State estimation is necessary in diagnosing anomalies in Water Demand Systems (WDS). In this paper we present a neural network performing such a task. State estimation is performed by using optimization, which tries to reconcile all the available information. Quantification of the uncertainty of the input data (telemetry measures and demand predictions) can be achieved by means of robust estate estimation. Using a mathematical model of the network, fuzzy estimated states for anomalous states of the network can be obtained. They are used to train a neural network capable of assessing WDS anomalies associated with particular sets of measurements.
Water companies use telemetry systems for control and operation purposes. By considering the data provided by telemetry, the engineer on duty makes operation decisions trying to optimize the system utilization. Nevertheless, the system complexity does not permit but to take a few real-time measures, which only incompletely represent the network state. They give indication of only certain aspects of the system, leaving out other more specific or "less relevant" ones. Thus, suitable techniques that allow for more accurate network health estimation are necessary so that anomalies can be detected more rapidly, and light anomalies, which develop progressively and insidiously, can be identified. This will enable to control their consequences in earlier stages, thus avoiding, among other things, losses of water, which can be of great importance.
The state of a WDS is obtained by interrelating different measures within a mathematical model of the network, [Mar95]. Different tools to analyze water networks have been developed in the last years, SARA [GMF98], and EPANET [Ros97], among others. But state estimation cannot be accurately performed if there are missing or uncertain data. Thus, system operators need error limits for the state variables. Yet, data are abundant since they are permanently received. Therefore, operators cannot evaluate errors easily or in real time. It is expected that suitable techniques borrowed from Artificial Intelligence (AI) could encapsulate the necessary knowledge to assess the network state.
In this paper, we present an approach for the diagnosis and decision making process which is necessary on a neural network for clustering and pattern classification. First, the mathematical model, a state estimation procedure and a mechanism for treating uncertainties, already presented in [Izq04] and [Izq05], are briefly presented. The state estimator, together with the error limits will be used as a surrogate of the real WDS to generate data to train and check the neural network (NN). Then, the inherent procedures to neural techniques will be described. Specifically, the NN architecture, the classification and clustering mechanisms of both, crisp and fuzzy, patterns and the training technique will be presented.
Analyzing pressurized water systems is a complex task, especially for big systems. But even for moderately sized cities, it involves solving a big number of non-linear simultaneous equations. The complete set of equations may be written by using blockmatrix notation,
where A 12 is the so-called connectivity matrix describing the way demand nodes are connected through the lines. Its size is L × N p , N p being the number of demand nodes and L the number of lines; q is the vector of the flow rates through the lines, H the vector of unknown heads at demand nodes; A 10 is an L × N f matrix, N f being the number of fixed-head nodes with known head H f , and Q is the N p -dimensional vector of demands. Finally, A 11 (q) is an L × L diagonal matrix. System (1) is a non-linear problem whose solution is the state vector x = (q, H) t of the system. The non-linear relations describing the system balances are complemented by the specific telemetry measurements. These measurements are integrated into the model by expanding system (1) to a new system, typically overdetermined:
The components A 31 and A 32 in system (2) were introduced to account for additional telemetry measurements M t with uncertainties in the demand predictions. System (2) is usually solved using least-square methods for a state estimation by an over-relaxation iterative process applied to a linearized version of (2):
where A ′ 11 is the Jacobian matrix corresponding to A 11 .
Error limit analysis is a process to determine uncertainty bounds for the state estimation originated by the lack of precision of measurements and data. To put it in a nutshell, the question is what is the reliability of the estimated state x * , if measurement vectors y are not crisp but may vary in some region, [yδy, y + δy]? Different techniques may be used to estimate this unknown but bounded error, [Mil96], [Nor86], [Kur97]. We use a variant of the so-called sensitivity matrix analysis, [Bar03], which uses the state estimator presented above.
In [Izq05], it is proved that a component by component bound, e * , for δx * can be obtained by means of
where W is a diagonal matrix that weights the equations according to the nature of the right-hand sides, and the vertical bars indicate absolute values of all matrix and vector entries. Because of linearity, the bounds calculated by (4) are symmetrical and the error limit may be expressed as a multidimensional interval (see cell definition in next section) [x * ] in the state space
A neural network for clustering and classification is a mechanism for pattern recognition. Here, we use multidimensional cells, [Sim92], [Lik94]. Voronoi diagrams are used in Ref. [Ble97].
A cell C is a region of the pattern
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