Dynamic Modeling of Pressure Reducing Valves

Models are currently available for representing the dynamical behavior of most water network components. Such models are relatively simple, accurate and can be easily solved. However, there is no generally accepted dynamic model of a pressure reducing valve (PRV). The key contributions of this paper are the development of several dynamic models—two phenomenological, one behavioral, and one linear—to represent the behavior of PRVs. The models vary in complexity but perform similarly. Experimental data is used to assess the accuracy of the models. The phenomenological models are derived from physical laws and provide an excellent but complex representation of a PRV. The behavioral model is simpler and sufficient for most practical purposes. The linear model does not take the needle valve setting (which controls the valve’s speed) into account and therefore has limited use.     

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.     

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.     

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.     

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.     

Assessing Pressure Changes in an On-Demand Water Distribution System on Drip Irrigation Performance—Case Study in Italy

The hydrant pressure head in an on-demand water distribution system can be subject to high fluctuation depending on the discharge flowing inside the pipes, with consequent impacts on the performance of on-farm irrigation systems. In this work, an Italian water distribution system was analyzed using the AKLA model at upstream discharges of 1,200 and 600?L?s?1 to estimate the range of hydrant pressure variation. A computer model was developed, calibrated, and used to evaluate the performance of a drip irrigation system by relating the on-farm network with the hydrant characteristic curve at a certain operating status. The flow regulator within the hydrant played an important role in stabilizing the performance of the network at hydrant pressures higher than 27 m. At lower hydrant pressures, to apply the same amount of water, irrigation time must be extended by 17 and 95% for pressure heads of 20 and 12 m, respectively. These approaches described have great utility to ensure adequate irrigation management when water is delivered by pressurized on-demand systems.     

Leaks and Water Use Representation in Water Distribution System Models: Finding a Working Equivalence

The challenge of water demand representation in water distribution systems is revisited with a brief exploration of the relationship between a pressure-dependent leak and a fixed legitimate demand. Specifically, the idea that a leak can be modeled as an increment to legitimate demand in such a way that it entails an equivalent impact on both water loss and energy consumption is explored. Conversely, the representation of demands as leaks is briefly considered. The effectiveness of pressure reduction and demand curtailment as leak management schemes are compared for a single pipe system. The influence of pipe resistance on this relationship is assessed, suggesting that such schemes are more important in rougher pipes. In general, the notion that leakage and demand analysis/management are two sides of the same coin, and that pressure/demand management is essentially conservation, is put forth.     

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.     

Tank Simulation for the Optimization of Water Distribution Networks

In this paper an original approach to the simulation of floating-on-the-system tanks as decision variables for water distribution system design optimization is presented, aiming to bridge the gap between traditional engineering practice and mathematical considerations needed for genetic algorithms (GAs). The paper includes a systematic and detailed critical overview of various mathematical approaches in literature, as well as a novel, more “engineering oriented” approach to the simulation of tanks as decision variables for water distribution system design optimization, describing in detail assumptions and impacts to the evaluation of potential solutions. Tank simulation is based on two decision variables: capacity and minimum normal operational level, omitting risers. Shape and ratio between emergency/total capacities are taken into consideration as design parameters. Assessment of tank performance is carried out by four criteria for the normal daily operational cycle, differentiating between operational and filling capacity, as well as two further criteria for emergency flows. The original design and operational mathematical assumptions are implemented in a fuzzy multiobjective GA model, which is applied to the well-known example from literature “Anytown” water distribution network to benchmark the results.     

Modeling the Behavior of Flow Regulating Devices in Water Distribution Systems Using Constrained Nonlinear Programming

Currently the modeling of check valves and flow control valves in water distribution systems is based on heuristics intermixed with solving the set of nonlinear equations governing flow in the network. At the beginning of a simulation, the operating status of these valves is not known and must be assumed. The system is then solved. The status of the check valves and flow control valves are then changed to try to determine their correct operating status, at times leading to incorrect solutions even for simple systems. This paper proposes an entirely different approach. Content and co-content theory is used to define conditions that guarantee the existence and uniqueness of the solution. The work here focuses solely on flow control devices with a defined head discharge versus head loss relationship. A new modeling approach for water distribution systems based on subdifferential analysis that deals with the nondifferentiable flow versus head relationships is proposed in this paper. The water distribution equations are solved as a constrained nonlinear programming problem based on the content model where the Lagrangian multipliers have important physical meanings. This new method gives correct solutions by dealing appropriately with inequality and equality constraints imposed by the presence of the flow regulating devices (check valves, flow control valves, and temporarily closed isolating valves). An example network is used to illustrate the concepts.     

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.     

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.     

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.     

Analysis of Clément’s First Formula for Irrigation Distribution Networks

The aim of this article is to determine with real data to what extent the hypotheses on which Clément’s first formula is based are fulfilled, and to compare the results of applying this formula. To this end the flow demand in the peak period was studied in two distribution networks with different irrigation methods and crops located in the Ebro River basin (northeast Spain). The calibration procedure for this formula proposed by the Centre Technique du Génie Rural des Eaux et des Forets (CTGREF) in 1977 was also analyzed. The result was that most of the hypotheses were not fulfilled. Furthermore, the discharge distributions obtained in the period of study did not correspond to a normal distribution. However, comparing the real accumulated probability curve and that calculated by Clément’s formula, it was found that the differences between the two curves for probabilities greater than 90% (a wide range of application of the formula) were lower than 9.4%. The reason for this result was found. It was shown also, that the CTGREF adjustment procedure did not provide substantial improvement in the estimation of flows because the aim of the fit was to achieve a normal distribution rather than an accumulated distribution function.     

