Effect of the sampling interval on the measurement accuracy of electronic water meters under real consumption conditions Open Access

Digitization is advancing in all areas worldwide and is also becoming increasingly important in the water supply sector. Since the launch of the first ultrasonic water meter in 2008 (Diehl Stiftung & Co. KG 2021), the share of electronic water meters installed, e.g., in households, has increased steadily in recent years (Doucet 2019). This trend is also visible in the global smart water metering market. The market was valued at $4.91 billion in 2020, and is projected to reach $9.73 billion by 2030, growing at a CAGR of 7.5% from 2021 to 2030 (Rake et al. 2021). However, it should be noted that in market analyses on smart water metering, electronic meters and mechanical meters with electronic add-on devices are often jointly considered. In addition, electronic water meters and flow meters with the same measuring principle are often grouped together. A general trend towards increasing use can be derived from this and is also reflected in the growing number of type approvals for electronic water meters. The reasons for an increased use of those meters are for example integrated additional functionalities such as leakage detection, a wider measuring range or less susceptibility to certain types of disturbances and installation situations. An up-to-date and comprehensive overview of electronic water meters, their measuring principles, aspects of remote reading and additional functionalities can be found in a brochure by DVGW (German association of the gas and water industry) (DVGW 2022).

Both measuring principles used by electronic meters measure at discrete points in time in contrast to mechanical meters, which measure continuously. The consumed volume, which is needed for billing purposes, is calculated internally based on the integration of each determined flow through the cross-sectional area over time.

Independent from the measurement principle the manufacturers of water meters must demonstrate that the water meters are capable to measure the consumption correctly according to legal requirements. For this, amongst others the battery life of electronic water meters must be ensured for a defined period of time. A typical value is 15 years. However, since each measurement consumes energy, an attempt is made to keep the number of measurements at an absolutely necessary level to ensure a longer battery life. In normative documents such as ISO 4064:2014 or OIML R49:2013 no specifications are included for the frequency of measurements because the current versions were foremost developed for mechanical meters. Since electronic water meters came up at a later stage there are no regulations or recommendations for a sampling interval so far. The sampling interval can be set up by the manufacturer in dependence of the measurement principle for instance. For the magnetic-inductive method a quite short sampling interval could be defined because it is possible to measure the voltage, which is proportional to the flow, with a high temporal resolution. In contrast, the ultrasonic method has a restriction regarding the minimum sampling interval. Based on information by two water meter manufacturers (one among them Kamstrup A/S) and a water supplier (bnNETZE GmbH), the ultrasonic system requires at least 2 seconds for each calculation, which means that the highest feasible sampling interval for this type of meter is 2 seconds.

The sampling interval may have an influence on the measurement accuracy (measurement error) of the meter under real consumption conditions and depending on the considered time period. It is currently unclear how large the impact may be on the total captured volume. Currently, the measurement errors of water meters are determined (e.g. according to ISO 4064:2014 resp. OIML R49: 2013) from constant flow rates. For a constant flow, the effect of the sampling interval is negligible (van der Wiel et al. 2021). An error can arise because the start and end of the test do not need to be equal with the first or last discrete measuring point. This means that it is possible that at the beginning and at the end some volume would not be detected. However, a constant flow does not correspond to real conditions. Water meters operating under real consumption conditions where dynamic water flows are typical may have a larger measurement error due to the sampling interval. Under real conditions there may be periods with large deviations from the mean flow rate that will not be detected if they occur between single measurements. This would result in an under- or over-registration of the volume.

Till now there is no study about the impact of the sampling interval of a water meter on the measurement error at operating conditions. So far, a simple study on a washing machine fill cycle (66 s with 3.35 L) was done by Holmes-Higgin (2019). He shows that in this case due to a sampling interval of 4 seconds an error between −31.34 and +25.37% occurs in the measured volume depending on the start of the first measurement. For comparison the maximum permissible error (MPE) of a water meter at lower flow rates is maximum ±5% (OIML R 49 2013). Holmes-Higgin (2019) demonstrates in his study that there is a significant difference if the first measured value is equal to the start of the filling cycle, or if it is shifted between 1 and 3 seconds.

This simple study already illustrates that there may be a significant impact of the sampling interval and starting point on the accuracy of a discrete measurement. In a further study Eff (2020) shows something similar. In his study, a simple flow profile of a dispensing process is used to show the dependence of the measurement error by sampling interval on the starting point of the sampling. Additionally, he demonstrates that the measurement error is getting smaller with a longer observation period. For this, he used the simple flow profile as before but replicated it up to 500 times. He also carried out this investigation for the three test profiles of the EMPIR project 17IND13 ‘Metrology for real-world domestic water metering’ (MetroWaMet 2018), for which he obtained comparable results.

