Direct assessment of background leakage levels for individual district metered areas (DMAs) using correspondence of demand characteristics between DMAs Open Access

Leakage continues to be a major challenge for water companies in many parts of the world (global average 30% with large regional variations, Liemberger & Wyatt 2019). With continuing population growth and urbanization, and a changing climate that is currently already affecting the demand for and the availability of water, but is likely to do so to a much larger degree in the coming decades and beyond, it is more important than ever to minimize losses of water through leakage.

Lambert (1994) distinguishes three types of leakage: background leakage (e.g. at joints, with low flow rates that are hard to detect), unreported leakage (potentially significant flows, but nobody noticed, and the water disappears into the soil or a waterway), and reported leakage. This paper focuses on the former two categories, that is to say pre-existing leakage that is difficult to detect and quantify from flow signals. This distinguishes the current approach from burst detection, which focuses on the third type of leakage. In this paper, the term background leakage is intented to encompass both unreported and true background leakage as defined by Lambert (1994).

The most commonly used methods to assess (background) leakage at the level of district metered areas (DMAs) or supply areas in practice are the minimum night flow method (Puust et al. 2010) and the water balance method (Farley & Trow 2003). The former hinges on an estimate for the minimum demand in a DMA during the quiet hours of the night or an assumption of zero consumption for sufficiently small DMAs. The latter requires, among other things, registration or estimates of actual demand. A third approach, the burst and background estimates method (BABE, Lambert 1994; Lambert & Morrison 1996), aims to evaluate individual components of leakage using a number of characteristics of the network and its operation. Finally, a hydraulic-model-based approach can be applied (e.g. Giustolisi et al. 2008; Marzola et al. 2022; Romero et al. 2022; Steffelbauer et al. 2022; Wu et al. 2022), though this does not appear to be common practice among water utilities for estimating leakage levels, and obviously this requires the availabilty of a hydraulic model of adequate quality.

This paper proposes a new approach to rank a group of DMAs in terms of background leakage rate and to quantify background leakage levels for individual DMAs in this group. This is done using an extension of the comparison of flow pattern distributions or CFPD method (Van Thienen 2012; Van Thienen & Vertommen 2016). The approach presented is based on an assumption of similarity in demand behavior between different DMAs. It requires no other data than net inflow for the DMAs or supply areas under consideration, and no assumptions other than that of similarity of demand. As such, it provides a low-data-requirements method for the evaluation of background and unreported leakage that does not share underlying assumptions with the commonly used mininum night flow method and may potentially present a supplement or alternative to it.

The paper is structured as follows. After a brief introduction of the original CFPD method, its extension to background leakage identification is described. The method is then illustrated and tested on synthetic DMA flow data, and a sensitivity analysis is presented. Next, the validity of the CFPD approach for non-uniform and non-constant leakage behavior is determined using hydraulic models. The combined set of methods is subsequently applied to actual DMA flow data from a set of supply areas in the Netherlands. Finally, the results are discussed and conclusions are presented.

In this section, we give a summary of the method presented by Van Thienen (2012). Consider a supply area for which the flow rate into the area (accounting for all inflow, outflow and storage) is observed for a period of time (e.g., a day, a week, a month or an entire year) and again for a comparable period of the same length in another year. The observed patterns are likely to be similar in shape but not exactly the same. The simple CFPD procedure allows a quantitative comparison of these patterns, taking the following steps (see Figure 1):

• 1.

  Sort both data sets by magnitude from small to large. Sorted measurement ranks, scaled to a 0–1 range, are on the horizontal axis, while flow rates are shown on the vertical axis.

• 2.

  Plot one observed data set against the other on a CFPD plot.

• 3.

  Determine a linear best fit with slope a and intercept b.

The slope a is a scaling factor of which the physical interpretation is a change in demand consistent with the pre-existing demand pattern. Hence this change is called a consistent change. The intercept b is a shift in the total pattern, which is a change in demand that is not consistent with the pre-existing demand pattern. Hence this change is called an inconsistent change. This can be physically interpreted in terms of a change in the (constant) leakage rate. A more elaborate discussion on the interpretation of these factors is presented by Van Thienen (2012).

