1 Introduction

WDNs are critical infrastructures that deliver potable water to consumers. Proper design and operation of these networks is essential to guarantee a reliable and safe water supply, guaranteeing public health and economic growth Adedeji et al. (2018). Hydraulic models are used to simulate and analyze WDNs, and are important tools to assist decisions. Model results can be used to design and operate WDNs, ensuring reliability and efficiency. Traditionally, these models have been built and operated using historical data from sensors. However, the digital transformation of the water sector is promoting Advanced Metering Infrastructure (AMI), such as smart water meters and Supervisory Control and Data Acquisition (SCADA) systems that open the door to real-time modelling of WDNs, which is becoming increasingly important (Grievson et al. 2022). The use of real-time sensor data in models can help capturing the complexity and variability of real-world systems, leading to improved and timely decision-making (Rossman 1993; Antonowicz et al. 2018). Although there are different ways to use real-time data in modelling, the integration of technology and digitalization has given rise to new approaches to updating model states. One of them is Data Assimilation (DA), which has the potential of improving model accuracy in real-time by utilizing long-term measurement data (Hill et al. 2014). DA synthesizes prior knowledge of model states with available measurements to provide an optimized estimate of current model states and reduce uncertainties. However, these measurements can be unstable and contain larger errors. The ability to address measurement errors using calibration methods and efficiently utilizing a large amount of data is challenging Zhou et al. (2018).

The use of Kalman Filters (KF) for WDNs was first introduced by Todini (1999) for calibrating pipe roughness coefficients in WDNs with a simple linear structure. As KF can only be used for linear systems, the Extended Kalman Filter (EKF) was applied by Shang et al. (2006, 2008) to estimate nodal demands in a small hypothetical network by approximating nonlinear systems with tangent linear operators. These studies showed good results with KF and EKF in cases of limited nonlinearity and uncertainty, but their efficacy may be limited in highly looped networks (Van Den Bossche 2013) or the presence of large measurement errors Shang et al. (2006, 2008).

The effectiveness of the EnKF was proven in updating water demands and water demand model parameters for a Water Demand Forecasting Model under the assumption of known pipe roughness values and no leakage in the system (Okeya et al. 2014). They explored the possibility of burst detection using Kalman filtering of flow observations and forecasts from the hydraulic model, and an extension of this study by Okeya et al. (2014) showed that the applied methodology was effective in detecting bursts in real-time and estimating the leak flow. Ruzza (2017) carried out a similar leak detection study in WDNs using KF, EnKF, Ensemble Smoothing, and Normal-Score EnKF to identify nodal leakages. Ensemble-based methods are also effective in providing stable calibration results to ensure the long-term accuracy of models as demonstrated by Zhou et al. (2018, 2022).

EnKF avoids model linearization by simulating model states using an ensemble of parameters derived from Monte Carlo perturbations. Particle Filter (PF), which extends the use of the ensemble to non-Gaussian models and increases the ensemble size, was successfully used by Do et al. (2017a, 2017b) to estimate nodal demand patterns in WDN models using measurements with specific errors. A recent study by Bragalli et al. (2016) tested the use of EnKF in WDNs using an innovative 3-step EnKF for a small WDN which showed promising results for the capabilities of a multi-step DA in WDNs.

EnKF is an ideal and optimal method for applying DA for WDN as EnKF is stable with large nonlinear systems and a low probability of divergence from the true value. The computational demand of EnKF is also lower than PF (Simon 2006; Gillijns et al. 2006; Van Den Bossche 2013).

Despite the previous research efforts, the application of DA techniques in WDNs is still limited. In particular, the extent to which model errors can be reduced under measurement uncertainty is still unknown. Additionally, extended-period simulations have not been carried out for a multi-step DA algorithm. In previous studies, performed by Bragalli et al. (2016) and Okeya et al. (2014), Demand Driven Analysis (DDA) was used.

