Implicit state-estimation technique for water network monitoring

doi:10.1016/S1462-0758(00)00050-9

Urban Water

Volume 2, Issue 2, June 2000, Pages 123-130

https://doi.org/10.1016/S1462-0758(00)00050-9 Get rights and content

Abstract

This paper presents a new implicit formulation of the standard weighted least squares (WLS) state-estimation problem for water networks with very low measurement redundancy. The formulation is based on the loop equations and the state variables are the unknown nodal demands. The minimisation problem is solved using a Lagrangian approach. It is shown that the method can be applied to weak demand indicators, such as property counts, rather than direct demand information which is often unreliable. Furthermore, a new leak detection scheme is derived from the method. This method is investigated under idealised noise-free conditions in order to demonstrate the principles and to gauge the potential of the scheme. The application of the scheme in more realistic circumstances of uncertainty is also discussed.

Introduction

The method of weighted least squares (WLS) state-estimation for water distribution networks is well known, e.g. Bargiela (1984), Powell, Irving, and Sterling (1988) and Brdys and Ulanicki (1994). The best fit of hydraulically consistent nodal heads is found by a series of normal projections. Although the numerical problem is well defined, it may lead to practical problems for particular networks, e.g. Hartley and Bargiela (1993). A network calibration determines the static network parameters, while state-estimation determines current values of pressures and flows, given fixed network parameters. The latter is in principle a dynamic procedure, but it is often carried out at snapshot instances in time due to computational limitations, especially for large realistic networks. For this reason there is some mathematical similarity between state-estimation and network calibration. Ormsbee (1989) used implicit system constraints for network calibration, but due to the type of constraints imposed the least-squares fit was carried out by using a non-gradient optimisation method by Box (1965). Lingireddy and Ormsbee (1999) used a genetic algorithm (GA) for the minimisation procedure, but in common with other calibration studies, e.g. Lansey and Basnet (1991), and Greco and Del Guidice (1999), the algorithm constitutes a two-level hierarchy where the optimisation part is controlling a network simulation module. Although such approaches may be modular, they may not lead to the best convergence. Niranjan Reddy, Sridharan, and Rao (1996) carried out a parameter estimation study using a WLS technique. Their method was based on explicit calculation of the sensitivity coefficients for satisfaction of the combined non-linear system of equations as a function of the system parameters. The parameters also included loading conditions as variables; hence their method may be applied to state-estimation for demand as unknown variables.

In this paper, a new implicit formulation of the WLS method is considered; the WLS state-estimation is regarded as an integrated problem and solved using an implicit Lagrangian approach. The aim for the proposed method is to benefit from the smaller solution matrices which relate to the loop structure rather than the nodal structure of the network. In this way the method benefits from the advantages of the looped formulation when controlling elements are present, see also Andersen and Powell (1999a).

Section snippets

Formulation of the network equations

A simple piped network example is used for a demonstration of the general network equations. See Fig. 1.

The example (Fig. 1) has n=4 demand nodes and two fixed head datum nodes, these are reservoirs, (a) and (b). The network is analysed into tree-links of a spanning tree rooted at a fixed head datum node. The remaining links become chord-links. This analysis, however, is not unique. Here, reservoir (a) is arbitrarily chosen as the root node. Since all datum nodes in a sense belong to the same

The simulation problem

A simulation is defined here as a solution of the network model for given nodal demands. Assuming that the flow function $Ψ(δ h)$ is monotonic, its inverse function is then written as $δ h =Φ(q)$ . In the case of the Hazen–Williams flow model, the inverse is easily obtained. From Eq. (5), the nodal heads can be formally written $h =−(U^{T})^{−1} Φ (q_{T})− a .$ Substitution into (6) gives $q_{C} =Ψ V^{T} (U^{T})^{−1} Φ (q_{T})− a + b .$ Furthermore, substituting $q_{T}$ via (1) and operating Φ on both sides leads to $Φ(q_{C})= V^{T} (U^{T})^{−1} Φ (U^{−1} d − U^{−1} Vq_{C})− a + b .$ The

Principles of implicit state-estimation

Let the vector $x (n×1)$ be the unknown state variable. The classical WLS state-estimation minimises the objective function $Min : Ω=(z − g (x))^{T} W (z − g (x)).$ Here g(x) represents the hydraulic model equation for the heads, flows and demands as a function of the state variable x. The vector $z (m×1)$ is the measurement vector and the matrix $W (m×m)$ is a diagonal matrix of measurement weights.