Note that the word pattern is used here in the sense of a time series that is generally repetitive to a significant degree with some variations. In general, it is preferable to construct the CFPD plot with the first period on the horizontal axis and the second on the vertical. In this case a>1 and/or b>0 corresponds to an increase in flow rate. Note that comparison of periods of different lengths or sampling frequencies is also possible but requires an additional interpolation step, as depicted in Figure 1(e).

However, this does not yet give us absolute background leakage levels for individual DMAs. This requires an additional step. If we postulate a hypothetical DMA 0 that has no background leakage, the parameters of interest, i.e. the absolute background leakage levels b_0j for DMAs j are introduced.

This results in a clear matrix structure for C, D, E, and F, see Appendix I in the supplementary material online.

As for each DMA, the net flow (i.e. inflows minus outflows) is considered, this also works for cascading DMAs.

We solve for these unknowns by applying a numerical optimization scheme (implemented in SciPy, Virtanen et al. 2020), minimizing the residual between model and data. We impose the boundary condition that all b_0j≥0, i.e. no negative background leakage rates. Note that the parameters a_0j have no physical meaning, since there is no natural pattern scale for the postulated DMA 0. The parameters a_0j are constrained by requiring that a₀₁=1. This sets the scale for the postulated undisturbed pattern, and thus fixes the other a_0j parameters as well.

There are several sources of error in this analysis that can have a significant impact on its results. The effects of DMA patterns not showing the same behavior (in the sense that CFPD analyses result in a linear relation) are investigated by adding noise to the flow signals. Also, the effects of choosing a wrong value for the total background leakage level L are determined.

First, we look at leaks that have a more or less constant flow rate. This situation is representative of networks that have small pressure variations throughout the day due to implemented pressure management. It has been shown for simple test cases (Van Thienen 2012) that introducing leakage pressure dependence does not invalidate the CFPD approach. This will be further validated for a range of relevant conditions using hydraulic models below.

We start by generating a large number of flow patterns from a single, unperturbed flow signal that is postulated to be leak free (Figure 2). In this case, we have chosen a pattern with 0 minimum night flow, but this is not necessarily the case. These patterns are then transformed using the CFPD paradigm, i.e., first scaled and then shifted, the latter operation representing the addition of a leak that has a constant flow rate. A large number of different scaling factors a (range 1–10) and offset factors b (range 0–100) are chosen randomly. Normally distributed noise is added. Clusters of DMAs are analyzed using the procedure described above, as a function of noise level and error in the estimated DMA cluster background leakage. The complete procedure is described in the pseudocode in Figure 3.

The results of this analysis are shown in Figure 4 through Figure 7, for DMA clusters of 2, 3, 5 and 8 DMAs, respectively. The mean error in the recovered background leakage for the DMA cluster and its standard deviation are plotted in Figures 4–7 subfigures a and b. These are presented as a percentage of the actual DMA cluster background leakage rate. The mean error in the recovered background leakage for individual DMAs and its standard deviation are plotted in Figures 4–7 subfigures c and d. These are presented as a percentage of the mean DMA flow, since presenting them in the same manner as previously described would result in extreme magnification of errors in low background leakage DMAs.

The resulting images are all quite similar for the varying number of DMAs. Any error that is included in the estimate for the total background leakage is directly reflected in the sum of the recovered background leakage levels (subfigures a, c). This means that the multiple DMA CFPD (mCFPD) method is able to reconstruct the actual background leakages in all DMAs provided that the estimated total background leakage is accurate. If this is not the case, i.e. if the summed non-revenue water (NRW) error is nonzero, this error propagates directly into the recovered values for the background leakage levels of the DMAs. The robustness of this recovery declines somewhat for greater errors in the total NRW estimate, in both directions, but remains relatively small, showing a standard error which is mostly below 10% of the actual value for the total recovered background leakage levels for a cluster (subfigures b) and mostly within 5% of the mean flow for the individual DMA background leakage levels (subfigures d). None of the tests shows any significant sensitivity to the noise that was added to the signals. It must be noted, however, that using other probability density functions for the noise signal, in particular skewed ones (e.g., resulting from lower sensitivity and larger error at low flow rates), is likely to result in a degradation of the performance for increasing noise levels.