In this paper, a three-step Ensemble Kalman Filter-based DA for WDNs (3-EnKF-WDN) approach is presented. The approach is innovative as the hydraulic modelling involves extended period simulation and Pressure-Dependant Demand (PDD). The objective is to understand to which extent model errors can be reduced under measurement uncertainty, in particular due to sensor precision and noise, when incrementally assimilating the system states of pressure (step 1), flow (step 2) and demand (step 3). We also propose a new evaluation metric, Combined Total Variance Ratio, to quantify the overall effectiveness of this DA process. Additional analyses include the effect of the number of ensembles in the EnKFs, and the computational demand of 3-EnKF-WDN.

The remainder of the paper presents the methodology section, outlining the approach used in this study. Two case studies are used to demonstrate the application of the proposed DA method. Afterwards, the results are presented and discussed. Conclusions and findings are drawn in the last section.

2 Methodology

The methodology consists of three parts. The first part details the implementation of the improved DA algorithm, which starts with an initialization, and moves incrementally by assimilating pressure, flow and demand data. The second part presents the new evaluation metric, Combined Total Variance Ratio, to quantify the overall effectiveness of the DA process. Finally, the third part includes an experimental setup to evaluate the effect of the measurement uncertainties on the effectiveness of the DA process, the effect of different numbers of ensembles in the EnKF and the computational demand of the DA.

2.1 Three-step Ensemble Kalman Filter-based DA for WDNs (3-EnKF-WDN)

The structure of the 3-EnKF-WDN algorithm is shown in Fig. 1, and further detailed in the sections below.

The multi-step EnKF for WDNs involves initializing the ensemble of state estimates and updating the ensembles with measurements of head, flow, and demands. This process is repeated at each time step of the simulation to estimate the hydraulic state of the network over time. The \(q_j\) after the state symbol refers to the “known” demand which is used to initialize the 3-step DA.

Fig. 1
figure 1

Step-by-step implementation of the 3-Step DA Algorithm

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2.1.1 Initialization

Before proceeding with the 3-steps, it is necessary to generate the initial ensemble of states describing our prior knowledge, using the following procedure:

  1. 1.(a)

    Generate an ensemble of demands (q) with a mean \(\mu _{q_j}\) (base demand of each node) and variance \(\sigma ^2_{q_{j}}\)

  2. (b)

    Using EPANET 2.2 modelling system and the WNTR Python library (Klise et al. 2017b, a) we compute matrices of pressure (\(H_{q_j}\)) and flowrate (\(Q_{q_j}\)) initialized in the network with the ensembles of demands (\(q_{meas}\)) and their averages \(H_{|q_j}, Q_{|q_j}\), with |, denoting the average of the respective state being calculated

  3. (c)

    The number of ensembles “m” must be large enough for the estimated co-variance matrices to be inverted

Once initialised, data assimilation is carried out for up to 3-steps depending on the available type of measurements, as follows.

2.1.2 Step One - Assimilation of Pressure Head

Update the ensemble of state estimates with head measurements by calculating the Kalman gain, assimilating these measurements and estimating the flow and demand, as follows:

  1. 1.(a)

    Calculate the ensemble mean \(\mu _{H}\) and ensemble prior variance of Head \(P_H\), using Eqs. 1 and 2.

    $$\begin{aligned} P_H=\frac{1}{m-1}\sum _{j=1}^{m}\left[ \left( H_{|q_j}-\mu _H\right) \left( H_{|q_j}-\mu _H\right) ^T\right] \end{aligned}$$
    (1)
    $$\begin{aligned} \mu _H=\ \frac{1}{m}\sum _{j=1}^{m}H_{|q_j} \end{aligned}$$
    (2)
  2. (b)

    Calculate the Kalman Gain \(K_H\) for the head using the error in the estimate and the errors in the measurement of the head (Eq. 3)

    $$\begin{aligned} K_H=P_HM_H^T{(M_HP_HM_H^T+R_{Z_H})}^{-1} \end{aligned}$$
    (3)

    where \(R_{z_H}\) is the precision of head sensors and \(v_{z_H}\) is the noise in head sensors.

