Keywords: calibration, optimization, response surface, Levenberg-Marquardt, Epanet, water distribution system.

Any water distribution system (WDS) model needs to be calibrated. The accuracy of the hydraulic model depends on how well it has been calibrated. Calibration parameters usually include pipe roughness, pipe diameters and demands, where the two first belong to flow conditions and the last one to boundary conditions. Economic development in Estonia and the accompanying increase of water price have led to the situation in which practically each consumer has a water flow meter. Therefore, only roughness of pipes is considered in this paper.

Calibration of pipe roughness is a difficult task if the WDS contains pipes of different ages. In an ideal case roughness coefficients for all pipes must be calibrated. However, measurements are quite expensive and therefore we always have much fewer measurements than we require for calibration. Even with an extensive data collection effort, pipe roughnesses for all links cannot be determined exactly [1]. Thus, a major problem associated with model calibration is the need to calibrate a large number of pipes using only a few measurements. Therefore, grouping of pipes is widely used at calibration [1]. Clustering of pipes is a difficult task for old WDS because great age differences of pipes lead to a large number of clusters. Therefore, instead of grouping of pipes with different age we propose to describe dependence of roughness on age by some flexible function with a small number of parameters. It may be as follows for the pipe roughness used to determine the friction factor for the Darcy-Weisbach formula

where is the roughness for pipe i; , and are the minimal and maximal roughness values, which correspond to minimal and maximal ages of pipes; and are the minimal and maximal ages of pipes in the WDS.

Approximation (87) is quite flexible and describes both the linear and nonlinear dependencies with different degree of nonlinearity.

Parameters , and b may be found by minimizing the objective function (the sum of the squares of the differences between the measured and simulated values). The simplest way to find the minimal value of the objective function (OF) is the trial-and-error method. This method requires significant computer time to examine all possible variants hence it is recommended to use it with the coarse grid. The investigation showed that the trial-and-error method is useful for the analysis of results and may be even used to obtain parameters if the number of them is small.

Our calculations for a real WDS, containing thousands of pipes, showed that the Levenberg-Marquardt algorithm failed because of oscillations of the objective function. Calculations of the objective function showed that its surface looked like emery paper with a large number of small local maximums and minimums. At the same time, the surface would be very good for the Levenberg-Marquardt algorithm (one minimum with good gradients in any direction) if we could eliminate these small variations. The simplest way to do it is to select a long enough step size for the parameters, when calculating partial derivatives for the Levenberg-Marquardt method [2]. Two methods have been used to estimate the step. One of them is based on the number of reliable digits in the objective function values [2]. The other method, proposed in the paper, is based on trying different step sizes to find the step size, the Levenberg-Marquardt method can work with.

A WDS representing a part of the whole WDS of the City of Tallinn was used as an example of the real WDS. This WDS includes thousands of pipes. The age of pipes varies from 0 to 41 years. Calculations showed that the Levenberg-Marquardt method worked very well with a long enough step size for calculations of the gradient of the objective function. The surface of the objective function obtained by the trial-and-error method facilitates calibration significantly. Visualization of the response surface discovers lack of data and helps to plan measurements.

It must also be mentioned that we used the TOOLKIT developed by Rossman [3] for the EPANET2 to complete the calculations automatically. The TOOLKIT permits the creation of external programs, which govern the calculations and therefore is much better for calibration than commercial programs.

1

Mallick K.N., Ahmed I., Tickle K.S., and Lansey K.E., "Determining Pipe Groupings for Water Distribution Networks", J. Water Resour. Plng. And Mgmt., ASCE, 128(2), 130-139, 2002. doi:10.1061/(ASCE)0733-9496(2002)128:2(130)

2

Dennis Jr., J.E. and Schnabel, R.B., "Numerical methods for unconstrained optimization and nonlinear equations", SIAM, Philadelphia, 1996.

3

Rossman, L.A., "EPANET 2 User's Manual", Water Supply and Water Resources Division, National Risk Management Research Laboratory, Cincinnati,OH 45268, September, 2002.

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £105 +P&P)