Quality Modeling of Water Distribution Systems Using Sensitivity Equations

In this paper, unsteady water quality modeling and the associated sensitivity equations are solved for water distribution systems. A new solution algorithm is proposed, designed for slow varying velocity and based on a time splitting method to separate and solve efficiently each phenomenon such as advection and chemical reaction. This numerical approach allows simultaneous solution of both the direct problem and the sensitivity equations. Special attention is given to the treatment of advection, which is handled with a total variation diminishing scheme. The general model presented in this study permits global sensitivity analysis of the system to be performed and its efficiency is illustrated on two pipe networks. The importance of the sensitivity analysis is shown as part of the calibration process on a real network.     

Adjusting Nodal Demands in SCADA Constrained Real-Time Water Distribution Network Models

This paper describes the proportional demand method and the target demand method, two techniques for adjusting estimated demands in hydraulic models of water distribution networks to produce solutions that are consistent with available supervisory control and data acquisition (SCADA) data. The two techniques assume that pipe resistances and SCADA data are accurate and that the combination of SCADA data and demand estimates produce overdetermined problems. Nodal demands are regarded as stochastic variables which fluctuate about an estimated mean value. The method of weighted least squares is used to obtain solutions that satisfy all of the constraints imposed by SCADA data with adjusted nodal demands that most closely resemble the estimates. The methods are intended for use in real-time modeling but are limited to quasi-steady state flow. The paper demonstrates the methods on two example problems.     

Optimization of Water Distribution Networks Using Integer Linear Programming

In this study optimum design of municipal water distribution networks for a single loading condition is determined by the branch and bound integer linear programming technique. The hydraulic and optimization analyses are linked through an iterative procedure. This procedure enables us to design a water distribution system that satisfies all required constraints with a minimum total cost. The constraints include pipe sizes, which are limited to the commercially available sizes, reservoir levels, pipe flow velocities, and nodal pressures. Accuracy of the developed model has been assessed using a network with limited solution alternatives, the optimal solution of which can be determined without employing optimization techniques. The proposed model has also been applied to a network solved by others. Comparison of the results indicates that the accuracy and convergence of the proposed method is quite satisfactory.     

Real-Time Connectivity Modeling of Water Distribution Networks to Predict Contamination Spread

This paper presents a new approach to analyzing water distribution networks during a contamination event. Previous computer models for predicting the extent of contamination spread in water distribution networks are demand-driven models. The new approach makes use of supervisory control and data acquisition (SCADA) data to create connectivity matrices, which encapsulate the worst-case projection of the potential spread of contamination obtained by combining the effects of all possible scenarios. Two methods for creating connectivity matrices are described, the first based on operating modes, and the second on fundamental paths. Both methods produce identical results, although the method of fundamental paths is more efficient computationally. The connectivity- and hydraulic-based approaches are compared using an example problem.     

Water Distribution Network Analysis Using Excel

The analysis of water distribution networks has been and will continue to be a core component of civil engineering water resources curricula. Since its introduction in 1936, the Hardy Cross method has been used in virtually every water resources engineering text to introduce students to network analysis. The technique gained widespread popularity primarily because it is amenable to manual calculation techniques. However, the same subtle elegance that facilitates manual calculations often obscures the primary engineering and physical principles of water distribution systems relative to the nuances of algorithm implementation. Herein, the authors illustrate the application of commonly available spreadsheet software (MicroSoft Excel) to more concisely and effectively solve typical undergraduate network distribution problems using linear theory. Application development is much more efficient and straightforward than the corresponding Hardy Cross implementation enabling students to concentrate upon the engineering system and relevant design issues. The technique presented utilizes commonly available technology and is presented as a supplement to alternatives discussed in recent literature.     

Modeling Discoloration in Potable Water Distribution Systems

Discoloration of potable water supplied to customer taps is one of the biggest causes of water quality related customer complaints. At present, understanding of the fundamental processes that cause discoloration is limited and the modeling of events unfeasible. This paper describes the development, verification, and validation of a novel cohesive transport modeling approach to simulate discoloration within distribution systems. The model is based on the principal that strength characteristics of fine particulate material accumulated at the pipe walls are conditioned by the shear stress of the usual daily hydraulics. Discoloration occurs when the flow through the systems changes, exceeding the peak daily value. Fieldwork results from two sites are presented in detail: Site 1 demonstrates model application including sensitivity and parameter dependence, while data from Site 2 is used to investigate the hypothesis that daily hydraulic forces condition the material layers within the pipes. Model simulations are also presented for a selection of other field sites to demonstrate the wider applicability of the model.     

