Implicit state-estimation technique for water network monitoring

Urban Water 2 (2000) 123±130

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Implicit state-estimation technique for water network monitoring Johannes H. Andersen *, Roger S. Powell Water Operational Research Centre (WORC), Department of Systems Engineering, Brunel University, Uxbridge UB8 3PH, UK Received 16 December 1999; received in revised form 18 August 2000; accepted 11 September 2000

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. Ó 2000 Elsevier Science Ltd. All rights reserved. Keywords: State-estimation; Water distribution systems; Leakage detection

1. Introduction The method of weighted least squares (WLS) stateestimation for water distribution networks is well known, e.g. Bargiela (1984), Powell, Irving, and Sterling (1988) and Brdys and Ulanicki (1994). The best ®t of hydraulically consistent nodal heads is found by a series of normal projections. Although the numerical problem is well de®ned, 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 ¯ows, given ®xed 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 ®t 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 *

Corresponding author. Tel.: +44-1895-203-296; fax: +44-1895812-556. E-mail address: [email protected] (J.H. Andersen).

(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 coecients 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 bene®t from the smaller solution matrices which relate to the loop structure rather than the nodal structure of the network. In this way the method bene®ts from the advantages of the looped formulation when controlling elements are present, see also Andersen and Powell (1999a).

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

1462-0758/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 1 4 6 2 - 0 7 5 8 ( 0 0 ) 0 0 0 5 0 - 9

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J.H. Andersen, R.S. Powell / Urban Water 2 (2000) 123±130

2

3 02 qT1 ÿ1 6 qT 2 7 B6 1 6 7 B6 4 qT 5 ˆ W@4 1 3 qT 4 0

0 0 ÿ1 0 0 ÿ1 1 0

32 3 2 31 0 h1 ha 6 h2 7 6 0 7 C 0 7 76 7 ‡ 6 7C 0 54 h3 5 4 0 5A h4 0 ÿ1 …3†

and 

qC1 qC2



0



B 0 0 ˆ WB @ 0 0

1 0

1 2 3  h1   C 7 ÿ1 6 6 h2 7 ‡ 0 C : 4 5 h3 ÿhb A 1 h4

…4†

Again, these equations can be written in compact matrix forms: Fig. 1. A simple water network.

The example (Fig. 1) has n ˆ 4 demand nodes and two ®xed head datum nodes, these are reservoirs, (a) and (b). The network is analysed into tree-links of a spanning tree rooted at a ®xed 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 virtual node, reservoir (b) cannot be part of the spanning tree; the chord C2 completes a pseudo-loop via the virtual datum node. For the general case it is assumed that the corresponding graph is connected. In the above example, thin arrows indicate the reference directions for the tree links (T) and thick arrows are used for the chords (C). The chords are also referred to as co-tree links. The number of chords is equal to the number l of fundamental loops. In this network we have l ˆ 2 . The network equations are as follows: Flow continuity: 2 3 2 32 3 2 3 qT 1 1 ÿ1 ÿ1 0 0 0  d1  6 d2 7 6 0 1 6 7 6 0 ÿ1 7 0 7 6 7ˆ6 76 qT2 7 ‡ 6 0 7 qC1 ; 4 d3 5 4 0 0 1 0 54 qT3 5 4 ÿ1 0 5 qC2 qT 4 d4 0 0 0 1 1 ÿ1 …1† where di are the nodal demands, qTi the tree ¯ows and qCi are the chord ¯ows. Eq. (1) is conveniently written in a compact matrix form as: d ˆ UqT ‡ VqC :

…2†

Here d …n  1† is the vector of the nodal demands, qT …n  1† and qC …l  1† are the vector forms of the tree and chord ¯ows. The matrix U …n  n† is the node±link incidence matrix for the tree, the network labelling is chosen such that U becomes upper triangular. The matrix V …n  l† is the node±link incidence matrix for the chord ¯ows. The next equations relate the ¯ows to the pressure drops over the links:

qT ˆ W…ÿUT h ‡ a†;

