ exp( kδt) 3. If particle reaches node u before time T, split particle via a call to Algorithm 2 When a particle reaches its upstream node u, as determined in Algorithm 1, the particle is split along M different flow paths corresponding to M pipes discharging into node u. The particle splitting algorithm, including the correction to impact coefficients for dilution at mixing nodes, is presented in the following algorithm 2. The characteristics of the particle reaching its upstream node (to be split) are designated γ p and t p. PBA Algorithm 2: Split Particle at Upstream Node 0. Given γ p, t p for particle to be split at node u 1. Calculate M, the number of pipes with inflow to node u at algorithm time t p Calculate Q, the total discharge into node u at time t p 2. If M > 0 then for each inflow pipe of length L and discharge q, add to list A a new particle associated with the inflow pipe and having characteristics: x = L, t = t p, γ = γ p (q/q). Delete the split particle from the active list, A 3. If M = 0(particle reaches the water source) then move the split particle from the active list, A, to the completed list, C 27

39 Algorithms 1 and 2 are used by the backtracking algorithm described below in algorithm 3. This algorithm first calls the single pipe backtracking algorithm during each hydraulic period, until t = T for all particles in the active list A (i.e. they reached the beginning of the hydraulic period). Then the same is done for the previous hydraulic period, and so on, until all particles arrive at the water source. Since a single water quantity source is assumed (for the moment) to be the only water quality source, the particle information in list C then includes the input - output information for all paths between source and output node. If the hydraulic simulation duration is too short then algorithm 3 will terminate prematurely before all particles reach a water source; in such a case the active particle characteristics are the linkage between network water quality initial conditions (at the final active particle locations) and the output water quality. PBA Algorithm 3: PBA Without Tanks 0. Simulate network hydraulics for required simulation time interval; Set current hydraulic solution equal to that for the desired output time (T o ); Set Algorithm time T a = 0 and T = δt h, the current hydraulic period duration; Initialize active particle list, A, with one particle where γ = 1, t = 0, x = 0, and upstream node u equal to output node O; Call Algorithm 2 to split the particle in A at output node O; Initialize completed particle list C =. 1. While t < T for some particle in A, backtrack this particle via a call to Algorithm 1 28

40 2. If the list A is not empty, then (a) Set current hydraulic solution to that for the previous hydraulic period with period time step δt h ; (b) T = T + δt h ; (c) Repeat step1. 3. List C contains the complete input-output information, including the time delays, t, and associated impact coefficients, γ, for each path between the input (the water source) and output Example Calculation A B S t a0, γ a0, x a0 a O 4 t a1, γ a1, x a1 a C D t a2, γ a2, x a2 a b t b0, γ b0, x b0 E F t a3, γ a3, x a3 a b t b1, γ b1, x b1 t a4, γ a4, x a4 a b t b2, γ b2, x b2 ta5, γ a5, x a5 a b t b3, γ b3, x b3 Figure 3.2: PBA Example for Network Without Tanks Figure 3.2 illustrates the particle backtracking algorithm for a simple network without tanks. The network has four pipes indexed from 1 to 4 (see panel A). O is 29

41 the output node, S is the source node, and q i, µ i and k i are flow rate, velocity and first order decay constant in pipe i, respectively, which depend on the hydraulic period. The hydraulic simulation duration is 12 (arbitrary units), divided into two hydraulic periods of lengths 10 (period 1) and 2 (period 2). The output time is the end of hydraulic period 2 (simulation time equal to 12), and thus hydraulic changes occur at algorithm time T = 2(beginning of period 2) and T = 12 (beginning of simulation). The necessary data is provided in Table 3.1. In the following computations, these data are used to illustrate six algorithm steps corresponding to panels A-F in Figure 3.2. These steps compute the time delays and impact coefficients for the two paths between source and output (represented by particles a and b ). A. Particle a is generated at the output node and added to active list A: γ a0 = 1, t a0 = 0, x a0 = L 4. B. Algorithm 1 transports particle a upstream in pipe 4 to its upstream node (δt = x a0 /µ 4 = 2/2 = 1), and updates particle characteristics: γ a1 = e k4δt γ a0 = e 0.1(1) 1 = , t a1 = t a0 + δt = = 1, Table 3.1: Data for PBA Example Calculations Hydraulic Period 1 Hydraulic Period 2 Pipe Length µ q k µ q k

42 x a1 = 0. C. Algorithm 2 splits particle a into two ( a and b ) and updates particle characteristics: γ a2 = (q 2 /(q 2 + q 3 ))γ a1 = (0.5/( )) = , t a2 = t a1 = 1, x a2 = L 2 = 3, γ b0 = (q 3 /(q 2 + q 3 ))γ a1 = (1.5/( )) = , t b0 = t a1 = 1, x b0 = L 3 = 3. D. Hydraulic conditions change. Algorithm 1 transports particle a upstream in pipe 2 and particle b upstream in pipe 3 until the beginning of the second hydraulic period (δt = T t a2 = T t b0 = 2 1 = 1), and updates particle characteristics: γ a3 = e k2δt γ a2 = e 0.1(1) = , t a3 = t a2 + δt = = 2, x a3 = x a2 δt(µ 2 ) = 3 1(1) = 2, γ b1 = e k3δt γ b0 = e 0.1(1) = , t b1 = t b0 + δt = = 2, x b1 = x b0 δt(µ 3 ) = 3 1(2) = 1, Pipe flow rates, velocities, and decay rate constants are updated to reflect new hydraulic conditions during period 1. E. Algorithm 1 transports particle a in pipe 2 to its upstream node (δt = 31

43 x a3 /µ 2 = 2/0.5 = 4), and particle b in pipe 3 to its upstream node (δt = x b1 /µ 3 = 1/1 = 1). Algorithm 2 splits each particle. The net effect of both operations on particle characteristics is: γ a4 = (q 1 /q 1 )e k2δt γ a3 = (1/1)e 0.1(4) = , t a4 = t a3 + δt = = 6, x a4 = L 1 = 2, γ b2 = (q 1 /q 1 )e k3δt γ b1 = (1/1)e 0.1(1) = , t b2 = t b1 + δt = = 3, x b2 = L 1 = 2. F. Algorithm 1 transports particles a and b in pipe 1 to its upstream node (δt = x a4 /µ 1 = x b2 /µ 1 = 2/1 = 2), and updates particle characteristics: γ a5 = e k1δt γ a4 = e 0.1(2) = , t a5 = t a4 + δt = = 8, x a5 = 0, γ b3 = e k1δt γ b2 = e 0.1(2) = , t b3 = t b2 + δt = = 5, x b3 = 0. Upon completion of the PBA, the output concentration at time T o = 12 can be calculated as a simple linear function of the source concentrations at times 4 (= 12 8) and 7 (= 12 5) (see equation 3.2.1): c o (12) = c i (4) c i (7). 32

44 3.4 Particle Backtrack Algorithm with Storage Tanks The above algorithm is relatively simple due to the absence of storage tanks. A tank with plug flow mixing regime can be considered a large pipe and the same algorithm applies without change. If the tank contents are completely mixed, however, then the situation is more complicated. Here a modified version of the PBA is presented for completely mixed single inlet/outlet tanks; the same algorithmic approach could be used for flow-through completely mixed tanks, after appropriate changes to the basic mass balance equations. Figure 3.3 illustrates how the PBA is used, in combination with a tank water quality model, to model input-output behavior when tanks are present. Clearly, the Tank Water Quality Model TANK OUTFLOW PBA TANK INFLOW (B) (C) (D) OUTPUT (A) PBA SOURCE INPUT PBA Figure 3.3: PBA for Networks With Tanks PBA described above can provide the input-output information for all flow paths that do not include a storage tank (between (A) and (D) in Figure 3.3). Other paths, however, flow from source to tank during the fill cycle, and then from tank to output during the drain cycle ((A) to (B) to (C) to (D) in Figure 3.3). The same PBA can 33

45 provide the input-output information between source and tank inflow ((A) to (B)) and between tank outflow and the output ((C) to (D)) - by considering the tank inflow as an output, and the tank outflow as an input. The missing piece is the input-output relationship between the tank outflow and inflow concentrations ((B) to (C)). If a particle reaches a draining completely mixed tank at time t, a quantitative relationship is required between the completely mixed tank concentration at time t and the tank inlet concentrations during previous fill periods. Strictly speaking, this relationship involves the concentrations of every water particle entering the tank during all previous fill periods. As a practical compromise, a tank water quality model is used to develop such a relationship for time discretized tank inflows. This water quality model is described next, followed by algorithms that incorporate it within a modified PBA Water Quality in Common Inlet/Outlet Completely Mixed Storage Tank Consider a completely mixed storage tank with a common inlet/outlet pipe that alternately drains and fills. The water quality dynamics are described by a species mass balance and a flow balance: d(vc s ) dt = q si c si q so c s k s c s v (3.4.1) dv dt = q si q so (3.4.2) where c s is the completely mixed species concentration in the storage tank, v is the tank water volume, q si is the tank inlet flow, c si is the tank inlet species concentration, 34

46 q so is the tank outlet flow, and k s is a tank first order decay rate coefficient. During the fill period q so = 0, while during the drain period q si = 0. During the drain period, the above water quality dynamics describe first-order species decay within the tank: c s (T s + t q ) = c s (T s ) exp( k s t q ) (3.4.3) where T s is a simulation time, and t q is a water quality time step. During the filling period, the inflow rate(q si ) and concentration (c si ) are assumed to be constant during a water quality time step; under this assumption the water quality dynamics during the filling period is a function of the inflow concentration and the completely mixed tank concentration: c s (T s + t q ) = c si q si k s v(t s + q) [1 exp( k s t q )] + c s(t s )v(t s ) v(t s + t q ) exp( k s t q ) (3.4.4) Equation is valid for any time step, while equation requires a sufficiently small water quality time step so that the assumption of constant inflow concentration is valid. To derive the input-output relationship between the completely mixed tank concentration and the water source, new particles are generated to represent the discretetime tank inputs, and these particles are then tracked back to the source. This generation of particles is terminated when the impact of the tank concentration on the output is negligible. 35

