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Sensitivity of calibration results to uncertainties in input data: Case of operational water distribution system
Abstract
The data used for calibration of a water distribution system (WDS) model contain uncertainties. These uncertainties include, in particular, errors in measurements of pressures and water flows, errors in demands, demands pattern, erroneous node elevations, limited sample size, etc. Moreover, some assumptions influence the results of calibration as well. For example, water inflow into WDS is often higher than sum of registered demands. This difference between water inflow and demands includes real and apparent leakages with unknown proportionality. The results of calibration depend on the assumed proportion between real and apparent leakages and on the assumed distribution of them over WDS. Weight coefficients, assumed for pressure and flow measurements in objective function, used in the optimization procedure, are another assumption influencing calibration results. WDS models contain also patterns of demands. But real demands are usually measured for quite long periods of time (e.g. a month) and give information only on the average base demand. Experience in the calibration of WDS models showed large uncertainties of hourly demands estimated by this manner. The effect of these types of uncertainties on calibration results has been considered in the paper.
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Calibration of a water distribution network is intended to develop a model that mimics field conditions under a range of demand distributions. In this paper, a three-step calibration procedure is developed that considers the uncertainties in measurement and estimation and provides a measure of the quality of the calibration. The approach can also be used to identify preferable conditions for data collection. The procedure's steps are parameter estimation, calibration assessment, and data collection design. Parameter estimation considers input uncertainty and the resulting uncertainty in model parameters. Calibration assessment analyzes the propagation of the parameter errors on model predictions. The trace of the covariance matrix of the predictive heads is used to measure the model uncertainty. Based on this uncertainty and using a sensitivity-based heuristic analysis, data collection experiments can be designed for systemwide tests and critical pipe for individual pipe tests. An example system is analyzed to select calibration demand conditions.
The accuracy of a Water Distribution System (WDS) model depends on how well it has been calibrated. Analysis of measurements showed that measured pressures often contain systematic errors. Another type of errors is inaccurate estimation of the elevations of junctions, which will lead to differences between measured and modeled pressures. In this paper we propose to use pressure differentials for the calibration of operational water distribution system containing thousands of pipes with different age. The pump speed is regulated in a way that pressure in the junction next to pump, is constant. Pressures in other junctions depend on junction elevation and head loss in pipes. Head loss, in turn, depends on water consumption and roughness of pipe. Pressure differentials in a junction at times with high and low consumption do not contain systematic errors of loggers and errors due to inaccurate elevation of junction. It is shown also that this approach eliminates significantly the influence of leakages as well.
The aim of the paper is to demonstrate that error dynamics may be used for calibration of water distribution system models. Our suggestions are: to use an optimization procedure to obtain proportions of leakages and unaccounted demands; to rank pressure measurements according to closeness of the measurement to water source; to analyze error dynamics to control whether roughness or demand are over/underestimated; it is assumed that the absence of error increase/decrease with the increase/decrease of water flow will show that pressure was modelled correctly even if some constant error (e.g. wrong elevation of junction) was present.
This paper aims at presenting the historical evolution and the present status of water distribution network design approaches in the light of water losses reduction. Although pressure management has been recognised as one of the four actions aimed at reducing water losses, very little work is currently available in the literature on design criteria that would allow improving pressure management in operation. Only recent works have aimed at improving design of new systems or expansions and rehabilitation of the existing networks by formulating a multi objective problem, where the first objective is clearly the reduction of costs (network, non revenue water, maintenance), while additional objectives, aim at improving resilience of the system by reducing the internally dissipated power and at redistributing more evenly the pressure field over the entire water distribution network. The new techniques have the advantage of producing networks that can be operated under lower pressure fields, which is the objective of pressure management. The new multi objective optimization techniques are generally based on genetic algorithms, which make extensive use of network analysis and simulation. Therefore the paper also discusses the representation of leakage losses in the current generation of water distribution networks simulation packages. The presently available approaches are shown to be unsatisfactory in that they produce considerable errors in terms of energy and head losses. The paper concludes by presenting a set of possible alternatives and future perspectives.
The paper describes methodology and results of calibration of the network model proposed in The Battle of the Water Calibration Networks (BWCN). The task includes calibration of roughness and demand patterns for each DMA. Fire flow tests data were used first for the calibration of pipe roughness. Calibration was accomplished using optimization procedure to minimize the sum of the errors squared (differences between measured and modelled pressures and flows). Errors in pressure and water flow were used in optimization with different weight. Weight coefficients were obtained to ensure the same influence of errors in pressures and water flow on the objective function. Optimization procedure was used for calibration of demands as well. The objective was to find such time demand which would minimize errors of the modelled tanks levels. The Levenberg-Marquardt algorithm with proper selection of low limit for step along parameters for the calculation of partial derivatives was used for optimization. This limit is necessary since the surface of the objective function obtained by the Epanet2 oscillates, and the correct gradient may be found only if steps along parameters are large enough to avoid local minimums generated by oscillations. The main problem of demands calibration is that the SCADA data contain momentary measurements. In such conditions, the solution may have at least two answers if the state of pump has been changed during the time interval. Therefore, the optimization procedure is sensitive to initial values. The paper describes the methodology used for calibration.
The weighted-least-squares method based on the Gauss-Newton minimization technique is used for parameter estimation in water distribution networks. The parameters considered are: element resistances (single and/or group resistances, Hazen-Williams coefficients, pump specifications) and consumptions (for single or multiple loading conditions). The measurements considered are: nodal pressure heads, pipe flows, head loss in pipes, and consumptions/inflows. An important feature of the study is a detailed consideration of the influence of different choice of weights on parameter estimation, for error-free data, noisy data, and noisy data which include bad data. The method is applied to three different networks including a real-life problem.
The aim of the paper is to demonstrate that the Levenberg–Marquardt algorithm can give successful results when operational water distribution systems are calibrated with the proper selection of parameter increment for the calculation of partial derivatives. The functional dependence of pipe roughness on age, which describes linear and nonlinear dependences, is proposed for the calibration of a model of a water distribution system containing pipes of different age. It is also shown that the visualization of response surface on a coarse grid is very useful for the analysis of the results.