Introduction
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
. Another approach is the Kalman filter (KF) method which
provides a solution for the network state based on the available
measurements. The standard KF performs poorly in nonlinear looped WDN due to
the use of a linearized system model . Overall, the above
methods generate a point in state space and are referred to as point state estimation .
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 interval state estimation.
In this work a hydraulic interval state estimation methodology is described and its use
is demonstrated with a case study of a modified transport network of a water utility in Cyprus.
An application of this method for determining the existence of unaccounted-for water in the network is presented.
The use of measurement bounds for the representation of measurement
uncertainty and their incorporation into the state estimation cost function
was introduced by . Interval state estimation was
developed by as the so-called set-bounded state estimation
problem. An implicit state estimation technique for leakage detection for an
idealized grid network under steady conditions was presented by
. A straightforward method for interval state estimation
is the use of Monte Carlo simulations (MCS), which under some assumptions
converge to the true uncertainty bounds by randomly generating and evaluating
a large number of parameter sets or realizations . The
interval-based approach used in this paper has the advantage of calculating
algorithmically the bounded state estimates in a way that guarantees the
inclusion of the true state. MCS, even with a large number of simulations,
cannot guarantee that all possible cases will be simulated. The applicability
of the proposed algorithm is thus suitable for event and fault-detection
methodologies that require strict bounds on state estimates.
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 . When
additionally this bounded estimate is generated in real time, it helps to
reduce the time of detecting water leakages and prevent catastrophic
scenarios such as water contamination.
The paper is organized as follows:
Sect. formulates the problem of hydraulic state estimation and describes a methodology to solve this problem based on the Iterative Hydraulic Interval State Estimation (IHISE) algorithm.
In Sect. , a case study is presented in which this method is applied to a modified real transport network.
Finally, we discuss the application of this method for determining the existence of unaccounted-for water in the network.
Hydraulic interval state estimation
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 j, where j∈{1,..,np} and np is the number of pipes. These are
characterized by pipe length, diameter and roughness coefficient, parameters
which are generally assumed known. Pipe parameters are used to compute the
Hazen–Williams (H–W) resistance coefficients rj, which are in turn used
to formulate the energy conservation equations of a water network
.
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 j will be included
in r̃j∈rjl,rju. The
interval parameter rj̃ is the uncertain H–W coefficient for pipe
j, with rjl and rju being the lower and upper bounds
of each coefficient, respectively.
Nodes are indicated by the index i, where i∈{1,..,nu} and
nu is the number of nodes with an unknown head, thus excluding the
nodes that represent water sources. In this work we consider water transport
networks in which sensors measure all the water demands at nodes, which,
typically, are the inflows of district metered areas (DMAs). Measurements
arrive at a fixed time interval from sensors that may not be accurate, and
each measurement is associated with a certain measurement error. The
uncertainty of each measurement is given as a percent error of the
measurement, and it is modeled as an interval with the measurement being the
mean value of the interval. Measured water demand at node i is then given
by the interval q̃ext,i=qext,il,qext,iu, where
qext,il is the lower bound on water demand and
qext,iu is the upper bound.
The unknown state vector of the WDN is denoted by x=[q⊤h⊤]⊤∈Rn, where h∈Rnu are the
unknown heads at nodes, q∈Rnp are the water
flows in pipes and n=np+nu. These are computed by
formulating the conservation of energy and mass equations, as formulated by
. The matrix formulation for a general looped water
distribution system, which also includes the uncertain parameters and
variables as intervals (denoted with a tilde), is given by
Ã11(q̃)A12A210q̃h̃=-hextq̃ext,
where Ã11(q̃)∈Rnp×np is a diagonal matrix containing the nonlinear terms
r̃j|q̃j|ν-1, A12=A21⊤∈Rnp×nu is the incidence flow matrix
that indicates the connectivity of nodes with links, ν=1.852 is a
constant associated with the H–W coefficient and hext∈Rnp is a vector that contains the known heads in each
equation. For simplicity, we assume that measurements of the tank levels are
available; thus, hext is known.