Occasionally, the set of flow signals is such that the optimizer cannot find a good solution, even for synthetic examples for which a true solution is available and known. The main expression of this behavior is in large SSR (sum of squares of residuals) values, which should therefore be monitored and considered in relation to plausible background leakage level numbers. This behavior is further illustrated in Appendix II in the supplementary material.

As can be seen in the results of the previous section, both the estimated background leakage levels for individual DMAs and consequently the sum of these estimates depend strongly on the estimate of the background leakage level L that is fed into the problem as prior information (Equation (9)). Handling and propagation of this uncertainty is discussed in Appendix III in the supplementary material.

Before applying the mCFPD method described above to real-world data, first a number of background leakage scenarios are investigated in a set of hydraulic models. The purpose of this is to investigate to what degree the CFPD model holds when a more complex leakage model is applied, that is to say, whether we can apply and interpret the mCFPD approach that is built on the CFPD paradigm.

A set of network models for real-world systems with different characteristics was obtained, see Table 1. The first is a modification of an existing network, in which a number of looped structures have been opened up to create a more branched topology. This had originally been proposed to improve the network self-cleaning. For the present application, this variant of the network has been selected because it is expected to show larger pressure variations and thus variations in background leakage through the day. The model represents part of a Dutch town, with about 1,000 connections.

The second is the network of a city of approximately 77,000 inhabitants. These models have a relatively flat topography and little pressure variations in the network throughout the day.

In order to investigate the effects of more significant spatial and temporal pressure variations, for each of these models, a series of alternatives with synthetic topography (sloping west-east with varying gradients) and synthetic head profiles at the source (cosine pattern with varying extreme values at midnight and midday) were generated. In each of these modified models, a synthetic background leakage scenario was generated by randomly adding emitters whose characteristics were selected randomly within a specified range to a specified fraction of 10% of the junctions in the network model. Hydraulic simulations were performed using EPAnet 2.2 (Rossman et al. 2020). In each case, a CFPD analysis was performed comparing hydraulic simulation with and without added leakage (but using the same topography and head function scenario).

Both figures show a similar situation as a function of the elevation profile and head function amplitudes. Both mean background leakage levels over a 24 hour period (subfigures a) and CFPD reconstructed mean background leakage levels (subfigures c) indicate a noisy signal. This is because the background leakage levels are randomly generated for each instance of the hydraulic model with a different combination of elevation profile and head function. However, the important observation is that they represent more or less the same noisy pattern. This shows that even in a case where there is significant variation in the background leakage level through the day (represented by a higher standard error in subfigure b), the CFPD recovery of the mean background leakage (b factor in subfigure c) is still good. The observed errors (subfigure d) are well within +/−20% for the cases studied here, and show some asymmetry with respect to the elevation profile. This is presumably related to asymmetry in the network layout, i.e., high topography related pressures on the extensive east side of the network will have a higher proportion of randomly distributed leakages than the more sparse western side. The larger number of leakages in a high pressure part of the network will result in a disproportionate increase in leakage volume. Error parameters for the CFPD fit (subfigures e–g) visually correlate well with the variability of the daily changes in the background leakage level (subfigure b).

From these diagrams, we can conclude that for the hydraulic models studied and within the bounds of the prescribed parameter range, the CFPD approach catches compounded leakage signals in complex networks to well within +/−20% of their true values even when topography and source head variations through the day cause significant variations in the instantaneous background leakage level. This justifies the use of the CFPD methodology as a basis for the mCFPD approach.

Application of the basic CFPD method only gives meaningful results if the input data are not disturbed by anomalies, such as occasional fillings of swimming pools, erratic behavior by large volume customers, and other exceptional events. Therefore, in order to obtain a successful multi DMA CFPD analysis, we need to find a time window in a long time series that avoids these anomalies and gets the best CFPD fit (assumed to be the most reliable). This is done by evaluating all possible time windows of length l in the dataset, starting from the first window of length l in the timeseries and progressively moving the window one time step. For each of these windows CFPD analyses are performed between all DMAs. By assessing which time window results in the greatest mean R², that is for which the CFPD fits are the best, the most appropriate time window is identified.