The studies by Holmes-Higgin (2019) and Eff (2020) are a good starting point, but in order to make reliable estimates of the effects of the sampling interval on the measurement accuracy of electronic water meters, further research with longer consumption profiles without cyclicities is needed. This was the motivation to address the issue more comprehensively. Several aspects were investigated in this study using real consumption profiles. The first part of the study focuses on the influence of the interpolation method used to calculate the total volume that passed the meter. Two simple interpolation types (rectangular and linear) are considered. In the second part the influence of the sampling interval itself on the measured volume is investigated. In addition, the influence of the observation period on the measurement error is addressed. Finally, the influence of the starting point of the sampling is studied.

Two different types of data sets, stochastic consumption data and consumption data measured in households, were used for the study. The consumption data directly measured in households are important, because for getting information about the effect of the sampling interval under real-world conditions without restrictions, only real consumption profiles can show the impact. The data used comes from a DVGW study from 2017 (Martin et al. 2017). For this study the water consumption in different houses (detached houses (DH) and apartment houses (AH)) of different German cities, were measured between 2014 and 2016. The consumption was monitored for time periods between 23 and 84 days with a resolution of 1 Hz. For the measurement in the DHs ultrasonic water meters with a permanent flow rate (Q₃) of 4 m³/h and a ratio R of permanent flow rate to minimum flow rate (Q₁) of 160 were used. Due to higher flow rates in AHs ultrasonic water meters with Q₃ = 16 m³/h and R = 160 were deployed. This results in a good database of in total 58 consumption time series of different lengths (Schumann 2019). Out of the consumption of all DHs a real-world profile with a duration of one year can be derived. Since the data was recorded in different cities and in different years there exists no measured database for one house for one year. However, the database is sufficient for the aim of the investigations carried out here. Previous studies showed that there is neither a significant difference between countries nor a relevant seasonal effect on the water consumption (Schumann et al. 2021). For this reason, it is possible to put together a one year-long profile from different houses measured at different times.

For the AH it is possible to realize a data set of nine months (except January, February, and July). This means that a shorter observation period is considered, but this should be sufficient to show basic effects of the sampling interval on the measurement accuracy.

The second type of data is stochastic data. Schumann (2019) developed an algorithm, which can be used to derive stochastically assured consumption profiles out of measured consumption data. The stochastic data sets are based on the previously mentioned measured consumption data. With the algorithm an approximately 1 day-long profile (86,472 s) and an approximately 1 week-long profile (604,920 s) were generated. The day-long profile comprises flow data with a minimum flow duration of 3 s and a maximum of 108.33 s. The 1 with the length of approximately 1 week has also a maximum flow duration of 108.33 s but a minimum of 1 s. All stochastic data provides the duration of every flow point up to three digits, which, in contrast to the real data set, would allow an examination of the sampling interval smaller than 1 second in further studies.

The benefit of the stochastic profiles is that not only one house is considered, because the data reflects the consumption of all 58 houses. Furthermore, those stochastic profiles can be made accessible for everyone in contrast to the data directly measured at households and as such are classified as personal data. With those stochastic profiles novel water meters or measuring techniques can easily be tested for the influence of the sampling interval on the measuring accuracy. In addition, it is possible to conduct simple studies on the influence of changes in consumer behaviour, e.g., due to climate change with a reasonable effort.

In case of the linear progression the volumes correspond to the surface area under the curve of the measuring points, too. This volume can be determined by the integral of the function, in which the measuring points are connected linearly. This integral is calculated by the data analysis and visualization program Origin (OriginLab Corporation 2022).

Due to the recording frequency of 1 Hz this data has already a small error, which is unknown. In the study the 1 Hz recording is used as real consumption. Also, the stochastic data has this unknown error because the stochastic data results out of the 1 s data.

Table 1

Investigated sampling intervals

Sampling interval /s .
1
5
10
30
60
90
120
150
180
210
240
270
300
600

Sampling interval /s .
1
5
10
30
60
90
120
150
180
210
240
270
300
600

View Large

In contrast to mechanical meters, where typical reasons for measurement deviations are the friction and abrasion of mechanical components, measurement deviations in ultrasonic meters can be caused by the sampling interval in addition to deposition and abrasion effects.

The maximum error is used to identify the sampling intervals where the error exceeds defined limits, e.g. 1%, in the data set under consideration for the first time. For this purpose, the use of is very suitable, because for the same sampling interval smaller errors than can occur when considering several profiles. These smaller errors are not relevant for an evaluation in this study. The maximum error is used to investigate the influence of the observation period, and the start of the sampling on the error.