47 3.4.2 Algorithm for System with One Tank For simplicity, the backtrack algorithm is described for a single storage tank. The extension of this algorithm to multiple tanks does not require new concepts and thus is omitted. A new particle list, the stored list, S, is introduced to hold those particles that reach the tank during the drain period. The stored list particle characteristics are the time delay t s and impact coefficient γ s. When hydraulic conditions are updated, the algorithm first checks the status of the storage tank. If the tank is draining and the list S is not empty, the time delay and impact coefficient of each particle in S is updated consistent with equation If filling, new particles are generated to represent the effects of discrete-time inputs on the particles in S, and particle characteristics are updated, consistent with equation PBA Algorithm 4: Tank Algorithm For Single Hydraulic Period 0. Given flow rate in inlet/outlet pipe, q si, first order tank decay coefficient, k s 1. If tank is filling, then for each particle in list S, do the following: (a) Given γ s, t s for current particle, and tank water volume,v, at time, t s (b) While t s is less than T i. Generate new particle at downstream node of inlet/outlet pipe with time delay t s, and γ = q si k sv [1 exp( k s t q )]γ s ii. Add the new particle to active list A 36

48 iii. Adjust γ s, t s and v: γ s = v q si t q v t s = t s + t q v = v q si t q exp( k s t q )γ s 2. If tank is draining, for each particle in list S, do the following: (a) γ s = exp( k s (T t s ))γ s (b) t s = T (c) if γ s is negligible, delete the particle from list S The time step t q in algorithm 4 is the parameter that regulates the generation of new particles from completely mixed storage tanks and should be sufficiently small that inflow concentration is approximately constant during one time step. The efficiency of the PBA also depends on this parameter, as the computational work depends on the number of particles. Algorithms 2 and 3 are now revised to accommodate the storage tank. Revised PBA Algorithm 2: Split Particle at Upstream Node 0. Given: γ p, t p, u for particle to be split at node u 1. Calculate M, the number of links with inflow to node u at time t Calculate Q, the total discharge into node u at time t 2. If M > 0 then for each inflow link of length L carrying flow q, add to list A a new particle with characteristics: 37

49 (a) x = L (b) t = t p (c) γ = γ p (q/q) Delete the split particle from the active list, A 3. If M = 0 and the upstream node is a water source,then move the split particle from the active list, A, to the completed list, C Else if M = 0 and the upstream node is a tank, then move the split particle from the active list, A, to the stored list, S Revised PBA Algorithm 3: PBA With Completely Mixed Storage Tanks 0. Simulate network hydraulics for required simulation time interval; Set current hydraulic solution equal to that for the desired output time; Set Algorithm time T a = 0 and T = δt h, the current hydraulic period time step; Initialize completed particle list C = ; Select time step t q. 1. If the output is not a tank, then (a) Initialize active particle list A with one particle with γ = 1, t = 1, x = 0, and upstream node equal to output node O; (b) Call Algorithm 2 to split the particle in A at output node O; (c) Initialize stored list S =. 2. If the output is a tank, then 38

50 (a) Initialize stored particle list S with one particle with γ s = 1, t s = 0; (b) Initialize active particle list A = ; 3. If the stored list S is not empty, call Algorithm 4 to update characteristics of particles in S and, if tank is filling, to generate new particles. 4. While t < T for some particle in A, backtrack this particle via a call to Algorithm If either list A or list S is not empty, then (a) Set current hydraulic solution to that for the previous hydraulic step of length δt h (b) T = T + δt h (c) Repeat steps 3 and List C contains the complete input-output information, including the time delays, t, and associated impact coefficients, γ, for each path between input and output. 3.5 Implementation Issues Efficiency and Accuracy Traditional Eulerian and Lagrangian approaches for modeling water quality in distribution systems require both spatial and temporal discretization. Typical assumptions 39

51 include uniform concentration within discrete pipe segments, and constant concentration over a discrete time interval for flows exiting storage tanks (the EPANET network model, for example, employs both these assumptions within its water quality algorithms ([45]). In the sense that discretization and approximation are needed, the PBA is no different than these traditional forward models. If no tanks exist along the paths between input and output, however, the PBA is exact and thus more accurate than traditional forward modeling; each particle is tracked individually without the lumping of close particles. The presence of completely mixed storage tanks between input and output introduces temporal discretization errors that are related to the magnitude of t q (Equation 3.4.4), over which the water quality entering the tank is assumed to be constant. As t q is made smaller the effect of a time varying tank input concentration (or, time varying time delay between input and storage tank) is more accurately captured using more particles. The addition of particles adds to the computational burden in two ways: the computations associated with tracking each particle, and those associated with memory management required to store particle information. If several tanks are present between input and output then the number of particles generated could be large, and a larger t q may be required. Fortunately, the I/O analysis can be run sequentially for different output locations and times, and the information from one I/O analysis is not needed for another. Consequently, the memory and computational requirements can be managed for practical network models. The memory demand also depends on the I/O information that must be saved 40

52 for each particle. For example, more memory will be required to record detailed information about the pipes belonging to each path and the travel time in each pipe, compared to recording only the time delays and impact coefficients of each path Multiple Water Quantity and Quality Sources The description of the PBA allows for one water source, and one water quality input located at the water source. In practice, there often exist multiple water sources modeled as fixed head reservoirs or known input flow rates (each associated with a water quality input), plus secondary water quality inputs that are not associated with a water source (e.g., booster chlorinators). In addition, the secondary inputs can be modeled to mimic the various types of local control that are available (e.g., known mass input rate, flow proportional mass input rate, or local feedback control of mass input rate to achieve a desired set point concentration). The method used to model secondary inputs must be considered when modifying the PBA to account for their influence. If a set point secondary input model is used, the linear superposition of individual path impacts will not, in general, accurately describe the output concentration, although the path information provided by the PBA may still be useful. Networks with multiple water sources (but without secondary water quality inputs) are straightforward to incorporate into the above algorithms via a bookkeeping procedure. No matter their number, the backtracking of a particle always terminates when it reaches some water source (although the particular source is not known a priori). If water at an output node and time is a blend from multiple water sources, 41

53 the PBA can identify the impacts and delays for all flow paths originating at each water source, by continuing to backtrack each particle (as needed) after it reaches a water source. The output concentration is (by analogy to equation 3.2.1): c o (T o ) = M N k γ kj c ik (T o t kj ) (3.5.1) k=1 j=1 where c ik = strength at source k; M = the number of water sources, and N k = the number of paths between source k and the output. One completed list C is kept for each water source, and the appropriate list is modified whenever a particle reaches a water source. If the water source node has zero inflow links (such as a reservoir node), the particle is removed from the active list as described in the above algorithm; otherwise, the particle is split among the source and other input flows, and the impact coefficient associated with the source flow is added to the appropriate list C. The remaining split particles remain active. Secondary water quality inputs (i.e.those not associated with a water source) present an additional challenge to the PBA. Three types of secondary input models can be defined [45]: known mass rate addition ( mass ), flow proportional mass rate addition ( flow paced ), and local feedback control of mass input rate to achieve a desired concentration ( set point ). Each of these models obeys the same mass balance at the secondary input node: 1 c o = ( i IN q )ṁ + q ( i i i IN q )c i (3.5.2) i where, in this context, c o is the completely mixed concentration leaving the secondary input node, IN is the set of all pipes incident to the secondary node and flowing into i IN 42

54 the node at the time the particle arrives, q i is the flow rate in pipe i IN, c i is the concentration in pipe i adjacent to the node, and ṁ[m/t ] is the mass rate of addition by the secondary water quality input. The first term in equation shows how the PBA must modify the particle impact coefficient prior to adding the particle to the completed list C for the secondary input node. The second term is simply the usual split of the particle among the inflow pipes (these particles must remain active). In the case of a mass secondary input, the input model specifies the mass rate ṁ and the particle impact coefficient is multiplied by 1/ i IN q i and then recorded on the completed list C. In this case the impact coefficient for ṁ has units[t/l 3 ] and the source strength has units [M/T ], yielding the proper units of concentration when multiplied together. In the case of flow paced secondary input, the input model specifies that the mass rate is proportional to the total flow through the node, or ṁ = α i IN q i, so that the source strength is the proportionality factor α, which is interpreted as the concentration added to that resulting from mixing of the pipes. Substitution into shows that the particle impact coefficient is multiplied by unity and then recorded on the completed list C for a flow paced source. Thus the impact coefficient remains dimensionless in this case, and the source strength has concentration units [M/L 3 ]. The set point secondary input must be treated as a separate case. As noted, the set point input attempts to keep the output concentration c o = c o, where c o is the desired set point. This attempt will fail if the concentration resulting from only the inflows (the second term in 3.5.2) is greater than c o. During a typical 43

55 network simulation this situation presents no difficulty, as the set point input model will effectively set ṁ = 0(this correctly mimics the behavior of a local feedback controller). The PBA, however, has no way of neatly handling this discrete on/off behavior that depends on upstream inputs, due to the assumed linear form of the input/output relationship. Conceptually, this occurs because the intelligence of the implied local feedback controller (which is making decisions about the mass rate of addition) is not included in the linear input/output relationship. Note that ṁ = c o i IN q i i IN q ic i (from solution of for ṁ), so long as c o i IN q i i IN q ic i 0 (otherwise ṁ = 0). We are confronted with a need to assume whether this term is positive or negative, or whether the concentration resulting from only the inflows is less than the desired set point, or exceeds it. This choice boils down to choosing between a set point input that completely governs its downstream concentration, or one which doesn t impact at all its downstream concentration. In the former case, substituting ṁ = c o i IN q i i IN q ic i in shows that the particle impact coefficient is multiplied by unity and is removed from the active to the completed list C without splitting the particle (and thus the particle dies at the set point input); in the latter case, ṁ = 0 and the particle is split as usual without being added to the completed list C. Unfortunately there can be no a priori basis for making this decision one way or the other, and thus the PBA is limited in its ability to accurately characterize input/output behavior for multiple water quality inputs where set point secondary inputs are present. A third, and perhaps the best, choice is to assume ṁ = 0, while also recording the particle impact coefficients on 44

56 the completed list C for each set point input. Such an approach will provide a complete set of impact coefficients for each input, which would accurately describe the influence of each set point source provided that c o i IN q i i IN q ic i 0 and that no downstream set point sources exist whose set point concentration exceeds its concentration resulting from inflow alone. 3.6 Practical Utility of Input-Output Characteristics In this section we briefly highlight a few situations where input-output information, and thus the PBA, may prove useful Disinfectant Residual Feedback Control System Design A feedback control system for automatic regulation of chlorine residual concentration confronts a system that is characterized by multiple travel paths between a regulated source input and a sensed output. Each of these paths has an associated time delay and impact coefficient, and these I/O characteristics affect the ability of feedback control algorithms to effectively control chlorine residual at the output [46]. Therefore a prior understanding of I/O characteristics is critical in the design of a feedback control system - specifically in the analysis of alternative actuator (chlorine input) and sensor locations. Given a selected input and output location pair, the I/O analysis can estimate the range of time delays and impact coefficients. The controller must update an internal 45