…5†

qC ˆ W…ÿVT h ‡ b†:

…6†

The vector h …n  1† is the vector of nodal heads, h ˆ ‰h1 ; . . . ; hn ŠT . The vector function W…dh† is the component wise hydraulic ¯ow model giving the ¯ow as a function of head drop dhi for each link. Notice that the node±link incident matrices U and V from Eq. (2) also appear in (5) and (6). The link±node relationships are found by transposing U and V. The matrix formulation is useful for the analytical study of the simulation and state-estimation problems, but the matrices are normally not used directly in an ecient software implementation. The vectors a and b are necessary boundary vectors. The vector a of length n has a single entry corresponding to the ®xed-head root node. The vector b of length l has entries for each subsequent ®xed-head node which gives rise to a virtual fundamental loop. 3. The simulation problem A simulation is de®ned here as a solution of the network model for given nodal demands. Assuming that the ¯ow function W…dh† is monotonic, its inverse function is then written as dh ˆ U…q†. In the case of the Hazen±Williams ¯ow model, the inverse is easily obtained. From Eq. (5), the nodal heads can be formally written ÿ1

h ˆ ÿ…UT † …U…qT † ÿ a†:

…7†

Substitution into (6) gives   ÿ1 qC ˆ W VT …UT † …U…qT † ÿ a† ‡ b :

…8†

Furthermore, substituting qT via (1) and operating U on both sides leads to ÿ
…9† U…qC † ˆ VT …UT †ÿ1 U…Uÿ1 d ÿ Uÿ1 VqC † ÿ a ‡ b: The only unknown variable present in (9) is the chord ¯ow vector qC , though this vector appears on both leftand right-hand side. Since the length of qC is the same as

J.H. Andersen, R.S. Powell / Urban Water 2 (2000) 123±130

the number of fundamental loops, which may be considerably smaller than the number of nodes, there is scope for ®nding an ecient algorithm for solving (6). Once qC is known, the network state follows readily via qT from (2) and h from (5). Eq. (9) is referred to as the `chord-equation'. The equation states that the head losses over the chords equals the head losses calculated the other way round the loops. This is the well-known Kircho€'s second law of energy conservation in loops. Eq. (9) can be cast in a residue form by de®ning the residue function F…qC † as the di€erence between the leftand the right-hand sides of (9) ÿ
F…qC †ˆU…qC †ÿVT …UT †ÿ1 U…Uÿ1 dÿUÿ1 VqC †ÿa ÿb: …10† Hence the solution of the simulation problem amounts to solving the system of non-linear equations: F…qC † ˆ 0. The system can be solved eciently by using the Newton±Raphson iteration method, but the details are outside the scope of this paper. As will be shown below, the chord equation in this form becomes a useful tool for the development of the implicit state-estimation method.

4. 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 :

T

X ˆ …z ÿ g…x†† W…z ÿ g…x††:

…11†

Here g(x) represents the hydraulic model equation for the heads, ¯ows 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 Hessian matrix Dx ˆ …JT WJ†ÿ1 JT W…z ÿ g…x††;

…12†

where J is the Jacobian of g(x). The choice of physical variables to represent the state vector x is yet unspeci®ed. If the nodal heads represent the state vector …x ˆ h† then g(x) may be readily calculated for head, ¯ow and demand measurements. However, the solution matrix JT WJ may become large and dicult to solve eciently. Another possibility for the state vector is the demand vector …x ˆ d†. This may have certain advantages, but the non-linear hydraulic model g(x) can no longer be readily calculated for the head and ¯ow measurements. Intuitively, a secondary series of iterations may then be necessary. However, by using a Lagrangian approach such double-iteration is avoided.