  3. (c)

    Assimilate the measurements of Head (\(Z_H\)) and update the Head values (\(H_{{q_j}{z_H}}\)), using Eq. 4:

    $$\begin{aligned} H_{|q_jz_H}=H_{|q_j}+K_H(z_H-M_HH_{|q_j}-{vz}_H) \end{aligned}$$
    (4)
  4. (d)

    Estimate Flow \((Q_{{q_j}{z_H}})\) using hydraulic head losses

  5. (e)

    Estimate Demand \((q_{{q_j}{z_H}})\) using the Pipe-Node Incidence Matrix (\(A_{21}\)) as defined by Todini and Pilati (1988), Eq. 5.

    $$\begin{aligned} q_{|q_jz_H}=A_{21}Q_{|q_jz_H} \end{aligned}$$
    (5)

2.1.3 Step Two - Assimilation of Flow

Update the ensemble of state estimates with measurements of flow by assimilating the measurements, estimating the head and demand, and calculating the Kalman gain.

  1. 1.(a)

    Calculate the ensemble mean \(\mu _Q\) and ensemble prior variance of the Flow \(P_Q\). Where;

    $$\begin{aligned} P_Q=\frac{1}{m-1}\sum _{j=1}^{m}\left[ \left( Q_{|q_jz_H}-\mu _Q\right) \left( Q_{|q_jz_H}-\mu _Q\right) ^T\right] \end{aligned}$$
    (6)
    $$\begin{aligned} \mu _Q=\ \frac{1}{m}\sum _{j=1}^{m}Q_{|q_jz_H} \end{aligned}$$
    (7)
  2. (b)

    Calculating the Kalman Gain \(K_F\) for flow using the error in the estimate and the errors in the measurement of flow (Precision of flow sensors; \(R_{zQ}\), noise in flow sensors; \(v_{zQ}\))

    $$\begin{aligned} K_Q=P_QM_Q^T{(M_QP_QM_Q^T+R_{Z_Q})}^{-1} \end{aligned}$$
    (8)
  3. (c)

    Assimilate the measurements of Flow (\(Z_Q\)) and update the Flow values (\(Q_{q_jz_Hz_Q}\))

    $$\begin{aligned} Q_{|q_jz_Hz_Q}=Q_{|q_jz_H}+K_Q(z_Q-M_QQ_{|q_jz_H}-{vz}_Q) \end{aligned}$$
    (9)
  4. (d)

    Estimation of Demand (\(q_{q_jz_Hz_Q}\)) using Pipe-Node Incidence Matrix(A21 )

    $$\begin{aligned} q_{|q_jz_Hz_Q}=A_{21}Q_{|q_jz_Hz_Q} \end{aligned}$$
    (10)
  5. (e)

    Estimation of Head (\(H_{q_jz_Hz_Q}\)) using hydraulic head losses and Pipe-Node Incidence Matrices (\(A_{11}\), \(A_{12}\) and \(A_{21}\)) as detailed in Bragalli et al. (2016)

2.1.4 Step Three - Assimilation of Demand

Update the ensemble of state estimates with measurements of demands by assimilating the measurements, estimating the flow and head, and calculating the Kalman gain.

  1. 1.(a)

    Calculate the ensemble mean \(\mu '_Q\) and ensemble prior variance of the \(Q_{q_jz_Hz_Q}\)

    $$\begin{aligned} P_Q^\prime =\frac{1}{m-1}\sum _{j=1}^{m}\left[ \left( Q_{|q_jz_Hz_Q}-{\mu \prime }_Q\right) \left( Q_{|q_jz_Hz_Q}-{\mu \prime }_Q\right) ^T\right] \end{aligned}$$
    (11)
    $$\begin{aligned} \mu _Q=\ \frac{1}{m}\sum _{j=1}^{m}Q_{|q_jz_Hz_Q} \end{aligned}$$
    (12)
  2. (b)

