**5. Conclusions**

The data for the Hanoi network were taken from the literature (Fujiwara and Khang, 1990), and the pipe diameters were assumed to be the ones obtained by Cunha and Sousa (2001). The demand data from the literature was used to estimate the number of users at each node, assuming a single-user mean demand of 0.002 l/s. All the users in the network were as‐ sumed to be residential and having the same characteristics. The standard deviation of de‐ mand was assumed to be 0.06 l/s. The Multivariate Streamflow model [28] was used to generate synthetic stochastic demands with different levels of cross-correlation between the single-users. The nodal demands were then introduced in the network and the performance of the network was simulated using EPANET [32]. For each considered degree of cross-cor‐ relation between demands, 100 simulations were performed, resulting in series of pressure

The first aspect that emerges from the simulations, is that the number of nodes that fail, i.e., which do not satisfy the minimum pressure requirements, increase when the cross-correla‐ tion degree increases. Higher correlations imply more synchronous consumptions, leading

**Figure 6.** Total number of nodes that do not satisfy minimum pressure requirements in 100 simulations.

Observing Figure 6 it is clear that the cross-correlation between demands significantly af‐ fects the outcome pressure heads. The number of nodes that do not satisfy the minimum pressure requirements in the network increase from 194 nodes (total nodes in the network that fail in 100 simulations) when the cross-correlation between demands is equal to 0.001, to a total of 543 nodes when the cross-correlation between demands is 0.999. In other words, the probability of failure increases from 6.3% to 17.5% between the minimum and maximum

heads for each node and for each correlation level.

124 Water Supply System Analysis - Selected Topics

to pressure failures. Figure 6 illustrates this result.

levels of cross-correlation that were considered.

Understanding and modelling the stochastic nature of water demand represents a challeng‐ ing field for researchers. Stochastic modelling faces difficulties like scarce availability of data for calibration purposes, high computational efforts associated to simulations, and the com‐ plexity of the problem itself. Moreover, the statistical properties of water demand change with the spatial and temporal scales that are used, which makes it even more difficult to ac‐ curately model the stochastic structure of demand. The proposed scaling laws represent a step forward in understanding the relation between the parameters that describe probabilis‐ tic demands and the spatial and temporal scales in which demands are measured and in which they should be modelled for WDS design or management purposes. The use of scal‐ ing laws allow a more accurate quantification of the statistical parameters, like variance and correlation, based on the real demand patterns, number of users at each node and the sam‐ pling time that is used. The scaling laws also allow to easily change the scale of the problem, since the statistical parameters and levels of uncertainty can be derived for any desired time or spatial scale.

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The scaling laws were derived analytically and validated using synthetically generated sto‐ chastic demands and real demand data from Latina, Italy. A good agreement was found be‐ tween the theoretical expressions, the synthetic demand data and the real demand data. Results show that the mean increases linearly with the number of aggregated users. The var‐ iance increases with spatial aggregation according to an exponent that varies between 1 and 2. In theory, for spatially uncorrelated demands the scaling laws is linear and for perfectly correlated demands the scaling law is quadratic. This aspect is clearly verified by the syn‐ thetic data. The scaling law for the covariance between 2 groups of users increases according to the product between the numbers of users in each group. The cross-correlation coefficient depends separately on the number of users in each group, and increases towards a limit val‐ ue. Even for small values of cross-correlation between single-user demands, this parameter cannot be ignored since it significantly increases with the aggregation of consumers.

The performed network simulation considering stochastic demands with different pre-de‐ fined levels of correlation show a clear influence of the degree of correlation on the outcome pressure heads: higher levels of correlation lead to larger fluctuations of the pressure heads and to more frequent pressure failures. At this point, the stochastic correlated demands were only used for simulation purposes. However, in future work a similar approach, can be used for design and management purposes. The consideration of correlated stochastic de‐ mands will result in more realistic and reliable water distribution networks.
