Dynamic Memory Computation of Impedance Matrix Method

The computational efficiency of the impedance matrix method has been greatly improved for large pipe networks with various dimensions and complexity. Several numerical methods for solving linear system were modified to deal with the complex domain operation and used into impedance evaluation. Two different memory reduction schemes were developed based on one-dimensional storage and implemented with the biconjugate gradient method and the Gaussian elimination scheme, respectively. A new implementation of the impedance matrix method, namely, the dynamic memory allocation scheme, was introduced to efficiently model hydraulic transients in pipeline systems that have large topological structures. Three hypothetical pipe networks, the multiseries system, the multilooped system, and the multiblock system, were used to test the performance of the developed schemes. The impact of randomizing pipeline parameters, i.e., friction factor, length, and wave speed, on computation efficiency was evaluated and compared. The dynamic memory allocation scheme not only reduces costs substantially in CPU execution time and memory space compared to other schemes but also shows significant potential as a real-time unsteady flow predictor for large pipe networks.     

Simultaneous Zonation and Calibration of Pipe Network Parameters

A procedure for simultaneously zoning and calibrating pipe network parameters is proposed and applied to the determination of pipe resistance coefficients. The methodology is aimed at grouping the parameters of all the pipes into a small number of zone parameters, constrained to keep the difference between the computed and the measured water heads below a given tolerance. It is shown that, in the case of nonlooped networks, the methodology leads to a linear minimization problem where the objective function is a measure of the heterogeneity of the estimated parameters. In the case of looped networks an iterative procedure, where the linear problem is coupled with a nonlinear problem having a restricted number of decision variables, is proposed and demonstrated. Application of the procedure to a hypothetical example is shown. In a lab experiment, the model of a pipe network made by two different types of links has been calibrated using two measurement points and three different measured data sets, each of which was obtained by adjusting the valves located in the network to modify the pressure field. Comparison of the measured and the estimated resistance coefficients shows good correlation.     

Calibration of On-Demand Irrigation Network Models

In this study, a new procedure for calibrating on-demand irrigation network models was developed. This procedure used a new objective function called maximum data with a reasonable error (MDRE) for calibrating the network. It was compared with the two more commonly used objective functions in calibration procedures that are the simple least squares (SLS) and the maximum likelihood estimator for the heteroscedastic error case (HMLE). In order to carry out the calibration, a quasi-Newton optimization method was used having as variable the Hazen-Williams head losses coefficient (C). This procedure was applied to an on-demand irrigation network located in Tarazona de La Mancha (Albacete, Spain) where flow and pressure at hydrant level was measured. The calibration procedure using the MDRE objective function was applied considering all the pressure control points simultaneously and the obtained results were compared with the results of considering the pressure control points independently. Therefore, the effect of the location of the pressure control point was studied. Results showed that, when the proposed objective function was used, the root mean squared error (RMSE) comparing the measured and simulated data after calibration was lower than when the SLS or HMLE objective functions were used. The location of the pressure control points throughout the irrigation network could affect the results; therefore, it was more accurate to use all the control points simultaneously than independently in the calibration process.     

Steady-State Water Quality Analysis for Pipe Network Systems

In 1993 P. F. Boulos and T. Altman developed an efficient explicit scheme for determining steady state water quality in a distribution system for conservative and zero-order reacting constituents. This approach is extended here to first- and second-order reactions and a general reaction relationship is discussed. Mass balance relationships and nonconservative reaction kinetics lead to a general matrix for constituent analysis. The directed graph that results in steady flow conditions permits single equations to be solved sequentially providing the water quality distribution throughout the system. The method can be used to solve for linear and nonlinear conditions and is demonstrated for first-order decay and growth and second-order decay on a 13-pipe system.     

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.     

Optimal Sampling Design Methodologies for Water Distribution Model Calibration

Sampling design (SD) for water distribution systems (WDS) is an important issue, previously addressed by various researchers and practitioners. Generally, SD has one of several purposes. The aim of the methodologies developed and presented here is to find the optimal set of network locations for pressure loggers, which will be used to collect data for the calibration of a WDS model. First, existing SD approaches for WDS are reviewed. Then SD is formulated as a multiobjective optimization problem. Two SD models are developed to solve this problem, both using genetic algorithms (GA) as search engines. The first model is based on a single-objective GA (SOGA) approach in which two objectives are combined into one using appropriate weights. The second model uses a multiobjective GA (MOGA) approach based on Pareto ranking. Both SD models are applied to two case studies (literature and real-life problems). The results show several advantages and one disadvantage of the MOGA model when compared to SOGA. A comparison of the MOGA SD model solution to the results of several published SD models shows that the Pareto optimal front obtained using MOGA acts as an envelope to the Pareto fronts obtained using previously published SD models.     