57 input-output model of the system, and such prior information can help controller performance by reducing the number of parameters required by the model (for example, by ignoring time delays with negligible impact). The I/O characteristics would also assist in identifying good input/output location pairs, from a feedback control design perspective. Input/output locations characterized by long transport delays and small impact coefficients would imply weak control of the output by the input, with consequent large controller effort required. In the control system literature, this situation is usually referred to as weakly controllable. Similarly, input/output characteristics that vary significantly over time (e.g., for different output times over the course of a day) will require that the controller learn these changes and adapt rapidly, and this may be difficult if time delays are long. So input nodes should be selected such that the travel paths to the output nodes of interest are relatively simple, with a small range of time delays and large impact coefficients. Our experience suggests that such insights are not always easy to develop from a basic knowledge of network topology or hydraulics, as illustrated below Water Quality Fault Diagnosis Water distribution systems are complicated systems and water can travel through a large number of paths before reaching an output node. Each travel path includes different pipes and tanks and may even start from different water sources. Therefore, it can be a challenge to diagnose the causes of water quality problems at specific locations and times. The I/O characteristics provided by the PBA include not only the composition of the flow paths, but also the impact of those paths on the output 46

58 water quality. As illustrated below, the I/O information can be used for quantitative analysis of an individual pipe or tank contribution to output water quality. With the help of well calibrated network model and I/O analysis, it should be possible to focus the diagnosis effort on important pipes and tanks, and to identify network elements that cause water quality deterioration Chlorine Wall Decay Study Typically, field studies of chlorine wall decay involve defining the input and output location and characterizing the chlorine decay and pipe flow between such two points. As a practical matter, the input and output locations should be designed such that the flow paths and time delays are known. This I/O information is, however, difficult to identify using traditional network models, which often leads to a straightforward field test design consisting of a single pipe segment with no junctions between input and output. As discussed before, in such a design the pipe segment must meet certain requirements. It must be long enough so that measurement errors do not corrupt the estimation of residual decay, and it must be relatively homogeneous in its residual decay characteristics. Unfortunately, such requirements are difficult to satisfy in typical distribution systems. The I/O analysis can alleviate some of these practical difficulties, by clarifying the relationship between input and output and thus freeing the field study design to consider a wider variety of input and output measurement locations. The parameter estimation algorithm that uses I/O model developed in this chapter will be discussed in the next chapter. 47

59 3.7 Model Applications Performance of The Input-Output Model The performance of the proposed algorithm is tested using the network topology shown in Figure 3.4, that of a utility in the western United States. Reservoir Out In3 In2 In1 Tank2 Tank1 Figure 3.4: Pipeline Topology for the Network Application The network includes two tanks (tank 1 and 2) and chlorine is added only at the reservoir, which is the only water source in the network and is shown at the top of the figure. Water flow from the reservoir is constant and equal to the average water demand of the network. All pumps and control rules have been removed from the network model and the hydraulic dynamics of the network is completely controlled by 48

60 the daily cycle of water demands. When the demand is higher than the water supply rate, the tanks are draining, otherwise they are filling. Thus the tanks fill and drain on a cycle governed only by the diurnal system curve. A constant chlorine concentration of 1.0 mg/l was modeled at the reservoir source. The hydraulic information needed for water quality modeling was retrieved using the EPANET Programmer s Toolkit [41] and the water quality results from the PBA are compared in figures 3.5 and 3.6 with EPANET water quality simulations for Tank 1 and node Out, respectively Concentration(mg/L) I/O Model EPANET Time(hours) Figure 3.5: I/O Model and EPANET Comparison for Tank1 The results over one 24 hour cycle show that when the I/O water quality modelling parameters, such as the water quality time step, are selected appropriately, the input - output water quality modelling results are virtually identical with EPANET simulations. The EPANET algorithm also includes a water quality time step parameter that is similar in concept to the PBA quality time step, and this parameter was set conservatively for these comparisons. Thus these results validate the conceptual 49

61 0.52 Concentration(mg/L) I/O Model EPANET Time(hours) Figure 3.6: I/O Model and EPANET Comparison for Node Out approach behind the PBA I/O Characteristics and Feedback Control System Design For feedback control of chlorine residual within the distribution system, the selection of input locations is of critical importance because the characteristics of the flow paths between input and output can significantly affect the performance of the controller. In figure 3.7, the I/O relationships between the node Out and four candidate input nodes (Reservoir, In1, In2 and In3) are shown in four subplots. The input is assumed to be of set point type and therefore the impact coefficient is unit less. It is assumed that the set point concentration is always larger than the inflow concentration and therefore the backtrack particle dies when it reaches the input. (Thus these data are most appropriate to a design situation where the various input locations are 50

62 being evaluated against one another, but only one of them would be constructed.) Each subplot is a checkerboard plot showing the relationship between transport delays and their impact coefficients through one sampling period(24 hours), using a sampling step of 1 hour. The horizontal and vertical axes represent the sampling time and transport delay, respectively. The cell shading density indicates the sum of the impact coefficients for all paths with delays in the given interval. Darker shading indicates a larger impact of the corresponding travel paths on the output water quality, and no shading indicates the absence of any travel paths within that interval of time delays (note that the shading and time delay scales depend on the input location). Time Delay(hours) Input:Reservoir Time(hours) Time Delay(hours) Input:In Time(hours) Time Delay(hours) Input:In Time(hours) Time Delay(hours) Input:In Time(hours) Figure 3.7: I/O Characteristics between node Out and Candidate Input 51

63 Node In3 is close to node Out and there is only one flow path between them at any output sampling time. However, the transport delay of this single path changes significantly during one hydraulic cycle, from less than 2 hours to more than 7 hours. Therefore a model used by a feedback controller for this input/output node pair should allow for time delays within such a range. Changing the input location to the reservoir or node In1 may have little effect on controller performance because the key I/O characteristics (the range and variation of dominant time delays) are similar, although chlorine dosing at the reservoir will require larger control effort (chlorine dosage) to achieve the same target output concentration (note difference in magnitude of the summed dominant impact coefficients). Although the results show a variety of different flow paths and wide range of delays for the reservoir and In1 input nodes, most paths are weak and it may be reasonable to ignore them from the perspective of feedback control. It is interesting that although nodes In1 and In2 are close to one another on the same network main, their I/O relationships with the output node are quite different. Indeed, there exists no flow path between nodes In2 and Out during an interval of output times, and impact during the remaining times is weak; thus the I/O analysis would suggest that node In2 is not a good candidate for chlorine input to regulate residual at node Out Flow Path Analysis The PBA can identify all paths between input and output, including the composition of each path. When tanks are draining and the number of flow paths is large, their analysis could be tedious, and perhaps less useful to study individually. Instead, the 52

64 effects of different pipes and tanks on output water quality can be computed using the PBA information. As particles travel through a pipe or stay inside a storage tank, the disinfectant consumption (for example) in this specific pipe or tank can be recorded and summed to quantify each pipe or tank s effect on output water quality. Note that a similar analysis can not be performed using typical network water quality modelling software. Such codes typically allow computation of, say, chlorine loss within particular network elements, but that information is distinctly different from the effect of network elements on chlorine loss at a particular sampling location and time. A simple network is used to illustrate the type of results that can be obtained. The network is shown in Figure 3.8 and includes 12 pipes (p1 to p12) and one tank. The constant chlorine concentration at the source is 1 mg/l, and is the only chlorine addition. If there was no chlorine consumption in the pipes and the tank, the output concentration would be also 1 mg/l. The PBA can calculate the chlorine concentration loss in the pipes and tank, the sum of which is the difference between input (1 mg/l) and output concentrations. Figures 3.8 and 3.9 show the dependence of chlorine concentration loss at the output node on specific network elements at two different sampling times, one during the tank filling period and the other during the draining period. In both Figures 3.8 and Figure 3.9, no chlorine concentration loss is associated with pipes p3, p6 or p10, because no water travels through these pipes before reaching the output. The chlorine concentration loss in p8 and p11 are significant, because all 53

65 Tank mg/l 0.02 mg/l p12 p1 p2 p mg/l p mg/l p5 p mg/l p9 p mg/l p mg/l p mg/l p11 Output = mg/l Figure 3.8: Flow Path Analysis in Tank Filling Period mg/l mg/ L mg/ L mg/l mg/l mg/ L mg/l mg/l mg/l mg/ L Output = mg/l Figure 3.9: Flow Path Analysis in Tank Draining Period 54

66 water reaching the output travels through one or the other and the decay is significant in both of the pipes. In Figure 3.8, no water passes through the tank and the chlorine concentration loss in the tank is zero. In Figure 3.9, because of residence time in the tank, the water reaching the output has significantly lower chlorine concentration. The chlorine concentration loss in the tank is mg/l, which accounts for the majority of the output difference (0.289 mg/l) between the two cases. 55

67 Chapter 4 Wall Demand Coefficient Estimation with I/O Model 4.1 Mathematical Problem Statement and Key Assumptions A traditional least squares approach to parameter estimation is taken, with the objective function to be minimized: N s Φ(K w ) = i=1 e i 2 (4.1.1) where K w = 1 N p vector of the wall demand coefficients; e i = the difference between the ith pair of measured (c m ) and modeled (c o ) chlorine concentration; N s = number of samples (measurements); and N p = number of wall demand coefficients to be estimated. The methods and algorithms described below could be extended directly to other objective functions for parameter estimation, such as weighted least square algorithm or, more generally, maximum likelihood estimation. 56

68 Data collection in WDS will not allow estimation of a unique K w for each pipe. Rather the pipes should be grouped and those in the same group are assumed to have the same wall demand coefficient. We use G i (1 i N p ) to represent the sets of pipe indices that share the same wall demand coefficient K wi. Chlorine first order decay is a key assumption here and the linear relationship between upstream and downstream chlorine concentration can not be established without this assumption. Other assumptions made in this stage of research are: During the process of wall demand coefficient estimation, the hydraulic model is assumed to be accurate, i.e. hydraulic model is calibrated before the estimation of wall demand coefficients. Other parameters in the chlorine decay model, such as chlorine bulk decay coefficient, mass transfer coefficient and hydraulic radius are assumed to be known or estimated independently before the calibration of wall demand coefficient. The proposed estimation method is to calibrate only one kind of water quality model parameter: pipe wall demand coefficient for chlorine decay. Pipes are assumed to be grouped accurately based on their wall demand coefficient, i.e. pipes in the same group have the same true values of the wall demand coefficients. 57