125

In the framework of the loop equation method for a water network with given nodal demands, it was shown that solving the system amounts to ®nding the correct chord-¯ows which satis®es Kircho€'s second law. The independent variables for the proposed Lagrangian method are the nodal demand vector …x ˆ d† and the chord ¯ow vector (y ˆ qC ). Hence, the hydraulic model for heads, ¯ows and demands now become a function of both x and y. The hydraulic model, however, is only ``physically realised'' when Kircho€'s second law is satis®ed, this imposes the Lagrangian constraint F…x; y† ˆ 0 for the chord-equation (10). The term ``extended hydraulic model'' is used for the hydraulic model g(x,y) as function of x and y. This introduction of the extended model is the salient step in the solution process; while the hydraulic model g(x) require a full network simulation to calculate, the extended model g(x,y) can be calculated explicitly in terms of the basic head-loss/¯ow relationships and its partial derivatives are easily found. The detailed derivations is outside the scope of this paper, see Andersen and Powell (1999b), and Powell, Andersen, Nagar, and Hindi (1999). The solution for vectors x and y is found by a Newton±Raphson iteration scheme. The above formulation, however, is general in the sense that any combination of reliable network measurements such as heads, ¯ows and demands may be used. In practical situations the demand information is `weak'; that is the demands are in most cases based on estimates rather that direct measurements. If such demand data were used as substitutes for genuine measurements in the scheme, then the result of the state-estimation would be dominated by the demand data and there would be a risk of circular conclusions. This point follows from two reasons: (1) The number of real measurements is generally much smaller than the dimension of the state-space, therefore a large amount of estimated demand data or `pseudodemands' are required in order to satisfy the data redundancy condition for the state estimator. (2) The `standard' assumption of uncorrelated measurements reinforces the demand data. Hence in this paper an alternative method of using demand information is proposed. 5. 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 pro®les are available inside the DMAs. Traditionally, state-estimation methods use pseudo-measurements in the form of

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nominal demand pro®les in order to obtain sucient data redundancy. But as discussed above, such demand pro®les add little new information since the nominal demand pro®les are found from statistics of globally measured DMA consumptions. The demand pro®les are usually composed of the product of static distribution factors by a common time-factor from a nominal timepro®le. In the DMA-structured state-estimation a different approach is followed; the assumed demand information is relaxed such that only static distribution factors are used. These factors depend on demand indicators such as property counts and consumer type. Although the distribution factors are only proportional guesses, this information is presumably more reliable than the nominal demands; at least one source of uncertainty is eliminated.

6. DMA structured implicit state-estimation theory Using the demand variable as state vector, the total demand for a DMA can be simply written as X xk ; i ˆ 1; . . . ; nDMA : …13† Di ˆ k2DMAi

Let the matrix R ˆ brji c be a matrix of static demand distribution factors of size (ns  nDMA ); 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: rji Di are close to the state vector values, but not necessarily identical since the state vector x must satisfy the hydraulic model. The information contained in R may be utilised by minimising the squared di€erences between these values. Let the matrix S …nDMA  ns † be de®ned such that each row of S is an indicator function for a particular DMA, that is S is a summation matrix for the state vector. Hence brji Di ÿ xj c ˆ …RS ÿ I†x

…14†

is the intrinsic deviations from the redistributed demands. The WLS contribution to cost function can be expressed as the quadratic matrix form: xT Qx ˆ xT …RS ÿ I†T WDMA …RS ÿ I†x;

…15†

where WDMA is a diagonal matrix of relative weight factors for the DMAs. Minimising the cost function (15) subject to hydraulic consistency has almost no in¯uence on the total ¯ow into the DMAs. By using demand indicators, the possible feedback loop between the successive use of pseudo-measurements and total DMA demands has been broken.