Nitrification Modeling in Pilot-Scale Chloraminated Drinking Water Distribution Systems

A suspended growth nitrification model was developed to describe nitrification dynamics in terms of chloramine, ammonia, nitrite, nitrate, and nitrifying bacteria concentrations in pilot-scale chloraminated drinking water systems. The model provided a semimechanistic base to study the regrowth and persistence of nitrifiers in chloraminated distribution systems. Results showed that the developed suspended growth model, without a biofilm nitrification component, was able to simulate and predict nitrification episodes in the pilot-scale systems. In the restricted low nutrient drinking water environment, growth kinetic parameters for nitrifiers were estimated to be significantly lower than ranges reported in the literature. The maximum specific growth rate and ammonia half-saturation constant for ammonia oxidizing bacteria were estimated to be 0.46?day?1and 0.023?mg NH3–N/L, respectively. In addition, an estimated reaction rate of 70±32?L/(mg?HPC?day) between chloramines and soluble microbial products suggests that heterotrophic growth can be a significant contributor to chloramine decay in some chloraminated distribution systems.     

Nonlinear Control of Open-Channel Water Flow Based on Collocation Control Model

This paper is devoted to the nonlinear control of open-channel water flow dynamics via a one-dimensional collocation control model for irrigation canals or dam-river systems. Open channel dynamics are based on the well-known Saint-Venant nonlinear partial differential equations. In order to obtain a finite-dimensional model an orthogonal collocation method is used, together with functional approximation of the solutions of Saint-Venant equations based on Lagrange polynomials. This method can give a more tractable model than those obtained from classical finite-difference or finite-element methods (from the viewpoint of both state dimension and structure), and is well suited for control purposes. In particular it is shown how such a model can be used to design a nonlinear controller by techniques of dynamic input–output linearization with the goal of controlling water levels along an open-channel reach. Controller performance and robustness are illustrated in simulations, with a simulated model for the canal chosen as more accurate than the one used for control design.     

Improved Control of Pressure Reducing Valves in Water Distribution Networks

The behavior of transients in water pipe networks is well understood but the influence of modulating control valves on this behavior is less well known. Experimental work on networks supplied through pressure reducing valves (PRVs) has demonstrated that, in certain conditions, undesirable phenomena such as sustained or slowly decaying oscillation and large pressure overshoot can occur. This paper presents results from modeling studies to investigate interaction between PRVs and water network transients. Transient pipe network models incorporating random demand are combined with a behavioral PRV model to demonstrate how the response of the system to changes in demand can produce large or persistent pressure variations, similar to those seen in practical experiments. A proportional-integral-derivative (PID) control mechanism, to replace the existing PRV hydraulic controller, is proposed and this alternative controller is shown to significantly improve the network response. PID controllers are commonly used in industrial settings and the methods described are easy to implement in practice.     

Adjoint Sensitivity Analysis of Contaminant Concentrations in Water Distribution Systems

Sensitivity analysis is used to determine how a system state or a model output changes due to a change in the value of a system parameter or a model input. We present the adjoint approach for determining the sensitivity of the concentration of a contaminant in a water distribution system to a change in a system parameter such as the location of the source of contamination, the reaction rate of the contaminant, and others. With the adjoint method, the sensitivity of the model output to any number of parameters can be obtained with one simulation of the adjoint model. If the number of parameters of interest exceeds the number of model outputs for which the sensitivity is desired, the adjoint method is more efficient than traditional direct methods of calculating sensitivities. We develop the adjoint equations for water quality in a water distribution system, verify the adjoint-based sensitivity equation using an analytical example, and demonstrate the numerical calculation of adjoint sensitivities using EPANET.     