Equation () represents a system of nonlinear
equations, which include interval parameters, and it is referred to in the
literature as a nonlinear interval parametric (NIP) problem
. The objective is to find the smallest interval state
vector x̃=[q̃⊤h̃⊤]⊤ that contains all the
solutions of this system of equations for every value contained in the
interval parameters. To solve the NIP problem given in
Eq. (), an algorithmic technique named
iterative hydraulic interval state estimation (IHISE) was developed
by the authors. The IHISE method comprises five steps: (1) find initial
bounds on the state vector x. (2) Use interval linearization to remove
nonlinear terms from Eq. () and transform them into a
system of linear interval parametric (LIP) equations. (3) Formulate a linear
program (LP) using the system of LIP equations. (4) Solve the LIP problem.
(5) Iteratively tighten the bounds on x and approximate the solution of the
NIP problem.
A diagram illustrating how the IHISE algorithm works in a real-time framework.
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Figure illustrates how this technique is implemented
in a real-time framework. At discrete time instant k, the measurements from
the sensors in the network are received, which include the water outflow
qext(k) and the water level in tanks. The measured tank level at
each time instant is used to calculate the known head vector
hext(k) of the network. Since these equations only depend on the
current time instant k, the discrete time notation is omitted. The
uncertainty of these measurements is inserted by converting them into
intervals with the measurement as the mean value. The hydraulic equations of
Eq. () are then formulated using the new measurements.
Modeling uncertainty is represented by including the interval parameters
r̃j in the equations.
The first step of the IHISE algorithm is to impose initial bounds on the state vector x.
The initial bounds should be an outer interval solution of Eq. () .
An outer interval solution includes all the point solutions of Eq. (), but it is not the smallest possible interval.
Bounds on the unknown head vector h can be chosen using physical properties of the network such as the minimum head of each node and the maximum head that pumps and water sources can add to the network.
After finding an initial interval for the unknown heads h̃(0), the special structure of Eq. (),
in which each equation contains only one flow state qi, allows us to use h̃(0) and interval arithmetic to find the initial bounds on the flows q̃(0).
In the second step, the nonlinear terms present in Eq. () need to be linearized in
order for the system of equations to be transformed into a LIP problem and solved .
This is achieved using interval linearization .
Given a range of values for the state x̃ in which interval linearization will be performed, each
of the nonlinear functions is enclosed between two lines and an interval term represents the linearization uncertainty.
In the third step, the LIP equations are formulated into a LP with constraints.
The interval terms in these equations are transformed into constraints of the LP and a suitable cost
function ensures that the solution of this problem will give either the minimum or maximum of a certain state.
To get an interval solution of the whole state vector x, in step four, the
LP formulated is solved for all the states by changing the cost function. At
the end of this step, an interval solution for the linearized system of
equations is derived. The new bounds on state vector x are then checked for
convergence in step five. The criterion for convergence is that the relative
change in bounds e(m) at iteration m must be smaller than a specified
small number ϵ, where ϵ defines the largest allowable
absolute error for the calculated bounds of each state, e.g., for flow states
ϵ=0.01 (m3 h-1). The relative change in bounds e(m)
is computed as follows:
e(m)=xu(m)-xl(m)-xu(m-1)-xl(m-1)1,
where xu and xl indicate the upper and lower bounds of
the state x, respectively. The algorithm then gives the final state bounds
calculated as the result. Otherwise, the new bounds are used as initial
bounds and the algorithm re-iterates from step two.
Case study
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. . The modified network contains three loops and
comprises 9 demand nodes, one water tank and 12 links which represent
pipes. Flow sensors (F) are installed at demand nodes, which represent
aggregated real measurements at entrances to DMAs, and a water level sensor
(L) is installed in the tank. Sensor measurements arrive at fixed 5 min
intervals. The tank's water input originates from four water sources, of
which three are water dams and one is a desalination unit. The water inflow
q0 coming from these sources is measured with a flow sensor. The water
outflow q1 of the tank is not directly measured.
Illustrative diagram of the water transport network of this case study.
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Real-time hydraulic interval state estimation
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 ±2 % of the actual flow at those locations. Modeling uncertainty
is also present in the form of pipe parameter uncertainty. For this case
study we assumed a total uncertainty of ±2 % in the Hazen–Williams
coefficient. The value of this uncertainty may vary, as it is calculated
using expert-elicited bounds on the modeled pipe parameters. It is assumed
that the network graph is known, and thus structural uncertainty is
neglected. This is a valid assumption in transport networks where the
structure is known, as it is the network in this case study.