Thus the Lagrangian for the DMA-structured stateestimation problem becomes L ˆ …z ÿ g…x; y††T W…z ÿ g…x; y†† ‡ xT Qx ‡ kT F…x; y†; …16† where k 2 Rl is the vector of Lagrangian multipliers. The ®rst-order conditions are    T oL Qx T ˆ ÿ2J W…z ÿ g…x; y†† ‡ 2 ‡ KT k ˆ 0; 0 o…x; y† …17† where J and K are Jacobians of g and F, respectively. Hence ÿ1

k ˆ 2…KTy † JTy W…z ÿ g…x; y††;

…18†

which gives  T ÿ1 Jx ÿ Jy …Ky † Kx W…z ÿ g…x; y†† ÿ Qx ˆ 0:

…19†

7. The Newton step Let G…x; y† ˆ 0 comprise the total system of nonlinear equations for which the solution is sought, that is   ÿ1 T …J x ÿ Jy …Ky † Kx † W…z ÿ g…x; y†† ÿ Qx ˆ 0: G…x; y† ˆ F…x; y† …20† This leads to the following Newton±Raphson iteration scheme:  ÿ1   oG…x; y† Dx G…x; y†: …21† ˆ ÿa Dy o…x; y† The factor a serves to stabilise the convergence of scheme, 0 < a 6 1. The maximum factor depends on the various round-o€ errors of the many steps involved in a compact numerical scheme for the solution of (21), hence this factor can only be found by experimentation. 8. 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 (¯ow meters) situated at the corners of the grid. The supply is sourced from four ®xed head nodes (a, b, c, d) having identical real heights of 100 m. Four pressure transducers placed

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127

Table 1 Estimated demands

Fig. 2. A symmetric grid network.

at the nodes (a, b, c, d) measure these heights. All the pipes in this network are assumed to have identical parameters as follows: length ˆ 300 m, diameter ˆ 150 mm, and the Hazen±Williams coecient equal to 100. The ``external'' information about the network comprises the four ¯ow and pressure measurements at the supply pipes together with the proportional demand indicators, these are assumed to be identical. The ``internal'' information is the ``true'' or ``real'' state of the network given by the real nodal demands. The ¯ow measurements for the example are derived by network simulation from the real nodal demands. If all real demands were identical, then the problem of estimating the state from the external information becomes trivial. In the following tests, the real demand of a single node is changed considerably relative to the other nodes. Such a change in demand could be due to leakage; the identi®cation of such a node would be of considerable interest. For the leakage con®guration tests, all the real demands are set to 1 l/s except for the single leak node, here the real demand is set to 10 l/s. Due to the symmetry of the network there are essentially only six di€erent leakage con®gurations with a single leak node. The nodes 1, 2, 3, 21, 22, 33 are used as representative nodes, these nodes are circled (Fig. 2). It is easily veri®ed that a single node leakage con®guration using any other node in the network will produce identical results as for one of the representative nodes. By varying the ``leak node'', the nodal demands in Table 1 were estimated using the weights: …0:2 l=s†ÿ2 for the ¯ow measurements, …0:1 m†ÿ2 for the head measurements and …10 l=s†ÿ2 for the demand factors. Further decreasing the weights for the demand factors has little in¯uence on the result; this

Leak node (10 l/s)

Estimated demand l/s at leak node

Estimated demand (l/s) at node 1

Total estimated demand (l/s)

1 2 3 21 22 33

4.43 1.49 1.29 1.33 1.27 1.25

4.43 2.19 1.66 1.85 1.53 1.37

45.00 45.00 45.00 45.00 45.00 45.00

state estimator produces an asymptotic solution for the demand factor weights approaching zero. The estimation results show that there is some considerable smoothing of information towards the centre of the grid. At node 33 the information of the leak ¯ow is lost, the demand value of 1.25 l/s coincides with the average demand. Furthermore, for leak nodes in the upper leaf quarter of the network, the corner node (node 1) always shows the largest estimated demand. The state estimator shows elevated demands at the leak nodes, but does not identify the leak nodes particularly well. However, a shortlist of potential leak nodes may be gathered for further examination. The response of the state estimator, however, is as expected; the estimator is minimising the statistical error over a large range of possible realisations. The only information indicating a non-uniform demand distribution is the four ¯ow measurements of the supply ¯ows. Notice also that the total estimated demand is conserved at 45.00 l/s in all the experiments. 9. 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 pro®le according to the demand indicators and that a single node causes this departure. The suspicion of leakage may be derived from increased water usage or night ¯ows. In any case a leakage detection test could also be performed just as a screening procedure, the performance of the test would then depend on the degree of validity of this additional mathematical assumption. For the leakage detection scheme, a potential leak node (test node) is selected for the state-estimation. The state estimator must ignore the cost of accepting a possible excessive demand for the test node; hence the