Lagrangian Modeling of Weakly Nonlinear Nonhydrostatic Shallow Water Waves in Open Channels

A Lagrangian, nonhydrostatic, Boussinesq model for weakly nonlinear and weakly dispersive flow is presented. The model is an extension of the hydrostatic model—dynamic river model. The model uses a second-order, staggered grid, predictor-corrector scheme with a fractional step method for the computation of the nonhydrostatic pressure. Numerical results for solitary waves and undular bores are compared with Korteweg-de Vries analytical solutions and published numerical, laboratory, and theoretical results. The model reproduced well known features of solitary waves, such as wave speed, wave height, balance between nonlinear steepening and wave dispersion, nonlinear interactions, and phase shifting when waves interact. It is shown that the Lagrangian moving grid is dynamically adaptive in that it ensures a compression of the grid size under the wave to provide higher resolution in this region. Also the model successfully reproduced a train of undular waves (short waves) from a long wave such that the predicted amplitude of the leading wave in the train agreed well with published numerical and experimental results. For prismatic channels, the method has no numerical diffusion and it is demonstrated that a simple second-order scheme suffices to provide an efficient and economical solution for predicting nonhydrostatic shallow water flows.     

Dealing with Zero Flows in Solving the Nonlinear Equations for Water Distribution Systems

Three issues concerning the iterative solution of the nonlinear equations governing the flows and heads in a water distribution system network are considered. Zero flows cause a computation failure (division by zero) when the Global Gradient Algorithm of Todini and Pilati is used to solve for the steady state of a system in which the head loss is modeled by the Hazen-Williams formula. The proposed regularization technique overcomes this failure as a solution to this first issue. The second issue relates to zero flows in the Darcy-Weisbach formulation. This work explains for the first time why zero flows do not lead to a division by zero where the head loss is modeled by the Darcy-Weisbach formula. In this paper, the authors show how to handle the computation appropriately in the case of laminar flow (the only instance in which zero flows may occur). However, as is shown, a significant loss of accuracy can result if the Jacobian matrix, necessary for the solution process, becomes poorly conditioned, and so it is recommended that the regularization technique be used for the Darcy-Weisbach case also. Only a modest extra computational cost is incurred when the technique is applied. The third issue relates to a new convergence stopping criterion for the iterative process based on the infinity-norm of the vector of nodal head differences between one iteration and the next. This test is recommended because it has a more natural physical interpretation than the relative discharge stopping criterion that is currently used in standard software packages such as EPANET. In addition, it is recommended to check the infinity norms of the residuals once iteration has been stopped. The residuals test ensures that inaccurate solutions are not accepted.     

Explicit Integration Method for Extended-Period Simulation of Water Distribution Systems

Extended-period simulation of incompressible and inertialess flow in water distribution systems is normally done using numerical integration techniques, although regression methods are also sometimes employed. A new method for extended-period simulation, called the explicit integration (EI) method, is proposed. The method is based on the premise that a complex water distribution system can be represented by a number of simple base systems. The simple base systems are selected in such a way that their dynamic equations can be solved through explicit integration. In this paper a simple base system consisting of a fixed-head reservoir feeding a tank through a single pipeline is analyzed. It is then illustrated how a complex water distribution system can be decoupled into simple base systems and its dynamic behavior simulated using a stepwise procedure. The EI method is compared to the commonly used Euler numerical integration method using two example networks. It is shown that the accuracy of the EI method is considerably better than that of the Euler method for the same computational effort.     

Pseudotransient Continuation Method in Extended Period Simulation of Water Distribution Systems

This paper presents and discusses a new static solver that implements the pseudotransient continuation method for the quasi-steady state analysis, or extended-period simulation of water distribution systems. The implementation is based on the concept of virtual tanks and has a clear physical meaning. The steady state solver described in this paper can analyze a pipe network under pressure deficient conditions and is free from some convergence problems that occur in the Newton-Raphson method-based solvers when analyzing a pipe network with control devices. The numerical examples considered in the paper demonstrate the convergence of the proposed method in cases where existing static solvers (e.g., that of the EPANET 2 hydraulic simulator) fail.     

Modeling and Control of Vinyl Chloride in Drinking Water Distribution Systems

In water distribution systems containing PVC pipe manufactured in the “early era” (prior to 1977), vinyl chloride can leach into drinking water resulting in vinyl chloride concentrations exceeding the 2 μg?L?1 maximum contaminant level. Field testing of dead-end segments of water distribution systems consisting of early-era PVC pipe was conducted to examine their initial intrapipe vinyl chloride monomer (VCM) concentrations based on a Fickian-diffusion-based leaching model. The experiments showed a wide range of VCM concentrations within early-era PVC pipe ranging from less than 50 to more than 600 mg?kg?1. Based on the diffusion modeling approach, a protocol was designed that provides a means for utility managers to calibrate the model for specific dead-end lines. The paper delineates procedures to determine which dead ends require flushing to control vinyl chloride, examines the effects of system parameters such as temperature on vinyl chloride leaching, and provides a method to devise flush schedules and volumes. Through a properly designed, tested, and maintained flush protocol such as that developed in this research, public water systems with dead-end lines consisting of early-era PVC pipe can control vinyl chloride concentrations using either manual or automatic flush valves.     

