Hydraulic state estimation in water distribution networks is the task of estimating water flows and pressures in the pipes and nodes of the network based on some sensor measurements. This requires a model of the network as well as knowledge of demand outflow and tank water levels. Due to modeling and measurement uncertainty, standard state estimation may result in inaccurate hydraulic estimates without any measure of the estimation error. This paper describes a methodology for generating hydraulic state bounding estimates based on interval bounds on the parametric and measurement uncertainties. The estimation error bounds provided by this method can be applied to determine the existence of unaccounted-for water in water distribution networks. As a case study, the method is applied to a modified transport network in Cyprus, using actual data in real time.

Hydraulic state estimation in water distribution networks (WDNs) is a
challenging task due to the presence of modeling uncertainties, such as
structural uncertainty introduced by skeletonization of the network,
parameter uncertainty of pipe roughness coefficients and uncertainty in water
demands. While this last uncertainty can be reduced by the use of real-time
flow measurements, these measurements come with their own instrument
uncertainties and noise

In standard state estimation techniques, statistical characterization of
sensor measurement error is needed to give more weight to measurements
originating from more accurate sensors. Using the weighted least squares
method, the nodal demands are adjusted to fit the constraints imposed by the
measurements and produce the most probable state estimate

Most point state estimation methods assume a known statistical characterization of the measurement error.
This could lead to significant estimation errors, especially in the case when pseudo-measurements are used, which are estimates determined from population densities and historical data.
The use of pseudo-measurements may be necessary when there are not enough sensors to guarantee the observability of the network.
In this case, no measure of the estimation error is available.
Additionally, in order for point state estimation methods to produce feasible solutions, model calibration is required a priori or during state estimation

An alternative approach for the representation of measurement and model parameter uncertainty is the use of bounds.
In contrast to traditional point state estimation methods, the use of bounding uncertainty
can provide upper and lower bounds on the state variables.
This method is referred to as

The use of measurement bounds for the representation of measurement
uncertainty and their incorporation into the state estimation cost function
was introduced by

In many applications, such as leakage detection and contamination detection,
the derivation of a range of possible values for the state of the WDN
provides useful information for event and fault-detection methodologies.
Hydraulic state bounds can be used to generate bounds on chlorine
concentration in the water network or other chemicals in the water, by taking
into consideration the uncertainty in decay rate

The paper is organized as follows:
Sect.

A water transport network is modeled using a directed graph, for which nodes
represent water sources, junctions of pipes and water demand locations, and
the links represent pipes. Each pipe is indicated by the index

Modeling uncertainty in a WDN is considered in this work to arise from
insufficient knowledge of pipe parameters. The uncertain parameters are
represented using intervals, with the actual value of the parameter being
within a corresponding interval. For notational convenience, the parameters
representing intervals will be denoted with a tilde.
Any uncertain parameters in pipe

Nodes are indicated by the index

The unknown state vector of the WDN is denoted by

Equation (

A diagram illustrating how the IHISE algorithm works in a real-time framework.

Figure

The first step of the IHISE algorithm is to impose initial bounds on the state vector

In the second step, the nonlinear terms present in Eq. (

To get an interval solution of the whole state vector

This study uses data from a real water transport sub-network in Cyprus. A modified version of the
transport network is used, of which an illustrative diagram is shown in
Fig.

Illustrative diagram of the water transport network of this case study.

The implementation of this case study in real time is based on a platform for real-time monitoring of WDN against hydraulic and quality events. A model of the transport network was created as an EPANET input file. Using the platform, one can select the dates with available sensor data and request a state estimation. The available measurements from demand nodes and the level of the tank are then retrieved and a data validation process takes place which replaces missing data.

Sensor measurements have an uncertainty which is defined by the installed
sensor's specifications. The measurements given by the flow sensors are
within

Using the IHISE algorithm, bounds on water flows and pressures in the network
are generated using the flow measurements at demand nodes and the tank level
measurements, by taking into account measurement and modeling uncertainty.
The algorithm needs approximately

State estimate (black line) and bounds on this estimate using the IHISE algorithm
(blue area) for the water flow in pipe

A common practice in water utilities is to use mass balance to determine
whether there is unaccounted-for water exiting the network. In this case
study, since there is no sensor measuring the tank outflow

Using data from the case study network, the two tank outflow estimates were
calculated for a period of 2 days, from “24 August 2016 23:10” to
“26 August 2016 23:10”. The two estimates are compared in
Fig.

Using the IHISE algorithm and the given model and measurement uncertainties, bounds on these same estimates can be calculated:
the bounds on tank outflow by simulating the network using the network outflow and tank level measurements are indicated by

In this section the potential of the IHISE algorithm to be used for event detection in water distribution systems is demonstrated.
An artificial leakage is added to the network model, an approximate location of which is indicated in Fig.

In order to determine the location of the leak, additional measurements should be available. Assuming the existence of pressure sensors in the network, a comparison of the measured pressure with the estimated pressure could indicate the presence of a leak. However, in this case, the measurements are affected by not only the sensor uncertainty (as when calculating mass balance), but also by the network modeling uncertainty, which may greatly affect the pressure estimates. Using the IHISE algorithm, the effect of both measurement and modeling uncertainties is considered in calculating the bounding estimates, and the existence of a leak can be determined with greater certainty.

We assume the existence of a pressure sensor at node

The effect of a leakage occurring in the network at time “26 August 2016 00:10” on a pressure

In this work we described a methodology for real-time hydraulic interval
state estimation, to monitor water transport networks. Using real-time
uncertain measurements from a real transport network, the proposed

The EPANET file for the network depicted in Fig. 2, with realistic water demands,
can be found in

The authors declare that they have no conflict of interest.

This research is funded by the European 80 Research Council under ERC Advanced Grant ERC-2011-AdG-291508 (FAULT-ADAPTIVE) and the European Union Horizon 2020 programme under grant agreement no. 739551 (KIOS CoE). Edited by: Ran Shang Reviewed by: two anonymous referees