Calibration of a model of an operational water distribution system containing pipes of different age

https://doi.org/10.1016/j.advengsoft.2008.11.015Get rights and content

Abstract

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.

Introduction

All models of water distribution systems (WDS) require calibration. The precision of hydraulic models depends on how accurately they have been calibrated. The calibration parameters usually include pipe roughness, pipe diameters and demands, where the first two relate to flow conditions and the last one to boundary conditions. The corrosion and deposition processes, which occur over time after the pipe instalment, make it more difficult to determine the actual pipe diameter. Therefore, in the absence of another value, nominal pipe diameters are generally used for model development, and the roughness coefficient is adjusted to compensate the change in diameter due to the pipe wall build-up [1], [2]. Economic developments in Estonia and the resulting increased water costs require that each consumer be supplied with a water flow meter, which in turn facilitates calibration.

Calibration of pipe roughness is a difficult task if the WDS contains pipes of different age. In an ideal situation roughness coefficients for all pipes should be calibrated. WDS model calibration relies upon field measurement data (junction pressures, pipe flows, etc. [3]). However, measurements are quite expensive and usually much fewer measurements are available than required for calibration. Because of a large number of unknown values it is impossible to calibrate the model of a real system precisely [4]. Pipe roughness for all links cannot be determined accurately even when considerable efforts have been made to collect data [5]. 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 [5]. All pipes are clustered into groups of identical estimated roughness and calibration is used to find these values. Roughness of old pipes depends on age, pipe material, water quality in WDS and some other factors.

Clustering of pipes is a difficult task for old WDS because great age differences of pipes lead to a large number of clusters and consequently to a large number of parameters to be calibrated. Therefore, first the aim was to show that a flexible function, consisting of a small number of parameters to describe the dependence of roughness on age may be used instead of clustering. The parameters of the function may be found by an implicit method [6], [7], [8] that minimizes the objective function (OF), the sum of the squares of the differences between the measured and the simulated values.

The simplest way to find the minimal value of the OF is the trial-and-error algorithm. However, this algorithm would require a long period of time for all the possible variants to be examined. Several researchers have reported that standard algorithms (e.g. the Levenberg–Marquardt algorithm (LMA)) often find local minimums and they have proposed to use the Genetic Algorithm (GA) for calibration [9]. However, the GA also consumes much computer time. Kapelan et al. [10] proposed to use the combination of the GA and LMA algorithms in order to decrease computer time. A comprehensive review of available water distribution calibration models can be found in [11].

The calculations for an operational WDS containing thousands of pipes showed that the LMA had failed because of oscillations of the OF. The calculations of the OF indicated that its surface looks like emery paper with a large number of small local maximums and minimums. However, that surface would be very good for the LMA (one minimum with good gradients) if we could eliminate these small variations. The simplest way to do it is to select a long enough step along the parameters, when calculating partial derivatives for the LMA [12]. This approach was tested and the second goal of the paper is to prove that the LMA algorithm works in this case quite well. The LMA requires only 1–2 min to find three parameters for a network containing more than 2000 pipes (computer with CPU 3.4 GHz). Short computer time will enable one to do more sophisticated analysis and to test many different variants of calibration.

This investigation showed that the LMA is still useful for the calibration of operational WDS but the LMA needs information on the precision of calculations of the OF to work successfully. That is a significant result because the LMA requires much less computer time than the popular GA. The flexible function proposed in the article allows the dependence of roughness on age to be described for old WDS using a small number of parameters. It is shown also that the visualization of the OF surface is useful for the analysis of calibration results.

Section snippets

Method of calibration

Optimization methods are used to search for a solution of how to describe the unknown calibration parameters which will minimize the OF [4], [13], which usually is the sum of the residuals (the difference between the modelled and the observed values) squared. Since the number of measurements is usually much smaller than the number of pipes and nodes, it is clear that we cannot calibrate each single pipe. If a WDS contains pipes of different age, it will be reasonable to approximate the value of

Results and discussion

A WDS representing a part of the whole WDS of the City of Tallinn (Fig. 1) was used as an example of the operational WDS. The characteristics of pipes installed are presented in Table 1. It can be seen that the age of pipes varies from 0 to 41 years. Cast iron is the material of the overwhelming majority of pipes (81% of the total length of pipes). Other materials used are plastic and steel (11% and 8% of the total length, respectively).

Calibrations were performed on the basis of pressures

Conclusions

  • The Levenberg–Marquardt algorithm requires a low limit for the step along parameters to calculate partial derivatives in order to work on the operational WDS model containing thousands of pipes.

  • The approximation of the dependence of pipe roughness on age by the function with a low number of parameters is easier than pipe grouping according to age.

  • The analysis showed that the response surface is very useful to understand the results. The combination of the Levenberg–Marquardt algorithm with the

Acknowledgements

This work was financially supported by Target Financing of Estonia (grant SF0140072s08) at Tallinn University of Technology and by the Estonian Science Foundation (grant G7646).

References (22)

  • Vassiljev A, Koppel T. Calibration of the model of an operational water distribution system. In: Topping BHV, Montero...
  • K.E. Lansey et al.

    Calibration assessment and data collection for water distribution networks

    J Hydraul Eng

    (2001)
  • M. Greco et al.

    New approach to water distribution network calibration

    J Hydraul Eng

    (1999)
  • T.M. Walski

    Model calibration data: the good, the bad, and the useless

    J AWWA

    (2000)
  • K.N. Mallick et al.

    Determining pipe groupings for water distribution networks

    J Water Resour Plan Manage, ASCE

    (2002)
  • L.E. Ormsbee

    Implicit network calibration

    J Water Resour Plan Manage, ASCE

    (1989)
  • K.E. Lansey et al.

    Parameter estimation for water distribution networks

    J Water Resour Plan Manage, ASCE

    (1991)
  • J.A. Liggett et al.

    Inverse transient analysis in pipe networks

    J Hydraul Eng, ASCE

    (1994)
  • J.P. Vitkovsky et al.

    Leak detection and calibration using transients and genetic algorithms

    J Water Resour Plan Manage, ASCE

    (2000)
  • Z.S. Kapelan et al.

    A hybrid inverse transient model for leakage detection and roughness calibration in pipe networks

    J Hydraul Res

    (2003)
  • Haestad. Methods et al.

    Advanced water distribution modeling and management

    (2003)
  • Cited by (55)

    • Automatic calibration toolbox for SWMM5

      2023, Advances in Engineering Software
    • Real-time demands and calibration of water distribution systems

      2015, Advances in Engineering Software
      Citation Excerpt :

      The time with water flow 193 l/s and sumdev 0.99 was selected for calibration. The usual set of measurements for calibration in this area includes only pressures in 18–20 junctions and water flow at pump station [14,18]. In the year 2012 water flow has been measured additionally in 13 pipes of the WDS.

    • Estimation of real-time demands on the basis of pressure measurements by different optimization methods

      2015, Advances in Engineering Software
      Citation Excerpt :

      The possibility to find real-time demands within acceptable time was investigated for both optimization approaches. The results presented in this paper are based on the WDS model calibrated by the methods described in [3,11,13,15]. The influence of the pressure sensor’s errors on the results of the calculations was studied as well.

    View all citing articles on Scopus
    View full text