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J.H. Andersen, R.S. Powell / Urban Water 2 (2000) 123±130

minimisation cost function is altered such that the test node only in¯uences the cost of the ®t to the real measurement data. In Eq. (16), the cost function can be split into three parts: L ˆ …Dz†T W…Dz† ‡ xT Qx ‡ kT F: …I†

…II†

…III†

…22†

The ®rst term is the WLS ®t to the real ¯ow and pressure transducers, the second term is the intrinsic WLS ®t to the redistributed demands as in Eq. (15). The third term is the Lagrangian cost of violating the constraint F…x; y† ˆ 0; this term converges to zero as the solution is approached. The changes for the leakage detection a€ects the second term; the Q matrix is changed in a consistent way such that the in¯uence of the test node is removed while the term still measures departures from the redistributed demands. (The demands are redistributed from the total demand less the leak demand). This is achieved by removing the column of the matrix …RS ÿ I† and the row for the transpose in Eq. (15) corresponding to the test node. Furthermore, the corresponding weight for the test node is set to zero. To test this idea, a real demand of 10 l/s was speci®ed for the leak node in the six cases as examined above. In each of these six representative cases, the state-estimation was performed for each of the 36 network nodes in turn as a test node. In this experiment, the weights used were …0:2 l=s†ÿ2 for the ¯ow measurements and …1 l=s†ÿ2 for the demand factors. A larger weight was given to the demand factors in this experiment as this does not e€ect the deviation for the test node. The resulting sum of squares ®t to the ¯ow measurements are shown in Fig. 3(a)±(f). The contributions from the real measurements in the WLS objective function are shown in Fig. 3. In each of the six cases the leak node is uniquely identi®ed by this measure. As discussed above, the six cases are representative for all the cases of a single node leakage due to the symmetric con®guration of the network. Furthermore, the signal is very strong in all cases. Leak node 3 is identi®ed by test node 3, but the signal for test node 22 comes in second place. These nodes are neighbours and located such that their relative hydraulic distances from the ¯ow meters are relatively similar. It is possible to imagine network con®gurations where two nodes are located at equal hydraulic distance from the transducers, in which case the nodes becomes indistinguishable for the measurements. The signal for node 33 is relatively weaker but the node is still clearly identi®ed. Another possibility is to look at the estimated demands for all the cases in the experiment. Where the test node coincides with the leak node, a generally accurate distribution of demand was found. When the leak node and the test node were di€erent, the picture was much

Fig. 3. Total sum of squares ®t of ¯ow and head measurements for leak node tests.

less clear; large ¯uctuations occurred. For this experiment, it was not possible to identify a particular node as the unique leak node by just looking at the demand estimates.

10. 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 insucient meter coverage is

J.H. Andersen, R.S. Powell / Urban Water 2 (2000) 123±130

bound to dilute the leakage signal. Some of the expected e€ects of the departures from ideal circumstances are examined in this section. Symmetry, as seen in the grid network, may not be a common feature in real water networks. The symmetric grid network was chosen here to reduce the number of essentially di€erent cases of single node leakage con®gurations. The grid represents a single DMA, measured at the boundary. Water is supplied deep into the interior and there is no through ¯ow across the DMA. Furthermore, as a result of the symmetry, there are nodes with approximately equal hydraulic distances from some of the measured supplies. These features often make both state-estimation and leakage detection problems more dicult to solve than for an inhomogeneous water network. The only measured variables which are dependent on the location of the leak node, are the four supply ¯ows, the real ¯ow values are shown in Table 2 for the six di€erent leakage scenarios. The simulated measurement for the idealised example is assumed noise free; it is clear that a ¯ow measurement error of 0.2 l/s would be inadequate for resolving the supply ¯ows in the leakage detection examples. Likewise, the height measurement errors of 0.1 m at the four measurement locations (a, b, c, d) are also inadequate; these height errors corresponds to approximate ¯ow errors of 1.5 l/s at the supply links. However, both height and ¯ow data are required since the demand indicators are only proportional. In the theory of maximum likelihood estimation, the optimal weights for the WLS are related to the statistical measurement errors by: W ˆ