Improved Pressurized Pipe Network Hydraulic Solver for Applications in Irrigation Systems

GESTAR is an advanced computational hydraulic software tool specially adapted for the design, planning, and management of pressurized irrigation networks. A summary is given of the most significant characteristics of GESTAR. The hydraulic solver for quasi-steady scenarios uses specific strategies and incorporates several new features that improve the algorithms for pipe network computation, overcoming some of the problems that arise when attempting to apply drinking water software, using the gradient method, to irrigation systems. It is shown that the gradient method is a nodal method variant, where flow rates are relaxed using head loss formula exponents. Although relaxation produces a damping effect on instabilities, it is still unable to solve some of the numerical problems common to the nodal methods. In this contribution the results of the research on computational strategies capable of dealing with low resistance elements, hydrant modelling, multiple regulation valves, numerous emitters, and pumps with complex curves are presented, obtaining accurate results even in conditions where other software fails to converge. GESTAR incorporates all these computational techniques, achieving a high convergence rate and robustness. Furthermore, GESTAR’s solver algorithm was easily adapted to incorporate inverse analysis options for optimum network control and parameter calibration. Illustrative examples are provided, documenting the improved numerical techniques and examples of GESTAR’s performance in comparison with EPANET2, a widely used gradient method-based hydraulic solver.     

Two Methods for the Computation of Commercial Pipe Friction Factors

Two methods are proposed for the computation of friction factors of commercial pipes. The first method applies the mean value of the zero velocity point (MZVP) to a theoretical friction factor equation, and the other directly computes the mean friction factor (MFF) by averaging the friction factor of both the smooth and rough walls while considering their relative contribution. The MFF method is preferred, because it is simple but covers all the flow characteristics of commercial pipes. Both MFF and MZVP methods consider two parts of a wall with different roughness heights: One part is rough and the other is smooth. A regression analysis was performed to determine optimum values of the roughness height and probability of encountering each part, using several sets of field data, including galvanized iron, wrought iron, cast iron, concrete, riveted steel, and concrete. The analysis showed that both the roughness height and the relative contribution of the rough part are strongly dependent on the pipe diameter. The MFF method gave an average error of less than 3%, whereas the traditional Colebrook–White equation gave an average error of more than 11% when compared with Colebrook’s data.     

Pipeline Network Features and Leak Detection by Cross-Correlation Analysis of Reflected Waves

This paper describes progress on a new technique to detect pipeline features and leaks using signal processing of a pressure wave measurement. Previous work (by the present authors) has shown that the analysis of pressure wave reflections in fluid pipe networks can be used to identify specific pipeline features such as open ends, closed ends, valves, junctions, and certain types of bends. It was demonstrated that by using an extension of cross-correlation analysis, the identification of features can be achieved using fewer sensors than are traditionally employed. The key to the effectiveness of the technique lies in the artificial generation of pressure waves using a solenoid valve, rather than relying upon natural sources of fluid excitation. This paper uses an enhanced signal processing technique to improve the detection of leaks. It is shown experimentally that features and leaks can be detected around a sharp bend and up to seven reflections from features/leaks can be detected, by which time the wave has traveled over 95?m. The testing determined the position of a leak to within an accuracy of 5%, even when the location of the reflection from a leak is itself dispersed over a certain distance and, therefore, does not cause an exact reflection of the wave.     

Time-Line Interpolation Errors in Pipe Networks

An exact method of assessing numerical errors in analyses of unsteady flows in pipe networks is introduced. The assessment is valid for fixed-grid method of characteristics analyses using time-line interpolations. A pipe polynomial transfer matrix is developed and is analogous to transfer function matrices used in free oscillation theory. The influence of reachback is assessed by comparing exact numerical predictions using a polynomial transfer matrix with exact analytical predictions obtained using free oscillation theory. The investigation is part of a long-term project aimed at automating the selection of numerical grid sizes in unsteady flow analyses. The eventual goal is to enable users of unsteady flow software to prescribe required degrees of accuracy instead of specifying the numerical grid itself. This paper is only a first step toward the long-term aim, but it is a big step toward an intermediate objective of providing exact benchmarking data for the assessment of approximate methods of automatic grid selection.     

Transient Modeling of Arbitrary Pipe Networks by a Laplace-Domain Admittance Matrix

An alternative to the modeling of the transient behavior of pipeline systems in the time-domain is to model these systems in the frequency-domain using Laplace transform techniques. Despite the ability of current methods to deal with many different hydraulic element types, a limitation with almost all frequency-domain methods for pipeline networks is that they are only able to deal with systems of a certain class of configuration, namely, networks not containing second-order loops. This paper addresses this limitation by utilizing graph theoretic concepts to derive a Laplace-domain network admittance matrix relating the nodal variables of pressure and demand for a network comprised of pipes, junctions, and reservoirs. The adopted framework allows complete flexibility with regard to the topological structure of a network and, as such, it provides an extremely useful general basis for modeling the frequency-domain behavior of pipe networks. Numerical examples are given for a 7- and 51-pipe network, demonstrating the utility of the method.     