Two-Point Linearization Method for the Analysis of Pipe Networks

This note proposes a new method for snapshot analysis of water distribution systems based on the commonly used gradient method. The proposed method uses a secant (intersecting the head-loss function in two points) instead of a tangent to approximate the pipe head-loss function. A theoretical model is developed for the flow range in which the secant approximates the head-loss function without exceeding a given allowable error. This scheme allows a tradeoff to be made between the allowable error and the number of iterations required to achieve convergence. The proposed method is applied to an example network to illustrate its application and benefits. It is argued that the number of iterations required to find a solution can be reduced significantly in both snapshot and extended-period simulations.     

Irrigation Distribution Networks’ Vulnerability to Climate Change

Climate change will lead to changed demands on existing irrigation systems. This paper presents a methodology for investigating the performance of irrigation networks under climate change, and applies this to an irrigation network in Cordoba, southern Spain. The methodology uses emission scenarios (A2 and B2) developed by the Intergovernmental Panel on Climate Change. A global climate model (HadCM3) is used with downscaling to predict climate variables for 2050 and 2080 under the emission scenarios. European agricultural policy scenarios are used to predict future cropping patterns. Irrigation water requirements are then estimated for various combinations of these climate and cropping pattern scenarios, and the performance of the irrigation network is evaluated in terms of the equity and adequacy of pressure at the outlets, using EPANET. The methodology was applied to the Fuente Palmera irrigation district, which supplies water on-demand for drip irrigation. The results show that climate change would have a major impact on network performance with the existing cropping pattern, but that expected changes in cropping pattern would reduce this impact.     

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.     

Eulerian–Lagrangian Method for Constituent Transport in Water Distribution Networks

The transport and mixing of contaminants in conduits is governed by advection, dispersion, and decay. Several models are available to trace the transport of such constituents and most assume that the principal mechanisms for transport are advection and reaction only. However in pipes where low velocities prevail, longitudinal dispersion is significant and models that neglect the dispersion effects fail to properly simulate the observed concentrations in low velocity pipes. This work presents a method for simulating the advection-dispersion-reaction process of constituent transport in water networks. A Eulerian–Lagrangian method is employed whereby the dispersion term in the governing equation is approximated using finite differences and the resulting first-order partial differential equation is then integrated using the method of characteristics. Analytical solutions of the transport equation are also derived to quantify the effect of neglecting dispersion at pipe junctions and to assess the accuracy of the proposed method. The Eulerian-Lagrangian method is tested on benchmark networks and on the field study at the Cherry Hill/Brushy Plains network. Results show that the model developed is capable of simulating transport with equal accuracy for low and high velocity flows with and without significant dispersion effects. It also performs better than other models because of the nonuniform grid distribution and the interpolation schemes used.     

Demand Forecasting for Irrigation Water Distribution Systems

One of the main problems in the management of large water supply and distribution systems is the forecasting of daily demand in order to schedule pumping effort and minimize costs. This paper examines methodologies for consumer demand modeling and prediction in a real-time environment for an on-demand irrigation water distribution system. Approaches based on linear multiple regression, univariate time series models (exponential smoothing and ARIMA models), and computational neural networks (CNNs) are developed to predict the total daily volume demand. A set of templates is then applied to the daily demand to produce the diurnal demand profile. The models are established using actual data from an irrigation water distribution system in southern Spain. The input variables used in various CNN and multiple regression models are (1) water demands from previous days; (2) climatic data from previous days (maximum temperature, minimum temperature, average temperature, precipitation, relative humidity, wind speed, and sunshine duration); (3) crop data (surfaces and crop coefficients); and (4) water demands and climatic and crop data. In CNN models, the training method used is a standard back-propagation variation known as extended-delta-bar-delta. Different neural architectures are compared whose learning is carried out by controlling several threshold determination coefficients. The nonlinear CNN model approach is shown to provide a better prediction of daily water demand than linear multiple regression and univariate time series analysis. The best results were obtained when water demand and maximum temperature variables from the two previous days were used as input data.     

Intrusion within a Simulated Water Distribution System due to Hydraulic Transients. II: Volumetric Method and Comparison of Results

A pilot-scale test rig was used to simulate intrusion behavior associated with hydraulic transient initiated by rapid valve closure in a water distribution system. In Part I, the test rig apparatus and operating conditions were described and intrusion volumes were reported based on a chemical tracer and mass balance calculations. In this paper, the experimental study is extended to determine intrusion volumes by a volumetric method that used video recordings of water fluctuations in the observation column. The results obtained using the volumetric and chemical tracer methods were compared to theoretical calculations. Intrusion volumes associated with a 12.7-mm (1/2-in.) diam orifice were evaluated in addition to 3.2 (1/8-in.) and 6.4-mm (1/4-in.) orifices. The impact of the external head on the intrusion volume was also assessed by comparing results using 0.91 (3 ft) versus 1.37 m (4.5 ft) of external head. The average intrusion volumes obtained using the volumetric approach ranged from 47.3 to 550.2 mL. These volumes were 64–298% greater than intrusion volumes determined by the chemical tracer method reported in Part I. However, the theoretical calculations indicate that the volumetric approach could underestimate intrusion volumes by as much as 50%.