    Calculate the Kalman Gain \(K'_Q\) for flow prime using the error in the estimate and the errors in the measurement of demands (Precision of demand sensors; \(R_{z_q}\), noise in demand sensors; \(v_{z_q}\)).

    $$\begin{aligned} {K\prime }_Q={P\prime }_QA_{21}M_q^T{(M_qA_{21}{P\prime }_QM_q^T+R_{Z_q})}^{-1} \end{aligned}$$
    (13)
  3. (c)

    Assimilate the measurements of demands(\(z_q\)) and update flow values (\(Q_{q_jz_Hz_Qz_q}\))

    $$\begin{aligned} Q_{|q_jz_Hz_Qz_q}=Q_{|q_jz_Hz_Q}+{K\prime }_Q(z_q-M_qA_{21}Q_{|q_jz_Hz_Q}-{vz}_q) \end{aligned}$$
    (14)
  4. (d)

    Estimate Demand (\(q_{q_jz_Hz_Qz_q}\)) using Pipe-Node Incidence Matrix(\(A_{21}\))

    $$\begin{aligned} q_{|q_jz_Hz_Qz_q}=A_{21}Q_{|q_jz_Hz_Qz_q} \end{aligned}$$
    (15)
  5. (e)

    Estimate Head (\(H_{q_jz_Hz_Qz_q}\)) using hydraulic head losses and Pipe-Node Incidence Matrices (\(A_{11}\), \(A_{12}\) and \(A_{21}\)) as detailed in Bragalli et al. (2016)

2.2 Evaluation Metric

The effectiveness of the DA can be estimated using the Total Variance (TV), Eq. 16, as suggested by Bragalli et al. (2016).

$$\begin{aligned} TV\{\overline{\otimes }\} = \frac{1}{S} \sum _{i=1}^{S} \left( (\overline{\otimes }_i - \otimes _i^{true}) ^2 + \frac{1}{S} \sum _{i=1}^{m} \left[ \frac{1}{m(m-1)} \sum _{j=1}^{m} \left( \otimes _i^j - \overline{\otimes }_i \right) ^2 \right] \right) \end{aligned}$$
(16)

where TV is the Total Variance, \(\otimes \) is the state variable (either H, Q or q), \(\overline{\otimes }\) is the ensemble mean, S is the number of state variables (i.e., number of nodes or pipes), m is the number of ensembles, i is the iterator for the state variable and j is the iterator for the ensembles.

However, for extended period simulation, we use the daily average TV value, obtained by dividing TV by the number of time steps used for the DA.

$$\begin{aligned} TVR\left\{ \overline{\otimes }\right\} =\frac{TV\left\{ \overline{\otimes }\right\} }{TV\otimes } \end{aligned}$$
(17)

where \(TVR\left\{ \overline{\otimes }\right\} \) is the Total Variance Ratio of the system state, \({TV\left\{ \overline{\otimes }\right\} }\) is the posterior system state assimilation (either 1 step, 2 steps assimilated), and \({TV\otimes }\) is the prior system state \(\otimes \) without the assimilation of measurements from the current step.

To quantify the overall effectiveness of the implemented DA method, the TV values for each system state are normalized to obtain a Total Variance Ratio (TVR), which are averaged to obtain a Combined Total Variance Ratio (CTVR), which indicates the overall effectiveness of all system states (head, demand and flow) of all assimilation steps.

$$\begin{aligned} CTVR=\frac{1}{N}\left[ \frac{1}{t}\ \sum _{i=1}^{N}{\ {TVR{\overline{\otimes }}_H}+\ {TVR{\overline{\otimes }}_Q}+\ {TVR{\overline{\otimes }}_q}}\right] \end{aligned}$$
(18)

where CTVR is the Combined Total Variance Ratio, N is the number of system states assimilated and being combined, \({TVR{\overline{\otimes }}_k}\) is the Total Variance Ratio for System State, and k is either head (H), flow(Q) or demand (q).