Also, when comparing a larger set of DMAs, it is possible or even likely that not all will exhibit exactly the same demand behavior (meaning that not all CFPD curves will provide very good linear fits). Within a larger set of DMAs, smaller subsets of DMAs may be present that do exhibit very similar demand behavior. Therefore, all possible combinations of DMAs in the datasets in up to n subsets are considered individually as described above, and the combination of subsets that has the lowest mean of mean standard errors in the b coefficient is selected. This will be illustrated in the applications to real DMA data section below. Note that the determination of optimal clusters requires the use of the same time window for all (based on an analysis for the complete set of DMAs) if the analysis results at different clustering levels are to be compared. This will be illustrated in the field data case below.

In order to further validate the approach described in this paper (in addition to the validation on synthetic flow data described above), one would want to have a flow dataset for which the true background leakage levels are known. Estimates from minimum night flow analyses may provide an indication, which may be used to verify that the results of the mCFPD analysis are plausible, but do not provide a ground truth. Application of the method in many different DMAs in addition to traditional methods may help to build confidence in the mCFPD approach. In the following paragraphs, we describe the application of the mCFPD method to a dataset from the Netherlands. These analyses are not presented as a validation, but as an illustration of the approach and associated challenges when dealing with real data.

Flow data into nine anonymized supply areas (which we designate WA…WI) were kindly provided for the year 2019 by the WML water company, which supplies the southeast of the Netherlands. Data from two DMAs consist of mutually dependent derived values that have for this reason been excluded from the analysis (see Table 2). The data have a time resolution of one measurement per hour.

In this case, since we have seven DMAs available, the consideration of the degree to which the flow patterns in the different DMAs are comparable in the sense of CFPD is considered implicitly employing the clustering method that has been described above. The complete set of DMAs has been considered as a single cluster, two clusters or three clusters. A 14 day period which has the best correspondence in the CFPD framework was identified (November 19 – December 2, 2020), as described above. The results for the single cluster analysis are presented in Figure 10; those for the two cluster analysis in Figures 11 and 12; and those for the three cluster analysis in Figure 13–15. These figures show the allocation of the total background leakage estimate by the mCFPD analysis for the entire cluster to the different DMAs that comprise the cluster. Also, the SSR is shown for the complete range (from zero to the summed minimum night flow for the period under consideration). The point at which this curve flattens out or bounces back up in the lower left of each of these diagrams corresponds to the model in which one (or theoretically possibly more than one) of the DMAs in the cluster has a zero background leakage level. We can consider this model to be our estimate for the minimum background leakage levels for all the DMAs in the cluster. These numbers have been compiled in Table 3.

The overview provided in Table 3 shows us a number of findings. To start with, the approach applied here gives a minimum background leakage estimate for each subcluster of DMAs with the smallest value for each subcluster being 0. Any existing offset between subclusters will not be expressed here.

This means that results for different clustering within a set cannot be directly compared (Table 3a), but must be corrected.

To illustrate this effect, let us consider the following simplified example: DMAs W, X, Y and Z each have different background leakage levels, of which X's is smallest. The mCFPD analysis will give a lower bound estimate for X's background leakage of 0. If we subsequently consider two subclusters W,Y and X,Z separately, the lower bound estimate for X will again be 0, but also that of Y in the other subcluster if that DMA has a smaller background leakage level than W. Therefore, in order to compare the numbers of the two subclusters, the offset between the subclusters has to be determined from the full cluster.

The offset correction term can be calculated (Table 3b) as the difference of the mean background leakage level of all the DMAs in a cluster and that of the same DMAs in the full set. This results in mostly comparable background leakage levels (Table 3c) for the different clustering levels (similar numbers in each column of Table 3c). It follows from the approach that at a higher level of subdivision, the similarities between the DMA patterns within the cluster must be greater, and therefore the determination of the b factors in the CFPD analysis more reliable. We observe that the differences between the single-cluster, two-cluster and three-cluster analyses are relatively small, i.e., the analysis is internally consistent for the three subdivisions.