To assess the impact of the observation period the stochastic profiles and a directly measured consumption profile of an AH were multiplied several times. The stochastic profiles were replicated up to 120 times to simulate different observation lengths and to investigate if the maximum error becomes smaller when longer observation intervals are considered as found by Eff (2020). Although the stochastic profile has a duration of slightly longer than 1 day by 72 s), cyclical effects can still occur if the sampling interval has no residual at the end of the profile. To avoid those cyclical effects the profile was shifted at each repetition for 1/120 of the sampling interval. Sampling intervals between 2 and 120 s in 1 s steps were used. The profile of the AH was replicated up to 20 times. Due to the exact length of one month, there are more cases with no balancing effects. To avoid those cases, the profile was shifted at each replication for 1/120 of the sampling interval, as was done for the stochastic profile.

In a next step the real-world profile was considered with different lengths of the observation time (one month up to one year) to assess the influence of the observation period under real consumption conditions.

Additionally to the length of the period under consideration and the length of the sampling interval itself, the point in time of the sampling may also affect the determined volume and thus provide an additional error source (Holmes-Higgin 2019). To gain more insights into this error source under real consumption conditions the points in time at which samples were taken were changed and the impact on the total volume considered. For this, the starting point of the profile was shifted. In all cases the consumption data before the new starting point was added at the end. This has the consequence that in all instances the total consumption is identical independent from the starting point. For the day-long stochastic profile the starting point of the sampling was shifted exemplarily once by 1803 s (∼30 min) and 43,341 s (∼12 h). Analogously the stochastic week-long profile was obtained. Due to the longer profile, the starting point was additionally shifted by 86,168 s (∼24 h). For the investigation of directly measured consumption the real-world profile of a DH was used. The real-world profile was shifted three times by three months (one quarter) to start the sampling at a different season of the year. To gain insights into the effect of only small time shifts, the time at which the samples are taken was shifted for 1 s, 2, 5 and 10 s for the real-world profile.

Table 2 provides an overview of which investigations were conducted using which data in the present study.

From Figure 3(b) it can be seen that the type of interpolation has a negligible influence on the measurement error at timesteps smaller than 60 s (⁠⁠). At timesteps greater than 60 s the difference between linear and rectangular interpolation becomes slightly larger (up to 0.93% difference at a sampling interval of 600 s). The difference between linear and rectangular interpolation is becoming almost linearly larger with larger sampling intervals.

Furthermore, the influence of the interpolation method on a directly measured consumption profile of an AH over 55 days was also considered. The error obtained for the real-world profile is less affected by the interpolation type as the stochastic profiles. In this case even at larger timesteps the influence of the interpolation is quite low. At the largest timestep of 600 s the difference between the interpolations is only 0.25%. As before, the difference is becoming almost linearly larger with a larger sampling interval. One possible reason for the smaller difference could be the longer observation time of 55 days instead of 1 day or 1 week. This assumption is confirmed, as a measured profile of 1 week shows comparable differences as the stochastic profile with the same length.

Based on these results, the rectangular function is used in the subsequent investigations, because the calculations can be performed much faster using the rectangular function as interpolation method. Furthermore, the focus is on sampling intervals below 120 s in the error determination, since in this range the differences between the interpolation types are very small.

The results shown in Figure 4(b) confirm the expectation that due to averaging the error becomes smaller for longer observation periods.

The overall occurring maximum error based on the sampling interval is getting smaller with each repetition. After 120 repetitions the maximum error due to the sampling interval is still larger than 0.4%. The maximum error is becoming smaller but there seems to be a limit so that the error never gets close to zero or many more repetitions are needed for getting an error close to zero.

Analogously, a profile of an AH with a duration of one month was considered. The errors of all sampling intervals, which have a residual at the end of the profile, are again getting smaller with a higher number of repetitions.

Figure 5 shows that in this case for a sampling interval of 28 s the error exceeds 1% for the first time, but even at a sampling interval of 600 s the error based on the sampling interval amounts to only 0.08% in this example. However, at a sampling interval in between the error can reach an order of magnitude of several percent. Overall, the errors fluctuate quite strongly depending on the sampling interval. For example, a sampling interval of 565 s causes an error of 8.23% and an interval of 566 s results in an error of 0.24%. The linear regression, which is a crude approach shows the trend, that with larger sampling intervals the error is generally getting larger.

Sampling intervals larger than a few seconds seem not conceivable, because the error in previous cases exceeds 1% for a sampling interval of 28 s for the first time. In order to exclude that in other cases possible compensatory effects have a significant influence, e.g., the 1% limit is exceeded the first time for a longer sampling interval, and the following investigations are nevertheless carried out for sampling intervals up to 120 s.