Solution for the Closed-Loop Problem in Pressurized Multipipe Systems

The fundamental background of the solution for the steady-state flow in pressurized water closed-loop pipe systems, without any reservoirs in-between, is presented in this paper. The use of the steady-state mass balance and energy equations to calculate discharges and heads in this type of hydraulic system leads to an undetermined problem. The way to solve this indeterminacy is to consider an additional continuity equation associated with the difference between initial and final conditions, taking into account fluid compressibility and pipe-wall deformability. A complete formulation is derived considering pressure and temperature changes in the hydraulic system. Simplified formulae are presented for isothermal flows in simple systems and multiple closed-loops with pipes in series and in parallel. This problem can also be solved by a pseudotransient analysis technique applied to steady-state conditions. Proposed solutions for this problem are applied to steady-state flows and tested for different system configurations.     

Pipe Index Vector: A Method to Improve Genetic-Algorithm-Based Pipe Optimization

This paper proposes a methodology for the optimal design of water distribution systems based on genetic algorithms. The objective of the optimization is to minimize the capital cost, subject to ensuring adequate pressures at all nodes during peak demands. The proposed method is novel in that it involves the use of a pipe index vector to control the genetic algorithm search. The pipe index vector is a measure of the relative importance of pipes in a network in terms of their impact on the hydraulic performance of the network. By using the pipe index vector it is possible to exclude regions of the search space where impractical and infeasible solutions exist. By reducing the search space it is possible to generate feasible solutions more quickly and hence process much healthier populations than would be the case in a standard genetic algorithm. This results in optimal solutions being found in a fewer number of generations resulting in a substantial saving in terms of computational time. The method has been tested on several networks, including networks used for benchmark testing least cost design algorithms, and has been shown to be efficient and robust.     

Frequency-Domain Modeling of Transients in Pipe Networks with Compound Nodes Using a Laplace-Domain Admittance Matrix

An alternative to modeling the transient behavior of pipeline systems in the time domain is to model these systems in the frequency domain using Laplace transform techniques. A limitation with traditional frequency-domain pipeline models is that they are only able to deal with systems of a limited class of configuration. Despite the development of a number of recent Laplace-domain network models for arbitrarily configured systems, the current formulations are designed for systems comprised only of pipes and simple node types such as reservoirs and junctions. This paper presents a significant generalization of existing network models by proposing a framework that allows not only complete flexibility with regard to the topological structure of a network, but also, encompasses nodes with dynamic components of a more general class (such as air vessels, valves, and capacitance elements). This generalization is achieved through a novel decomposition of the nodal dynamics for inclusion into a Laplace-domain network admittance matrix. A symbolic example is given demonstrating the development of the network admittance matrix and numerical examples are given comparing the proposed method to the method of characteristics for 11-pipe and 51-pipe networks.     

Godunov-Type Solutions for Water Hammer Flows

First- and second-order explicit finite volume (FV) Godunov-type schemes for water hammer problems are formulated, applied, and analyzed. The FV formulation ensures that both schemes conserve mass and momentum and produce physically realizable shock fronts. The exact solution of the Riemann problem provides the fluxes at the cell interfaces. It is through the exact Riemann solution that the physics of water hammer waves is incorporated into the proposed schemes. The implementation of boundary conditions, such as valves, pipe junctions, and reservoirs, within the Godunov approach is similar to that of the method of characteristics (MOC) approach. The schemes are applied to a system consisting of a reservoir, a pipe, and a valve and to a system consisting of a reservoir, two pipes in series, and a valve. The computations are carried out for various Courant numbers and the energy norm is used to evaluate the numerical accuracy of the schemes. Numerical tests and theoretical analysis show that the first-order Godunov scheme is identical to the MOC scheme with space-line interpolation. It is also found that, for a given level of accuracy and using the same computer, the second-order scheme requires much less memory storage and execution time than either the first-order scheme or the MOC scheme with space-line interpolation. Overall, the second-order Godunov scheme is simple to implement, accurate, efficient, conservative, and stable for Courant number less than or equal to one.     

Simulation of Transcritical Flow in Pipe/Channel Networks

Using finite difference methods in conjunction with the reduced momentum equation and applying boundary condition structure inherent to subcritical flow to all regimes, is an approach that enables efficient numerical simulation of supercritical and transcritical flows in pipe/channel systems. However, as well as certain errors within a single channel due to incomplete equations, this technique also may introduce unwanted effects propagating across a network in both upstream and downstream directions. These may include: unrealistic backwater effects due to improper boundary conditions, nonamplifying oscillations due to jerky jump movement, and other computational instabilities. Practical implications of these are analyzed in detail and are illustrated using a set of examples. Sensitivity analyzes and comparisons with analytical solutions and laboratory experiments are made. The measures to reduce the inaccuracies inevitable in simulation of transcritical flows are discussed.     