69 4.2 Multi Input Single Output Water Quality Model Input and Output In order to calibrate pipe wall demand coefficients, chlorine needs to be sampled at different locations. The I/O model developed in last chapter is capable of establishing the water quality relationship between upstream and downstream sampling locations. Output refers to sampling location where chlorine measurements are used to compare with water quality modeling results for the purpose of wall demand coefficient estimation, while inputs are locations upstream of an output and chlorine concentrations sampled there are treated as input variables in water quality modeling (i.e. as boundary conditions). One kind of input, for example, is a treatment plant where chlorine is added. Inputs are, however, not limited to water quality sources. They can be any location that water passes before reaching an output. For example, in subplot a of Figure 4.1, water flows from treatment plant S, passes node I1 and arrives at node O1. Both S and I1 can be treated as input corresponding to O1. The output, inputs and network components connecting them constitute a sub network in which inputs serve as both water and chlorine sources Flow Path in WDS In WDS, water leaving the treatment plants or other sources travels through different components, such as pipes, pumps and storage tanks, before reaching a certain consumer node. Chlorine concentrations at a specific location are therefore affected by the reactions within these components, which constitute the flow paths between 58

70 water source and the consumer node. Considering the complexity in WDS topology and operation, flow paths change both spatially (for different consumer nodes) and temporarily. In Figure 4.1, water leaves the water treatment plant S and reaches sampling location O1 (subplot a) and O2 (subplot b). The highlighted pipes are all the pipes through which water may travel before arriving at the sampling locations. It can be seen that although these two sampling locations are physically close to each other, the flow paths connecting them to the water treatment plant are quite different. If two designs are implemented to estimate wall demand coefficients: one design samples chlorine concentration at S and O1 and the other samples at S and O2, different pipes will be targeted. The flow path concept is applicable to any output and corresponding input set. Knowing the flow paths between output and inputs can help exclude pipes that do not affect output concentrations, and construct a simpler water quality model that targets wall demand coefficient estimation for specific pipes. For a single pipe segment, the single flow path from input (upstream end) to output (downstream end) is obvious. However, in a WDS, chlorine concentration can be sampled anywhere and the identification of flow paths between different output and input locations requires the I/O model I/O Model for Parameter Estimation In the problem of estimating wall demand coefficients, the I/O model is used to establish the analytical relationship between several inputs and a single output node. The input set can include any node upstream of the output, and includes the constraint that all water reaching the output must flow through one of the input nodes. 59

71 S S I1 O1 O2 I2 I3 (a) (b) Figure 4.1: Flow Path for Two Sampling Locations in a WDS Water storage tanks constitute one set of candidate input nodes. In water quality modeling, tanks may be assumed to be completely mixed, partially mixed, or plug flow. In reality, however, there are many types of tank geometries and operating characteristics, and tank modeling remains an uncertain element of a WDS water quality model. Since the objective of sampling (in this work) is to calibrate pipe wall demand coefficients, it is ideal to avoid such tank modeling errors. If water reaching the output travels through one or more tanks, each such tank can be treated as an input location and chlorine concentrations sampled at the tank outlet. In that all the tanks are excluded from the wall demand coefficient estimation, uncertain water quality processes in tanks do not offset estimation accuracy. In the development that follows we assume that any tank on a path to a output node will be treated as an 60

72 input sampling location. In order to estimate wall demand coefficients, the PBA that constructs the I/O model must be revised to collect detailed travel path information for each backtracking particle. Each travel path j is described by a sequence of travel segment T S j, l, such that segment l is associated with flow in a single pipe, under constant hydraulic conditions. Thus the first order decay coefficient K within a travel segment is constant. Let K j, l and t j, l be defined as the decay coefficient and travel time in travel segment T S j, l, respectively, and p j, l be the pipe index for travel segment T S j, l. If p j, l G i, then travel segment T S j, l is associated with the grouped wall demand coefficient K wi. The simple network shown in Figure 4.2 has one input I, one output O and two pipes 1 and 2. Flow direction in the two pipes is always from I to O, while the flow velocity and therefore overall chlorine decay coefficient K changes, caused by water demand change at O. It is obvious that there exist two flow path between the input and output. For a single output time T 1, a water particle reaching O is split to two according to flow rates in pipe 1 and 2 and both of the particles a and b are backtracked. At time T 2 (T 2 < T 1 because the algorithm is backtracking), hydraulic condition in the pipes changes and the first travel segment in both flow paths is recorded. Under new hydraulic condition, a and b are backtracked and b first reaches input I at time T 3, ending one flow path and its second travel segment. At time T 4, hydraulic conditions change again and another new travel segment is generated for the remaining flow path, which stops at time T 5 at node I. The I/O model constructed by PBA then includes one input and two flow paths: one with 61

73 2 travel segments and the other with 3. The impact of wall demand coefficients on output chlorine concentration at time T 1 is reflected in the overall decay coefficients for the 5 travel segments. Since there are no tanks in the flow paths, the impact (T1) (T2) a 1 I O 2 b b a (T3) a (T4) a b (T5) a Figure 4.2: Travel Segments in Flow Path coefficient of each path can be expressed explicitly as a function of travel segment chlorine decay coefficients and travel times. γ j = f j exp( K j, l t j, l ) = f j exp( K j, l t j, l ) (4.2.1) l P j l P j where P j = set of travel segment indices that constitute travel path j between the output and the inputs (obtained using the PBA ) and f j = dilution factor equaling to the flow ratio of path j. When p j, l G i, K j, l = K b + K f, j, l K wi R H, j, l (K f, j, l + K wi ) (4.2.2) 62

74 where R H, j, l and K f, j, l are the hydraulic radius and mass transfer coefficient of the travel segment T S j,l, respectively. Note that these quantities are known from the hydraulic simulation and the PBA. Combining and 4.2.1, a multiple input single output (MISO) model of chlorine concentration can be expressed as: c o (T ) = N γ j c ij (T t j ) = j=1 N f j exp( K j, l t j, l )c ij (T t j ) (4.2.3) l P j j=1 Notably, equations and express the modeled chlorine concentration explicitly in terms of pipe wall demand coefficients, and an iterative nonlinear parameter estimation method can be implemented to calibrate the coefficients efficiently. The difference between the PBA revised for wall demand coefficient estimation and that described in last chapter is that the former collects and stores information necessary for the calculation of path impact coefficients (i.e., f j, t j,l for all segments l in path j, and pipe index information associated with each segment), while the latter accumulates those coefficient values during the particle backtracking process. 4.3 Gauss-Newton Method for Parameter Estimation The Gauss-Newton method [47] is proposed to solve the least square problem 4.1.1, with and The ith iteration of the method is: K w (i + 1) = K w (i) N(i) 1 q(i) (4.3.1) 63

75 where N and q are the approximated Hessian matrix H and gradient vector of the function Φ(K w ), respectively. Each element of the Hessian matrix is defined H αβ = 2 Φ K wα K wβ = 2 N s i=1 e 2 e i i K wα K wβ + 2 N s i=1 ( e i K wα )( e i K wβ ). In the Gauss-Newton method, the first term of H αβ is neglected and N is used to approximate H, where N is defined N αβ = 2 N s i=1 ( e i K wα )( e i K wβ ). Each element of the gradient vector is defined q α = Φ K wα = 2 N s i=1 e e i i K wα. 4.4 Model Sensitivity analysis In the description of the iterative Gauss-Newton method, the derivative of the difference between measured and modeled chlorine concentration e with respect to wall demand coefficient need to be calculated. From the definition of e and equation 4.2.3, From equation 4.2.1, e(t ) K wi = c o(t ) K wi = N j=1 γ j K wi c ij (T t j ) (4.4.1) γ j K j, l = γ j t j, l (4.4.2) K wi K wi l P j Combine equation and 4.4.2, e(t ) N = γ j K j, l t j, l c ij (T t j ) (4.4.3) K wi K j=1 wi l Pj From the definition of overall decay coefficient 3.4.1, K j, l K wi = K 2 f, j, l R H, j, l (K f, j, l +K wi ) 2 for p j, l G i 0 for p j, l / G i (4.4.4) 64

76 Whenever the estimates of wall demand coefficients are updated, the model sensitivity analysis is performed, since e(t )/ K wi depends on K wi. All the parameters in equation and other than K wi are collected using the particle backtracking process in a preprocessing step. Therefore the PBA is required only once for each measurement, and the update of model sensitivity during each Gauss-Newton iteration is computationally efficient. 4.5 Example Calculation The example shown in this section is only for the purpose of parameter estimation algorithm illustration. The units are arbitrary and the hydraulic conditions used in the example were set to be simple without solving the system equations, so they may not be consistent with mass/energy conservation principle. But such limitations will not hurt the purpose of showing how the parameter estimation algorithm works. I P1 P2 O I P1 P2 O Hydraulic condition A: f1 = 0.5 f2 = 0.5 t1 = 1.0 t2 = 2.0 Hydraulic condition B: f1 = 0.6 f2 = 0.4 t1 = 1.5 t2 = 2.5 Figure 4.3: Parameter Estimation Example A very simple network shown in Figure 4.3 has two pipes (P 1 and P 2), one input node I and one output node O. Both pipe has hydraulic radius of 1. The 65

77 wall demand coefficients of pipe P 1 and P 2 are 0.5 and 1.0, respectively. The input chlorine strength is constantly 1 and two measurements are taken at O under different steady state hydraulic conditions. Travel time from I to O and flow ratio of the path through pipe P i are t i and f i respectively, which are shown in the figure. Although mass transfer coefficients are affected by hydraulic conditions, it is assumed in this example that they are constantly 1.0 in both pipes. It is also assumed that there is no bulk decay and no measurement error. The measurements under hydraulic conditions A and B are calculated using true wall demand coefficients and the results are and 0.479, respectively. Based on the pipe characteristics and assumptions made above, the total decay coefficient in pipe P i, K i, is K i = K wi 1 + K wi (4.5.1) The sensitivity of total decay coefficient to wall demand coefficient can be simplified as K i K wj = 1 (1+K wi ) 2 for i = j 0 for i j (4.5.2) Therefore the sensitivity of error to wall demand coefficient, i.e. Equation can be simplified as e = f i exp( K wit i t i ) (4.5.3) K wi 1 + K wi (1 + K wi ) 2 The initial estimates of the two wall demand coefficients are set to be K w1 = K w2 = 0.0. The steps to update the wall demand coefficient estimates within one iteration are: 66