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 4 s to calculate bounds for each
hydraulic step, on a personal computer with Intel Core i5-2400 CPU at
3.10 GHz. The bounds converge after eight iterations. The size of bounds
does not increase over time because it depends only on the measurements of
that specific time step. The effect of accumulating uncertainty due to the
dynamic calculation of tank levels does not affect the size of the bounds in
this case study, because the tank level is measurable. For illustration
purposes, flow and pressure estimates using a real-time EPANET-based state
estimator are also generated. The state estimates for a selected pipe and
node, accompanied by its corresponding uncertainty bounds generated by the
IHISE algorithm, are shown in Fig. .
State estimate (black line) and bounds on this estimate using the IHISE algorithm
(blue area) for the water flow in pipe 3 (a) and the head at node 9 (b).
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Determining the existence of unaccounted-for water using bounds on state estimates
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 q1, mass
balance can be checked by generating an estimate of q1 using two different
sets of data: the first is by calculating the sum of all the measured outflow
(demands), indicated here by q1a(k); the second is the calculation of
the tank outflow using the measured tank inflow q0(k) and tank water
level measurement ht(k), indicated here by q1b(k), as
follows:
q1b(k)=q0(k)-(αt/Δt)Δht(k),Δht(k)=ht(k)-ht(k-1),
where αt is the base area of the tank and Δt=5 min is the measurement time step.
(a) Comparison of two estimates of the tank outflow,
q1a(k) and q1b(k). (b) Comparison of the uncertainty
bounds generated by the IHISE algorithm for the same estimates.
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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. a. It can be observed that the two estimates have a
non-zero difference at almost all time steps. This can be due to noisy data,
and thus it cannot be determined with certainty whether there is
unaccounted-for water. A way to deal with this is to calculate the average
difference ed between these data for the given period of time,
i.e., ed = mean(q1a(k)-q1b(k)),∀k∈{1…ks}, where ks is the total number of
measurements from each sensor. This calculation gives a constant
unaccounted-for flow ed=18.82 (m3 h-1), which may be due to background leakages, or it
may be due to non-uniformly distributed measurement errors. Checking the
water utility leakage reports of the examined period, there was no recorded
leakage for the sub-network of this study.
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 q̃1a(k),
and bounds on tank outflow using the tank inflow and tank level measurements are indicated by q̃1b(k).
The comparison of these two sets of bounds is shown in Fig. b.
The two sets of bounds overlap, indicating that the variance can be explained by measurement and modeling uncertainty.
There are some specific time steps that the bounds do not overlap, which may be due to noisy data that can be eliminated using a suitable data validation strategy.
It can also be observed that bounds generated by the tank level and tank inflow measurements are wider.
This is because these bounds are calculated using the dynamic equation Eq. () which uses three uncertain measurements for the calculation of the bounds.
Determining the existence of an artificial leakage using bounds on state estimates
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. .
The leakage has a magnitude of 20 (m3 h-1) and its time profile is described by an abrupt constant outflow starting at “26 August 2016 00:10”.
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 9 of the network as
shown in Fig. . The pressure sensor reading is compared with
the IHISE bounding estimates, as shown in Fig. a.
The error between the pressure sensor reading and the estimated bounds, which
is defined as the distance of the reading from the bounds when the reading is
outside the bounds, is also calculated, and is shown in
Fig. . It is observed that there is a pressure
sensor reading error after the leakage occurs. The error presents only during
the night hours, when the pressure is higher due to the low demand and thus a
pressure drop due to a leakage is more apparent. Similarly, if we assume the
existence of a flow sensor on pipe 12, the same effect can be observed when
the flow reading is compared with the IHISE bounds, as shown in
Fig. b. There is a flow sensor error during the
night hours, while the error persists in smaller magnitude for the rest of
the day. These observations indicate the existence of the leakage despite the
measurement and modeling uncertainties in the network.
The effect of a leakage occurring in the network at time “26 August 2016 00:10” on a pressure (a) and a flow
(b) state, compared to the estimated uncertainty bounds for the same states calculated by the IHISE algorithm. Below each
graph, the corresponding error of the state compared to the calculated bounds is presented.
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