An alternative approach to estimating background (i.e., true background and unreported) leakage levels that relies solely on flow measurements at DMA inlets has been presented in this paper. Its main advantages with respect to other methods are the use of the complete 24 h signal (which in theory should provide a stronger statistical significance to the findings), its insensitivity to unbiased noise, and the absence of the need for any estimations or assumptions except the central assumption of the method. This central assumption is that all DMAs within a set or subsets of DMAs have similar demand behavior. In practice it will be easy to find both situations in which this assumption holds very well (e.g. neighboring areas in a city with similar economic and demographic characteristics), as well as situations where it does not (e.g. comparing a mostly commercial area to a mostly residential area or comparing areas with non-neglegible constant or erratic industrial consumption) and therefore DMAs for which the mCFPD approach may be suitable and those for which it is clearly not. An approach to verify this correspondence and minimize its effect on the analysis has been presented as well.

The low data requirements (sensors on DMA inflows/outflows only, low time resolution) of the method and the absence of any assumptions other than the central one may make the method particularly useful for developing countries. That is, provided that they do not have intermittent supply, which is still not uncommon in the developing world (Charalambous & Laspidou 2017), and which will strongly affect flow measurements and patterns (Totsuka et al. 2004; Criminisi et al. 2009; Weston et al. 2022). At this point, it is not clear which degree of intermittency of supply, if any, is acceptable, and to what degree secondary effects of intermittent water supply, such as the use of personal water storage tanks, affect the applicability of the mCFPD method. This requires further investigation.

Even though the approach presented here does not provide the information on where the background leakage is in a DMA, it does provide a basis for prioritizing DMAs for leakage localization and network rehabilitation efforts for water utilities.

The minimum night flow (MNF) method appears to be the most commonly used approach to determine background and unreported leakage levels, and allows its users to do the same things as described in the previous paragraph. MNF does hinge on assumptions with respect to night consumption, which may include a regional component (Amoatey et al. 2018). The mCFPD method is based on a completely different and independent assumption, as described above. The combined application of both MNF and mCFPD may strengthen the credibility of the outcomes of both analyses (in case they align) or give reason for a further investigation of the applicability of the underlying assumption. This will also give insight into which of the assumtions (i.e. those for MNF and mCFPD, respectively) may be critical in specific practical situations.

It must be noted that the quality of the output of this method relies on the quality of the input data. That is to say, systematic errors in the flow measurements are directly reflected in the analysis results. Even though noise with a non-skewed distribution may be expected to average out in the CFPD analysis and therefore not significantly affect (m)CFPD outputs, offsets in the input data will be directly reflected in the b factors (background leakage estimates). More complex measurement error behavior may be reflected in both the a and b factors. As with any method, this is to be taken into consideration when interpreting the results.

Sensitivity tests and application to synthetic signals from hydraulic models demonstrate the validity of the method in ideal circumstances. Subsequent application of the method to real data shows the feasibility of the analysis. However, due to the lack of reliable true background leakage values, this does not suffice as a validation. The presentation of the method validated on synthetic data in this paper reflects the author's hope and expectation that attempts at validation will be done by researchers in the field with access to a range of suitable datasets.

This paper describes a newly proposed method and provides a basic analysis of its sensitivity and applicability. It is quite clear, however, that more experience in applying this method is necessary to grow confidence in its applicability and generate insights into the practical bounds of the method. In addition to application to a range of field data sets for validation purposes, a number of questions and ideas remain for further investigation:

• As flow meters are sometimes selected for the expected flow during peak hours rather than the minimum night flow, accuracy issues in the lower flow range may provide an additional source of uncertainty for minimum night flow analyses. Even though these issues may also arise in the mCFPD approach, it would be worthwhile to investigate if excluding the parts of the flow data that are known to be within the low accuracy range of the flow meters used can improve the quality of the analysis results.

• It would be very valuable to obtain a better understanding of what makes flow patterns similar or dissimilar in the CFPD framework, in order to better understand the bounds of application of the mCFPD method.

• Using statistical filtering on the input data (e.g., using the 95% most conforming data) or other types of data preprocessing may make the method more robust.

• Understanding needs to be built on to what degree the method is applicable under conditions of intermittent supply.

The author would like to thank Brabant Water and WML water companies for supplying the data and models presented in this paper and Dick Bos (Brabant Water) and Henk Vogelaar (WML) for their support in understanding the data and models. Also, fruitful comments on an early version of the paper and discussions with Dragan Savić, as well as constructive reviews by two anonymous reviewers are gratefully acknowledged.

Data cannot be made publicly available; readers should contact the corresponding author for details.

The author declares there is no conflict.