ACCEPTED VERSION
Vitkovsky, John; Liggett, James A.; Simpson, Angus Ross; Lambert, Martin Francis
Optimal measurement site locations for inverse transient analysis in pipe networks Journal of
Water Resources Planning and Management, 2003; 129 (6):480-492
© ASCE 2003
http://hdl.handle.net/2440/1022
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28 March 2014
1
OPTIMAL MEASUREMENT SITE LOCATIONS FOR INVERSE
TRANSIENT ANALYSIS IN PIPE NETWORKS
John P. Vítkovský
1
, James A. Liggett
2
, Angus R. Simpson, M. ASCE
3
,
and Martin F. Lambert
4
ABSTRACT
The quality of leak detection and quantification, and calibration for friction coefficients, in
pipelines and networks by the inverse transient method are dependent on the quantity and
location of data measurement sites. This paper presents an approach for determining the
configuration of measurement sites that produces optimal results. Three performance
indicators, two that are based on A- and D-optimality criteria and one that is based on the
sensitivities of the heads with respect to the parameters, show which configurations are
superior. These are illustrated by two case studies, the first of which is a small pipe network
in which all configurations are considered directly (fully enumerable) and the second is a
larger pipe network in which statistics are drawn from a sampling of configurations. For the
large network, a genetic algorithm—with a new crossover operator—performs a search of
1
Research Associate, School of Civil and Environmental Engineering, University of
Adelaide, Adelaide SA 5005, Australia. (Corresponding Author)
Email: jvitkovs@civeng.adelaide.edu.au; Tel: +61 8 8303 4324; Fax: +61 8 8303 4324
2
Professor Emeritus, School of Civil and Environmental Engineering, Cornell University,
Ithaca, NY 14853-3501, USA. Email: jal8@cornell.edu
3
Associate Professor, School of Civil and Environmental Engineering, University of
Adelaide, Adelaide SA 5005, Australia. Email: asimpson@civeng.adelaide.edu.au
4
Senior Lecturer, School of Civil and Environmental Engineering, University of Adelaide,
Adelaide SA 5005, Australia. Email: mlambert@civeng.adelaide.edu.au
2
possible measurement site configurations to determine the optimal measurement locations.
The number of sites as well as time length of data at each site are considered also.
INTRODUCTION
Inverse analysis has been applied in a variety of fields to determine parameters of problems,
boundary conditions and even the basic equations governing a process. By definition an
inverse problem is one where measurements of one or more events are known but the
parameters defining the physical condition, the boundary or initial conditions, and/or the
governing equations are unknown. Inverse analysis has been applied under transient
conditions to leak detection and friction factor calibration in pipelines and pipe networks by
Liggett and Chen (1994) and Vítkovský (2001). It typically requires a large quantity of data
for accurate calculation, and an unsteady event provides much more data than a steady event.
The basic objective of inverse analysis in a piping system is to find leaks, but Liggett and
Chen (1994) noted that unless the frictional properties are well knownwhich is seldom the
caseleak detection and quantification could not be carried out with sufficient precision.
Thus, the analysis requires a simultaneous calibration for friction factors and leak areas. In
addition, wave speed in a pipe is seldom known accurately and that factor is often included in
the sought-for parameters. Although the primary objective has been leak detection and
quantification, the calibration aspect forms a major side benefit as the frictional properties are
required for the analysis of a network, design of additional infrastructure and maintenance of
networks.
3
A transient event in a pipeline system can be generated using a change in valve or pump
conditions. The measured data are the pressures observed periodically and simultaneously at
various locations in a pipe or pipe network during the transient event. Since flow rates are
more difficult and expensive to measure they are not generally used.
There are two methods to obtain an inverse solution, a direct and an indirect method. Each
has advantages and disadvantages (Neuman 1973). The direct method treats the model
parameters as the dependent variables in a formal inverse boundary value problem from
which a direct solution of the parameters is made. The direct method requires that the data
(and derivatives of the data) are exact and complete. Errors in the data may cause the
problem to become improperly posed and solutions might only exist for certain restricted
conditions. The indirect method minimizes the difference between measured and calculated
data. Essentially, the indirect method uses a “guided search” by a minimization algorithm.
An advantage of the indirect method is that data (and derivatives of the data) need not be
known at all points and times in the network. In pipe networks, the data are measured at
particular locations and certainly not over the entire domain of the dependent variables
(required by the direct method). Hence, the indirect method is used in this research.
Accuracy of the inverse method is very dependent on the quality and quantity of
measurements. However, all measurements are not equally effective. The objective of this
paper is to explore data collection methods that make transient inverse analysis effective and
economical. Questions include:
1. How many measurement sites are necessary in a network for adequate results?
2. Where should these measurement sites be placed to produce the best results?
3. What (time) length of data is needed to produce accurate results?
4
4. What degree of confidence is associated with the results?
The optimum location of measurement sites is a combinatorial problem, i.e., for a given
number of measurement sites there are many combinations of site configurations. Two case
studies are considered. One is a completely enumerable network (meaning that all possible
measurement sites and measurement configurations are considered); the other is a large
network where full enumeration is not practicable.
The “optimal sampling design” consists of a plan of measurement sites that optimize the
inverse solution. Historically, the field of groundwater monitoring has made considerable
inroads into optimal sampling design. Carrera and Neuman (1986) suggested the reduction of
parameter variances (A-optimality criterion) be used to determine the optimal locations to
make measurements. Knopman and Voss (1989) optimized the accuracy to which the
parameters are determined, cost of sampling and even the type of model used. Their
optimization contained multiple objectives and produced an optimal front of solutions.
Loaiciga et al. (1992) give a review of groundwater sampling design.
In the context of this paper, sampling design is applied to water distribution systems. Walski
(1983) suggested some rules-of-thumb for the steady state calibration of water distribution
systems based on practical experience. Yu and Powell (1994) determined optimal sampling
designs using a decision-tree technique for optimization based on the A-optimality criterion,
sampling cost and distance from the sampling locations to a control center. Bush and Uber
(1998) used a ranking of three different criteria based on sensitivities to generate near optimal
sampling designs for calibration. Their results compared well with the D-optimality criterion.
Meier and Barkdoll (2000) considered the calibration problem using a number of different