78 1. Model chlorine output using K w1 and K w2. The modeled results are C A = 1.0 (under hydraulic condition A) and C B = 1.0 (under condition B). 2. Calculate difference between measured and modeled results. e A = = and e B = = Calculate sensitivity of error to wall demand coefficients. Since K wi = 0.0, e K wi = f i t i. S A1 = e A K w1 = = 0.5. S A2 = e A K w2 = = 1.0. S B1 = e B K w1 = = 0.9. S B2 = e B K w2 = = Calculate matrix N and vector q. N 11 = 2 (S A1 S A1 + S B1 S B1 ) = 2 ( ) = N 12 = 2 (S A1 S A2 + S B1 S B2 ) = 2 ( ) = N 21 = N 12 = 2.8. N 22 = 2 (S A2 S A2 + S B2 S B2 ) = 2 ( ) = 4. q 1 = 2 (e A S A1 + e B S B1 ) = 2 ( ) = q 2 = 2 (e A S A2 + e B S B2 ) = 2 ( ) = Update estimates according to Equation K w1 = K w2 =

79 The same procedures 1-5 are repeated to keep updating the wall demand coefficients estimate. In this example, after 10 iterations, K w1 = 0.500, K w2 = 0.998, e A = and e B = Estimate Uncertainties Analysis under Chlorine Measurement Error Both chlorine measurements and the hydraulic model, which is the base of water quality modeling, may contain errors. Hydraulic model calibration is not perfect due to errors in water demands, pipe roughnesses and storage tank levels. Such errors will propagate to pipe flow velocity and to simulation of chlorine transport within the WDS. The impact of hydraulic error on wall demand coefficient estimate will be considered in Chapter 5, where sampling network design (input/output selection) methods are proposed to reduce its effect. The uncertainty in parameter estimates caused by measurement errors on model output variables has been discussed widely in the parameter estimation literature. When the output measurement errors are iid (independent and identically distributed), the estimate uncertainty matrix can be approximated as [47] V o = 2σ 2 N 1 (4.6.1) where V o = covariance matrix of the estimates under output measurement error; and σ 2 = variance of chlorine measurement error. In I/O model, both output and input concentrations contain measurement error and the estimate uncertainties caused by measurement errors at inputs need to 68

80 be quantified. The following analysis is analogous to that commonly performed for measurement errors [47]. Under estimates K w, assume the objective function Φ(K w ) reaches a local minimum. When input concentration measurements vary slightly from c i to c i + δc i, where c i is a 1 N ci vector that contains input concentrations having impact on output water quality, minimization of Φ(K w ) causes a change in the estimates to K w + δk w. Φ(c i, K w ) K w = 0 (4.6.2) Φ(c i + δc i, K w + δk w ) K w = 0 (4.6.3) Equation is expanded in Taylor series up to first order. The resulting Taylor expansion subtracting equation leads to, ( 2 Φ K 2 w ) ( ) 2 Φ δk w + δc i 0 (4.6.4) K w c i Therefore, δk w = H 1 ( 2 Φ K w c i ) δc i (4.6.5) where the Hessian matrix can be H is approximated as N. Under the influence of input measurement errors, the estimate covariance matrix V i is V i E(δK w δk T w) (4.6.6) Combining equations and 4.6.6, the estimate variance only under input measurement errors, V i is V i E(N 1 ( 2 Φ/ K w c i )δc i δc i T ( 2 Φ/ K w c i ) T N 1 ) (4.6.7) 69

81 When measurement errors on input concentration also have iid distribution with variance σ 2, Define the matrix L Then V i = σ 2 N 1 ( 2 Φ/ K w c i )( 2 Φ/ K w c i ) T N 1 (4.6.8) L = 2 Φ/ K w c i (4.6.9) V i = σ 2 N 1 LL T N 1 (4.6.10) If f β is the βth element in input concentration vector c i, then the element L αβ at row α and column β of matrix L is L αβ = 2 Φ K wα f β = 2 Assume the error e i is small, N s e i i=1 2 e i N s + 2 K wα f β i=1 i=1 ( e i K wα )( e i f β ) (4.6.11) N s L αβ 2 ( e i )( e i ) (4.6.12) K wα f β The element LL T xy at row x and column y of matrix LL T is then N ci [ Ns ] (LL T ) xy = 4 ( e i )( e N s i ) ( e i ei )( ) K wx f β K wy f β β=1 i=1 i =1 (4.6.13) Elements of the vector c i are chlorine concentrations sampled at input locations at different times. When the sampling frequency at outputs is much lower than that at inputs, it can be reasonable to assume that input concentration at a given time and input location will affect only one sampled output concentration. This assumption simplify the equation since e i f β e i f β = 0 when i i. N s (LL T ) xy = 4 e i e i K i=1 wx K wy β=1 70 N ci ( ) 2 ei (4.6.14) f β

82 Based on the multi input single output I/O model, N ci β=1 ( e i f β ) 2 in equation equals to the sum of the squared impact coefficients for the output concentration c o corresponding to e i. Equations and together provide an easy way to calculate the estimate uncertainties caused by measurement errors on input concentration. Assuming input and output measurement errors are not correlated, the estimated variance matrix under both kinds of error is then, V = V o + V i (4.6.15) The calculation of V o and V i requires impact coefficients of flow paths and sensitivities of modeled output concentration to wall demand coefficients. Both sets of information can be obtained together with parameter estimates based on calibrated wall demand coefficients. 4.7 Application Network I A simple network is used to illustrate wall demand coefficient estimation using inputoutput modelling. The network is shown in figure 4.4. The characteristics of the pipes are summarized in table 4.1, in which the wall demand coefficient K w is in the unit f t/day. There are three pipe groups in terms of the wall demand coefficient. Pipe s ID and the group to which a pipe belong are shown together in the figure. The node labelled Source is the only chlorine input point and the input strength 71

83 switches between 1 mg/l and 3 mg/l every hour. The accurate chlorine bulk decay coefficient is assumed to be known before calibration, which is 0.5 1/day. The chlorine concentration is measured at one output and two inputs. The hydraulic model is assumed to be accurate and the chlorine measurement error at both inputs and output is assumed to be normally distributed with zero mean and standard deviation of 0.05 mg/l. The design of the simulation cases are shown in table 4.2 and in all the simulation cases, Monte Carlo simulation with 500 realizations is implemented to gather the statistics of the calibration performance. In each case, 80 measurements are gathered at the output nodes. The simulation results are shown in table 4.3, where µ and σ (ft/day) are the sample mean and standard deviation of the estimated wall demand coefficient, respectively, and ˆσ is a linear estimate of the same standard deviation. For each realization, the sensitivity coefficients are used to calculate ˆσ and table 4.3 gives the range of these values. Model fit to the sampled data is measured by the sample mean µ r and standard deviation σ r of the root mean square error(rmse). with units mg/l. In simulation case 1, the output node is n7 and input nodes are source and tank. Input-output analysis shows that pipes p1, p2, p4, p7 and p12 can be isolated from the network and all of these pipes belong to pipe group 1. Monte-carlo simulation results show a reliable estimate with mean 1.00 f t/day and standard deviation f t/day. When the output is located at node n8 in case 2, pipe p5 and p8 are also included in the subnetwork for estimation. Comparing to the results in case 1, the estimate in case 2 in much less reliable, wall demand coefficient estimates having standard 72

84 Tank Source Pump p12 (Group1) p1 n1 p2 n2 p3 n3 (Group1) (Group1) (Group3) p4 p5 p6 (Group1) (Group2) (Group3) n4 p9 n5 p10 n6 (Group1) (Group3) p7 (Group1) p8 (Group2) n7 p11 (Group1) n8 Figure 4.4: Network I for Chlorine Wall Demand Coefficient Estimation Table 4.1: Pipe Characteristics of the Example Network Pipe Length(ft) Diameter(inch) K w (ft/day) Pipe Group p p p p p p p p p p p p

85 Table 4.2: Monte Carlo Simulation Cases for Wall Demand Coefficient Estimation Case No Input Output Sampling Time Sampling Interval 1 Source, Tank n (hr) 0.25 (hr) 2 Source, Tank n (hr) 0.25 (hr) 3 Source, Tank n (hr) 0.25 (hr) 4 n2, n5 n (hr) 0.25 (hr) Table 4.3: Simulation Results for Wall Demand Coefficient Estimation Case µ r σ r Pipe µ σ ˆσ (mg/l) (mg/l) Group (ft/day) (ft/day) (ft/day)

86 deviation of ft/day and 0.18 ft/day for pipe group 1 and 2 respectively. In simulation case 3, the output is at node n6 and the demand coefficient of all the three pipe groups can be estimated. However, when there are three parameters to be estimated, the estimation accuracy is poor, especially for pipe group 1 and 2. In simulation case 4, keeping the output at n6 and putting the input nodes at n2 and n5 isolate the pipe group 3 for the purpose of wall demand coefficient estimation. In this case with only one parameter to be estimated, the estimate is more reliable than that in the case 3: standard deviation of the estimate is reduced from 0.11 ft/day to 0.05 f t/day. In all of the four simulation cases, the mean value of the RMSE is always about 0.05 mg/l with very little variation. Therefore it is hard to tell the magnitude of the uncertainty in the wall demand coefficient estimates from the fitness between observed and modeled results. However, the linear estimates of the wall demand coefficient standard deviation approximates the uncertainty of the estimates very well in this case Network II The proposed parameter estimation approach to calibrate pipe wall demand coefficients is also tested in a real network shown in Figure 4.5. Node S, which is the treatment pant, provides water continuously to the network with flow rate from 2375 to 3790 gallon per minute. There are six elevated tanks (T1 to T6) and 4 pumps in the network. The filling and draining of the tanks are controlled by the operation of pumps, according to the water levels in the tanks. A hydraulic model of the network 75

87 was calibrated for a period of 16 days (384 hours) using pressure measurements, tank water levels and water meter readings [48]. Water leaving the treatment plant, S, was measured for free chlorine concentration and the data are plotted in Figure 4.5. In the simulation study, pipes in the application network are assigned to one of the C2 A1 B1 C1 A2 B2 Figure 4.5: Network II for Chlorine Wall Demand Coefficient Estimation two pipe groups generated by a Markov Random Field model (see Appendix A). The pipes bolded in the figure 4.5 (group 1) are assumed to have true wall demand coefficient 2.5 ft/day, while the other pipes are grouped together (group 2) with wall demand coefficient 1.5 f t/day. Three designs (A, B, and C) are tested for wall demand coefficient estimation. 76