The corresponding Newton iteration step solves a normal equation problem using the Gauss–Newton approximation to the

Nodal demand information in DMA-structured networks

District metered areas (DMAs) are increasingly being introduced into water distribution networks for improved demand management and the potential for leakage detection. The real transducers are situated at the DMA boundaries and consumer profiles are available inside the DMAs. Traditionally, state-estimation methods use pseudo-measurements in the form of nominal demand profiles in order to obtain sufficient data redundancy. But as discussed above, such demand profiles add little new information

DMA structured implicit state-estimation theory

Using the demand variable as state vector, the total demand for a DMA can be simply written as $D_{i} =∑_{k∈DMA_{i}} x_{k}, i=1,…,n_{DMA} .$ Let the matrix $R =⌊r_{ji} ⌋$ be a matrix of static demand distribution factors of size (n_s×n_DMA); the number of state variables by the number of DMAs. Each column vector of R comprises the demand factors for a particular DMA; the sum of the distribution factors for a DMA should be equal to one. Then the expectation would be that the redistributed demands: r_jiD_i are close to the state

The Newton step

Let $G (x, y)=0$ comprise the total system of non-linear equations for which the solution is sought, that is $G (x, y)= (J_{x} − J_{y} (K_{y})^{−1} K_{x})^{T} W (z − g (x, y))− Qx F (x, y) = 0 .$ This leads to the following Newton–Raphson iteration scheme: $Δ x Δ y =−α ∂ G (x, y) ∂ (x, y)^{−1} G (x, y).$ The factor α serves to stabilise the convergence of scheme, 0<α⩽1. The maximum factor depends on the various round-off errors of the many steps involved in a compact numerical scheme for the solution of (21), hence this factor can only be found by

State-estimation of demands for a grid network

A network example has been constructed in order to study the properties of the state-estimation and the possibility of leak detection, see Fig. 2. To facilitate the analysis, the example has some idealised features. The network consist of a single DMA, but the generalisation to several DMAs is fairly straightforward (Powell et al., 1999). The nodes are at the crossing points between the pipes of a square grid. Water is fed into the network from four metered supply lines (flow meters) situated

State-estimation with enhanced leakage detection

The state estimator can focus on a leakage problem by a subtle change to the external information about the network. A mathematical solution is indeed possible provided that one and only one node in the network has a real demand with a substantially larger deviation from the redistributed demands than for the other nodes. This additional information does not appear to be in a very precise form, but it indicates that a departure is expected from a statistically balanced demand profile according

Discussion

This section deals with some practical questions that may arise, in a real life application of the state-estimation method. The main focus of the paper is on the principles and the ultimate mathematical possibility of detecting leakage by using a novel state-estimation method. Network uncertainty combined with measurement errors as well as insufficient meter coverage is bound to dilute the leakage signal. Some of the expected effects of the departures from ideal circumstances are examined in

Conclusions

An implicit WLS state-estimation method has been outlined; the method is based on the loop equation framework for network simulation. As the method lends itself to the use of nodal demands as the state variables, it becomes particularly suited for water networks with weak demand information. The paper also proposes a new mathematical solution to the leakage detection problem; this particular solution follows naturally from the implicit state-estimation. The ultimate possibility of leak node

References (13)

Andersen, J. H., & Powell, R. S. (1999a). Simulation of water networking containing controlling elements. Journal of...
Andersen, J. H., & Powell, R. S. (1999b). A loop based simulator and implicit state-estimator. In B. Coulbeck, & B....
Bargiela, A. B. (1984). On-line monitoring of water distribution networks. Ph.D. Thesis, University of...
M.J. Box
A new method of constrained optimization and a comparison with other methods
Computer Journal
(1965)
M.A. Brdys et al.
Operational control of water systems: structures, algorithms and applications
(1994)
M. Greco et al.
New approach to water distribution network calibration
Journal of Hydraulic Engineering, ASCE
(1999)

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