Typically, one year is used as billing period for potable water consumption, e.g. in Germany. During this time the error of the meter based on the sampling interval must be negligible, because additionally each meter has always an unknown measurement error. If the error based on the sampling interval is close to the permitted limit it cannot be guaranteed that the water meter is measuring the volume within the tolerance range correctly. Based on the examples shown in Figure 6 the error of the sampling interval exceeds the limits in Table 3 at different times. With longer observation periods, the limit values are exceeded first at a larger sampling interval.

So far only the effect of the sampling interval was considered, but the starting point was kept identical.

Depending on the starting point of the sampling a notable difference in the error is arising. For example, a sampling interval of 87 s causes for the original profile an error of −3.46% and for the shifted profile an error of 5.34%. For the considered cases the maximum occurring error is increasing with the length of the sampling interval. As expected, the stochastic 1 week-long profile yields comparable results to the 1 day-long profile.

The impact of the time shift on the real-world profile is shown in Figure 7. The influence of small time shifts is shown in Figure 7(a). For an easier interpretation the points are connected in Figure 7(a) and 7(c) analogous to Figure 7(a).

That even small time shifts of a few seconds cause a different error becomes visible in Figure 8(b). The difference between the errors of the original one-year profile with a sampling start on 1st of January at midnight and a 1 s shifted profile causes a change of the error up to 0.79% for a sampling interval of 117 s. However, even at small sampling intervals the difference is not negligible (0.19% at a sampling interval of 10 s). The time shifts of 2, 5 and 10 s have a similar influence. Depending on the limit value to be complied with, a shift in the start time can already lead to errors outside the tolerance range of Table 3, but in comparison to Holmes-Higgin (2019) the differences due to small time shifts are quite small. A possible reason could be the longer observation time and the associated higher water consumption.

The analysis concerning the starting point of the sampling was repeated for consumption profiles of AHs. Here, comparable results were obtained to those discussed previously.

This study was carried out to get first insights into the influence of the sampling interval of electronic water meters on the determined consumption volume and the associated error. For this purpose, consumption profiles based on real measured data and stochastic consumption profiles were investigated regarding the influence of the interpolation between two samples, the influence of the starting point of the sampling and the length of the observation period (=billing period). The investigations are intended to help clarify the question of whether requirements should be placed on the interpolation type or the length of the sampling interval when data from electronic water meters are used for consumption billing.

In principle, stochastic and real data lead to comparable results in all investigations as should be expected.

The comparison of the interpolation types shows that linear and rectangular interpolation between samples lead to comparable results for timesteps lower than 120 s. The difference between those interpolation types grows linearly with the duration of the sampling interval and becomes smaller with longer observation times. Regarding the interpolation method, no further requirements seem to be necessary.

A clear dependence on the start time of the sampling is found. Already with small time shifts (even 1 s) the error due to the sampling interval can change significantly. Thus, by looking at a single start time, no conclusion can be drawn about the error caused by the discrete measurement. From this it follows that for an evaluation of different profiles it is necessary to look at different start times of the sampling.

With longer observation times the sampling related error is generally getting smaller, although it can happen that for individual sampling intervals due to compensation effects the resulting error is smaller for short observation periods than for longer. However, it turns out that the considered limit values of the error between 0.25 and 2% are exceeded earlier for short observation periods than for longer ones. From the cases studied it follows that for an observation period of one year and a sampling interval of 14 s or below, the error is less than 0.5%. For an observation period of one month, the sampling interval must be less than 4 s to ensure an error of less than 0.5%. Measuring 0.5% too much or too little has different effects on billing depending on the country. The average water price per cubic meter was 3.4 € for a consumption of 120 m³ per year. However, the price in Europe varies widely, from free water in Ireland to 9 €/m³ in Denmark. Additionally, water consumption per capita per day varies widely, ranging from 85 L in Lithuania to 240 L in Italy (EurEau 2017). Considering the average values, an error of 0.5% means that the customer would pay 2.04 € less. At the moment, the impact on costs seems relatively small, but it is likely to increase in the coming years due to rising energy costs, further increasing the importance of accurate billing.

All the investigations carried out here show the fundamental influences of the sampling interval on the total volume recorded by the meter based on European consumption behaviour. The results cannot be applied to the world market without further verification. Actions are needed in defining the maximum sampling interval. When determining the limit value, it must be considered that the assumption of an ideal water meter was made in this study. Under real conditions other influencing factors such as temperature, air bubbles or particle load also affect the meter's measurement accuracy. The future challenge will be to define a sampling interval that ensures that the maximum error that occurs never exceeds a predefined limit. Which limit for the maximum error is acceptable must be determined by the relevant standardization bodies.

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

The authors declare there is no conflict.