88 In each design, chlorine concentrations are measured at two output locations (for example, A1 and A2 for design A) and the corresponding input locations: S and the tanks that are in the flow path from S to the output locations. As mentioned earlier, sampling chlorine concentration at the tanks can exclude tank modeling error from wall demand coefficient estimation. Chlorine is sampled at the outputs from time 301 to time 324 with sampling interval of 15 minutes. Therefore we have 96 chlorine measurements at each output location. The sampling durations at the input locations (S and tanks) are assumed to be long enough to close the I/O relationship between any output and input measurements. In other words, all water reaching the output from time 301 to 324 must be measured for chlorine concentration when leaving the inputs. It is assumed that the hydraulic model does not contain any error. Both of the input and output chlorine measurements are assumed to be subject to errors with independent identical distribution (iid). The standard deviation of the iid error added to chlorine measurements is 0.05 mg/l. Monte Carlo simulation with 500 realizations is implemented for each design to study the uncertainties in parameter estimates and the results are shown in Table 4.4, in which µ i and σ i represent the mean and standard deviation of the wall demand coefficient estimate for pipe group i respectively. The ˆσ i is the estimated standard deviation for parameter i and reflects the uncertainty in wall demand coefficient estimation caused by the assumed chlorine measurement errors. The calculation of ˆσ i is described in last section and ˆσ 2 i = V ii, where V ii are the diagonal elements of matrix V in Equation Model fit to the sampled data is measured by the mean µ r and standard deviation σ r of the root mean 77

89 square error (RMSE). Both µ r and σ r are with unit mg/l. Table 4.4: Simulation Results for Wall Demand Coefficient Estimation Design µ 1 µ 2 σ 1 σ 2 ˆσ 1 ˆσ 2 µ r σ r (ft/day) (mg/l) A B C For this application example, three designs produce different estimate uncertainties. The standard deviation of wall demand coefficient estimate ranges from (1.6% of true value) to (2.4% of true value) ft/day for pipe group 1, and (2.1% of true value) to (2.9% of true value) ft/day for group 2. In fact, each of the three designs are relatively good designs selected using a metric that is based on sensitivity analysis. The details of the sampling design procedure will be discussed in next chapter. The following example illustrates a bad design for the application network can generate wall demand coefficients estimates with such high uncertainties that the estimates should not be used in chlorine residual modeling. From the Table 4.4, it can be seen that design A is the best of the three. If output A2 is replaced with location B1, the Monte-Carlo simulation results are shown in Table 4.5. For this design, the standard deviation (0.57 f t/day) of the estimates for wall demand coefficient of pipe group 1 is as high as one third of the true value (1.50 ft/day). In order to compare the impact of input measurement error and that of output measurement error, the above designs are tested when there are no input measurement errors, i.e. the chlorine concentrations at the inputs are assumed to be accurate. 78

90 Table 4.5: Simulation Results for Bad Design Output Locations µ 1 µ 2 σ 1 σ 2 µ r σ r (ft/day) (mg/l) A1,B The results are shown in Table 4.6, together with the estimation results when both input and output measurement errors exist. The µ i and σ i respectively represent the estimate mean and standard deviation of pipe group i. It can be seen that under the assumption that measurement errors have the same standard deviation, output measurement errors in these cases are the dominant factor in parameter estimate uncertainty, which therefore could be approximated well if only output measurement errors are considered in uncertainty analysis. Table 4.6: Impact of Input and Output Measurement Errors Design µ 1 µ 2 σ 1 σ 2 µ 1 (ft/day) µ 2 σ 1 σ 2 A B C Observation The simulation results in the two examples show that 1) the estimation is almost unbiased under the assumption that the chlorine measurement error at inputs and output has normal distribution with zero mean; 2) water quality parameter estimation with input-output model can reliably exclude pipes that have no effect on output water quality, given an accurate hydraulic model; 3) pipe grouping and sampling design can affect the estimation accuracy; 4) under the assumption that measurement 79

91 error variance is known, the linear estimates of the standard deviation of wall demand coefficient estimates agree well with those calculated from Monte Carlo Simulation, at least for the error standard deviation (0.05 mg/l) considered in this study ; 5) similar sampling effort results in less reliable estimates when there are more pipe groups. 80

92 Chapter 5 Sampling Design for Wall Demand Coefficient Estimation 5.1 Introduction Within a WDS, there are numerous locations where chlorine concentration can be measured for the purpose of water quality model calibration. Where to take chlorine measurements is an issue of sampling design, which can significantly affect the accuracy of model calibration. As seen in the applications of last chapter, poor estimation of model parameters can occur despite a good fit between measurements and model prediction. Since calibrated water quality models will be used to predict water quality at locations other than where measurements are taken and be applied to a variety of scenarios, significant uncertainty in parameter estimates can lead to significant model prediction errors. Therefore, parameter confidence is as important as a good match between sampled and modeled system responses. Minimization of the difference between model prediction and measurement is usually the objective of 81

93 model calibration, while sampling design focuses on increasing the confidence of parameter estimates or reducing the variance of the predicted response using estimated parameters. Although a limited number of studies considered WDS water quality model calibration algorithms, the problem of sampling design for model calibration has not been addressed. In this chapter, statistical experiment design theory is used to assess the sampling designs for WDS water quality model calibration. Specifically, a sampling design method is proposed to estimate reliable pipe chlorine wall demand coefficients. This sampling method aims to generate a design that is robust to both chlorine measurement errors and hydraulic model uncertainties. It is observed that sampling at locations where chlorine concentration is insensitive to hydraulic disturbance leads to reliable parameter estimation when hydraulic errors exist in the model. Based on this observation, a heuristic based on sensitivities of predicted chlorine concentrations to pipe flow velocities over sampling period are used to compare sampling design s robustness to hydraulic model errors. Sensitivities of chlorine concentration to wall demand coefficients are considered and an experiment design algorithm is applied to generate an approximate D-optimal design. 5.2 Problem Statement and Assumptions Water quality can be sampled at certain locations, for example treatment plant, tank outlet and hydrant. In the practice of calibrating pipe wall demand coefficients, chlorine concentration should be sampled such that a water quality model between 82

94 output (downstream) nodes and their corresponding input (upstream) nodes can be established and the parameter estimation algorithm can be applied to calibrate wall demand coefficients. In the simplest situation, the chlorine sources (for example, treatment plants) are treated as the only inputs and a trial and error method is implemented to fit the chlorine measurements to model predictions at sampling locations (outputs) within a WDS. As described in the last chapter, the Input-output water quality model developed in this work is used to establish such a water quality relationship that is explicit in terms of wall demand coefficients. The sampling design procedure selects a set of output and input nodes so that all outputs are covered by the inputs (i.e. water reaching the output must pass one of the inputs) and the parameter estimates are robust to both chlorine measurement and hydraulic errors. All the output nodes must be covered by inputs in order to define a complete input-output relationship. Notice that lack of input/output relationship can occur regardless of the approach to parameter estimation; the I/O model approach makes this relationship explicit and thus specifies the subnetwork that participate in transport between input and output nodes. Although it is not difficult to apply the parameter estimation algorithm once a sampling design is known, the sampling design is challenging because of the large decision space and the need for statistical models for the source of errors (e.g. the node water demands). Many locations can be sampled as output and for a certain output node, there can exist many upstream nodes for which the coverage criterion is met. There are design parameters in addition to sampling locations, such as sampling 83

95 duration and frequency. To manage the complexity of the sampling design problem, certain simplifying assumptions are made. 1. There exists a hydraulic model that represents the expected hydraulic condition of the network during the field test. A hydraulic model is the basis of water quality simulation and the sampling design is based on a certain hydraulic scenario. 2. Like all experimental design methods, this one depends on a priori estimates of the model parameters. Therefore although the wall demand coefficients are to be estimated, initial estimates of their values must be assumed. In this study, true model parameters are used to generate sampling design to estimate these parameters. Discussions on this paradox of sampling design can be seen elsewhere [34]. 3. Treatment plants and storage tanks are treated as input locations in water quality model and chlorine concentration is assumed to be sampled at their outlets. Sampling at tank outlets can exclude the tank modeling errors and simplify the input-output water quality relationship. 4. Chlorine measurement errors are uncorrelated and identically distributed. 5. Sampling duration and sampling frequency are assumed to be known, with the only decision variable being output sampling locations. This assumption is consistent with sampling duration and frequency being dictated by practical concerns of measurement techniques and sampling effort (e.g. continuous 84

96 monitoring versus grab samples). 5.3 Sampling Design Methodology Sensitivity and D Optimality in Parameter Estimation Consider a very simple case in which model response y (chlorine concentration in this study) is the function of one parameter θ (wall demand coefficient) and another model input variable x (flow velocity, for example). y = η(θ, x) (5.3.1) When both y and x are accurate and the model is perfect, the true value of parameter θ can be calculated. However, in reality both y and x contain errors: δy and δx. These errors cause the error in parameter estimate: δθ. y + δy = η(θ + δθ, x + δx) (5.3.2) A first order Taylor series expansion of Equation 5.3.2, after subtracting 5.3.1, is δy = η η δθ + δx (5.3.3) θ x From equation 5.3.3, the magnitude of the parameter estimate error can be calculated. δθ = 1 η θ η δy x δx 1 η δy + η x δx (5.3.4) Equation shows that the reliability of parameter estimates can be controlled by selecting sampling points where model response y is sensitive to θ and insensitive to x. 85 θ

97 When there is more than one parameter to be estimated and parameter sensitivity coefficients are correlated, the volume of a model parameter confidence region can be used to judge the accuracy of parameter estimates. The volume is proportional to the square root of the determinant of the covariance matrix of the parameter estimates, det V θ. Mathematically, the criteria of minimizing the volume of parameter confidence region is equivalent to maximizing det V 1 θ and called D optimality [47]. Consider the case in which two measurements y 1 and y 2 are used to estimate two parameters θ 1 and θ 2 and other than the model parameters, there are two model inputs x 1 and x 2. y i = η i (θ 1, θ 2, x 1, x 2 ); i = 1, 2 (5.3.5) Through analysis similar to the that for single parameter case, the relationship between the errors δθ, δy, and δx can be derived. δy i η i x 1 δx 1 η i x 2 δx 2 = η i θ 1 δθ 1 + η i θ 2 δθ 2 (5.3.6) We define δy i η i x 1 δx 1 η i x 2 δx 2 as combined error ɛ i, with the resulting parameter inverse covariance matrix V 1 θ. V 1 θ = [ ] T η Vɛ 1 θ [ ] η θ (5.3.7) where [ η θ ] in this example is the 2 2 matrix of sensitivity coefficients, and Vɛ is the covariance of combined errors ɛ. When the number of measurements, N m, is larger than the number of parameters to be estimated, N p, the equation is still valid, with [ η θ ] being a Nm N p sensitivity matrix. 86