1 : r2

…23†

As stated earlier, the weights used for the real measurements are in accordance with this relationship for a statistical errors of rflow ˆ 0:2 l/s and rhead ˆ 0:1 m. The weights for the demand factors, as in (15), do not necessarily conform to this model as the factors are not measurements in an ordinary sense; very large weights leads to a strict distribution of the measured supply ¯ows proportional to the demand factors. On the other hand, very small weights for the demand factors lead to Table 2 Supply ¯ows for the six leakage scenarios Leak node

qa

qb

qc

qd

1 2 3 21 22 33

12.47 11.58 11.39 11.46 11.35 11.29

10.86 11.18 11.29 11.20 11.28 11.25

10.81 11.10 11.15 11.14 11.18 11.21

10.86 11.14 11.17 11.20 11.20 11.25

129

a ®nite asymptotic solution, although in practice some numerical diculties may arise in this case. Nevertheless, asymptotic solutions have been found for some simple network examples. The existence of an asymptotic solution is interesting; since the state-estimation equations for fewer measurements than state variables cannot be solved in principle, yet a solution may be found if the demand factors are included with weights approaching zero. The interpretation of such a solution would the most extreme departure from strict demand factor distribution, the real distribution is found somewhere between the two extremes. Thus reducing the weights on the demand factors beyond a certain limit has relatively little e€ect on the state-estimation. The state-estimation is not performed for nodes with zero demand factors; such nodes are not included as state variables. In real networks, the demand indicators and therefore the demand factors may vary according to consumer type. If the local demands are generated by random unit events, then the statistical variance of the departure from the mean can be expected to be proportional to the mean. Thus the statistical error of the demand factors is assumed proportional to the square root of the demand factors and hence the weights become proportional to the inverse demand factors. A simple test of this model was performed using three levels of demand factors as shown in Table 3. The three node ranges are divided according to closeness to the centre of the grid, see also Fig. 2. As before, this con®guration conserves the symmetry of the state-estimation results such that only six single node leakage con®gurations need to be considered. The leakage detection scans were performed for the six representative cases. The resulting WLS ®t to the real measurements identi®es the leak nodes uniquely except for node 33. In this case, the large demand factor for the node and its neighbouring nodes obscures the identi®cation since the leak of 10 l/s is only twice the real demand. In their paper on WLS parameter estimation, Niranjan Reddy et al. (1996) have investigated the e€ects using di€erent approaches for deriving the weights. They recommend the standard form (23) where the statistical error is known, but a re-weighted scheme for problems with noisy or bad measurements. However, the implication of choosing a scheme for deriving the weights is a subject in its own right, see e.g. Powell et al. (1988). Table 3 Experiment with variable demand factors Node range Demand indicator Demand factor Real demand (l/s)

1±20 200 0.0236 1.00

21±32 40 0.0047 0.20

33±36 1000 0.1179 5.00

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J.H. Andersen, R.S. Powell / Urban Water 2 (2000) 123±130

All the state-estimation runs were initialised from a ¯at demand distribution: x ˆ 1 and zero chord ¯ows y ˆ 0. An initial state with positive demands is likely to produce a non-zero ¯ow in most, but not necessary all the links in a network. A potential problem comes from the hydraulic head-loss/¯ow relationship. It is well known that the derivative of this function for any realistic hydraulic model is zero for zero ¯ow. In the loop equation framework, however, this only poses a problem if the sum of head-losses around a particular loop becomes insensitive to a change of ¯ow in that loop. Apart from exceptionally contrived cases, this will only happen if the ¯ows in all the links of the loop are zero. However, this could well happen if such a loop is situated in an area of the network with zero demand. The software automatically detects the zero ¯ow condition for a loop or a group of interconnected loops; such loops are temporarily excluded from the solution process. A partial solution for this excluded part of the network is trivially obtained. The starting method is similar to the method used in Andersen and Powell (1999a). The Newton±Raphson iteration steps (21) sometimes require scaling by a factor a in order to stabilise the convergence of scheme. This factor was found by experimentation such that the stopping criterion of MAX…kDxkMAX ; kDykMAX † < 10ÿ10 l=s

