**5. Conclusions**

The proposed analysis provides a methodological framework to investigate the main features of water demand hourly peak factors based on hourly consumption data. The main objective is the estimation of the sample mean of hourly peak factors, the associated standard error (allowing for the definition of confidence bands), and its probability distribution. Those quantities are investigated in a perspective of spatial aggregation: For each considered aggregation level, artificial populations are created by aggregating multiple consumption time series and analysing the related statistics.

Theoretical expressions for the sample mean and for the standard error are provided (Equations (6) and (9), respectively), where the standard error expression accounts for the cross-correlation among samples. Moreover, empirical relations of the sample mean and standard error as a function of the number of aggregated households or meters (or users) are also provided (Equations (7) and (11), respectively). Concerning the probability distribution, sample means can be considered normally distributed, with model parameters effectively estimated by Equations (6) and (9).

The outcomes of the research in terms of mean peak factor are consistent with previous literature analyses focusing on similar or higher-resolution consumption datasets. In addition, the confidence band suggests a high accuracy of its estimation. The structure of the dependence on the aggregation level suggests the presence of an asymptotic value for a high number of users, as also suggested by some recent literature works.

The research confirms the possibility of using 1 h-aggregation consumption datasets for the analysis of water demand peak factors and provides a general framework to perform the stochastic analysis for aggregated consumption data. The empirical relation for the estimation of the expected value of the hourly peak factor has a general validity, although regression parameters' values are a reflection of the specific consumptions of the pilot area. General validity can be also extended to Equation (11) for the estimation of the standard deviation if the effect of a finite population is neglected. Indeed, results showed that the finite population condition does not affect the probability distribution of sample means, which remains normal, but it may affect the amplitude of the confidence bands, which could be underestimated. The proposed methodology will be further applied on other distribution systems. Moreover, additional investigations about the effect of spatial correlation on the coefficient of variation of peak discharges, as well as the quantification of the peak factor variance, will be the object of future research.

As a final remark, the structure and the coefficients of the empirical relationship described by Equation (7) for the expected value of the hourly peak water demand factor allows formulating the following general considerations, that can be of significant aid in the design and verification of water distribution networks.


**Author Contributions:** Conceptualization, G.D.G., C.D.C., and R.P.; data curation, G.D.G., C.D.C., and R.P.; formal analysis, G.D.G., C.D.C., and R.P.; investigation, G.D.G., C.D.C., and R.P.; methodology, G.D.G., C.D.C., and R.P.; software, G.D.G., C.D.C., and R.P.; supervision, G.D.G., C.D.C., and R.P.; validation, G.D.G., C.D.C., and R.P.; visualization, G.D.G., C.D.C., and R.P.; writing—original draft, G.D.G., C.D.C., and R.P.; writing—review and editing, G.D.G., C.D.C., and R.P. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **References**


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