98 The determinant of the product of two square matrices of the same order is equal to the product of the determinants [49]. Therefore when N m = N p ( [ det η ] T [ η ] ) detv 1 θ θ θ = (5.3.8) detv ɛ The matrix M = [ ] η T [ η ] θ θ is called the information matrix in sampling design theory. From equation 5.3.8, in order to achieve the D optimality, i.e. maximize the determinant of V 1 θ, the ratio between the determinants of the information matrix and the covariance matrix of the combined measurement errors needs to be maximized. In parameter estimation problems, N m is usually larger than N p. However, it is still reasonable to assume that increasing the determinant of the information matrix and decreasing the magnitude of combined measurement errors can improve the reliability of parameter estimates Sampling Design Method In this study, the impact of hydraulic model errors on water quality model calibration is controlled by prohibiting candidate output locations that can potentially generate matrices V ɛ with large determinants; then a D optimal algorithm is used to select sampling locations (from remaining locations) that maximize the determinant of the information matrix. Heuristic that compares the impact of hydraulic errors. Here we focus on diagonal elements in the covariance matrix V ɛ that affect its determinant. For the case of two hydraulic values x, the ith diagonal element is (δy i η i x 1 δx 1 η i x 2 δx 2 ) 2, with representing the mean value. Such focus on diagonal elements is a simplification that 87

99 ignores correlation among the ɛ i, but note that we are not attempting to accurately estimate V ɛ but rather to develop computationally efficient design heuristics. In this study, it is assumed that the standard deviation σ of the measurement errors δy is constant. It is also reasonable to assume that the chlorine measurement errors and hydraulic model errors are not correlated. Based on these assumptions, (δy i η i x 1 δx 1 η i x 2 δx 2 ) 2 = σ 2 +( η i x 1 ) 2 var(x 1 )+( η i x 2 ) 2 var(x 2 )+2( η i x 1 )( η i x 2 )cov(x 1, x 2 ), where var() and cov() means variance and covariance respectively. If the standard deviation of the hydraulic errors are proportional to hydraulic data, i.e. var(x i ) = a 2 x 2 i and a is a constant, then η i x 1 x 1 + η i x 2 x 2 is proposed as a heuristic to measure the impact of hydraulic model errors on parameter estimates. Small values of η i x 1 x 1 + η i x 2 x 2 are preferred based under the assumption that they lead to small value of detv ɛ. Errors in a hydraulic model can be measured as errors in flow velocity in network links over the simulation period. A sampling design heuristic for a chlorine measurement c at location i at time t is defined, N l N h H i (t) = l=1 t =1 c i(t) v lt (5.3.9) v lt where N l = number of pipes in the network; N h = number of hydraulic simulation periods; and v lt = flow velocity in pipe l and hydraulic simulation period t. H i (t) has the same unit as c i (t) and is proposed to represent the magnitude of model prediction error possibly caused by errors in the hydraulic model. Since the sampling design problem is to select sampling locations, and more than one measurement is typically obtained at each location, H is modified to be the mean 88

100 over all samples at a location. H i = 1 H(c i (t)) (5.3.10) N i t T s where N i is the number of chlorine measurement at location i. All locations are ranked using H and the chlorine concentrations at those locations associated with larger values of H are assumed to be more sensitive to hydraulic model errors. The sampling location with large H value should therefore be avoided. The threshold value of H should ultimately depend on experience and the magnitude of hydraulic model errors; due to the approximate nature of H and lack of information about velocity error statistics, a precise value of this threshold is not possible at this time. The value of H requires sensitivities of model-predicted chlorine concentration to pipe flow velocities. For computational simplicity, the sensitivity of a measurement to flow velocity in specific pipe and at specific time is done without considering the correlation between changes in flow velocities through conservation of mass and energy. The objective here is to get a heuristic that represents the impact of hydraulic errors on parameter estimation rather than to accurately calculate the sensitivity of chlorine measurements to pipe velocity. The computational details of the sensitivity calculation will be discussed in next section. D optimal design to minimize the impact of chlorine measurement errors. If there are N m measurements and N p parameters to be estimated, the information matrix can be written as [35], M = N m i=1 89 f T i f i (5.3.11)

101 where f i = [ η i θ 1, η i η θ 2,..., i θ Np ]. The information matrix of the chlorine measurements at a sampling location i, M i, can be calculated as: M i = t T s f T (c i (t))f(c i (t)) (5.3.12) f(c i (t)) = [ c i(t) θ 1, c i(t),..., c i(t) ] (5.3.13) θ 2 θ Np If a design ξ contains more than one sampling location, the information matrix M is the sum of the information matrices of all the sampling locations. M = i ξ M i (5.3.14) In this study, the number of the total output sampling locations S, sampling duration and frequency are specified. Because of the nonlinearity of this optimal sampling design problem, an approximate enumeration method is required to get the global maximum of detm. Considering the combinatorial nature of the optimization problem, it is difficult to generate an optimal design even with a moderate number of sampling and candidate locations. An iterative algorithm to generate sub-optimal designs [35] (Fedorov exchange algorithm) has been implemented widely. This approach selects S sample locations randomly, then exchange one location in the design with another not in the design. The rule of exchange is that it maximizes the increase of the determinant of the information matrix. The algorithm stops when the determinant of the information matrix can not be improved further using this simple pair-wise exchange. This algorithm also provides an efficient way to calculate the increase of the determinant caused by exchange of two locations. However this algorithm still contains combinatorial problem that can be computationally expensive. 90

102 In this study, a non-sequential algorithm [50] is used to generate the optimal design efficiently. Instead of considering the addition and deletion of the samples together, this algorithm separates the two processes and avoids computing all possible pairwise combination at each step. After randomly selecting S sampling locations, the algorithm adds the location which gives a maximum increase in the determinant of the information matrix. The design is then recovered by deleting the location to cause minimum decrease in the determinant. The process iterates until the same location is entered and then deleted. More specifically, the non-sequential D-optimal design procedure for S sampling locations is: 1. Calculate M i for all sampling locations. 2. Randomly select S sampling locations for the initial design. 3. Calculate M of the initial design. 4. Select location i from candidates not in the design to maximize the determinant of M + M i. 5. Add i to the design, M = M + M i. 6. Select location j in the design to maximize the determinant of M M j. 7. Delete j from design, M = M M j. 8. If i = j, stop. Otherwise repeat from 4. 91

103 Since the problem is not combinatorial, it is affordable to carry out the design procedures several times, beginning with different initial random designs. If the determinant of the information matrices of the final designs coincide, there is more confidence that they approximate global optimal design. Otherwise, the design with the largest information matrix determinant is selected Sensitivity of Model Response to Flow Velocity In equation 5.3.9, the value c i(t)v lt 0 when part or all of the water reaching loca- v lt tion i at time t travels through pipe l in hydraulic period t. There are multiple travel paths between inputs and the node i and each travel path is comprised of multiple travel segments. A travel segment represents the hydraulic condition of a pipe within a hydraulic period; in each travel segment both the overall decay coefficient and flow velocity are constant. Without tanks in the travel path (recall that tanks are input locations), the Input-Output model, 4.2.3, can be expressed here as, c i (t) = N i (t) j=1 f j exp( s P j K j, s t j, s )c ij (t t j ) (5.3.15) Where N i (t) = number of travel paths for sampling location i at time t; f j flow ratio of path j; K j,s = overall first order decay coefficient in travel segment T S j,s ; t j,s = travel time in T S j,s ; t j = travel time in path j; P j = set of pipe indices of path j; and c ij = input concentration of path j. Based on the Input-Output model, H i (t) can be expressed in terms of the velocities that affect the model response, N l N h H i (t) = l=1 t =1 c N i (t) i(t) v lt = v lt 92 j=1 s P j c i(t) v j,s v j,s (5.3.16)

104 where v j,s is the flow velocity in travel segment T S j,s for measurement at location i and time t j, and without considering the correlation between changes in flow velocity, c i(t) v j,s v j,s = f j exp( s P j K j, s t j, s )c ij (t t j ) (K j,st j,s ) v j,s v j,s (5.3.17) The relationship between v j,s and t j,s through the length of travel segment L j,s is t j,l = L j,s /v j,l. When a travel segment is terminated spatially at the end of a pipe, L j,s is not affected by v j,s. (K j,st j,s ) v j,s = v j,s t j,s K j,s K j,s (5.3.18) v j,s v j,s v j,s When travel segment is terminated temporally at the end of a hydraulic period, t j,s is not affected by v j,s. (K j,st j,s ) v j,s = v j,s t j,s K j,s (5.3.19) v j,s v j,s In backtracking algorithm that constructs the Input-output model, the flow velocity, travel time and overall decay coefficients for each travel segment are all calculated dynamically. For sampling design, the sensitivity of the decay coefficient to flow velocity within a travel segment is calculated by perturbation analysis during backtracking process and recorded as additional travel segment information. With all these information, equation can be evaluated at the end of particle backtracking process. 5.4 Application The Network II of the chapter 4 is used here for sampling design application. The network with certain node IDs is shown here in Figure 5.1. The water source and 93

105 all the tanks are treated as input locations. Chlorine is still sampled at the outputs from time 301 to time 324 hour with sampling interval of 15 minutes. In the following Figure 5.1: Network for Sampling Design Application simulations, the standard deviation of the iid error added to chlorine measurements is 0.05 mg/l. Hydraulic model errors are assumed to be caused by the base demand errors at consumer nodes, which are iid and have standard deviation 10 or 20 per cent of the node base demand. Since usually the total demand can be measured accurately, after the demand errors are imposed to consumer nodes, the resulted demand are adjusted proportionally so that the total demand of the system does not change. 94

106 5.4.1 Relationship of H and M with Wall Demand Coefficient Estimation Uncertainty It is assumed that all the pipes in the network have the same wall demand coefficient, which is 1.5 ft/day. In such situation, M is a single value rather than a matrix and detm = M, which is the sum of squared sensitivity coefficients (chlorine concentration with respect to wall demand coefficient, in the unit of mg.day/l.f t) over sampling period. The values of H and M for all the nodes in the pipe network are shown in Figure 5.2. It can be seen that both H and M have quite wide range, with H from almost zero to 1.5 mg/l and M from almost zero to 6 (mg.day/l.ft) 2. H M Figure 5.2: Value of M and H 95