…24†

could be reached. Some diculties in convergence were experienced in isolated cases of the leakage detection state-estimation experiments. To run all cases uniformly, the factor a ˆ 0:1 was used. Furthermore, the condition of zero weight for the test node in the leakage detection runs was relaxed; the ordinary weight derived from the demand factors for the test node was scaled down by a factor of 0.01. 11. 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 identi®cation in the case of no network uncertainties and measurement errors has been demon-

strated for an idealised network example. Nevertheless, the example captures many typical features of a DMA with multiple metered supplies at the boundaries. In practical circumstances, network uncertainties and measurement errors are bound to obscure the signal, but the method may pinpoint demand inconsistencies and as a result, indicate a ranked shortlist may of possible leakage problem areas. The paper provides the mathematical tools for further case studies of practical demand monitoring and leakage detection. References Andersen, J. H., & Powell, R. S. (1999a). Simulation of water networking containing controlling elements. Journal of Water Resources Planning and Management, ASCE, 125(3), 162±169. Andersen, J. H., & Powell, R. S. (1999b). A loop based simulator and implicit state-estimator. In B. Coulbeck, & B. Ulanicky (Series Ed.) & D. A. Savic & G. A. Walters (Eds.), Water industry systems: modelling and optimisation applications (Vol. 1, pp. 75±87). Research Studies Press. Bargiela, A. B. (1984). On-line monitoring of water distribution networks. Ph.D. Thesis, University of Durham. Box, M. J. (1965). A new method of constrained optimization and a comparison with other methods. Computer Journal, 8, 45±52. Brdys, M. A., & Ulanicki, B. (1994). Operational control of water systems: structures, algorithms and applications. UK: Prentice Hall. Greco, M., & Del Guidice, G. (1999). New approach to water distribution network calibration. Journal of Hydraulic Engineering, ASCE, 125(8), 849±854. Hartley, J., & Bargiela, A. (1993). Utilization of smoothing formulae for describiong hydraulic relationships in water networks. In B. Coulbeck (Ed.), Integrated computer applications in water supply (Vol. 1, pp. 35±48). Research Studies Press. Lansey, K. E., & Basnet, C. (1991). Parameter estimation for water distribution networks. Journal of Water Resources Planning and Management, ASCE, 117(1), 126±144. Lingireddy, S., & Ormsbee, L. E. (1999). Optimal network calibration model based on genetic algorithm. In Proceedings of the 26th water resources management and planning conference, Tempe, Az, USA. Niranjan Reddy, P. V., Sridharan, K., & Rao, P. V. (1996). WLS method for parameter estimation in water distribution networks. Journal of Water Resources Planning and Management, ASCE, 122(3), 157±644. Ormsbee, L. E. (1989). Implicit network calibration. Journal of Water Resources Planning and Management, ASCE, 115(2), 243±257. Powell, R. S., Irving, M. R., & Sterling, M. J. H. (1988). A comparison of three real-time state-estimation methods for on-line monitoring of water distribution systems. In D. Coulbeck (Ed.), Computer applications in water supply (Vol. 1, pp. 333±348). Research Studies Press. Powell, R. S., Andersen, J. H., Nagar, A. K., & Hindi, K. H. (1999). DMA-structured state-estimation and structured uncertainty analysis for water distribution networks. In Monitoring, modelling & leakage management in water distribution networks. EPSRC-UKWIR Project, Proceedings of Workshop 1, Exeter, UK.