2.3 Evaluating the effect of measurement uncertainty

Measurements are always affected by a degree of uncertainty. In the case of WDNs, measurement uncertainty depends on the sensors used for measuring the system’s states. The precision and noise of these sensors are important in determining how well the sensors can capture the true states of the system.

Therefore, it is important to identify the limit of applicability of the proposed 3-step DA algorithm under uncertain observations To this end, we propose a number of experiments to test the effect of uncertainty due to sensor precision and uncertainty due to sensor noise, applied to the measurements of head, flow and demand. On the one hand, to investigate the effect of the uncertainty due to noise, six different levels of noise were applied to each state measurement. The selected noise values were varied using a normal distribution with a \(5\%\) standard deviation. In total 600 simulations were carried out for each sensor type. On the other hand, the effect of the uncertainty due to sensor precision was investigated applying six different precision values for each state of sensor, and for all the possible combinations of sensor precision values.

3 Case Studies

Two networks of different sizes which are representative of real-world WDNs are taken for this study.

The first case study is the Modena network which is the same WDN used by Bragalli et al. (2016); Han et al. (2020); Bhave and Gupta (2006) and in many other similar studies. The network consists of 317 pipes, 268 nodes and 4 reservoirs with a fixed head between 72.0 m and 74.5 m. The network has a total length of 71. 8 km of pipes with diameters between 100 mm and 400 mm. Although the network of Modena is small, the topology and distribution of the network make it suitable for the proposed research as the network is comparable to real world small WDNs as seen in Fig. 2.

Fig. 2
figure 2

Layout of the Modena network

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The second case study is the Five Reservoir network (FiveRes), which is much larger than Modena. The network consists of 1278 pipes, 935 nodes and 5 reservoirs Zheng and Zecchin (2014). The layout of the network is given in Fig. 3. The network has a total length of 253.7 km of pipes with a diameter of 600 mm. The FiveRes network provides a suitable comparison of how the DA algorithm can handle larger and more complex WDN models.

Fig. 3
figure 3

Layout of the FiveRes network

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The monitoring network in Modena is more distributed compared to the FiveRes Network as seen in Fig. 4. The number of sensors is also much less in FiveRes compared to the size of the network as seen in Table 1. Hence, it may not provide a good representation of the hydraulic states within the WDN for FiveRes. As such the experiments for measurement uncertainty were repeated for the FiveRes network with sensors located at all the nodes and links.

Fig. 4
figure 4

Monitoring Networks for Head, Demand and Flow Sensors in Modena (Left) and FiveRes WDN (Right)

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4 Results and Discussion

The methodology in Fig. 1 was applied to the networks of Modena and FiveRes, modifying precision and noise of the measurements and evaluating their effect on the models’ error.

4.1 Uncertainty Due to Noise

In the case of Modena, as seen in Fig. 5 the DA method is most sensitive to noise in the flow measurements, and any noise beyond one litre per second results in the DA algorithm being ineffective, as CTVR exceeds one. The threshold of noise for the head is between 0.1 and 0.2 meters of noise. Noise in the measurement of demand, on the other hand, is resilient to an increase in noise up to 0.5 litres per second. Therefore, in the case of Modena, both flow and head sensors must be calibrated regularly to ensure that their accuracy remains within the effective threshold for DA. However, demand sensors require less maintenance and calibration as they can be effective to a higher threshold of noise compared to the other state measurement sensors.

The results for the FiveRes network show that increasing noise in the measurement of the systems states results in an increase in CTVR. However, analyzing the results for FiveRes in Fig. 6, we observed both head and demand exhibit significant variation in results when a suboptimal monitoring network in the FiveRes WDN is used compared to the fully monitored network as shown in Table 1.

  1. 1.

    Head Sensors:

    1. (a)

      The DA remains effective for the entire range of heads tested based on the minimum CTVR

    2. (b)

      CTVR exceeds one even at the lowest noise levels, as indicated by the maximum CTVR

    3. (c)

      When the network is fully monitored, noise is not acceptable in head sensors for successful DA.