Calibration of Water Distribution Hydraulic Models Using a Bayesian-Type Procedure

Estimating model parameters is a difficult, yet critical step in the use of water distribution system models. Most of the optimization-based approaches developed so far concentrate primarily on efficient and effective ways of obtaining optimal calibration parameter values. At the same time, very little effort has been made to determine the uncertainties (i.e., errors) associated with those values (and related model predictions). So far, this has typically been done using the first-order second moment (FOSM) method. Even though reasonably computationally efficient, the FOSM approach relies on several restrictive assumptions and requires computationally demanding calculation of derivatives. To overcome these limitations, the recently developed shuffled complex evolution metropolis (SCEM-UA) global optimization algorithm is linked to the Epanet2 hydraulic model and used to solve a least-squares-type calibration problem. The methodology is tested and verified on the Anytown literature case study. The main advantage of the SCEM-UA algorithm over existing approaches is that both calibration parameter values and associated uncertainties can be determined in a single optimization model run. In addition, no model linearity or parameter normality assumptions have to be made nor any derivatives calculated. The main drawback of the SCEM-UA methodology is that it could, potentially, be computationally demanding, although this is not envisaged as a major problem with current computers.     

Numerical Sensitivity Study of Unsteady Friction in Simple Systems with External Flows

This paper investigates the importance of unsteady friction effects when performing water hammer analyses for pipe systems with external fluxes due to demands, leaks, and other system elements. The transient energy equation for a system containing an orifice-type external flow is derived from the two-dimensional, axial momentum equation. A quasi-two-dimensional flow model is used to evaluate the relative energy contribution of total friction, unsteady friction, and the external flow, in a 1,500?m pipeline, with orifice flows ranging from steady-state flows of 2–70% of the mean pipe flow, and a Reynolds number of 600,000. It is found that for initial lateral flows larger than around 30% of the mean flow, unsteady friction effects can probably be neglected, whereas for external flows smaller than this, unsteady friction should generally be considered. Overall, the relative role of unsteady friction is found to diminish as the external flux increases, implying that unsteady friction is not critical for systems with large external flows. These results imply that unsteady friction may have a significant impact on the validity of transient leak detection techniques that have been derived assuming quasi-steady friction. To demonstrate this point, an existing transient leak detection method, originally derived under quasi-steady conditions, is tested with unsteady friction included.     

Calibration Method of Current Meters

This paper presents a rapid method for propeller current meter calibration, enabling calibration of current meters in their actual working conditions using simpler equipment than what is currently used in traditional calibrations, and exploiting the uniform velocity profile present through submerged outflows (e.g., flow nozzles and orifices). Experiments were performed to confirm the fundamental hypothesis of uniform horizontal and vertical velocity distribution downstream of a submerged jet. Two experimental velocities were adopted to determine the calibration curve: one based on the discharge and the outflow area; the other derived from Torricelli’s formula, which relies on the head difference between two reservoir levels. Because a current meter measures local velocity, the influence on the measurements’ reliability as a function of current meter position in the submerged outflow jet was investigated. An uncertainty analysis was also performed, and a comparison of the results with the preexisting calibration lines obtained by towing tank is presented.     

Head- and Flow-Based Formulations for Frequency Domain Analysis of Fluid Transients in Arbitrary Pipe Networks

Applications of frequency-domain analysis in pipelines and pipe networks include resonance analysis, time-domain simulation, and fault detection. Current frequency-domain analysis methods are restricted to series pipelines, single-branching pipelines, and single-loop networks and are not suited to complex networks. This paper presents a number of formulations for the frequency-domain solution in pipe networks of arbitrary topology and size. The formulations focus on the topology of arbitrary networks and do not consider any complex network devices or boundary conditions other than head and flow boundaries. The frequency-domain equations are presented for node elements and pipe elements, which correspond to the continuity of flow at a node and the unsteady flow in a pipe, respectively. Additionally, a pipe-node-pipe and reservoir-pipe pair set of equations are derived. A matrix-based approach is used to display the solution to entire networks in a systematic and powerful way. Three different formulations are derived based on the unknown variables of interest that are to be solved: head-formulation, flow-formulation, and head-flow-formulation. These hold significant analogies to different steady-state network solutions. The frequency-domain models are tested against the method of characteristics (a commonly used time-domain model) with good result. The computational efficiency of each formulation is discussed with the most efficient formulation being the head-formulation.