107 M and Estimation Uncertainty under Chlorine Measurement Error Certain sampling locations with different M values are selected and Monte-Carlo simulations with 500 realizations are implemented to calculate the parameter estimate uncertainty if chlorine concentrations are measured at these sampling locations. When it is assumed that there is no hydraulic error and both input and output measurements are subject to errors described above, the results are shown in Table 5.1 and the relationship between the M and standard deviation of the parameter estimation is plotted in Figure 5.3. In the table, µ and σ (ft/day) are the sample mean and standard deviation of the estimated wall demand coefficient, respectively. It can be seen that although input measurement errors are not considered in calculating M, it serves as a very good criteria in selecting sampling locations. Node 208 has the maximum value of M, which is 5.55 (mg.day/l.ft) 2 and sampling at this nodes leads to most reliable estimate of the single parameter (with standard deviation ft/day). H and Estimation Uncertainty under Hydraulic Model Error As shown in Equation 5.3.8, even if there are only hydraulic model errors and no chlorine measurement errors, the value M still affects the reliability of wall demand coefficient estimation. Therefore, in order to test the impact of H values on parameter estimation, sampling locations selected for comparison should have very close, if not the same, M value. Monte-Carlo simulations with 500 realizations are implemented for selected sampling locations. No chlorine measurement error is imposed. Base demand errors that are iid with standard deviations which are 10 per cent of the 96

108 Table 5.1: Relationship of M and Parameter Estimation Uncertainty under Chlorine Measurement Error Node Id M µ σ (mg.day/l.ft) 2 (ft/day)

109 Estimate Stadard Deviation M value Figure 5.3: Relationship of M and Parameter Estimation Uncertainty under Chlorine Measurement Error node base demands are added before demand normalization to conserve the total demand. The results are shown in Table 5.2. Again, µ and σ (ft/day) are the sample mean and standard deviation of wall demand coefficient estimate. It is observed that with the exception of location 53 and 152, it is always the case that the smaller the value H, the smaller the standard deviation of the parameter estimate. When there are hydraulic model errors, sampling at locations with relative large H values (68, 65, 85 and 71) leads to unreliable estimate of the wall demand coefficient. This observation justify the restriction of sampling locations with H value in the sampling design procedure. The situation should be avoided is selecting a sampling location is associated with both small M and large H, like node 68. The combination of small M and large H may make a sampling design especially vulnerable to hydraulic model 98

110 errors. Table 5.2: Relationship of H and Parameter Estimate Uncertainty under Hydraulic Model Error Node Id M H µ σ (mg.day/l.ft) 2 mg/l (ft/day) Two Parameters Application In this situation, pipes are assigned to belong to one of the two pipe groups generated by a Markov Random Field model (see Appendix A). The pipes bolded in the figure 4.5 (group 1) are assumed to have true wall demand coefficient 2.5 ft/day, while the other pipes are grouped together (group 2) with wall demand coefficient 1.5 f t/day. The objectives of the design is to select 2 sampling locations and measure chlorine concentrations there from time 301 to 324 hour, with sampling interval of 0.25 hour. Each sampling location has its own H value and therefore two Hs are associated with each design. In this study, the larger of the two is specified as the H value for the design. 99

111 Global D-Optimal Design Enumeration method is used first and the determinant of the matrix M and H is calculated for each combination, or design. A Total of 396 nodes are selected to be candidate sampling locations and ( /2) designs are generated. The detm and H for all these designs are plotted in Figure 5.4. The design with the largest detm, which is 4.89 (mg.day/l.ft) 4, consists of node 208 and 162. The H value of this design is 0.69 mg/l. Implementation of Design Approach From the figure 5.4, it can be seen that by setting the threshold value of H, H, to be greater than or equal to 0.69 will result in the same design as that generated only by D-optimal criteria. However, setting the H to be less than 0.69 generates different designs. If there is no hydraulic model error, node 208 and 162 should certainly be selected for chlorine measurement. When the hydraulic model contains significant uncertainty, the best value of H is not obvious. When the hydraulic uncertainties are higher, H should be set relatively lower. However, since both small value of detm and large hydraulic uncertainties contribute to unreliable wall demand coefficient estimates, setting small H should be avoided if that will result in final design with very small detm. Using four different H s (0.69, 0.60, 0.50 and 0.40), four optimal designs are generated using proposed sampling design approach. The design details and the impact of detm on parameter estimation when there are only chlorine measurement errors are shown in Table 5.3. It is obvious that design with larger detm tends to 100

112 det M of design H value of design Figure 5.4: Relationship between H Value and detm generate more reliable wall demand coefficient estimates. Base demand errors are Table 5.3: Relationship of detm and Parameter Estimate Reliability Design ID H node detm µ 1 µ 2 σ 1 σ 2 mg/l (mg.day/l.ft) 4 (ft/day) O , O , O , O , then introduced to all the water consumption nodes and these errors are iid and have standard deviation that is 10 (test I) or 20 (test II) per cent of the base demand. Monte-Carlo simulation results for wall demand coefficients estimation when both chlorine measurement errors and hydraulic model uncertainties exist are shown in Table 5.4. When the hydraulic model uncertainties are relatively low (test I for 101

113 all the designs), design O1 is still the best design that generates most reliable wall demand coefficient estimates. However, in test II, design O2 and O3 get more reliable wall demand coefficient estimates. Although design O4 has smallest H value, its detm is so small that compared to other designs, it is not robust to either chlorine measurement or hydraulic model errors. Table 5.4: Sampling Design Results and Estimation Reliability Design ID Test detm H µ1 µ 2 σ 1 σ 2 (mg.day/l.ft) 4 mg/l (ft/day) O1 I II O2 I II O3 I II O4 I II The enumeration method is computationally expensive, especially when there are lots of candidate sampling locations. The D-optimal design algorithm discussed in sampling design methodology is used to get the design with maximum detm. Started with different initial designs, 100 trials are run to get the D-optimal design. It is found that this algorithm always find the global optimal design in this particular application. The results shown in Figure 5.5 show the results when this is no limit on H. No matter what the initial design is, this algorithm always stops at the design with maximum detm, which is

114 det M of initial design det M of final design 3 det M Trial index Figure 5.5: D-Optimal Design Algorithm Results with Different Initial Designs Observation The results of this particular sampling design application show that 1)the values of both detm and H that are associated with sampling designs have wide range; 2)it is reasonable to use H to qualitatively compare the impact of potential hydraulic errors on wall demand coefficient estimation reliability under different design; 3)it is hard to select a threshold value for H because of the difficulty to define the statistics about hydraulic model errors and correlate such statistics and H with parameter estimation error; 4)the D-optimal algorithm implemented is computationally efficient and can locate global solution successfully. 103

115 Chapter 6 Pilot Field Study 6.1 Overview A pilot field study was implemented to collect data for calibration of network water quality models that predict chlorine residual. A sodium chloride tracer test was applied as a series of four pulses over a 24 hour period and the normal fluoride feed was shutdown during the same period. The NaCl signals were measured with 9 continuous conductivity meters to provide flow and velocity information. Ten million gallons of treated water, produced during the 24 hour tracer injection period, was followed as it travelled through the distribution system. Free chlorine concentration, conductivity, temperature, and ph were monitored for 24 hours at the treatment plant during application of the tracer pulses, and at each of the nine conductivity stations during passage of the conductivity signal. Three water quality stations were equipped with continuous analyzers for free chlorine. The pilot experiment is the first known distribution system water quality study 104

116 attempt to follow a large volume of finished water through an extensive portion of the distribution system, and to measure in-situ water quality changes occurring in this same water volume. Such data provide unique information about processes that affect water quality in the distribution system, including chlorine consumption due to pipe wall demand. 6.2 Study Area Description A schematic of the study area piping is shown in Figure 6.1 (this area represents roughly one-half the treatment plant service area). Figure 6.1 also shows sampling locations (circles), and the pipe size distribution with thicker lines representing the larger, and thinner lines representing the smaller piping. Distribution system flow paths begin at the single treatment plant source (AA), and continue through two main transmission lines (AF and AG), leading to two residential/commercial districts (areas 1 and 2). Flow paths continue through area 2 to feed area 1, the ultimate destination for the water quality study. Flow path splitting and recombination through areas 1 and 2 contribute to more complex flow paths than those associated with locations AF and AG. The system includes two pumped ground storage tanks, but they are operated infrequently and used as a supplement for fire demand. These two pumped storage tanks were off-line for the entire duration of the study, according to a request by the team to simplify data interpretation and analysis. Subareas 1 and 2 are shown in Figures 6.2 and 6.3, including the sampling locations 105

117 Figure 6.1: Network Pilot Study Area within each area and the pipe size distribution. Sampling locations AH and AI monitor the two main flow paths through subarea 2, which in turn feed subarea 1. Sampling locations AB and AE monitor the two flow paths feeding subarea 1, and AC and AD are interior locations that are affected by more complex sets of flow paths, due to flow splitting and recombination, and temporal variation in water demand. All sampling locations were located in the field at fire hydrants using standard hydrant adaptors, and a continuous flow rate of approximately 1.5 GPM was maintained to reduce residence time in the hydrant barrel. Each sampling location was monitored by a continuous conductivity and temperature analyzer, which is housed in a security box and contains a grab sampling port (see Figure 6.4). Sample locations AC, AD, and AE also were monitored by a continuous chlorine and ph analyzer, 106

118 Figure 6.2: Network Subarea 1 Figure 6.3: Network Subarea 2 107

119 Figure 6.4: Continuous Conductivity and Temperature Analyzer and Grab Sample Tap used to provide greater temporal resolution at these destination output locations. The analyzers used are Emerson MODEL SOLU COMP II CHLORINE/pH ANALYZER and ATI A15 Chlorine monitor. Grab samples were collected at all sampling locations and analyzed in the field for free chlorine (using DPD method with HACH DR/850 colorimeter) and ph. 6.3 Field Test Protocol A saturated NaCl solution was added to the treatment plant finished water beginning at 11:30AM on 9 September The addition was designed to produce a series of 100 mg/l brine pulses after mixing, spaced in time over a 24 hour period. The 100 mg/l pulse was selected to more than double the background conductivity (400 µs/cm), and is a factor of three less than the maximum allowable NaCl increase 108