  2. 2.

    Demand Sensors:

    1. (a)

      Demand sensors become completely ineffective beyond a noise level of 0.8 litres per second.

    2. (b)

      Demand sensors show the maximum CTVR exceeding the threshold of one even at very low noise levels.

    3. (c)

      When the network is fully monitored, the DA becomes less dependent on demand sensors and the noise in demand sensors does not have a significant impact on the effectiveness of the DA

  3. 3.

    Flow Sensors:

    1. (a)

      The DA remains effective until approximately 2 lps based on the minimum CTVR.

    2. (b)

      The maximum CTVR shows that the DA is not effective at all ranges similar to head and demand.

    3. (c)

      When the network is fully monitored, noise is not acceptable in flow sensors for successful DA. The results consistently exceed the CTVR threshold of one for all noise levels beyond zero.

Table 1 Configuration of sensors in the tested case studies
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Fig. 5
figure 5

CTVR against Noise in state measurements. Head (Left), Demand (Middle), Flow (Right), - for Modena Network

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Fig. 6
figure 6

CTVR against Noise in state measurements. Head (Top), Demand (Middle), Flow (Bottom), - for FiveRes (left) and Fully Monitored FiveRes (Right)

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In general, it is observed that an increase in the noise in state measurements results in reduced effectiveness of the 3-EnKF-WDN, as the CTVR increases when noise increases. Recall that \(CTVR>1\) indicates that the prior state yields less error compared to the assimilated states, and therefore 3-EnKF-WDN is ineffective.

4.2 Uncertainty Due to Sensor Precision

Figure 7 shows how the average TV of flow varies with varying precision of sensors for Modena. It can be seen from the sub-plots that, in general, an increase in the precision value of flow sensors results in an increase in the average TV of flow. Figure 8 shows a close-up of one of the sub-plots from Fig. 7, when the precision of demand sensors (\(R_{z_{q}}\)) is fixed at 0.1 litres/second and the precision of head sensors (\(R_{z_{H}}\)) is fixed at 0.1 meters. It shows that the average total variance of flow states increases with the increased precision value of flow sensors.

These findings suggest that an additional step of DA has a greater impact on decreasing the average TV of flow (i.e., on reducing the model error), than increasing the precision of flow sensors. In practical cases, this implies that operators may opt for less precise sensors, but a variety of sensors, to perform a multi-step DA and achieve better results. As seen in Fig. 8, carrying out a multi-step DA seems to be much more effective in reducing model error than assimilating just one system state. Also, the precision of the sensor does not improve the results more considerably than the improvement obtained by an additional DA step. Similar results were obtained for both FiveRes and Modena WDNs with slight variations which may be due to factors such as the network topology, and hydraulic state of the WDN, among others.

Fig. 7
figure 7

Average Total Variance of Flow against precision of Sensors for Modena Network

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Fig. 8
figure 8

Average TV of Flow when Rzq: 0.1, RzH: 0.1 against precision of Flow Sensors for Modena WDN

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Fig. 9
figure 9

Average TV for different numbers of ensembles for Modena WDN

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Fig. 10
figure 10

Simulation time against the number of ensembles for Modena and FiveRes Network

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4.3 Discussion on Number of Ensembles and Computational Demand

As each member of the ensembles in the EnKF is an independent realization of the model, we discuss how the number of ensembles affect the performance of the proposed 3-step EnKF. Generally, the higher the number of ensembles, the better the estimation of uncertainty, and computational demand is greater (Mulder 2014). Hence the implemented 3-step EnKF was simulated with ensembles between 5 members up to 100 members.

Figure 9 show that the increase in the number of ensembles improves the results of the DA as the average TV decreases in all cases with the increase in the number of ensembles used by the EnKF. It can also be seen that the consecutive steps of assimilation result in a reduction in the model error as well. The asymptotic behaviour also indicates that few ensembles yield high average TV, but that it reduced rapidly as more ensembles are added. However, the rate of reduction of TV starts to be marginal after 30 to 50 ensembles, indicating that more ensembles are not necessary. This behaviour was seen for both Modena and FiveRes networks.