Validation and Extension of Image Velocimetry Capabilities for Flow Diagnostics in Hydraulic Modeling

In the last 3 decades, quantitative image velocimetry has considerably grown in popularity in the fluid mechanics research community. More recently, image-based techniques have been extended to hydraulic applications for mapping and quantifying free-surface velocities spanning large areas in free-surface laboratory and natural-scale flows. During the adaptation process it has been proven the great potential that image velocimetry holds for qualitative and quantitative observations hydraulic applications. Current efforts are directed toward evaluating the technique performance, perfecting implementation aspects, and expanding its flow diagnostic capabilities. Presented here are new laboratory measurements that estimate the accuracy of image velocimetry by comparison with alternative instruments, recent developments targeting enhancement of several technique components, and proof-of-concept experiments that demonstrate new measurement and operational capabilities. The ultimate goals of the new experimental evidence are to demonstrate that image velocimetry possesses full-grown capabilities for laboratory hydraulic investigations and has the potential to be successfully implemented in important river and coastal engineering applications.     

Self-Adaptive Fitness Formulation for Evolutionary Constrained Optimization of Water Systems

In design of water distribution networks, there are several constraints that need to be satisfied; supplying water at an adequate pressure being the main one. In this paper, a self-adaptive fitness formulation is presented for solving constrained optimization of water distribution networks. The method has been formulated to ensure that slightly infeasible solutions with a low objective function value remain fit. This is seen as a benefit in solving highly constrained problems that have solutions on one or more of the constraint bounds. In contrast, solutions well outside the constraint bounds are seen as containing little genetic information that is of use and are therefore penalized. In this method, the dimensionality of the problem is reduced by representing the constraint violations by a single infeasibility measure. The infeasibility measure is used to form a two-stage penalty that is applied to infeasible solutions. The performance of the method has been examined by its application to two water distribution networks from literature. The results have been compared with previously published results. It is shown that the method is able to find optimum solutions with less computational effort. The proposed method is easy to implement, requires no parameter tuning, and can be used as a fitness evaluator with any evolutionary algorithm. The approach is also robust in its handling of both linear and nonlinear equality and inequality constraint functions. Furthermore, the method does not require an initial feasible solution, this being an advantage in real-world applications having many optimization variables.     

Jacobian Matrix for Solving Water Distribution System Equations with the Darcy-Weisbach Head-Loss Model

The widely used Todini and Pilati method for solving the equations that model water distribution systems was originally developed for pipes in which the head loss is modeled by the Hazen-Williams formula. The friction factors in this formula are independent of flow. Rossman’s popular program EPANET implements elements of the Todini and Pilati algorithm, but when the Darcy-Weisbach head-loss formula is used, it does not take into account the dependence of the friction factors on the Reynolds number, and therefore flow, in computing the Jacobian. We present the correct Jacobian matrix formulas, which must be used in order to fully account for the friction factor’s dependence on flow when the Todini and Pilati method is applied with the Darcy-Weisbach head-loss formula. With the correct Jacobian matrix the Todini and Pilati implementation of Newton’s method has its normally quadratic convergence restored. The new formulas are demonstrated with an illustrative example.     

Modeling Residual Chlorine Response to a Microbial Contamination Event in Drinking Water Distribution Systems

Changes in chlorine residual concentrations in water distribution systems could be used as an indicator of microbial contamination. Consideration is given on how to model the behavior of chlorine within the distribution system following a microbial contamination event. Existing multispecies models require knowledge of specific reaction kinetics that are unlikely to be known. A method to parameterize a rate expression describing microbially induced chlorine decay over a wide range of conditions based on a limited number of batch experiments is described. This method is integrated into EPANET-MSX using the programmer’s toolkit. The model was used to simulate a series of microbial contamination events in a small community distribution system. Results of these simulations showed that changes in chlorine induced by microbial contaminants can be observed throughout a network at nodes downstream from and distant to the contaminated node. Some factors that promote or inhibit the transport of these chlorine demand signals are species-specific reaction kinetics, the chlorine concentration at the time and location of contamination, and the system’s unique demand patterns and architecture.