The simulation time was compared for the different number of ensembles using various configurations of computer systems. It is observed that an increase in the number of ensembles from 5 to 100 results in an increase in simulation time from 37 seconds to 593 seconds for Modena and 269 seconds to 4464 seconds for FiveRes. With the increase in the size of the network from Modena (268 nodes, 317 links) to FiveRes (935 Nodes, 1278 Links) the increase in simulation time is exponential. This can be seen by the increase in the gradient of the graph in Fig. 10. Although the increase in the size of the network is \(\approx \) 3.5 times, the increase in simulation time is by \(\approx \) 7.5 times.

Table 2 Computer systems used for testing the proposed DA algorithm
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Fig. 11
figure 11

Simulation time against the number of ensembles for Modena and FiveRes WDNs using different computer systems

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In addition, 3-EnKF-WDN was tested on three different computer systems with different computational resources. The specifications of these systems are given in Table 2.

From the three different computer systems tested, the processors and their respective clock speeds show the most significant effect on the computational time. The current implementation of the algorithm runs serially without any parallel components, as such the computation time depends on the single-core clock speeds of the processors. Hence the results seen from Fig. 11 are representative of the base and boosted clock speeds of the processors used in the systems in Table 2 If the ensembles are generated in parallel, it will bring about a significant improvement in the computation time of the 3-EnKF-WDN. This will allow for the use of the 3-EnKF-WDN for larger WDNs, for running it for more time-steps without a significant computational burden.

5 Conclusions and Recommendations

In this paper, 3-EnKF-WDN, a 3-step DA method that assimilates pressure, flow and demand data, running a hydraulic model in extended-period simulation and under PDA was presented, along with a new evaluation metric called Combined Total Variance Ratio. The method was applied to two networks to evaluate its effectiveness in reducing the error in the hydraulic model under uncertain measurements.

The study demonstrated the importance of considering the effect of measurement uncertainty when using the 3-step DA algorithm. Two sources of uncertainty in the measurements were explored, namely precision and noise. It was found that the precision of sensors and the noise in measurements affect the efficacy of the 3-step DA.

When noise is added to the measurements, 3-EnKF-WDN becomes generally ineffective, within a small range of variation. The effect of the noise is significant in extensively monitored WDN. The findings also confirm the importance of maintaining the sensors with noise as small as possible. This could be achieved by carrying out regular maintenance and calibration of sensors. In practical applications, it is recommended to carry out simulations like the experiments with noise-in-state measurements used in this study to determine the respective thresholds of noise up to which the 3-step DA is still effective for the respective WDN.

It was also found that having high-precision sensors measuring one variable brings less reduction in model error than having less precise sensors measuring more variables.

The study also demonstrated that 30 to 50 ensembles are enough for the 3-EnKF-WDN to perform well, on the two studied networks, and that increasing ensembles beyond this number only introduces unnecessary computational burden.

It was also found that sensor data of demand do not improve the model error when applying 3-EnKF-WDN when the WDN is fully monitored (i.e., with head sensors in all the nodes and flow sensors in all the links). This is similar to the results obtained by Bragalli et al. (2016) where the TVR(q) of demand was found to be the least sensitive to reduction in the TVRs for the multi-objective optimization carried out in their study.

The proposed method has the potential to be applied to diverse WDN problems such as leak detection, anomaly detection, demand estimation, and water quality evaluation. This can be achieved by adapting the multi-step DA algorithm for the required purpose.

Some limitations of the study include the heavy computational time required. Parallelization of the algorithm using a method that can run hydraulic simulations in parallel is a solution to be explored in future research. In addition, the effect of the order and synchronicity of the assimilated data needs to be established. Other explorations to be made include the effect of the standard deviation or variation of the ensembles of demands and the effect of measurement uncertainty on the Kalman Gain.